Research: dirty echo, verifier repair, and priced bit-sharing boundaries

research/bit-paired-boundary-gate/CLAIM.json

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+{
+  "continuation_of": "https://github.com/CrocSwap/integer-mult-bounds/pull/673",
+  "base_commit": "b3917936aafbd0d680a954a86e25c4ce7e28a9f9",
+  "goal_status": "ACTIVE_UNACHIEVED",
+  "success_gate": "A valid local supplier and chronological physical hosting, at least 100 net paid rank units saved per invocation, complete exact moment below one at c>1/99, and all finite interfaces priced.",
+  "result": "Paired endpoint algebra and global composition are exact. The explicit wrapper is too costly. A three-dimensional common boundary with 296 whole helpers has a verified owner plan and a passing conditional rank-one invoice with 31 repair units per invocation. No matching local scalar word has been constructed; monotone reframing of the retained word is excluded.",
+  "selected_frontier": "unchanged",
+  "coordination": "Existing PR673 open at expected b391793 head, with no comments or review comments. PR770 proof read at pinned 48d027a664288ea1d4e987de63cb79588b8b936b; its Gaussian floor is an external conditional premise, not a freshly replayed dependency. Final metadata refresh through PR802; PR801, PR791 and updated PR717 bodies inspected for overlap. They do not supply the paired local word."
+}

