diff --git a/research/paired-cube-diagonal-bit-168/NOTICE b/research/paired-cube-diagonal-bit-168/NOTICE
new file mode 100644
index 0000000000..c2d7935d7f
--- /dev/null
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+Prepared by Chafik Boukhalfa with substantial Anthropic Claude and OpenAI
+Codex assistance. Apache-2.0. Inherited files retain their original notices
+and assistance disclosures. This package claims no exclusive priority over
+concurrent work.
+
+Retained sources and lineage:
+- icekylinx: PR144 paired cubes, signed decoder, shared-core accounting,
+  original compiler, configurable local channels and finite bridge;
+  PR130/generalized Clifford frames.
+- eumemic: PR117 module infrastructure; PR155/157 scalar modules, physical
+  frames/reuse and partner-pair bit word; PR161 output merging and searched
+  p12 module; PR168 annealed pair, triple and all-but-one modules on both
+  words, nested-prefix schedules, the L1 local circuits this package varies,
+  fused outputs, bundle descent, both physical layers, the terminal-sink
+  gate and the physical bit word, pinned at
+  4a3c769e5c5430e7114c4d3e099ff34664677f17. Their Anthropic Claude and
+  OpenAI Codex disclosures are retained. PR131/143 supply frame/reuse and
+  late-read lineage; PR154 supplies isolated snapshot verification.
+- DaysSky: PR162 generalized carrier extension (`compile_closure` and the
+  extension rule reused here on the new graph) and full-intersection
+  compiler lineage. The selected graph retains the PR168 nested schedule.
+- jamesyc / James Chang: PR124 compensated birth cuts, PR34 balanced
+  assembly, and PR166 terminal-output accumulation and exact simultaneous
+  target-chain checking. The PR166 proof is retained in references/pr166.
+- GamingPuzzled / Joel Pulikkan: PR160 searched-bit-module composition.
+- gupt1156 PR148 and Abhinav Ramachandran/geckods PR158 (Hermes assistance):
+  paid atom-toll re-instantiation, including PR158 at
+  6e086603f936dbb95c6d7171132ed943cc024db9.
+- jacklightChen / Zhihao Chen: PR23/29 ordinary supplier; Rohan Arun
+  (rohanarun): PR100/103 paid positional layout and deferred signed words.
+- Earlier bit lineage retained by the prerequisite: Zhihao Chen PR97,
+  Swapnil Jain's round-seven checkers, an664 PR128, DaysSky PR150, hpst3r
+  PR147, SovereignSteak PR122, and the source-notice contributors.
+
+This package contributes, on the complex side, a random-restart and
+coordinate search over PR144's configurable local channel circuit on the
+PR168 v4 word, selecting face diagonals for A[0,0], A[0,1] and A[1,1] and
+fixed-coordinate differences for three G channels so that the carrier
+closure gains 1,485 arcs; our own legal-edge Hopcroft-Karp carrier matching with PR162's
+extension; endpoint frame descent and compensated handoffs for the new
+graph; independent complete admission of the resulting physical word and
+its 44 terminal deletions; independent physical-bit admission and exact
+prime witnesses for every used bit frame; and, on the bit side, a fixed-coordinate/face-diagonal variant of PR168's bit
+local circuit (A[0,0], A[0,1], A[2,0], A[2,1] face diagonals; A[1,*]
+direction-0 edges; G[0,1,0] fixed-coordinate differences), our own carrier
+matching, gauge selection and complete physical layer (endpoint frame descent and 1,760 compensated handoffs) for
+that graph, with 34 PR166 terminal deletions on face0 roots, independently
+admitted with exact prime witnesses for every used bit frame;
+and the fully paid balanced composition with finer positive backoffs. Scalar virtual stock, physical
+handoffs, terminal removals and every external row reserve remain
+distinct. The complete 47-inequality body is source-normalized to PR141.
+Prerequisite constructions, the configurable circuit itself and the
+terminal-compilation concept are not claimed as original here.
+
+The package is based on main (3b6b66891c0ac888521cf591fe306c6286601d4f),
+which integrates Dugongue's PR186 entrance-bank construction and already
+carries PR175's retired-register update. references/pr168-v4 holds
+byte-identical copies of the twelve PR168 v4 files this package executes or
+pins (icekylinx/eumemic/DaysSky lineage as noted in those files), with
+their digests in references/pr168-v4/SOURCE.json.
diff --git a/research/paired-cube-diagonal-bit-168/PROOF.md b/research/paired-cube-diagonal-bit-168/PROOF.md
new file mode 100644
index 0000000000..8c710b395b
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/PROOF.md
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+# Construction and retained proof contracts
+
+The package is based on `main` at `3b6b66891c0ac888521cf591fe306c6286601d4f`; its construction inputs are PR168 at `4a3c769e5c5430e7114c4d3e099ff34664677f17`, vendored byte-identically in `references/pr168-v4`. PR144 supplies the paired-coordinate/shared-core framework and the configurable local channel circuit; PR161/168 supply the annealed modules, fused outputs, scalar words, L1 bit association, nested schedules and the bit physical layer; PR162 supplies the carrier extension rule; PR166 supplies terminal-output accumulation. This package changes the complex local circuit again (keeping PR189's bit side) and rederives the complete finite composition under the inherited all-size interfaces.
+
+## Face-diagonal local circuit
+
+Each of the 165 cubes computes the same 13 local outputs: the six single-bit sums $A[i,a]$ of the four ports with bit $i=a$, the six signed channels $G[j,k,m]$ and $F=A[f,0]+A[f,1]$. PR144's circuit lets every $A[i,a]$ be assembled from two edge sums along a direction $d\ne i$ or from the two face diagonals, and every $G[j,k,m]$ from edge differences along the third direction, long-diagonal differences or fixed-coordinate differences. PR168's L1 circuit assembles $A[1,\cdot]$ from direction-2 edges, $A[2,\cdot]$ from direction-1 edges and $G[1,2,0]$ from direction-0 edge differences.
+
+This package assembles $A[0,0]$, $A[0,1]$ and $A[1,1]$ from the two face diagonals, $A[1,0]$ from direction-2 edges and $A[2,\cdot]$ from direction-1 edges; $G[0,1,0]$, $G[0,1,1]$ and $G[0,2,1]$ are edge differences and $G[0,2,0]$, $G[1,2,0]$, $G[1,2,1]$ are fixed-coordinate differences; $F=A[2,0]+A[2,1]$. The configuration was found by 240 random restarts over the $3^{13}$ options followed by two coordinate-descent rounds, scored by the numerical physical root. The circuit has 33 additions per cube instead of 27. Every addition still has support-disjoint operands and every root support equals its contract; the graph builder and the independent scalar audit check this. The modules, the `f8:00111100` fold order and the nested schedule of PR168 are unchanged.
+
+The new signed graph has c=21,909, q=3,157 and M=33,466 mixer operations. Carrier arcs are chosen by our own legal-edge Hopcroft–Karp matching on the coordinate-padded frames and extended by PR162's greedy rule; the compiled closure is acyclic and span-contained. There are 12,189 arcs instead of 10,704, so the complete virtual auxiliary stock is R_scalar=c+q−12,189=12,877. Gauges are selected by PR144's chronological greedy rule: 2,310 roles of rank 18.
+
+## Physical word, handoffs and frames
+
+Operation frames are descended from the compiler frames by exact endpoint moves of single operations, equal-frame components and connected bundles: each move takes the join of incoming frames and value span or the meet of outgoing frames. 13,364 operations differ from the compiler frames. The 2,310 compensated handoffs pair dead ungauged non-root donors, whose last frames lie in the recipient gauge, with gauged recipients read at their first-use deadline; they leave 10,567 roles before terminal deletion. Only the explicit final frames and pairs are mathematical inputs.
+
+The verifier reconstructs every signed source/target identity, binary value span, carrier, center closure, physical role chain and actual target read. Compensation reads the current inherited dirty value with independently derived exact adjoint coefficients. The original physical word is checked on all 13,207 formal source, target and dirty columns under both shear signs. Every signed gate a←c_a a+c_b b is undone by a←c_a a−c_a c_b b, in reverse chronology; gates whose carried operand is the minuend are compiled with the sign on the destination coefficient and are audited as such.
+
+## Terminal-output accumulation
+
+The terminal compiler selects 44 ungauged, destination-only, disjoint-root roles. Each has eight targets, rank-18 common read frame and coefficient 1/2. No removed role is a source, gauge, alias endpoint, retained consumer input or another root. All sink writes are additive in the final frame. Target groups are disjoint, and each pivot's first old nonzero-frame target read follows its sink's last write. Of the 165 disjoint roots, 119 have retained producer consumers, 13 are written in the center phase and 4 have no uncorrected pivot.
+
+Retain every surviving coordinate's original dirty-response correction, including response through a deleted sink. At the common zero frame after the center phase, subtract each pivot from its seven other targets. Replace r←r+c b by y_p←y_p+c b/2. After the last write, add y_p back to those seven targets. Remove the old sink read and register. The independent target checker includes all intervening compensation reads and every new target-frame event; source-role frames at replaced writes remain as before.
+
+The modified executor checks all 1,320 source, 1,320 target and 10,523 dirty columns in both directions. An independently computed absolute residual coefficient bound of 394,518 makes 24-bit packed digits injective. The actual literal reverse word restores all retained dirty coordinates. Missing a pivot write, deleting an ancestor's original response and omitting the initial target shears are rejected. The scalar addition delta is −202, so the complete earlier virtual scalar charge remains conservative; the fixed odd divisor remains three.
+
+For each terminal sink, the pivot copies the old rank increments and the remaining seven targets enter the shared rank-18 frame. The sink's final rank-four increment and one target rank-18 increment disappear. The three-core histogram loses three children of rank 4 and three of rank 18; W loses one. The final profile is m=66, W=13,163, rank=867,438, deficit=1,320 and largest child 20. The exact stock identity is 12,877−10,523=2,310 handoffs+44 terminal sinks. Outward rational intervals prove b=665489485337/10^15; its next 10^−15 grid point fails this fixed profile.
+
+## Bit local circuit, physical word, terminal sinks, exact decoder and prime range
+
+The bit graph (p=12, h=24, v=1,760) is PR168's: its annealed pair module, its all-but-one module with n=10, its L1 association of the G channels, its partner-pair source mixing and its output merge are unchanged. Only the per-cube local circuit differs: $A[0,0]$, $A[0,1]$, $A[2,0]$ and $A[2,1]$ are assembled from the two face diagonals, $A[1,0]$ and $A[1,1]$ from direction-0 edges, and $G[0,1,0]$ from two fixed-coordinate differences instead of two direction-2 edges; the other $G$ channels add two edges. Every addition keeps support-disjoint operands and every output keeps its contract. The configuration was found by 100 random restarts over the bit circuit options followed by three coordinate-descent rounds, each candidate priced through the full physical pipeline. The carrier arcs are a fresh maximum matching by the retained generator's Hopcroft–Karp rule, and the gauges are PR144's chronological greedy selection with the G-nondegeneracy filter: 3,960 gauges of ranks 20 and 21. The virtual stock is R=18,908. Admission runs the retained package checker on this graph: exact rational frames, the mod-2 decoder with stars and side parts, source lines and target caps, operand nesting, role and target chains, the partner chronology, G-nondegeneracy, an F2 dirty-scratch replay, the ledger recount and five mutation controls.
+
+**Physical layer.** Operation frames are descended from the compiler frames by moving equal-frame connected groups to the join of their incoming frames and node spans or to the meet of their outgoing frames, repairing degenerate endpoints by one row; gauge, root and target frames are fixed. 7,282 operation frames differ from the compiler frames, including 84 enlargements; each is nondegenerate for G and contains its node's full value span. The 1,760 aliases pair dead ungauged non-root donors, whose last frames lie in the recipient gauge, with gauged recipients read immediately before their first operation. All physical role chains and retimed target reads are recounted independently.
+
+**Terminal sinks.** Apply James Chang's PR166 substitution to 34 face0 side roots. Each selected root is a destination-only, ungauged, non-alias, non-source register with a single root read, no retained consumers, no center-phase writes, all writes inside its rank-21 root cap and a pivot target with no earlier gauge correction. Its register is deleted; at the common zero frame the pivot is subtracted from the other targets, every write goes to the pivot, and after the last write the pivot is added back at the root frame. The compiler recomputes every condition from the physical frames and the actual gauge read chronology, replays the complete modified target chronology with every retained partner delivery, and checks the whole modified word on all 1,760 source, 1,760 target and 17,114 dirty columns over F2 and over the integers with an independently computed residual bound of 39,780 and 24-bit packed digits. Omitting a write, an ancestor response or the initial target shear is rejected. The scalar addition delta is −158, so the retained virtual scalar inventory remains conservative. Per core the histogram loses three children of rank 21 and three of rank 3 per sink.
+
+The final bit profile has virtual R=18,908, physical R=17,114, W=20,634, rank=1,483,712, deficit=1,936 and largest child 60. Independent literal execution checks every one of the 20,668 formal columns of the physical word over F2 before deletion, restores every source/dirty coordinate and verifies the identity on targets. Separate integer execution verifies the defining integer decoder, whose F2 reduction is the identity; it does not assert an integer identity. Missing compensation, missing partner delivery, stale dirty reads and a zero operation frame are rejected.
+
+Exact Bareiss witnesses cover all 24,401 used bit frame IDs and their distinct integer bases B. Each nonzero d=det(9BGB^t) factors into powers of primes at most 31 times a positive residual below 2^80; every factor identity is checked. Determinants reach 118 bits. Every prime q>2^80 therefore avoids every determinant, preserving all Gram inverses/projectors over Z/q^w and the original fixed-prime range. Sources, gauges, roots, operation frames, partner mixes/deliveries and ambient frames are included. Zero determinants, incorrect factors and oversized residuals are rejected.
+
+Every ideal bit edge pays the full 32m^2 fallback on fraction 10^−16, and 2m^3/2^80<10^−16. The contaminated moment and rank mass contract. Outward rational intervals certify a0=677773948354561/10^18; the next 10^−18 grid point fails this fixed envelope. The true bad fraction may be smaller; no optimality claim is inferred.
+
+Use the retained standard ordinary wrapper with a_old=384599/10^10. Select the first 10^−24 grid point above a0/(1+a0−a_old): theta=677340914792209011107/10^24. Its ordinary saving is A_B=(1−theta)a0+theta a_old. Both adapter and restored-row tolls are paid by A_B<theta<1−A_B; the previous grid point fails the strict lower condition. The full uniform weighted local-ring and restoration contracts remain inherited assumptions.
+
+## Full finite scalar, group, router and row bills
+
+Retain R_scalar=12,877 and M=33,466 despite physical handoffs and terminal deletion. The complete local scalar charge is
+
+    L=4(c+v)+10v+4hv+4h^2+8h+8+2h+8 R_scalar v(M+16)+32v
+     =4,552,920,126,520.
+
+Decoder denominators divide six, signed gate coefficients are ±1, dirty numerators use M+16 bits and 3q<2^15. The fixed odd divisor is three. Let V=|O(66,2)|, W=V·13,163, N=V·1,320 and s=V·867,438. Group cardinality cancels only in the normalized moment. It remains in K=3VL+8W+4N+8·66·R_scalar·V, G_route=64·67^3(K+1)(W+1)^2 and E=64(W+66+G_route+1)^3. The verifier rederives literal charges, B=s+E, C0=32·66·B^2, C1=1 and exact-child induction.
+
+The 66→20 halving depth is one. Full W has 2,159 bits. The separate reserves 9,909 and 252 give total row coefficient 12,320, polynomial degree 70,000, positive gap 224336/5 and suffix slope 280,000. Bit selector rows are borrowed and restored internally. Dropping any reserve, replacing full W by local W, charging only physical roles for the scalar word, omitting terminal removals or counting them as aliases is rejected.
+
+## Paid balanced transfer
+
+Use the PR34/RaD paid positional layout as applied to signed words in PR100/103 and retained in PR141. Both axis-major/group-major permutations are paid by the ordinary arbitrary-coordinate router. Active, spectator, scratch, phase, control and restored-row fields move together. The additional long-axis top bit is paid, with twiddle order and inverse alignment preserved. The geometric inequality remains strict. The application retains ordinary complete interchange/restoration and completed exact complex tensor contracts.
+
+Set b=665489485337/10^15, beta=eta=10^−24, zeta=10^−30 and a=min(A_B,(1−beta)b−zeta). The complex branch binds: a=(1−beta)b−zeta. With q=a(1−2eta), choose
+
+    c=q+eta/4, epsilon=(1−eta)/(1+q), lambda'=1−q,
+    lambda=((1−a)+lambda')/2, g=epsilon q,
+    r=(g+1−epsilon)/2, delta=eta/8.
+
+The seven margins are 1−epsilon, a, g, a, min(1−epsilon−delta,r−delta), 1−epsilon−delta and epsilon; their minimum is g. The identities 1−epsilon−g=eta, 1−epsilon−r=eta/2 and 1−epsilon(1+c)=eta−epsilon eta/4 preserve strict gaps. All 47 inequalities remain, including epsilon−a>0. Source normalization proves the PR141 arithmetic body changes only bridge plumbing and the explicit guard a≤A_B.
+
+The certified point is kappa=166261725903847/(25·10^16). Its next 10^−18 grid point fails only for these fixed supplier certificates and parameters. Matched comparisons change the positive outer backoffs while holding the same suppliers; the published parent certificate is reported separately.
+
+Finite checks establish supplied words, profiles, exact moments, prime witnesses, source closure, finite bills and arithmetic. General Clifford lifts, common generic bases, uniform weighted compilation, arbitrary-width routing, restored internal rows, layout applicability, prime supply, precision/recovery, uniform setup and analytic fixed-tape transfer remain inherited all-size contracts.
diff --git a/research/paired-cube-diagonal-bit-168/README.md b/research/paired-cube-diagonal-bit-168/README.md
new file mode 100644
index 0000000000..8589a5919f
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/README.md
@@ -0,0 +1,55 @@
+# A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 paired cubes
+
+Under the retained multiplication interfaces, this finite witness certifies
+
+$$
+T(n)=O\left(n(\log n)^{1-\kappa}\right),\qquad
+\kappa=\frac{166261725903847}{25\cdot10^{16}}
+=0.000665046903615388.
+$$
+
+The package applies to `main` at `3b6b66891c0ac888521cf591fe306c6286601d4f` (which integrates Dugongue's #186 on top of my #181 and eumemic's #168). Its construction inputs are eumemic's PR168 at `4a3c769e5c5430e7114c4d3e099ff34664677f17`; the twelve PR168 v4 files it executes or pins (compiler, physical checkers, package bit checker, parent certificate, frame-theorem note) are carried byte-identically in `references/pr168-v4` with their digests, so the package is self-contained on `main`. On the complex side this package replaces the per-cube local channel circuit by a configuration found by random restarts and coordinate descent over PR144's circuit options: the sums $A[0,0]$, $A[0,1]$ and $A[1,1]$ are assembled from the two face diagonals, $A[1,0]$ from direction-2 edges and $A[2,\cdot]$ from direction-1 edges; the channels $G[0,1,\cdot]$ and $G[0,2,1]$ use edge differences and $G[0,2,0]$, $G[1,2,0]$, $G[1,2,1]$ use fixed-coordinate differences; $F=A[2,0]+A[2,1]$. The circuit has 33 additions per cube instead of 27, but the carrier closure admits 12,189 arcs instead of 10,704, so the virtual auxiliary stock falls from 13,372 to 12,877 roles. The modules, fused outputs, nested schedules and all other inputs of PR168 are retained. A new carrier matching, gauge selection, endpoint frame descent and compensated handoffs are regenerated for the new graph, and 44 terminal-output deletions are selected by this package's own checker. On the bit side it goes beyond PR189: a random restart and three coordinate-descent rounds over the bit circuit options select face diagonals for $A[0,0]$, $A[0,1]$, $A[2,0]$ and $A[2,1]$, direction-0 edges for $A[1,\cdot]$ and fixed-coordinate differences for $G[0,1,0]$ (edges elsewhere), so the bit virtual stock falls from 19,788 to 18,908 roles; our producer rebuilds the carrier matching, gauges and physical layer (7,282 descended frames, 1,760 compensated handoffs) and 34 face0 terminal sinks are deleted. Both suppliers and the paid balanced composition are independently verified. **The bit supplier's effective saving, about 0.000677340914792, now exceeds every published complex saving but icekylinx/ikeboy's source-assisted word; with this package's own complex word the complex branch binds, so the headline bound is unchanged from PR196 and the contribution is the certified bit profile for stronger complex suppliers to compose with.**
+
+| Complete paid profile, per group vertex | Complex | Bit |
+|---|---:|---:|
+| Local / ambient dimension | 22 / 66 | 24 / 72 |
+| Scalar virtual auxiliary roles | 12,877 | 18,908 |
+| Physical auxiliary roles | 10,523 | 17,114 |
+| Compensated handoffs | 2,310 | 1,760 |
+| Terminal deletions | 44 | 34 |
+| Total persistent roles | 13,163 | 20,634 |
+| Rank mass | 867,438 | 1,483,712 |
+| Rank deficit | 1,320 | 1,936 |
+| Largest child | 20 | 60 |
+
+The complex saving is $b=665489485337/10^{15}=0.000665489485337$. The bit coarse saving is $a_0=677773948354561/10^{18}=0.000677773948354561$. Its established ordinary wrapper uses
+
+$$
+\theta=\frac{677340914792209011107}{10^{24}},\qquad
+A_B=(1-\theta)a_0+\theta\frac{384599}{10^{10}}
+\approx0.000677340914792209.
+$$
+
+Both supplier moments use outward rational intervals. The bit moment includes the complete worst-case rare-class fallback; the strict atom and internally borrowed-row tolls are paid. **The complex supplier limits the selected bound:** the transfer saving is $a=(1-\beta)b-\zeta<A_B$.
+
+Run the read-only finite verifier with Python 3.11 or newer:
+
+```sh
+python3 -B research/paired-cube-diagonal-bit-168/verify.py
+```
+
+It verifies the full source closure, signed graph, original and terminal-modified complex words in both directions, every formal source/target/dirty column, complete target chronology, the new bit graph through the retained package checker, our physical bit layer, the terminal-modified bit word's F2 identity and its defining integer decoder, all used-frame prime witnesses, exact paid moments, full scalar/group/router/row bills, 47 strict inequalities, seven margins and adverse controls. It reproduces `certificate.json` and rejects source drift. No foreign producer or aggregate checker runs for package admission. `--write` is an authoring operation before source freeze.
+
+All 24,401 used bit frame IDs, including every descended operation frame, have distinct integer Gram witnesses. Their largest cleared determinant has 118 bits; exact factor identities leave every prime factor below $2^{80}$. The original range $q>2^{80}$ is therefore retained.
+
+The full repository gate applies PR154's isolated snapshot technique:
+
+```sh
+python3 -B research/paired-cube-diagonal-bit-168/verify_parallel.py --output /new/path/verification --jobs 8
+```
+
+It discovers all 16 native `verify` groups of `main` and runs those plus this package in separate clones of one frozen source snapshot. Every group must preserve every input byte; nested jobs are limited to one. The package workflow also verifies Python 3.11, 3.13 and 3.14. The verification baseline is `main`, which already carries PR175's retired-register update. No new benchmark or whole-suite speed claim is made.
+
+[PROOF.md](PROOF.md) states the construction and retained obligations; [NOTICE](NOTICE) records provenance and assistance. `SOURCE.json` pins every package and prerequisite byte. Matched arithmetic comparisons keep the same suppliers while changing the positive outer backoffs. Adjacent grid exclusions concern only these fixed certificates and parameters.
+
+This is a conditional finite witness. General Clifford/tensor interfaces, uniform weighted bit compilation, internally borrowed rows, paid positional layout, precision/recovery and analytic fixed-tape transfer remain source-specified contracts. It is neither an unconditional multiplication theorem nor a practical runtime measurement.
diff --git a/research/paired-cube-diagonal-bit-168/SOURCE.json b/research/paired-cube-diagonal-bit-168/SOURCE.json
new file mode 100644
index 0000000000..49a9dcd03b
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/SOURCE.json
@@ -0,0 +1,2272 @@
+{
+  "canonical_command": "python3 -B research/paired-cube-diagonal-bit-168/verify.py",
+  "files": {
+    ".gitattributes": "672dd6ef1e36d19844f103f05bfb6843d462e35f98946fe86a557896bfee97ae",
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+    ".github/pull_request_template.md": "8585cd6f0442cdaddd26061f5b09502743a9806f182027ab908f9b8ad426920d",
+    ".github/workflows/bit-elim-144.yml": "266a6e3c88c7932d50445105408ec65fb213b1e160b0400a8e76c4ba09a6c5bf",
+    ".github/workflows/bit-reuse-147.yml": "0dc62d83988d6e8a6a0f3818357dd984dbbb585f1b41dd34c98d722db72c10f1",
+    ".github/workflows/parallel-announcements.yml": "d9692959037fd253c0f7602b6a425bcdf8acedfea6d27e7371d0c971feed904f",
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+    "CITATION.cff": "551e27b08d995d60e39575cd17566bcd8db1e07a67be8685d473b4019224dac5",
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+    "CONTRIBUTORS.md": "b68a5c7cc97e3fedb061676721472b755175b8102351353e5487dc7ff17389fd",
+    "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+    "Makefile": "768131bd4ce18433f6abccdc9812a38c92e11220e3a1019c6d95334b42f529d1",
+    "NOTICE": "b49022cb4104d90b182e4c335f670faf45529025b15db77bf0291835083c28bd",
+    "PATCHING.md": "fa0ac0e3fb608f1bc4567f5ed3ec9dbda1fa428d2cfd16907b1bd0fef3c759a0",
+    "README.md": "d104c31d5b7e0d3a8d011a0e3517ab615e5b499b3dc05684f7ad3cadcd28bdae",
+    "SOURCES.json": "e4e785a201a8345e871abf6753ba7a10d57654ac4b62fc007bcdddabe6311441",
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+    "tests/test_layer_preparation.py": "212778ced6c1aac05bbf1f26da5b4f0f9ddf420ad9378daa2e96f98704d564be",
+    "tests/test_lean_sources.py": "b5219815172e6fd81c63322ae022a5cc6041fc93de1736e7d3a4b483311beef8",
+    "tests/test_mixed_point.py": "bb279af4bf072e9709d70cefe0cdf0230688d865f9c5e902350c4aade9218452",
+    "tests/test_network_research.py": "42fd73f32a89703577be4dcdacfbc803ebb96312a385fa8a66067c0147da760f",
+    "tests/test_network_search.py": "45b60cbefb7de4ef037bcdc46472b5f8077853db8c954aba6bcf22c8d30f4995",
+    "tests/test_network_variants.py": "642cd409f9fae0abac7c26a532fa4767b1e87905fd78cc822dac7513200fe5d3",
+    "tests/test_nonadjacent.py": "2cbfca88aea2ff395c2ef332d33528cccecbcb28b8a40146bd49fad3187eeb0f",
+    "tests/test_open_pr_lean.py": "da44ffedcbb2c7c5e88d2ba530ebbeb53e2c9f598c53bb8d2103401ae4ecb9b3",
+    "tests/test_packed_audit.py": "5b56769de910cc4dda45db08ba5b29dbb7dd3c7f21bd9a5daffa8092a74b0cdd",
+    "tests/test_pair_assembly.py": "61e77567942f174018ed4572b7684fb07a9220e2f8ab750269c1c5abc2230353",
+    "tests/test_paired_circuit_check.py": "323ba370869dc08776ac7e6e68a9d3ef171f975a8c54a769a07c6db4f5d547be",
+    "tests/test_paired_cube_entrypoints.py": "94195d3c0d2029f4917a52e6e85bdd37b909cc88bb1feec5fefaf89ba35df563",
+    "tests/test_paired_network.py": "efd9fb982491dce52e0c53cd38af3e45139a2c6a1d373e816fab6328ec78418d",
+    "tests/test_parity_side.py": "4b6e78dd62e53defc2aeb29dd87df941b975fba376fbe868ad197216468bbfdb",
+    "tests/test_partial_swap.py": "31447fbf450d98499255951bbe3024c96dc20b3492fb3f3d411498f02b4765ee",
+    "tests/test_positive_side_core.py": "46bd4173c6a3fd9295fc2682bf7f572161e95abc6c61e0859bb2f8eee7fb6319",
+    "tests/test_positive_skip.py": "794b30d985d0859eb422e18b762582f3b4050a5fff5b97540bcf14109be00389",
+    "tests/test_prime_core.py": "0b8c38f4b71aa65ade0720d8286e871c426865c4c6e6b45cc2eb2dc7f192e1fc",
+    "tests/test_prime_subset_limits.py": "6c6d7733bedaf016ea5e74d3ee367984eb464db9027859d0b6f4fd766fc50e4c",
+    "tests/test_quadratic_affine.py": "6f2fdf0dff1b8e4d97b045edee1d645aaf0f17c649834f615f4fc6aee2637c20",
+    "tests/test_rank_product_core.py": "501c277c785a6fac39ab8f8c758ed12ffba7a6c0bda3fba957a105287224d7d6",
+    "tests/test_recycled_bit_integration.py": "a335266f2a35690e759d1bfe9368467857c1653b0c8bcbf8fd15383cb3d9ad01",
+    "tests/test_reuse_network.py": "035b5daaa98a73bc39666f4c0aca3d4d180e8ef4fd08f1941874d010ea309043",
+    "tests/test_scratch_pooling.py": "0558a23444202bd6099226501c10e95f01389f553df349a5e89929a725735e31",
+    "tests/test_selected_result.py": "6aad4b7ad3ba8ef1452851886fddaffad9225b0e1b832017211f3c8e402e6b1b",
+    "tests/test_shared_core.py": "523398ca2caeba570f978d022ae443c67b2f9992996597bafa7c67b858c4afe7",
+    "tests/test_shared_point_network.py": "94704721cdf97c8173f4dd5448aa4e82b83b705c3ecfe9769d38aaef4e74975f",
+    "tests/test_short_guards.py": "0febf170c355b335a0102a1d38ae351507b54031666ac431e1deff6247fd7d0e",
+    "tests/test_signed_sparse_core.py": "2518f4919d6f39ff41ac5c9e10c422de8b29803d79eec94983c73d34009dd585",
+    "tests/test_single_intersection.py": "333f14dee70ddffaf266a8041d0ee3a0bf64794e28c1f05cc8a6e8953a37543e",
+    "tests/test_skip_clones.py": "a2189256bafa2a07ccd1f3342d696a1fd4d27fd598a4a12ce1d97942dd661bf8",
+    "tests/test_skip_strips.py": "657f21cacac80bd598fe811ed2a113b7418904548095c2000e2faf3f10eff44f",
+    "tests/test_skip_suffix.py": "4aa6334e4b466f3006b8b961f8f3ff147a347fd2b6defc4d88d615d3a418a4ed",
+    "tests/test_small_block_witness.py": "8025d87026ea15763ecdd64811dffdc52eca1f0e67ee9bee4e9a6d6c125e4ea3",
+    "tests/test_sparse_fusion.py": "ad078502dbbcff04147665aed61609b31dc09a8e9a404d4e344aa5d462f568eb",
+    "tests/test_split_skip.py": "6d274ff66bb5465fc8a58958ecead7d27ec282717f9445ee1a2b02ef72731c50",
+    "tests/test_stage_pair.py": "746aaf91e95495644e0e89f81955d02a3a0d029934090e07a3607c207ab0f2b1",
+    "tests/test_stronger_screens.py": "059e3535f9e610aafbd159897516c299d8d15b9a55806f5197042e19bb4b3e55",
+    "tests/test_subset_core_family.py": "3d776e0d60aa226059a6dc6705ccef1b46f941d3aacd22fd60b5025fa7e45f90",
+    "tests/test_ternary_patch.py": "19e7a05d7f9b3cc7d32336173ce456fc63ff85ff231dbae5a89dcbf3b1b0a684",
+    "tests/test_ternary_side.py": "05b9b82f5bb308010d7663841a90f430326891f601a92af5602ac88a8109f26b",
+    "tests/test_tune_routing.py": "004106143616a82be9ba6cf7525ba1772328a47e75aae6047a0559afb6038a30",
+    "upstream/LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+    "upstream/README.md": "8d1ceff898a4e0e681679f91c4def485820be9e99f04e4861f2ce601a364723e",
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+    "upstream/build/figures/proof-map.tex": "76d4deacf07bea355412abcfd798c7cbe6dcccef29a4a420a9e2a7b158f23848",
+    "upstream/build/main.tex": "42f0f506976bc2084374a71e97edc1549689a7accf35dda4db2a18af4f13c1aa",
+    "upstream/build/references.bib": "33cc2df13ccf85330849e8fb34d4ac7613b23f625787457a5b7b9aa6d7369476",
+    "upstream/build/sections/00-introduction.tex": "98721f03682ccf1cc7d0850b07adabaa2d8f3afe552e3861753eba9d284d0da0",
+    "upstream/build/sections/01-history.tex": "798956c8150c31e3e2f5f29c47110d25ce619faad1374f4d3a0f15fa3ee33ead",
+    "upstream/build/sections/02-streams.tex": "606c80db61cad13aa0c3b6060dc4b88dbbe7ac60cdd831aa93f28d908fd09dab",
+    "upstream/build/sections/03-motifs.tex": "ed76761194988a01af77ead0ea6469cb5dcd29690ab985a0b5a4006b0f2b69bb",
+    "upstream/build/sections/04-swap.tex": "412b170ccaab38dba96e95e3cbfae09f4876a9818f8d278776a746b0f8044906",
+    "upstream/build/sections/05-layers.tex": "20cfc8d12099d4e4ed9e3e7eeb6162edd9b6e0e076f24693364cd24377252594",
+    "upstream/build/sections/06-transforms.tex": "924316e4bc905066281d8f9e8746a49d8829aeb393688608deb1e61c3c87853c",
+    "upstream/build/sections/07-resampling.tex": "8a6b91670f9163afb54892489fa2210c77fe1b058f0325ec7d18b44b94be6e2c",
+    "upstream/build/sections/08-assembly.tex": "763d7945b1e1ec4ebae9b799bcefcc45fa9bae6e2ffad799868a9c3d513dbd4a",
+    "upstream/build/sections/09-exact-arithmetic.tex": "5714e5d0656c78c6a1fde8d01fd11f1ba856a7ea94149b283bd9a50fdbf516c3",
+    "upstream/build/sections/10-transposition.tex": "c14ad8d56112f6578678a58211bc8d63c3a4b60e55078ac73bc38f7348e713cb",
+    "upstream/build/sections/11-grouped-rectangles.tex": "6525cd0a01980e56accfeaf06e9d875bc699197ba42c6e22b69435e352275174",
+    "upstream/formalization.yaml": "2dcbd0d6e6a22475f53d49bfdfcac9b6d2b470653c332a9f56522f5be166edb9",
+    "upstream/manifest.json": "16ab13971694cabab09440c47ebf2fc0496a9d35fc3af33950fda3498f8e2392"
+  },
+  "inputs": [
+    {
+      "commit": "4a3c769e5c5430e7114c4d3e099ff34664677f17",
+      "files": 12,
+      "pr": 168,
+      "vendored": "research/paired-cube-diagonal-bit-168/references/pr168-v4"
+    },
+    {
+      "commit": "4dbc2af5f223a0fbf22542c9d6c3f71442b6a3a6",
+      "pr": 166
+    },
+    {
+      "commit": "f1081fabbfadaaaaf7292a40d428dbbb865ce920",
+      "note": "retained-register overlay as merged on main",
+      "pr": 175
+    }
+  ],
+  "main_commit": "3b6b66891c0ac888521cf591fe306c6286601d4f",
+  "prerequisite": {
+    "branch": "main",
+    "commit": "3b6b66891c0ac888521cf591fe306c6286601d4f",
+    "repository": "https://github.com/CrocSwap/integer-mult-bounds"
+  },
+  "scope": "Complete source closure on main plus byte-identical vendored PR168 v4 inputs (references/pr168-v4); selected finite words and paid arithmetic. Retained all-size contracts remain assumptions.",
+  "status": "Frozen selected conditional finite witness"
+}
diff --git a/research/paired-cube-diagonal-bit-168/arithmetic/balanced_shared.py b/research/paired-cube-diagonal-bit-168/arithmetic/balanced_shared.py
new file mode 100644
index 0000000000..e94c7dcaa9
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/arithmetic/balanced_shared.py
@@ -0,0 +1,71 @@
+"""Paid balanced assembly with an explicit uniform-bit/shared-complex bridge.
+
+The complete 47-constraint arithmetic body is PR141 balanced_stopped.py,
+with only bridge plumbing and an actual-bit-supplier bound added. Original
+PR34/RaD/PR23, icekylinx and PR100/103 attribution is retained in the pinned
+source. It does not assert a finite old coarse profile for the uniform bit
+supplier. General cutoff/analytic/word contracts remain external.
+Prepared with OpenAI Codex assistance; Apache-2.0.
+"""
+from fractions import Fraction as Q
+from shared_bridge import rational, require, validate_shared_bridge
+
+def assembly(finite_bridge, complex_row, a_bit, kappa, beta=Q(1,20), h=Q(1,10**12),
+             a_complex=Q(717,10**7), *, original_prefix=False,
+             old_guard=False, old_exposures=False):
+    """Return 47 strict slacks and seven margins, or raise InvalidAssembly.
+
+    The three keyword switches are negative controls: they impose the old
+    prefix, nonlinear guard, or old separate movement charges at these SAME
+    balanced parameters. They do not silently optimize an alternative family.
+    """
+    f = validate_shared_bridge(finite_bridge, complex_row)
+    a, b, kappa, beta, h = map(rational, (a_bit,a_complex,kappa,beta,h))
+    require(a <= f['bit_uniform']['ordinary_saving'], 'transfer saving exceeds actual uniform bit supplier')
+    require(0 < h < Q(1,2) and kappa > 0, 'positive backoff and saving required')
+    tau, sigma = 1-a, 1-b
+    q = a*(1-2*h)
+    require(1+q != 0, 'undefined balanced epsilon')
+    lp, c, eps = 1-q, q+h/4, (1-h)/(1+q)
+    lam = (tau+lp)/2
+    G = eps*q
+    r, delta = (G+1-eps)/2, h/8
+    C1 = Q(19991,10000) if old_guard else Q(1)
+    margins = dict(g1=1-eps, g2=a, g3=G, g4=a,
+                   g5=min(1-eps-delta,r-delta), g6=1-eps-delta, g7=eps)
+    if original_prefix:
+        margins['g1'] = 1-eps*(1+c)
+    if old_exposures:
+        margins.update(g2=eps*c*a, g4=a*(1-eps))
+    internal = tau+(1-beta)*max(sigma-tau,Q(0))
+    leaf = sigma+beta*(1-sigma)
+    slacks = dict(a_positive=a, a_below_b=b-a, b_below_one_over32=Q(1,32)-b,
+        beta_positive=beta, beta_below_one=1-beta, phase_leaf_above_bit=(1-beta)*b-a,
+        q_positive=q, q_below_internal=1-internal-q, q_below_leaf=1-leaf-q,
+        c_positive=c, c_below_one=1-c, q_below_reservations=c-q,
+        lambda_above_tau=lam-tau, lambda_above_sigma=lam-sigma,
+        lambda_above_internal=lam-internal, lambda_prime_above_lambda=lp-lam,
+        compact_leaf=lp-leaf, compact_reservations=lp-(1-c), lambda_prime_below_one=q,
+        epsilon_positive=eps, epsilon_below_one=1-eps, guard_width=1-eps*C1,
+        K_geometry=1-eps*(1+c), K_dominates_log=eps*c,
+        record_suffix=1-eps, phase_local=1-eps-delta, phase_boundary=r-delta,
+        gamma_sublinear=1-eps-r, cell_above_band=eps-(1-r)/2,
+        prime_interval_packing=1-eps, alpha_positive=r, alpha_below_one=1-r,
+        alpha_below_one_fourth=Q(1,4)-r, delta_positive=delta,
+        delta_below_one_eighth=Q(1,8)-delta, short_record_fallback=eps-a,
+        small_field_exposure=1-eps-G, artificial_boundary=8-eps+r-delta-G,
+        literal_scalar_guard=Q(f['semantic']['strict_literal_gap']),
+        row_product_gap=f['rows']['degree_gap'])
+    slacks.update({name+'_above_kappa': value-kappa for name,value in margins.items()})
+    require(len(slacks) == 47 and len(margins) == 7, 'constraint list incomplete')
+    failed = {name:str(value) for name,value in slacks.items() if value <= 0}
+    require(not failed, str(failed))
+    require(min(margins.values()) == G, 'unexpected controlling margin')
+    require(1-eps-G == h and 1-eps-r == h/2, 'balanced identities failed')
+    require(1-eps*(1+c) == h-eps*h/4, 'geometric identity failed')
+    parameters = dict(a_bit=a, a_complex=b, tau=tau, sigma=sigma, beta=beta,h=h,
+        q=q,c=c,epsilon=eps,lambda_=lam,lambda_prime=lp,alpha_squared_power=r,
+        delta=delta,C0=f['semantic']['C0'],C1=C1,kappa=kappa)
+    return dict(parameters=parameters,constraints=slacks,margins=margins,
+        minimum_margin=G,absorption_gap=G-kappa,scoped_limit=a/(1+a),
+        recurrence=dict(internal=internal,leaf=leaf,reservations=1-c),finite_bridge=f)
diff --git a/research/paired-cube-diagonal-bit-168/arithmetic/certificate.py b/research/paired-cube-diagonal-bit-168/arithmetic/certificate.py
new file mode 100644
index 0000000000..1badb50751
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/arithmetic/certificate.py
@@ -0,0 +1,168 @@
+#!/usr/bin/env python3
+"""Exact paid bridge, unchanged balanced arithmetic body, and strict controls.
+
+Finite bridge formulas: icekylinx PR144/130, eumemic PR152, Apache-2.0.
+Balanced layout: PR23/29 Zhihao Chen, RaD, James Chang PR34, and Rohan
+Arun PR100/103. This adapter preserves their separately stated contracts.
+Prepared for chafreaky with substantial OpenAI Codex assistance.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+from hashlib import sha256
+from math import prod
+from pathlib import Path
+import copy
+import gzip
+import importlib.util
+import json
+import sys
+
+HERE=Path(__file__).resolve().parent;PACKAGE=HERE.parent;ROOT=HERE.parents[2]
+sys.path.insert(0,str(HERE));sys.dont_write_bytecode=True
+if hasattr(sys,'set_int_max_str_digits'):sys.set_int_max_str_digits(0)
+from shared_bridge import require,halving,validate_shared_bridge,nofloat
+from balanced_shared import assembly
+
+AC=Q(665489485337,1000000000000000)
+COARSE=Q('677773948354561/1000000000000000000')
+ATOM=Q('677340914792209011107/1000000000000000000000000')
+OLD=Q(384599,10**10)
+AB=(1-ATOM)*COARSE+ATOM*OLD
+BETA=ETA=Q(1,10**24)
+WEAKENING=Q(1,10**30)
+GRID=10**18
+
+
+def js(x):
+    if isinstance(x,Q):return str(x)
+    if isinstance(x,dict):return {str(k):js(v) for k,v in x.items()}
+    if isinstance(x,(tuple,list)):return [js(v) for v in x]
+    return x
+
+
+def bridge(row,coarse=COARSE,atom=ATOM):
+    """Retained finite formulas with the actual new local inventory supplied."""
+    h,v,R,c,M=(row[k] for k in ('h','v','R','c','total_M_operations'))
+    scalar_R=row['scalar_role_reserve']
+    m=3*h;half=m//2
+    V=2**(m-1+(half-1)**2)*prod(2**(2*i)-1 for i in range(1,half))
+    W=V*(2*v+R);s=V*row['rank_per_vertex'];N=V*v
+    r=max(int(k) for k,n in row['child_histogram'].items() if int(k) and n)
+    local=4*(c+v)+10*v+4*h*v+4*h*h+8*h+8+2*h+8*scalar_R*v*(M+16)+32*v
+    logical=3*V*local+8*W+4*N+8*m*scalar_R*V
+    G=64*(m+1)**3*(logical+1)*(W+1)**2
+    E=64*(W+m+G+1)**3;charge=2*G*W*W+8*s+4*W+4+32*m
+    B=s+E;C0=32*m*B*B;depth=halving(m,r);coefficient=depth*W.bit_length()+9909+252;degree=70000
+    actual=(1-atom)*coarse+atom*OLD
+    result=dict(bit_uniform=dict(coarse_saving=coarse,atom_beta=atom,old_atom_saving=OLD,
+        ordinary_saving=actual,selector_stock='Internally borrowed and restored q-adic rows; complete spectators preserved.',
+        external_stock='The bit group cardinality is not inserted into external polynomial row stock.'),
+        conservative_old_coarse_row_reserve=9909,ordinary_leaf_row_degree=252,
+        complex=dict(m=m,W=W,s=s,N=N,maxchild=r,invocations_per_stage=V,stages=3,
+            auxiliary_banks_after_sharing=1,halving_degree=depth,wire_bits=W.bit_length(),
+            original_X_involution_scalar_group_upper=32*v,local_group_upper=local,
+            logical_group_upper=logical,finite_group_router_upper=G,scalar_group_upper=G,
+            coefficient_bound=h+4,coefficient_denominator_divides=6,
+            physical_roles_per_vertex=R,scalar_role_reserve=scalar_R,reuse_pairs=row['reuse_pairs'],terminal_sinks=row.get('terminal_sinks',0)),
+        semantic=dict(E=E,literal_charge=charge,strict_literal_gap=E-charge,B=B,C0=C0,C1=1,
+            induction_gap=2*B*(m-r)-s-E,fixed_odd_divisor=3,
+            exact_grid='One dyadic grid times 3^(-K), K=G*(D_complex+1); completed children preserve the incoming odd exponent.'),
+        rows=dict(coefficient=coefficient,complex_coefficient=depth*W.bit_length(),degree=degree,
+            suffix_slope=4*degree,degree_gap=Q(degree)-Q(51*coefficient,25)))
+    validate_shared_bridge(result,row)
+    return result
+
+
+def below(value,grid=GRID):
+    # Largest grid point strictly below a rational, including exact boundaries.
