Research circuit: lower historical dyadic center bill with one basis shear (no new kappa)

research/dyadic-center-basis-shear/README.md

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+# One center-basis shear lowers the historical dyadic scalar bill
+
+**Finite complex-supplier circuit improvement; no new kappa.** A single
+unimodular reparameterization of the historical rank-h dyadic central factor
+lowers its complete scalar-operation bound at h=26. The center count, side
+circuit, all-incidence frame groups, recursive children and dirty restoration
+contract are retained.
+
+| Count | Original | Replacement | Saved |
+|---|---:|---:|---:|
+| Gather G | 14,100 | 12,352 | 1,748 |
+| Scatter H | 31,281 | 31,557 | −276 |
+| G + H | 45,381 | 43,909 | 1,472 |
+| Complete signed invocation | 786,938 | 783,994 | 2,944 |
+| Complete three-stage scalar bound | 15,959,102,640,000 | 15,899,398,320,000 | 59,704,320,000 |
+
+These are the historical circuit's addition/subtraction/halving charges,
+not runtime measurements, selected JP8/CK primitive counts or new recursive
+rank savings. The unchanged rank ledger means this improvement supplies no
+larger multiplication exponent. It is useful as a fully priced local circuit
+rewrite and an exact example of center-basis optimization.
+
+## Reproduce
+
+From the repository root with Python 3.11+ and its standard library:
+
+```sh
+python3 -I -B research/dyadic-center-basis-shear/verify.py
+```
+
+Expected final line:
+
+```text
+PASS_DYADIC_CENTER_BASIS_SHEAR h=26 moves=1300 admissible=650 G+H=45381->43909 full_scalar_saved=59704320000 no_new_kappa
+```
+
+The command recomputes every field of [CERTIFICATE.json](CERTIFICATE.json).
+`--out /fresh/path.json` emits the recomputed certificate and refuses an
+existing output. It verifies [SOURCES.json](SOURCES.json) before inherited
+imports. An assertion-enabled isolated subprocess runs the original
+constructors even when the outer verifier is invoked with `-O`.
+
+The actual adapter is `circuit.shear(original_factor, 1, 2, -1)`, consumed
+by `source_worker.py` through the unchanged original `invoke` executor.
+No source patch or selected-record change is required to replay it.
+
+## Base, selection and dependency boundary
+
+Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`. The OS-random assignment
+seed was **20**, complex-supplier track. [SELECTION.json](SELECTION.json)
+records the three candidates, scope, initial optimistic budget and bounded
+search. Current CK/JP8 coefficient reuse overlaps #769/#787/#793; fixed-order
+A/B frame freedom is covered by #776. Seed `20 % 3 = 2` selected the historical
+center factor. Seed 20 also chooses the first of its two tied one-shear
+minimizers. No pinned sprint issue existed, so the claim is local.
+
+The selected certificate was followed to the actual
+[#626 construction](../complex-k492-owner-repair/README.md),
+[review](../../docs/research/community-k492-review.md) and
+[assembly](../community-k492-audit/pr626/replay/ASSEMBLY.json).
+The current accepted conditional value at this base remains
+`51110230636731/31250000000000000 = 0.001635527380375392`.
+Its source head is `660de893d27bb7abd1d5b01b1640207528a5c851`, with reviewed
+#622 `f69477f4f6d3c7eb1f740b166881bffed4ffa9f6` and retained #543
+`2065fe4f72a914d907f1046603f5a36f17018535`. This package changes none of them.
+
+The object improved here is Douglas R. Colkitt's
+[historical dyadic center](../../docs/research/dyadic-central-factor.md),
+[written factor/phase argument](../../notes/dyadic-central-factor.tex), and
+[implementation](../../scripts/experiments/complex_centers.py), retained in
+commit `d55bcd62ceb8f322bcf311a5a66e94c3ac7724f0`. The old note's h=26 saving
+and `2^-31` language describe its historical checkpoint. They are not the
+current frontier. There is no pending-PR code dependency, and no claim to
+supersede contemporary E8 suppliers or their coefficient optimizations.
