Research: restored CK helper pairs save 35,872 primitives per orientation (no new κ)

research/complex-ck-helper-basis/CLAIM.md

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+# Sprint claim
+
+Seed: 24 (OS randomness, secrets.randbelow(100)). Track: complex supplier.
+Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60.
+Claim: investigate disjoint signed-unit helper-basis shears in the selected CK
+frame-zero compensation matrix. For each ordered helper pair and sign ±1,
+price preparation, changed rational additions, and restoration under the
+unchanged binary-Horner primitive invoice; retain a deterministic disjoint set.
+Verify the complete arbitrary-input block map in both orientations, source pins,
+frame boundaries, prefix bound and mutation controls. No new kappa claim.
+Expected deliverable: finite circuit research note, exact certificate, verifier,
+and a focused patch.
+
+Candidates considered:
+1. CK frame-zero helper-basis factorization (chosen; seed 24 modulo 3 = 0).
+2. ATM3 scalar coefficient microcircuits (not selected; inactive supplier).
+3. CK multi-scale relocation (not selected; overlaps pending #750).
+
+Overlap: read all 585 open PRs through #764 and the central review ledger.
+#749 factors JP8 target rows through a restored target basis; this instead
+factors CK helper columns, with explicit dirty-helper restoration. #719 concerns
+fixed-middle cleanup pairs; #750 relocates a cleanup scale. #696/#734 change
+coefficient lowering. No pending implementation is imported or additive savings
+claimed. Only open non-PR issue is #650; no sprint coordination issue found, so the
+claim was recorded here.
+
+Result: finite 124-pair CK helper-basis construction saves 35,872 real primitives
+per orientation; no new kappa. All 716 columns and native frames/wrappers pass.
+Final overlap refresh through #773 credits #769's distinct CK coefficient
+sharing. Main remains at the base.