+    x=value*grid
+    return Q(-(-x.numerator//x.denominator)-1,grid)
+
+
+def old_assembly():
+    spec=importlib.util.spec_from_file_location('retained_prefix_assembly',ROOT/'scripts/structured_bulk_assembly.py')
+    module=importlib.util.module_from_spec(spec);spec.loader.exec_module(module)
+    return module.assembly
+
+
+def price(row,b,coarse,beta,eta,weakening,layout,atom=ATOM):
+    finite=bridge(row,coarse,atom)
+    actual=finite['bit_uniform']['ordinary_saving']
+    a=min(actual,(1-beta)*b-weakening)
+    q=a*(1-2*eta)
+    value=(1-eta)*q/(1+q) if layout=='balanced' else (1-eta)*q/(1+q*(1+eta)+q)
+    k=below(value)
+    if layout=='balanced':result=assembly(finite,row,a,k,beta=beta,h=eta,a_complex=b)
+    else:result=old_assembly()(a,b,finite,k,eta=eta,beta=beta)
+    assert len(result.get('constraints',result.get('strict_constraints')))==47 and len(result['margins'])==7
+    assert result['minimum_margin']==value
+    return dict(kappa=k,a=a,actual_bit_saving=actual,complex_saving=b,beta=beta,eta=eta,
+        weakening=weakening,layout=layout,minimum_margin=value,assembly=result,finite_bridge=finite)
+
+
+def normalized_body_check():
+    source=(PACKAGE/'references/pr141/balanced_stopped.py').read_text()
+    old=source[source.index('def assembly('):source.index('\n\ndef cutoffs(')].rstrip()
+    body=(HERE/'balanced_shared.py').read_text();body=body[body.index('def assembly('):].rstrip()
+    body=body.replace('def assembly(finite_bridge, complex_row, a_bit, kappa,','def assembly(finite_bridge, ordinary_leaf, a_bit, kappa,',1)
+    body=body.replace('f = validate_shared_bridge(finite_bridge, complex_row)','f = validate_bridge(finite_bridge, ordinary_leaf)',1)
+    body=body.replace("    require(a <= f['bit_uniform']['ordinary_saving'], 'transfer saving exceeds actual uniform bit supplier')\n",'',1)
+    require(body==old,'The retained 47-slack body changed outside bridge plumbing and supplier support')
+
+
+def certificate(row,physical):
+    nofloat(row);normalized_body_check()
+    final=price(row,AC,COARSE,BETA,ETA,WEAKENING,'balanced')
+    require(final['kappa']==Q('166261725903847/250000000000000000'),'Selected final kappa differs')
+    counts=physical['scalar_bound']
+    require(counts['terminal_sinks']==row['terminal_sinks']==physical['terminal_compiler']['eligible_count'],'Terminal sink inventory mismatch')
+    require(counts['terminal_scalar_gate_delta']<=0 and counts['terminal_original_scalar_reserve_retained'],'Terminal compiler must fit the retained scalar reserve')
+    require(final['finite_bridge']['complex']['local_group_upper']==counts['local_group_upper'],'Scalar bill detached from physical word')
+    for key,field in (('physical_R','R'),('logical_R','scalar_role_reserve'),('handoffs','reuse_pairs'),('c','c'),('q','q'),('M_operations','total_M_operations')):
+        require(counts[key]==row[field],'Physical scalar inventory mismatch '+key)
+    require(counts['fixed_odd_divisor']==3 and counts['scalar_denominators_divide']==6,'Exact odd-grid contract')
+    require(sha256((PACKAGE/'references/pr168-v4/notes/general-clifford-frames.tex').read_bytes()).hexdigest() in physical['source_sha256'].values(),'Frame theorem source drift')
+    parent_path=PACKAGE/'references/pr168-v4/certificates/paired-cube-network.json'
+    require(sha256(parent_path.read_bytes()).hexdigest()=='05b253a21dc526b6039f56d1068162a19a8a4f29291ad94eb892656b88e9142d','Published prerequisite certificate changed')
+    parent=json.loads(parent_path.read_text())
+    published_parent=dict(source_commit='4a3c769e5c5430e7114c4d3e099ff34664677f17',certificate_sha256=sha256(parent_path.read_bytes()).hexdigest(),kappa=parent['kappa'],ordinary_saving=parent['bit']['effective_saving'])
+    rows=[('Same suppliers with earlier positive outer backoffs',row,AC,COARSE,Q(1,10**9),Q(1,10**8),Q(1,10**10),'balanced',ATOM),
+          ('Same suppliers with finer positive outer backoffs',row,AC,COARSE,BETA,ETA,WEAKENING,'balanced',ATOM)]
+    controls_table=[]
+    for name,*args in rows:
+        result=price(*args);controls_table.append(dict(name=name,**{k:v for k,v in result.items() if k not in ('assembly','finite_bridge')}))
+    accepted=[]
+    def reject(name,operation):
+        try:operation()
+        except (KeyError,ValueError,AssertionError):accepted.append(name)
+        else:raise ValueError('Adverse control accepted: '+name)
+    finite=final['finite_bridge'];a=final['a'];k=final['kappa']
+    def mutate(name,change):
+        bad=copy.deepcopy(finite);change(bad);reject(name,lambda:validate_shared_bridge(bad,row))
+    mutate('full group stock replaced by local W',lambda x:x['complex'].update(W=row['W_per_vertex']))
+    mutate('old ordinary reserve omitted',lambda x:x.update(ordinary_leaf_row_degree=0))
+    mutate('conservative old coarse reserve omitted',lambda x:x.update(conservative_old_coarse_row_reserve=0))
+    mutate('uniform supplier replaced by fictional finite profile',lambda x:(x.pop('bit_uniform'),x.update(bit_coarse={'m':529,'W':108516254})))
+    mutate('actual ordinary saving overstated',lambda x:x['bit_uniform'].update(ordinary_saving=AB+Q(1,10**24)))
+    mutate('unpaid previous atom grid',lambda x:x['bit_uniform'].update(atom_beta=ATOM-Q(1,10**24),ordinary_saving=(1-ATOM+Q(1,10**24))*COARSE+(ATOM-Q(1,10**24))*OLD))
+    mutate('scalar role reserve collapsed to physical stock',lambda x:x['complex'].update(scalar_role_reserve=row['R']))
+    mutate('terminal removals omitted',lambda x:x['complex'].update(terminal_sinks=0))
+    mutate('terminal removals counted as aliases',lambda x:x['complex'].update(reuse_pairs=row['reuse_pairs']+row['terminal_sinks']))
+    mutate('odd divisor silently changed to 21',lambda x:x['semantic'].update(fixed_odd_divisor=21))
+    mutate('C0 detached from scalar charge',lambda x:x['semantic'].update(C0=1))
+    mutate('scalar group charge omitted',lambda x:x['complex'].update(scalar_group_upper=0))
+    mutate('stale total row coefficient',lambda x:x['rows'].update(coefficient=1))
+    mutate('inadequate polynomial row degree',lambda x:x['rows'].update(degree=1))
+    mutate('float in otherwise unused metadata',lambda x:x['bit_uniform'].update(extra=0.5))
+    mutate('noncontracting maximum child',lambda x:x['complex'].update(maxchild=row['m']))
+    reject('transfer saving exceeds actual uniform supplier',lambda:assembly(finite,row,AB+Q(1,10**24),k,beta=BETA,h=ETA,a_complex=AC))
+    reject('old prefix charged at balanced parameters',lambda:assembly(finite,row,a,k,beta=BETA,h=ETA,a_complex=AC,original_prefix=True))
+    reject('old nonlinear scalar guard',lambda:assembly(finite,row,a,k,beta=BETA,h=ETA,a_complex=AC,old_guard=True))
+    reject('old separate movement exposures',lambda:assembly(finite,row,a,k,beta=BETA,h=ETA,a_complex=AC,old_exposures=True))
+    reject('next final 1e-18 kappa grid point',lambda:assembly(finite,row,a,k+Q(1,GRID),beta=BETA,h=ETA,a_complex=AC))
+    return js(dict(status='PASS',kappa=k,actual_uniform_bit_saving=AB,coarse_bit_saving=COARSE,
+        complex_saving=AC,weakened_transfer_saving=a,beta=BETA,eta=ETA,strict_weakening=WEAKENING,atom_exponent=ATOM,
+        bridge=finite,assembly=final['assembly'],matched_comparisons=controls_table,published_parent=published_parent,
+        unchanged_47_slack_body=True,adverse_controls=accepted,adverse_control_count=len(accepted),
+        next_grid_scope='Rejection at these fixed certified supplier values and stated parameters; no global or true-profile optimality claim.',
+        scope='Conditional finite witness retaining the source-specified uniform bit, exact tensor, row-restoration, paid layout and analytic/fixed-tape contracts.'))
diff --git a/research/paired-cube-diagonal-bit-168/arithmetic/interval_moment.py b/research/paired-cube-diagonal-bit-168/arithmetic/interval_moment.py
new file mode 100644
index 0000000000..c135d98d86
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/arithmetic/interval_moment.py
@@ -0,0 +1,76 @@
+"""Independent exact interval moments; no floating point or producer execution.
+
+Positive atanh series, range reduction by 4/3, explicit geometric tails,
+80 logarithm terms and degree-12 exponential enclosures. The final rounding
+is outward on the 10^-45 grid. Prepared with OpenAI Codex assistance.
+"""
+from fractions import Fraction as Q
+from functools import lru_cache
+from math import factorial
+GRID = 10**45
+LOG_TERMS = 80
+EXP_DEGREE = 12
+
+def require(ok, reason):
+    if not ok: raise ValueError(reason)
+
+def lower(x):return Q((x*GRID).numerator//(x*GRID).denominator,GRID)
+def upper(x):
+    y=x*GRID
+    return Q(-((-y.numerator)//y.denominator),GRID)
+
+@lru_cache(None)
+def log_interval(x):
+    x=Q(x);require(x>=1,'log interval domain')
+    base=Q(4,3);k=0
+    while x>base:x/=base;k+=1
+    def short(y):
+        z=(y-1)/(y+1);power=z;s=Q(0)
+        for j in range(LOG_TERMS):s+=power/Q(2*j+1);power*=z*z
+        lo=2*s;tail=2*power/(Q(2*LOG_TERMS+1)*(1-z*z))
+        return lo,lo+tail
+    lo,hi=short(x);blo,bhi=short(base)
+    return lower(k*blo+lo),upper(k*bhi+hi)
+
+def exp_interval(lo,hi):
+    require(0<=lo<=hi<Q(1,4),'exp enclosure domain')
+    def poly(x):
+        value=term=Q(1)
+        for j in range(1,EXP_DEGREE+1):term=term*x/j;value+=term
+        return value
+    low=poly(lo);high=poly(hi)
+    tail=hi**(EXP_DEGREE+1)/factorial(EXP_DEGREE+1)/(1-hi/Q(EXP_DEGREE+2))
+    return lower(low),upper(high+tail)
+
+def prepare(profile):
+    m,W=profile['m'],profile['W'];require(type(m) is int and type(W) is int and m>1 and W>0,'profile dimensions')
+    raw=profile['child_multiplicities'];hist={int(t):n for t,n in raw.items()}
+    require(len(hist)==len(raw),'aliased histogram keys')
+    require(all(0<t<m and type(n) is int and n>0 for t,n in hist.items()),'invalid child or multiplicity')
+    mass=sum(t*n for t,n in hist.items());require(mass==profile['total_rank']==W*m-profile['N']+profile['L'],'rank mass')
+    require(max(hist)==profile['maxchild'],'maximum child')
+    require(0<mass<W*m,'rank-moment contraction')
+    return [(t,n,*log_interval(Q(m,t))) for t,n in sorted(hist.items())]
+
+def moment(profile,a,details=False):
+    a=Q(a);require(a>=0,'nonnegative saving');lo=hi=Q(0);rows=[]
+    for t,n,L,U in prepare(profile):
+        E,F=exp_interval(a*L,a*U);w=Q(t*n,profile['m']*profile['W']);lo+=w*E;hi+=w*F
+        if details:rows.append(dict(child=t,multiplicity=n,log_lower=L,log_upper=U,exp_lower=E,exp_upper=F,contribution_lower=w*E,contribution_upper=w*F))
+    result=dict(saving=a,lower=lo,upper=hi,strict_gap_lower=1-hi)
+    if details:result['terms']=rows
+    return result
+
+def saving_grid(profile,denominator=10**24):
+    require(type(denominator) is int and denominator>0,'positive saving grid')
+    prepare(profile);lo=0;hi=denominator//32;iterations=0
+    require(moment(profile,Q(lo,denominator))['upper']<1,'initial contraction')
+    require(moment(profile,Q(hi,denominator))['lower']>1,'initial exclusion')
+    while hi-lo>1:
+        mid=(hi+lo)//2;m=moment(profile,Q(mid,denominator));iterations+=1
+        if m['upper']<1:lo=mid
+        elif m['lower']>1:hi=mid
+        else:raise ValueError('Moment interval overlaps one; increase enclosure precision')
+    accepted=moment(profile,Q(lo,denominator),True);rejected=moment(profile,Q(hi,denominator),True)
+    require(accepted['upper']<1<rejected['lower'],'adjacent saving proof failed')
+    return dict(accepted=accepted,rejected=rejected,denominator=denominator,iterations=iterations,method=dict(log_base='4/3',log_terms=LOG_TERMS,exp_degree=EXP_DEGREE,outward_grid=GRID))
diff --git a/research/paired-cube-diagonal-bit-168/arithmetic/shared_bridge.py b/research/paired-cube-diagonal-bit-168/arithmetic/shared_bridge.py
new file mode 100644
index 0000000000..b6cea18a19
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/arithmetic/shared_bridge.py
@@ -0,0 +1,93 @@
+"""Exact shared-core bridge validator for a paid balanced-layout adapter.
+
+The numerical bridge follows icekylinx/eumemic PR152 at 627089e85badda0b450048c29307786716d51037.
+This does not replay or establish a scalar word, weighted selector theorem,
+moment enclosure, or analytic/fixed-tape interface. Those are distinct inputs.
+Prepared with OpenAI Codex assistance; Apache-2.0.
+"""
+from fractions import Fraction as Q
+from math import prod
+
+def require(test,reason):
+    if not test:raise ValueError(reason)
+def rational(value):
+    require(type(value) in (int,str,Q),'Exact rational required')
+    return Q(value)
+def integer(value):
+    q=rational(value);require(q.denominator==1,'Integer required');return q.numerator
+def halving(m,r):
+    require(0<r<m,'Strictly contracting positive child')
+    k=1
+    while m**k<=2*r**k:k+=1
+    return k
+def nofloat(value):
+    require(not isinstance(value,float),'Binary float not permitted')
+    if isinstance(value,dict):
+        for x in value.values():nofloat(x)
+    if isinstance(value,(list,tuple)):
+        for x in value:nofloat(x)
+
+def validate_shared_bridge(bridge,row):
+    """Recompute numeric PR152 bridge, without inventing a finite bit profile.
+
+    `row` is the explicit complex local inventory. Its scalar/frame proof and
+    the supplier moment proofs remain separate source-bound obligations.
+    The supplied bit_uniform fields describe an already completed ordinary
+    interchange with borrowed-and-restored q-adic rows, not old PR141's
+    finite coarse-profile contract. The inherited conservative external
+    reserves 9909 and 252 are retained as separate stated proof inputs.
+    """
+    nofloat(bridge);nofloat(row)
+    h,v,R,c,M=(integer(row[k]) for k in ('h','v','R','c','total_M_operations'))
+    scalar_R=integer(row['scalar_role_reserve'])
+    reuse=integer(row['reuse_pairs']);terminal=integer(row.get('terminal_sinks',0))
+    require(scalar_R==integer(row['c'])+integer(row['q'])-integer(row['matched']),
+            'Scalar role reserve must retain the complete pre-reuse inventory')
+    require(reuse>=0 and terminal>=0 and scalar_R-R==reuse+terminal,'Physical role count must match reuse and terminal removals')
+    require(h>=2 and h%2==0 and min(v,R,c,M)>0,'Positive even-dimensional complex family')
+    m=3*h
+    hist={integer(k):integer(n) for k,n in row['child_histogram'].items() if integer(n)}
+    require(all(0<t<m and n>0 for t,n in hist.items()),'Proper positive child histogram')
+    r=max(hist);localW=2*v+R;localS=sum(t*n for t,n in hist.items())
+    require((m,localW,localS)==tuple(integer(row[k]) for k in ('m','W_per_vertex','rank_per_vertex')),'Complex inventory totals')
+    require(m*localW-localS==integer(row['deficit_per_vertex'])==2*v-3*integer(row['loss']),'Shared-core deficit identity')
+    half=m//2
+    vertices=2**(m-1+(half-1)**2)*prod(2**(2*i)-1 for i in range(1,half))
+    W,s,N=vertices*localW,vertices*localS,vertices*v
+    original_X=32*v
+    local=4*(c+v)+10*v+4*h*v+4*h*h+8*h+8+2*h+8*scalar_R*v*(M+16)+original_X
+    logical=3*vertices*local+8*W+4*N+8*m*scalar_R*vertices
+    G=64*(m+1)**3*(logical+1)*(W+1)**2
+    E=64*(W+m+G+1)**3
+    literal=2*G*W**2+8*s+4*W+4+32*m
+    B=s+E;C0=32*m*B**2;depth=halving(m,r)
+    expected_complex=dict(m=m,W=W,s=s,N=N,maxchild=r,invocations_per_stage=vertices,stages=3,
+      auxiliary_banks_after_sharing=1,halving_degree=depth,wire_bits=W.bit_length(),
+      original_X_involution_scalar_group_upper=original_X,local_group_upper=local,
+      logical_group_upper=logical,finite_group_router_upper=G,scalar_group_upper=G,
+      coefficient_bound=h+4,coefficient_denominator_divides=6,
+      physical_roles_per_vertex=R,scalar_role_reserve=scalar_R,reuse_pairs=reuse,terminal_sinks=terminal)
+    expected_semantic=dict(E=E,literal_charge=literal,strict_literal_gap=E-literal,B=B,C0=C0,C1=1,
+      induction_gap=2*B*(m-r)-s-E,fixed_odd_divisor=3)
+    for name,expected in [('complex',expected_complex),('semantic',expected_semantic)]:
+        for k,value in expected.items():require(rational(bridge[name][k])==value,'Stale '+name+'/'+k)
+    require(E>literal and 2*B*(m-r)>=s+E and C0>2*B+18,'Completed exact-child semantic induction')
+    # These are conservative, retained external reserves. The varying q-adic
+    # group cardinality is deliberately not interpreted as an external stock.
+    old_coarse=integer(bridge['conservative_old_coarse_row_reserve'])
+    old_leaf=integer(bridge['ordinary_leaf_row_degree'])
+    require(old_coarse==9909 and old_leaf==252,'Retained separate external row reserves')
+    coeff=depth*W.bit_length()+old_coarse+old_leaf
+    degree=integer(bridge['rows']['degree']);gap=Q(degree)-Q(51,25)*coeff
+    require(degree>0 and gap>0,'Insufficient external row reserve')
+    expected_rows=dict(coefficient=coeff,complex_coefficient=depth*W.bit_length(),
+      degree=degree,suffix_slope=4*degree,degree_gap=gap)
+    for k,value in expected_rows.items():require(rational(bridge['rows'][k])==value,'Stale row/'+k)
+    u=bridge['bit_uniform']
+    coarse,atom,old,actual=(rational(u[k]) for k in ('coarse_saving','atom_beta','old_atom_saving','ordinary_saving'))
+    require(0<coarse<1 and 0<atom<1 and old==Q(384599,10**10),'Uniform stopped supplier constants')
+    require(actual==(1-atom)*coarse+atom*old,'Uniform stopped saving identity')
+    require(actual<atom<1-actual,'Paid atom-adapter and internal-row tolls')
+    return dict(complex=expected_complex,semantic=expected_semantic,rows=expected_rows,
+      bit_uniform=dict(coarse_saving=coarse,atom_beta=atom,old_atom_saving=old,ordinary_saving=actual),
+      conservative_old_coarse_row_reserve=old_coarse,ordinary_leaf_row_degree=old_leaf)
diff --git a/research/paired-cube-diagonal-bit-168/bit/base_word.py b/research/paired-cube-diagonal-bit-168/bit/base_word.py
new file mode 100644
index 0000000000..044f278fbe
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/bit/base_word.py
@@ -0,0 +1,202 @@
+#!/usr/bin/env python3
+"""Reusable independent formal checks for a physical partner-pair bit word.
+
+The supplied frozen inputs are source-pinned. This never runs their producer.
+The Z check is of the defining integer decoder; only its F2 reduction is I.
+Prepared with OpenAI Codex assistance; Apache-2.0.
+"""
+from collections import Counter, defaultdict
+import importlib.util
+import json
+from pathlib import Path
+import sys
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[2]
+BASE = HERE.parent / 'references/pr168-v4/research/paired-cube-bit'
+P = 12
+if sys.flags.optimize: raise SystemExit('refusing -O')
+
+
+def need(ok, msg):
+    if not ok:
+        raise ValueError(msg)
+
+class Candidate:
+    def exact_frames(self):
+        C = self.C
+        C.decoder(); C.geometry()
+        for i in self.changed_frames:
+            f = self.opframe[i]; x = self.ops[i][2]; bits = C.sup[x]
+            need(C.nondeg(f), 'replacement operation frame nondegenerate')
+            while bits:
+                low = bits&-bits; s=low.bit_length()-1; bits-=low
+                need(C.in_frame(C.chi[s],f), 'value span in replacement operation frame')
+        for b, d in self.pairs:
+            need(C.sub(self.endframe[d], self.gauge[b]['frame']), 'exact Q donor-to-birth containment')
+            need(C.nondeg(self.endframe[d]) and C.nondeg(self.gauge[b]['frame']), 'handoff endpoints nondegenerate')
+
+    def register(self, rows):
+        C = self.C
+        B,_ = self.module.reduce_rows(rows,self.h if hasattr(self,'h') else C.h)
+        B = tuple(sorted(map(tuple,B),key=lambda r:next(i for i,x in enumerate(r) if x)))
+        if not hasattr(self,'frame_ids'):
+            self.frame_ids = {}
+        if B in self.frame_ids: return self.frame_ids[B]
+        f=max(C.B)+1; A,_=self.module.kernel(B,C.h)
+        C.B[f],C.A[f],C.dimf[f]=B,A,len(B);self.frame_ids[B]=f
+        return f
+
+
+    def row(self):
+        C,h,v=self.C,self.h,self.v
+        _,baseline_Y,src,_=C.chains()
+        seq=defaultdict(list)
+        for i,(a,b,x) in enumerate(self.ops):
+            seq[a].append(self.opframe[i]);seq[b].append(self.opframe[i])
+        for j,s in enumerate(self.w['rootroles']):seq[s].append(self.w['root_frame'][j])
+        start={b:self.w['source_frame'][x] for x,b in self.source.items()}
+        start.update({b:z['frame'] for b,z in self.gauge.items()})
+        recipient={d:b for b,d in self.pairs}
+        H=Counter()
+        for s in range(self.R):
+            if s in self.donor:continue
+            chain=list(seq[s])
+            if s in recipient:
+                b=recipient[s];chain += [self.gauge[b]['frame']]+seq[b]
+            chain.append(self.w['full_frame'])
+            prev=start.get(s);dp=0 if prev is None else C.dimf[prev]
+            if s in self.source.values():H[1]+=1
+            for f in chain:
+                need(prev is None or C.sub(prev,f),'physical role chain nested')
+                d=C.dimf[f]
+                if d>dp:H[d-dp]+=1
+                prev,dp=f,d
+        for j,r in enumerate(self.g['roots']):
+            if r['kind']=='center':H[C.dimf[self.w['root_frame'][j]]]+=1
+        Y=Counter();current=[None]*v
+        def read(t,f):
+            old=current[t];d=0 if old is None else C.dimf[old]
+            need(old is None or C.sub(old,f),'actual target frame chronology')
+            if C.dimf[f]>d:Y[C.dimf[f]-d]+=1
+            current[t]=f
+        for b in sorted(self.order,key=lambda b:self.readtime[b]):
+            for t in self.gauge[b]['targets']:read(t,self.gauge[b]['frame'])
+        deliveries=defaultdict(list)
+        for e in self.k['entries']:deliveries[e['deliver_after_root']].append(e)
+        for j,r in enumerate(self.g['roots']):
+            if r['kind']!='center':
+                for t in r['targets']:read(t,self.w['root_frame'][j])
+            for e in deliveries[j]:
+                for t in e['receivers']:read(t,e['deliver_frame'])
+        for t,f in enumerate(current):
+            need(f is not None and all(self.module.dot(C.cov[t],b)==0 for b in C.B[f]),'final target cap')
+            gap=h-1-C.dimf[f]
+            if gap:Y[gap]+=1
+        need(sum(r*n for r,n in Y.items())==v*(h-1),'target rank mass')
+        hist=Counter()
+        for part in (H,Y,src):
+            for r,n in part.items():
+                if r and n:hist[r]+=3*n
+        for b,z in self.gauge.items():
+            if b not in self.donor:hist[3*z['dim']]+=1
+        hist[2]+=2*v
+        hist={r:n for r,n in sorted(hist.items()) if n}
+        need(all(0<r<3*h and n>0 for r,n in hist.items()),'positive proper child bins')
+        R=self.R-len(self.pairs);W=2*v+R;mass=sum(r*n for r,n in hist.items());m=3*h
+        need(W*m-mass==C.prof['deficit_per_vertex'],'reuse preserves telescoping deficit')
+        return dict(h=h,v=v,R=R,virtual_R=self.R,reused_registers=len(self.pairs),W_per_vertex=W,m=m,
+            loss=C.prof['loss'],rank_per_vertex=mass,deficit_per_vertex=W*m-mass,
+            child_histogram=hist,maxchild=max(hist),selected_roles=len(self.gauge)-len(self.pairs),
+            selected_rank_histogram=dict(Counter(z['dim'] for b,z in self.gauge.items() if b not in self.donor)),
+            changed_operation_frames=len(self.changed_frames),
+            frames_not_contained_in_original=sum(not C.sub(self.opframe[i],self.original_opframe[i]) for i in self.changed_frames),
+            physical_target_histogram=dict(Y),physical_internal_histogram=dict(H),
+            foreign_producer_replays=0)
+
+    def adjoint(self):
+        adj = [dict() for _ in range(self.R)]
+        for r,s in zip(self.g['roots'], self.w['rootroles']):
+            for t in r['targets']: adj[s][t] = adj[s].get(t,0)+1
+        for a,b,_ in reversed(self.ops):
+            for t,c in adj[a].items(): adj[b][t] = adj[b].get(t,0)+c
+        return adj
+
+    def formal(self, ring, tamper=None):
+        """All source, target and dirty-register columns; no random sampling."""
+        v, phys = self.v, self.phys
+        regs = sorted(set(phys.values())); index = {r:2*v+j for j,r in enumerate(regs)}
+        if ring == 2:
+            unit = lambda i: 1<<i
+            def accum(dst,val,c): return dst ^ val if c&1 else dst
+        else:
+            unit = lambda i: {i:1}
+            def accum(dst,val,c):
+                out = dict(dst)
+                for i,z in val.items():
+                    k = out.get(i,0)+c*z
+                    if k: out[i]=k
+                    else: out.pop(i,None)
+                return out
+        x = [unit(s) for s in range(v)]; y = [unit(v+t) for t in range(v)]
+        a = {r:unit(index[r]) for r in regs}
+        adj = self.adjoint()
+        at = defaultdict(list)
+        for b in self.order: at[self.readtime[b]].append(b)
+        done = []
+        stale_used = False
+        def read(b):
+            nonlocal stale_used
+            if tamper == 'omit_compensation' and b == self.order[0]: return
+            value = a[phys[b]]
+            if tamper == 'stale' and b in self.donor and not stale_used:
+                value = unit(index[phys[b]]); stale_used = True
+            for t,c in adj[b].items(): y[t] = accum(y[t],value,-c)
+        def op(i,sign):
+            b,d,_ = self.ops[i]
+            a[phys[b]] = accum(a[phys[b]],a[phys[d]],sign)
+        for b in range(self.R):
+            if b not in self.gauge: read(b)
+        for s,b in self.source.items(): a[phys[b]] = accum(a[phys[b]],x[s],1)
+        for i in self.phase1: op(i,1); done.append(i)
+        for r,b in zip(self.g['roots'],self.w['rootroles']):
+            if r['kind']=='center':
+                for t in r['targets']:y[t]=accum(y[t],a[phys[b]],1)
+        for j,i in enumerate(self.rest):
+            for b in at[j]: read(b)
+            op(i,1); done.append(i)
+        for b in at[len(self.rest)]: read(b)
+        deliveries=defaultdict(list)
+        for e in self.k['entries']:deliveries[e['deliver_after_root']].append(e)
+        broken=False
+        for j,(r,b) in enumerate(zip(self.g['roots'],self.w['rootroles'])):
+            if r['kind']!='center':
+                for t in r['targets']: y[t] = accum(y[t],a[phys[b]],1)
+            for e in deliveries[j]:
+                c,d = e['carrier'],e['passive']
+                x[c] = accum(x[c],x[d],1)
+                for t in e['receivers']:
+                    if tamper == 'missing_partner' and not broken: broken=True;continue
+                    y[t] = accum(y[t],x[c],1)
+        for e in self.k['entries']:
+            x[e['carrier']] = accum(x[e['carrier']],x[e['passive']],-1)
+        for i in reversed(done): op(i,-1)
+        for s,b in self.source.items(): a[phys[b]] = accum(a[phys[b]],x[s],-1)
+        need(all(a[r] == unit(index[r]) for r in regs), 'all dirty register columns restored')
+        need(all(x[s] == unit(s) for s in range(v)), 'all source columns restored')
+        if ring == 2:
+            want = [accum(unit(v+t),unit(t),1) for t in range(v)]
+        else:
+            # Integer lift: the exact defining decoder, whose reduction is I over F2.
+            support = [{s:1} for s in range(v)]
+            for aa,bb in self.g['args'][v:]: support.append(accum(support[aa],support[bb],1))
+            want = [unit(v+t) for t in range(v)]
+            for r in self.g['roots']:
+                for t in r['targets']: want[t] = accum(want[t],support[r['node']],1)
+            for e in self.k['entries']:
+                for t in e['receivers']:
+                    want[t] = accum(want[t],unit(e['carrier']),1)
+                    want[t] = accum(want[t],unit(e['passive']),1)
+        need(y == want, 'all target columns equal defining decoder')
+        return dict(ring=str(ring),formal_variables=2*v+len(regs),
+            target_contract='F2 identity' if ring==2 else 'integer defining decoder',
+            identity=ring==2,defining_decoder=True,all_dirty_and_source_columns_restored=True)
diff --git a/research/paired-cube-diagonal-bit-168/bit/prime-witnesses.json.gz b/research/paired-cube-diagonal-bit-168/bit/prime-witnesses.json.gz
new file mode 100644
index 0000000000..de58a8d53e
Binary files /dev/null and b/research/paired-cube-diagonal-bit-168/bit/prime-witnesses.json.gz differ
diff --git a/research/paired-cube-diagonal-bit-168/bit/prime_witnesses.py b/research/paired-cube-diagonal-bit-168/bit/prime_witnesses.py
new file mode 100644
index 0000000000..7f16d93030
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/bit/prime_witnesses.py
@@ -0,0 +1,99 @@
+"""Exact finite prime exclusions for the selected new rational bit frames.
+
+For integer basis B and G=I-J/9, det(9 B G B^t) is a nonzero integer.
+Excluding its prime divisors and 2,3,5 makes the reduced frame nondegenerate
+over every odd local ring Z/q^w. Projector denominators divide this determinant
+times fixed powers of 3. Retained frames retain their old exclusion witnesses.
+The selected q is an arbitrarily large fixed prime chosen after this finite
+set; this is not verification at a preselected numeric q.
+"""
+from pathlib import Path
+import argparse,gzip,hashlib,json,sys
+
+HERE=Path(__file__).resolve().parent
+
+def det(A):
+    A=[list(r) for r in A];n=len(A)
+    if not n:return 1
+    sign=1;previous=1
+    for k in range(n-1):
+        pivot=next((i for i in range(k,n) if A[i][k]),None)
+        if pivot is None:return 0
+        if pivot!=k:A[k],A[pivot]=A[pivot],A[k];sign=-sign
+        p=A[k][k]
+        for i in range(k+1,n):
+            for j in range(k+1,n):
+                v=A[i][j]*p-A[i][k]*A[k][j]
+                assert v%previous==0
+                A[i][j]=v//previous
+            A[i][k]=0
+        previous=p
+    return sign*A[-1][-1]
+
+PRIMES=(2,3,5,7,11,13,17,19,23,29,31)
+
+def factor_witness(determinant):
+    assert type(determinant) is int and determinant != 0
+    residual=abs(determinant);powers={}
+    for prime in PRIMES:
+        power=0
+        while residual%prime==0:residual//=prime;power+=1
+        powers[str(prime)]=power
+    validate_factor(determinant,powers,residual)
+    return powers,residual
+
+def validate_factor(determinant,powers,residual):
+    assert type(determinant) is int and determinant != 0
+    assert set(powers)==set(map(str,PRIMES))
+    assert all(type(x) is int and x>=0 for x in powers.values())
+    assert type(residual) is int and 0<residual<2**80
+    reconstructed=residual
+    for p in PRIMES:reconstructed*=p**powers[str(p)]
+    assert reconstructed==abs(determinant)
+
+def certificate(word=None):
+    if word is None:
+        from word import Candidate
+        word=Candidate()
+    C=word.C;h=word.h
+    used=set(word.opframe)|set(word.w['source_frame'])|set(word.w['root_frame'])|{word.w['full_frame']}
+    used.update(z['frame'] for z in word.w['gauges'])
+    for e in word.k['entries']:used.update((e['mix_frame'],e['deliver_frame']))
+    groups={}
+    for frame in sorted(used):
+        B=C.B[frame]
+        key=json.dumps(B,separators=(',',':'));groups.setdefault(key,[]).append(frame)
+    records=[]
+    for key,frames in sorted(groups.items()):
+        B=json.loads(key);s=list(map(sum,B))
+        gram=[[9*sum(a*b for a,b in zip(x,y))-s[i]*s[j] for j,y in enumerate(B)] for i,x in enumerate(B)]
+        determinant=det(gram);powers,residual=factor_witness(determinant)
+        records.append(dict(basis_sha256=hashlib.sha256(key.encode()).hexdigest(),
+            frame_ids=frames,dimension=len(B),cleared_gram_determinant=determinant,
+            small_prime_powers=powers,remaining_factor=residual,gram_denominator_power_of_9=len(B)))
+    controls=[];zeros={str(p):0 for p in PRIMES}
+    for name,args in (('zero determinant',(0,zeros,1)),('incorrect factor identity',(15,zeros,1)),
+        ('residual above retained prime lower bound',(2**80+7,zeros,2**80+7))):
+        try:validate_factor(*args)
+        except AssertionError:controls.append(name)
+        else:raise ValueError('Invalid prime witness accepted: '+name)
+    return dict(status='PASS exact prime witnesses for every used physical frame',h=h,
+        source_commit='4a3c769e5c5430e7114c4d3e099ff34664677f17',
+        input_sha256={str(p.relative_to(HERE.parents[2])):hashlib.sha256(p.read_bytes()).hexdigest() for p in word.input_paths},
+        script_sha256=hashlib.sha256(Path(__file__).read_bytes()).hexdigest(),
+        stripped_primes=list(PRIMES),frame_witnesses=records,unique_used_bases=len(records),
+        total_used_frames=len(used),total_operation_frames=len(word.changed_frames),
+        maximum_determinant_bits=max(abs(z['cleared_gram_determinant']).bit_length() for z in records),
+        all_remaining_factors_below_retained_prime_lower_bound=True,adverse_controls=controls,
+        rule='Every q>2^80 avoids every nonzero used-frame determinant, so the original prime range is unchanged.',
+        scope='Exact integer cleared-Gram factors for all physical frames including every changed operation, gauge, source, root, partner-mix, delivery and ambient frame. Uniform split-rank and fallback arguments remain the retained local-ring interface.')
+
+def main():
+    assert not sys.flags.optimize
+    ap=argparse.ArgumentParser();ap.add_argument('--write',action='store_true');args=ap.parse_args()
+    record=certificate();target=HERE/'prime-witnesses.json.gz'
+    if args.write:target.write_bytes(gzip.compress((json.dumps(record,sort_keys=True,separators=(',',':'))+'\n').encode(),mtime=0))
+    else:assert record==json.loads(gzip.decompress(target.read_bytes())),'Exact prime witnesses changed'
+    print(json.dumps({k:v for k,v in record.items() if k!='frame_witnesses'}))
+
+if __name__=='__main__':main()
diff --git a/research/paired-cube-diagonal-bit-168/bit/prove.py b/research/paired-cube-diagonal-bit-168/bit/prove.py
new file mode 100644
index 0000000000..e8eda4df5a
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/bit/prove.py
@@ -0,0 +1,107 @@
+#!/usr/bin/env python3
+"""PR168-v4 physical bit word with 556 frame overrides and 34 terminal sinks, exact paid moment and positive atom toll.
+
+Checks supplied graph/frame inputs and all formal columns without executing
+the foreign producer. The analytic weighted-compiler contracts are retained.
+"""
+from fractions import Fraction as Q
+from pathlib import Path
+import contextlib,gzip,json
+import sys
+
+HERE=Path(__file__).resolve().parent
+sys.path.insert(0,str(HERE));sys.path.insert(0,str(HERE.parent/'arithmetic'))
+sys.dont_write_bytecode=True
+from word import Candidate,need
+from interval_moment import moment,log_interval,exp_interval
+from prime_witnesses import certificate as prime_certificate
+
+
+def js(x):
+    if isinstance(x,Q):return str(x)
+    if isinstance(x,dict):return {str(k):js(v) for k,v in x.items()}
+    if isinstance(x,(tuple,list)):return [js(v) for v in x]
+    return x
+
+
+def paid_moment(p,a,details=False):
+    raw=moment(p,a,details);m,w=p['m'],p['W']
+    count=sum(p['child_multiplicities'].values());bad=Q(1,10**16);fallback=32*m*m
+    ll,lu=log_interval(Q(m));el,eu=exp_interval(a*ll,a*lu)
+    weight=bad*Q(fallback*count,w*m)
+    return dict(saving=a,lower=raw['lower']+weight*el,upper=raw['upper']+weight*eu,
+        strict_gap_lower=1-raw['upper']-weight*eu,raw=raw,fallback_lower=weight*el,
+        fallback_upper=weight*eu,edge_count=count,fallback_children_per_edge=fallback,
+        bad_fraction=bad)
+
+
+def certify(row):
+    p=dict(m=row['m'],W=row['W_per_vertex'],N=row['deficit_per_vertex'],L=0,
+        total_rank=row['rank_per_vertex'],maxchild=row['maxchild'],child_multiplicities=row['child_histogram'])
+    grid=10**18;lo=0;hi=grid//100
+    need(paid_moment(p,Q(lo,grid))['upper']<1,'zero-saving rank contraction')
+    need(paid_moment(p,Q(hi,grid))['lower']>1,'upper bracket')
+    while hi-lo>1:
+        mid=(lo+hi)//2;r=paid_moment(p,Q(mid,grid))
+        if r['upper']<1:lo=mid
+        elif r['lower']>1:hi=mid
+        else:raise ValueError('increase moment precision')
+    coarse=Q(lo,grid);accepted=paid_moment(p,coarse,True);rejected=paid_moment(p,Q(hi,grid),True)
+    need(accepted['upper']<1<rejected['lower'],'adjacent grid paid-moment proof')
+    old=Q(384599,10**10);threshold=coarse/(1+coarse-old);atomgrid=10**24
+    floor=(threshold*atomgrid).numerator//(threshold*atomgrid).denominator
+    atom=Q(floor+1,atomgrid);actual=(1-atom)*coarse+atom*old
+    need(actual<atom<1-actual,'paid atom and row adapter toll')
+    need(Q(2*p['m']**3,2**80)<Q(1,10**16),'fixed prime rare-class bound')
+    need(Q(p['total_rank'])+accepted['bad_fraction']*accepted['fallback_children_per_edge']*accepted['edge_count']<p['W']*p['m'],'contaminated mass contracts')
+    previous=atom-Q(1,atomgrid)
+    need(previous<=(1-previous)*coarse+previous*old,'previous atom grid does not pay strict toll')
+    return dict(coarse_saving=coarse,accepted=accepted,rejected=rejected,coarse_grid=grid,
+        atom_beta=atom,old_atom_saving=old,ordinary_saving=actual,atom_threshold=threshold,
+        atom_grid=atomgrid,atom_lower_gap=atom-actual,atom_upper_gap=1-actual-atom,
+        controls=dict(next_coarse_grid_rejected=True,previous_atom_grid_rejected=True),
+        scope='Adjacent coarse exclusion for this fixed worst-case bad-class envelope and least atom on the stated grid; no true bad-fraction or global optimality claim.')
+
+
+def main():
+    need(not sys.flags.optimize,'assertions enabled')
+    word=Candidate();word.exact_frames();row=word.row()
+    need(row['reused_registers']==len(word.pairs),'physical aliases counted')
+    # The package word itself (node frames, no pairs) passes the retained package checker with its five mutation controls.
+    Checker=word.module.Checker;checked=Checker(HERE.parent/'selected/bit',12).run()
+    reference=json.loads((HERE.parent/'selected/bit/profile_p12.json').read_text())
+    need((checked['roles'],checked['W'],checked['m'],checked['deficit'])==(reference['R'],reference['W_per_vertex'],reference['m'],reference['deficit_per_vertex']),'checked package word profile')
+    need(row['virtual_R']==reference['R'] and row['deficit_per_vertex']==reference['deficit_per_vertex'] and row['loss']==reference['loss'],'physical layer over the checked package word')
+    parent_row=dict(package_profile={k:reference[k] for k in ('R','W_per_vertex','m','deficit_per_vertex','loss','rank_per_vertex')},package_checker=dict(roles=checked['roles'],W=checked['W']))
+    formal=[word.formal(r) for r in (2,0)]
+    controls=[]
+    for tamper in ('omit_compensation','missing_partner','stale'):
+        try:word.formal(2,tamper)
+        except ValueError:controls.append(tamper)
+        else:raise ValueError('Adverse word control accepted: '+tamper)
+    i=word.changed_frames[0];old=word.opframe[i];word.opframe[i]=word.register([])
+    try:word.exact_frames()
+    except ValueError:controls.append('zero_operation_frame')
+    else:raise ValueError('Zero frame accepted')
+    word.opframe[i]=old
+    from terminal import prove as prove_terminal
+    selection=json.loads((HERE.parent/'selected/bit/sinks.json').read_text())
+    with contextlib.redirect_stdout(sys.stderr):terminal=prove_terminal(word,selection)
+    terminal.pop('seconds');terminal.pop('maxrss')
+    overridden_row=row;row=terminal['profile']
+    row['child_histogram']={int(k):n for k,n in row['child_histogram'].items()}
+    need(row['terminal_sinks']==len(selection) and terminal['scalar_addition_delta']<=0,'terminal sink count and retained scalar bill')
+    need(row['W_per_vertex']==overridden_row['W_per_vertex']-len(selection) and row['R']==overridden_row['R']-len(selection),'terminal removals reduce stock by the sink count')
+    coarse=certify(row)
+    primes=prime_certificate(word)
+    need(primes==json.loads(gzip.decompress((HERE/'prime-witnesses.json.gz').read_bytes())),'source-bound all-frame prime witness reproduction')
+    need(primes['total_operation_frames']==row['changed_operation_frames'] and primes['h']==row['h'],'prime witnesses cover changed frames')
+    from hashlib import sha256
+    prime_summary={k:v for k,v in primes.items() if k!='frame_witnesses'}
+    prime_summary['witness_file']='bit/prime-witnesses.json.gz'
+    prime_summary['witness_sha256']=sha256((HERE/'prime-witnesses.json.gz').read_bytes()).hexdigest()
+    print(json.dumps(js(dict(status='PASS',profile=row,overridden_profile=overridden_row,parent_profile=parent_row,terminal=terminal,formal=formal,coarse=coarse,prime_witnesses=prime_summary,word_controls=controls,
+        foreign_producer_replays=0,scope='Complete F2 identity, integer defining decoder, exact physical frame geometry and retained weighted-compiler contracts.')),sort_keys=True))
+
+
+if __name__=='__main__':main()
diff --git a/research/paired-cube-diagonal-bit-168/bit/terminal.py b/research/paired-cube-diagonal-bit-168/bit/terminal.py
new file mode 100644
index 0000000000..da99c9e305
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/bit/terminal.py
@@ -0,0 +1,224 @@
+"""Exact terminal-target compiler on a frozen physical paired-cube bit word.
+
+James Chang's PR166 terminal substitution, specialized and independently
+verified by Chafik Boukhalfa with OpenAI Codex assistance. All inherited
+word, frame, gauge, reuse and partner-pair credits remain in the adapter.
+No source producer is executed. The original adjoint includes deleted roots.
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+from hashlib import sha256
+import argparse,importlib.util,json,time,math,sys,gc,resource,signal
+
+def need(x,msg):
+ if not x:raise AssertionError(msg)
+def compact(c):return {str(k):v for k,v in sorted(c.items()) if v}
+
+def prove(word,selection):
+ t0=time.monotonic();C=word.C;h,v,R=word.h,word.v,word.R;ops=word.ops
+ baseline=word.row();phase=set(word.phase1);order=word.phase1+word.rest;position={i:j for j,i in enumerate(word.rest)}
+ sources=set(word.source.values());aliasroles=set(word.donor)|set(word.donor.values());writes=defaultdict(list);uses=defaultdict(list);rootids=defaultdict(list)
+ for i,(a,b,n) in enumerate(ops):writes[a].append(i);uses[b].append(i)
+ for j,s in enumerate(word.w['rootroles']):rootids[s].append(j)
+ at=defaultdict(list)
+ for s in word.order:at[word.readtime[s]].append(s)
+ first=[len(word.rest)+1]*v
+ for s in word.order:
+  for target in word.gauge[s]['targets']:first[target]=min(first[target],word.readtime[s])
+ chosen=[];seen=set();targets=set()
+ for entry in selection:
+  j,pivot=entry['root'],entry['pivot'];need(type(j) is int and 0<=j<len(word.g['roots']),'root index')
+  r=word.g['roots'][j];s=word.w['rootroles'][j];U=word.w['root_frame'][j]
+  need(j not in seen and r['kind']=='side','distinct side roots');seen.add(j)
+  need(s not in sources|aliasroles|set(word.gauge),'zero-gauge independent scratch root')
+  need(rootids[s]==[j] and not uses[s] and writes[s],'one terminal root with writes and no retained consumers')
+  need(all(i not in phase for i in writes[s]),'root untouched in center phase')
+  need(all(C.sub(word.opframe[i],U) for i in writes[s]),'all writes within root cap')
+  need(type(pivot) is int and pivot in r['targets'],'pivot target')
+  need(first[pivot]>max(position[i] for i in writes[s]),'pivot untouched by compensation before post')
+  need(not targets.intersection(r['targets']),'simultaneous disjoint target groups');targets.update(r['targets'])
+  path=[word.opframe[i] for i in writes[s]]+[U,word.w['full_frame']];prev=None;prefix=[]
+  for f in path:
+   need(prev is None or C.sub(prev,f),'deleted root trajectory nested')
+   d=C.dimf[f]-(0 if prev is None else C.dimf[prev])
+   if d:prefix.append(d)
+   prev=f
+  need(sum(prefix)==h,'complete deleted root path')
+  chosen.append(dict(root=j,role=s,targets=r['targets'],pivot=pivot,root_frame=U,root_rank=C.dimf[U],writes=writes[s],deleted_path=prefix))
+ removed={e['role'] for e in chosen};deletedroots={e['root'] for e in chosen};bywrite={i:e for e in chosen for i in e['writes']};after={e['writes'][-1]:e for e in chosen}
+ need(all(a not in removed and b not in removed for i,(a,b,n) in enumerate(ops) if i not in bywrite),'removed roles only occur in their own writes')
+ # Complete modified target chronology, including every retained partner delivery.
