Audit selected k8 helper endpoint rigidity and rank-preserving failures

research/k8-helper-endpoint-audit/README.md

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+# Selected k8 helper endpoints: fixed-schedule rigidity and rank camouflage
+
+**Scoped adversarial audit; no new κ and no selected acceptance bug.** All
+480 helper final frames of the selected PR1094 k8 word are minimal under its
+unchanged gates, entrances, gate frames and early/full cleanup schedule.
+Consequently no helper end can be replaced by a smaller subspace, and an
+equal-dimensional replacement must be the original subspace. This also holds
+for simultaneous endpoint-only replacements. Changing the schedule or gate
+frames can escape the statement.
+
+The added evidence is a complete endpoint-forcing certificate and 78 concrete
+same-rank adversaries. It makes explicit why a correct scalar map and a valid
+rank-only owner allocation cannot replace local frame admission. The existing
+local audit already enforces that admission; this package corroborates it.
+
+## Reproduce and evidence
+
+From the repository root, Python 3.12+ and the standard library:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/k8-helper-endpoint-audit
+# Direct read-only replay, including all 78 retained local rejection runs:
+python3 -I -O -B research/k8-helper-endpoint-audit/verify.py
+```
+
+Expected status: `PASS_K8_HELPER_ENDPOINT_BOUNDARY_AUDIT`. Normal verification
+compares its deterministic result to [RESULT.json](RESULT.json), writes only
+temporary files, and runs six additional small-domain/tampering tests through
+the registered command. `--write PATH` explicitly regenerates the receipt.
+
+| Check | Evidence |
+|---|---|
+| Restricted endpoints | 32 rank-four and 46 rank-eight ends; every last native frame equals the committed end |
+| Other endpoints | All 402 helpers occur in an ordinary full-frame cleanup gate |
+| Positive local replay | Retained exact rational audit: all 704 columns, both orientations, frames, entrances and cones |
+| Positive owner replay | Retained PR1091 auditor: all 69,120 roles, 144 calls and 288 windows; profile agrees with current PR1094 SPEC/PLAN/PROFILE |
+| Endpoint adversaries | One per restricted helper, 78 actual retained local rejection runs; each passes both scalar endpoint tests first |
+| Owner diagnostic | Every adversary has the identical ordered vector of 480 residual widths; all other owner-auditor inputs are unchanged |
+| Counted exclusion | 105,927,652 alternative single-end same-rank caps containing their original entrances; excluded by span inclusion, not enumerated |
+
+Only **one** complete owner replay is executed. The owner auditor is a function
+of geometry, widths, profile, roles and plan, so unchanged inputs give the same
+result for every adversary. We do not claim 78 separate global or owner replays.
+The retained local and owner implementations are explicitly reused comparison
+targets. The new span certificate and explicit-set checks do not import a
+supplier builder or the retained auditors' span functions.
+
+## Exact boundary and proof
+
+Let `E_q` be a helper's entrance label subspace and `F_q` its committed final
+subspace, inside `F₂⁹`. These binary spaces describe **labels**; the retained
+scalar endpoint tests use exact rational coefficients.
+
+For a helper absent from ordinary full cleanup, propagate its frames through
+the literal A/B and early-cleanup chain. Every allowed final cap `G_q` must
+contain the last frame `L_q`, by monotonicity. The receipt supplies the last
+gate and exact basis witnessing `L_q = F_q` for all 78 restricted helpers.
+For each of the other 402 helpers it supplies an ordinary cleanup gate index.
+The inherited local contract requires every register in such a gate to have
+a full final frame, so its cap is forced to `F₂⁹`.
+
+Thus every admissible `G_q` contains `F_q`. Smaller dimension is impossible;
+equal dimension implies equality. The argument is coordinatewise, so changing
+several endpoints together cannot evade it while the schedule and gate frames
+remain fixed. This is a statement about endpoint subspaces, **not** a global
+moment optimum, an arbitrary frame reassignment bound or a cleanup-resynthesis
+lower bound. Enlarging endpoints is not excluded or priced here.
+
+For an end of dimension `d` containing a fixed entrance of dimension `s`, the
+number of dimension-`d` caps containing the entrance is the binary Gaussian
+coefficient `[9-s choose d-s]₂`. Exactly one equals `F_q`; the others fail the
+necessary inclusion above. The restricted ends comprise 32 `(s,d)=(0,4)`,
+16 `(0,8)` and 30 `(1,8)`, giving
+
+```text
+32 * (3309747 - 1) + 16 * (511 - 1) + 30 * (255 - 1)
+= 105927652 alternative single-end caps.
+```
+
+The six tests independently enumerate all subspaces up to ambient dimension
+four as point sets, check the Gaussian and quotient counts, compare elimination
+with explicit XOR closure, exercise invalid labels and frame witnesses, and
+reject a corrupted source input. No floating-point acceptance is used.