research/complex-ck-helper-basis/README.md

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+# Restored helper pairs in CK frame-zero compensation
+
+**Status: finite complex-supplier circuit construction; no new κ.**
+
+The selected #626/#622 CK word's compensation can use 124 disjoint temporary
+helper shears. Its complete scalar program saves **35,872 real-component
+primitives per orientation**, including preparation, rational coefficient work,
+source reads and restoration. All 716 arbitrary live input columns are checked
+in both orientations, with separate primitive replay of the replacement and
+arbitrary incoming coefficient-work input. Every helper is restored before
+entry adapters or any paid recursive child.
+
+| Quantity | Selected CK | Replacement |
+|---|---:|---:|
+| Compensation additions (including basis changes) | 8,704 | 5,804 |
+| Compensation real primitives | 99,248 | 63,376 |
+| Full scalar macros, either orientation | 21,936 | 19,036 |
+| Forward real primitives | 224,938 | 189,066 |
+| Backward real primitives | 224,986 | 189,114 |
+| Forward conservative prefix bits | 40 | 41 |
+| Backward conservative prefix bits | 62 | 63 |
+
+This is a different finite circuit, not a faster implementation of its checker.
+The conservative positive prefix envelopes grow, but remain below the inherited
+192-bit allowance. No practical runtime or improved outer exponent is asserted.
+
+## Reproduce
+
+From the repository root, with Python 3.11+ and no third-party packages:
+
+```sh
+python3 -I -B research/complex-ck-helper-basis/verify.py --out /tmp/ck-helper-replay
+```
+
+The output directory must be fresh. Expected final line:
+
+```text
+PASS_CK_RESTORED_HELPER_BASIS pairs=124 real_saved=35872 columns=716+1 orientations=2 controls=6+25 kappa=unchanged
+```
+
+The command validates pinned inputs, exhausts the search, reconstructs both
+programs with the retained emitter, runs the retained frame/wrapper auditor and
+its 25 negative controls, and checks all proof-relevant fields against
+[CERTIFICATE.json](CERTIFICATE.json). Generated words, programs, audit receipts,
+logs and `EVIDENCE.json` go only in the chosen directory. Existing output is
+never overwritten. `--write-certificate` is an explicit maintainer regeneration
+option; normal verification never changes the committed certificate.
+
+The independent search census and mathematical failure-mode tests are:
+
+```sh
+TEST_SKIP_LOG=/tmp/test-skips.log python3 -B research/complex-ck-helper-basis/test_circuit.py
+```
+
+Eleven tests; zero expected skips. The runner writes the actual skipped tests
+and reasons to `TEST_SKIP_LOG`. The certificate replay also works under `-I -O`
+and `PYTHONOPTIMIZE=2`: new checks use explicit exceptions, and retained native
+subprocesses run with assertions enabled (`-E`, without optimization flags).
+
+## Exact local identity and cost
+
+Write the frame-zero compensation as `y <- y + C h`. For a chosen ordered
+helper pair `(a,b)` and `s` in `{−1,+1}`, replace its contribution by:
+
+1. Prepare `h_a <- h_a + s h_b`.
+2. For every target, use coefficients `C_a` and `C_b − s C_a` on the new
+   `h_a` and unchanged `h_b`, omitting zero coefficients.
+3. Restore `h_a <- h_a − s h_b`.
+
+For arbitrary dirty helpers and arbitrary targets,
+`C_a(h_a+s h_b) + (C_b−s C_a)h_b = C_a h_a + C_b h_b`.
+The preparation/restoration product is the identity on helpers. Pairs are
+disjoint, so the column substitutions compose without cross terms. The six
+unpaired eligible helpers keep their original coefficients. Unused entering
+and cold-ended helpers are untouched.
+
+Every used helper has `start=0` and `final=FULL`. In particular it is not a
+wrapped helper, so this temporary change occurs entirely before all entry
+adapters and wrappers. All other canonical-word fields and every emitted
+non-scalar event are unchanged. The native auditor rechecks the common frames,
+wrapper lifetimes, paid children, private centres and complete scalar endpoints.
+
+For the backward orientation, apply inverse transpose to each gate, in the
+same chronological order, exactly as the native emitter does. This is valid
+because inverse transpose is a group homomorphism. Forward helper preparation
+becomes its dual helper shear, and target updates become source/target-swapped
+updates with negated coefficients. Both full emitted programs pass exact replay;
+independent dense three-coordinate maps check every pair/target in both
+orientations. No commutation through the unchanged intervening word is needed.
+
+For nonzero reduced dyadic `q=n/d`, the retained binary-Horner real price is
+
+```text
+P(q) = 2 [3 + (bit_length(|n|)−1) + popcount(|n|) + (bit_length(d)−1)].
+P(0) = 0.
+```
+
+The factor two is for the real and imaginary components. The three constant
+operations are the initial work clear, signed target accumulation and final
+work clear. The remaining terms charge all source additions, doubles and
+halves. Each unit preparation or restoration costs eight real primitives;
+every chosen pair pays sixteen. The primitive interpreter independently