+ Y=Counter();current=[None]*v;events=0
+ def move(t,f):
+  nonlocal events
+  prev=current[t];need(prev is None or C.sub(prev,f),'new target chain nesting')
+  need(all(word.module.dot(C.cov[t],b)==0 for b in C.B[f]),'target movement stays inside cap')
+  d=C.dimf[f]-(0 if prev is None else C.dimf[prev]);need(d>=0,'target rank monotonicity')
+  if d:Y[d]+=1
+  current[t]=f;events+=1
+ for j,i in enumerate(word.rest):
+  for s in at[j]:
+   for t in word.gauge[s]['targets']:move(t,word.gauge[s]['frame'])
+  if i in bywrite:move(bywrite[i]['pivot'],word.opframe[i])
+  if i in after:
+   for t in after[i]['targets']:move(t,after[i]['root_frame'])
+ for s in at[len(word.rest)]:
+  for t in word.gauge[s]['targets']:move(t,word.gauge[s]['frame'])
+ deliveries=defaultdict(list)
+ for e in word.k['entries']:deliveries[e['deliver_after_root']].append(e)
+ for j,r in enumerate(word.g['roots']):
+  if r['kind']=='side' and j not in deletedroots:
+   for t in r['targets']:move(t,word.w['root_frame'][j])
+  for e in deliveries[j]:
+   for t in e['receivers']:move(t,e['deliver_frame'])
+ for t,f in enumerate(current):
+  need(f is not None and C.dimf[f]<=h-1,'final target cap')
+  gap=h-1-C.dimf[f]
+  if gap:Y[gap]+=1
+ need(sum(r*n for r,n in Y.items())==v*(h-1),'full target rank mass')
+ local=Counter({int(r):n for r,n in baseline['physical_internal_histogram'].items()})
+ for e in chosen:local.subtract(e['deleted_path'])
+ need(all(n>=0 for n in local.values()),'positive remaining local ledger')
+ _,_,source,_=word.C.chains();child=Counter()
+ for part in (local,Y,source):
+  for r,n in part.items():
+   if r and n:child[r]+=3*n
+ for s,z in word.gauge.items():
+  if s not in word.donor:child[3*z['dim']]+=1
+ child[2]+=2*v;physical_R=baseline['R']-len(chosen);W=2*v+physical_R;rank=sum(r*n for r,n in child.items())
+ need(W*3*h-rank==baseline['deficit_per_vertex'],'terminal rank deficit unchanged')
+ delta=Counter(child);delta.subtract({int(r):n for r,n in baseline['child_histogram'].items()});expected=Counter()
+ for e in chosen:expected[e['root_rank']]-=3;expected[h-e['root_rank']]-=3
+ need(compact(delta)==compact(expected),'independent complete ledger agrees with exact terminal cancellation')
+ # Literal local sandwich, with arbitrary target values and non-pivot corrections.
+ for e in chosen:
+  ts=e['targets'];p=ts.index(e['pivot']);n=len(ts);k=len(e['writes']);N=2*n+k
+  initial=[[int(a==b) for b in range(N)] for a in range(n)];state=[r[:] for r in initial];program=[]
+  def apply(event,direction=1):
+   kind,a,b,c=event;c*=direction
+   if kind=='shear':state[a]=[x+c*y for x,y in zip(state[a],state[b])]
+   else:state[a][b]+=c
+  for a in range(n):
+   if a!=p:program.append(('shear',a,p,-1))
+  program.extend(('basis',p,n+j,1) for j in range(k))
+  program.extend(('basis',a,n+k+a,1) for a in range(n) if a!=p)
+  for a in range(n):
+   if a!=p:program.append(('shear',a,p,1))
+  for event in program:apply(event)
+  for a in range(n):
+   want=initial[a][:]
+   for j in range(k):want[n+j]+=1
+   if a!=p:want[n+k+a]+=1
+   need(state[a]==want,'integer arbitrary-target sandwich')
+  for event in reversed(program):apply(event,-1)
+  need(state==initial,'literal inverse of integer sandwich')
+ # Full original integer adjoint, including the removed root responses.
+ adj=word.adjoint()
+ def without_deleted():
+  out=[{} for _ in range(R)]
+  for j,(r,s) in enumerate(zip(word.g['roots'],word.w['rootroles'])):
+   if j not in deletedroots:
+    for t in r['targets']:out[s][t]=out[s].get(t,0)+1
+  for a,b,n in reversed(ops):
+   for t,c in out[a].items():out[b][t]=out[b].get(t,0)+c
+  return out
+ badadj=None;live=sorted(set(word.phys.values())-removed);index={r:2*v+i for i,r in enumerate(live)};columns=2*v+len(live)
+ need(len(live)==physical_R,'terminal physical inventory')
+ def execute(mode,direction=1,bits=None,mutation=None):
+  nonlocal badadj
+  major=mode=='bound';binary=mode=='F2'
+  unit=(lambda i:1) if major else (lambda i:1<<i) if binary else (lambda i:1<<(bits*i))
+  def add(x,c,y):return x+abs(c)*y if major else x^y if binary and c&1 else x if binary else x+c*y
+  X0=[unit(i) for i in range(v)];Y0=[unit(v+i) for i in range(v)];Z0={r:unit(index[r]) for r in live}
+  x=X0[:];y=Y0[:];z=Z0.copy();done=[];top=1
+  def track(value):
+   nonlocal top
+   if major:top=max(top,value)
+   return value
+  responses=adj
+  if mutation=='omit-ancestor-response':
+   if badadj is None:badadj=without_deleted()
+   responses=badadj
+  def read(s):
+   value=z[word.phys[s]]
+   for t,c in responses[s].items():y[t]=track(add(y[t],-direction*c,value))
+  def gate(i,sign):
+   a,b,n=ops[i];a,b=word.phys[a],word.phys[b];z[a]=track(add(z[a],sign,z[b]))
+  for s in range(R):
+   if s not in word.gauge and s not in removed:read(s)
+  for n,s in word.source.items():z[word.phys[s]]=track(add(z[word.phys[s]],1,x[n]))
+  for i in word.phase1:gate(i,1);done.append(i)
+  for r,s in zip(word.g['roots'],word.w['rootroles']):
+   if r['kind']=='center':
+    for t in r['targets']:y[t]=track(add(y[t],direction,z[word.phys[s]]))
+  for e in chosen:
+   if mutation=='omit-pre-target' and e is chosen[0]:continue
+   for t in e['targets']:
+    if t!=e['pivot']:y[t]=track(add(y[t],-1,y[e['pivot']]))
+  omitted=False
+  for j,i in enumerate(word.rest):
+   for s in at[j]:read(s)
+   if i in bywrite:
+    e=bywrite[i];a,b,n=ops[i]
+    if mutation=='omit-write' and not omitted:omitted=True
+    else:y[e['pivot']]=track(add(y[e['pivot']],direction,z[word.phys[b]]))
+   else:gate(i,1);done.append(i)
+   if i in after:
+    e=after[i]
+    for t in e['targets']:
+     if t!=e['pivot']:y[t]=track(add(y[t],1,y[e['pivot']]))
+  for s in at[len(word.rest)]:read(s)
+  for j,(r,s) in enumerate(zip(word.g['roots'],word.w['rootroles'])):
+   if r['kind']=='side' and j not in deletedroots:
+    for t in r['targets']:y[t]=track(add(y[t],direction,z[word.phys[s]]))
+   for e in deliveries[j]:
+    a,b=e['carrier'],e['passive'];x[a]=track(add(x[a],1,x[b]))
+    for t in e['receivers']:y[t]=track(add(y[t],direction,x[a]))
+  for e in word.k['entries']:
+   a,b=e['carrier'],e['passive'];x[a]=track(add(x[a],-1,x[b]))
+  need(done==[i for i in order if i not in bywrite],'exact retained forward operation list')
+  for i in reversed(done):gate(i,-1)
+  for n,s in word.source.items():z[word.phys[s]]=track(add(z[word.phys[s]],-1,x[n]))
+  if binary:want=[Y0[t]^X0[t] for t in range(v)]
+  else:
+   # Independent defining integer decoder from source-disjoint graph supports.
+   # It includes all original roots and every original partner delivery.
+   want=Y0[:];cache={}
+   for r in word.g['roots']:
+    n=r['node']
+    if n not in cache:
+     mask=C.sup[n];value=0
+     while mask:
+      bit=mask&-mask;mask-=bit;value=add(value,1,X0[bit.bit_length()-1])
+     cache[n]=value
+    for t in r['targets']:want[t]=add(want[t],direction,cache[n])
+   for e in word.k['entries']:
+    value=add(X0[e['carrier']],1,X0[e['passive']])
+    for t in e['receivers']:want[t]=add(want[t],direction,value)
+  if major:return max(top,max(a+b for a,b in zip(y,want)),max(z[r]+Z0[r] for r in live),max(a+b for a,b in zip(x,X0)))
+  need(y==want,'all output columns equal the independent defining decoder')
+  need(z==Z0 and x==X0,'all dirty and source columns restored')
+  return dict(mode=mode,direction=direction,formal_columns=columns,source_columns=v,target_columns=v,dirty_columns=len(live),all_outputs=True,all_sources_and_dirty_restored=True,partner_mix_unmix_at_original_anchors=True,retained_cleanup_literal_reverse=True)
+ bound=execute('bound');bits=8*((bound.bit_length()+2+7)//8);need((1<<bits)>2*bound,'injective integer residual digit bound')
+ print('bound',bound,'bits',bits,'columns',columns,flush=True)
+ formal=[execute('F2')];gc.collect();formal.append(execute('Z',1,bits));gc.collect();formal.append(execute('Z',-1,bits));gc.collect()
+ controls={}
+ for mutation in ['omit-ancestor-response','omit-pre-target','omit-write']:
+  try:execute('F2',mutation=mutation)
+  except AssertionError:controls[mutation]='REJECTED'
+  else:raise AssertionError('accepted mutation '+mutation)
+ original_hist={int(r):n for r,n in baseline['child_histogram'].items()}
+ profile=dict(baseline,R=physical_R,terminal_sinks=len(chosen),active_virtual_R=R-len(chosen),scalar_role_reserve=R,W_per_vertex=W,rank_per_vertex=rank,child_histogram=compact(child),physical_internal_histogram=compact(local),physical_target_histogram=compact(Y))
+ return dict(status='PASS complete terminal bit word, exact geometry/ledger/F2/integer columns',profile=profile,baseline_profile=baseline,selected=chosen,target_events=events,child_delta=compact(delta),formal=formal,integer_residual_bound=bound,packed_digit_bits=bits,base_exceeds_twice_bound=True,controls=controls,literal_sandwich_inverse=True,all_original_adjoint_responses_retained=True,original_partner_delivery_anchors_retained=True,new_frames_added=0,scalar_addition_delta=-sum(len(e['writes'])+2 for e in chosen),scalar_bill='Retain the original conservative full-word scalar inventory: removed old root reads/old root corrections and deleted inverse gates dominate the new pre/post target shears.',foreign_producer_replays=0,seconds=time.monotonic()-t0,maxrss=resource.getrusage(resource.RUSAGE_SELF).ru_maxrss)
+
+if __name__=='__main__':
+ if sys.flags.optimize:raise SystemExit('run without -O')
+ p=argparse.ArgumentParser();p.add_argument('--adapter',type=Path,required=True);p.add_argument('--selection',type=Path,required=True);p.add_argument('--out',type=Path,required=True);a=p.parse_args()
+ resource.setrlimit(resource.RLIMIT_CPU,(180,180));signal.alarm(180)
+ spec=importlib.util.spec_from_file_location('checked_bit_adapter',a.adapter);m=importlib.util.module_from_spec(spec);spec.loader.exec_module(m);w=m.Candidate();w.exact_frames()
+ pins={str(p):sha256(p.read_bytes()).hexdigest() for p in [Path(__file__),a.adapter,a.adapter.with_name('base_word.py'),Path(w.module.__file__),a.selection,*w.input_paths]}
+ a.out.with_suffix('.inputs.json').write_text(json.dumps(pins,sort_keys=True,indent=2)+'\n')
+ data=json.loads(a.selection.read_text());selected=data['selected'] if isinstance(data,dict) else data
+ result=prove(w,selected)
+ for path,digest in pins.items():need(sha256(Path(path).read_bytes()).hexdigest()==digest,'source changed during proof: '+path)
+ result['input_pins']=pins;a.out.write_text(json.dumps(result,sort_keys=True,indent=2)+'\n');print(json.dumps({k:result[k] for k in ['status','seconds','integer_residual_bound','packed_digit_bits','controls','maxrss']}),flush=True)
diff --git a/research/paired-cube-diagonal-bit-168/bit/word.py b/research/paired-cube-diagonal-bit-168/bit/word.py
new file mode 100644
index 0000000000..6b8d96043a
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/bit/word.py
@@ -0,0 +1,104 @@
+"""Independent complete formal admission of the PR168-v4 physical bit word with our own physical layer (endpoint descent and reuse pairs) and terminal sinks.
+
+Retains eumemic's exact integer linear algebra and original decoder/geometry
+predicates. Physical chains, chronological formal columns and paid moments
+use our separate verifier. No producer or foreign aggregate checker runs.
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+from hashlib import sha256
+import gzip,importlib.util,json,sys,time
+HERE=Path(__file__).resolve().parent
+SOURCE=HERE.parent/'references/pr168-v4'
+PKG=HERE.parent
+def module(name,path):
+    spec=importlib.util.spec_from_file_location(name,path);m=importlib.util.module_from_spec(spec);spec.loader.exec_module(m);return m
+def read(p):
+    raw=p.read_bytes();return json.loads(gzip.decompress(raw) if p.suffix=='.gz' else raw)
+old=module('own_frozen_word',PKG/'bit/base_word.py');need=old.need
+
+class Candidate(old.Candidate):
+    def __init__(self):
+        self.module=backend=module('exact_integer_backend',SOURCE/'research/paired-cube-bit/check_paired_cube_bit.py')
+        C=backend.Checker.__new__(backend.Checker);self.C=C;C.mut=None
+        B=F=PKG/'selected/bit'
+        self.input_paths=[B/'graph_p12.json',B/'profile_p12.json',F/'word_p12.json.gz',F/'frames_p12.json.gz',F/'kchron_p12.json']
+        C.g,C.prof,C.w,C.fr,C.k=map(read,self.input_paths)
+        C.h=C.g['h'];C.v=C.g['v'];C.frames()
+        self.h,self.v,self.R=C.h,C.v,C.prof['R'];self.w,self.g,self.k=C.w,C.g,C.k
+        self.ops=self.w['ops'];self.nf={int(n):f for n,f in self.w['node_frame'].items()}
+        self.source={int(n):s for n,s in self.w['sources'].items()};self.gauge={z['role']:z for z in self.w['gauges']}
+        need(len(self.gauge)==len(self.w['gauges']),'unique gauge roles')
+        self.original_opframe=tuple(self.nf[n] for a,b,n in self.ops);self.opframe=self.w['op_frame'][:]
+        need(len(self.opframe)==len(self.ops),'complete operation frame vector')
+        self.changed_frames=[i for i,(a,b) in enumerate(zip(self.original_opframe,self.opframe)) if a!=b]
+        self.phase1=sorted(self.w['phase1']);phase=set(self.phase1)
+        self.rest=[i for i in range(len(self.ops)) if i not in phase];order=self.phase1+self.rest
+        self.position={i:p for p,i in enumerate(order)};pos={i:p for p,i in enumerate(self.rest)}
+        self.order=[z['role'] for z in reversed(self.w['gauges'])]
+        self.role_ops=defaultdict(list)
+        for i in order:
+            for s in self.ops[i][:2]:self.role_ops[s].append(i)
+        need(all(xs==sorted(xs) for xs in self.role_ops.values()),'execution order preserves every role chronology')
+        self.first={s:pos.get(xs[0],-1) for s,xs in self.role_ops.items()}
+        self.last={s:pos.get(xs[-1],-1) for s,xs in self.role_ops.items()}
+        self.endframe={s:self.opframe[xs[-1]] for s,xs in self.role_ops.items()}
+        self.rootroles=set(self.w['rootroles']);self.rootkind={s:r['kind'] for s,r in zip(self.w['rootroles'],self.g['roots'])}
+        need(len(self.rootroles)==len(self.w['rootroles']),'unique root roles')
+        reads={int(s):p for s,p in self.w['reads'].items()}
+        need(set(reads)<=set(self.gauge),'only gauges have explicit reads')
+        self.readtime={s:reads.get(s,len(self.phase1))-len(self.phase1) for s in self.order}
+        need(all(0<=self.readtime[s]<=self.first[s] for s in self.order),'all gauge reads after centre phase and before first use')
+        self.pairs=[[b,a] for a,b in self.w['pairs']];self.donor=dict(self.pairs)
+        need(len(self.donor)==len(self.pairs)==len(set(self.donor.values())),'one-to-one aliases')
+        need(not set(self.donor)&set(self.donor.values()),'disjoint donor recipient sets')
+        for b,d in self.pairs:
+            need(d not in self.gauge and d not in self.rootroles and b in self.gauge,'ungauged non-root donor and gauged recipient')
+            need(self.last[d]<self.readtime[b],'donor dead before recipient read')
+        self.phys={s:self.donor.get(s,s) for s in range(self.R)}
+        need(len(set(self.phys.values()))==self.R-len(self.pairs),'physical slot count')
+        need(all(self.phys[a]!=self.phys[b] for a,b,n in self.ops),'distinct physical gate ports')
+        C.chains=self.base_chains
+
+    def base_chains(self):
+        """Reconstruct phase/gauge/source contracts; row() counts physical paths."""
+        C,w,h,v=self.C,self.w,self.h,self.v
+        order=self.phase1+self.rest
+        roleprev={};pred=[]
+        for i,(a,b,n) in enumerate(self.ops):
+            pred.append((roleprev.get(a,-1),roleprev.get(b,-1)));roleprev[a]=roleprev[b]=i
+        pending=[roleprev[s] for s in self.rootroles if self.rootkind[s]=='center' and s in roleprev];closure=set()
+        while pending:
+            i=pending.pop()
+            if i not in closure:closure.add(i);pending.extend(j for j in pred[i] if j>=0)
+        need(closure==set(self.phase1),'phase one is exactly centre closure')
+        content=[0]*self.R
+        for n,s in self.source.items():content[s]=1<<n
+        for i in order:
+            a,b,n=self.ops[i];args=self.g['args'][n]
+            if content[a]:need(args is not None and {content[a],content[b]}=={C.sup[t] for t in args},'actual addition operands')
+            else:need(content[b]==C.sup[n],'actual copied node value')
+            content[a]|=content[b]
+        co=[set() for _ in range(self.R)]
+        for r,s in zip(self.g['roots'],w['rootroles']):co[s].update(r['targets'])
+        for i in reversed(order):
+            a,b,n=self.ops[i];co[b].update(co[a])
+        touched={s for i in self.phase1 for s in self.ops[i][:2]}
+        for s,z in self.gauge.items():
+            need(s not in self.source.values() and s not in touched,'gauge untouched in phase one and source injection')
+            need(set(z['targets'])==co[s] and C.dimf[z['frame']]==z['dim']>0,'gauge dimension and complete response support')
+        src=Counter();members=Counter();labels=list(map(set,self.g['labels']))
+        for e in self.k['entries']:
+            c,d,f,cap=e['carrier'],e['passive'],e['mix_frame'],e['deliver_frame']
+            need(len(labels[c]&labels[d])==1,'orthogonal partner pair')
+            need(C.dimf[f]==2 and C.in_frame(C.chi[c],f) and C.in_frame(C.chi[d],f),'partner mix span')
+            need(C.sub(f,cap) and all(self.module.dot(C.cov[t],u)==0 for t in e['receivers'] for u in C.B[cap]),'partner mix delivery cap')
+            need(e['carrier_chain']==[w['source_frame'][c],f,cap,w['full_frame']] and e['passive_chain']==[w['source_frame'][d],f,w['full_frame']],'exact partner chains')
+            root=self.g['roots'][e['deliver_after_root']]
+            need(root['kind']=='side' and sorted(e['receivers'])==sorted(root['targets']),'partner delivery at own receiver root')
+            for chain in (e['carrier_chain'],e['passive_chain']):
+                for a,b in zip(chain,chain[1:]):
+                    need(C.sub(a,b),'nested source chronology');src[C.dimf[b]-C.dimf[a]]+=1
+            members[c]+=1;members[d]+=1
+        need(all(members[s]==1 for s in range(v)),'every source in one partner pair')
+        return Counter(),Counter(),src,self.R
diff --git a/research/paired-cube-diagonal-bit-168/certificate.json b/research/paired-cube-diagonal-bit-168/certificate.json
new file mode 100644
index 0000000000..7b80d2cd04
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/certificate.json
@@ -0,0 +1,4442 @@
+{
+  "arithmetic": {
+    "actual_uniform_bit_saving": "677340914792209011106157466449676890973/1000000000000000000000000000000000000000000",
+    "adverse_control_count": 21,
+    "adverse_controls": [
+      "full group stock replaced by local W",
+      "old ordinary reserve omitted",
+      "conservative old coarse reserve omitted",
+      "uniform supplier replaced by fictional finite profile",
+      "actual ordinary saving overstated",
+      "unpaid previous atom grid",
+      "scalar role reserve collapsed to physical stock",
+      "terminal removals omitted",
+      "terminal removals counted as aliases",
+      "odd divisor silently changed to 21",
+      "C0 detached from scalar charge",
+      "scalar group charge omitted",
+      "stale total row coefficient",
+      "inadequate polynomial row degree",
+      "float in otherwise unused metadata",
+      "noncontracting maximum child",
+      "transfer saving exceeds actual uniform supplier",
+      "old prefix charged at balanced parameters",
+      "old nonlinear scalar guard",
+      "old separate movement exposures",
+      "next final 1e-18 kappa grid point"
+    ],
+    "assembly": {
+      "absorption_gap": "14997626888680345751524011177060836353630220354194253091514663/500332244410424089575910423090842681151818848181152485337000000000000000000000000",
+      "constraints": {
+        "K_dominates_log": "332744409924090075910047758473270852729147270729394494318848181151818847514663/500332244410424089575910423090842681151818848181152485337000000000000000000000000",
+        "K_geometry": "375332369410299089700910423090842681151818848181152485337/500332244410424089575910423090842681151818848181152485337000000000000000000000000",
+        "a_below_b": "666489485337/1000000000000000000000000000000000000000",
+        "a_positive": "665489485336999999999999333510514663/1000000000000000000000000000000000000000",
+        "alpha_below_one": "999999000000999999000000500332244410424089575910423090842681151818848181152485337/1000664488820848179151820846181685362303637696362304970674000000000000000000000000",
+        "alpha_below_one_fourth": "249500633385363864636134865695980388696361303638694362837181151818848181152485337/1000664488820848179151820846181685362303637696362304970674000000000000000000000000",
+        "alpha_positive": "665488819848180151820345849440951879548120451881879831318848181151818847514663/1000664488820848179151820846181685362303637696362304970674000000000000000000000000",
+        "artificial_boundary": "28021267642263141736858268578810664824988675011325028334096043455456544543457456011/4002657955283392716607283384726741449214550785449219882696000000000000000000000000",
+        "b_below_one_over32": "30584510514663/1000000000000000",
+        "beta_below_one": "999999999999999999999999/1000000000000000000000000",
+        "beta_positive": "1/1000000000000000000000000",
+        "c_below_one": "499667255257331499999999875998734228005499999999999333510514663/500000000000000000000000000000000000000000000000000000000000000",
+        "c_positive": "332744742668500000000124001265771994500000000000666489485337/500000000000000000000000000000000000000000000000000000000000000",
+        "cell_above_band": "999999000000999998999999499667755589575910424089576909157318848181151818847514663/2001328977641696358303641692363370724607275392724609941348000000000000000000000000",
+        "compact_leaf": "665989485336999999999999333510514663/500000000000000000000000000000000000000000000000000000000000000",
+        "compact_reservations": "1/4000000000000000000000000",
+        "delta_below_one_eighth": "999999999999999999999999/8000000000000000000000000",
+        "delta_positive": "1/8000000000000000000000000",
+        "epsilon_below_one": "332744409924090075910423090842681151818848181152485337/500332244410424089575910423090842681151818848181152485337",
+        "epsilon_positive": "499999500000499999500000000000000000000000000000000000000/500332244410424089575910423090842681151818848181152485337",
+        "g1_above_kappa": "3749531805231189043903396771870981799077843043260608561/125083061102606022393977605772710670287954712045288121334250000000000000000",
+        "g2_above_kappa": "442581721611999999999333510514663/1000000000000000000000000000000000000000",
+        "g3_above_kappa": "14997626888680345751524011177060836353630220354194253091514663/500332244410424089575910423090842681151818848181152485337000000000000000000000000",
+        "g4_above_kappa": "442581721611999999999333510514663/1000000000000000000000000000000000000000",
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+        "external_stock": "The bit group cardinality is not inserted into external polynomial row stock.",
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+        "selector_stock": "Internally borrowed and restored q-adic rows; complete spectators preserved."
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+        "exact_grid": "One dyadic grid times 3^(-K), K=G*(D_complex+1); completed children preserve the incoming odd exponent.",
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+              1018,
+              1019,
+              1020,
+              1021,
+              1022,
+              1023
+            ],
+            "write_ranks": [
+              16,
+              18
+            ],
+            "writes": [
+              22489,
+              25211
+            ]
+          },
+          {
+            "first_any_old_read": 11146,
+            "first_pivot_old_read": 28247,
+            "last_write_chronology": 25299,
+            "pivot": 1024,
+            "post_after_op": 25123,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              16,
+              2
+            ],
+            "role": 10456,
+            "root_index": 1408,
+            "targets": [
+              1024,
+              1025,
+              1026,
+              1027,
+              1028,
+              1029,
+              1030,
+              1031
+            ],
+            "write_ranks": [
+              16,
+              18
+            ],
+            "writes": [
+              21356,
+              25123
+            ]
+          },
+          {
+            "first_any_old_read": 11146,
+            "first_pivot_old_read": 27997,
+            "last_write_chronology": 25229,
+            "pivot": 1088,
+            "post_after_op": 25053,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              15,
+              3
+            ],
+            "role": 10458,
+            "root_index": 1496,
+            "targets": [
+              1088,
+              1089,
+              1090,
+              1091,
+              1092,
+              1093,
+              1094,
+              1095
+            ],
+            "write_ranks": [
+              15,
+              18
+            ],
+            "writes": [
+              21365,
+              25053
+            ]
+          },
+          {
+            "first_any_old_read": 11146,
+            "first_pivot_old_read": 28177,
+            "last_write_chronology": 25241,
+            "pivot": 1128,
+            "post_after_op": 25065,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              15,
+              3
+            ],
+            "role": 10457,
+            "root_index": 1551,
+            "targets": [
+              1128,
+              1129,
+              1130,
+              1131,
+              1132,
+              1133,
+              1134,
+              1135
+            ],
+            "write_ranks": [
+              15,
+              18
+            ],
+            "writes": [
+              21362,
+              25065
+            ]
+          },
+          {
+            "first_any_old_read": 11146,
+            "first_pivot_old_read": 28213,
+            "last_write_chronology": 25028,
+            "pivot": 1136,
+            "post_after_op": 24852,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              12,
+              6
+            ],
+            "role": 10331,
+            "root_index": 1562,
+            "targets": [
+              1136,
+              1137,
+              1138,
+              1139,
+              1140,
+              1141,
+              1142,
+              1143
+            ],
+            "write_ranks": [
+              12,
+              18
+            ],
+            "writes": [
+              19736,
+              24852
+            ]
+          },
+          {
+            "first_any_old_read": 11146,
+            "first_pivot_old_read": 28059,
+            "last_write_chronology": 25182,
+            "pivot": 1184,
+            "post_after_op": 25006,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              4,
+              14
+            ],
+            "role": 8962,
+            "root_index": 1628,
+            "targets": [
+              1184,
+              1185,
+              1186,
+              1187,
+              1188,
+              1189,
+              1190,
+              1191
+            ],
+            "write_ranks": [
+              4,
+              18,
+              18
+            ],
+            "writes": [
+              13014,
+              24950,
+              25006
+            ]
+          },
+          {
+            "first_any_old_read": 11146,
+            "first_pivot_old_read": 28107,
+            "last_write_chronology": 25248,
+            "pivot": 1240,
+            "post_after_op": 25072,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              15,
+              3
+            ],
+            "role": 10351,
+            "root_index": 1705,
+            "targets": [
+              1240,
+              1241,
+              1242,
+              1243,
+              1244,
+              1245,
+              1246,
+              1247
+            ],
+            "write_ranks": [
+              15,
+              18
+            ],
+            "writes": [
+              20585,
+              25072
+            ]
+          },
+          {
+            "first_any_old_read": 11146,
+            "first_pivot_old_read": 28289,
+            "last_write_chronology": 25311,
+            "pivot": 1304,
+            "post_after_op": 25135,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              15,
+              3
+            ],
+            "role": 10463,
+            "root_index": 1793,
+            "targets": [
+              1304,
+              1305,
+              1306,
+              1307,
+              1308,
+              1309,
+              1310,
+              1311
+            ],
+            "write_ranks": [
+              15,
+              18
+            ],
+            "writes": [
+              21387,
+              25135
+            ]
+          }
+        ],
+        "status": "AUTHOR_EXACT_FINITE_COMPILER_CHECK_PASS",
+        "target_histogram": {
+          "0": 6270,
+          "1": 3960,
+          "10": 8,
+          "11": 2,
+          "12": 6,
+          "13": 2,
+          "14": 4,
+          "15": 12,
+          "16": 7,
+          "18": 1276,
+          "2": 23,
+          "3": 16,
+          "4": 7,
+          "5": 2,
+          "6": 7,
+          "7": 3,
+          "8": 5,
+          "9": 1
+        },
+        "targets_disjoint": true,
+        "v": 1320
+      },
+      "terminal_formal": {
+        "absolute_residual_bound": 394518,
+        "columns": [
+          {
+            "columns": 13163,
+            "direction": 1,
+            "dirty": 10523,
+            "outputs": true,
+            "restored": true
+          },
+          {
+            "columns": 13163,
+            "direction": -1,
+            "dirty": 10523,
+            "outputs": true,
+            "restored": true
+          }
+        ],
+        "controls": {
+          "omit-ancestor-response": "REJECTED",
+          "omit-first-write": "REJECTED",
+          "omit-pre-shear": "REJECTED"
+        },
+        "digit_bits": 24,
+        "input_pins": {
+          "baseline.json.gz": "5975dc6981aa18ac4ba4a1ca290d9163beedc8bc48e9600f4de5390758b57ef9",
+          "frames.json.gz": "05cf2b64cdc6142294512fcd963285ed535599b433691a547eaf775424f7b104",
+          "graph.json.gz": "4720aa3bf8960d89fca62cf8dc71dc7c5f8b3b85c59ecae727184ae12e1f7c18",
+          "physical-frames.json.gz": "bbbe32fb7a5d2415b7b2f130c46d1e20de392ea1cf6587fca6bc015879a3ca4f",
+          "physical-pairs.json.gz": "33b847e6f40e40809dce1c3eb25afe357b88b0b14c9a5ccca95b78553a595065",
+          "physical.py": "01961fad4b04aaa940d2dc99c87690bc861b3861200296974c5e065ce3446659",
+          "profile-before.json.gz": "e41fd008059d8a08d8f3078831e577201acd9f4aa115dd0197e28286cf281c80",
+          "profile.json.gz": "67de497a9c1ea64bb0cf9dc3ca655e7f9c704452a615f3e7eb45d124ed40c2c1",
+          "sinks.json": "d56196c9a6c4ed0d0eac660dbf5a7bdfbb118222e954cf2126a90ac0b1c84e6a",
+          "word.json.gz": "b70c03e2401df03f67bbba36e0c33704bde377902e446831a4df94833e935cc8"
+        },
+        "source_sha256": "d7fefe86e637584ad1bf53c8f7714b88d5abed92eaa29d110ef099536e51f516",
+        "status": "PASS full modified finite word"
+      }
+    },
+    "profile": {
+      "R": 10523,
+      "W_per_vertex": 13163,
+      "c": 21909,
+      "child_histogram": {
+        "1": 93600,
+        "10": 2589,
+        "11": 987,
+        "12": 3555,
+        "13": 642,
+        "14": 2142,
+        "15": 3255,
+        "16": 330,
+        "17": 132,
+        "18": 7788,
+        "19": 5829,
+        "2": 42555,
+        "20": 561,
+        "3": 29427,
+        "4": 10830,
+        "5": 7044,
+        "6": 4407,
+        "7": 2559,
+        "8": 3675,
+        "9": 1323
+      },
+      "deficit_per_vertex": 1320,
+      "h": 22,
+      "loss": 440,
+      "m": 66,
+      "matched": 12189,
+      "maxchild": 20,
+      "q": 3157,
+      "rank_per_vertex": 867438,
+      "reuse_pairs": 2310,
+      "scalar_role_reserve": 12877,
+      "terminal_sinks": 44,
+      "total_M_operations": 33466,
+      "v": 1320
+    },
+    "status": "PASS"
+  },
+  "kappa": "166261725903847/250000000000000000",
+  "scope": "Finite word, paid moments, full finite bridge and 47 strict arithmetic constraints; source-specified all-size contracts remain assumptions.",
+  "status": "PASS conditional finite witness"
+}
diff --git a/research/paired-cube-diagonal-bit-168/complex/check_sinks.py b/research/paired-cube-diagonal-bit-168/complex/check_sinks.py
new file mode 100644
index 0000000000..802e59c616
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/complex/check_sinks.py
@@ -0,0 +1,204 @@
+#!/usr/bin/env python3
+"""Author exact finite check for target-accumulating disjoint-root deletion.
+
+Portability-only adaptation by Chafik Boukhalfa with OpenAI Codex assistance.
+Original PR166 author James Chang. Uses frozen selected complex data; never imports upstream programs,
+regenerates a source graph, searches a new producer, or computes a new kappa.
+Python standard library. Copyright 2026; Apache-2.0.
+"""
+import argparse,gzip,hashlib,json,resource,signal,sys,time
+from collections import Counter
+from fractions import Fraction
+from pathlib import Path
+
+
+def main():
+ if sys.flags.optimize:raise SystemExit('Run without -O')
+ ap=argparse.ArgumentParser(description=__doc__)
+ ap.add_argument('--candidate',type=Path,required=True)
+ ap.add_argument('--output',type=Path,required=True)
+ a=ap.parse_args();start_time=time.monotonic()
+ resource.setrlimit(resource.RLIMIT_CPU,(60,60));signal.alarm(60)
+ pins=json.loads((a.candidate/'result.json').read_text())['pins'];data={}
+ for name in ['graph','word','frames','physical-frames','physical-pairs','profile-before','profile']:
+  name+='.json.gz';b=(a.candidate/name).read_bytes();assert hashlib.sha256(b).hexdigest()==pins[name]
+  data[name[:-8]]=json.loads(gzip.decompress(b))
+ g,w,fw,changes,pairs,row,profile=[data[n] for n in ['graph','word','frames','physical-frames','physical-pairs','profile-before','profile']]
+ H,V,R=g['h'],g['v'],row['R'];assert (H,V)==(22,1320)
+ def basis(rows):
+  b={}
+  for x in rows:
+   for p,y in sorted(b.items(),reverse=True):
+    if x>>p&1:x^=y
+   if x:
+    p=x.bit_length()-1
+    for k,y in list(b.items()):
+     if y>>p&1:b[k]=y^x
+    b[p]=x
+  return tuple(b[k] for k in sorted(b,reverse=True))
+ def perp(rows):
+  rows=basis(rows);p={r.bit_length()-1:r for r in rows};out=[]
+  for j in range(H):
+   if j in p:continue
+   x=1<<j
+   for k,r in p.items():
+    if r>>j&1:x|=1<<k
+   out.append(x)
+  return basis(out)
+ def within(A,B):return len(basis(tuple(A)+tuple(B)))==len(B)
+ def fmt(c):return {str(k):v for k,v in sorted(c.items()) if v}
+ FULL=basis(1<<i for i in range(H));sources={int(k):v for k,v in w['sources'].items()}
+ aliases={b:a for a,b,t in pairs};donors={a:b for a,b,t in pairs};deadline={b:t for a,b,t in pairs};alias=lambda s:aliases.get(s,s)
+ live=set(range(R))-set(aliases);assert len(live)==profile['physical_R']
+ phase=set(w['phase1']);chron=w['phase1']+[i for i in range(len(w['ops'])) if i not in phase];when={i:j for j,i in enumerate(chron)}
+ ofs=[perp(fw['annihilators'][n]) for aa,bb,n in w['ops']]
+ for i,F in changes:assert basis(F)==tuple(F);ofs[i]=tuple(F)
+ roleops=[[] for _ in range(R)];roots_of=[[] for _ in range(R)]
+ for i,(aa,bb,n) in enumerate(w['ops']):
+  assert alias(aa)!=alias(bb);roleops[aa].append(i);roleops[bb].append(i)
+ for j,s in enumerate(w['rootroles']):roots_of[s].append(j)
+ spans=[()]*len(g['args'])
+ for n,args in enumerate(g['args']):spans[n]=(g['inputs'][n],) if args is None else basis(spans[args[0]]+spans[args[1]])
+ rootframes=[spans[r['node']] if r['kind']=='center' else perp(g['inputs'][t] for t in r['targets']) for r in g['roots']]
+ selected={z['role']:z for z in w['selected']};first_read=[len(chron)+1]*V
+ for z in selected.values():
+  t=deadline.get(z['role']);position=len(phase) if t is None else when[t]
+  for y in z['targets']:first_read[y]=min(first_read[y],position)
+ # Recount the complete old physical row paths and compare every positive bin.
+ starts=[()]*R
+ for n,s in sources.items():starts[s]=(g['inputs'][n-1],)
+ for s,z in selected.items():starts[s]=perp(z['annihilator'])
+ rframe={s:rootframes[j] for j,s in enumerate(w['rootroles'])};rkind={s:g['roots'][j]['kind'] for j,s in enumerate(w['rootroles'])}
+ def chain(s):return [starts[s]]+[ofs[i] for i in roleops[s]]+([rframe[s]] if s in rframe else [])+[FULL]
+ local=Counter()
+ for s in live:
+  seq=chain(s)
+  if s in donors:seq=seq[:-1]+chain(donors[s])
+  assert all(within(A,B) for A,B in zip(seq,seq[1:]))
+  local.update(len(B)-len(A) for A,B in zip(seq,seq[1:]) if len(B)>len(A))
+  if s in sources.values():local[1]+=1
+  for t in (s,donors.get(s)):
+   if t is not None and rkind.get(t)=='center':local[len(rframe[t])]+=1
+ assert fmt(local)==profile['local_histogram']
+ source=Counter({int(k):v for k,v in profile['source_data_histogram'].items()});target=Counter({int(k):v for k,v in profile['target_data_histogram'].items()})
+ def children(L,T):
+  c=Counter({r:3*n for r,n in L.items() if r});c.update({r:3*n for r,n in T.items() if r});c.update({r:3*n for r,n in source.items() if r});c[2]+=2*V
+  for s,z in selected.items():
+   if s not in aliases:c[3*z['rank']]+=1
+  return c
+ oldchildren=children(local,target);assert fmt(oldchildren)==profile['child_histogram']
+ rejected=Counter();eligible=[]
+ for j,r in enumerate(g['roots']):
+  if r.get('channel')!='disjoint':continue
+  s=w['rootroles'][j];writes=[i for i in roleops[s] if w['ops'][i][0]==s];uses=[i for i in roleops[s] if w['ops'][i][1]==s];why=[];U=rootframes[j]
+  if s in aliases or s in donors:why.append('alias_or_donor')
+  if s in sources.values():why.append('source_injection')
+  if s in selected:why.append('selected_gauge')
+  if roots_of[s]!=[j]:why.append('multiple_roots')
+  if uses:why.append('retained_producer_consumer')
+  if not writes:why.append('no_writes')
+  if any(w['opcoeff'][i][0]!=1 for i in writes):why.append('nonadditive_destination')
+  if any(not within(ofs[i],U) for i in writes):why.append('outside_read_frame')
+  if any(i in phase for i in writes):why.append('center_phase_touch')
+  available_pivots=[t for t in r['targets'] if writes and first_read[t]>max(when[i] for i in writes)]
+  if not available_pivots:why.append('no_uncorrected_pivot')
+  if why:rejected.update(why);continue
+  assert len(r['targets'])==8 and len(U)==18 and r['coefficients']==['1/2']*8
+  # The sink's whole original trajectory is copied by only one target pivot.
+  # Every other target stays at0 until the common rank18 post-read.
+  path=[()]+[ofs[i] for i in writes]+[U];assert all(within(A,B) for A,B in zip(path,path[1:]))
+  prefix=[len(B)-len(A) for A,B in zip(path,path[1:]) if len(B)>len(A)]
+  assert sum(prefix)==18
+  eligible.append(dict(root_index=j,role=s,targets=r['targets'],pivot=available_pivots[0],writes=writes,write_ranks=[len(ofs[i]) for i in writes],prefix_children=prefix,pre_at_phase_cut_after_center_before_old_reads=True,post_after_op=writes[-1],last_write_chronology=max(when[i] for i in writes),first_pivot_old_read=first_read[available_pivots[0]],first_any_old_read=min(first_read[t] for t in r['targets'])))
+ assert len(set(t for x in eligible for t in x['targets']))==8*len(eligible)
+ removed={x['role'] for x in eligible};removed_ops={i for x in eligible for i in x['writes']}
+ assert all(aa not in removed and bb not in removed for i,(aa,bb,n) in enumerate(w['ops']) if i not in removed_ops)
+ # All formal source and dirty coefficients of every selected sink are checked
+ # by exact adjoint pullback of its current final value through actual aliases.
+ # Remaining source/dirty identity follows from literal inverse cancellation
+ # of the unchanged retained word; no sampler or packed-digit hypothesis.
+ def final_form(s,stop):
+  c={s:1}
+  for i in reversed(chron[:stop+1]):
+   aa,bb,n=w['ops'][i];aa,bb=alias(aa),alias(bb);ca,cb=w['opcoeff'][i];v=c.get(aa,0)
+   if v:
+    c[aa]=ca*v;c[bb]=c.get(bb,0)+cb*v
+    if not c[bb]:del c[bb]
+  fresh={n-1:c[alias(s)] for n,s in sources.items() if c.get(alias(s),0)}
+  return c,fresh
+ dirtyterms=[]
+ for item in eligible:
+  s=item['role'];c,fresh=final_form(s,item['last_write_chronology']);assert c[s]==1
+  assert not (set(c)&(removed-{s}))
+  cube={x//2 for x in g['labels'][item['targets'][0]]}
+  expected={n:1 for n,t in enumerate(g['labels']) if cube.isdisjoint(x//2 for x in t)}
+  assert fresh==expected and len(fresh)==448;dirtyterms.append(len(c)-1)
+  # Exact independent-target/sink/accumulation-symbol sandwich, both signs.
+  # Algebra includes arbitrary previous target corrections in y[t].
+  for direction in (1,-1):
+   N=16+len(item['writes']);ys=[[Fraction(int(i==t)) for i in range(N)] for t in range(8)]
+   for t in range(1,8):ys[t]=[a-b for a,b in zip(ys[t],ys[0])]
+   # Arbitrary intervening non-pivot corrections are explicit independent
+   # columns. The chosen pivot has none until after its post-read.
+   for t in range(1,8):ys[t][8+len(item['writes'])+t]+=1
+   for k,i in enumerate(item['writes']):ys[0][8+k]+=direction*Fraction(w['opcoeff'][i][1],2)
+   for t in range(1,8):ys[t]=[a+b for a,b in zip(ys[t],ys[0])]
+   for t in range(8):
+    expected=[Fraction(int(i==t)) if i<8 else direction*Fraction(w['opcoeff'][item['writes'][i-8]][1],2) if i<8+len(item['writes']) else Fraction(int(t>0 and i==8+len(item['writes'])+t)) for i in range(N)]
+    assert ys[t]==expected
+ # Whole simultaneous local and shared three-core ledger.
+ L=local.copy();T=target.copy()
+ for item in eligible:
+  L.subtract(Counter(item['prefix_children']+[4]));T[18]-=1;T.update(item['prefix_children'])
+ assert all(v>=0 for v in L.values()) and all(v>=0 for v in T.values())
+ # Replay every target event in the actual deadline order, interleaving the
+ # new pivot writes and post-reads. Pre-shears are at the common zero frame
+ # just after the center phase and before its first compensation read.