+
+## Exact construction
+
+Let `v=binom(h,3)` and `T0={0,1,2}`. For triple indicators `s,t`, the source
+has scatter numerators U and gather coefficients V:
+
+    U[S,0] = s0+s1+s2−1,     U[S,i] = si−s0  (i≥1),
+    V[0,T] = 1,              V[i,T] = ti−1(i∈T0) (i≥1).
+
+Its central map is `UV/2`, with `(UV)[S,T]=|S∩T|−1`.
+Use center-coordinate matrix `A=I−E12`, whose inverse is `I+E12`:
+
+    V' = A V,       U' = U A^−1.
+
+Only one gather row and one scatter column change:
+
+    V'[1,T] = t1−t2,
+    U'[S,2] = s1+s2−2s0.
+
+All other entries are unchanged. Since `E12²=0`, the product `U'V'=UV`
+is exact in every dimension. The new gather still uses only 0,±1, while
+scatter numerators lie in 0,±1,±2. Numerator −2 means a single subtraction
+with scalar coefficient −1. This extends the old `verify_factor` alphabet
+check (which excluded −2) but introduces no primitive operation: the old
+negative scatter phase already applies −1 when an original numerator is +2.
+The optional checker explicitly admits and prices both signs of 2; the
+original checker remains untouched and is tested to reject the extended
+alphabet for its original reason.
+
+## Dirty restoration and why no adapter is paid
+
+Let c be the arbitrary initial contents of the existing h center roles.
+The central subsequence acts as
+
+    y ← y − U'c/2,
+    c ← c + V'x,
+    y ← y + U'c/2,
+    c ← c − V'x.
+
+It leaves x and c unchanged and adds `U'V'x/2 = UVx/2` to y. The inverse
+sequence subtracts this map and also restores c. No actual register is
+converted from c to Ac: the new scatter cancels the same arbitrary dirty c
+on its two visits. Consequently there is no initial/final center-basis
+conversion and no additional helper or restoration work.
+
+The unchanged side subsequence supplies the complementary side map, so the
+complete twelve-operation invocation is still `y ← y+x`, or `y ← y−x`
+in reverse. Full exact formal-variable replay checks all 998 independent
+source/target/side/center columns at h=8 and all 2,767 at h=10, in both
+directions. Separately, the independent central implementation checks 356
+basis executions at h=4,6,8 in coordinates `(x,2y,c)`; h<8 is algebra-only
+coverage, not a claim of a complete historical supplier at those sizes.
+
+## Complete cost and frame accounting
+
+Each nonzero gather entry costs one signed addition. A scatter numerator
+±2 costs one signed addition; ±1 costs one halving and one signed addition.
+Every source access represented by those updates is counted. The side
+mixer M, copy C and weighted-injection J terms are freshly recounted from the
+unchanged h=26 side topology:
+
+    M=158444, C=10400, J=10400,
+    fixed side bill = 4M + 2C + 4J = 696176.
+
+The factors 4,2,4 include the inverse mixers, both copies and both weighted
+injections. The full invocation costs `4M+2C+4J+2G+2H`, including both
+central scatters and inverse gather. There are `3v²=20,280,000` invocations.
+Thus the full scalar-bound decrease is `6v²·1472 = 59,704,320,000`.
+
+For the chosen shear, the old gather row has `binom(h−1,3)` nonzeros; the new
+one has `2 binom(h−2,2)`. The old affected scatter column costs
+`4 binom(h−2,2)`. Counting whether s0 is present and how many of s1,s2 are
+present gives new cost `5 binom(h−2,2)`. Therefore
+
+    saving in G+H = binom(h−1,3) − 3 binom(h−2,2)
+                  = (h−10)(h−2)(h−3)/6.