+executes those operations and clears arbitrary incoming private work.
+
+There are 5,556 nonzero middle additions and 248 preparation/restoration gates.
+The ordinary Horner lowering is unchanged. No new scratch register, recursive
+child, helper allocation, frame transition or uncharged copy is introduced.
+The complete native prefix bounds, including coefficient work, are
+`1258556220792` forward and `153667580418596477857/32` backward. The native
+positive-bound method does not cancel restoration terms, so it is conservative.
+
+## Completed search and rejection evidence
+
+OS seed **24** assigns the complex-supplier track. Three candidates and the
+local-only claim are in [CLAIM.md](CLAIM.md). The seed also sets the cyclic
+helper-index tie break. This is a new helper-column search class, distinct
+from prior fixed-middle cleanup searches and JP8 target-row factorization.
+
+The domain is all **254 × 253 × 2 = 128,524** legal ordered helper pairs and
+signs. Exact integer quarter-unit coefficients suffice. **9,756** candidates
+have positive complete savings. Sort by descending saving, then helper indices
+cyclically shifted by 24, then sign; accept the first pair whose helpers are
+both unused. This one deterministic greedy pass selects 124 pairs. The
+certificate lists every selected pair and binds the entire sorted profitable
+list by SHA-256. No timeout, randomized sampling, additional rounds or matching
+optimality claim is involved. Different matchings and general bases remain open.
+
+The test census reconstructs sparse coefficient columns and prices all pairs
+using joint coefficient histograms and repeated integer division, independently
+of the production search and its `bit_length`/`bit_count` cost formula.
+It reproduces the complete positive-candidate digest and chosen pairs.
+
+Six new semantic controls reject omitted preparation/restoration, either wrong
+basis sign, a wrong middle coefficient and an aliased preparation. Tests also
+reject duplicate pairs, nonzero-frame and cold-wrapped helpers, favorable cost
+corruption, a wrong dual sign and corrupt source bytes. Gzip recompression is
+accepted by binding decompressed payloads. Twenty-five retained native controls
+remain active, including cold-wrapper and frame errors. Finite matrix replay
+is not a formal proof of the full multiplication theorem.
+
+## Source pins, attribution and limits
+
+Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+Selected #626: `660de893d27bb7abd1d5b01b1640207528a5c851`.
+Construction #622: `f69477f4f6d3c7eb1f740b166881bffed4ffa9f6`.
+[SOURCE-PINS.json](SOURCE-PINS.json) pins every consumed source, including the
+native emitter and auditor. Native word/program payloads are additionally
+checked against the selected #626 SPEC. The selected κ remains
+`51110230636731/31250000000000000 = 0.001635527380375392`.
+
+Credit to Jeremy Kothe and the #622 Claude Code contributors, Chafik Boukhalfa's
+#626 packaging, DreamingOfClouds' native cold-helper construction and checks,
+icekylinx's direct-delivery/emitter and Horner invoice lineage, and all retained
+predecessors. The inherited bit supplier is eumemic's #543. Original source
+notices and licenses remain in their packages.
+
+The conceptual relation to pending **#749** (Sricharan Vagvala / @theon199,
+Codex-assisted) is explicit: it factors JP8 target rows through target bases;
+this contribution factors CK helper columns and checks temporary helper dirt
+and restoration. **#719** proves a fixed-middle CK cleanup obstruction;
+**#750** moves a cleanup scale; **#696/#734** change individual coefficient
+lowering. None of their implementations is imported, and no additive saving
+relative to any pending PR is claimed. Initial overlap inspection covered all
+585 open PRs through #764; the only open non-PR issue was #650, not a sprint
+coordination issue. The final refresh reaches #773; new **#769**
+(`e891b2b19e3185ec91bcf001e255af87f42389ce`) shares varying-magnitude CK
+coefficient preparation while retaining the macro word. It is distinct from
+this helper-basis replacement; no combined saving is asserted. Main remains
+the exact base commit.
+
+This isolated package changes neither the selected certificate nor production
+supplier sources. Production use would require an admitted typed compensation
+rewrite, refreshed bindings and global owner/invoice/assembly receipts. The
+paid profile and rank deficit are unchanged, so this scalar saving alone does
+not supply a better recurrence exponent. The all-size compiler, common-label
+and mutable-basis realization, whole-call hosting, normalized residuals,
+precision/tape/recovery, bit and analytic assumptions remain inherited. Global
+assembly, full cold supplier regeneration, repository-wide `make verify`,
+remote CI and Lean are not claimed by this local result.
+
+Contributed by David Awad. Prepared with substantial AI assistance (OpenAI Codex)
+throughout. No independent human review is claimed. New work is Apache-2.0
+under the repository license.