+ by_write={i:item for item in eligible for i in item['writes']}
+ after={item['writes'][-1]:item for item in eligible}
+ read_at={}
+ for z in reversed(w['selected']):
+  d=deadline.get(z['role']);position=len(phase) if d is None else when[d]
+  read_at.setdefault(position,[]).append(z)
+ current=[() for _ in range(V)];actual_target=Counter();target_events=[]
+ def move_target(t,U,kind,index):
+  U=tuple(U);assert within(current[t],U),(kind,index,t,current[t],U)
+  actual_target[len(U)-len(current[t])]+=1
+  target_events.append([kind,index,t,list(current[t]),list(U)])
+  current[t]=U
+ for position,i in enumerate(chron):
+  for z in read_at.get(position,[]):
+   U=perp(z['annihilator'])
+   for t in z['targets']:move_target(t,U,'old-read',z['role'])
+  if i in by_write:
+   item=by_write[i];move_target(item['pivot'],ofs[i],'pivot-write',i)
+  if i in after:
+   item=after[i];U=rootframes[item['root_index']]
+   for t in item['targets']:move_target(t,U,'post-read',i)
+   assert all(current[t]==current[item['pivot']]==U for t in item['targets'])
+ for j,r in enumerate(g['roots']):
+  if r['kind']=='side' and w['rootroles'][j] not in removed:
+   for t in r['targets']:move_target(t,rootframes[j],'retained-root',j)
+ for t,q in enumerate(g['inputs']):move_target(t,perp((q,)),'target-finish',t)
+ assert fmt(Counter({r:n for r,n in actual_target.items() if r}))==fmt(Counter({r:n for r,n in T.items() if r}))
+ # Independent reverse of each scalar sandwich, including arbitrary inserted
+ # non-pivot corrections, yields the inverse rather than assuming a sign rule.
+ for item in eligible:
+  N=16+len(item['writes']);initial=[[Fraction(int(i==t)) for i in range(N)] for t in range(8)]
+  state=[r[:] for r in initial];events=[]
+  for t in range(1,8):events.append(('shear',t,0,Fraction(-1)))
+  for k,i in enumerate(item['writes']):events.append(('basis',0,8+k,Fraction(w['opcoeff'][i][1],2)))
+  for t in range(1,8):events.append(('basis',t,8+len(item['writes'])+t,Fraction(1)))
+  for t in range(1,8):events.append(('shear',t,0,Fraction(1)))
+  def apply(e,sign):
+   kind,t,s,c=e;c*=sign
+   if kind=='basis':state[t][s]+=c
+   else:state[t]=[a+c*b for a,b in zip(state[t],state[s])]
+  for e in events:apply(e,1)
+  for e in reversed(events):apply(e,-1)
+  assert state==initial
+ C=children(L,T);n=len(eligible);delta={str(k):C[k]-oldchildren[k] for k in C.keys()|oldchildren.keys() if C[k]!=oldchildren[k]}
+ assert delta==({'4':-3*n,'18':-3*n} if n else {});W=profile['W_per_vertex']-n;rank=sum(k*v for k,v in C.items());assert rank==profile['rank_per_vertex']-66*n and 66*W-rank==1320
+ result=dict(status='AUTHOR_EXACT_FINITE_COMPILER_CHECK_PASS',scope='Only all-eight disjoint roots; retained full proof contracts unchanged',input_pins={k:pins[k] for k in pins if k in [n+'.json.gz' for n in data]},h=H,v=V,old_physical_R=len(live),new_physical_R=len(live)-n,old_W=profile['W_per_vertex'],new_W=W,old_rank=profile['rank_per_vertex'],new_rank=rank,deficit=1320,eligible_count=n,rejected_conditions=dict(rejected),targets_disjoint=True,simultaneous_compatible=True,all_formal_source_and_dirty_coefficients=True,formal_original_source_dirty_target_columns=2*V+len(live),remaining_formal_columns=2*V+len(live)-n,chronological_inverse_by_literal_reverse=True,full_old_physical_histogram_verified=True,all_new_target_events_replayed=True,scalar_sandwich_literal_reverse_checked=True,shared_child_delta=delta,scalar_gate_delta=-sum(len(x['writes'])+2 for x in eligible),new_odd_denominator=False,new_kappa_claimed=False,local_histogram=fmt(L),target_histogram=fmt(T),child_histogram=fmt(C),sinks=eligible,dirty_coefficient_nonzero_range=[min(dirtyterms),max(dirtyterms)],seconds=time.monotonic()-start_time,rss_kib=resource.getrusage(resource.RUSAGE_SELF).ru_maxrss)
+ a.output.write_text(json.dumps(result,indent=2)+'\n');print(json.dumps({k:v for k,v in result.items() if k not in ['sinks','local_histogram','target_histogram','child_histogram','input_pins']},indent=2))
+if __name__=='__main__':main()
diff --git a/research/paired-cube-diagonal-bit-168/complex/formal_sinks.py b/research/paired-cube-diagonal-bit-168/complex/formal_sinks.py
new file mode 100644
index 0000000000..703d57f386
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/complex/formal_sinks.py
@@ -0,0 +1,137 @@
+"""Independent full execution of PR166 sink substitution on our fused168.
+
+James Chang's PR166 compiler; fused168 source lineage in its SOURCE.json.
+New full packed-column executor by Chafik Boukhalfa with OpenAI Codex.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from collections import defaultdict
+from hashlib import sha256
+import argparse,gzip,json,time,importlib.util,sys
+
+BASE=Path(__file__).resolve().parent
+ap=argparse.ArgumentParser(description=__doc__)
+ap.add_argument('--candidate',type=Path,default=BASE.parents[1]/'r13-search/fused168')
+ap.add_argument('--selection',type=Path,default=BASE/'fused168-screen.json')
+ap.add_argument('--formal-helper',type=Path)
+ap.add_argument('--output',type=Path,default=BASE/'formal-sinks.json')
+args=ap.parse_args();INPUT=args.candidate;HELPER=args.formal_helper or INPUT/'prove.py'
+if sys.flags.optimize:raise SystemExit('Run without -O')
+spec=importlib.util.spec_from_file_location('original_formal',HELPER)
+old=importlib.util.module_from_spec(spec);spec.loader.exec_module(old)
+def load(name):return json.loads(gzip.decompress((INPUT/(name+'.json.gz')).read_bytes()))
+g,w,row=[load(n) for n in ['graph','word','profile-before']]
+pairs=load('physical-pairs');selection=json.loads(args.selection.read_text())
+for name,digest in selection['input_pins'].items():assert sha256((INPUT/name).read_bytes()).hexdigest()==digest,name
+sinks=selection['sinks'];removed={s['role'] for s in sinks};sinkroots={s['root_index'] for s in sinks}
+alias={b:a for a,b,t in pairs};R=row['R'];v=g['v']
+live=[s for s in range(R) if s not in alias and s not in removed]
+slot={s:i for i,s in enumerate(live)}
+phys=lambda s:slot[alias.get(s,s)]
+D,adj=old.adjoint(g,w,R)
+ops=w['ops'];coef=w['opcoeff'];phase=w['phase1'];rest=[i for i in range(len(ops)) if i not in set(phase)]
+deadlines=dict((b,t) for a,b,t in pairs);at=defaultdict(list)
+for s,t in deadlines.items():
+ if t is not None:at[t].append(s)
+selected=[z['role'] for z in w['selected']];deferred=set(selected)
+sources={int(x):s for x,s in w['sources'].items()}
+bywrite={i:s for s in sinks for i in s['writes']};after={s['writes'][-1]:s for s in sinks}
+cnum=[[int(6*Q(c)) for c in r['coefficients']] for r in g['roots']]
+assert len(live)==selection['new_physical_R']
+
+def execute(direction,bits=None,mutation=None):
+ # With bits=None this is an absolute coefficient L1 majorant. With bits
+ # supplied it is the actual signed word on an injectively packed basis.
+ major=bits is None; top=6; count=2*v+len(live)
+ def basis(i):return 6 if major else 6<<(bits*i)
+ def add(x,c,y):return x+abs(c)*y if major else x+c*y
+ def track(xs):
+  nonlocal top
+  if major:top=max(top,max(xs,default=0))
+ X=[basis(i) for i in range(v)];Y=[basis(v+i) for i in range(v)];Z=[basis(2*v+i) for i in range(len(live))]
+ initial=Y[:];z=Z[:]
+ # Retain the ORIGINAL response of every retained dirty coordinate,
+ # including responses propagated through the deleted sinks.
+ virtual=[0 if s in deferred or s in alias or s in removed else Z[phys(s)] for s in range(R)]
+ def gate(values,i):
+  a,b,_=ops[i];ca,cb=coef[i]
+  values[a]=abs(ca)*values[a]+abs(cb)*values[b] if major else ca*values[a]+cb*values[b]
+ def decode(values,kind=None,skip=False):
+  out=[0]*v
+  for j,(root,s,cs) in enumerate(zip(g['roots'],w['rootroles'],cnum)):
+   if kind is not None and root['kind']!=kind:continue
+   if skip and j in sinkroots:continue
+   for t,c in zip(root['targets'],cs):out[t]=add(out[t],c,values[s])
+  assert all(x%6==0 for x in out)
+  return [x//6 for x in out]
+ for i in range(len(ops)):gate(virtual,i)
+ response=decode(virtual,skip=mutation=='omit-ancestor-response')
+ Y=[add(y,-direction,r) for y,r in zip(Y,response)];track(Y);del virtual,response
+ for n,s in sources.items():z[phys(s)]=add(z[phys(s)],1,X[n-1])
+ done=[]
+ def actualgate(i):
+  a,b,_=ops[i];ca,cb=coef[i];a,b=phys(a),phys(b)
+  z[a]=abs(ca)*z[a]+abs(cb)*z[b] if major else ca*z[a]+cb*z[b]
+  if major:
+   nonlocal top
+   top=max(top,z[a])
+  done.append(i)
+ def read(s):
+  for t,c in D[s].items():
+   assert z[phys(s)]%6==0
+   Y[t]=add(Y[t],-direction*c,z[phys(s)]//6)
+  track(Y)
+ for i in phase:actualgate(i)
+ virtual=[0 if s in removed else z[phys(s)] for s in range(R)]
+ center=decode(virtual,'center');Y=[add(y,direction,c) for y,c in zip(Y,center)];track(Y);del virtual,center
+ for sink in sinks:
+  pivot=sink['pivot']
+  for t in sink['targets']:
+   if t!=pivot and mutation!='omit-pre-shear':Y[t]=add(Y[t],-1,Y[pivot])
+ track(Y)
+ for s in reversed(selected):
+  if deadlines.get(s) is None:read(s)
+ for i in rest:
+  for s in at.get(i,[]):read(s)
+  if i in bywrite:
+   sink=bywrite[i];pivot=sink['pivot'];b=phys(ops[i][1]);assert z[b]%2==0
+   if mutation!='omit-first-write' or i!=sinks[0]['writes'][0]:Y[pivot]=add(Y[pivot],direction*coef[i][1],z[b]//2)
+   track(Y)
+  else:actualgate(i)
+  if i in after:
+   sink=after[i];pivot=sink['pivot']
+   for t in sink['targets']:
+    if t!=pivot:Y[t]=add(Y[t],1,Y[pivot])
+   track(Y)
+ virtual=[0 if s in removed else z[phys(s)] for s in range(R)]
+ side=decode(virtual,'side',True);Y=[add(y,direction,c) for y,c in zip(Y,side)];del virtual,side
+ for a in range(0,v,8):
+  for j,q in enumerate(g['inputs'][a:a+8]):
+   value=0
+   for r,x in zip(g['inputs'][a:a+8],X[a:a+8]):
+    c=1 if (q^r).bit_count()==6 else -1 if (q^r).bit_count()==2 else 0
+    value=add(value,c,x)
+   assert value%2==0;Y[a+j]=add(Y[a+j],direction,value//2)
+ track(Y)
+ if not major:assert Y==[y+direction*x for y,x in zip(initial,X)],'all formal output columns'
+ for i in reversed(done):
+  a,b,_=ops[i];ca,cb=coef[i];a,b=phys(a),phys(b)
+  z[a]=z[a]+abs(cb)*z[b] if major else ca*(z[a]-cb*z[b])
+  if major:top=max(top,z[a])
+ for n,s in sources.items():z[phys(s)]=add(z[phys(s)],-1,X[n-1])
+ track(z)
+ if not major:assert z==Z,'retained arbitrary dirty columns restored'
+ # Include the expected map in a safe final residual bound.
+ return top+12 if major else dict(direction=direction,columns=count,dirty=len(live),outputs=True,restored=True)
+
+if __name__=='__main__':
+ t=time.monotonic();bound=execute(1);bits=8*((bound.bit_length()+2+7)//8);assert 2**bits>2*bound
+ print('majorant',bound,'digit bits',bits,flush=True)
+ columns=[execute(d,bits) for d in [1,-1]]
+ controls={}
+ for mutation in ['omit-first-write','omit-ancestor-response','omit-pre-shear']:
+  try:execute(1,bits,mutation)
+  except AssertionError:controls[mutation]='REJECTED'
+  else:raise AssertionError('accepted '+mutation)
+ result=dict(status='PASS full modified finite word',absolute_residual_bound=bound,digit_bits=bits,columns=columns,controls=controls,source_sha256=sha256(Path(__file__).read_bytes()).hexdigest(),input_pins={p.name:sha256(p.read_bytes()).hexdigest() for p in list(INPUT.glob('*.json.gz'))+[args.selection,HELPER]},seconds=time.monotonic()-t)
+ args.output.write_text(json.dumps(result,indent=2)+'\n');print(json.dumps(result))
diff --git a/research/paired-cube-diagonal-bit-168/complex/physical.py b/research/paired-cube-diagonal-bit-168/complex/physical.py
new file mode 100644
index 0000000000..5874de8cb8
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/complex/physical.py
@@ -0,0 +1,168 @@
+#!/usr/bin/env python3
+"""Exact audit of the extended-link fused PR161/162 complex physical word, including every input.
+
+Chafik Boukhalfa with OpenAI Codex assistance. Apache-2.0.
+The scalar signed graph, binary frames and original compiler are credited to
+icekylinx (PR144); searched modules and physical reuse baseline to eumemic
+(PR157 and PR161), extended carrier closure by DaysSky (PR162), with the lineage retained there. New checks independently execute
+all physical aliases, deadline reads and inverse cleanup on an injective
+packed formal basis, using a separately derived absolute coefficient bound.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from collections import defaultdict,Counter
+from hashlib import sha256
+import argparse,gzip,json,sys,time,copy,gc,resource
+
+
+def adjoint(g,word,R):
+    D=[{} for _ in range(R)]
+    for root,s in zip(g['roots'],word['rootroles']):
+        for t,c in zip(root['targets'],root['coefficients']):
+            c=Q(c)*6;assert c.denominator==1;D[s][t]=D[s].get(t,0)+int(c)
+    for (a,b,_),(ca,cb) in zip(reversed(word['ops']),reversed(word['opcoeff'])):
+        for t,c in D[a].items():D[b][t]=D[b].get(t,0)+cb*c
+        if ca<0:D[a]={t:-c for t,c in D[a].items()}
+    D=[{t:c for t,c in row.items() if c} for row in D]
+    for z in word['selected']:assert set(D[z['role']])<=set(z['targets'])
+    receipt=dict(nonzeros=sum(map(len,D)),max_numerator_over_6=max(abs(c) for row in D for c in row.values()),L1=sum(abs(c) for row in D for c in row.values()),gauge_supports_contained=True)
+    selected={z['role']:D[z['role']] for z in word['selected']}
+    return selected,receipt
+
+class Formal:
+    def __init__(self,g,word,row,pairs,D):
+        self.g,self.word,self.row,self.D=g,word,row,D;self.v=g['v'];self.R=row['R']
+        self.merge={b:a for a,b,t in pairs};assert len(self.merge)==len(pairs)==len(set(self.merge.values()))
+        assert not set(self.merge)&set(self.merge.values())
+        self.live=[s for s in range(self.R) if s not in self.merge];self.slot={s:i for i,s in enumerate(self.live)}
+        self.phys=[self.slot[self.merge.get(s,s)] for s in range(self.R)]
+        self.sources={int(x):s for x,s in word['sources'].items()}
+        self.selected=[z['role'] for z in word['selected']];self.deferred=set(self.selected)
+        self.phase=sorted(word['phase1']);pset=set(self.phase);self.rest=[i for i in range(len(word['ops'])) if i not in pset]
+        self.deadline={b:t for a,b,t in pairs};self.at=defaultdict(list)
+        for b,t in self.deadline.items():
+            if t is not None:self.at[t].append(b)
+        self.ops=[tuple(x) for x in word['ops']];self.coef=[tuple(x) for x in word['opcoeff']]
+        assert all(a in (-1,1) and b in (-1,1) for a,b in self.coef)
+        assert all(self.phys[a]!=self.phys[b] for a,b,x in self.ops)
+        self.cnum=[[int(6*Q(c)) for c in r['coefficients']] for r in g['roots']]
+        self.bound=self.coefficient_bound()
+        self.width=8*((self.bound.bit_length()+2+7)//8);self.base=1<<self.width
+        assert self.base>2*self.bound
+    def coefficient_bound(self):
+        # Unscaled formal variables: a physical slot has coefficients bounded
+        # by its absolute row sum b. Actual packed slots use 6 times the basis.
+        g,word,v,R=self.g,self.word,self.v,self.R
+        old=[int(s not in self.deferred and s not in self.merge) for s in range(R)]
+        for (a,b,_),(ca,cb) in zip(self.ops,self.coef):old[a]=abs(ca)*old[a]+abs(cb)*old[b]
+        y=[6]*v
+        for root,s,cs in zip(g['roots'],word['rootroles'],self.cnum):
+            for t,c in zip(root['targets'],cs):y[t]+=abs(c)*old[s]
+        b=[1]*len(self.live);top=max([6*x for x in old]+y)
+        for node,s in self.sources.items():b[self.phys[s]]+=1
+        done=[]
+        def gate(i):
+            nonlocal top
+            (a,c,_),(ca,cb)=self.ops[i],self.coef[i];a,c=self.phys[a],self.phys[c]
+            b[a]=abs(ca)*b[a]+abs(cb)*b[c];top=max(top,6*b[a]);done.append(i)
+        def read(s):
+            for t,c in self.D[s].items():y[t]+=abs(c)*b[self.phys[s]]
+        for i in self.phase:gate(i)
+        # All root/center positive readouts may be safely bounded by their
+        # value after the complete M: centers are closed in phase one.
+        for s in reversed(self.selected):
+            if self.deadline.get(s) is None:read(s)
+        for i in self.rest:
+            for s in self.at.get(i,[]):read(s)
+            gate(i)
+        for root,s,cs in zip(g['roots'],word['rootroles'],self.cnum):
+            for t,c in zip(root['targets'],cs):y[t]+=abs(c)*b[self.phys[s]]
+        # K has four +/-1/2 source contributions in every output row.
+        y=[x+12+6 for x in y];top=max(top,max(y))
+        for i in reversed(done):
+            (a,c,_),(ca,cb)=self.ops[i],self.coef[i];a,c=self.phys[a],self.phys[c]
+            b[a]=abs(ca)*b[a]+abs(ca*cb)*b[c];top=max(top,6*b[a])
+        for node,s in self.sources.items():b[self.phys[s]]+=1;top=max(top,6*b[self.phys[s]])
+        return top
+    def decode(self,z,kind=None):
+        out=[0]*self.v
+        for root,s,cs in zip(self.g['roots'],self.word['rootroles'],self.cnum):
+            if kind is not None and root['kind']!=kind:continue
+            for t,c in zip(root['targets'],cs):out[t]+=c*z[s]
+        assert all(x%6==0 for x in out);return [x//6 for x in out]
+    def run(self,direction,mutation=None):
+        v,R,w=self.v,self.R,self.width
+        X=[6<<(w*i) for i in range(v)];Y0=[6<<(w*(v+i)) for i in range(v)];Z=[6<<(w*(2*v+i)) for i in range(len(self.live))]
+        # Compute precisely the initial reads D[s]z_s without expanding
+        # nongauged D: exact transposition identity D=JM gives this cheaper
+        # independent algebraic evaluation. All late physical reads below
+        # use the explicitly computed integer adjoint rows.
+        old=[Z[self.phys[s]] if s not in self.deferred and s not in self.merge else 0 for s in range(R)]
+        for (a,b,_),(ca,cb) in zip(self.ops,self.coef):old[a]=ca*old[a]+cb*old[b]
+        old=self.decode(old);y=[y-direction*z for y,z in zip(Y0,old)];del old;gc.collect()
+        z=Z[:];done=[]
+        def read(s):
+            if mutation=='missing-read' and s==self.selected[0]:return
+            value=z[self.phys[s]];assert value%6==0
+            for t,c in self.D[s].items():y[t]-=direction*c*(value//6)
+        def gate(i):
+            (a,b,_),(ca,cb)=self.ops[i],self.coef[i];a,b=self.phys[a],self.phys[b]
+            z[a]=ca*z[a]+cb*z[b];done.append(i)
+        for node,s in self.sources.items():z[self.phys[s]]+=X[node-1]
+        for i in self.phase:gate(i)
+        physical_virtual=[z[self.phys[s]] for s in range(R)]
+        center=self.decode(physical_virtual,'center');del physical_virtual
+        for t,c in enumerate(center):y[t]+=direction*c
+        del center
+        late=[b for b,t in self.deadline.items() if t is not None];early=late[-1] if mutation=='early-read' and late else None
+        for s in reversed(self.selected):
+            if self.deadline.get(s) is None or s==early:read(s)
+        for i in self.rest:
+            for s in self.at.get(i,[]):
+                if s!=early:read(s)
+            gate(i)
+        physical_virtual=[z[self.phys[s]] for s in range(R)]
+        side=self.decode(physical_virtual,'side');del physical_virtual
+        for t,c in enumerate(side):y[t]+=direction*c
+        del side
+        def K(xs):
+            out=[]
+            for a in range(0,v,8):
+                block=xs[a:a+8];qs=self.g['inputs'][a:a+8]
+                for q in qs:
+                    value=sum((1 if (q^r).bit_count()==6 else -1 if (q^r).bit_count()==2 else 0)*x for r,x in zip(qs,block));assert value%2==0;out.append(value//2)
+            return out
+        mixed=K(X);assert K(mixed)==X
+        for t,x in enumerate(mixed):y[t]+=direction*x
+        assert y==[y+direction*x for y,x in zip(Y0,X)],'full formal output map'
+        for i in reversed(done):
+            (a,b,_),(ca,cb)=self.ops[i],self.coef[i];a,b=self.phys[a],self.phys[b]
+            z[a]=ca*(z[a]-cb*z[b])
+        for node,s in self.sources.items():z[self.phys[s]]-=X[node-1]
+        assert z==Z,'all physical dirty columns restored'
+        return dict(direction=direction,every_formal_column=True,source_columns=v,target_columns=v,dirty_columns=len(self.live),K_and_inverse=True,all_aliases_and_deadline_reads=True,all_physical_dirty_slots_restored=True)
+
+def main():
+    ap=argparse.ArgumentParser();ap.add_argument('--candidate',type=Path,required=True);ap.add_argument('--source',type=Path,required=True);ap.add_argument('--out',type=Path,required=True);a=ap.parse_args();sys.path.insert(0,str(a.source/'scripts'))
+    from verify_fused import verify
+    from paired_cube_physical import physical
+    start=time.time();pins=json.loads((a.candidate/'result.json').read_text())['pins'];data={}
+    for name in ['graph','baseline','frames','word','profile-before','physical-frames','physical-pairs','profile']:
+        p=a.candidate/(name+'.json.gz');assert sha256(p.read_bytes()).hexdigest()==pins[p.name];data[name]=json.loads(gzip.decompress(p.read_bytes()))
+    g,b,w,word,row=[data[n] for n in ['graph','baseline','frames','word','profile-before']]
+    finite=verify(g,b,w,word,row);print('PASS exact signed scalar pairs and pre-reuse complete physical frame word',flush=True)
+    actual=physical(g,w,word,row,data['physical-frames'],data['physical-pairs']);assert json.loads(json.dumps(actual))==data['profile'];print('PASS all descended frames, nested chains, aliases and target read chronology',flush=True)
+    D,adj=adjoint(g,word,row['R']);formal=Formal(g,word,row,data['physical-pairs'],D);print('PASS exact adjoint; formal coefficient bound',formal.bound,'digit bits',formal.width,flush=True)
+    columns=[formal.run(d) for d in (1,-1)];print('PASS every formal column under both shear signs',flush=True)
+    mutations={}
+    for tag in ['missing-read','early-read']:
+        try:formal.run(1,tag)
+        except AssertionError:mutations[tag]='REJECTED'
+        else:raise AssertionError('accepted mutation '+tag)
+    cs=[Q(c) for r in g['roots'] for c in r['coefficients']]
+    assert set(cs)<={Q(1,2),Q(-1,2),Q(1,3),Q(-1,6)} and 3*row['q']<2**15
+    local=4*(row['c']+row['v'])+10*row['v']+4*row['h']*row['v']+4*row['h']**2+8*row['h']+8+2*row['h']+8*row['R']*row['v']*(row['total_M_operations']+16)+32*row['v']
+    scalar=dict(c=row['c'],q=row['q'],M_operations=row['total_M_operations'],logical_R=row['R'],physical_R=actual['physical_R'],handoffs=actual['pairs'],scalar_denominators_divide=6,fixed_odd_divisor=3,gate_and_inverse_coefficients=[-1,1],decoder_coefficients=sorted(map(str,set(cs))),three_q_less_than_2pow15=True,binary_dirty_read_bits=row['total_M_operations']+16,local_group_upper=local,scope='Conservative inherited virtual-word bill retained after aliasing, including all compensated reads. Alias and frame route choices are finite permutations/Clifford adapters; their separate global movement bill remains an assembly obligation.')
+    result=dict(status='PASS exact fused physical candidate, all formal source/target/dirty columns',source_sha256={str(p):sha256(p.read_bytes()).hexdigest() for p in [Path(__file__),Path(__file__).with_name('verify_fused.py'),a.source/'scripts/paired_cube_physical.py',a.source/'scripts/paired_cube/frames.py',a.source/'notes/general-clifford-frames.tex']},input_sha256=pins,pre_reuse_finite=finite,physical=actual,exact_adjoint=adj,formal=dict(columns=columns,absolute_coefficient_bound=formal.bound,packed_digit_bits=formal.width,base_exceeds_twice_bound=True),mutations=mutations,scalar_bound=scalar,seconds=time.time()-start,maxrss=resource.getrusage(resource.RUSAGE_SELF).ru_maxrss,foreign_producer_replays=0,scope='Finite candidate audit only. General exact Clifford and shared cover contracts, analytic supplier bridge, and paid global multiplication assembly remain separate obligations.')
+    a.out.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n');print(json.dumps({k:result[k] for k in ['status','seconds','maxrss','scalar_bound']}),flush=True)
+if __name__=='__main__':main()
diff --git a/research/paired-cube-diagonal-bit-168/complex/prove.py b/research/paired-cube-diagonal-bit-168/complex/prove.py
new file mode 100644
index 0000000000..a3bf2ab75c
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/complex/prove.py
@@ -0,0 +1,74 @@
+#!/usr/bin/env python3
+"""Portable exact physical and moment certificate for the selected fused word."""
+from fractions import Fraction as Q
+from pathlib import Path
+import gzip
+import json
+import subprocess
+import sys
+import tempfile
+
+HERE=Path(__file__).resolve().parent;PACKAGE=HERE.parent;ROOT=HERE.parents[2];SRC=PACKAGE/'references/pr168-v4'
+sys.path.insert(0,str(PACKAGE/'arithmetic'));sys.dont_write_bytecode=True
+from interval_moment import saving_grid
+
+
+def js(x):
+    if isinstance(x,Q):return str(x)
+    if isinstance(x,dict):return {str(k):js(v) for k,v in x.items()}
+    if isinstance(x,(tuple,list)):return [js(v) for v in x]
+    return x
+
+
+def main():
+    assert not sys.flags.optimize
+    candidate=PACKAGE/'selected/complex'
+    with tempfile.TemporaryDirectory(prefix='paired-cube-exact-') as tmp:
+        out=Path(tmp)/'physical.json'
+        run=subprocess.run([sys.executable,'-B',str(HERE/'physical.py'),'--candidate',str(candidate),'--source',str(SRC),'--out',str(out)],text=True,capture_output=True)
+        if run.returncode:raise ValueError(run.stdout+run.stderr)
+        physical=json.loads(out.read_text())
+    physical.pop('seconds',None);physical.pop('maxrss',None)
+    physical['source_sha256']={str(Path(k).resolve().relative_to(ROOT)):v for k,v in physical['source_sha256'].items()}
+    with tempfile.TemporaryDirectory(prefix='paired-cube-sinks-') as tmp:
+        checked=Path(tmp)/'selection.json';formal_path=Path(tmp)/'formal.json'
+        command=[sys.executable,'-B',str(HERE/'check_sinks.py'),'--candidate',str(candidate),'--output',str(checked)]
+        run=subprocess.run(command,text=True,capture_output=True)
+        if run.returncode:raise ValueError(run.stdout+run.stderr)
+        selection=json.loads(checked.read_text());frozen=json.loads((candidate/'sinks.json').read_text())
+        for value in (selection,frozen):
+            value.pop('seconds',None);value.pop('rss_kib',None)
+        assert selection==frozen,'Frozen simultaneous terminal selection differs'
+        command=[sys.executable,'-B',str(HERE/'formal_sinks.py'),'--candidate',str(candidate),
+            '--selection',str(candidate/'sinks.json'),'--formal-helper',str(HERE/'physical.py'),'--output',str(formal_path)]
+        run=subprocess.run(command,text=True,capture_output=True)
+        if run.returncode:raise ValueError(run.stdout+run.stderr)
+        sink_formal=json.loads(formal_path.read_text());sink_formal.pop('seconds',None)
+    raw=json.loads(gzip.decompress((candidate/'profile-before.json.gz').read_bytes()))
+    before=physical['physical'];paid=dict(before)
+    n=selection['eligible_count'];assert n==44
+    assert (selection['old_physical_R'],selection['old_W'],selection['old_rank'])==(before['physical_R'],before['W_per_vertex'],before['rank_per_vertex'])
+    paid.update(physical_R=selection['new_physical_R'],W_per_vertex=selection['new_W'],
+        rank_per_vertex=selection['new_rank'],child_histogram=selection['child_histogram'],
+        local_histogram=selection['local_histogram'],target_data_histogram=selection['target_histogram'],terminal_sinks=n)
+    assert all(z['dirty']==paid['physical_R'] and z['columns']==paid['W_per_vertex'] for z in sink_formal['columns'])
+    physical['before_terminal_sinks']=before;physical['physical']=paid
+    physical['original_formal']=physical['formal'];physical['formal']=sink_formal['columns']
+    physical['terminal_compiler']=selection;physical['terminal_formal']=sink_formal
+    physical['scalar_bound'].update(physical_R=paid['physical_R'],terminal_sinks=n,
+        terminal_scalar_gate_delta=selection['scalar_gate_delta'],terminal_original_scalar_reserve_retained=True)
+    from hashlib import sha256
+    for name in ('check_sinks.py','formal_sinks.py','prove.py'):
+        physical['source_sha256'][str((HERE/name).relative_to(ROOT))]=sha256((HERE/name).read_bytes()).hexdigest()
+    row={k:raw[k] for k in ('h','v','c','q','matched','total_M_operations','loss')}
+    row.update({k:paid[k] for k in ('m','W_per_vertex','rank_per_vertex','deficit_per_vertex','child_histogram')})
+    row.update(R=paid['physical_R'],scalar_role_reserve=raw['R'],reuse_pairs=paid['pairs'],terminal_sinks=n,maxchild=max(map(int,paid['child_histogram'])))
+    p=dict(m=row['m'],W=row['W_per_vertex'],N=row['deficit_per_vertex'],L=0,total_rank=row['rank_per_vertex'],maxchild=row['maxchild'],child_multiplicities=row['child_histogram'])
+    moment=saving_grid(p,10**15)
+    assert moment['accepted']['saving']==Q(665489485337,1000000000000000)
+    moment['complex_saving']=moment['accepted']['saving']
+    moment['scope']='Strict contraction and adjacent point exclusion for this fixed complete complex profile.'
+    print(json.dumps(js(dict(status='PASS',profile=row,physical=physical,moment=moment)),sort_keys=True))
+
+
+if __name__=='__main__':main()
diff --git a/research/paired-cube-diagonal-bit-168/complex/verify_fused.py b/research/paired-cube-diagonal-bit-168/complex/verify_fused.py
new file mode 100644
index 0000000000..47f73cf035
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/complex/verify_fused.py
@@ -0,0 +1,156 @@
+"""R13 candidate specialization: remove only PR152 selected-count pin. All finite checks retained. New candidate wrapper by Chafik Boukhalfa with OpenAI Codex; inherited credits below."""
+# Copyright 2026 icekylinx. Apache-2.0.
+# Developed with GPT-6 Astra assistance; integrated with Codex assistance.
+"""Independent finite checks of the signed scalar word and binary geometry.
+
+Copyright 2026 icekylinx, Apache-2.0; OpenAI GPT-6 Astra assistance. Binary Lagrangian
+intersection ranks use L_U=U^perp in X plus U in Z, including degenerate U.
+"""
+from collections import Counter
+from functools import lru_cache
+from paired_cube.frames import basis, perp, contained
+
+
+def verify(g, baseline, witness, word, record):
+    h,v=g['h'],g['v'];args=[None]+[None if a is None else [y+1 for y in a] for a in g['args']]
+    roots=[dict(r,node=r['node']+1) for r in g['roots']]
+    n=len(args);ann=witness['annihilators'];order=witness['order'];arcs=dict(witness['matching_arcs'])
+    active=set(order)
+    inputs=g['inputs']; spans=[()]*n;positive=[0]*n;negative=[0]*n
+    @lru_cache(None)
+    def frame(A):
+        U=perp(A,h)
+        assert len(U)+len(A)==h and all((a&b).bit_count()%2==0 for a in A for b in U)
+        return U
+    def each(bits):
+        while bits:
+            low=bits&-bits;bits^=low;yield low.bit_length()-1
+    for x in range(1,n):
+        if args[x] is None:
+            positive[x]=1<<(x-1);spans[x]=(inputs[x-1],)
+        else:
+            a,b=args[x];assert 0<a<x and 0<b<x
+            assert not (positive[a]|negative[a])&(positive[b]|negative[b])
+            sign=g['signs'][x-1];assert sign in (-1,1)
+            positive[x]=positive[a]|(positive[b] if sign==1 else negative[b])
+            negative[x]=negative[a]|(negative[b] if sign==1 else positive[b])
+            spans[x]=basis(spans[a]+spans[b])
+        if x in active:assert contained(spans[x],frame(tuple(ann[x])))
+    # Recount every actual signed broadcast, with integer twice-coefficients.
+    Hrows=[[0]*v for _ in range(v)]
+    rootann=[];rootframes=[]
+    for r in roots:
+        x=r['node']
+        if r['kind']=='center':
+            assert not negative[x]
+            expected=sum(1<<s for s,q in enumerate(inputs) if q>>r['coordinate']&1)
+            assert positive[x]==expected
+            U=spans[x];A=perp(U,h)
+            assert len(U)==h-2
+            assert r['coefficients']==['1/3' if q>>r['coordinate']&1 else '-1/6' for q in inputs]
+        else:
+            A=basis(inputs[t] for t in r['targets']);U=perp(A,h)
+            for t,c in zip(r['targets'],r['coefficients']):
+                assert c in ('1/2','-1/2');sign=1 if c=='1/2' else -1
+                for s in each(positive[x]):Hrows[t][s]+=sign
+                for s in each(negative[x]):Hrows[t][s]-=sign
+        rootann.append(A);rootframes.append(U)
+        assert contained(spans[x],U)
+    cube=[tuple(a//2 for a in label) for label in g['labels']]
+    scalar_pairs=0
+    for t,q in enumerate(inputs):
+        for s,u in enumerate(inputs):
+            overlap=(q&u).bit_count();same=cube[t]==cube[s]
+            expected=0 if same else 1-overlap
+            assert Hrows[t][s]==expected, ('Signed H mismatch',t,s)
+            distance=(q^u).bit_count()//2
+            K2=(1 if distance==3 else -1 if distance==1 else 0) if same else 0
+            B2=overlap-1
+            assert Hrows[t][s]+K2+B2==2*(t==s)
+            scalar_pairs+=1
+    # Actual local K matrix, inverse, and destructive-source frame itinerary.
+    for start in range(0,v,8):
+        qs=inputs[start:start+8]
+        K2=[[1 if (a^b).bit_count()==6 else -1 if (a^b).bit_count()==2 else 0 for b in qs] for a in qs]
+        assert all(sum(K2[i][k]*K2[k][j] for k in range(8))==4*(i==j) for i in range(8) for j in range(8))
+        for parity in (0,1):
+            source=[i for i in range(8) if i.bit_count()%2==parity];target=[i for i in range(8) if i.bit_count()%2!=parity]
+            U=basis(qs[i] for i in source);assert len(U)==3
+            assert all(contained((qs[i],),U) for i in source)
+            assert all(contained(U,perp((qs[j],),h)) for j in target)
+            assert [3-1,(h-1)-3,h-(h-1)]==[2,h-4,1]
+            assert all(sum(K2[i][k]*K2[j][k] for k in source)==4*(i==j) for i in target for j in target)
+    # Full backward intersections, including every carrier edge and root cap.
+    required=[[] for _ in args];position={x:i for i,x in enumerate(order)}
+    for x in order:
+        if args[x]:
+            for y in args[x]:required[y].extend(ann[x]);assert position[y]<position[x]
+    for r,A in zip(roots,rootann):required[r['node']].extend(A)
+    assert len(arcs)==len(set(arcs.values()))==record['matched']
+    for x,u in arcs.items():
+        assert args[x]
+        if u>>31:
+            j=u&0x7fffffff;value=roots[j]['node'];A=rootann[j]
+        else:
+            t=u//2;value=args[t][u&1];A=ann[t];assert position[x]<position[t]
+        assert value in args[x];required[x].extend(A)
+    for x in order:assert basis(required[x])==tuple(ann[x]), ('Incomplete intersection',x)
+    # Replay actual signed physical operations and every positive frame movement.
+    R=record['R'];values=[(0,0)]*R;full=basis(1<<i for i in range(h));roleann=[full]*R
+    first=[None]*R;last=[-1]*R;H=Counter();phase=set(word['phase1']);touched=set(word['sources'].values())
+    for x,s in word['sources'].items():
+        x=int(x);values[s]=(positive[x],negative[x]);roleann[s]=perp((inputs[x-1],),h);H[1]+=1
+    for i,((a,b,x),(ca,cb)) in enumerate(zip(word['ops'],word['opcoeff'])):
+        assert a!=b and ca in (-1,1) and cb in (-1,1)
+        if i in phase:
+            assert all(last[s]<0 or last[s] in phase for s in (a,b));touched.update((a,b))
+        for s in (a,b):
+            if first[s] is None:first[s]=x
+            A=tuple(ann[x]);assert contained(A,roleann[s])
+            H[len(roleann[s])-len(A)]+=1;roleann[s]=A;last[s]=i
+        pa,na=values[a];pb,nb=values[b];assert not (pa|na)&(pb|nb)
+        if ca<0:pa,na=na,pa
+        if cb<0:pb,nb=nb,pb
+        values[a]=(pa|pb,na|nb);assert values[a]==(positive[x],negative[x])
+    for r,s,A,U in zip(roots,word['rootroles'],rootann,rootframes):
+        assert values[s]==(positive[r['node']],negative[r['node']])
+        assert contained(tuple(A),roleann[s]);H[len(roleann[s])-len(A)]+=1;roleann[s]=tuple(A)
+        if r['kind']=='center':assert last[s]<0 or last[s] in phase;H[len(U)]+=1
+    for A in roleann:H[len(A)]+=1
+    assert [H[r] for r in range(1,h+1)]==baseline['remaining_internal_histogram'][1:]
+    co=[0]*R
+    for r,s in zip(roots,word['rootroles']):co[s]|=sum(1<<t for t in r['targets'])
+    for a,b,x in reversed(word['ops']):co[b]|=co[a]
+    limits=[None]*v
+    for r,A in zip(roots,rootann):
+        if r['kind']=='side':
+            for t in r['targets']:
+                if limits[t] is None:limits[t]=A
+    seen=set();birth=Counter()
+    for z in word['selected']:
+        s=z['role'];A=tuple(z['annihilator']);d=z['rank'];targets=list(each(co[s]))
+        assert s not in touched and s not in seen;seen.add(s)
+        assert targets==z['targets']
+        assert A==basis(tuple(ann[first[s]])+tuple(u for t in targets for u in limits[t]))
+        assert d==h-len(A)>0 and contained(tuple(ann[first[s]]),A)
+        r=h-len(ann[first[s]]);H[r]-=1;H[r-d]+=1;birth[d]+=1
+        for t in targets:limits[t]=A
+    assert [H[r] for r in range(1,h+1)]==record['remaining_internal_histogram'][1:]
+    assert dict(birth)=={int(k):c for k,c in record['selected_rank_histogram'].items()}
+    current=[full]*v;Y=Counter()
+    for z in reversed(word['selected']):
+        A=tuple(z['annihilator'])
+        for t in z['targets']:
+            assert contained(A,current[t]);Y[len(current[t])-len(A)]+=1;current[t]=A
+    for r,A in zip(roots,rootann):
+        if r['kind']=='side':
+            for t in r['targets']:
+                assert contained(A,current[t]);Y[len(current[t])-len(A)]+=1;current[t]=A
+    for t,q in enumerate(inputs):assert contained((q,),current[t]);Y[len(current[t])-1]+=1
+    assert dict(Y)=={int(k):c for k,c in record['target_data_histogram'].items()}
+    assert record['selected_roles']==len(seen) and birth==Counter({int(k):v for k,v in record['selected_rank_histogram'].items()})
+    return dict(signed_scalar_pairs_checked=scalar_pairs,scalar_H_exact=True,K_involution_and_inverse_exact=True,
+                decoder_identity_exact=True,conservative_source_supports=True,K_source_itinerary_verified=True,
+                full_backward_intersections=True,physical_signed_M_replayed=True,
+                center_phase_closed=True,full_gauge_intersections=True,selected_sources_untouched=True,
+                reverse_target_chains=True,positive_rank_ledger_recounted=True,selected_roles=len(seen))
diff --git a/research/paired-cube-diagonal-bit-168/references/pr141/ORIGINAL-PROOF.md b/research/paired-cube-diagonal-bit-168/references/pr141/ORIGINAL-PROOF.md
new file mode 100644
index 0000000000..2cca3a12e1
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr141/ORIGINAL-PROOF.md
@@ -0,0 +1,101 @@
+# Balanced positional transfer of the stopped product-ring composition
+
+This is an additional paid layout composition. It is not an omission of the old prefix cost or a claim that a favorable child histogram by itself proves an all-size multiplication theorem.
+
+## Contracts that have to survive the composition
+
+The supplied finite word must return every dirty auxiliary coordinate, compute the exact signed complex source map, use legal common frames on both ports, and account for every transition, source/target front, exterior, center copy and endpoint correction. Its signed reflected word reverses the literal operation order, negates shears, exchanges data banks and complements every frame. The complete paid histogram and literal scalar upper bound are inputs to this arithmetic layer. Reflection and scalar-identity evidence are separately hash-bound.
+
+PR104's opposite-bank product-ring construction makes each proper projector residual of rank r one reversed child of width r. Its global conjugation, common generic basis, full involution factorization and paid atom adapters remain required. It supplies an ordinary interchange only after the constant-size outer wrapper. That completed wrapper has arbitrary-width, complete-spectator and restored-row guarantees. Its constant number of calls is outside the recursive moment and does not multiply the child histogram. The atom exponent remains theta=1/1000, with old ordinary saving 384599/10^10. Thus the round-seven coarse bit saving 620523/(5*10^9) gives ordinary saving 1240189553/10^13, and theta exceeds this saving, so paid atom loops are lower order.
+
+The complex primitive retains the completed exact tensor contract, translated endpoint gauges, inverse phase correction, and common odd-21 grid. It must preserve a completed child's incoming odd denominator. No boundary rounding, characteristic-two substitution for signed arithmetic, free address reversal, or uncharged coefficient multiplication is introduced.
+
+Readout n/42 may be realized by |n| consecutive same-frame shears with coefficient sign(n)/42. Each has magnitude at most one, their exact sum is n/42, and their inverse negates the pieces in reverse order. They require no new source copy or frame transition. The supplied conservative L1 scalar bill may include unused copy/scan allowance, but its local/global bound must still cover the actual selected literal word. This observation does not authorize ignoring repeated scalar occurrences or a wrong inverse sign.
+
+## Why the balanced layout applies
+
+Use the inherited PR34/RaD paid positional layout as made explicit for deferred/signed networks in PR100 and PR103. Separate the extra top bit of each long FFT axis, and use common named low-coordinate groups K_j in [K,2K). The axis-major/group-major permutation and inverse are paid by the ordinary arbitrary-coordinate router. The resulting extra prefix work is O(T p d), with saving margin 1-epsilon. Complete active, spectator, scratch, phase, control and row fields travel with their descriptors; FFT order, twiddle order and forward/inverse alignment are preserved. No local scalar operation or physical residual is moved across another operation by this outer regrouping.
+
+The stopped ordinary wrapper exposes precisely the ordinary interchange contract consumed by that router. The complex word exposes the same completed tensor contract required by the semantic C1=1 induction. Therefore the stopped product-ring implementation can instantiate the same balanced layout, under the inherited all-size contracts. The original geometric condition 1-epsilon(1+c)>0 remains in the constraint list; only the work margin changes from the old repeated-prefix charge to the proved paid layout's 1-epsilon. A negative control reinstates the old work margin at the new parameters and rejects them.
+
+## All three row stocks remain present
+
+Along any active nested call chain, reserve
+
+    W_complex^D_complex * W_coarse^D_coarse * W_old^D_old.
+
+After a complex and coarse division, the old ordinary stock remains available to every stopped leaf or peeled atom. Sequential wrappers and copies reuse restored rows. They create no additional simultaneous factor, and also do not remove the old factor.
+
+For each family, d is the least positive integer with m^d > 2*r_max^d, so its depth is at most d*ceil(log2(max(2,e))) plus a fixed constant. The selected dimensions give:
+
+| Family | m | largest child | role bits | d | row coefficient |
+|---|---:|---:|---:|---:|---:|
+| Coarse bit | 529 | 528 | 27 | 367 | 9909 |
+| Complex | 576 | 574 | 27 | 200 | 5400 |
+| Old ordinary leaf | 575 | 529 | 28 | 9 | 252 |
+
+The total is 15561. A p^32000 stock leaves strict gap
+
+    32000 - (51/25)*15561 = 6389/25 > 0.
+
+The suffix slope is 128000. The validator recomputes all three minimal halving degrees, every wire-bit count, the old-stock contribution, the total coefficient, the degree gap and the suffix slope. It does not reuse the original balanced helper's two-family row formula. Semantic E, literal charge, B, C0, C1=1 and the completed-child induction are also recomputed from the supplied scalar upper bound.