research/bit-paired-boundary-gate/PROOF.md

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+# A paired boundary contract and its pre-construction cost gate
+
+This continues PR673 at `b3917936aafbd0d680a954a86e25c4ce7e28a9f9`.
+**The requested construction remains unachieved.** This note verifies a
+different composable endpoint, a concrete but unprofitable realization, and
+an exact owner plan and cost target for a replacement local word. No new
+supplier or kappa is admitted. The selected frontier is unchanged.
+
+## 1. Change both incoming boundaries
+
+Work in an eight-dimensional rational Euclidean address space. Choose
+orthogonal projections P and Q with PQ=QP=0, rank(P)=1 and rank(Q)=k-1.
+Put R=I-P-Q. For a projector U, D_U exchanges the head and tail address
+components on U, fixes its complement, and is an involution. On orthogonal
+subspaces, D_U D_V=D_(U+V). Here these are ADDRESS permutations; addition
+of their actions on Boolean payloads is a different operation.
+
+The proposed local scalar word still computes `(x,y,z) -> (x,y+x,z)` for
+every incoming dirty helper vector z, but with different frame boundaries:
+
+| Role | Incoming frame | Outgoing frame | Monotone climb |
+|---|---|---|---:|
+| source | P | I-Q | 8-k |
+| target | Q | I-P | 8-k |
+| each helper | 0 | I | 8 |
+
+Conjugating the scalar shear by these frames gives the exact payload
+operator
+
+    M = [[D_R, 0], [D_I, D_R]].
+
+Indeed the source diagonal is D_(I-Q)D_P=D_R, the target diagonal is
+D_(I-P)D_Q=D_R, and the off-diagonal is D_(I-P)D_P=D_I.
+The reflected reverse word, with data roles exchanged, has the matching
+upper-triangular operator. Thus separate source and target entrances can
+produce a common boundary of rank k. This does not establish a scalar word
+with those frames, nor a new orthogonality representation of the E8 graph.
+
+The checker constructs rational orthogonal coordinates at each of the 111
+actual PR589 ports. A Householder map sends 2e_0 to its norm-four port q;
+P is the first coordinate, Q the next k-1 coordinates, and R the remaining
+8-k. All inner products and port images are checked exactly for k=2 and 3.
+This supplies geometric candidates for Q, not compatible scalar gates.
+
+## 2. The global endpoint composes exactly
+
+Let a=8-k. Add eight exterior blocks E_1,...,E_8 of dimension a to the
+initial eight-space. A rational involution rho exchanges R with E_s and
+fixes P+Q. It sends the local diagonal D_R to D_Es and the local off-diagonal
+D_I8 to D_(P+Q+E_s). Its incoming/outgoing frame identities are verified
+in the exact orthogonal coordinates above.
+
+Use the retained nine-stage B_4 bank schedule and its chronological idle
+advances, with every old width 7 replaced by a. The full endpoint is D_full
+on all eight banks in dimension
+
+    m_core = k+9(8-k).
+
+The verifier proves this identity in the group algebra F2[C2^10], retaining
+formal sums of permutations. Its ten commuting involutions represent P+Q
+and the nine residual blocks. The result is the single full-address exchange
+on every output bank, with no off-diagonal contribution. The regular action
+is faithful and covers all 8*1024 formal basis columns. Removing any of the
+18 bridge shears changes the endpoint. This check does not replace the
+missing local word by an assertion that such a word exists.
+
+For comparison with the actual m=64, W=28416 invoice, pad the core with
+64-m_core ordinary directions and apply one paid child on those directions
+to each data family after the completed core. The two exchanges commute.
+This retains the productive denominator and the desired D_64 endpoint.
+Padding is charged; it is not extra productive volume or free helper space.
+
+## 3. A concrete wrapper exists, but is too expensive
+
+The old local word has operator `[[D_(I-P),0],[D_I,D_(I-P)]]`.
+Apply D_Q to the target before it and D_Q to the source after it. This
+exactly produces M. Each of these native children has rank k-1. They
+are real frame descents: target Q -> 0 and source I -> I-Q. They cannot
+be subtracted from the old invoice as if they were absorbed climbs.
+
+For k=3 the wrapper adds 222 rank-two children per invocation. A simple
+physical plan allocates 439 owned helper rows, initializes them to zero,
+and reuses them after each completed local invocation. The inherited exact
+local replay restores all helper scalars; applying their final ADDRESS
+permutation to zero still gives zero. The nine existing center rows remain
+typed. Stock is 28416+439+9=28864 full rows, with productive W still 28416.
+No helper work in this plan advances an idle data owner.
+
+The complete recursive histogram for this explicit owned-row plan includes:
+the actual PR589 local profile 288 times; 288*222 rank-two wrappers; every
+full idle advance of the new five-coordinate schedule; and 28416 rank-16
+padding children. The retained fallback shape is charged per child. Its
+rank mass is 3018720 and its moment at c=1/99+10^-9 exceeds **1.7015**.
+It increases the paid mass by 4202 2/3 per invocation. It fails even if new
+chart preparation and routing have zero recursive cost, so we do not build
+its supplier/controller.
+
+The scalar/source-access bill still includes 288*24305 aligned XOR scans,
+18*111*32 bridge XOR scans, and 288*9 inherited center copy/clear episodes.
+Owned helper initialization costs 439 full-row zero scans. Each added
+wrapper retains its native child and the inherited source/route interface.
+New descriptor, chart and tape head-return constants have not been emitted;
+their omission favors a candidate that already fails. This is not a finite
+compiler admission or a claim that the new cover's fallback proof is done.
+
+## 4. A target that does not depend on perfect consolidation
+
+The meaningful next target uses k=3 directly, without those wrappers:
+source P -> I-Q, target Q -> I-P, both with rank-five monotone paths. Keep
+v=111, nine stages, 32 replicas, and at most nine rank-seven COPY episodes.
+Require no more than 296 width-eight helper paths, each with arbitrary
+incoming dirt restored in scalar coordinates.
+
+The per-replica idle widths are
+
+    10,5,20,5,30,5,30,20,10.
+
+Each has 222 physical copies and the same untouched-stage windows as the
+pinned chronology. There are exactly
+
+    222 * (1+0+2+0+3+0+3+2+1) = 2664 = 9*296
+
+whole width-eight slots. An integral flow supplies all 296 helpers at every