+
+## A minimal same-rank adversary
+
+Register **273** is a cold helper with final basis `[388,96,24,2]`. Replace
+only that final basis by `[388,96,24,1]`. Both spaces have dimension four,
+the entrance remains zero, and every scalar instruction and claimed width is
+unchanged. Its last early-cleanup gate, original cleanup index **40**, requires
+label `2`, which the replacement does not contain. Both exact scalar maps pass;
+the retained local audit then rejects `frame not monotone (273, ...)`.
+
+Each of the other 77 fixtures similarly replaces one complementary basis
+direction by a direction outside the old end, while preserving its entrance.
+The receipt names a missing required label and the observed rejection site.
+These are invalid **frame contracts**, not counterexamples to the scalar
+identity or the selected construction.
+
+## Source binding, enforcement and scope
+
+Base: `bce54fb3f895cbe5cd4c0eb477c3afeed100190f`. Selected PR1094 audited
+head: `0fc329d0bec93ca7cd288eaea1295172e8789810`. [SOURCES.json](SOURCES.json)
+pins each consumed input by Git blob and SHA256, with the selected record and
+task catalogue retained as historical provenance. The PR1091 diagnostic word
+copy has decompressed hash
+`c2b294300760e1fd6eeab8c22c4d13e4633dca40e52cac8da0d68163a76e3ca8`,
+which is checked against the selected-source-bound k8 recipe's output pin.
+
+The retained source shows two separate defenses. `verify_slots.py` binds the
+rebuilt word to its content hash and explicitly executes `independent/audit_local.py`;
+the latter checks monotonic final climbs before computing width inventories.
+`scripts/verify_frontier1042.py` runs that slot layer; PR1094 inherits the checked
+local word. By contrast PR1091's `word_widths()` only subtracts endpoint and
+entrance ranks, as appropriate for an owner audit that assumes local admission.
+Our mutated raw words deliberately enter **below** the immutable word binding.
+They are rejected locally and do not match the frozen recipe payload.
+
+This package does not rerun the selected cold replay, native compiler, phase
+recipes, precision/tape proof, bit H1–H5, outer moment/assembly or analytic
+transfer. It establishes no all-size theorem or physical supplier admission.
+Source pins and the selected construction are unchanged. [TASK.json](TASK.json)
+records the assignment, bounded domain and overlap check; coordination is in
+[Discussion #1378](https://github.com/CrocSwap/integer-mult-bounds/discussions/1378).
+
+## Open next questions
+
+Can a changed tail schedule permit a smaller endpoint for one current rank-eight
+helper after paying every partner's frame change and restoration? This matters
+because endpoint-only reductions are excluded here, while PR1030/PR1090 show
+that changed gate frames and tails can achieve rank-seven ends. First inspect
+one receipt's forcing gate and propagate a proposed lower frame backward
+through all its operands. The blocker is constructing and pricing the complete
+new local word and hosting it, not obtaining another unchanged-width owner run.
+The structured question is in [TASK.json](TASK.json); no search for this next
+question was conducted.
+
+## Credit and overlap
+
+Rohan Arun's PR1094 supplies the selected owner map; Jeremy Kothe's leaver/k8
+lineage and the retained gauge44 contributors supply the word and local audit.
+Josie Elliston's PR1091 (`b4c13a228acb80f74c682b7097f0ce9bbf51bc4e`) supplies
+the owner-audit comparison and word copy; Sagar Thakrar's PR1049
+(`6d0a145e8b4774b1b0d4e3da56890af5adc9b04f`) already independently audits the
+word's scalar/frame semantics. Douglas Colkitt's batch1099 composition applies
+the owner auditor to PR1094. This package claims no priority for monotone frame
+inclusion, Gaussian subspace counting or their existing admission checks.
+
+PR1358 fixes endpoints and studies singleton B-gate enlargements; PR1312 studies
+X2 cross-child owner-axis collisions; PR1238 studies the source/target end-cap
+wall. PR892 (`9778ba2d891468996b81eb50c389bb4169df0025`) already proves a
+rank-eight endpoint floor for a different historical coded helper using its
+inverse row, with a broader changed-cleanup argument. This package instead
+supplies the selected word's fixed-schedule **helper** endpoint
+census and paired scalar/rank-blind versus frame-sensitive fixtures. Live
+overlap was checked across open PR bodies (including drafts), discussions,
+local notes and the ledger; this is not an exhaustive search of every code diff.
+
+Prepared for Arham Tahir with substantial OpenAI Codex assistance in code,
+investigation and writing. No independent human peer review is claimed.
+New work is Apache-2.0; predecessor files and notices are retained unchanged.