+
+## Exact parameters and all retained inequalities
+
+Let b be the certified complex saving, A the proved ordinary bit saving, beta>0 the complex stopping parameter, eta>0 the balanced backoff, and zeta>0 a strict weakening. Set
+
+    a = min(A, (1-beta)*b-zeta),
+    q = a*(1-2*eta),
+    c = q+eta/4,
+    epsilon = (1-eta)/(1+q),
+    lambda' = 1-q,
+    lambda = ((1-a)+lambda')/2,
+    g = epsilon*q,
+    r = (g+1-epsilon)/2,
+    delta = eta/8.
+
+The seven margins are
+
+    1-epsilon, a, g, a,
+    min(1-epsilon-delta, r-delta),
+    1-epsilon-delta, epsilon.
+
+The minimum is g. The identities
+
+    1-epsilon-g = eta,
+    1-epsilon-r = eta/2,
+    1-epsilon*(1+c) = eta-epsilon*eta/4 > 0
+
+retain the scalar, Gaussian and geometric gaps. The stopped internal exponent and complex leaf exponent remain exactly those in PR104; (1-beta)*b>a ensures the leaf ordering. All 47 strict inequalities are retained. Source normalization shows the full adapter differs from the pinned balanced implementation only in three-stock validation, explicit old-leaf argument plumbing and inclusion of the old leaf in cutoff maxima. No constraint or cost margin is deleted.
+
+## Selected finite arithmetic and grid ceiling
+
+The selected joint-expansion profile retains 2108 birth reuses, eliminates 381 nonaliased terminal roles, and has R=26216, m=576, W=114315520, rank=65843877440 and deficit=1862080. Its scalar group bound is G=3116113870848. The actual emitted physical profile, both-orientation frame/scalar audits, and literal word are separate SHA-bound construction evidence. Its entire histogram is supplied, not inferred from role reduction or an unrestricted majorization claim. Two separate rational enclosures accept the greatest 10^-24-grid complex saving
+
+    b = 112814910866013477738/10^24
+
+and reject b+10^-24. The first enclosure reduces logarithms by 4/3 with 80 atanh terms and uses degree-12 exponentials with outward 10^-45 rounding. The independent pinned PR65/100 method reduces by 2 with 32 atanh terms and uses a degree-eight exponential bound with outward 10^-40 rounding. Every first-method log/exponential interval lies inside the second, and both separate the accepted and rejected moments from one.
+
+Taking beta=eta=10^-24 and zeta=10^-30 gives
+
+    kappa = 112802185097556/10^18.
+
+The next 10^-18 point is rejected. More strongly, let b_plus be the rejected complex point. The moment is strictly increasing in b, so the true profile root is below b_plus. Every member of this balanced family has a<min(A,b_plus) and g<a/(1+a). Therefore
+
+    g < b_plus/(1+b_plus) < (112802185097556+1)/10^18.
+
+This proves grid optimality within this fixed-profile, fixed-bit-supplier balanced transfer family. It proves no global optimum over other profiles, networks or transfer architectures.
+
+The separate same-beta/eta comparison retains beta=10^-6, eta=10^-8 and zeta=10^-14 in both layouts. It gives old-prefix kappa 112789346042450/10^18 and balanced kappa 112802068914290/10^18. Their difference 12722871840/10^18 is caused by the paid balanced layout. Finer positive backoffs give an additional 116183266/10^18; the two effects are reported separately.
+
+## Checks and boundary
+
+The arithmetic checks include old-prefix rejection, omitted/zero old-leaf stock, a two-stock coefficient, stale maximum-child/halving data, stale or absent scalar charge, stale C0, insufficient row reserve, zero leaf-ordering slack, excess kappa and deletion of a paid singleton. Each rejection is checked separately. The fixed-profile next-kappa exclusion uses the rejected complex moment, not merely an assertion that a parameter search ended.
+
+The optional explicit arithmetic cutoff for the selected tiny backoffs is log2(input) >= 14000000000000000000000000. This is only a sufficient numeric-comparison threshold. Prime supply, fixed-basis setup, descriptor/catalogue domination, logarithm absorption and exact recovery retain their inherited eventual conditions; this value is not a practical runtime threshold.
+
+Original authorship remains attached: icekylinx for stopped product-ring accounting and the ordinary wrapper; PR23/29 Zhihao Chen, RaD/hipotures and James Chang/PR34 for the semantic/balanced layout; PR100/103 Rohan Arun for its deferred/signed integration; Swapnil Jain and the relevant complex producer/frame/recycling authors for the finite inputs. This three-stock adapter and independent certificate were prepared with OpenAI Codex assistance. The supplied physical word, all-size frame implementation, simultaneous basis, atom streaming, routing, prime/precision/recovery and analytic fixed-tape transfer remain separate inherited obligations.
+
+## Portable exact boundary
+
+The portable certificate generator admits only explicit integer mathematical profile fields and canonical integer child counts. It derives h, v, m, N, L, W and deficit relations before arithmetic, recursively rejects floating-point values before serialization and after readback, and binds the raw physical profile and scalar-bill bytes by SHA256. Discovery timings stay outside the mathematical certificate. The independent secondary enclosure executes only three verbatim arithmetic function bodies from the pinned source. The selected validation applies 18 adverse controls and compares every controlling quantity, moment enclosure, assembly and cutoff against the separately generated prepackage witness. The two certificates agree exactly on these mathematical fields.
diff --git a/research/paired-cube-diagonal-bit-168/references/pr141/balanced_stopped.py b/research/paired-cube-diagonal-bit-168/references/pr141/balanced_stopped.py
new file mode 100644
index 0000000000..5b02fb6948
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr141/balanced_stopped.py
@@ -0,0 +1,204 @@
+# Three-reserve stopped-product adaptation of the inherited PR34/RaD/PR23 balanced
+# positional assembly retained in PR104 and proved applicable to deferred words
+# in PR100/103. Original attribution below is preserved. This module verifies
+# arithmetic only; scalar-word/profile provenance and all-size interfaces remain
+# separate obligations. Prepared with OpenAI Codex assistance, Apache-2.0.
+"""Exact, import-safe balanced copied-center semantic assembly arithmetic.
+
+The PR36 copied-center guard uses E=64*(W+m+G+1)^3, including G.
+Its complex saving is 717/10^7. All balanced parameter formulas are unchanged.
+
+Adapted from Zhihao Chen PR23/29 semantic assembly (Apache-2.0), RaD's
+balanced transform / arbitrary routing / bulk resampling composition, and
+jamesyc PR34's check_composition.py at 7fecbe3. PR34 source is research/reversed-two-stage/check_composition.py in that
+pinned upstream commit. Original parameterized transcription by the
+research workers is in rectangular-dimensions/next/assembly_parameterized.py.
+
+This checks arithmetic, not physical corner existence, producer correctness,
+or inherited all-size tape/analytic hypotheses. No imports execute producers,
+parse argv, read files, change globals, or mutate the supplied finite bridge.
+"""
+from fractions import Fraction as Q
+
+
+class InvalidAssembly(AssertionError):
+    """A named exact input validation or strict inequality failed."""
+
+
+def require(condition, message):
+    if not condition:
+        raise InvalidAssembly(message)
+
+
+def rational(value):
+    # Fraction(float) silently preserves binary floating error; forbid it.
+    require(type(value) in (int, str, Q), 'expected integer or exact rational string/Fraction')
+    return Q(value)
+
+
+def integer(value):
+    x = rational(value)
+    require(x.denominator == 1, 'nonintegral graph constant')
+    return x.numerator
+
+
+def ceil(value):
+    return -(-value.numerator // value.denominator)
+
+
+def halving_degree(m, child):
+    require(0 < child < m, 'child must strictly contract')
+    degree = 1
+    while m**degree <= 2 * child**degree:
+        degree += 1
+    return degree
+
+
+def validate_bridge(finite_bridge, ordinary_leaf):
+    """Recompute rather than trust stored semantic and product-stock totals.
+
+    scalar_group_upper remains a supplied, proof-certified local scalar upper
+    charge. Its producer provenance is external to this arithmetic interface.
+    """
+    result = {}
+    for name in ('bit_coarse', 'complex', 'ordinary_leaf'):
+        data = ordinary_leaf if name == 'ordinary_leaf' else finite_bridge[name]
+        m, W, child = (integer(data[k]) for k in ('m', 'W', 'maxchild'))
+        require(m > 1 and W > 0, 'positive arity and role volume required')
+        degree, bits = halving_degree(m, child), W.bit_length()
+        require(integer(data['halving_degree']) == degree, name + ' stale halving degree')
+        require(integer(data['wire_bits']) == bits, name + ' stale role logarithm')
+        result[name] = dict(m=m, W=W, maxchild=child,
+                            halving_degree=degree, wire_bits=bits)
+    phase = finite_bridge['complex']
+    s, scalar = (integer(phase[k]) for k in ('s', 'scalar_group_upper'))
+    m, W, child = (result['complex'][k] for k in ('m', 'W', 'maxchild'))
+    require(0 < s < m*W and scalar > 0, 'invalid complex rank/scalar bound')
+    E = 64*(W+m+scalar+1)**3
+    B = s+E
+    literal = 2*scalar*W**2+8*s+4*W+4+32*m
+    C0 = 32*m*B**2
+    semantic = dict(E=E, B=B, C0=C0, C1=1, literal_charge=literal,
+                    strict_literal_gap=E-literal,
+                    induction_gap=2*B*(m-child)-(s+E))
+    require(E > literal and semantic['induction_gap'] >= 0 and C0 > 2*B+18,
+            'semantic completed-child bound failed')
+    for key, value in semantic.items():
+        require(rational(finite_bridge['semantic'][key]) == value,
+                'stale semantic constant: '+key)
+    result['complex'].update(s=s, scalar_group_upper=scalar)
+    result['semantic'] = semantic
+    rows = finite_bridge['rows']
+    leaf_degree = result['ordinary_leaf']['halving_degree']*result['ordinary_leaf']['wire_bits']
+    require(integer(finite_bridge['ordinary_leaf_row_degree']) == leaf_degree,
+            'ordinary leaf stock missing or stale')
+    coefficient = sum(result[name]['halving_degree']*result[name]['wire_bits']
+                      for name in ('bit_coarse', 'complex', 'ordinary_leaf'))
+    degree = integer(rows['degree'])
+    gap = degree-Q(51,25)*coefficient
+    require(degree > 0 and gap > 0, 'insufficient product row degree')
+    expected = dict(coefficient=coefficient, degree=degree,
+                    degree_gap=gap, suffix_slope=4*degree)
+    for key, value in expected.items():
+        require(rational(rows[key]) == value, 'stale product row constant: '+key)
+    result['rows'] = expected
+    return result
+
+
+def assembly(finite_bridge, ordinary_leaf, a_bit, kappa, beta=Q(1,20), h=Q(1,10**12),
+             a_complex=Q(717,10**7), *, original_prefix=False,
+             old_guard=False, old_exposures=False):
+    """Return 47 strict slacks and seven margins, or raise InvalidAssembly.
+
+    The three keyword switches are negative controls: they impose the old
+    prefix, nonlinear guard, or old separate movement charges at these SAME
+    balanced parameters. They do not silently optimize an alternative family.
+    """
+    f = validate_bridge(finite_bridge, ordinary_leaf)
+    a, b, kappa, beta, h = map(rational, (a_bit,a_complex,kappa,beta,h))
+    require(0 < h < Q(1,2) and kappa > 0, 'positive backoff and saving required')
+    tau, sigma = 1-a, 1-b
+    q = a*(1-2*h)
+    require(1+q != 0, 'undefined balanced epsilon')
+    lp, c, eps = 1-q, q+h/4, (1-h)/(1+q)
+    lam = (tau+lp)/2
+    G = eps*q
+    r, delta = (G+1-eps)/2, h/8
+    C1 = Q(19991,10000) if old_guard else Q(1)
+    margins = dict(g1=1-eps, g2=a, g3=G, g4=a,
+                   g5=min(1-eps-delta,r-delta), g6=1-eps-delta, g7=eps)
+    if original_prefix:
+        margins['g1'] = 1-eps*(1+c)
+    if old_exposures:
+        margins.update(g2=eps*c*a, g4=a*(1-eps))
+    internal = tau+(1-beta)*max(sigma-tau,Q(0))
+    leaf = sigma+beta*(1-sigma)
+    slacks = dict(a_positive=a, a_below_b=b-a, b_below_one_over32=Q(1,32)-b,
+        beta_positive=beta, beta_below_one=1-beta, phase_leaf_above_bit=(1-beta)*b-a,
+        q_positive=q, q_below_internal=1-internal-q, q_below_leaf=1-leaf-q,
+        c_positive=c, c_below_one=1-c, q_below_reservations=c-q,
+        lambda_above_tau=lam-tau, lambda_above_sigma=lam-sigma,
+        lambda_above_internal=lam-internal, lambda_prime_above_lambda=lp-lam,
+        compact_leaf=lp-leaf, compact_reservations=lp-(1-c), lambda_prime_below_one=q,
+        epsilon_positive=eps, epsilon_below_one=1-eps, guard_width=1-eps*C1,
+        K_geometry=1-eps*(1+c), K_dominates_log=eps*c,
+        record_suffix=1-eps, phase_local=1-eps-delta, phase_boundary=r-delta,
+        gamma_sublinear=1-eps-r, cell_above_band=eps-(1-r)/2,
+        prime_interval_packing=1-eps, alpha_positive=r, alpha_below_one=1-r,
+        alpha_below_one_fourth=Q(1,4)-r, delta_positive=delta,
+        delta_below_one_eighth=Q(1,8)-delta, short_record_fallback=eps-a,
+        small_field_exposure=1-eps-G, artificial_boundary=8-eps+r-delta-G,
+        literal_scalar_guard=Q(f['semantic']['strict_literal_gap']),
+        row_product_gap=f['rows']['degree_gap'])
+    slacks.update({name+'_above_kappa': value-kappa for name,value in margins.items()})
+    require(len(slacks) == 47 and len(margins) == 7, 'constraint list incomplete')
+    failed = {name:str(value) for name,value in slacks.items() if value <= 0}
+    require(not failed, str(failed))
+    require(min(margins.values()) == G, 'unexpected controlling margin')
+    require(1-eps-G == h and 1-eps-r == h/2, 'balanced identities failed')
+    require(1-eps*(1+c) == h-eps*h/4, 'geometric identity failed')
+    parameters = dict(a_bit=a, a_complex=b, tau=tau, sigma=sigma, beta=beta,h=h,
+        q=q,c=c,epsilon=eps,lambda_=lam,lambda_prime=lp,alpha_squared_power=r,
+        delta=delta,C0=f['semantic']['C0'],C1=C1,kappa=kappa)
+    return dict(parameters=parameters,constraints=slacks,margins=margins,
+        minimum_margin=G,absorption_gap=G-kappa,scoped_limit=a/(1+a),
+        recurrence=dict(internal=internal,leaf=leaf,reservations=1-c),finite_bridge=f)
+
+
+def cutoffs(finite_bridge, ordinary_leaf, result):
+    """PR34 sufficient numeric comparisons, retaining its all-interval padding.
+
+    These are eventual arithmetic thresholds, not practical running times or
+    substitutes for additional native setup / prime / recovery thresholds.
+    """
+    f = validate_bridge(finite_bridge, ordinary_leaf)
+    p = result['parameters']
+    e,c,r,beta = (p[k] for k in ('epsilon','c','alpha_squared_power','beta'))
+    ka,km,kb,ks = (ceil(1/x) for x in (r,e*c,1-e,e*beta))
+    C0 = f['semantic']['C0']
+    largest = max(f[name]['m'] for name in ('bit_coarse','complex','ordinary_leaf'))
+    cuts = dict(guard=ceil(Q((2*C0).bit_length())/(1-e)),
+        normalization=ceil(7/(1-e-r)), alpha=16*ka*ka+1, compact=64*km*km+1,
+        geometry=ceil(3/(1-e*(1+c))),phase_cell=ceil(9/(e-(1-r)/2)),
+        period=128*kb*kb+1,stopped_leaf=ks*(4*largest).bit_length(),log_p=25,reservoir=14)
+    common = max(cuts.values())
+    require(common >= max(2*ka,2*km,2*kb,25), 'cutoff does not cover monotone regime')
+    checkpoints=[]
+    for j in range(6):
+        z=common*2**j
+        powers=dict(alpha=(z//ka,8*(z+ka)+64),compact=(z//km,32*(z+km)+192),
+            period=(z//kb,16*(z+kb)+56),
+            rows=(z//kb,f['rows']['suffix_slope']*(z+kb+8)),
+            leaf=(z//ks,4*largest))
+        require(all(lhs >= rhs.bit_length() for lhs,rhs in powers.values()),
+                'compressed-power threshold failed')
+        checkpoints.append(dict(log2_input=z,powers={name:dict(exponent=lhs,rhs=rhs,
+             rhs_bits=rhs.bit_length()) for name,(lhs,rhs) in powers.items()}))
+    return dict(cutoff_log2_input=cuts,common_cutoff=common,power_checkpoints=checkpoints,
+        all_interval_argument='For [Z,Z+k), bound RHS at Z+k and LHS at floor(Z/k). '
+          'Successive intervals double LHS; the positive affine RHS grows at most '
+          'linearly, so 2^j >= j+1 proves persistence.',
+        additional_eventual_conditions=['BHP prime threshold and distinct-prime packing',
+          'Fixed rational basis, eligible common prime and native table setup',
+          'Catalogue/descriptor domination, logarithm absorption, exact recovery'],
+        practical_runtime_claim=False)
diff --git a/research/paired-cube-diagonal-bit-168/references/pr166/PROOF.md b/research/paired-cube-diagonal-bit-168/references/pr166/PROOF.md
new file mode 100644
index 0000000000..4b4341c091
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr166/PROOF.md
@@ -0,0 +1,221 @@
+# Dirty-target accumulation removes terminal producer rows
+
+Status: author-checked finite compiler lemma and 67-row substitution on the
+current PR163 complex word. This is not a new assembled multiplication bound.
+Independent review and any advertised supplier/assembly update remain separate.
+
+Scientific inputs are the selected complex files introduced at
+15c702a929b7d640107a95e196186ad74e876c82. They are unchanged at publication
+base e1813796ef5c3ca38c5dd7b9e8d81b3908ff5997; that later commit changes only
+attribution NOTICE and its SOURCE digest. The checker verifies the existing
+result.json SHA256 pins for every finite input it consumes.
+
+## Relation to earlier constructions
+
+Dumas and Grenet's *In-place accumulation of fast multiplication formulae*
+(https://arxiv.org/pdf/2307.12712) motivates using arbitrary-valued output
+storage. Its RAM results do not establish this frame-constrained compiler.
+No novelty claim is made for the scalar pre/post cancellation identity.
+
+PR161's P±Q merge and PR163's singleton-channel fusion create internal producer
+sums. This substitution instead removes independent auxiliary coordinates by
+using existing independently valued data targets. The paired-cube identity,
+Clifford representatives, physical reuse, copied-center interfaces and all-size
+compiler assumptions retain their upstream attribution and proof obligations.
+
+## Local lemma with chronological hypotheses
+
+Let z_s be an independently dirty, zero-entrance-gauge producer role. Assume:
+
+1. It is neither a source-injection role nor an aliased recipient/donor.
+2. Its only producer incidences are additive destination writes
+   z_s <- z_s + b_j a_j, with b_j in {1,-1}. It never controls another
+   producer role. It has exactly one root read, uniformly z_s/2 into a
+   target group T.
+3. Its literal write-frame chain is 0 <= U_1 <= ... <= U_k <= U,
+   where U is the common root read frame. Every write uses the identical
+   exact T_(U_j) on its control and destination.
+4. All these writes occur after the copied-center phase. Before the first
+   deferred target corrections, the targets in T are at frame0.
+5. Some pivot c in T receives no target correction between that common-zero
+   point and the last write/post-read. Other targets may receive corrections
+   during this interval, provided those corrections do not depend on target
+   contents and their existing frames are contained in U.
+6. Later target reads continue their old nested chains from U. No other
+   target-controlled producer operation is introduced.
+
+All targets and retained dirty roles have arbitrary independent initial values.
+No target is assumed clean, and the deleted dirty variable is not set to zero:
+the new construction has one fewer independent auxiliary input coordinate.
+
+Keep the ORIGINAL initial and deferred old-response coefficient columns for
+EVERY retained role. Omit only the deleted role's column. In particular, do
+not recompute the old correction from a decoder which simply drops its root;
+that would omit the root response of its retained ancestors. The inherited
+expanded scalar old-response implementation and its conservative charge remain
+available. The sink's own old-response column is exactly the uniform1/2
+column on T because it never controls another producer row.
+
+At the common-zero point, after the copied-center contribution and before
+its deferred corrections, perform
+
+    y_t -= y_c                 for t in T minus {c}.
+
+At the original time of each write, raise the pivot along the deleted role's
+actual chain and replace that write by
+
+    y_c += (b_j/2) a_j.
+
+Immediately after the last such write, raise the pivot and the other targets
+to the original U and perform
+
+    y_t += y_c                 for t in T minus {c}.
+
+Delete the old positive root read, all writes and inverse-cleanup writes on
+z_s, its initial old-response column, its physical stream and its full-frame
+ending. Every retained producer operation, original source operation, source
+return, copied-center step and old-response column stays in its original order.
+
+Write f=sum_j b_j a_j for the ACTUAL current controls, including all their
+retained dirty dependence. The sandwich adds f/2 to all targets and preserves
+every old target value. Any intervening correction e_t on a non-pivot is
+unchanged:
+
+    (y_t-y_c+e_t) + (y_c+f/2) = y_t+e_t+f/2.
+
+An intervening correction to the pivot would be broadcast to every target;
+this is why the pivot chronology condition is essential. The test does not
+incorrectly require every non-pivot correction to wait.
+
+The retained producer is the same invertible word with destination-only sink
+writes omitted. It contains no reference to the deleted coordinate, so its
+literal inverse restores every retained dirty input. Its response from each
+retained source/dirty column is unchanged at the targets, and the retained
+old-response columns cancel those dirty parts as before. Earlier target data,
+center contributions and later target contributions are all preserved.
+
+## Exact frames, inverse and simultaneous use
+
+The pre-shears use T_0 on both targets. Each accumulating write uses the
+original exact T_(U_j) on pivot and control. Every post-shear uses one
+identical T_U on its two operands. Rank-zero reconciliation is actual inherited
+scalar/movement work, not equality up to phase.
+
+Reverse the COMPLETE literal new word and invert every shear. Use
+D_U=T_U F^(-1) in the reverse/complement orientation. Then
+
+    D_U D_V^(-1) = T_U T_V^(-1)
+
+exactly. All reversed gate incidences are still identical, all transition
+ranks are preserved, and sources/retained dirty values are restored. This is
+the chronological inverse, not the invalid source/target-swapped transpose.
+The data-bank and rank-zero endpoint conventions remain the inherited ones.
+
+For sinks with disjoint target groups, perform each pre-sandwich at the same
+common-zero cut and each post-sandwich at its own last-write time. There is
+no target collision or producer feedback, so all substitutions coexist.
+
+## Paid ledger
+
+Let r=dim(U) and let P be the complete positive-width prefix of the deleted
+role from0 through every write to U. Its old local row bill is P+[h-r].
+The pivot's old first target child [r] is replaced by P. Other target paths
+and all retained row paths are unchanged. Hence the EXACT local child change is
+
+    -[r] -[h-r],
+
+with one independent row removed. This accounts for every prefix, endpoint and
+full-frame cleanup; fragmentation inside P cancels rather than disappearing
+without payment. The same delta holds for the complete inverse orientation.
+
+In the existing three completed orthogonal cores, for each zero-gauge sink:
+
+    delta W = -1;
+    delta children = -3*[r] -3*[h-r];
+    delta rank = -3h;
+    delta deficit = 0.
+
+There is no exterior tail for a zero-gauge physical row. The full group factor
+and retained routing/row reserves are not reduced by fiat. For moment exponent
+p in (0,1), the unnormalized slack improvement is
+
+    3 r^p + 3(h-r)^p - (3h)^p > 0,
+
+because the six positive pieces sum to3h. This is a strict finite-profile
+improvement, not a newly certified assembled kappa.
+
+For k writes and n targets, the old affected scalar gates are k forward
+writes, k inverse writes, n initial dirty reads and n positive root reads.
+The new ones are k accumulating writes and2(n-1) target shears. Thus the
+scalar-gate count decreases by k+2. All new coefficients already belong to
+the Gaussian-dyadic alphabet; no odd divisor is introduced. Keep the original
+conservative virtual R, M, expanded-correction scalar charge, full finite
+stock and precision/routing reserves rather than shrinking them without proof.
+
+## Exact current finite instance
+
+The checker examines ONLY the165 all-eight disjoint roots of the selected
+PR163 word. It finds67 eligible roots on disjoint target cubes. The other98
+have retained producer consumers; two of those also touch the center phase.
+It does not search arbitrary face, singleton or cross-cube combinations.
+
+For example, root0 uses unaliased role12354. Its only incidences are additive
+writes21902 and24308, at ranks15 and18. Target0 can be the pivot. Targets6
+and7 receive early compensation, but their independent additive corrections
+are valid while encoded. The pivot's first compensation occurs after the
+post-read. This concrete distinction is verified in the actual target-event
+replay rather than inferred from a scalar channel formula.
+
+The selected simultaneous change is:
+
+    physical R: 13041 -> 12974
+    W per vertex: 15681 -> 15614
+    rank per vertex: 1033626 -> 1029204
+    deficit: 1320 -> 1320
+    child multiplicity18: -201
+    child multiplicity4: -201
+    scalar gates: -300
+
+selected.json contains every selected role, its original writes, pivot and
+chronological boundary, as well as the complete new local/target/shared
+histograms and source pins. Its decimal runtime is a diagnostic, not a
+mathematical certificate input.
+
+## What the author check actually establishes
+
+check_sink_accumulator.py performs these exact checks from existing frozen
+inputs, without importing an upstream executable or rebuilding a producer:
+
+- original source pins and full positive physical histogram equality;
+- all role classifications, aliases, source injections, literal consumers,
+  write-frame chains and original root destinations;
+- an available late-corrected pivot for every selected group;
+- every new target event interleaved with all original deadline reads and
+  retained root reads, including all endpoint changes;
+- exact adjoint pullbacks through the actual aliased producer for every
+  selected sink, covering all1320 original source columns and all13041
+  original dirty columns, including zero columns;
+- each fresh disjoint channel has exactly its448 required source terms;
+- arbitrary target and intervening non-pivot corrections, both signs, and
+  actual reversal of each scalar sandwich;
+- retained source/dirty identity by destination-only deletion and literal
+  inverse cancellation; no Monte Carlo or injective-digit assumption;
+- disjoint simultaneous support and complete child/row deltas.
+
+Together with the unchanged inherited complete word, this covers its15681
+original source/target/dirty coordinates and the15614-coordinate new word.
+The original saved full proof contracts remain dependencies. The package does
+not update the bit supplier, balance parameters, all47 assembly inequalities,
+precision constants or advertised multiplication exponent.
+
+## Reproduce
+
+From the repository root:
+
+    python research/paired-cube-target-accumulator/check_sink_accumulator.py \
+      --candidate research/paired-cube-balanced-161/selected/complex \
+      --output /tmp/target-accumulator-check.json
+
+The standard-library checker has60-second CPU/wall and768MiB address-space
+caps. The author run completed in approximately5.5seconds and85MiB. Compare
+mathematical fields, not runtime/RSS, with selected.json.
diff --git a/research/paired-cube-diagonal-bit-168/references/pr166/README.md b/research/paired-cube-diagonal-bit-168/references/pr166/README.md
new file mode 100644
index 0000000000..0adb5c653a
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr166/README.md
@@ -0,0 +1,15 @@
+# Target-accumulating terminal compiler
+
+This author-checked research packet replaces67 actual PR163 producer sinks
+with existing independently valued target accumulators. The complete finite
+profile removes67 rows and201 children of each width18 and4; deficit1320 is
+preserved. It makes no new assembled multiplication-bound claim.
+
+Read PROOF.md for the exact chronological lemma, retained dirty corrections,
+full inverse, scalar/endpoint bill and inherited obligations. selected.json is
+the concrete current-word specification. Run check_sink_accumulator.py against
+the existing paired-cube-balanced-161/selected/complex data to reproduce it.
+
+The motivating scalar idea is output accumulation; no novelty claim is made.
+Existing scalar-producer, frame, physical-reuse and all-size interfaces retain
+the attribution in the source package's current NOTICE.
diff --git a/research/paired-cube-diagonal-bit-168/references/pr168-v4/SOURCE.json b/research/paired-cube-diagonal-bit-168/references/pr168-v4/SOURCE.json
new file mode 100644
index 0000000000..c57edcae25
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr168-v4/SOURCE.json
@@ -0,0 +1,20 @@
+{
+  "commit": "4a3c769e5c5430e7114c4d3e099ff34664677f17",
+  "files": {
+    "certificates/paired-cube-network.json": "05b253a21dc526b6039f56d1068162a19a8a4f29291ad94eb892656b88e9142d",
+    "notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54",
+    "research/paired-cube-bit/check_paired_cube_bit.py": "4d7c86cd999a1519e0dfb6dcf33a8f26e442a353c3665a0f06881216a260fbf3",
+    "scripts/certify.py": "2348d95008276c5d5764f7e028c3918e484582cd589bd4d9b82919161745e53e",
+    "scripts/paired_cube/__init__.py": "b5fb17c79fd41a75c0648e7a5f99076f9f42a619857d5e34e71699bf00ef0c91",
+    "scripts/paired_cube/closure.py": "e4b3086f4b5663297fe06f4646e1349f10d8647997989f17eef6ec4e84aec693",
+    "scripts/paired_cube/frames.py": "7d897418dbfee47de0723ea4983098a982610ccba45e1bac9000a79626d8421c",
+    "scripts/paired_cube/gauges.py": "8a501c66e1330f054e0e11a35e9321d2deea7b51cd03ca56b6a41765e43fe061",
+    "scripts/paired_cube/graph.py": "d10478ba509c7d8abcefcaaf7c5e0b580b9cd9cbe7777b3e49b2fe24620985c8",
+    "scripts/paired_cube/modules.py": "6c134f3d24c6cd9aca03763050aaf600005c126c09b35ac0ff5f040e0e29db87",
+    "scripts/paired_cube/verify.py": "b1eb42a91381ae031d97879f7c43e6c00fc4787e132ed60eb3e124994bdb7d35",
+    "scripts/paired_cube_physical.py": "9c9ea1ea29e551ab6116d7c8ca09f4703ff9e2fa102032d5055327fde07ba319"
+  },
+  "pr": 168,
+  "repository": "https://github.com/CrocSwap/integer-mult-bounds",
+  "scope": "Byte-identical copies of the PR168 v4 files this package executes or pins (compiler frames/closure/gauges/graph/modules, physical checkers, package bit checker, parent certificate, frame theorem note), so the package is self-contained on main."
+}
diff --git a/research/paired-cube-diagonal-bit-168/references/pr168-v4/certificates/paired-cube-network.json b/research/paired-cube-diagonal-bit-168/references/pr168-v4/certificates/paired-cube-network.json
new file mode 100644
index 0000000000..a8f20e3388
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr168-v4/certificates/paired-cube-network.json
@@ -0,0 +1,374 @@
+{
+  "assembly": {
+    "absorption_gap": "4998206273112780939153/781762749911107251572755000000000",
+    "margins": {
+      "Gaussian_arithmetic": "16408090802882068106709685449/25016407997155432050328160000000",
+      "balanced_prefix": "102551544721450314551/156352549982221450314551",
+      "bulk_exposure": "2050999685449/3125000000000000",
+      "compact_phase_layer": "10254998119595049233649685449/15635254998222145031455100000000",
+      "coordinate_movement": "2050999685449/3125000000000000",
+      "dimension": "156249998437500000000000/156352549982221450314551",
+      "scalar_work": "82041079424610269419349685449/125082039985777160251640800000000"
+    },
+    "minimum_margin": "10254998119595049233649685449/15635254998222145031455100000000",
+    "parameters": {
+      "C0": 97669291764581971727897399340796439745427398210756942058757461462193164381719625313532819641527363257083473637043456750425771043504865994302864994759107018349829953359732688723068163038925026590380085470313869884447701723662500698566932240436548003786859894761106413356476366586106109469625284711919844994268260907139466275551818412625594842265089162982132961175931200937276563214353402097618110667697564201492547804122096641643712240930548654635079762582961119708632798563862699522914741883780352066531794750409773941726470397224815013473786448614481463972302260116019476602387301387998033936381378363985420155321324021051217265364825381036409646946303604226012319469038869990774521314263986670126395129035577392484871922159919747230004216213955269142463210781157978154845511026951699019884511470320531522268000113056842981689884261761807080598763976575950640266275780068455496492275923446299732975375170859839166341448809500036332225895647989964790587320990661294824351859968748973099076142479751635296458442033739330211900873561812913557293923274738191711972471927301764013221965521902122061348965875895078911630750576829042909141411939265816456675838208795595679410798324336442893554636786736308730026236354443440370156408647340230099425508737237503488969309258270272151907469676715658281070207153036478607653516831845851507749671110595470029032298087855160812872240130666986644609245679270274923288516749615933506319605403346830817542503292795968876374524030915165270155659774233936971680428653709662912958368879262551841000448557426513945100644597007962787411573731401717418536555995738727396417952305945747581594846033308802262282453489132631861701749178611732272308053128649255694113918473578875634109321459793840068738343369543640062073667213699026254353101126746896818632579260751957853410082657763910943060760303463367932658548925530442463896398756447104770753338984075680719146805659857499457356402987417225556477555136007460799174775163846037447536277634665355671666443476074258853409836369016399087120362545693892006051928205410124728510012260102742700823385231701494486348114985395491670816737513129078779860227601904390565645820967711471770897686453889198996886623455341219043554171531872829847060570056917313621519389637569222565728168683079710468133671330605430350655245646556973228777903135129669808588075222956485331924114464730473728450873709141173584653147473535508093646323039463062161943118572048378428173109675405355357089281175632403300951365828760586147366564219659268164536890557142236653672558498120766888906574243426492377998542065242549654629737128573164078668309206001039728402847903705092522562415996927492189375779469352711886069811006076008309626734055208818921108289813223425465521904246523888762231274246207445718204784843844895507235203092165592774877142408906853132960566104762005225817573346855143300118471735977009352851185300942432649943761095004221170646305283330831228148322879736166532226849120164597939647084205950787650264881142953423445362941242275436711616325402697285500207827461538058523488078073024898276339843555931995111998484689768319637303437867334601571730808670492557478546235975801891253986972305936729946938339286173423869671489992904944540558370191863926955188471757191500323530975718410219219872160492722970470307653503018766389816812098136764299518730869985094811968202111488652327554983739537240269417335493102026705575483842984673255579091738471843904909258104470197846304474898004967061856178586619745290394346366364108415009442561993325120948732968278503678086667045030414972423047949552607365739558576475695919903019561961987291189558274475525715972244098417813629970633906031664078663295738216991875595466675374577106146650969538432630170656032908139165458348387183406134291912025740163533531938484757651161652355943077253469679633609553205045569468576599739405661296403394098968757750713684856788328469143548153606523346173629398835781213382858078322067027045282824359304010625813873822928656100311881401998855676497453533669135388589051521785075410451692906675169612763741350076112567366082293032327405743542462346609809698919975354172795405815566000038014499914860506977186145275006129333000120345086339606517412705101401419867646844123396985820917540637839276825815820216377033462645559599305437114813114023153337223629090216188707857880687573618674782242852605892749584939935589009905515713883126454422778154170025173726287722015939517726440817431538145667538762769512612391967628131117784201684430305705294584400102053704180109895405635140445911898216860851542691550983764425811550377760786625735201794509238012875254486886684905192652831046205812633527195355116324752846914666828833584005991684906019121197451757764253921589624262497534692803153569314783411832866166263788359010901323577865338928738901255262898076857480641157094834565047655128514554907040704765372522976877294764111381039750132892400346911979165386727725887650793487450877293997798310199728494773414041213243231251374917747058942218908975831500146287200025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+      "C1": 1,
+      "a_bit": "2050999685449/3125000000000000",
+      "a_complex": "2051/3125000",
+      "actual_bit_saving": "1649038264871199969/2500000000000000000000",
+      "alpha_squared_power": "20510152591740080688749685449/31270509996444290062910200000000",
+      "beta": "1/1000000000",
+      "c": "102550372846450314551/156250000000000000000000",
+      "delta": "1/800000000",
+      "epsilon": "156249998437500000000000/156352549982221450314551",
+      "eta": "1/100000000",
+      "kappa": "3279447/5000000000",
+      "lambda_": "312294900033506099685449/312500000000000000000000",
+      "lambda_prime": "156147450017778549685449/156250000000000000000000",
+      "q": "102549982221450314551/156250000000000000000000",
+      "sigma": "3122949/3125000",
+      "tau": "3122949000314551/3125000000000000"
+    },
+    "recurrence": {
+      "internal": "3122949000314551/3125000000000000",
+      "leaf": "3122949000002051/3125000000000000",
+      "reservations": "156147449627153549685449/156250000000000000000000"
+    },
+    "strict_constraints": {
+      "Gaussian_arithmetic_above_kappa": "11746434073759059897347937/3127050999644429006291020000000000",
+      "K_dominates_log": "10255037182094658608649685449/15635254998222145031455100000000",
+      "K_geometry": "117290050372846450314551/15635254998222145031455100000000",
+      "alpha_below_one": "31249999843852549982221450314551/31270509996444290062910200000000",
+      "alpha_below_one_fourth": "7797117346519332435038800314551/31270509996444290062910200000000",
+      "alpha_positive": "20510152591740080688749685449/31270509996444290062910200000000",
+      "artificial_boundary": "875656321605274931959790750943653/125082039985777160251640800000000",
+      "balanced_prefix_above_kappa": "7822625705384185296666703/781762749911107251572755000000000",
+      "beta_below_one": "999999999/1000000000",
+      "beta_positive": "1/1000000000",
+      "bit_positive": "2050999685449/3125000000000000",
+      "bulk_exposure_above_kappa": "1345310449/3125000000000000",
+      "c_below_one": "156147449627153549685449/156250000000000000000000",
+      "c_positive": "102550372846450314551/156250000000000000000000",
+      "cell_above_band": "31249999531147450017778549685449/62541019992888580125820400000000",
+      "compact_leaf": "17675999685449/156250000000000000000000",
+      "compact_phase_layer_above_kappa": "4998206273112780939153/781762749911107251572755000000000",
+      "compact_reservations": "1/400000000",
+      "complex_above_bit": "314551/3125000000000000",
+      "complex_below_one_over32": "382421/12500000",
+      "coordinate_movement_above_kappa": "1345310449/3125000000000000",
+      "delta_below_one_eighth": "99999999/800000000",
+      "delta_positive": "1/800000000",
+      "dimension_above_kappa": "780737242286518453811430296666703/781762749911107251572755000000000",
+      "epsilon_below_one": "102551544721450314551/156352549982221450314551",
+      "epsilon_positive": "156249998437500000000000/156352549982221450314551",
+      "gamma_sublinear": "1/200000000",
+      "guard_width": "102551544721450314551/156352549982221450314551",
+      "lambda_above_internal": "2050999685449/312500000000000000000000",
+      "lambda_above_sigma": "33506099685449/312500000000000000000000",
+      "lambda_above_tau": "2050999685449/312500000000000000000000",
+      "lambda_prime_above_lambda": "2050999685449/312500000000000000000000",
+      "lambda_prime_below_one": "102549982221450314551/156250000000000000000000",
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+      "contract": "Finite complex role stock plus unchanged conservative bit reserves; q-adic selectors internally borrowed and restored; one prefix and one padding; sequential reuse",
+      "degree": 70000,
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+  "scope": "Exact inventories, supplier moments, full fallback/router charge and semantic assembly. Local scalar/frame verification is separate; paired source scheduling, completed-core sharing and inherited uniform-recursion/analytic/tape contracts are written proof dependencies.",
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+    "scripts/paired_cube/verify.py": "b1eb42a91381ae031d97879f7c43e6c00fc4787e132ed60eb3e124994bdb7d35",
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+    "scripts/paired_cube_bit_physical.py": "2c1f8268f6bfb8026f83995376806e4ad1c2b3bb3173d05d9970e290652c98a2",
+    "scripts/paired_cube_network.py": "7e9dca7de0e0fa843c642ed50f97c84a2c676a1b9129ea9263e5394b145d44f7",
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+    "scripts/paired_cube_producer.py": "97efa5842c040537f05072e26b7ba1851a1965eca6ec2472e75aa452cf7a7256",
+    "scripts/partial_gauge_bit.py": "7d58e86061da2d73991531e9bc431589019359391adea866c61c7dd1f99ac855",
+    "scripts/structured_bulk_assembly.py": "e99467e14211fba699878ed036e1e2362eb8e395dc22f9ecdcecc7fbf2cc736d",
+    "scripts/three_stage_cover_network.py": "7e0ad6fd1921b6905a53ce1eeaa03d3bb3a8a8bbaa0382516ea98b636f46335e"
+  },
+  "status": "Conditional paired-cube multiplication witness"
+}
diff --git a/research/paired-cube-diagonal-bit-168/references/pr168-v4/notes/general-clifford-frames.tex b/research/paired-cube-diagonal-bit-168/references/pr168-v4/notes/general-clifford-frames.tex
new file mode 100644
index 0000000000..0367eeb005
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr168-v4/notes/general-clifford-frames.tex
@@ -0,0 +1,102 @@
+\subsection{Preimage frames for arbitrary binary subspaces}
+This frame construction is due to icekylinx with OpenAI GPT-6 Astra assistance;
+the present exposition and integration use Codex.
+Let $H=\mathbb F_2^h$ with the standard dot product. In the binary
+symplectic space $H\oplus H$ put
+\[
+ L_0=\{(0,z)\},\qquad
+ F=\begin{pmatrix}I&I\\0&I\end{pmatrix},\qquad
+ L_F=FL_0=\{(z,z)\}.
+\]
+We also write $F$ for the exact Gaussian-dyadic operator $C^{\otimes h}$.
+Its exact square is a Pauli translation; only its binary symplectic square
+is the identity. For an exact Clifford operator $T$ the frame label is
+$L(T)=T^{-1}L_0$. The physical transition from $T$ to $S$ is $ST^{-1}$,
+and its Fourier rank is
+$d(L(S),L(T))$, where $d(L,L')=h-\dim(L\cap L')$.
+This follows by transporting the intersection by $S$: the intersection
+with $L_0$ computes the kernel dimension of the $B$ block of $ST^{-1}$.
+
+\begin{lemma}
+For every subspace $U\subseteq H$, including degenerate subspaces,
+\[
+ L_U=\{(u,u+v):u\in U,\ v\in U^\perp\}
+\]
+is Lagrangian, and
+\[
+ d(L_U,L_V)=\dim U+\dim V-2\dim(U\cap V),\qquad
+ FL_U=L_{U^\perp}.
+\]
+\end{lemma}
+\begin{proof}
+The displayed parametrization is injective and has dimension $h$.
+Its symplectic pairing is $u\cdot v'+v\cdot u'=0$.
+In an intersection, $u$ belongs to $U\cap V$ and $v$ to
+$(U+V)^\perp$, giving the asserted dimension and distance.
+Finally $F(u,u+v)=(v,u+v)$ proves the complement identity.
+\end{proof}
+
+\subsection{Exact representatives and nested children}
+Choose a deterministic invertible binary matrix $G$ carrying the first
+$r=\dim U$ coordinate space $E_r$ to $U$, and set
+\[
+ K_G=\begin{pmatrix}G&0\\G+G^{-T}&G^{-T}\end{pmatrix},\qquad
+ T_U=K_G T_{E_r}K_G^{-1}.
+\]
+Here $T_{E_r}$ is the exact coordinate $C$ operator, and the $K_G$ are
+exact Gaussian-dyadic Clifford lifts. The matrix $K_G$ is symplectic
+because $G^TG+I$ is symmetric. It fixes both $L_0,L_F$ and maps $L_V$
+to $L_{GV}$. Thus both $K_G$ and $FK_GF^{-1}$ have Fourier rank zero;
+their lifts are fixed binary address permutations with unit quadratic
+phases. In particular,
+\[
+ L(T_U)=L_U,\quad \operatorname{rank}B(T_U)=r,\quad
+ \operatorname{rank}B(FT_UF^{-1})=r,\quad T_U^2\text{ is Pauli}.
+\]
+Set $V_U=GE_r^\perp$, a direct complement of $U$. The binary action gives
+$T_U^{-1}L_F=L_{V_U}$ and hence
+$\operatorname{rank}B(FT_U^{-1})=h-r$.
+No assertion $V_U=U^\perp$ is needed.
+For the norm-one source lines and their orthogonal target hyperplanes,
+retain the inherited exact representatives; set $T_0=I,T_H=F$.
+At every common scalar gate use one identical exact representative on all
+incident registers. If separately transported representatives have the
+same preimage, their physical ratio has rank zero and is implemented as
+an actual left adapter before the gate.
+
+For $U\subseteq V$, the exact transition $T_VT_U^{-1}$ has rank
+$\dim V-\dim U$. Clifford elimination expresses it as two rank-zero
+adapters surrounding that many coordinate Fourier factors. These factors
+are a single $C^{\otimes r}$ child: the scaled Hadamard satisfies
+\[
+ \frac{1+i}{2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}
+   =SCS,\qquad S=\operatorname{diag}(1,i).
+\]
+Thus no new irrational field is introduced. All adapters are charged
+finite movement and scalar operations. Exact transitions telescope;
+commutativity between distinct $T_U$ is unnecessary.
+
+\subsection{Dirty tails and reverse/complement words}
+A dirty role entering with gauge $T_\sigma$ and whose last active frame
+is $T_U$, with $\sigma\subseteq U$, is assigned terminal gauge
+$FT_\sigma$. Its logical value is restored by the complete transparent
+scalar word, so its physical action is exactly $F$. The tail is
+$FT_\sigma T_U^{-1}$. Its rank is
+\[
+ d(L_U,L_{V_\sigma})=h-\dim U+\dim\sigma.
+\]
+Indeed $H=\sigma\oplus V_\sigma$ and $U\supseteq\sigma$ imply
+$\dim(U\cap V_\sigma)=\dim U-\dim\sigma$.
+An inactive exterior adds $m-h$, giving $m-\dim U+\dim\sigma$.
+
+For the complete time-reversed word use
+\[
+ D_U=T_UF^{-1},\qquad L(D_U)=L_{U^\perp},\qquad
+ D_UD_V^{-1}=T_UT_V^{-1}.