research/complex-ck-helper-basis/SOURCE-PINS.json

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+{
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "selected_source_head": "660de893d27bb7abd1d5b01b1640207528a5c851",
+  "construction_source_head": "f69477f4f6d3c7eb1f740b166881bffed4ffa9f6",
+  "files": {
+    "word": {
+      "path": "research/community-k492-audit/pr626/replay/mixed-global/base/providers/ck/CANONICAL-WORD.json.gz",
+      "payload_sha256": "9bbc75dede090cfd7a0a0252424ea9d9f6211e2350d2a801d4e4f2f0784167c7"
+    },
+    "programs": {
+      "path": "research/community-k492-audit/pr626/replay/mixed-global/base/providers/ck/LOCAL-PROGRAMS.json.gz",
+      "payload_sha256": "854f17274092263dc25e60367f4cd0d29f60a8446becd17cc1883ebed7a97900"
+    },
+    "checks": {
+      "path": "research/community-k492-audit/pr626/replay/mixed-global/base/providers/ck/LOCAL-CHECKS.json",
+      "payload_sha256": "059c4e59142bdf5c19dc7c0afebcddfec28e6e60cbc7299b8bdcc624afbc0cbe"
+    },
+    "spec": {
+      "path": "research/complex-k492-owner-repair/data/SPEC.json",
+      "payload_sha256": "2a2577ac526535826fed9d5f7f83f6e2e0ec058a146c64f7286720b995fa2323"
+    },
+    "emitter": {
+      "path": "research/complex-k492-owner-repair/code/hpe_emit_local.py",
+      "payload_sha256": "7e4a6fbc38c8228157b6311457d1496bf460291393043932d2c365f004f77805"
+    },
+    "auditor": {
+      "path": "research/complex-k492-owner-repair/code/hpe_audit_local.py",
+      "payload_sha256": "6bd9a1e279c9c9e4f3767d9f82e537b559292906feb2ddab02f380fb6523a810"
+    }
+  }
+}

research/complex-ck-helper-basis/VALIDATION.md

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+# Local validation
+
+Linux x86_64, CPython 3.13.5; base
+`66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+
+- Full certificate generation: PASS; both complete emitted scalar programs,
+  all 716 arbitrary live columns, frame incidence and wrapper ordering.
+- Independent primitive block replay: PASS in both orientations, all 716 live
+  columns plus one arbitrary incoming coefficient-work column.
+- New semantic mutations: 6/6 rejected. Retained native mutations: 25/25
+  rejected. These are rejection controls, not additional unittest test cases.
+- `PYTHONOPTIMIZE=2 python3 -I -O -B .../verify.py --out <fresh-dir>`:
+  PASS, exact committed certificate match. Retained subprocess assertions stay
+  enabled. No optimized false PASS is accepted.
+- Focused mathematical and packaging unittest suite under isolated optimized
+  Python: **11 passed, 0 failed, 0 skipped**. `TEST_SKIP_LOG` is empty (0 lines,
+  no skip reasons). Includes all 128,524 pair/sign prices by a separate census,
+  27,776 pair/target/orientation dense maps, source corruption rejection,
+  gzip-container portability and dirty private work clearing.
+- `make selected-record-check`: PASS; selected result unchanged.
+- Staged whitespace, Python syntax and package-relative documentation links:
+  PASS before local commit.
+
+The initial discovery run completed mathematical checks but encountered a
+report-packaging KeyError (`negative_controls` versus the retained receipt's
+`negative_controls_rejected`). This was corrected, then complete certificate
+generation and the independent optimized replay passed. No failed or skipped
+check is counted as a pass.
+
+No repository-wide `make verify`, full selected cold regeneration, global
+owner/invoice/assembly regeneration, remote CI or Lean was run. Only the new
+research package is changed.
+
+Initial read-only overlap review covered 585 open PRs through #764, the review
+ledger and open issues. Final refresh reaches #773 and credits #769's distinct
+CK coefficient reuse. The relevant #749/#750/#719/#696/#734 heads remain pinned
+in the overlap snapshots; upstream main remains at the exact base.
+No independent human review is claimed.