+stage. The checker expands it into 2664 reservations per replica, 85248
+globally, including family, window, coordinate offset, stage and helper ID.
+It checks disjoint absolute coordinates and chronological membership.
+Each replica translates the family by 888. All unused coordinates are
+returned at their original idle boundaries; their per-replica histogram is
+
+    {2:444, 4:444, 5:666, 6:444}.
+
+No new helper storage is needed for this plan; physical stock would be
+W+9=28425. The rank-16 padding is performed after the core, separately on
+every productive family. It supplies no unproved extra hosting slots.
+297 whole width-eight helpers per stage cannot fit this particular plan:
+the total slot capacity is already exhausted. Partial-width entrances,
+new banking, or different chronological schedules are outside that fence.
+
+If the missing local word had these properties, its data, helper and COPY
+rank mass per invocation would be
+
+    222*5 + 296*8 + 63 = 3541.
+
+Charge **every one of these units as its own rank-one child**, including
+all helper restoration. This dominates any partition into proper children
+of the same rank mass at 0<c<1, including the per-child fallback. Retain the
+literal owner remainder and padding. The complete conditional profile is
+
+    {1:1019808, 2:14208, 4:14208, 5:21312, 6:14208, 16:28416}.
+
+Two exact rational interval engines give the following strict results at
+**c=1/99+10^-9**, which is strictly larger than the necessary rate:
+
+| Conditional local bill | Net rank saving per invocation | Moment, approximately |
+|---|---:|---:|
+| one child per monotone role/copy | 197 1/3 | 0.9835051889234464 |
+| every local unit at rank one | 197 1/3 | 0.9947928245644905 |
+| above plus 31 rank-one repair children per invocation | 166 1/3 | 0.9999126541717095 |
+| above plus 32 repair children per invocation | 165 1/3 | 1.0000778099654908 |
+
+Thus this target pays consolidation pessimistically and leaves a precise
+31-unit repair margin at the existing per-role child volume. Any actual
+repair must also account for its source reads, storage, routing and companion
+operations; this allowance does not provide those mechanisms. It clears
+both the 100-net-unit gate and the bit-rate
+gate **only under the missing local-word hypothesis**. It is not an achieved
+improvement. Source access, all dirty columns, COPY rank, actual frame paths,
+finite cover routing, stock constants and analytic interfaces of that word
+must still be verified. The unchanged complex supplier also prevents a
+claim of kappa=0.01 from this bit-only conditional table.
+
+The boundary accounting explains the change. For general k with the charged
+padding and complete hosted remainder,
+
+    deficit = 32*(8*111*k - 9*C).
+
+With C=63, moving from k=1 to k=2 would save only 98 2/3 units per invocation,
+slightly below the requested 100. k=3 saves 197 1/3, before excess motions or
+other repairs. Neither helper deletion nor increasing the denominator is
+responsible for this conditional credit.
+
+## 5. The old scalar order cannot simply be re-framed
+
+Start every old source at its actual line and every other role at zero.
+Ignore the supplied frame table. At each scalar gate assign the smallest
+possible monotone frame: the rational span of every operand's current frame.
+Grant each copied arrival frame zero, which only makes this test easier.
+Any monotone assignment for the same ordered scalar word must contain these
+spans. Nonzero target/helper entrances could only enlarge them.
+
+On the actual PR589 word, after B the source ranks are `{4:3, 7:108}`.
+After C, all **111** sources have forced rank eight. They cannot finish in
+I-Q of dimension six (or seven for k=2) along a monotone path. Omitting C
+changes the closure and provides a nontrivial control. This is a topology
+obstruction for monotone re-framing of this word, not a lower bound for
+other scalar words or carefully paid descents. The next construction must
+change the source/read/restoration topology or pay for enough resets within
+the cost gate; a frame optimizer on the unchanged stream cannot realize it.
+
+## 6. Gaussian E8 is a distinct, conditionally closed shortcut
+
+Gaussian E8 has 60 complex lines in complex dimension four, or rational
+rank-two ports in dimension eight. PR770's weighted-clique certificate
+reports a copy lower bound of **40 complex rank units** in its stated
+whole-line scalar-copy model. That is 80 rational rank units, not 40.
+The floor is an external premise here, not independently re-proved.
+
+Under the B_K accounting and slot-domination hypotheses of the inherited
+format analysis, put k=2, h=8 and C/v=80/60=4/3. Let s=2K+1. The deficit
+per port is D=(4K-4)/3. The checker uses exact logarithm intervals to prove
+D/Cost < 0.0078 for K=1,...,42. For K>=43, the pre-stretch idle slots alone
+give
+
+    Cost >= 24 sum_(p=1)^(K-1) p log(K/p)
+         >= 24 sum p(1-p/K) = 4(K^2-1),
+    D/Cost <= 1/(3(K+1)) <= 1/132 < 0.0078.
+
+The inequality exp(t)-1>=t then bounds every saving in that format below
+0.0078<1/99. Helpers, fallback and remaining costs cannot improve this
+favorable bound. This excludes the shortcut only with the stated COPY
+floor and format hypotheses; vector-valued interacting delivery or changed
+geometry remains outside it. The two-sided boundary target above does not
+assume that its paired planes retain the old E8 orthogonality graph.
+
+## Attribution and scope
+
+The native D_U calculus, compressed B_K recipe and existing scalar word are
+inherited from the PR370/386/436/443/510/545/589 lineages, with their existing
+icekylinx, Zhihao Chen, eumemic, chafreaky, DreamingOfClouds, OpenAI and Claude
+notices. The owner windows build on DaysSky's hosting work, independently
+checked in this contribution's prior gate. The format slot bounds are
+DreamingOfClouds' PR595/717 work. Gaussian context comes from GamingPuzzled's
+PR620 and chinmayarvind23's PR770 (which credits thakrarsagar's PR738 and
+Mister-Meeseeks' PR703). No priority claim is made for the elementary
+endpoint identities, flow allocation or rank/moment inequalities.
+
+Prepared for @androolloyd with substantial OpenAI Codex assistance,
+Apache-2.0. Finite algebra and exact prices are not a formal proof of the
+multiplication theorem. The success goal remains active and unachieved.