+\]
+The last equality is exact, so reversal preserves every inverse transition
+rank. Reverse the scalar signs and the whole transparent word, including
+old-value corrections and final cleanup. The endpoints $D_0=F^{-1}$
+and $F$ differ by a Pauli, and the inherited norm-one endpoints are
+reconciled by actual rank-zero adapters. Neither equality of binary
+matrices nor equality of frame cosets substitutes for these operators.
diff --git a/research/paired-cube-diagonal-bit-168/references/pr168-v4/research/paired-cube-bit/check_paired_cube_bit.py b/research/paired-cube-diagonal-bit-168/references/pr168-v4/research/paired-cube-bit/check_paired_cube_bit.py
new file mode 100644
index 0000000000..69d535a6c2
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr168-v4/research/paired-cube-bit/check_paired_cube_bit.py
@@ -0,0 +1,465 @@
+#!/usr/bin/env python3
+"""Standalone checker for subset-port bit words with broadcast roots (paired-cube bit word with partner-pair
+source mixing).  Standard library only; independent of the generator (own exact integer linear algebra, own
+register simulation).  Refuses to run under python -O.
+
+Inputs (plain JSON written by paired_cube_bit_word.py): graph_pP.json, word_pP.json, frames_pP.json,
+kchron_pP.json, profile_pP.json.  Checks:
+  1. exact frames: every frame record (integer annihilator rows 'a' or integer basis rows 'b') is completed to
+     the other representation by exact integer elimination; A.B^T = 0 exactly and rank A + rank B = h;
+  2. the mod-2 decoder: supports of all additions are disjoint; for every target the deliveries (star roots,
+     side roots, partner singles, partner-pair sums) XOR to {T}; stars read are exactly the c in T and the
+     side part is exactly sum_{|S cap T|=1} x_S;
+  3. geometry: source frames are the lines <chi_S>; side root frames lie in every receiver's cap
+     ker(3 chi_T - 1) and equal the common cap of the group; centre frames are the star spans; the frame of
+     every addition contains its operands' frames; every root node's frame lies in its root frame;
+  4. every role chain (start -> each op frame -> root frame -> full) is nested, including gauged starts
+     (sigma inside the first op frame, gauged roles untouched in phase one); every target chain (gauge reads
+     ascending, side root caps in root order, partner-pair delivery at the level-2 cap, final cap) is nested;
+     the partner-pair chronology: mix frame span{chi_c,chi_d} with H0-Gram I_2, contained in the receivers'
+     level-2 cap, carrier chain [<chi_c>, mix, cap, full] and passive chain [<chi_d>, mix, full] nested;
+  5. G = I - J/9 nondegenerate on every frame used (exact integer determinant test);
+  6. literal F2 replay with arbitrary dirty scratch: y = y0 + x, scratch and sources restored;
+  7. independent ledger recount from the role chains equals the saved child histogram (PR #144 bit ledger);
+  8. mutation controls (must all be REJECTED): broken partner mix, delivery outside the shared cap (full
+     frame), delivery below the level (face cap without span{chi_c,chi_d}), dropped star, non-nested role chain.
+Usage: python3 check_paired_cube_bit.py --dir out --p 12
+"""
+import argparse, json, random, sys
+from collections import Counter, defaultdict
+from math import gcd
+from operator import mul
+from pathlib import Path
+
+if sys.flags.optimize:
+    raise SystemExit('refusing to run under python -O: assertions are part of the check')
+
+
+class Fail(Exception):
+    pass
+
+
+def need(cond, msg):
+    if not cond:
+        raise Fail(msg)
+
+
+# ------------------------------------------------------------------ exact integer linear algebra
+def prim(v):
+    g = 0
+    for x in v:
+        g = gcd(g, abs(x))
+    if g > 1:
+        v = [x // g for x in v]
+    f = next((x for x in v if x), 0)
+    return tuple(-x for x in v) if f < 0 else tuple(v)
+
+
+def reduce_rows(rows, h):
+    """Integer reduced echelon form: returns (rows, pivot columns); each pivot column is zero outside its row."""
+    rows = [list(r) for r in rows]
+    out, pivs = [], []
+    for r in rows:
+        for pr, pc in zip(out, pivs):
+            if r[pc]:
+                a, b = pr[pc], r[pc]
+                r = [x * a - y * b for x, y in zip(r, pr)]
+        r = list(prim(r))
+        c = next((j for j in range(h) if r[j]), None)
+        if c is None:
+            continue
+        for i, (pr, pc) in enumerate(zip(out, pivs)):
+            if pr[c]:
+                a, b = r[c], pr[c]
+                out[i] = list(prim([x * a - y * b for x, y in zip(pr, r)]))
+        out.append(r); pivs.append(c)
+    return out, pivs
+
+
+def kernel(rows, h):
+    red, pivs = reduce_rows(rows, h)
+    ps = set(pivs)
+    basis = []
+    for f in range(h):
+        if f in ps:
+            continue
+        D = 1
+        for r, c in zip(red, pivs):
+            if r[f]:
+                D = D * abs(r[c]) // gcd(D, abs(r[c]))
+        v = [0] * h
+        v[f] = D
+        for r, c in zip(red, pivs):
+            if r[f]:
+                v[c] = -r[f] * D // r[c]
+        basis.append(prim(v))
+    return basis, len(red)
+
+
+def rank(rows, h):
+    return len(reduce_rows(rows, h)[0])
+
+
+def dot(a, b):
+    return sum(map(mul, a, b))
+
+
+# ------------------------------------------------------------------ checker
+class Checker:
+    def __init__(self, d, p, mutate=None):
+        self.mut = mutate
+        rd = lambda name: json.loads((d / ('%s_p%d.json' % (name, p))).read_text())
+        self.g, self.w, self.fr, self.k, self.prof = rd('graph'), rd('word'), rd('frames'), rd('kchron'), rd('profile')
+        self.h, self.v = self.g['h'], self.g['v']
+        if mutate == 'dropped_star':
+            j = next(i for i, r in enumerate(self.g['roots']) if r['kind'] == 'center')
+            self.g['roots'][j]['targets'] = []
+        if mutate == 'broken_mix':
+            self.k['entries'][0]['skip_mix'] = True
+        if mutate == 'outside_cap':
+            # raise the mixed carrier past the receivers' shared cap (to the full frame) and deliver there
+            e = self.k['entries'][0]
+            e['deliver_frame'] = self.w['full_frame']
+            e['carrier_chain'][2] = e['deliver_frame']
+        if mutate == 'below_level':
+            # deliver at the receivers' level-1 (face) cap, which does not contain span{chi_c, chi_d}
+            e = self.k['entries'][0]
+            face = next(j for j, r in enumerate(self.g['roots']) if r['kind'] == 'side' and set(e['receivers']) < set(r['targets']))
+            e['deliver_frame'] = self.w['root_frame'][face]
+            e['carrier_chain'][2] = e['deliver_frame']
+        if mutate == 'non_nested':
+            # move one addition's frame down to the frame of its smallest operand's source line
+            x = next(int(k) for k, f in sorted(self.w['node_frame'].items(), key=lambda kv: int(kv[0])) if int(k) >= self.v and str(int(k)) not in set(map(str, self.w['plain'])))
+            self.w['node_frame'][str(x)] = self.w['source_frame'][0]
+
+    # ---------------- frames
+    def frames(self):
+        h = self.h
+        self.A, self.B, self.dimf = {}, {}, {}
+        for key, rec in self.fr['frames'].items():
+            f = int(key)
+            if 'a' in rec:
+                A = [tuple(r) for r in rec['a']]
+                need(rank(A, h) == len(A), 'independent annihilator rows %d' % f)
+                B, _ = kernel(A, h)
+            else:
+                B = [tuple(r) for r in rec['b']]
+                need(rank(B, h) == len(B), 'independent basis rows %d' % f)
+                A, _ = kernel(B, h)
+            need(len(A) + len(B) == h and len(B) == rec['dim'], 'dimensions of frame %d' % f)
+            need(all(dot(a, b) == 0 for a in A for b in B), 'A.B = 0 for frame %d' % f)
+            self.A[f], self.B[f], self.dimf[f] = A, B, len(B)
+        self.cc = {}
+        return len(self.A)
+
+    def sub(self, f1, f2):
+        """frame f1 contained in frame f2 (exact)"""
+        if f1 == f2:
+            return True
+        key = (f1, f2)
+        r = self.cc.get(key)
+        if r is None:
+            B1, A2 = self.B[f1], self.A[f2]
+            r = len(B1) <= len(self.B[f2]) and all(dot(a, b) == 0 for a in A2 for b in B1)
+            self.cc[key] = r
+        return r
+
+    def in_frame(self, vec, f):
+        return all(dot(a, vec) == 0 for a in self.A[f])
+
+    def nondeg(self, f):
+        h = self.h
+        B, A = self.B[f], self.A[f]
+        if not B or len(B) == h:
+            return True
+        if len(B) <= len(A):
+            s = [sum(b) for b in B]
+            M = [[9 * dot(B[i], B[j]) - s[i] * s[j] for j in range(len(B))] for i in range(len(B))]
+            return rank(M, len(B)) == len(B)
+        s = [sum(a) for a in A]
+        M = [[(9 - h) * dot(A[i], A[j]) + s[i] * s[j] for j in range(len(A))] for i in range(len(A))]
+        return rank(M, len(A)) == len(A)
+
+    # ---------------- decoder
+    def decoder(self):
+        g, v = self.g, self.v
+        args = g['args']
+        lab = [frozenset(l) for l in g['labels']]
+        sup = [1 << i for i in range(v)] + [0] * (len(args) - v)
+        for x in range(v, len(args)):
+            a, b = args[x]
+            need(not sup[a] & sup[b], 'support-disjoint addition %d' % x)
+            sup[x] = sup[a] | sup[b]
+        self.sup = sup
+        acc, stars, side = [0] * v, [set() for _ in range(v)], [0] * v
+        for r in g['roots']:
+            need(r.get('coefficient', 1) == 1, 'mod-2 unit coefficients')
+            for t in r['targets']:
+                acc[t] ^= sup[r['node']]
+                if r['kind'] == 'center':
+                    stars[t].add(r['coordinate'])
+                else:
+                    side[t] ^= sup[r['node']]
+        for e in self.k['entries']:
+            for t in e['receivers']:
+                acc[t] ^= (1 << e['carrier']) | (1 << e['passive'])
+                side[t] ^= (1 << e['carrier']) | (1 << e['passive'])
+        for t in range(v):
+            need(acc[t] == 1 << t, 'decoder identity at target %d' % t)
+            need(stars[t] == set(lab[t]), 'stars read by target %d' % t)
+            orth = sum(1 << s for s in range(v) if len(lab[s] & lab[t]) == 1)
+            need(side[t] == orth, 'side part = sum over |S cap T| = 1 at target %d' % t)
+        for r in g['roots']:
+            if r['kind'] == 'center':
+                c = r['coordinate']
+                need(sup[r['node']] == sum(1 << s for s in range(v) if c in lab[s]), 'star support %d' % c)
+
+    # ---------------- geometry and chains
+    def geometry(self):
+        g, w, h, v = self.g, self.w, self.h, self.v
+        lab = g['labels']
+        chi = [tuple(1 if j in l else 0 for j in range(h)) for l in lab]
+        cov = [tuple((3 if j in l else 0) - 1 for j in range(h)) for l in lab]
+        nf = {int(k): f for k, f in w['node_frame'].items()}
+        rf = w['root_frame']
+        full = w['full_frame']
+        need(self.dimf[full] == h, 'full frame')
+        for s in range(v):
+            f = w['source_frame'][s]
+            need(self.dimf[f] == 1 and self.in_frame(chi[s], f), 'source line <chi_S>')
+        args = g['args']
+        for x, f in nf.items():
+            if x < v:
+                need(self.in_frame(chi[x], f), 'leaf label inside its frame')
+            else:
+                for y in args[x]:
+                    need(self.sub(nf[y], f), 'operand frame inside node frame (%d <- %d)' % (x, y))
+        loss = 0
+        for j, r in enumerate(g['roots']):
+            f = rf[j]
+            need(self.sub(nf[r['node']], f), 'root node frame inside root frame %d' % j)
+            if r['kind'] == 'center':
+                c = r['coordinate']
+                S = [chi[s] for s in range(v) if c in lab[s]]
+                need(all(self.in_frame(x, f) for x in S) and rank(S, h) == self.dimf[f], 'centre frame = star span')
+                loss += self.dimf[f]
+            else:
+                need(all(dot(cov[t], b) == 0 for t in r['targets'] for b in self.B[f]), 'root frame inside caps')
+                need(self.dimf[f] == h - rank([cov[t] for t in r['targets']], h), 'root frame = common cap')
+        need(loss == self.prof['loss'], 'copied-centre loss')
+        self.chi, self.cov, self.nf, self.rf = chi, cov, nf, rf
+
+    def chains(self):
+        """Role chains of the physical word and target chains; returns the recounted child histogram."""
+        g, w, h, v = self.g, self.w, self.h, self.v
+        nf, rf, full = self.nf, self.rf, w['full_frame']
+        ops, rootroles = w['ops'], w['rootroles']
+        R = 1 + max(max(a, b) for a, b, _ in ops)
+        R = max(R, 1 + max(rootroles), 1 + max(w['sources'].values()))
+        need(R == self.prof['R'], 'role count')
+        start = {}
+        for leaf, s in w['sources'].items():
+            start[s] = w['source_frame'][int(leaf)]
+        phase1 = set(w['phase1'])
+        touched_p1 = set()
+        for i in phase1:
+            touched_p1.update(ops[i][:2])
+        # phase one = closure of the centre roots' last ops under per-role precedence (recomputed)
+        prev_op, pred = {}, []
+        for i, (a, b, _) in enumerate(ops):
+            pred.append((prev_op.get(a, -1), prev_op.get(b, -1)))
+            prev_op[a] = prev_op[b] = i
+        stack = [prev_op[s] for r, s in zip(g['roots'], rootroles) if r['kind'] == 'center' and s in prev_op]
+        closure = set()
+        while stack:
+            i = stack.pop()
+            if i not in closure:
+                closure.add(i); stack.extend(j for j in pred[i] if j >= 0)
+        need(closure == phase1, 'phase one is the centre closure')
+        # conservative response support of every role (union incidence through the XOR word)
+        co = [0] * R
+        for j, (r, s) in enumerate(zip(g['roots'], rootroles)):
+            for t in r['targets']:
+                co[s] |= 1 << t
+        for a, b, _ in reversed(ops):
+            co[b] |= co[a]
+        gauge_of = {}
+        for z in w['gauges']:
+            need(sum(1 << t for t in z['targets']) == co[z['role']], 'gauge targets = response support of its role')
+            s = z['role']
+            need(s not in start, 'gauged role is not a source role')
+            need(s not in touched_p1, 'gauged role untouched in phase one')
+            need(self.dimf[z['frame']] == z['dim'] and z['dim'] > 0, 'gauge dimension')
+            need(self.nondeg(z['frame']), 'gauge frame nondegenerate')
+            start[s] = z['frame']; gauge_of[s] = z
+        seq = defaultdict(list)
+        for i, (a, b, x) in enumerate(ops):
+            f = nf[x]
+            seq[a].append(f); seq[b].append(f)
+        for j, s in enumerate(rootroles):
+            seq[s].append(rf[j])
+        H = Counter()
+        for s in range(R):
+            chain = [start.get(s)] + seq[s] + [full]
+            prev = chain[0]
+            dprev = 0 if prev is None else self.dimf[prev]
+            if prev is not None and s not in gauge_of:
+                H[1] += 1          # source role: injection frame <chi_S> (entrance child of width 1)
+            for f in chain[1:]:
+                if prev is not None:
+                    need(self.sub(prev, f), 'nested role chain (role %d)' % s)
+                d = self.dimf[f]
+                need(d >= dprev, 'ascending role chain (role %d)' % s)
+                if d > dprev:
+                    H[d - dprev] += 1
+                prev, dprev = f, d
+        for j, r in enumerate(g['roots']):
+            if r['kind'] == 'center':
+                H[self.dimf[rf[j]]] += 1   # copied-centre transform
+        # partner-pair chronology
+        src = Counter()
+        lab = [set(l) for l in g['labels']]
+        for e in self.k['entries']:
+            c, d = e['carrier'], e['passive']
+            need(len(lab[c] & lab[d]) == 1, 'carrier and passive orthogonal (H0-Gram I_2)')
+            mf, cf = e['mix_frame'], e['deliver_frame']
+            need(self.dimf[mf] == 2 and self.in_frame(self.chi[c], mf) and self.in_frame(self.chi[d], mf), 'mix frame = span{chi_c, chi_d}')
+            need(self.nondeg(mf), 'mix frame nondegenerate')
+            need(self.sub(mf, cf), 'mix frame inside the receivers shared cap')
+            need(all(dot(self.cov[t], b) == 0 for t in e['receivers'] for b in self.B[cf]), 'delivery frame inside receiver caps')
+            need(e['carrier_chain'][0] == w['source_frame'][c] and e['passive_chain'][0] == w['source_frame'][d], 'chains start at source lines')
+            need(e['carrier_chain'][1:] == [mf, cf, w['full_frame']] and e['passive_chain'][1:] == [mf, w['full_frame']], 'chain shape')
+            for ch in (e['carrier_chain'], e['passive_chain']):
+                for f1, f2 in zip(ch, ch[1:]):
+                    need(self.sub(f1, f2), 'nested source-register chain')
+                for f1, f2 in zip(ch, ch[1:]):
+                    src[self.dimf[f2] - self.dimf[f1]] += 1
+        # target chains: gauge reads (time order = reverse selection order), side caps in root order,
+        # partner-pair delivery at the level-2 cap right after its root, final cap
+        events = defaultdict(list)
+        for z in reversed(w['gauges']):
+            for t in z['targets']:
+                events[t].append(z['frame'])
+        deliveries = defaultdict(list)
+        for e in self.k['entries']:
+            deliveries[e['deliver_after_root']].append(e)
+        for j, r in enumerate(g['roots']):
+            if r['kind'] == 'center':
+                continue
+            for t in r['targets']:
+                events[t].append(rf[j])
+            for e in deliveries.get(j, []):
+                need(sorted(e['receivers']) == sorted(r['targets']), 'delivery after the receivers own level-2 root')
+                for t in e['receivers']:
+                    events[t].append(e['deliver_frame'])
+        Y = Counter()
+        cap = {}
+        for t in range(v):
+            capf = None
+            prev, dprev = None, 0
+            for f in events[t]:
+                if prev is not None:
+                    need(self.sub(prev, f), 'nested target chain (target %d)' % t)
+                need(all(dot(self.cov[t], b) == 0 for b in self.B[f]), 'target frame inside cap (target %d)' % t)
+                if self.dimf[f] > dprev:
+                    Y[self.dimf[f] - dprev] += 1
+                prev, dprev = f, self.dimf[f]
+            need(dprev <= h - 1, 'target chain below the cap')
+            Y[h - 1 - dprev] += 1 if h - 1 - dprev else 0
+        # every source register is in exactly one mixing pair
+        members = Counter()
+        for e in self.k['entries']:
+            members[e['carrier']] += 1; members[e['passive']] += 1
+        need(all(members[s] == 1 for s in range(v)), 'each source in exactly one partner pair')
+        for f in set(self.nf.values()) | set(self.rf):
+            need(self.nondeg(f), 'G-nondegenerate frame %d' % f)
+        return H, Y, src, R
+
+    # ---------------- replay
+    def replay(self, seeds=3):
+        g, w, v = self.g, self.w, self.v
+        ops, rootroles = w['ops'], w['rootroles']
+        R = self.prof['R']
+        srcmap = {int(k): s for k, s in w['sources'].items()}
+        roots = g['roots']
+        for seed in range(seeds):
+            rng = random.Random(1000 + seed)
+            x = [rng.getrandbits(1) for _ in range(v)]
+            y0 = [rng.getrandbits(1) for _ in range(v)]
+            z0 = [rng.getrandbits(1) for _ in range(R)]
+            X, Yv, z = x[:], y0[:], z0[:]
+            # y <- y - J M z (old-value response of the dirty scratch)
+            zd = z0[:]
+            for a, b, _ in ops:
+                zd[a] ^= zd[b]
+            for j, r in enumerate(roots):
+                for t in r['targets']:
+                    Yv[t] ^= zd[rootroles[j]]
+            for leaf, s in srcmap.items():      # z <- z + V x  (at the rank-one source lines)
+                z[s] ^= X[leaf]
+            for a, b, _ in ops:                  # z <- M z
+                z[a] ^= z[b]
+            for j, r in enumerate(roots):        # y <- y + J z
+                for t in r['targets']:
+                    Yv[t] ^= z[rootroles[j]]
+            for e in self.k['entries']:          # partner-pair sums on the source registers
+                if not e.get('skip_mix'):
+                    X[e['carrier']] ^= X[e['passive']]
+                for t in e['receivers']:
+                    Yv[t] ^= X[e['carrier']]
+            for e in self.k['entries']:          # undo at the full frame
+                if not e.get('skip_mix'):
+                    X[e['carrier']] ^= X[e['passive']]
+            for a, b, _ in reversed(ops):        # z <- M^-1 z
+                z[a] ^= z[b]
+            for leaf, s in srcmap.items():       # z <- z - V x
+                z[s] ^= X[leaf]
+            need(all(Yv[t] == y0[t] ^ x[t] for t in range(v)), 'replay: y = y0 + x (seed %d)' % seed)
+            need(z == z0 and X == x, 'replay: scratch and sources restored (seed %d)' % seed)
+
+    def run(self):
+        nfr = self.frames()
+        self.decoder()
+        self.geometry()
+        H, Y, src, R = self.chains()
+        self.replay()
+        h, v = self.h, self.v
+        m, W = 3 * h, 2 * v + R
+        C = Counter()
+        for part in (H, Y, src):
+            for r, c in part.items():
+                if r and c:
+                    C[r] += 3 * c
+        for z in self.w['gauges']:
+            C[3 * z['dim']] += 1
+        C[2] += 2 * v
+        saved = {int(k): c for k, c in self.prof['child_histogram'].items()}
+        need(dict(C) == saved, 'independent recount equals the saved child histogram')
+        mass = sum(r * c for r, c in C.items())
+        need(W == self.prof['W_per_vertex'] and W * m - mass == 2 * v - 3 * self.prof['loss'] == self.prof['deficit_per_vertex'],
+             'W = 2v + R and deficit 2v - 3 ell')
+        return dict(frames=nfr, roles=R, W=W, m=m, deficit=W * m - mass, children=sum(C.values()))
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__)
+    ap.add_argument('--dir', type=Path, default=Path(__file__).resolve().parent / 'out')
+    ap.add_argument('--p', type=int, default=12)
+    ap.add_argument('--no-controls', action='store_true')
+    a = ap.parse_args()
+    res = Checker(a.dir, a.p).run()
+    print('PASS p=%d: %s' % (a.p, res))
+    if a.no_controls:
+        return
+    for m in ('broken_mix', 'outside_cap', 'below_level', 'dropped_star', 'non_nested'):
+        try:
+            Checker(a.dir, a.p, mutate=m).run()
+        except Fail as e:
+            print('control %-13s REJECTED: %s' % (m, e))
+            continue
+        raise SystemExit('FAIL: control %s was accepted' % m)
+    print('PASS: all 5 mutation controls rejected')
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/certify.py b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/certify.py
new file mode 100644
index 0000000000..a773311b72
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/certify.py
@@ -0,0 +1,283 @@
+#!/usr/bin/env python3
+"""Exact certificates for the numerical implications in manuscript #109.
+
+These checks do not certify the upstream algorithm or its time bounds.
+No floating-point arithmetic is used in a certificate.
+"""
+from dataclasses import dataclass
+from fractions import Fraction as Q
+import hashlib
+import json
+from math import comb, factorial
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[1]
+
+
+def dyadic(k):
+    return Q(1, 2**k)
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+@dataclass(frozen=True)
+class Parameters:
+    tau: Q
+    sigma: Q
+    epsilon: Q
+    c: Q
+    lam: Q
+    lamp: Q
+    kappa: Q
+    beta: Q = Q(1, 2)
+    delta: Q = Q(1, 16)
+    C1: int = 20
+
+
+def proposed(k):
+    """The user's family: k=50,42,36 gives kappa=2^-154,-130,-112."""
+    return Parameters(1-dyadic(k), 1-dyadic(k), dyadic(k+1),
+                      dyadic(k+1), 1-dyadic(2*k+1),
+                      1-dyadic(2*k+2), dyadic(3*k+4))
+
+
+def balanced(k):
+    """Balance g2 and g3, gaining a factor two with the same epsilon and c."""
+    a = dyadic(k)
+    return Parameters(1-a, 1-a, a/2, a/2, 1-3*a*a/4,
+                      1-a*a/2, a**3/8)
+
+
+def near_supremum(k, relative_gap=Q(1, 100)):
+    """Approach the strict-constraint supremum while retaining 2*kappa slack."""
+    require(0 < relative_gap < 1, "Choose a positive gap smaller than one")
+    a = dyadic(k)
+    q = (1-relative_gap)*a*a/(2-a)
+    c = q/a
+    e = a/(1+a+q)
+    lamp = 1-q
+    lam = (lamp+(1-a)*(1+2*c))/2
+    return Parameters(1-a, 1-a, e, c, lam, lamp, e*q/2)
+
+
+BASELINE = Parameters(1-dyadic(50), 1-dyadic(50), dyadic(75),
+                      dyadic(56), 1-dyadic(52), 1-dyadic(54), dyadic(182))
+
+A = Q(9, 500000000000)
+LOG_BOUND = Q(5743, 500)
+
+
+def pushed(variant):
+    """Strict-margin witnesses; variable beta uses the generalized layer proof."""
+    a = A
+    if variant == "109":
+        return Parameters(1-a, 1-a, 3*a/4, a/2, 1-3*a*a/4,
+                          1-a*a/2, dyadic(109))
+    if variant == "108":
+        return Parameters(1-a, 1-a, 3*a/4, 3*a/4, 1-7*a*a/8,
+                          1-3*a*a/4, dyadic(108), beta=Q(3, 4))
+    require(variant == "rational", "Unknown pushed variant")
+    beta, e, c = Q(999, 1000), Q(999, 1000)*a, Q(998, 1000)*a
+    lamp = 1-a*c
+    lam = (lamp+(1-a)*(1+c/beta))/2
+    return Parameters(1-a, 1-a, e, c, lam, lamp, Q(58, 10**34), beta=beta)
+
+
+def layout_degree(layout_model):
+    require(layout_model in ("adjacent", "nonadjacent"), "Unknown layout model")
+    return 2 if layout_model == "adjacent" else 1
+
+
+def constraints(p, *, layout_model="adjacent", assembly_model="original"):
+    """Positive slacks, transcribed from the cited labelled source statements."""
+    require(assembly_model in ("original", "tight-gaussian"), "Unknown assembly model")
+    tight = assembly_model == "tight-gaussian"
+    e, c, t, s, b = p.epsilon, p.c, p.tau, p.sigma, p.beta
+    return {
+        # prop:simultaneous-layer and its recurrence proof.
+        "tau_positive": t,
+        "tau_below_one": 1-t,
+        "sigma_positive": s,
+        "sigma_below_one": 1-s,
+        "c_positive": c,
+        "epsilon_positive": e,
+        "beta_positive": b,
+        "beta_below_one": 1-b,
+        "lambda_above_tau": p.lam-t,
+        "lambda_above_sigma": p.lam-s,
+        "lambda_below_one": 1-p.lam,
+        "packed_overhead": p.lam-t*(1+c/b),
+        "lambda_prime_above_lambda": p.lamp-p.lam,
+        "leaf_cost": p.lamp-(s+b*(1-s)),
+        "lambda_prime_below_one": 1-p.lamp,
+        "guard_width": 1-e*p.C1,
+        # eq:fixed-parameters and setup proof in sec:assembly.
+        "dimension_upper_bound": (Q(1, 3) if tight else Q(1, 12))-e,
+        "crt_layout": 1-t-e*(layout_degree(layout_model)-t),
+        "gaussian_cost": Q(1, 4)-p.delta-(Q(5, 4) if tight else Q(3, 2))*e,
+        "prefix_cost": 1-e*(1+c),
+        "scalar_cost": 1-p.delta-e,
+        "delta_positive": p.delta,
+        "delta_below_one_eighth": Q(1, 8)-p.delta,
+        # Derived conditions used outside the final displayed system.
+        "prime_interval_growth": 1-2*e,
+        "alpha_below_sqrt_p": Q(1, 4)-(e/4 if tight else e/2),
+        "gamma_sublinear": Q(1, 2)-(Q(3, 2) if tight else 2)*e,
+        "K_smaller_than_ell": 1-e-e*c,
+        "K_dominates_log_p": e*c,
+        "r_superpolynomial": 1-e,
+        "kappa_positive": p.kappa,
+    }
+
+
+def margins(p, *, layout_model="adjacent", assembly_model="original"):
+    require(assembly_model in ("original", "tight-gaussian"), "Unknown assembly model")
+    e, c, t = p.epsilon, p.c, p.tau
+    return {
+        "g1": 1-e*(1+c),
+        "g2": e*c*(1-t),
+        "g3": e*(1-p.lamp),
+        "g4": 1-t-e*(layout_degree(layout_model)-t),
+        "g5": Q(1, 4)-p.delta-(Q(5, 4) if assembly_model == "tight-gaussian" else Q(3, 2))*e,
+        "g6": 1-p.delta-e,
+        "g7": e,
+    }
+
+
+def certify_parameters(p, *, generalized_beta=False, strict_margin=False,
+                       layout_model="adjacent", assembly_model="original",
+                       guard_model="original"):
+    require(generalized_beta or p.beta == Q(1, 2),
+            "Upstream layer proof fixes beta=1/2; select the audited generalization")
+    require(0 < p.beta < 1, "Beta must lie strictly between zero and one")
+    require(guard_model in ("original", "stopping"), "Unknown guard model")
+    if guard_model == "original":
+        require(p.C1 >= 20, "C1 must cover the upstream guard bound")
+    else:
+        # Caller must also certify m>=3 and 2<=s_c<m^5 for its complex network.
+        require(p.C1 >= 2 and p.beta >= Q(9, 10),
+                "Stopping guard requires C1>=2 and beta>=9/10")
+    slacks = constraints(p, layout_model=layout_model, assembly_model=assembly_model)
+    for name, slack in slacks.items():
+        require(slack > 0, f"Failed strict constraint: {name} ({slack})")
+    gs = margins(p, layout_model=layout_model, assembly_model=assembly_model)
+    if strict_margin:
+        require(min(gs.values()) > p.kappa, "Need a positive absorption gap")
+    else:
+        require(min(gs.values()) >= 2*p.kappa,
+                "The proposed patch requires every g_i >= 2*kappa")
+    result = {"parameters": {k: str(v) for k, v in vars(p).items()},
+            "constraint_slacks": {k: str(v) for k, v in slacks.items()},
+            "margins": {k: str(v) for k, v in gs.items()},
+            "minimum_margin": str(min(gs.values())),
+            "limiting_margins": [k for k, v in gs.items() if v == min(gs.values())],
+            "all_margins_at_least_twice_kappa": min(gs.values()) >= 2*p.kappa,
+            "absorption_gap": str(min(gs.values())-p.kappa),
+            "slack_rule": "strict" if strict_margin else "half",
+            "generalized_beta": generalized_beta}
+    if assembly_model != "original" or guard_model != "original":
+        result.update(assembly_model=assembly_model, guard_model=guard_model)
+    return result
+
+
+def network(h):
+    """Generalized counts only; these do not prove the network interface."""
+    require(h > 6 and h != 9, "Need spare coordinates and nondegenerate rational form")
+    v, m = comb(h, 3), h**3
+    N, I = v**3, 3*v**2
+    zb, zc = 3*comb(h-3, 2), comb(h-3, 3)+3*(h-3)
+    Wb, Wc = 2*N+I*(v*zb+h), 2*N+I*(v*zc+h+1)
+    Lb, Lc = I*h*h, I*(h+1)*h
+    sb, sc = Wb*m-N+2*Lb, Wc*m-2*N+2*Lc
+    eb, ec = Q(Wb*m-sb, Wb*m), Q(Wc*m-sc, Wc*m)
+    E = 64*(Wc+m+1)**3
+    B = sc+E
+    nu = 1
+    while m**nu < B:
+        nu += 1
+    return dict(h=h, v=v, m=m, N=N, I=I, zb=zb, zc=zc,
+                Wb=Wb, Wc=Wc, Lb=Lb, Lc=Lc, sb=sb, sc=sc,
+                eta_b=eb, eta_c=ec, B=B, nu=nu, C1=max(20, nu+3))
+
+
+def certify_network(h, k, log_bound):
+    n = network(h)
+    # A positive Taylor partial sum below exp(log_bound) proves m<exp(log_bound).
+    total = Q(0)
+    terms = 0
+    while total <= n["m"] and terms < 1000:
+        total += Q(log_bound**terms, factorial(terms))
+        terms += 1
+    require(total > n["m"], "Could not certify the logarithm bound")
+    for key in ("eta_b", "eta_c"):
+        require(n[key] > log_bound*dyadic(k), f"Insufficient deficit: {key}")
+    require(2*n["Lb"] < n["N"] and 2*n["Lc"] < n["N"],
+            "The generalized residual lemma requires L<N/2 for both networks")
+    require(n["C1"] == 20, "Changed network needs a revised guard exponent")
+    return {"counts": {key: str(value) for key, value in n.items()},
+            "exponent_deficit": str(dyadic(k)), "log_bound": log_bound,
+            "exp_lower_bound": str(total), "exp_series_terms": terms,
+            "scope": "Counts and exponent comparison; not a construction certificate"}
+
+
+def verify_sources():
+    manifest = json.loads((ROOT / "upstream/manifest.json").read_text())
+    for local, info in manifest["files"].items():
+        data = (ROOT / "upstream" / local).read_bytes()
+        require(hashlib.sha256(data).hexdigest() == info["sha256"],
+                f"Modified upstream source: {local}")
+    return manifest["commit"]
+
+
+def certify_rational_network():
+    # Import here to avoid a module-level cycle with search_network.
+    from search_network import log_integer_bounds
+    n = network(46)
+    lo, hi = log_integer_bounds(n["m"])
+    require(hi < LOG_BOUND, "Log bound too small")
+    require(all(n[key] > A*LOG_BOUND for key in ("eta_b", "eta_c")),
+            "Rank deficit does not support the requested rational saving")
+    certify_network(46, 36, 12)  # Residual and guard conditions.
+    return {"h": 46, "saving": str(A), "log_bound": str(LOG_BOUND),
+            "log_lower": str(lo), "log_upper": str(hi),
+            "deficit_slacks": {k: str(n[k]-A*LOG_BOUND) for k in ("eta_b", "eta_c")}}
+
+
+def certificates():
+    return {
+        "upstream_commit": verify_sources(),
+        "scope": "Exact numerical certificates, conditional on the upstream proof interfaces",
+        "cases": {name: certify_parameters(p) for name, p in {
+            "baseline_182": BASELINE, "frozen_154": proposed(50),
+            "same_network_130": proposed(42), "h46_112": proposed(36),
+            "balanced_frozen_153": balanced(50),
+            "balanced_same_network_129": balanced(42),
+            "balanced_h46_111": balanced(36),
+            "near_supremum_frozen": near_supremum(50),
+            "near_supremum_same_network": near_supremum(42),
+            "near_supremum_h46": near_supremum(36),
+        }.items()} | {
+            f"pushed_{v}": certify_parameters(pushed(v), generalized_beta=v != "109",
+                                              strict_margin=True)
+            for v in ("109", "108", "rational")
+        },
+        "networks": {
+            "baseline_h100": certify_network(100, 50, 20),
+            "improved_h100": certify_network(100, 42, 14),
+            "improved_h46": certify_network(46, 36, 12),
+            "rational_h46": certify_rational_network(),
+        },
+    }
+
+
+if __name__ == "__main__":
+    result = certificates()
+    out = ROOT / "certificates/parameters.json"
+    out.parent.mkdir(exist_ok=True)
+    out.write_text(json.dumps(result, indent=2, sort_keys=True) + "\n")
+    for name, case in result["cases"].items():
+        print(f"PASS {name}: minimum {case['minimum_margin']} ({', '.join(case['limiting_margins'])})")
+    print(f"Wrote {out.relative_to(ROOT)}")
diff --git a/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/__init__.py b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/__init__.py
new file mode 100644
index 0000000000..8521197b71
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/__init__.py
@@ -0,0 +1 @@
+"""Selected round-nine paired-cube complex producer."""
diff --git a/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/closure.py b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/closure.py
new file mode 100644
index 0000000000..8591d6a030
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/closure.py
@@ -0,0 +1,147 @@
+#!/usr/bin/env python3
+"""Carrier-arc compilation under PR162's sufficient conditions.
+
+Copyright 2026 DaysSky (PR162, research/extended-links/cxlinks.py), Apache-2.0; moved here unchanged apart from
+this header and the imports (eumemic, Claude assistance). compile_graph() in frames.py accepts a carrier arc only
+when the donor's coordinate-padded maximal frame already lies inside the receiving frame, which is sufficient but
+not necessary. compile_closure() accepts any arc set whose generalized dependency graph is acyclic and whose full
+backward intersections still contain every value span; scripts/paired_cube/verify.py checks those conditions
+independently. With arcs that pass the pre-test and repository_order=True it returns compile_graph's record.
+"""
+import heapq
+from collections import Counter
+
+from paired_cube.frames import basis, perp, contained
+
+
+def compile_closure(g, arcs, repository_order=False):
+    """compile_graph() of scripts/paired_cube/frames.py for an arbitrary legal arc list [[donor node, use code], ...].
+
+    Frames are the full backward intersections (as in the repository) and the positive-rank ledger is the
+    repository's.  The operation order is (frame rank, generalized topological index); with repository_order it is
+    the repository's own key (dimension of the coordinate-padded frame, node), which is a linear extension only for
+    arcs that pass the repository's pre-test.  Raises AssertionError if the arc set is cyclic, if the order is not a
+    linear extension, or if a value span leaves its frame."""
+    g = dict(g)
+    g['args'] = [None] + [None if a is None else [x + 1 for x in a] for a in g['args']]
+    g['roots'] = [dict(r, node=r['node'] + 1) for r in g['roots']]
+    h, inputs, args, roots = g['h'], g['inputs'], g['args'], g['roots']
+    v, n, q = len(inputs), len(args), len(roots)
+    spans = [()] * n
+    for i, u in enumerate(inputs): spans[i + 1] = (u,)
+    for x in range(v + 1, n):
+        a, b = args[x]
+        assert 0 < a < x and 0 < b < x
+        spans[x] = basis(spans[a] + spans[b])
+    rframe, rann, Y, targetH, ell = [], [], [() for _ in inputs], Counter(), 0
+    for r in roots:
+        x = r['node']
+        if r.get('kind', 'side') == 'center':
+            U = spans[x]; A = perp(U, h); ell += len(U)
+        else:
+            A = basis(inputs[t] for t in r['targets']); U = perp(A, h)
+            for t in r['targets']:
+                assert contained(Y[t], U), ('target retreat', t)
+                targetH[len(U) - len(Y[t])] += 1; Y[t] = U
+        assert contained(spans[x], U), ('root physical incompatibility', x)
+        rframe.append(U); rann.append(A)
+    for t in range(v):
+        cap = perp((inputs[t],), h)
+        assert contained(Y[t], cap)
+        targetH[h - 1 - len(Y[t])] += 1
+    active = set(range(1, v + 1)); todo = [r['node'] for r in roots]
+    while todo:
+        x = todo.pop()
+        if x in active: continue
+        active.add(x)
+        if args[x]: todo.extend(args[x])
+    arcs = dict(arcs)
+    def use_node(code): return None if code >> 31 else code // 2
+    def use_value(code): return roots[code & 0x7fffffff]['node'] if code >> 31 else args[code // 2][code & 1]
+    succ, direct, indeg = [[] for _ in args], [[] for _ in args], [0] * n
+    for x in sorted(active):
+        if args[x]:
+            for y in args[x]: succ[y].append(x); indeg[x] += 1
+    for j, r in enumerate(roots): direct[r['node']].extend(rann[j])
+    assert len(set(arcs.values())) == len(arcs), 'one donor per use'
+    for x, code in arcs.items():
+        assert args[x] and use_value(code) in args[x] and code not in (2 * x, 2 * x + 1), 'a donor carries one of its own operands'
+        t = use_node(code)
+        if t is None: direct[x].extend(rann[code & 0x7fffffff])
+        else: succ[x].append(t); indeg[t] += 1
+    heap = [x for x in sorted(active) if indeg[x] == 0]; heapq.heapify(heap); topo = []
+    while heap:
+        x = heapq.heappop(heap); topo.append(x)
+        for y in succ[x]:
+            indeg[y] -= 1
+            if indeg[y] == 0: heapq.heappush(heap, y)
+    assert len(topo) == len(active), 'the generalized dependency graph is acyclic'
+    ann, rank = [None] * n, [0] * n
+    for x in reversed(topo):
+        ann[x] = basis(direct[x] + [z for y in succ[x] for z in ann[y]]); rank[x] = h - len(ann[x])
+        assert contained(spans[x], perp(ann[x], h)), ('value span outside its frame', x)
+    index = {x: i for i, x in enumerate(topo)}
+    if repository_order:
+        dag = [[] for _ in args]; constraint = [[] for _ in args]; padded = {}
+        for x in sorted(active):
+            if args[x]:
+                for y in args[x]: dag[y].append(x)
+        for j, r in enumerate(roots): constraint[r['node']].extend(rann[j])
+        pre = [None] * n
+        for x in sorted(active, reverse=True):
+            pre[x] = basis(constraint[x] + [z for y in dag[x] for z in pre[y]])
+            cover = g.get('side_padding_mask', 0)
+            for u in spans[x]: cover |= u
+            padded[x] = perp(pre[x] + tuple(1 << j for j in range(h) if not cover >> j & 1), h)
+        order = sorted(active, key=lambda x: (len(padded[x]), x))
+    else:
+        order = sorted(active, key=lambda x: (rank[x], index[x]))
+    position = {x: i for i, x in enumerate(order)}
+    for x in order:
+        for y in succ[x]: assert position[y] > position[x]
+    uses = [[] for _ in args]
+    for x in order:
+        if args[x]:
+            for j, y in enumerate(args[x]): uses[y].append(2 * x + j)
+    for j, r in enumerate(roots): uses[r['node']].append((1 << 31) | j)
+    donors = [x for x in order if args[x]]
+    H = Counter()
+    for x in order:
+        r, degree = rank[x], len(uses[x]); assert degree > 0, ('unused input', x)
+        H[r] += degree - 1
+        if args[x]:
+            H[h - r] += 1
+            for y in args[x]:
+                assert contained(ann[x], ann[y])
+                H[r - rank[y]] += 1
+        else: H[1] += 1; H[r - 1] += 1
+    for j, rt in enumerate(roots):
+        x = rt['node']; r = rank[x]; top = len(rframe[j]); assert r <= top
+        if rt.get('kind', 'side') == 'center':
+            assert r == top
+            H[r] += 1; H[h - r] += 1
+        else: H[top - r] += 1; H[h - top] += 1
+    for x, code in arcs.items():
+        value, t = use_value(code), use_node(code); rv, rd = rank[value], rank[x]
+        top = rank[t] if t is not None else len(rframe[code & 0x7fffffff])
+        assert top >= rd >= rv
+        H[h - rd] -= 1; H[rv] -= 1; H[top - rv] -= 1; H[top - rd] += 1
+    assert min(H.values()) >= 0
+    R = len(donors) + q - len(arcs)
+    mass = sum(r * c for r, c in H.items()); assert mass == h * R + ell, (mass, h * R + ell)
+    source = Counter({int(r): c for r, c in g['source_data_histogram'].items()})
+    assert sum(r * c for r, c in source.items()) == v * (h - 1)
+    m, W = 3 * h, 2 * v + R
+    children = Counter({r: 3 * c for r, c in H.items() if r})
+    children.update({r: 3 * c for r, c in source.items() if r})
+    children.update({r: 3 * c for r, c in targetH.items() if r})
+    children[2] += 2 * v
+    cmass = sum(r * c for r, c in children.items()); deficit = W * m - cmass
+    assert deficit == 2 * v - 3 * ell
+    profile = dict(h=h, v=v, R=R, q=q, c=len(donors), matched=len(arcs), loss=ell, selected_roles=0,
+                   selected_rank_histogram={}, remaining_internal_histogram=[H[r] for r in range(h + 1)],
+                   source_data_histogram=dict(source), target_data_histogram=dict(targetH), copied_centers_already=True,
+                   m=m, W_per_vertex=W, rank_per_vertex=cmass, deficit_per_vertex=deficit,
+                   child_histogram=dict(children), matching_frames='coordinate')
+    witness = dict(matching_arcs=[[x, c] for x, c in arcs.items()], annihilators=ann, order=order)
+    return profile, witness
diff --git a/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/frames.py b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/frames.py
new file mode 100644
index 0000000000..2621ef817b
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/frames.py
@@ -0,0 +1,180 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Developed with GPT-6 Astra assistance; integrated with Codex assistance.
+"""Actual monotone carrier matching and binary frame profile for a signed DAG.
+
+Signs are recorded in the construction's scalar ledger. This compiler uses
+the safe union of operand address spans, never cancellation of a fresh value
+to justify a physical frame. New work for icekylinx, OpenAI GPT-6 Astra assistance.
+"""
+from pathlib import Path
+from collections import Counter
+import json, math, sys, functools
+
+def basis(rows):
+    b={}
+    for x in rows:
+        for p,y in sorted(b.items(),reverse=True):
+            if x>>p&1:x^=y
+        if x:
+            p=x.bit_length()-1
+            for k,y in list(b.items()):
+                if y>>p&1:b[k]=y^x
+            b[p]=x
+    return tuple(b[p] for p in sorted(b,reverse=True))
+
+def perp(rows,h):
+    rows=basis(rows); piv={r.bit_length()-1:r for r in rows}; out=[]
+    for j in range(h):
+        if j in piv:continue
+        x=1<<j
+        for p,r in piv.items():
+            if r>>j&1:x|=1<<p
+        out.append(x)
+    return basis(out)
+
+@functools.lru_cache(maxsize=500000)
+def contained(A,B):
+    for x in A:
+        for y in B:x=min(x,x^y)
+        if x:return False
+    return True
+
+def compile_graph(g, frozen_arcs):
+    if g['args'][len(g['inputs'])] is not None:
+        # Accept the construction's compact zero-based node convention.