+
+This improves the scalar bill exactly when h>10, is neutral at h=10, and
+loses 10 operations at h=8. The replay checks the chosen formula against
+actual factors for every even h from 8 through 40. This is a proof for the
+chosen shear, not an unrestricted optimality theorem.
+
+Crucially, retain **all** original data and center incidences in each common
+gather/scatter frame group, including incidences with zero scalar coefficient.
+The inherited phase argument explicitly permits scalar factor changes under
+this rule. No gate is split into new frames and no incidence is silently
+removed. Every center retains its downward residual of dimension h; the
+first/third-stage reuse still preserves center indices. Thus all helpers,
+source corrections, endpoint repairs and recursive children keep their old
+accounting:
+
+| Ledger item | Unchanged value at h=26 |
+|---|---:|
+| Center roles per invocation | 26 |
+| Side roles per invocation | 89,622 |
+| m | 17,576 |
+| W | 1,247,192,960,000 |
+| L | 13,709,280,000 |
+| s | 21,920,655,731,520,000 |
+| Wm−s | 7,733,440,000 |
+
+The source's scalar node allowance `E=64(W+m+1)³` is unchanged. The replay
+checks `E > 4 steps_new + 8s + 8W + 8m`, and reproduces its conditional
+stopped recurrence guard `A(e)≤s A(e/m)+E` with the same leaf rule. All
+coefficients are still realized using the inherited signed-addition/halving
+alphabet; its operation-count-based guard remains applicable. This does
+not supply a new tight numerical-growth, tape or all-size realization proof.
+The paid rank profile and its saving are unchanged. Another historical
+supplier/assembly constraint cannot be improved by this scalar reduction
+alone, and no assembly rerun or new exponent is claimed.
+
+## Bounded search and independent evidence
+
+The primary census exhausts exactly `2h(h−1)=1300` single elementary shears
+`V_i←V_i+εV_j`, `U_j←U_j−εU_i`, with `i≠j`, `ε∈{−1,+1}`, at h=26.
+It accepts only gather magnitudes ≤1 and scatter numerator magnitudes ≤2.
+Exactly 650 moves meet both restrictions. The minimum change in G+H is
+−1472, attained by `(i,j,ε)=(1,2,−1)` and `(2,1,−1)`. The source-bound
+census records all coverage counts, the complete cost histogram and a
+canonical transcript hash. The stopping rule is exhaustive completion of
+this one-shear neighborhood; no multi-step search is being claimed.
+
+The independent implementation imports no producer or repository code.
+It also checks the one-shear census by 16 coordinate-type/sign orbits. The
+source factors are invariant, up to triple relabeling, under exchanging
+coordinates 1 and 2 and permuting coordinates 3 through h−1. Thus the
+classes {0}, {1,2}, {3,...,h−1} give eight ordered distinct-coordinate types
+and two signs, with exact orbit multiplicities. Their admissibility, full
+cost histogram and minimum must match the 1,300-move implementation.
+This is a different finite counting method, not a larger search.
+
+The independent implementation contracts the three distinguished coordinates and the remaining intersection
+cardinality separately, checks all 6,760,000 coefficients, and hashes the
+same complete row-major coefficient stream as the sparse-multiplication
+checker. Both costs and coefficient hashes must agree. Its separately
+executed integer central circuits preserve all dirty inputs and detect
+omitted restoration, omitted dirty cancellation, a gather-only rewrite and
+the wrong sign on a −2 numerator.
+
+Additional checks, exact failure controls, overlap refresh and environment
+are recorded in [VALIDATION.md](VALIDATION.md). No failed search is promoted
+to an impossibility theorem. Optimality is limited to the declared finite
+one-shear alphabet; general unimodular bases, multiple shears, new helpers,
+new frame schedules and current supplier constructions remain open here.
+
+## Attribution and remaining obligations
+
+Prepared by **Curtis Dransfield with substantial OpenAI Codex assistance** in
+construction, proof, code, verification and writing, including a separate
+Codex worker's independent implementation and review. This is methodological
+independence, not independent human peer review. New files use the repository's
+Apache-2.0 license.