research/bit-paired-boundary-gate/README.md

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+# Paired boundary preflight: an exact endpoint and a missing local word
+
+**No supplier or new kappa.** The 100-net-unit goal is still open.
+
+This follow-up changes the data boundary rather than withdrawing helpers
+from fixed obligations. It verifies a composable paired endpoint, rejects
+an explicit expensive wrapper, and gives a chronological 296-helper owner
+plan for a proposed rank-three common boundary. A hypothetical matching
+word passes even when all 3541 local rank units are charged separately; it
+can afford 31 additional rank-one repairs per invocation. The retained
+scalar word cannot realize it by monotone re-framing.
+
+Read [PROOF.md](PROOF.md) for the operator, full conditional invoice,
+concrete wrapper costs, source access, storage, and missing construction.
+Exact values and flow allocation are in [evidence/PRICE.json](evidence/PRICE.json).
+
+From the repository root, Python 3.11+ with the standard library:
+
+```sh
+sh research/bit-paired-boundary-gate/replay.sh /tmp/paired-boundary-new-output
+```
+
+Expected final line: `PASS_PAIRED_BOUNDARY_GATE_REPLAY`.
+The output directory must not exist. There is no solver, network access,
+package installation or historical Git dependency in replay. The checker
+supports `python3 -I -O -B`; admission checks do not rely on assertions.
+Input hashes bind the reused local replay and both interval engines.
+
+The exact checks cover 111 rational port charts, the full eight-bank
+operator in a faithful 1024-state permutation algebra, all 18 omitted
+bridge controls, all 661 inherited local scalar columns, 85248 global
+reservations, owner collision/timing controls, two price engines, the
+31/32 repair fence and the unchanged-word frame obstruction. The Gaussian
+appendix prices an external COPY-floor premise; it does not re-prove it.
+
+Nothing in `upstream/` or the selected frontier is changed. This is part of
+[PR673](https://github.com/CrocSwap/integer-mult-bounds/pull/673), not a new
+contribution branch or a claimed frontier result.