+        g=dict(g)
+        g['args']=[None]+[None if a is None else [x+1 for x in a] for a in g['args']]
+        g['roots']=[dict(r,node=r['node']+1) for r in g['roots']]
+    h=g['h']; inputs=g['inputs']; v=len(inputs); args=g['args']; n=len(args)
+    roots=g['roots']; q=len(roots); spans=[()]*n
+    # Node zero is reserved. Input node i+1 carries the corresponding source.
+    for i,u in enumerate(inputs):spans[i+1]=(u,)
+    for x in range(v+1,n):
+        a,b=args[x]
+        assert 0<a<x and 0<b<x
+        spans[x]=basis(spans[a]+spans[b])
+    rframe=[]; rann=[]; Y=[() for _ in inputs]; targetH=Counter(); ell=0
+    for r in roots:
+        x=r['node']
+        if r.get('kind','side')=='center':
+            U=spans[x]; A=perp(U,h); ell+=len(U)
+        else:
+            A=basis(inputs[t] for t in r['targets'])
+            U=perp(A,h)
+            for t in r['targets']:
+                assert contained(Y[t],U),('target retreat',t,Y[t],U)
+                targetH[len(U)-len(Y[t])]+=1
+                Y[t]=U
+        assert contained(spans[x],U),('root physical incompatibility',x)
+        rframe.append(U);rann.append(A)
+    for t in range(v):
+        cap=perp((inputs[t],),h)
+        assert contained(Y[t],cap)
+        targetH[h-1-len(Y[t])]+=1
+    active=set(range(1,v+1)); todo=[r['node'] for r in roots]
+    while todo:
+        x=todo.pop()
+        if x in active:continue
+        active.add(x)
+        if args[x]:todo.extend(args[x])
+    # The scalar word may list dead intermediates, but they allocate no role.
+    initial=spans
+    if g.get('matching_frames','maximal') in ('maximal','coordinate'):
+        initial=[()]*n;preann=[None]*n;successors=[[] for _ in args];constraints=[[] for _ in args]
+        for x in sorted(active):
+            if args[x]:
+                for y in args[x]:successors[y].append(x)
+        for j,r in enumerate(roots):constraints[r['node']].extend(rann[j])
+        for x in sorted(active,reverse=True):
+            preann[x]=basis(constraints[x]+[z for y in successors[x] for z in preann[y]])
+            initial[x]=perp(preann[x],h)
+            if g.get('matching_frames')=='coordinate':
+                # Explicit unused orthogonal address directions may be added
+                # to side caps. Center annihilators still exclude them.
+                cover=g.get('side_padding_mask',0)
+                for u in spans[x]:cover|=u
+                initial[x]=perp(preann[x]+tuple(1<<j for j in range(h) if not cover>>j&1),h)
+            assert contained(spans[x],initial[x])
+    order=sorted(active,key=lambda x:(len(initial[x]),x)); position={x:i for i,x in enumerate(order)}
+    uses=[[] for _ in args]; usevalue=[]; usetarget=[]; useframe=[]; usecode=[]
+    for x in order:
+        if args[x]:
+            for j,y in enumerate(args[x]):
+                uses[y].append(len(usevalue));usevalue.append(y);usetarget.append(x)
+                useframe.append(initial[x]);usecode.append(2*x+j)
+    for j,r in enumerate(roots):
+        x=r['node'];uses[x].append(len(usevalue));usevalue.append(x);usetarget.append(n+j)
+        useframe.append(rframe[j]);usecode.append((1<<31)|j)
+    donors=[x for x in order if args[x]]
+    bycode={code:u for u,code in enumerate(usecode)}
+    arcs={x:bycode[code] for x,code in frozen_arcs}
+    assert len(arcs)==len(frozen_arcs)==len(set(arcs.values()))
+    donors_set=set(donors)
+    for x,u in arcs.items():
+        assert x in donors_set and usevalue[u] in args[x]
+        t=usetarget[u]
+        assert t>=n or position[t]>position[x]
+        assert contained(initial[x],useframe[u])
+    succ=[[] for _ in args]; direct=[[] for _ in args]
+    for x in order:
+        if args[x]:
+            for y in args[x]:succ[y].append(x)
+    for j,r in enumerate(roots):direct[r['node']].extend(rann[j])
+    for x,u in arcs.items():
+        t=usetarget[u]
+        if t<n:succ[x].append(t)
+        else:direct[x].extend(rann[t-n])
+    ann=[None]*n; rank=[0]*n
+    for x in reversed(order):
+        ann[x]=basis(direct[x]+[z for y in succ[x] for z in ann[y]])
+        rank[x]=h-len(ann[x])
+        assert contained(spans[x],perp(ann[x],h))
+    H=Counter()
+    for x in order:
+        r=rank[x];degree=len(uses[x]);assert degree>0,('unused input',x)
+        H[r]+=degree-1
+        if args[x]:
+            H[h-r]+=1
+            for y in args[x]:
+                assert contained(ann[x],ann[y])
+                H[r-rank[y]]+=1
+        else:H[1]+=1;H[r-1]+=1
+    for j,root in enumerate(roots):
+        x=root['node'];r=rank[x];rt=len(rframe[j]);assert r<=rt
+        if root.get('kind','side')=='center':
+            assert r==rt
+            H[r]+=1;H[h-r]+=1
+        else:H[rt-r]+=1;H[h-rt]+=1
+    for x,u in arcs.items():
+        value=usevalue[u];t=usetarget[u];rv=rank[value];rd=rank[x]
+        rt=rank[t] if t<n else len(rframe[t-n])
+        assert rt>=rd>=rv
+        H[h-rd]-=1;H[rv]-=1;H[rt-rv]-=1;H[rt-rd]+=1
+    assert min(H.values())>=0
+    R=len(donors)+q-len(arcs)
+    mass=sum(r*c for r,c in H.items());assert mass==h*R+ell,(mass,h*R+ell)
+    source=Counter({2:v,h-4:v,1:v})
+    if g.get('source_data_histogram') is not None:source=Counter({int(r):c for r,c in g['source_data_histogram'].items()})
+    assert sum(r*c for r,c in source.items())==v*(h-1)
+    m=3*h;W=2*v+R
+    children=Counter({r:3*c for r,c in H.items() if r})
+    children.update({r:3*c for r,c in source.items() if r})
+    children.update({r:3*c for r,c in targetH.items() if r})
+    children[2]+=2*v
+    cmass=sum(r*c for r,c in children.items());deficit=W*m-cmass
+    assert deficit==2*v-3*ell
+    lo=0.;hi=.1
+    if deficit>0:
+        for _ in range(75):
+            a=(lo+hi)/2
+            if math.fsum(c*r*math.exp(a*math.log(m/r)) for r,c in children.items())<W*m:lo=a
+            else:hi=a
+    profile=dict(h=h,v=v,R=R,q=q,c=len(donors),matched=len(arcs),loss=ell,selected_roles=0,
+        selected_rank_histogram={},remaining_internal_histogram=[H[r] for r in range(h+1)],
+        source_data_histogram=dict(source),target_data_histogram=dict(targetH),copied_centers_already=True,
+        m=m,W_per_vertex=W,rank_per_vertex=cmass,deficit_per_vertex=deficit,
+        numerical_complex_root=lo if deficit>0 else None,child_histogram=dict(children),
+        matching_frames=g.get('matching_frames','maximal'),
+        status='Concrete zero-gauge signed DAG, actual monotone carrier matching and complete rank profile; numerical moment, not yet a strict assembled supplier')
+    witness=dict(matching_arcs=[[x,usecode[u]] for x,u in arcs.items()],annihilators=ann,order=order)
+    return profile,witness
diff --git a/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/gauges.py b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/gauges.py
new file mode 100644
index 0000000000..a86ef6bfb5
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/gauges.py
@@ -0,0 +1,121 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Developed with GPT-6 Astra assistance; integrated with Codex assistance.
+"""One exact chronological partial-gauge selection on a concrete binary word.
+
+Uses the retained initial old-response cancellation and center-closure lemma.
+The signed matrix is specified by the literal word, while union incidence is
+only a conservative test of which target frames must contain each gauge.
+"""
+from pathlib import Path
+from collections import Counter
+import json,sys,math
+from .frames import basis,perp,contained
+
+def select(g,p,w):
+    if g['args'][len(g['inputs'])] is not None:
+        g=dict(g);g['args']=[None]+[None if a is None else [x+1 for x in a] for a in g['args']]
+        g['roots']=[dict(r,node=r['node']+1) for r in g['roots']]
+    h=p['h'];v=p['v'];args=g['args'];roots=g['roots'];n=len(args)
+    order=w['order'];ann=w['annihilators'];arcs=dict(w['matching_arcs']);incoming=set(arcs.values())
+    uses=[[] for _ in args]
+    for x in order:
+        if args[x]:
+            for j,y in enumerate(args[x]):uses[y].append(2*x+j)
+    for j,r in enumerate(roots):uses[r['node']].append((1<<31)|j)
+    assign={};sources={};ops=[];opcoeff=[];R=0
+    for x in order:
+        if args[x]:
+            aa,bb=args[x];dest,control=assign[2*x],assign[2*x+1]
+            swapped=False
+            if x in arcs:
+                u=arcs[x];value=roots[u&0x7fffffff]['node'] if u>>31 else args[u//2][u&1]
+                if value==aa:dest,control=control,dest;swapped=True
+                else:assert value==bb
+                assert u not in assign;assign[u]=control
+            ops.append((dest,control,x))
+            sign=g['signs'][x-1]
+            opcoeff.append((sign,1) if swapped else (1,sign))
+        else:dest=R;R+=1;sources[x]=dest
+        free=[u for u in uses[x] if u not in incoming];assert free
+        for j,u in enumerate(free):
+            assert u not in assign
+            if j==0:assign[u]=dest
+            else:
+                target=R;R+=1;assign[u]=target;ops.append((target,dest,x));opcoeff.append((1,1))
+    assert R==p['R'],(R,p['R'])
+    rootroles=[assign[(1<<31)|j] for j in range(len(roots))]
+    prev=[-1]*R;pred=[];first=[None]*R
+    for i,(a,b,x) in enumerate(ops):
+        pred.append((prev[a],prev[b]));prev[a]=prev[b]=i
+        if first[a] is None:first[a]=x
+        if first[b] is None:first[b]=x
+    stack=[prev[s] for r,s in zip(roots,rootroles) if r.get('kind','side')=='center' and prev[s]>=0]
+    phase=set()
+    while stack:
+        i=stack.pop()
+        if i in phase:continue
+        phase.add(i);stack.extend(j for j in pred[i] if j>=0)
+    touched=set(sources.values())
+    for i in phase:touched.update(ops[i][:2])
+    co=[0]*R
+    for r,s in zip(roots,rootroles):co[s]|=sum(1<<t for t in r['targets'])
+    for a,b,x in reversed(ops):co[b]|=co[a]
+    inputs=g['inputs'];limit=[None]*v
+    for r in roots:
+        if r.get('kind','side')=='center':continue
+        A=basis(inputs[t] for t in r['targets'])
+        for t in r['targets']:
+            if limit[t] is None:limit[t]=A
+    for t in range(v):
+        if limit[t] is None:limit[t]=(inputs[t],)
+    selected=[];gauges=Counter();H=Counter({r:c for r,c in enumerate(p['remaining_internal_histogram'])})
+    candidates=sorted((s for s in range(R) if s not in touched and first[s] is not None),key=lambda s:(len(ann[first[s]]),co[s].bit_count(),s))
+    trial_a=g.get('gauge_trial_saving',0.00065)
+    def excess(t):return t*math.expm1(trial_a*math.log(3*h/t)) if t else 0.
+    rejected_cost=0
+    for s in candidates:
+        A=tuple(ann[first[s]]);targets=[];bits=co[s]
+        while bits:
+            low=bits&-bits;t=low.bit_length()-1;bits^=low;targets.append(t)
+            A=basis(A+tuple(limit[t]))
+            if len(A)==h:break
+        if len(A)==h:continue
+        r=h-len(ann[first[s]]);d=h-len(A)
+        delta=3*(excess(r-d)-excess(r))+excess(3*d)
+        delta+=3*math.fsum(excess(d)+excess(h-len(limit[t])-d)-excess(h-len(limit[t])) for t in targets)
+        if delta>=-1e-12:
+            rejected_cost+=1
+            continue
+        H[r]-=1;H[r-d]+=1;gauges[d]+=1
+        assert H[r]>=0
+        for t in targets:limit[t]=A
+        selected.append(dict(role=s,annihilator=A,rank=d,targets=targets))
+    Y=Counter();current=[()]*v
+    for z in reversed(selected):
+        U=perp(z['annihilator'],h)
+        for t in z['targets']:
+            assert contained(current[t],U)
+            Y[len(U)-len(current[t])]+=1;current[t]=U
+    for r in roots:
+        if r.get('kind','side')=='center':continue
+        U=perp(basis(inputs[t] for t in r['targets']),h)
+        for t in r['targets']:
+            assert contained(current[t],U)
+            Y[len(U)-len(current[t])]+=1;current[t]=U
+    for t in range(v):Y[h-1-len(current[t])]+=1
+    m=3*h;W=2*v+R;C=Counter({r:3*c for r,c in H.items() if r})
+    C.update({int(r):3*c for r,c in p['source_data_histogram'].items() if int(r)})
+    C.update({r:3*c for r,c in Y.items() if r});C.update({3*r:c for r,c in gauges.items()})
+    C[2]+=2*v;mass=sum(r*c for r,c in C.items());assert W*m-mass==2*v-3*p['loss']
+    lo=0.;hi=.1
+    for _ in range(75):
+        a=(lo+hi)/2
+        if math.fsum(c*r*math.exp(a*math.log(m/r)) for r,c in C.items())<W*m:lo=a
+        else:hi=a
+    p=dict(p,selected_roles=len(selected),selected_rank_histogram=dict(gauges),
+        remaining_internal_histogram=[H[r] for r in range(h+1)],target_data_histogram=dict(Y),
+        phase1_operations=len(phase),total_M_operations=len(ops),untouched_non_source_roles=R-len(touched),
+        numerical_complex_root=lo,child_histogram=dict(C),rank_per_vertex=mass,
+        gauge_cost_rejections=rejected_cost,gauge_trial_saving=trial_a,
+        gauge_selection='Descending first-frame dimension, narrow incidence first; exact local numerical moment-gain filter; intersections with target upper frames; old-value reads in reverse order')
+    return p,dict(selected=selected,phase1=sorted(phase),sources=sources,rootroles=rootroles,ops=ops,opcoeff=opcoeff)
diff --git a/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/graph.py b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/graph.py
new file mode 100644
index 0000000000..580ae3baa6
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/graph.py
@@ -0,0 +1,267 @@
+#!/usr/bin/env python3
+# Copyright 2026 icekylinx. Apache-2.0.
+# Developed with GPT-6 Astra assistance; integrated with Codex assistance.
+"""Construct the actual signed H=block(B)-B producer on paired triple cubes.
+
+The coarse positive query modules restrict the already adopted PR117 DAG
+(eumemic; Apache-2.0; Claude assistance; cbb05ce504d571546d9b7794c186a613c659c3bf).
+The paired-channel decomposition and this assembly are new work for icekylinx
+with OpenAI GPT-6 Astra assistance. This file constructs a circuit; it performs no audit.
+"""
+from itertools import combinations, product
+
+
+
+class Graph:
+    def __init__(self, p, local=None):
+        self.p = p
+        self.local = local
+        self.h = 2*p
+        self.cubes = list(combinations(range(p), 3))
+        self.labels = [tuple(2*i+b for i, b in zip(I, bits))
+                       for I in self.cubes for bits in product(range(2), repeat=3)]
+        self.a = [None] * len(self.labels)
+        self.signs = [1] * len(self.labels)
+        self.s = [1 << i for i in range(len(self.labels))]
+        self.intern = {}
+        self.source = {(I, bits): 8*j+k for j, I in enumerate(self.cubes)
+                       for k, bits in enumerate(product(range(2), repeat=3))}
+        self.F, self.A, self.G = {}, {}, {}
+        self.root = []
+        self.centers = []
+
+    def add(self, a, b, sign=1):
+        if a is None:
+            assert sign == 1
+            return b
+        if b is None:
+            return a
+        if sign == 1 and a > b:
+            a, b = b, a
+        key = a, b, sign
+        if key not in self.intern:
+            # Every channel sum combines disjoint original-source supports.
+            assert not self.s[a] & self.s[b], (a, b, sign)
+            self.intern[key] = len(self.a)
+            self.a.append([a, b])
+            self.signs.append(sign)
+            self.s.append(self.s[a] | self.s[b])
+        return self.intern[key]
+
+    def sum(self, xs):
+        xs = list(xs)
+        if not xs:
+            return None
+        while len(xs) > 1:
+            xs = [self.add(xs[i], xs[i+1]) if i+1 < len(xs) else xs[i]
+                  for i in range(0, len(xs), 2)]
+        return xs[0]
+
+    def module(self, data, inputs):
+        # Modules use either zero-based or one-based node indices.
+        args = data['args']
+        n = data.get('input_count', len(inputs))
+        one_based = args[0] == [0, 0] and len(args) > n and args[n] == [0, 0]
+        image = ([None] if one_based else []) + list(inputs)
+        start = n+1 if one_based else n
+        for pair in args[start:]:
+            a, b = pair
+            image.append(self.add(image[a], image[b]))
+        return [image[x] for x in data['roots']]
+
+    def local_channels(self):
+        if self.local is not None:
+            return self.configured_local_channels(self.local)
+        for I in self.cubes:
+            edges = {}
+            for ii, jj in combinations(range(3), 2):
+                kk = 3-ii-jj
+                for a, b in product(range(2), repeat=2):
+                    bits0 = [0]*3
+                    bits0[ii], bits0[jj] = a, b
+                    bits1 = bits0[:]
+                    bits1[kk] = 1
+                    edges[ii, jj, a, b] = self.add(
+                        self.source[I, tuple(bits0)], self.source[I, tuple(bits1)])
+                self.G[I, I[ii], I[jj], 0] = self.add(
+                    edges[ii, jj, 0, 0], edges[ii, jj, 1, 1], -1)
+                self.G[I, I[ii], I[jj], 1] = self.add(
+                    edges[ii, jj, 0, 1], edges[ii, jj, 1, 0], -1)
+            for ii in range(3):
+                jj = next(j for j in range(3) if j != ii)
+                for a in range(2):
+                    ids = []
+                    for b in range(2):
+                        key = (ii, jj, a, b) if ii < jj else (jj, ii, b, a)
+                        ids.append(edges[key])
+                    self.A[I, I[ii], a] = self.add(*ids)
+            self.F[I] = self.add(self.A[I, I[0], 0], self.A[I, I[0], 1])
+
+    def configured_local_channels(self, cfg):
+        """The same 13 outputs per cube (positions 0,1,2 of I) by a configured circuit:
+        A[i,a] = sum of the ports with bit i = a; 'e<d>': two edges along direction d, 'fd': two face diagonals.
+        G[j,k,m] = [bits (j,k) = (0,m)] - [bits (j,k) = (1,1-m)], [.] summing the third bit r; 'e': edges along r,
+        'l': long-diagonal differences, 's': differences at fixed r.  F = A[f,0] + A[f,1].
+        Signed pairs take the smaller port as minuend so equal differences are shared."""
+        A_cfg = {tuple(int(c) for c in k.split(',')): v for k, v in cfg['A'].items()}
+        G_cfg = {tuple(int(c) for c in k.split(',')): v for k, v in cfg['G'].items()}
+
+        def signed(xa, xb):
+            return (self.add(xa, xb, -1), 1) if xa < xb else (self.add(xb, xa, -1), -1)
+
+        def combine(t1, t2):
+            (n1, s1), (n2, s2) = t1, t2
+            if s1 == 1:
+                return self.add(n1, n2, s2)
+            assert s2 == 1, 'negated local output'
+            return self.add(n2, n1, -1)
+        for I in self.cubes:
+            def at(values):
+                bits = [0]*3
+                for q, b in values.items():
+                    bits[q] = b
+                return self.source[I, tuple(bits)]
+
+            def edge(d, fixed):
+                return self.add(at({**fixed, d: 0}), at({**fixed, d: 1}))
+            for j, k in combinations(range(3), 2):
+                r = 3-j-k
+                for mode in range(2):
+                    u, v = 0, mode
+                    kind = G_cfg[j, k, mode]
+                    if kind == 'e':
+                        node = self.add(edge(r, {j: u, k: v}), edge(r, {j: 1-u, k: 1-v}), -1)
+                    else:
+                        assert kind in ('l', 's')
+                        far = (1, 0) if kind == 'l' else (0, 1)
+                        node = combine(signed(at({j: u, k: v, r: 0}), at({j: 1-u, k: 1-v, r: far[0]})),
+                                       signed(at({j: u, k: v, r: 1}), at({j: 1-u, k: 1-v, r: far[1]})))
+                    self.G[I, I[j], I[k], mode] = node
+            for i in range(3):
+                j, k = [q for q in range(3) if q != i]
+                for a in range(2):
+                    kind = A_cfg[i, a]
+                    if kind == 'fd':
+                        n1 = self.add(at({i: a, j: 0, k: 0}), at({i: a, j: 1, k: 1}))
+                        n2 = self.add(at({i: a, j: 0, k: 1}), at({i: a, j: 1, k: 0}))
+                    else:
+                        d = int(kind[1])
+                        assert kind[0] == 'e' and d != i
+                        o = j if d == k else k
+                        n1, n2 = edge(d, {i: a, o: 0}), edge(d, {i: a, o: 1})
+                    self.A[I, I[i], a] = self.add(n1, n2)
+            f = cfg['F']
+            self.F[I] = self.add(self.A[I, I[f], 0], self.A[I, I[f], 1])
+
+    def center_channels(self):
+        # Route each cube total to maximal binary-tree intervals avoiding I.
+        nodes = []
+        def tree(lo, hi, parent):
+            x = len(nodes)
+            nodes.append(dict(lo=lo, hi=hi, parent=parent, children=[], inputs=[]))
+            if hi-lo > 1:
+                mid = (lo+hi)//2
+                nodes[x]['children'] = [tree(lo, mid, x), tree(mid, hi, x)]
+            return x
+        tree(0, self.p, None)
+        for I in self.cubes:
+            stack = [0]
+            while stack:
+                x = stack.pop()
+                node = nodes[x]
+                if all(not node['lo'] <= i < node['hi'] for i in I):
+                    node['inputs'].append(self.F[I])
+                else:
+                    stack.extend(node['children'])
+        total = [None]*len(nodes)
+        Z = {}
+        for x, node in enumerate(nodes):
+            inherited = None if node['parent'] is None else total[node['parent']]
+            total[x] = self.add(inherited, self.sum(node['inputs']))
+            if node['hi']-node['lo'] == 1:
+                Z[node['lo']] = total[x]
+        for i in range(self.p):
+            for e in range(2):
+                face = self.sum(self.A[I, i, 1-e] for I in self.cubes if i in I)
+                self.centers.append(self.add(Z[i], face))
+        return sum(len(node['inputs']) for node in nodes)
+
+    def finish(self, triple, pair, allbut, center_kind='star'):
+        self.local_channels()
+        base_nodes = len(self.a)
+        D = dict(zip(self.cubes, self.module(triple, [self.F[I] for I in self.cubes])))
+        P, Q = {}, {}
+        for i in range(self.p):
+            others = [a for a in range(self.p) if a != i]
+            pairs = list(combinations(others, 2))
+            for bit in range(2):
+                inputs = [self.A[tuple(sorted((i, *K))), i, bit] for K in pairs]
+                for K, node in zip(pairs, self.module(pair, inputs)):
+                    P[tuple(sorted((i, *K))), i, bit] = node
+        for i, j in combinations(range(self.p), 2):
+            others = [a for a in range(self.p) if a not in (i, j)]
+            for mode in range(2):
+                inputs = [self.G[tuple(sorted((i, j, k))), i, j, mode] for k in others]
+                for k, node in zip(others, self.module(allbut, inputs)):
+                    Q[tuple(sorted((i, j, k))), i, j, mode] = node
+        query_nodes = len(self.a)
+        def emit(I, bits_list, node, numerator=1, channel=''):
+            self.root.append(dict(node=node, targets=[self.source[I, b] for b in bits_list],
+                                  coefficients=[f'{numerator}/2']*len(bits_list),
+                                  kind='side', channel=channel))
+        for I in self.cubes:
+            all_bits = list(product(range(2), repeat=3))
+            emit(I, all_bits, D[I], channel='disjoint')
+            for a in range(2):
+                emit(I, [b for b in all_bits if b[0] == a],
+                     P[I, I[0], 1-a], channel='face0')
+            for a, b in product(range(2), repeat=2):
+                emit(I, [(a, b, c) for c in range(2)],
+                     P[I, I[1], 1-b], channel='face1')
+            for a, b in product(range(2), repeat=2):
+                emit(I, [(a, b, c) for c in range(2)],
+                     Q[I, I[0], I[1], a ^ b], 2*a-1, 'edge01')
+            for bits in all_bits:
+                emit(I, [bits], P[I, I[2], 1-bits[2]], channel='face2')
+            for ii, jj in [(0, 2), (1, 2)]:
+                for bits in all_bits:
+                    emit(I, [bits], Q[I, I[ii], I[jj], bits[ii] ^ bits[jj]],
+                         2*bits[ii]-1, f'edge{ii}{jj}')
+        if center_kind == 'star':
+            # Their scatter is 1/2 sum_(i in T) S_i - 1/6 sum_i S_i.
+            for i in range(self.p):
+                for e in range(2):
+                    self.centers.append(self.sum(self.A[I, i, e] for I in self.cubes if i in I))
+            incidences = 0
+            center_scatter = '1/2 sum_(i in T) S_i - 1/6 sum_i S_i'
+        else:
+            assert center_kind == 'disjoint'
+            incidences = self.center_channels()
+            center_scatter = 'sum_i E_i/(h-3) - 1/2 sum_(i in T) E_i'
+        from fractions import Fraction
+        for coordinate, node in enumerate(self.centers):
+            if center_kind == 'star':
+                coefficients = ['1/3' if coordinate in t else '-1/6' for t in self.labels]
+            else:
+                coefficients = [str(Fraction(1, self.h-3) -
+                                    (Fraction(1, 2) if coordinate in t else 0))
+                                for t in self.labels]
+            self.root.append(dict(node=node, targets=list(range(len(self.labels))),
+                                  coefficients=coefficients, kind='center', coordinate=coordinate))
+        return dict(p=self.p, h=self.h, v=len(self.labels),
+                    labels=self.labels, inputs=[sum(1 << x for x in t) for t in self.labels],
+                    args=self.a, signs=self.signs,
+                    roots=self.root, centers=self.centers,
+                    center_kind=center_kind, center_scatter=center_scatter,
+                    local_K='Two 4x4 Hadamard/2 blocks on opposite cube parities, using original X destructively',
+                    source_data_histogram={2: len(self.labels), self.h-4: len(self.labels),
+                                           1: len(self.labels)},
+                    target_data_histogram={self.h-4: len(self.labels), 1: 3*len(self.labels)},
+                    counts=dict(local_channel_additions=base_nodes-len(self.labels),
+                                query_additions=query_nodes-base_nodes,
+                                center_additions=len(self.a)-query_nodes,
+                                center_tree_incidences=incidences,
+                                signed_additions=sum(x < 0 for x in self.signs),
+                                side_root_uses=sum(r['kind'] == 'side' for r in self.root),
+                                center_roots=len(self.centers)),
+                    status='Exact signed scalar circuit and frozen-support geometry input; rank compiler and moment are separate.')
diff --git a/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/modules.py b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/modules.py
new file mode 100644
index 0000000000..85b1397ec5
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/modules.py
@@ -0,0 +1,352 @@
+#!/usr/bin/env python3
+# Copyright 2026 icekylinx. Apache-2.0.
+# Developed with GPT-6 Astra assistance; integrated with Codex assistance.
+"""Exact positive scalar DAG modules for the paired-cube construction.
+
+The pair-disjoint module is a specialization of the already adopted PR117
+positive DAG (eumemic, Apache-2.0, pinned cbb05ce504d571546d9b7794c186a613c659c3bf).
+The balanced all-but-one module is an explicit binary-tree construction.
+All module nodes are zero-based; the first input_count nodes are inputs,
+and args[node] is null on inputs or [left,right] on addition nodes.
+"""
+from pathlib import Path
+from itertools import combinations
+import gzip,json,argparse
+ROOT=Path(__file__).resolve().parents[2]
+HERE=Path(__file__).resolve().parent
+
+def restricted_pairs(n):
+ assert 2<=n<=22
+ w=json.loads(gzip.decompress((ROOT/'references/three-stage-cover/pr117/dag.json.gz').read_bytes()))
+ oldtriples=list(combinations(range(24),3));pairs=list(combinations(range(n),2));pid={p:i for i,p in enumerate(pairs)}
+ args=[None]*len(pairs);support=[1<<i for i in range(len(pairs))];by={s:i for i,s in enumerate(support)};image=[None]
+ for t in oldtriples:
+  p=tuple(x for x in t if x!=23)
+  image.append(pid[p] if 23 in t and len(p)==2 and p[-1]<n else None)
+ for aa,bb in zip(w['args'][::2],w['args'][1::2]):
+  a,b=image[aa],image[bb]
+  if a is None or b is None:image.append(a if b is None else b);continue
+  assert not support[a]&support[b]
+  s=support[a]|support[b]
+  if s not in by:by[s]=len(args);args.append([a,b]);support.append(s)
+  image.append(by[s])
+ oldroots={t:x for t,x in zip(oldtriples,w['D'])}
+ roots=[image[oldroots[(i,j,22)]] for i,j in pairs]
+ for (i,j),r in zip(pairs,roots):
+  expected=sum(1<<k for k,(a,b) in enumerate(pairs) if i not in (a,b) and j not in (a,b))
+  assert r is not None and support[r]==expected
+ active=set(range(len(pairs)));stack=list(roots)
+ while stack:
+  x=stack.pop()
+  if x in active:continue
+  active.add(x)
+  if args[x] is not None:stack.extend(args[x])
+ ids=sorted(active);ren={x:i for i,x in enumerate(ids)}
+ args=[None if args[x] is None else [ren[y] for y in args[x]] for x in ids];roots=[ren[x] for x in roots]
+ return dict(kind='pair_disjoint',n=n,input_count=len(pairs),input_labels=pairs,output_labels=pairs,args=args,roots=roots,
+  source='PR117 h24 witness specialization: input triple {a,b,23}; output D_{i,j,22}; exact support hash-consing and pruning',
+  source_commit='cbb05ce504d571546d9b7794c186a613c659c3bf')
+
+def all_but_one(n):
+ assert n>=3
+ args=[None]*n;support=[1<<i for i in range(n)];by={s:i for i,s in enumerate(support)}
+ def add(a,b):
+  if a is None:return b
+  if b is None:return a
+  assert not support[a]&support[b]
+  s=support[a]|support[b]
+  if s not in by:by[s]=len(args);args.append([a,b]);support.append(s)
+  return by[s]
+ def tree(lo,hi):
+  if hi-lo==1:return (lo,None,None)
+  mid=(lo+hi)//2;L=tree(lo,mid);R=tree(mid,hi)
+  return (add(L[0],R[0]),L,R)
+ T=tree(0,n);roots=[None]*n
+ def walk(t,outside):
+  x,L,R=t
+  if L is None:roots[x]=outside;return
+  walk(L,add(outside,R[0]));walk(R,add(outside,L[0]))
+ walk(T,None)
+ full=(1<<n)-1
+ assert all(support[r]==full^(1<<i) for i,r in enumerate(roots))
+ active=set(range(n));stack=list(roots)
+ while stack:
+  x=stack.pop()
+  if x in active:continue
+  active.add(x)
+  if args[x] is not None:stack.extend(args[x])
+ ids=sorted(active);ren={x:i for i,x in enumerate(ids)}
+ args=[None if args[x] is None else [ren[y] for y in args[x]] for x in ids];roots=[ren[x] for x in roots]
+ return dict(kind='all_but_one',n=n,input_count=n,input_labels=list(range(n)),output_labels=list(range(n)),args=args,roots=roots,
+  source='Explicit balanced binary-tree upward sums and complementary downward sums; all additions have disjoint supports')
+
+
+def restricted_triples(p):
+    w = json.loads(gzip.decompress(
+        (ROOT / 'references/three-stage-cover/pr117/dag.json.gz').read_bytes()))
+    labels = list(combinations(range(p), 3))
+    index = {t: i+1 for i, t in enumerate(labels)}
+    args = [(0, 0)] + [(0, 0)] * len(labels)
+    supports = [0] + [1 << i for i in range(len(labels))]
+    intern = {s: i for i, s in enumerate(supports) if s}
+    old_labels = list(combinations(range(w['h']), 3))
+    image = [0] + [index.get(t, 0) for t in old_labels]
+    for aa, bb in zip(w['args'][::2], w['args'][1::2]):
+        a, b = image[aa], image[bb]
+        if not a or not b:
+            image.append(a or b)
+            continue
+        s = supports[a] | supports[b]
+        if s not in intern:
+            intern[s] = len(args)
+            args.append((a, b))
+            supports.append(s)
+        image.append(intern[s])
+    roots = [image[x] for t, x in zip(old_labels, w['D']) if t[-1] < p]
+    active = set(range(len(labels)+1))
+    stack = roots[:]
+    while stack:
+        x = stack.pop()
+        if x in active:
+            continue
+        active.add(x)
+        stack.extend(args[x])
+    order = sorted(active)
+    rename = {x: i for i, x in enumerate(order)}
+    result = dict(
+        kind='disjoint_triples', p=p, input_labels=labels,
+        target_labels=labels, input_count=len(labels),
+        args=[[rename[a], rename[b]] for x in order for a, b in [args[x]]],
+        roots=[rename[x] for x in roots],
+        source='Already adopted PR117 positive DAG, exact zero restriction and D-only pruning',
+        source_commit='cbb05ce504d571546d9b7794c186a613c659c3bf',
+        source_author='eumemic',
+        contract='Root indexed by J sums input I exactly when I and J are disjoint.')
+    return result
+
+
+# ---- Generalized restrictions for other p / other #117-format sources (eumemic, Claude assistance; Apache-2.0).
+# Same contracts and checks as restricted_triples/restricted_pairs above; the defaults of those functions are
+# keep = 0..p-1 (triples) and keep = 0..n-1, pin = 23, out = 22 (pairs) on the PR117 h24 witness.
+def _source_witness(path):
+    return json.loads(gzip.decompress(Path(path).read_bytes()))
+
+def restricted_triples_from(path, keep):
+    w = _source_witness(path)
+    keep = sorted(keep); p = len(keep); rename = {x: j for j, x in enumerate(keep)}
+    labels = list(combinations(range(p), 3)); index = {t: i+1 for i, t in enumerate(labels)}
+    args = [(0, 0)] + [(0, 0)] * len(labels); supports = [0] + [1 << i for i in range(len(labels))]
+    intern = {s: i for i, s in enumerate(supports) if s}
+    old_labels = list(combinations(range(w['h']), 3))
+    def new(t): return tuple(rename[x] for x in t) if all(x in rename for x in t) else None
+    image = [0] + [index.get(new(t), 0) for t in old_labels]
+    for aa, bb in zip(w['args'][::2], w['args'][1::2]):
+        a, b = image[aa], image[bb]
+        if not a or not b:
+            image.append(a or b); continue
+        assert not supports[a] & supports[b]
+        s = supports[a] | supports[b]
+        if s not in intern:
+            intern[s] = len(args); args.append((a, b)); supports.append(s)
+        image.append(intern[s])
+    roots = [image[x] for t, x in zip(old_labels, w['D']) if new(t) is not None]
+    for J, r in zip(labels, roots):
+        assert r and supports[r] == sum(1 << i for i, I in enumerate(labels) if not set(I) & set(J))
+    active = set(range(len(labels)+1)); stack = roots[:]
+    while stack:
+        x = stack.pop()
+        if x in active: continue
+        active.add(x); stack.extend(args[x])
+    order = sorted(active); rename2 = {x: i for i, x in enumerate(order)}
+    return dict(kind='disjoint_triples', p=p, input_labels=labels, target_labels=labels, input_count=len(labels),
+        args=[[rename2[a], rename2[b]] for x in order for a, b in [args[x]]], roots=[rename2[x] for x in roots],
+        source='exact zero restriction of a pinned #117-format witness to kept points', kept=keep,
+        contract='Root indexed by J sums input I exactly when I and J are disjoint.')
+
+def restricted_pairs_from(path, keep, pin, out):
+    w = _source_witness(path)
+    keep = sorted(keep); n = len(keep)
+    assert pin not in keep and out not in keep and pin != out
+    rename = {x: j for j, x in enumerate(keep)}
+    oldtriples = list(combinations(range(w['h']), 3)); pairs = list(combinations(range(n), 2))
+    pid = {q: i for i, q in enumerate(pairs)}
+    args = [None]*len(pairs); support = [1 << i for i in range(len(pairs))]; by = {s: i for i, s in enumerate(support)}
+    image = [None]
+    for t in oldtriples:
+        rest = tuple(x for x in t if x != pin)
+        ok = pin in t and len(rest) == 2 and all(x in rename for x in rest)
+        image.append(pid[tuple(sorted(rename[x] for x in rest))] if ok else None)
+    for aa, bb in zip(w['args'][::2], w['args'][1::2]):
+        a, b = image[aa], image[bb]
+        if a is None or b is None:
+            image.append(a if b is None else b); continue
+        assert not support[a] & support[b]
+        s = support[a] | support[b]
+        if s not in by:
+            by[s] = len(args); args.append([a, b]); support.append(s)
+        image.append(by[s])
+    oldroots = {t: x for t, x in zip(oldtriples, w['D'])}
+    roots = [image[oldroots[tuple(sorted((keep[i], keep[j], out)))]] for i, j in pairs]
+    for (i, j), r in zip(pairs, roots):
+        expected = sum(1 << k for k, (a, b) in enumerate(pairs) if i not in (a, b) and j not in (a, b))
+        assert r is not None and support[r] == expected
+    active = set(range(len(pairs))); stack = list(roots)
+    while stack:
+        x = stack.pop()
+        if x in active: continue
+        active.add(x)
+        if args[x] is not None: stack.extend(args[x])
+    ids = sorted(active); ren = {x: i for i, x in enumerate(ids)}
+    args = [None if args[x] is None else [ren[y] for y in args[x]] for x in ids]
+    return dict(kind='pair_disjoint', n=n, input_count=len(pairs), input_labels=pairs, output_labels=pairs,
+        args=args, roots=[ren[x] for x in roots], kept=keep, pin=pin, out=out,
+        source='zero restriction of a pinned #117-format witness: inputs {a,b,pin}, outputs D_{i,j,out}')
+
+
+# ---- Direct pair-module source (PMOD lane; eumemic, Claude assistance; Apache-2.0).
+def pair_module_from(path, n):
+    """Zero-based pair-disjoint module given as JSON {input_count, args, roots}; same contract as restricted_pairs(n):
+    additions combine disjoint supports and root (i,j) sums exactly the inputs {a,b} disjoint from {i,j}."""
+    d = json.loads(Path(path).read_text())
+    pairs = list(combinations(range(n), 2))
+    assert d['input_count'] == len(pairs) == len(d['roots'])
+    support = []
+    for x, a in enumerate(d['args']):
+        if x < len(pairs):
+            assert a is None
+            support.append(1 << x)
+        else:
+            assert a is not None and 0 <= a[0] < x and 0 <= a[1] < x and not support[a[0]] & support[a[1]]
+            support.append(support[a[0]] | support[a[1]])
+    for (i, j), r in zip(pairs, d['roots']):
+        assert support[r] == sum(1 << k for k, (a, b) in enumerate(pairs) if i not in (a, b) and j not in (a, b))
+    return dict(kind='pair_disjoint', n=n, input_count=len(pairs), input_labels=pairs, output_labels=pairs,
+                args=[None if a is None else list(a) for a in d['args']], roots=list(d['roots']),
+                source='Direct pair module JSON; contract checked above')
+
+
+# ---- Direct triple-module source (TMOD lane; eumemic, Claude assistance; Apache-2.0).
+def triple_module_from(path, p):
+    """One-based disjoint-triples module given as JSON (the layout of restricted_triples(p)); same contract:
+    additions combine disjoint supports and the root indexed by J sums input I exactly when I and J are disjoint."""
+    d = json.loads(Path(path).read_text())
+    labels = list(combinations(range(p), 3))
+    assert d['input_count'] == len(labels) == len(d['roots'])
+    assert [tuple(x) for x in d['input_labels']] == labels == [tuple(x) for x in d['target_labels']]
+    args = d['args']
+    assert all(list(a) == [0, 0] for a in args[:len(labels) + 1])
+    support = [0] + [1 << i for i in range(len(labels))]
+    for x in range(len(labels) + 1, len(args)):
+        a, b = args[x]
+        assert 0 < a < x and 0 < b < x and not support[a] & support[b]
+        support.append(support[a] | support[b])
+    for J, r in zip(labels, d['roots']):
+        assert support[r] == sum(1 << i for i, I in enumerate(labels) if not set(I) & set(J))
+    return dict(kind='disjoint_triples', p=p, input_labels=labels, target_labels=labels, input_count=len(labels),
+                args=[list(a) for a in args], roots=list(d['roots']),
+                source='Direct triple module JSON; contract checked above',
+                contract='Root indexed by J sums input I exactly when I and J are disjoint.')
+
+
+# ---- Direct all-but-one source (QMOD lane; eumemic, Claude assistance; Apache-2.0).
+def all_but_one_from(path, n):
+    """Zero-based all-but-one module given as JSON {input_count, args, roots}; same contract as all_but_one(n):
+    additions combine disjoint supports and root i sums exactly the inputs other than i."""
+    d = json.loads(Path(path).read_text())
+    assert d['input_count'] == n == len(d['roots'])
+    support = []
+    for x, a in enumerate(d['args']):
+        if x < n:
+            assert a is None
+            support.append(1 << x)
+        else:
+            assert a is not None and 0 <= a[0] < x and 0 <= a[1] < x and not support[a[0]] & support[a[1]]
+            support.append(support[a[0]] | support[a[1]])
+    full = (1 << n) - 1
+    assert all(support[r] == full ^ (1 << i) for i, r in enumerate(d['roots']))
+    return dict(kind='all_but_one', n=n, input_count=n, input_labels=list(range(n)), output_labels=list(range(n)),
+                args=[None if a is None else list(a) for a in d['args']], roots=list(d['roots']),
+                source='Direct all-but-one module JSON; contract checked above')
+
+
+# ---- Output merging (pmerge.py variants; eumemic, Claude assistance; Apache-2.0).
+def merge_outputs(g, G, variant):
+    """Replace #144's single-target channels of each port T=(I,b) -- face2 P[I,I2,1-b2] (1/2), edge02
+    Q[I,I0,I2,b0^b2] ((2b0-1)/2), edge12 Q[I,I1,I2,b1^b2] ((2b1-1)/2) -- by signed sums read with coefficient 1/2.
+    w02: face2 + (2b0-1)*edge02, one node per (b0,b2) shared by the ports b1=0,1; edge12 stays single.  The decoder
+    contribution is unchanged and Graph.add asserts disjoint supports."""
+    from fractions import Fraction
+    side = [r for r in g['roots'] if r['kind'] == 'side']
+    centre = [r for r in g['roots'] if r['kind'] != 'side']
+    if variant.startswith('f8:'):
+        # per-class fold orders: port t folds its face2, edge02, edge12 singles in order PERMS[code[t % 8]]
+        perms = [(0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0)]
+        code = variant[3:]
+        assert len(code) == 8 and all(ch in '012345' for ch in code)
+        names = ('face2', 'edge02', 'edge12')
+        single = {}
+        keep = []
+        for r in side:
+            if r['channel'] in names:
+                assert len(r['targets']) == 1
+                single[r['targets'][0], r['channel']] = (r['node'], Fraction(r['coefficients'][0]))
+            else:
+                keep.append(r)
+        new = []
+        for t in range(g['v']):
+            parts = [single[t, ch] for ch in names]
+            order = perms[int(code[t % 8])]
+            node, c0 = parts[order[0]]
+            for k in order[1:]:
+                nd, c = parts[k]
+                sg = c / c0
+                assert sg in (1, -1)
+                node = G.add(node, nd, int(sg))
+            new.append(dict(node=node, targets=[t], coefficients=[str(c0)], kind='side', channel='fused'))
+        return dict(g, roots=keep + new + centre, args=G.a, signs=G.signs)
+    if variant == 'pf':
+        # PR #163's fold rule: each port's three single-target channels face2, edge02, edge12 become one signed
+        # sum, folded left with the positive-coefficient channels first (in that channel order), then the negatives.
+        names = ('face2', 'edge02', 'edge12')
+        single = {}
+        keep = []
+        for r in side:
+            if r['channel'] in names:
+                assert len(r['targets']) == 1
+                single[r['targets'][0], r['channel']] = (r['node'], Fraction(r['coefficients'][0]))
+            else:
+                keep.append(r)
+        new = []
+        for t in range(g['v']):
+            parts = [single[t, ch] for ch in names]
+            order = [k for k in range(3) if parts[k][1] > 0] + [k for k in range(3) if parts[k][1] < 0]
+            node, c0 = parts[order[0]]
+            for k in order[1:]:
+                nd, c = parts[k]
+                sg = c / c0
+                assert sg in (1, -1)
+                node = G.add(node, nd, int(sg))
+            new.append(dict(node=node, targets=[t], coefficients=[str(c0)], kind='side', channel='fused'))
+        return dict(g, roots=keep + new + centre, args=G.a, signs=G.signs)
+    specs = {'w02': [('face2', 'edge02')], 'w12': [('face2', 'edge12')],
+             's': [('face2', 'edge02', 'edge12')]}[variant]
+    chan = {c: k for k, spec in enumerate(specs) for c in spec}
+    found = {}
+    keep = []
+    for r in side:
+        if r['channel'] in chan:
+            assert all(Fraction(c) == Fraction(r['coefficients'][0]) for c in r['coefficients'])
+            found[chan[r['channel']], tuple(r['targets']), r['channel']] = (r['node'], Fraction(r['coefficients'][0]))
+        else:
+            keep.append(r)
+    new = []
+    for k, tg in sorted({(k, tg) for k, tg, _ in found}):
+        parts = [found[k, tg, ch] for ch in specs[k]]
+        node, c0 = parts[0]
+        for nd, c in parts[1:]:
+            sg = c / c0
+            assert sg in (1, -1)
+            node = G.add(node, nd, int(sg))
+        new.append(dict(node=node, targets=list(tg), coefficients=[str(c0)] * len(tg), kind='side', channel=variant))
+    new.sort(key=lambda r: -len(r['targets']))   # multi-target roots before singles (target chains ascend)
+    return dict(g, roots=keep + new + centre, args=G.a, signs=G.signs)
diff --git a/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/verify.py b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/verify.py
new file mode 100644
index 0000000000..000bd2d381
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube/verify.py
@@ -0,0 +1,155 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Developed with GPT-6 Astra assistance; integrated with Codex assistance.