+
+Credit Douglas R. Colkitt for the historical dyadic factor, disjoint-tensor
+side circuit, dirty transparent schedule, frame proof and recurrence ledger,
+with their existing AI disclosures. The selected construction's Jeremy Kothe,
+Chafik Boukhalfa, eumemic, DreamingOfClouds, Rohan Arun and retained predecessor
+credits remain in its unchanged sources. Contemporary coefficient-circuit
+work is credited as related scope, not imported or combined savings.
+
+This package freshly rebuilds h=26 scalar topology/counts and central
+coefficients. Its unchanged h=26 side coefficient/phase proofs are inherited;
+full dirty-input invocation replay is finite at h=8,10. The full tensorized
+network, upstream analytic transfer, fixed-tape/precision and all-size
+compiler assumptions are not newly proved. Neither the historical certificate
+nor the selected frontier is edited, and no Lean or full multiplication-theorem
+verification is asserted.

research/dyadic-center-basis-shear/SELECTION.json

@@ -0,0 +1,59 @@
+{
+  "assignment_seed": 20,
+  "seed_source": "OS randomness via secrets.randbelow(100)",
+  "started_utc": "2026-10-13T05:41:00.896452+00:00",
+  "expected_finish_utc": "2026-10-13T06:21:00.896452+00:00",
+  "verification_reserve_starts_utc": "2026-10-13T06:11:00.896452+00:00",
+  "track": "complex supplier",
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "claimed_utc": "2026-10-13T05:44:36.681013+00:00",
+  "coordination": "Pinned issue query returned no nodes; local claim",
+  "candidates": [
+    {
+      "object": "selected CK/JP8 coefficient reuse",
+      "decision": "declined: crowded by PR769/787/793 and predecessors"
+    },
+    {
+      "object": "selected fixed-order A/B frame reassignment",
+      "decision": "declined: closed by PR776 exact duals; no new mechanism"
+    },
+    {
+      "object": "historical rank-h dyadic center basis",
+      "decision": "chosen by seed20%3=2; no matching central-basis audit in612 open PR descriptions"
+    }
+  ],
+  "object": "One elementary center-basis shear in scripts/experiments/complex_centers.py factor(26), retaining all center/data incidences and common frames",
+  "method": "Enumerate all h(h-1)*2 single shears with coefficients +/-1 and retained unit gather/dyadic scatter alphabet; choose minimum complete gather+scatter bill, seed tie-break; prove product identity, price full transparent invocation and verify dirty restoration",
+  "hypothesis": "A unimodular center reparameterization can reduce gather+scatter operations without adding helpers or changing the paid frame profile",
+  "expected_deliverable": "One executable finite circuit improvement and exact scoped one-shear census, with proof, certificate, controls and no new kappa",
+  "domain": {
+    "main_h": 26,
+    "moves": 1300,
+    "signs": [
+      -1,
+      1
+    ],
+    "admissible_gather_abs_max": 1,
+    "admissible_scatter_numerator_abs_max": 2,
+    "stopping_rule": "Exhaust one-shear neighborhood; no multi-step or unbounded search"
+  },
+  "initial_budget_probe": {
+    "old_G": 14100,
+    "old_H": 31281,
+    "best_delta_G_plus_H": -1472,
+    "valid_moves": 650,
+    "note": "Small optimistic screen during selection; must verify complete cost and dirty/frame contracts"