research/bit-paired-boundary-gate/SOURCES.json

@@ -0,0 +1,25 @@
+{
+  "base_commit": "b3917936aafbd0d680a954a86e25c4ce7e28a9f9",
+  "files": {
+    "research/bit-role-sharing-budget/verify.py": "6bed74cccf0bdf6aaa4984f8ae130f2e3380acdf1525385a4a0be6da796d24e4",
+    "research/bit-sharing-hosting-gate/verify.py": "e050feb3f43dbb1b64e568f10ebc5f2561611b238c27220dbaf2384d318e4d49",
+    "research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz": "e3e301e62bc2b41dbf1ceac0acf3d35824ebeab1848c5a2142d4577b109da52f",
+    "research/complex-k492-owner-repair/vendor/multicold/engines/moment.py": "81cbc864e106418fc7854322746f72180cb8110ff39f74450a3b8d55a82c1482",
+    "research/e8-bit-p111-tuned/tune/EXPECTED.json": "b9f0e5ca038abe837b0007c8fe006f366a6f02009244ac57705a8a200e2301f4"
+  },
+  "external_context": {
+    "770": "48d027a664288ea1d4e987de63cb79588b8b936b",
+    "777": "eca5c317d8176401a1f71a5ea1e81384c2eab1e0"
+  },
+  "reviewed_external_proof": {
+    "url": "https://github.com/CrocSwap/integer-mult-bounds/blob/48d027a664288ea1d4e987de63cb79588b8b936b/research/gaussian-e8-copy-bound/PROOF.md",
+    "sha256": "0f4ba837c2337cc06d7d3315e6e6a8598cc3919baf4f8a90c2a03713c2556a2c",
+    "scope": "Read proof; did not rerun its exhaustive finite pairing certificate. The 80-rational-rank floor is an external premise in this package."
+  },
+  "late_overlap_metadata": {
+    "801": "63450b78e26b4b63641fa72c371010273d4d6866",
+    "791": "f80382efd4cd3e37c67b2a966398b55bb56af53c",
+    "717": "4150829a2cb66b823f4282da98985959ee448a90"
+  },
+  "late_overlap_scope": "PR bodies read at these heads; not mathematical dependencies. PR801 treats Witting/Hurwitz coordinate-total labels through 13 stages; PR791 arbitrary idempotent path penalties; updated PR717 fixed-word/decoder/format limits."
+}

research/bit-paired-boundary-gate/VALIDATION.json

@@ -0,0 +1,37 @@
+{
+  "status": "PASS_FOCUSED_PAIRED_BOUNDARY_GATE",
+  "python": "3.14.7",
+  "platform": "Darwin-25.1.0-arm64-arm-64bit-Mach-O",
+  "checked_at_utc": "2026-10-13T05:57:07.419687+00:00",
+  "checks": [
+    "Normal isolated exact verifier",
+    "Optimized isolated byte-identical receipt",
+    "Final relocated minimal-input replay in a path with spaces",
+    "Altered expected receipt rejected",
+    "Altered input bytes rejected under optimized Python",
+    "18 omitted bridge controls",
+    "Owner timing and absolute-coordinate collision controls",
+    "296/297 whole-helper capacity fence",
+    "31/32 rank-one repair fence at c strictly greater than 1/99",
+    "Two exact rational price engines on all five invoices",
+    "Actual 661-column inherited local-word replay",
+    "Independent least monotone frame closure on actual scalar stream",
+    "Selected-record and source-inventory integrity check"
+  ],
+  "not_repeated": [
+    "Unchanged aggregate cold replay and 47 outer inequalities last passed at b391793",
+    "PR770 exhaustive Gaussian pairing-rank proof certificate",
+    "Root-wide suite, remote CI and proof-assistant validation"
+  ],
+  "not_achieved": [
+    "No new local word for the paired endpoint",
+    "No full supplier/controller/compiler admission",
+    "No new kappa"
+  ],
+  "verified_file_sha256": {
+    "research/bit-paired-boundary-gate/verify.py": "c4b7ad22a1df72607b3b36faa00c7dcd8f42167e8000f3591564850c3eed114f",
+    "research/bit-paired-boundary-gate/SOURCES.json": "0796137d6d8b89b44a8cd41f60de548d8f4d6cf75aa194387cfc05d9cd8e1eb3",
+    "research/bit-paired-boundary-gate/evidence/PRICE.json": "b051c21240a55422acae9ba5d2aae3fd4178a19f43ba162b1f946d875d7f8610"
+  },
+  "metadata_note": "After the final replay, only external-overlap metadata was appended to SOURCES/CLAIM; the pinned input file set and executable evidence are unchanged."
+}