+"""Independent finite checks of the signed scalar word and binary geometry.
+
+Copyright 2026 icekylinx, Apache-2.0; OpenAI GPT-6 Astra assistance. Binary Lagrangian
+intersection ranks use L_U=U^perp in X plus U in Z, including degenerate U.
+"""
+from collections import Counter
+from functools import lru_cache
+from .frames import basis, perp, contained
+
+
+def verify(g, baseline, witness, word, record):
+    h,v=g['h'],g['v'];args=[None]+[None if a is None else [y+1 for y in a] for a in g['args']]
+    roots=[dict(r,node=r['node']+1) for r in g['roots']]
+    n=len(args);ann=witness['annihilators'];order=witness['order'];arcs=dict(witness['matching_arcs'])
+    active=set(order)
+    inputs=g['inputs']; spans=[()]*n;positive=[0]*n;negative=[0]*n
+    @lru_cache(None)
+    def frame(A):
+        U=perp(A,h)
+        assert len(U)+len(A)==h and all((a&b).bit_count()%2==0 for a in A for b in U)
+        return U
+    def each(bits):
+        while bits:
+            low=bits&-bits;bits^=low;yield low.bit_length()-1
+    for x in range(1,n):
+        if args[x] is None:
+            positive[x]=1<<(x-1);spans[x]=(inputs[x-1],)
+        else:
+            a,b=args[x];assert 0<a<x and 0<b<x
+            assert not (positive[a]|negative[a])&(positive[b]|negative[b])
+            sign=g['signs'][x-1];assert sign in (-1,1)
+            positive[x]=positive[a]|(positive[b] if sign==1 else negative[b])
+            negative[x]=negative[a]|(negative[b] if sign==1 else positive[b])
+            spans[x]=basis(spans[a]+spans[b])
+        if x in active:assert contained(spans[x],frame(tuple(ann[x])))
+    # Recount every actual signed broadcast, with integer twice-coefficients.
+    Hrows=[[0]*v for _ in range(v)]
+    rootann=[];rootframes=[]
+    for r in roots:
+        x=r['node']
+        if r['kind']=='center':
+            assert not negative[x]
+            expected=sum(1<<s for s,q in enumerate(inputs) if q>>r['coordinate']&1)
+            assert positive[x]==expected
+            U=spans[x];A=perp(U,h)
+            assert len(U)==h-2
+            assert r['coefficients']==['1/3' if q>>r['coordinate']&1 else '-1/6' for q in inputs]
+        else:
+            A=basis(inputs[t] for t in r['targets']);U=perp(A,h)
+            for t,c in zip(r['targets'],r['coefficients']):
+                assert c in ('1/2','-1/2');sign=1 if c=='1/2' else -1
+                for s in each(positive[x]):Hrows[t][s]+=sign
+                for s in each(negative[x]):Hrows[t][s]-=sign
+        rootann.append(A);rootframes.append(U)
+        assert contained(spans[x],U)
+    cube=[tuple(a//2 for a in label) for label in g['labels']]
+    scalar_pairs=0
+    for t,q in enumerate(inputs):
+        for s,u in enumerate(inputs):
+            overlap=(q&u).bit_count();same=cube[t]==cube[s]
+            expected=0 if same else 1-overlap
+            assert Hrows[t][s]==expected, ('Signed H mismatch',t,s)
+            distance=(q^u).bit_count()//2
+            K2=(1 if distance==3 else -1 if distance==1 else 0) if same else 0
+            B2=overlap-1
+            assert Hrows[t][s]+K2+B2==2*(t==s)
+            scalar_pairs+=1
+    # Actual local K matrix, inverse, and destructive-source frame itinerary.
+    for start in range(0,v,8):
+        qs=inputs[start:start+8]
+        K2=[[1 if (a^b).bit_count()==6 else -1 if (a^b).bit_count()==2 else 0 for b in qs] for a in qs]
+        assert all(sum(K2[i][k]*K2[k][j] for k in range(8))==4*(i==j) for i in range(8) for j in range(8))
+        for parity in (0,1):
+            source=[i for i in range(8) if i.bit_count()%2==parity];target=[i for i in range(8) if i.bit_count()%2!=parity]
+            U=basis(qs[i] for i in source);assert len(U)==3
+            assert all(contained((qs[i],),U) for i in source)
+            assert all(contained(U,perp((qs[j],),h)) for j in target)
+            assert [3-1,(h-1)-3,h-(h-1)]==[2,h-4,1]
+            assert all(sum(K2[i][k]*K2[j][k] for k in source)==4*(i==j) for i in target for j in target)
+    # Full backward intersections, including every carrier edge and root cap.
+    required=[[] for _ in args];position={x:i for i,x in enumerate(order)}
+    for x in order:
+        if args[x]:
+            for y in args[x]:required[y].extend(ann[x]);assert position[y]<position[x]
+    for r,A in zip(roots,rootann):required[r['node']].extend(A)
+    assert len(arcs)==len(set(arcs.values()))==record['matched']
+    for x,u in arcs.items():
+        assert args[x]
+        if u>>31:
+            j=u&0x7fffffff;value=roots[j]['node'];A=rootann[j]
+        else:
+            t=u//2;value=args[t][u&1];A=ann[t];assert position[x]<position[t]
+        assert value in args[x];required[x].extend(A)
+    for x in order:assert basis(required[x])==tuple(ann[x]), ('Incomplete intersection',x)
+    # Replay actual signed physical operations and every positive frame movement.
+    R=record['R'];values=[(0,0)]*R;full=basis(1<<i for i in range(h));roleann=[full]*R
+    first=[None]*R;last=[-1]*R;H=Counter();phase=set(word['phase1']);touched=set(word['sources'].values())
+    for x,s in word['sources'].items():
+        x=int(x);values[s]=(positive[x],negative[x]);roleann[s]=perp((inputs[x-1],),h);H[1]+=1
+    for i,((a,b,x),(ca,cb)) in enumerate(zip(word['ops'],word['opcoeff'])):
+        assert a!=b and ca in (-1,1) and cb in (-1,1)
+        if i in phase:
+            assert all(last[s]<0 or last[s] in phase for s in (a,b));touched.update((a,b))
+        for s in (a,b):
+            if first[s] is None:first[s]=x
+            A=tuple(ann[x]);assert contained(A,roleann[s])
+            H[len(roleann[s])-len(A)]+=1;roleann[s]=A;last[s]=i
+        pa,na=values[a];pb,nb=values[b];assert not (pa|na)&(pb|nb)
+        if ca<0:pa,na=na,pa
+        if cb<0:pb,nb=nb,pb
+        values[a]=(pa|pb,na|nb);assert values[a]==(positive[x],negative[x])
+    for r,s,A,U in zip(roots,word['rootroles'],rootann,rootframes):
+        assert values[s]==(positive[r['node']],negative[r['node']])
+        assert contained(tuple(A),roleann[s]);H[len(roleann[s])-len(A)]+=1;roleann[s]=tuple(A)
+        if r['kind']=='center':assert last[s]<0 or last[s] in phase;H[len(U)]+=1
+    for A in roleann:H[len(A)]+=1
+    assert [H[r] for r in range(1,h+1)]==baseline['remaining_internal_histogram'][1:]
+    co=[0]*R
+    for r,s in zip(roots,word['rootroles']):co[s]|=sum(1<<t for t in r['targets'])
+    for a,b,x in reversed(word['ops']):co[b]|=co[a]
+    limits=[None]*v
+    for r,A in zip(roots,rootann):
+        if r['kind']=='side':
+            for t in r['targets']:
+                if limits[t] is None:limits[t]=A
+    seen=set();birth=Counter()
+    for z in word['selected']:
+        s=z['role'];A=tuple(z['annihilator']);d=z['rank'];targets=list(each(co[s]))
+        assert s not in touched and s not in seen;seen.add(s)
+        assert targets==z['targets']
+        assert A==basis(tuple(ann[first[s]])+tuple(u for t in targets for u in limits[t]))
+        assert d==h-len(A)>0 and contained(tuple(ann[first[s]]),A)
+        r=h-len(ann[first[s]]);H[r]-=1;H[r-d]+=1;birth[d]+=1
+        for t in targets:limits[t]=A
+    assert [H[r] for r in range(1,h+1)]==record['remaining_internal_histogram'][1:]
+    assert dict(birth)=={int(k):c for k,c in record['selected_rank_histogram'].items()}
+    current=[full]*v;Y=Counter()
+    for z in reversed(word['selected']):
+        A=tuple(z['annihilator'])
+        for t in z['targets']:
+            assert contained(A,current[t]);Y[len(current[t])-len(A)]+=1;current[t]=A
+    for r,A in zip(roots,rootann):
+        if r['kind']=='side':
+            for t in r['targets']:
+                assert contained(A,current[t]);Y[len(current[t])-len(A)]+=1;current[t]=A
+    for t,q in enumerate(inputs):assert contained((q,),current[t]);Y[len(current[t])-1]+=1
+    assert dict(Y)=={int(k):c for k,c in record['target_data_histogram'].items()}
+    assert record['selected_roles']==len(seen)==2310 and birth==Counter({18:2310})
+    return dict(signed_scalar_pairs_checked=scalar_pairs,scalar_H_exact=True,K_involution_and_inverse_exact=True,
+                decoder_identity_exact=True,conservative_source_supports=True,K_source_itinerary_verified=True,
+                full_backward_intersections=True,physical_signed_M_replayed=True,
+                center_phase_closed=True,full_gauge_intersections=True,selected_sources_untouched=True,
+                reverse_target_chains=True,positive_rank_ledger_recounted=True,selected_roles=len(seen))
diff --git a/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube_physical.py b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube_physical.py
new file mode 100644
index 0000000000..187bd31c59
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr168-v4/scripts/paired_cube_physical.py
@@ -0,0 +1,328 @@
+#!/usr/bin/env python3
+"""Physical frame descent and compensated birth-cut reuse on the selected paired-cube complex word.
+
+Prepared by eumemic with Anthropic Claude assistance. Apache-2.0. The signed word, frozen arcs, gauges,
+chronology and shared-core accounting are icekylinx's PR144. Frame descent under PR130's general Clifford
+frames and compensated birth-cut reuse follow PR131 and jamesyc's PR124; deadline (late) compensation reads
+follow PR143. Frozen inputs: references/paired-cube/physical/{frames,pairs}.json.
+
+Checks, from the regenerated PR144 word and the frozen inputs only:
+  * every operation frame contains its node's value span, and every role chain
+      start -> operation frames -> root frame -> full space
+    is nested, with each recipient's chain spliced after its donor's last operation;
+  * every pair: donor ungauged, not a root, its last operation before the recipient's old-value read; the
+    recipient gauged and untouched before its read; donor last frame inside the recipient gauge; reads in
+    chronological order keep every target chain nested;
+  * the spliced positive-rank recount gives the shared-core profile with W = 2v + R - pairs and the unchanged
+    deficit 2v - 3 loss;
+  * an exact numeric replay mod 2^61-1 of the aliased signed word with arbitrary dirty scratch restores every
+    slot and adds x to y, for two seeds; omitting one recipient's read, reading a late recipient before its
+    donor's last operation, or flipping one signed coefficient is rejected.
+Writes certificates/paired-cube-physical-input.json with --write; otherwise checks it.
+"""
+import argparse
+from collections import Counter, defaultdict
+import json
+from pathlib import Path
+import random
+import sys
+import tempfile
+
+from paired_cube.frames import basis, perp, contained
+
+ROOT = Path(__file__).resolve().parents[1]
+REF = ROOT / 'references/paired-cube/physical'
+OUTPUT = ROOT / 'certificates/paired-cube-physical-input.json'
+P = (1 << 61) - 1
+
+
+def require(condition, message):
+    if not condition:
+        raise AssertionError(message)
+
+
+def regenerated_word():
+    from paired_cube_producer import regenerate
+    expected = json.loads((ROOT / 'certificates/paired-cube-complex-input.json').read_text())
+    with tempfile.TemporaryDirectory(prefix='paired-cube-physical-') as d:
+        regenerate(expected, Path(d))
+        return tuple(json.loads((Path(d) / name).read_text())
+                     for name in ('graph.json', 'frames.json', 'selection.json')) + (expected,)
+
+
+def physical(g, witness, word, record, frames_in, pairs_in):
+    h, v, R = g['h'], g['v'], record['R']
+    full = basis(1 << i for i in range(h))
+    args = [None] + [None if a is None else (a[0] + 1, a[1] + 1) for a in g['args']]
+    inputs = g['inputs']
+    roots = [dict(r, node=r['node'] + 1) for r in g['roots']]
+    ann = witness['annihilators']
+    ops = [tuple(o) for o in word['ops']]
+    coef = [tuple(c) for c in word['opcoeff']]
+    phase1 = sorted(word['phase1'])
+    pset = set(phase1)
+    rest = [i for i in range(len(ops)) if i not in pset]
+    sources = {int(x): s for x, s in word['sources'].items()}
+    require(len(ops) == record['total_M_operations'], 'Operation count')
+    spans = [()] * len(args)
+    for x in range(1, len(args)):
+        spans[x] = (inputs[x - 1],) if args[x] is None else basis(spans[args[x][0]] + spans[args[x][1]])
+    frames = [perp(tuple(ann[x]), h) for _, _, x in ops]  # PR144's full backward-intersection frames
+    moved = set()
+    for i, F in frames_in:
+        require(i not in moved and basis(F) == tuple(F) != frames[i], 'Duplicate, noncanonical or unmoved frame')
+        frames[i] = tuple(F)
+        moved.add(i)
+    changed = len(moved)
+    for i, (_, _, x) in enumerate(ops):
+        require(contained(spans[x], frames[i]), 'Value span outside operation frame')
+    start = [()] * R
+    for x, s in sources.items():
+        start[s] = (inputs[x - 1],)
+    gauge = {}
+    for z in word['selected']:
+        gauge[z['role']] = perp(tuple(z['annihilator']), h)
+        require(len(gauge[z['role']]) == z['rank'], 'Gauge rank')
+        start[z['role']] = gauge[z['role']]
+    rootframe, rootkind, rootann = {}, {}, {}
+    for r, s in zip(roots, word['rootroles']):
+        if r['kind'] == 'center':
+            U = spans[r['node']]
+            require(len(U) == h - 2, 'Centre frame rank')
+            A = perp(U, h)
+        else:
+            A = basis(inputs[t] for t in r['targets'])
+            U = perp(A, h)
+        require(s not in rootframe, 'Duplicate root role')
+        rootframe[s], rootkind[s], rootann[s] = U, r['kind'], A
+    role_ops = defaultdict(list)
+    for i, (a, b, _) in enumerate(ops):
+        require(a != b, 'Gate ports')
+        role_ops[a].append(i)
+        role_ops[b].append(i)
+    position = {i: k for k, i in enumerate(phase1 + rest)}
+
+    def chain(s):
+        return [start[s]] + [frames[i] for i in role_ops[s]] + ([rootframe[s]] if s in rootframe else []) + [full]
+
+    # Pairs and deadlines.
+    pairs = [(int(a), int(b)) for a, b, _ in pairs_in]
+    deadline = {int(b): (None if t is None else int(t)) for _, b, t in pairs_in}
+    donors = {a: b for a, b in pairs}
+    merge = {b: a for a, b in pairs}
+    require(len(donors) == len(merge) == len(pairs) and not set(donors) & set(merge), 'Pair aliases')
+    first = {s: l[0] for s, l in role_ops.items()}
+    for a, b in pairs:
+        require(a not in gauge and a not in rootframe, 'Donor kind')
+        require(b in gauge, 'Recipient kind')
+        last = role_ops[a][-1]
+        t = deadline[b]
+        if t is None:  # read at the phase cut, after all of phase one
+            require(last in pset and first[b] not in pset, 'Early pair chronology')
+        else:          # read immediately before operation t, the recipient's first operation
+            require(t == first[b] and t not in pset and position[last] < position[t], 'Late pair chronology')
+        require(contained(frames[last], gauge[b]), 'Donor frame outside recipient gauge')
+    # Chains, the spliced recount and the profile.
+    local = Counter()
+    for s in range(R):
+        seq = chain(s)
+        require(all(contained(A, B) for A, B in zip(seq, seq[1:])), 'Role chain nesting')
+        if s in merge:
+            continue
+        if s in sources.values():
+            local[1] += 1
+        if s in donors:
+            seq = seq[:-1] + chain(donors[s])
+        dims = [len(F) for F in seq]
+        local.update(d1 - d0 for d0, d1 in zip(dims, dims[1:]) if d1 > d0)
+        for t in (s, donors.get(s)):
+            if t is not None and rootkind.get(t) == 'center':
+                local[len(rootframe[t])] += 1
+    # Target chains along the actual read chronology (late reads move; sides last; inputs end).
+    cut = len(phase1)
+    when = {z['role']: (cut if deadline.get(z['role']) is None else position[deadline[z['role']]], k)
+            for k, z in enumerate(reversed(word['selected']))}
+    current = [full] * v
+    target = Counter()
+    for z in sorted(word['selected'], key=lambda z: when[z['role']]):
+        A = tuple(z['annihilator'])
+        for t in z['targets']:
+            require(contained(A, current[t]), 'Target chain order')
+            target[len(current[t]) - len(A)] += 1
+            current[t] = A
+    for r, s in zip(roots, word['rootroles']):
+        if r['kind'] == 'side':
+            for t in r['targets']:
+                require(contained(rootann[s], current[t]), 'Side target order')
+                target[len(current[t]) - len(rootann[s])] += 1
+                current[t] = rootann[s]
+    for t, q in enumerate(inputs):
+        require(contained((q,), current[t]), 'Target input')
+        target[len(current[t]) - 1] += 1
+    require(dict(target) == {int(k): n for k, n in record['target_data_histogram'].items()}, 'Target histogram')
+    source = Counter({int(k): n for k, n in record['source_data_histogram'].items()})
+    children = Counter({r: 3 * n for r, n in local.items() if r})
+    children.update({r: 3 * n for r, n in source.items() if r})
+    children.update({r: 3 * n for r, n in target.items() if r})
+    tails = Counter(len(U) for s, U in gauge.items() if s not in merge)
+    children.update({3 * d: n for d, n in tails.items() if d})
+    children[2] += 2 * v
+    m, W = 3 * h, 2 * v + R - len(pairs)
+    rank = sum(r * n for r, n in children.items())
+    require(W * m - rank == 2 * v - 3 * record['loss'] == record['deficit_per_vertex'], 'Telescoping deficit')
+    require(all(0 < r < m for r in children), 'Proper children')
+    replay = scalar_replay(g, word, R, pairs, deadline, ops, coef, phase1, rest, sources, roots)
+    return dict(h=h, v=v, R=R, physical_R=R - len(pairs), pairs=len(pairs),
+                late_pairs=sum(t is not None for t in deadline.values()), changed_operation_frames=changed,
+                m=m, W_per_vertex=W, rank_per_vertex=rank, deficit_per_vertex=W * m - rank, loss=record['loss'],
+                local_histogram=dict(sorted(local.items())), physical_gauge_histogram=dict(sorted(tails.items())),
+                source_data_histogram=dict(sorted(source.items())), target_data_histogram=dict(sorted(target.items())),
+                child_histogram=dict(sorted(children.items())), scalar_replay=replay,
+                checks=dict(value_spans_inside_frames=True, role_chains_nested=True, pair_chronology=True,
+                            donor_frames_inside_gauges=True, target_chains_nested_in_read_order=True,
+                            telescoping_deficit=True))
+
+
+def scalar_replay(g, word, R, pairs, deadline, ops, coef, phase1, rest, sources, roots):
+    h, v, inputs, labels = g['h'], g['v'], g['inputs'], g['labels']
+    inv = lambda a: pow(a % P, P - 2, P)
+    half, third, sixth = inv(2), inv(3), inv(6)
+    selected = [z['role'] for z in word['selected']]
+    deferred = set(selected)
+    cseed, dseed = {}, {}
+    for r, s in zip(roots, word['rootroles']):
+        if r['kind'] == 'center':
+            cseed[s] = [int(k == r['coordinate']) for k in range(h)]
+        else:
+            dseed[s] = {t: half if q == '1/2' else P - half for t, q in zip(r['targets'], r['coefficients'])}
+    cvec = [None] * R
+    dpart = [dict() for _ in range(R)]
+    for s, c in cseed.items():
+        cvec[s] = list(c)
+    for s, d in dseed.items():
+        dpart[s] = dict(d)
+    for (a, b, _), (ca, cb) in zip(reversed(ops), reversed(coef)):
+        if cvec[a] is not None:
+            cvec[b] = [(cb * u) % P for u in cvec[a]] if cvec[b] is None else \
+                [(w + cb * u) % P for w, u in zip(cvec[b], cvec[a])]
+        for t, u in dpart[a].items():
+            dpart[b][t] = (dpart[b].get(t, 0) + cb * u) % P
+        if ca != 1:
+            if cvec[a] is not None:
+                cvec[a] = [(ca * u) % P for u in cvec[a]]
+            dpart[a] = {t: (ca * u) % P for t, u in dpart[a].items()}
+    scatter = [[third if c in labels[t] else P - sixth for t in range(v)] for c in range(h)]
+    merge = {b: a for a, b in pairs}
+    late_at = defaultdict(list)
+    for b, t in deadline.items():
+        if t is not None:
+            late_at[t].append(b)
+    leaf = {s: x for x, s in sources.items()}
+    cubes = [range(k, k + 8) for k in range(0, v, 8)]
+
+    def run(seed, omit=None, early=None, flip=None):
+        rng = random.Random(seed)
+        live = sorted(set(range(R)) - set(merge))
+        slot = {s: i for i, s in enumerate(live)}
+        alias = lambda s: slot[merge.get(s, s)]
+        x = [rng.randrange(P) for _ in range(v)]
+        z = [rng.randrange(P) for _ in live]
+        y0 = [rng.randrange(P) for _ in range(v)]
+        a, y, yc, chron = list(z), list(y0), [0] * h, []
+
+        def read(s, sign, seed_only=False):
+            val = a[alias(s)]
+            c = cseed.get(s) if seed_only else cvec[s]
+            d = dseed.get(s, {}) if seed_only else dpart[s]
+            if c is not None:
+                for k, u in enumerate(c):
+                    yc[k] = (yc[k] + sign * u * val) % P
+            for t, u in d.items():
+                y[t] = (y[t] + sign * u * val) % P
+
+        def gate(i):
+            (dst, src, _), (ca, cb) = ops[i], coef[i]
+            cb = -cb if flip == i else cb
+            dd, ss = alias(dst), alias(src)
+            require(dd != ss, 'Aliased gate ports')
+            a[dd] = (ca * a[dd] + cb * a[ss]) % P
+            chron.append((dd, ss, ca, cb))
+        for s in range(R):
+            if s not in deferred and s not in merge:
+                read(s, -1)
+        for s, xn in leaf.items():
+            a[alias(s)] = (a[alias(s)] + x[xn - 1]) % P
+            chron.append((alias(s), None, xn - 1, None))
+        for i in phase1:
+            gate(i)
+        for s in cseed:
+            read(s, +1, True)
+        for s in reversed(selected):
+            if s != omit and (deadline.get(s) is None or s == early):
+                read(s, -1)
+        for i in rest:
+            for s in late_at.get(i, ()):
+                if s != omit and s != early:
+                    read(s, -1)
+            gate(i)
+        for s in dseed:
+            read(s, +1, True)
+        for cube in cubes:
+            qs = [inputs[t] for t in cube]
+            for i, t in enumerate(cube):
+                acc = sum(x[s] if (qs[i] ^ qs[j]).bit_count() == 6 else -x[s] if (qs[i] ^ qs[j]).bit_count() == 2
+                          else 0 for j, s in enumerate(cube))
+                y[t] = (y[t] + acc * half) % P
+        for k in range(h):
+            if yc[k]:
+                for t in range(v):
+                    y[t] = (y[t] + yc[k] * scatter[k][t]) % P
+        for dd, ss, ca, cb in reversed(chron):
+            if ss is None:
+                a[dd] = (a[dd] - x[ca]) % P
+            else:
+                a[dd] = (ca * (a[dd] - cb * a[ss])) % P
+        return a == z, all((y[t] - y0[t] - x[t]) % P == 0 for t in range(v))
+    result = {'seed1': run(1), 'seed2': run(2)}
+    require(all(r == (True, True) for r in result.values()), 'Aliased dirty-scratch replay')
+    controls = {}
+    if pairs:
+        controls['omitted_recipient_read'] = run(3, omit=pairs[0][1])
+    late = [b for b, t in deadline.items() if t is not None]
+    if late:
+        controls['late_read_before_donor_death'] = run(4, early=late[-1])
+    controls['flipped_signed_operation'] = run(5, flip=next(i for i, c in enumerate(coef) if c[1] == -1))
+    require(all(r[1] is False for r in controls.values()), 'Replay control accepted')
+    return dict(modulus='2^61-1', seeds_restored_and_y_plus_x=True,
+                rejected_controls=sorted(controls), slots=R - len(pairs))
+
+
+def checked_record():
+    g, witness, word, record = regenerated_word()
+    frames_in = json.loads((REF / 'frames.json').read_text())['frames']
+    pairs_in = json.loads((REF / 'pairs.json').read_text())['pairs']
+    result = physical(g, witness, word, record, frames_in, pairs_in)
+    require(json.loads(OUTPUT.read_text()) == json.loads(json.dumps(result)), 'Frozen physical input differs')
+    return result
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--write', action='store_true')
+    a = parser.parse_args()
+    g, witness, word, record = regenerated_word()
+    frames_in = json.loads((REF / 'frames.json').read_text())['frames']
+    pairs_in = json.loads((REF / 'pairs.json').read_text())['pairs']
+    result = physical(g, witness, word, record, frames_in, pairs_in)
+    text = json.dumps(result, indent=2, sort_keys=True) + '\n'
+    if a.write:
+        OUTPUT.write_text(text)
+    else:
+        require(OUTPUT.read_text() == text, 'Frozen physical input differs')
+    print('PASS physical paired-cube word: %d operation frames moved, %d pairs (%d late), W %d, rank %d, deficit %d'
+          % (result['changed_operation_frames'], result['pairs'], result['late_pairs'], result['W_per_vertex'],
+             result['rank_per_vertex'], result['deficit_per_vertex']), flush=True)
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/paired-cube-diagonal-bit-168/references/pr168/bit-physical-profile.json b/research/paired-cube-diagonal-bit-168/references/pr168/bit-physical-profile.json
new file mode 100644
index 0000000000..8014fc2a23
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr168/bit-physical-profile.json
@@ -0,0 +1,99 @@
+{
+  "R": 19788,
+  "W_per_vertex": 21548,
+  "changed_operation_frames": 2884,
+  "checks": {
+    "donor_frames_inside_gauges": true,
+    "gauge_read_window": true,
+    "operation_frames_nondegenerate_and_spanning": true,
+    "pair_chronology": true,
+    "partner_chronology": true,
+    "phase_one_closure": true,
+    "register_contents": true,
+    "role_chains_nested": true,
+    "target_chains_nested_in_read_order": true,
+    "telescoping_deficit": true
+  },
+  "child_histogram": {
+    "1": 125448,
+    "2": 62908,
+    "3": 29616,
+    "4": 8970,
+    "5": 7800,
+    "6": 3660,
+    "7": 3312,
+    "8": 2802,
+    "9": 1812,
+    "10": 2508,
+    "11": 1158,
+    "12": 4320,
+    "13": 1590,
+    "14": 3234,
+    "15": 1230,
+    "16": 3216,
+    "17": 1164,
+    "18": 1332,
+    "19": 918,
+    "20": 5280,
+    "21": 11160,
+    "22": 13272,
+    "60": 2200
+  },
+  "deficit_per_vertex": 1936,
+  "f2_replay": {
+    "lanes": 64,
+    "registers": 18028,
+    "rejected_controls": [
+      "alias_onto_live_register",
+      "omitted_recipient_read",
+      "read_before_donor_death"
+    ],
+    "seeds_restored_and_y_plus_x": true
+  },
+  "h": 24,
+  "late_pairs": 1760,
+  "local_histogram": {
+    "1": 35656,
+    "2": 18916,
+    "3": 9872,
+    "4": 2990,
+    "5": 2600,
+    "6": 1220,
+    "7": 1104,
+    "8": 934,
+    "9": 604,
+    "10": 836,
+    "11": 386,
+    "12": 1440,
+    "13": 530,
+    "14": 1078,
+    "15": 410,
+    "16": 1072,
+    "17": 388,
+    "18": 444,
+    "19": 306,
+    "21": 2840,
+    "22": 3544
+  },
+  "loss": 528,
+  "m": 72,
+  "operations": 41288,
+  "pairs": 1760,
+  "physical_R": 18028,
+  "physical_gauge_histogram": {
+    "20": 2200
+  },
+  "rank_per_vertex": 1549520,
+  "source_data_histogram": {
+    "1": 1760,
+    "2": 880,
+    "20": 880,
+    "22": 880
+  },
+  "target_data_histogram": {
+    "1": 4400,
+    "20": 880,
+    "21": 880
+  },
+  "v": 1760
+}
diff --git a/research/paired-cube-diagonal-bit-168/references/pr175/SOURCE.json b/research/paired-cube-diagonal-bit-168/references/pr175/SOURCE.json
new file mode 100644
index 0000000000..a1f3beeaca
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/references/pr175/SOURCE.json
@@ -0,0 +1,11 @@
+{
+  "commit": "f1081fabbfadaaaaf7292a40d428dbbb865ce920",
+  "files": {
+    "certificates/joint-dual-kappa.json": "4044f15abf49b1551193891836e2d20975b8246acc888180af47266c64f77af0",
+    "certificates/joint-dual-validation.json": "a7834f2dda430b4d28337faf86b74b2ea1259812bc4169520e141efb835ecd9d",
+    "research/joint-dual/SOURCE.json": "c8649e811e43b33a7281746fd61e9adbd1370690ef7c2ec114a6f2d61016704f",
+    "scripts/experiments/joint_dual_reclaim_compiler.py": "f0696953d057239ed5a75e28d857b814d43ca43deaaf7b7b8fd191aa115112ff"
+  },
+  "pr": 175,
+  "scope": "Exact four-file compiler runtime overlay; no mathematical fields or emitted words change."
+}
diff --git a/research/paired-cube-diagonal-bit-168/selected/bit/frames_p12.json b/research/paired-cube-diagonal-bit-168/selected/bit/frames_p12.json
new file mode 100644
index 0000000000..c21bfdfaa6
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/selected/bit/frames_p12.json
@@ -0,0 +1 @@
+{"P_used_for_ranks":"2^127-1","conventions":{"frame_ids":"frame id = id of the canonical annihilator; record \"a\" lists integer annihilator rows (frame = common kernel), record \"b\" lists integer basis rows; id 0 (empty \"a\") is the full space Q^h","gram":"G = I - J/9 on Q^h (H0 = G/2, H0(chi_S,chi_T) = (|S cap T|-1)/2); nondegeneracy tested for G","nodes":"zero-based graph nodes; nodes < v are the source leaves (port index)","ops":"[dest, control, node]: dest ^= control at the frame of node (copies: dest is the fresh role)","target_covector":"3*chi_T - 1 (target cap = {u : (3 chi_T - 1).u = 0} = H0-orthogonal complement of 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diff --git a/research/paired-cube-diagonal-bit-168/selected/bit/frames_p12.json.gz b/research/paired-cube-diagonal-bit-168/selected/bit/frames_p12.json.gz
new file mode 100644
index 0000000000..adc471b132
Binary files /dev/null and b/research/paired-cube-diagonal-bit-168/selected/bit/frames_p12.json.gz differ
diff --git a/research/paired-cube-diagonal-bit-168/selected/bit/graph_p12.json b/research/paired-cube-diagonal-bit-168/selected/bit/graph_p12.json
new file mode 100644
index 0000000000..1bee85f8e5
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/selected/bit/graph_p12.json
@@ -0,0 +1 @@
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48],[18211,25849],[25850,25851],[18221,25852],[25853,25854],[25855,25856],[25857,25858],[25860,25861],[25859,25862],[25863,25864],[3679,3919],[4129,4309],[4669,4729],[4759,4999],[5539,5659],[5749,5809],[5839,6049],[7459,7579],[7669,7729],[7759,7879],[7969,8029],[8059,8149],[18595,25866],[18656,25867],[25868,25869],[18829,25870],[25871,25872],[18540,18543],[25873,25874],[25875,25876],[18526,25877],[18580,18665],[25878,25879],[25880,25881],[18631,25882],[18653,25883],[25884,25885],[18663,25886],[25887,25888],[25889,25890],[25891,25892],[25894,25895],[25893,25896],[25897,25898]],"decoder":"mod 2: Y_T += sum of root nodes whose targets contain T (star roots: T reads star(c) for c in T; side roots; partner singles) + x_carrier + x_passive for each partner_mix entry receiving T; equals x_T = sum_{c in T} star(c) + sum_{|S cap T|=1} 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diff --git a/research/paired-cube-diagonal-bit-168/selected/bit/kchron_p12.json b/research/paired-cube-diagonal-bit-168/selected/bit/kchron_p12.json
new file mode 100644
index 0000000000..9c00fdf366
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/selected/bit/kchron_p12.json
@@ -0,0 +1 @@
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reads of non-gauged roles (frame 0); V injections at <chi_S>; ops in order (phase1 ops are those listed); centre copies read at 0; gauged old-value reads at their gauge frames (ascending per target); side root reads in root order; partial-K deliveries right after root deliver_after_root; undo at full frame; inverse ops; V 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+    },
+    {
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+    {
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+    {
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+        1129,
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+        1133,
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+        1141,
+        1142,
+        1143
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+    }
+  ],
+  "dirty_coefficient_nonzero_range": [
+    701,
+    721
+  ],
+  "seconds": 2.9162215419928543,
+  "rss_kib": 102481920
+}
diff --git a/research/paired-cube-diagonal-bit-168/selected/complex/word.json.gz b/research/paired-cube-diagonal-bit-168/selected/complex/word.json.gz
new file mode 100644
index 0000000000..805f74eefa
Binary files /dev/null and b/research/paired-cube-diagonal-bit-168/selected/complex/word.json.gz differ
diff --git a/research/paired-cube-diagonal-bit-168/verify.py b/research/paired-cube-diagonal-bit-168/verify.py
new file mode 100644
index 0000000000..240e2e4514
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/verify.py
@@ -0,0 +1,63 @@
+#!/usr/bin/env python3
+"""Read-only aggregate verification of the selected joint finite witness."""
+from hashlib import sha256
+from pathlib import Path
+import argparse
+import json
+import subprocess
+import sys
+
+HERE=Path(__file__).resolve().parent;ROOT=HERE.parents[1]
+sys.dont_write_bytecode=True
+if hasattr(sys,'set_int_max_str_digits'):sys.set_int_max_str_digits(0)
+
+
+def digest(path):return sha256(path.read_bytes()).hexdigest()
+
+
+def pins():
+    manifest=json.loads((HERE/'SOURCE.json').read_text())
+    assert manifest['files'],'Empty source closure'
+    values={name:digest(ROOT/name) for name in manifest['files']}
+    assert values==manifest['files'],'Pinned construction or dependency bytes changed'
+    return values
+
+
+def run(relative):
+    result=subprocess.run([sys.executable,'-B',str(HERE/relative)],text=True,capture_output=True)
+    if result.returncode:raise ValueError(relative+' failed:\n'+result.stdout+result.stderr)
+    return json.loads(result.stdout)
+
+
+def build():
+    print('Checking new signed complex word, all formal columns, frames and exact moment.',file=sys.stderr,flush=True)
+    complex_result=run('complex/prove.py')
+    print('Checking the fixed-coordinate/face-diagonal bit local circuit, our physical frames, aliases and 34 terminal sinks, full F2 identity and paid moment envelope.',file=sys.stderr,flush=True)
+    bit_result=run('bit/prove.py')
+    sys.path.insert(0,str(HERE/'arithmetic'))
+    from certificate import certificate
+    arithmetic=certificate(complex_result['profile'],complex_result['physical'])
+    assert bit_result['coarse']['coarse_saving']==arithmetic['coarse_bit_saving']
+    assert bit_result['coarse']['atom_beta']==arithmetic['atom_exponent']
+    assert bit_result['coarse']['ordinary_saving']==arithmetic['actual_uniform_bit_saving']
+    assert complex_result['moment']['complex_saving']==arithmetic['complex_saving']
+    return dict(status='PASS conditional finite witness',kappa=arithmetic['kappa'],
+        complex=complex_result,bit=bit_result,arithmetic=arithmetic,
+        scope='Finite word, paid moments, full finite bridge and 47 strict arithmetic constraints; source-specified all-size contracts remain assumptions.')
+
+
+def main():
+    parser=argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--write',action='store_true',help='Authoring only: write the canonical derived certificate before freezing SOURCE.json')
+    args=parser.parse_args()
+    assert not sys.flags.optimize,'Assertions must remain enabled'
+    before=None if args.write else pins()
+    result=build();target=HERE/'certificate.json'
+    if args.write:target.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n')
+    else:
+        assert result==json.loads(target.read_text()),'Canonical certificate does not reproduce'
+        assert pins()==before,'Source closure changed during verification'
+    print('PASS paired-cube-diagonal-bit-168 kappa='+result['kappa']+'; complete complex and bit words, 47 constraints, seven margins, and '+str(result['arithmetic']['adverse_control_count'])+' assembly controls.')
+
+
+if __name__=='__main__':main()
diff --git a/research/paired-cube-diagonal-bit-168/verify_parallel.py b/research/paired-cube-diagonal-bit-168/verify_parallel.py
new file mode 100644
index 0000000000..181e65095d
--- /dev/null
+++ b/research/paired-cube-diagonal-bit-168/verify_parallel.py
@@ -0,0 +1,154 @@
+#!/usr/bin/env python3
+"""Run every native verify group in isolated copies of one frozen source tree.
+
+Technique: PR154 by eumemic, pinned at
+d67773363075cc787b6f6ffa07191475c7c3fbec (Apache-2.0).
+Adaptation: Chafik Boukhalfa with OpenAI Codex assistance.
+
+The strict recipe parser rejects unaccounted commands, and source manifests
+are compared before/after snapshot creation and after all groups finish.
+The new finite package is an additional group. No producer/check is omitted.
+Unrelated PR154 source-pin and workflow policy changes are not adopted.
+"""
+import argparse
+from concurrent.futures import ThreadPoolExecutor, as_completed
+from hashlib import sha256
+import json
+import os
+from pathlib import Path
+import re
+import shutil
+import signal
+import subprocess
+import sys
+import time
+
+HERE=Path(__file__).resolve().parent;ROOT=HERE.parents[1]
+GIT=['git','-c','core.autocrlf=false','-c','commit.gpgsign=false',
+     '-c','user.name=isolated-verification','-c','user.email=verify@localhost']
+
+
+def need(ok,message):
+    if not ok:raise ValueError(message)
+
+
+def git(*args,cwd=ROOT):
+    return subprocess.check_output([*GIT,*args],cwd=cwd)
+
+
+def dump(path,value):
+    temp=path.with_suffix(path.suffix+'.tmp')
+    temp.write_text(json.dumps(value,indent=2,sort_keys=True)+'\n');temp.replace(path)
+
+
+def native_groups():
+    text=(ROOT/'Makefile').read_text();lines=text.splitlines();found=[];in_rule=False
+    need(lines.count('verify:')==1,'exactly one native verify recipe is required')
+    for line in lines:
+        if line=='verify:':
+            need(not in_rule and not found,'duplicate verify recipe');in_rule=True;continue
+        if not in_rule:continue
+        if line.startswith('\t'):
+            match=re.fullmatch(r'\t\$\(MAKE\) (verify-[a-z0-9-]+)',line)
+            need(match is not None,'unaccounted direct command in verify: '+line)
+            found.append(match.group(1));continue
+        if line.strip() and not line.startswith('#'):break
+    need(found and len(set(found))==len(found),'nonempty unique native verify groups')
+    need(len(found)==16,'review coverage before changing the native group count')
+    return found
+
+
+def source_names():
+    names=git('ls-files','-z','--cached','--others','--exclude-standard').split(b'\0')
+    return sorted({n.decode() for n in names if n and (ROOT/n.decode()).is_file()})
+
+
+def manifest(root,names):
+    result={}
+    for name in names:
+        path=root/name;need(path.is_file() and not path.is_symlink(),'regular source file required: '+name)
+        data=path.read_bytes();result[name]=dict(bytes=len(data),sha256=sha256(data).hexdigest())
+    return result
+
+
+def main():
+    pipeline_begin=time.monotonic()
+    ap=argparse.ArgumentParser(description=__doc__)
+    ap.add_argument('--output',type=Path,required=True)
+    ap.add_argument('--jobs',type=int,default=8)
+    ap.add_argument('--group-timeout',type=int,default=2400)
+    args=ap.parse_args();need(1<=args.jobs<=8,'group workers must be in [1,8]')
+    output=args.output.resolve();need(not output.exists(),'output directory must be new')
+    output.mkdir(parents=True);(output/'logs').mkdir();(output/'trees').mkdir()
+    groups=native_groups();names=source_names();before=manifest(ROOT,names)
+    need(len(names)>1500,'incomplete source tree')
+    template=output/'template';template.mkdir()
+    for name in names:
+        target=template/name;target.parent.mkdir(parents=True,exist_ok=True);shutil.copy2(ROOT/name,target)
+    need(manifest(template,names)==before==manifest(ROOT,names),'source drift during snapshot')
+    git('init','-q',cwd=template);git('add','-A',cwd=template)
+    git('commit','-q','--no-verify','-m','Frozen complete verification input',cwd=template)
+    commit=git('rev-parse','HEAD',cwd=template).decode().strip()
+    need(not git('status','--porcelain',cwd=template),'snapshot must be clean')
+    package=str(HERE.relative_to(ROOT)/'verify.py')
+    commands={g:['make',g] for g in groups};commands['selected-finite-package']=[sys.executable,'-B',package]
+    # Start historical finite searches and complete tests early; their slower
+    # tail otherwise dominates the run. The coverage record keeps native order.
+    priority=['verify-research','verify-tests','verify-joint','verify-producers',
+        'verify-community','verify-clones','verify-strips','verify-pair']
+    schedule=priority+[g for g in commands if g not in priority]
+    need(len(schedule)==len(commands) and set(schedule)==set(commands),'schedule coverage')
+    coverage=dict(technique_pr=154,technique_commit='d67773363075cc787b6f6ffa07191475c7c3fbec',
+        makefile_sha256=before['Makefile']['sha256'],native_recipe=groups,native_group_count=len(groups),
+        scheduled_commands=commands,execution_order=schedule,all_native_commands_covered=True,unaccounted_recipe_commands=0,
+        snapshot_commit=commit,source_files=before,group_workers=args.jobs,nested_jobs=1)
+    dump(output/'coverage.json',coverage)
+    env={k:v for k,v in os.environ.items() if k not in ('MAKEFLAGS','MFLAGS','MAKELEVEL')}
+    env.update(JOBS='1',MAKEFLAGS='-j1',OPENBLAS_NUM_THREADS='1',OMP_NUM_THREADS='1',PYTHONDONTWRITEBYTECODE='1')
+    setup_seconds=time.monotonic()-pipeline_begin;started=time.time();results={}
+    def run(label,command):
+        group_begin=time.monotonic()
+        tree=output/'trees'/label;git('clone','-q',str(template),str(tree),cwd=output)
+        need(git('rev-parse','HEAD',cwd=tree).decode().strip()==commit,'snapshot commit mismatch')
+        need(manifest(tree,names)==before,'group input manifest mismatch')
+        preparation_seconds=time.monotonic()-group_begin;begin=time.time();error=None;code=None
+        log=output/'logs'/(label+'.log')
+        with log.open('w') as stream:
+            try:
+                child=subprocess.Popen(command,cwd=tree,env=env,stdout=stream,stderr=subprocess.STDOUT,start_new_session=True)
+                code=child.wait(timeout=args.group_timeout)
+            except subprocess.TimeoutExpired:
+                error='group timeout';os.killpg(child.pid,signal.SIGTERM)
+                try:child.wait(timeout=5)
+                except subprocess.TimeoutExpired:os.killpg(child.pid,signal.SIGKILL);child.wait()
+        changed=git('status','--porcelain','--untracked-files=no',cwd=tree).decode()
+        try:unchanged=manifest(tree,names)==before
+        except (OSError,ValueError):unchanged=False
+        return label,dict(status='PASS' if code==0 and unchanged and not changed and not error else 'FAIL',
+            command=command,exit_code=code,error=error,elapsed_seconds=round(time.time()-begin,3),
+            clone_and_input_check_seconds=round(preparation_seconds,3),total_group_seconds=round(time.monotonic()-group_begin,3),
+            input_snapshot=commit,all_input_bytes_unchanged=unchanged,tracked_changes=changed,
+            log=str(log.relative_to(output)))
+    with ThreadPoolExecutor(max_workers=args.jobs) as pool:
+        pending={pool.submit(run,label,commands[label]):label for label in schedule}
+        for future in as_completed(pending):
+            label=pending[future]
+            try:label,result=future.result()
+            except Exception as exc:result=dict(status='FAIL',error=repr(exc),input_snapshot=commit)
+            results[label]=result;dump(output/'progress.json',dict(results=results,total=len(commands),snapshot_commit=commit))
+            print('[%d/%d] %s %s'%(len(results),len(commands),result['status'],label),flush=True)
+    after=manifest(ROOT,names)
+    stable=after==before and source_names()==names
+    complete=set(results)==set(commands)
+    ok=stable and complete and all(r['status']=='PASS' for r in results.values())
+    report=dict(status='PASS' if ok else 'FAIL',snapshot_commit=commit,
+        source_stable_before_and_after=stable,all_scheduled_groups_completed=complete,
+        coverage_sha256=sha256((output/'coverage.json').read_bytes()).hexdigest(),
+        setup_seconds=round(setup_seconds,3),elapsed_seconds=round(time.time()-started,3),
+        total_wall_seconds=round(time.monotonic()-pipeline_begin,3),results=results)
+    dump(output/'result.json',report)
+    print(report['status']+' complete isolated verification: '+str(len(results))+' groups',flush=True)
+    if not ok:raise SystemExit(1)
+
+
+if __name__=='__main__':main()