+  },
+  "overlap": {
+    "open_pr_count": 612,
+    "latest": 793,
+    "related": [
+      769,
+      776,
+      787,
+      793
+    ],
+    "pending_code_dependencies": []
+  },
+  "record_type": "Immutable selection snapshot; final local claim will carry PR result"
+}

research/dyadic-center-basis-shear/SOURCES.json

@@ -0,0 +1,31 @@
+{
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "sha256": {
+    "certificates/dyadic-central-factor.json": "ffae0f0a22863718e3134a924592dcf4b05e919223323ba536c587c592bfe6a1",
+    "certificates/selected-result.json": "b3bf402f31a69bde0629936a9b569bed48281568d7b16d5fba120b5311ea1f53",
+    "docs/research/community-k492-review.md": "3b731d30e80010b15408a289b0b32192700885474c8b36d48ee3320a7a8c5c47",
+    "docs/research/current-status.md": "197319a6edf04340ec3623d09d03714fcd6a981359f22bdcefdd06e7e2f4174d",
+    "docs/research/dyadic-central-factor.md": "e97b1dfe11b5eaebacf3f910131e13bcad586ea5fc1af05db23c00ae0eea5c11",
+    "notes/complex-compression.tex": "10312815290c7bb8ef4db153e30fba13c032bf17ccaf991a34f1aa6340405a0d",
+    "notes/disjoint-tensor-compression.tex": "528cba56872313b8cd63ff11f523a399dfb3c85cae1a764110ae9d26ac9151ac",
+    "notes/dyadic-central-factor.tex": "24c0543fb2fd815f114021f929c3efefa1ae79f29c2f7968585bcd4595a96158",
+    "research/community-k492-audit/pr626/replay/ASSEMBLY.json": "6127732e2fcdd470dab349c1c51d03cd9f0787ad0309b1b4a7c811a908028f4e",
+    "research/complex-k492-owner-repair/README.md": "ee5a1ace4828f7a073be97e95c2d931988774dd3a29a877e65abdf429aeec902",
+    "scripts/audit_complex_centers.py": "166b18f560cd0d520d23fa29ee8147e0ea056358c4268a466ec458195a97a960",
+    "scripts/binary_phase_frames.py": "4148cfa313cf050ed25296e8db36d35871520b8318fd086c7d1facf0c6af67d7",
+    "scripts/certify.py": "2348d95008276c5d5764f7e028c3918e484582cd589bd4d9b82919161745e53e",
+    "scripts/complex_compression.py": "9b6f758a5985ba5d9fdd726fe6556b3d3b4bd9d98aaf8ba06a7bb58093c4c504",
+    "scripts/exclusion_circuit.py": "b253f9d7318becda1c1f3b24e2287ed44a9c27fcea00cf9d531afb950d0457a0",
+    "scripts/experiments/complex_centers.py": "3744396587d729253d80980f6310b56e7bf0ea21b165bd178d0d6d4c49faf2e2",
+    "scripts/experiments/disjoint_tensor.py": "102dd59bf96033f96ba0dab4c2aba77b3a7a73b67b0f96929ff8b20a98b91cd8",
+    "scripts/general_network_guard.py": "790a2064449afdbffd6dbd4adbcf4a28eaf41931ce7134e4a4e8766f3764270c",
+    "scripts/incidence_rectangles.py": "5240fd33dcc3096f5685d10ca360186cad36b355a00d55686a4ea4449417d00e",
+    "scripts/paired_exclusion_circuit.py": "8f05f04cf06217befd6d0ee0f2d7d0c5c5eb32cfab8c9a8ca7917fd6d620b7d2",
+    "scripts/paired_network.py": "991223dc3c1374284c361ed017e42bf732dd971fa0bea2be2282cfb18257b3f7",
+    "scripts/prepare_layers.py": "a6e2adc612f264b1c0e51d18ce6315de9ae5e2ae2420aa22e99a22d8bf304947",
+    "scripts/search_network.py": "3a4faa5037d9b8e7ca5828afac193a96b6966a8b114f14937921795b24ec4f6a",
+    "scripts/shared_point_circuit.py": "c9bb7a5df92d260a4e3ad532b53e26936730d7c93f7facfb0784957c1e81c8ba",
+    "scripts/shared_point_network.py": "a85e7f0d8c42d6a27e3846308571ca324a1824ac5c2d59e4daa5c7673c33c697"
+  },
+  "upstream_frozen_commit": "adc7f1241b42e322a6451854ab7e4b4c146bf78a"
+}