Audit THREE-base supplier billing from native events (no new kappa)

research/three-base-paid-census/README.md

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+# THREE-base supplier: independent paid-event census
+
+**Finite accounting audit; no new kappa.** The supplier-census question left by
+[Ryanson Jonathan's #4426](https://github.com/CrocSwap/integer-mult-bounds/pull/4426)
+is answered for its pinned THREE-base word. Separate standard-library code
+recovers the unsplit paid histogram
+
+```text
+{1:2487, 2:713, 3:224, 4:102, 5:20, 8:8}
+```
+
+in both orientations: **3,554 paid calls, rank mass 5,157**. The 408 nested
+helper frame increments give `{5:28, 6:18, 7:56, 8:93, 9:213}`, rank mass
+**3,301**. These agree exactly with the supplier receipt used by #4426.
+
+This closes its finite census handoff, not the supplier's operator or physical
+correctness obligations. Evidence is an author-executed, separately implemented
+check; no independent human or proof-assistant review is claimed.
+
+## What is reconstructed
+
+The native input actually contains the **split** centers, with 22,633 events
+and paid profile `{1:2487, 2:713, 3:232, 4:102, 5:28}` in each direction.
+Its 16 center events form eight consecutive pairs. Their frame dimensions are
+`8 -> 3 -> 0` forward and `0 -> 3 -> 8` backward. Coalescing each pair recovers
+the unsplit program's 22,625 events without changing any scalar or warp event.
+
+The resulting paid census, identical in the two orientations, is:
+
+| Phase | Rank 1 | Rank 2 | Rank 3 | Rank 4 | Rank 5 | Rank 8 | Calls | Rank mass |
+| --- | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: |
+| A | 950 | 12 | 20 | 38 | 0 | 0 | 1,020 | 1,186 |
+| Centers | 0 | 0 | 0 | 0 | 0 | 8 | 8 | 64 |
+| B | 1,161 | 571 | 150 | 64 | 20 | 0 | 1,966 | 3,109 |
+| Cleanup tail | 72 | 53 | 0 | 0 | 0 | 0 | 125 | 178 |
+| Final | 304 | 77 | 54 | 0 | 0 | 0 | 435 | 620 |
+| **Total** | **2,487** | **713** | **224** | **102** | **20** | **8** | **3,554** | **5,157** |
+
+For the 15 native invocations inherited from Cinna Davis's #4416, the supplier
+part alone is `{1:37305, 2:10695, 3:3360, 4:1530, 5:300, 8:120}`, rank mass
+77,355. Controller idles are additional. This multiplication by 15 is arithmetic;
+#4426 retains credit for reconstructing the controller, its targets and idles.
+
+## Method and proof boundary
+
+1. Authenticate exactly three inputs against [SOURCES.json](SOURCES.json), then
+   require the supplier receipt's artifact hashes to name those actual native
+   and canonical bytes. They are already retained on the pinned main branch;
+   no unpublished Git object or network fetch is needed for this package.
+2. Independently eliminate all 1,159 nine-bit frame bases over F2. Every basis
+   must be independent. For each paid transition, compute containment by the
+   rank of the concatenated bases and derive the rank difference. The event's
+   declared rank is only a comparison target, never the rank calculation.
+3. Trace frames on 632 persistent fields and the owned C0 field. Check paid
+   entrances, warp entrances, scalar operands, final persistent frames, and
+   the final explicit center clear. `adapter`, `residual_adapter` and `phase`
+   are markers in the inherited event format; the paid moves they annotate
+   appear explicitly as `climb` or `center` events. Scalar and warp costs are
+   outside this recursive-child histogram.
+4. Verify both literal split programs and coalesce their adjacent center
+   pairs. Under the inherited common-frame convention, if
+   `T(a,b) = F_b F_a^-1`, then
+   `T(b,c) T(a,b) = F_c F_b^-1 F_b F_a^-1 = T(a,c)`.
+   Complex conjugation preserves this identity. The shared middle reference,
+   immediate adjacency, dimensions and nesting are checked; a scalar inserted
+   between the two pieces is rejected. This is telescoping of frame symbols,
+   not a fresh check of serialized Clifford phases or a regeneration of the
+   original unsplit word from its ancestor.
+5. For each helper `L224` through `L631`, independently verify the nested
+   entry/exit spaces and count their dimension difference. C0 receives no
+   productive credit. Compare the full distributions, not only rank masses,
+   to the receipt. Recheck frames and counts on the coalesced programs too.
+
+The verifier imports no repository module and executes no predecessor
+interpreter. The source interpreter was read to identify the native format
+and tariff. The elimination algorithm, census, center reconstruction, controls
+and receipt are separately written. The selected construction already had
+native replays; this package supplies additional independent accounting
+coverage requested by #4426, not a first proof of the supplier.
+
+## Verification
+
+From the repository root, Python 3.12, standard library only:
+
+```sh
+python -B scripts/research_package.py --package research/three-base-paid-census --check-only
+python -B scripts/research_package.py --package research/three-base-paid-census
+python -O -B research/three-base-paid-census/verify.py
+```
+
+The direct verifier recomputes and compares [RESULT.json](RESULT.json).
+`--write` regenerates only that package-local result. All assertions use explicit
+exceptions and remain active under `-O`.
+
+There are **16 rejecting controls**: source corruption, artifact misbinding,
+wrong paid rank, a deleted climb, a same-dimension nonnested edge, stale warp
+entrance, wrong scalar frame, borrowed-field clearing, missing/interrupted
+center pairs, a dependent basis, wrong final frame, unknown event, two
+rank-mass-preserving paid-profile changes, and a helper-distribution change
+preserving both count and rank mass. All 512 ordered triples in F2^3 additionally
+cross-check elimination against enumerated spans, including dependent triples.
+
+An explicit **coverage probe** changes an ADD sign in memory and obtains the
+same census. It demonstrates why these checks cannot certify the operator
+identity. Actual input-file changes are separately rejected by SHA256 binding.
+
+## Sources, overlap and limitations
+
+Base: canonical main `2a13eb1827784206202db90e5dc8327cf8a250db`, priority revision
+7, selected conditional kappa `23/10000`. Exact input blobs and SHA256 values
+are in [SOURCES.json](SOURCES.json); scope and freshness are in [TASK.json](TASK.json).
+Coordination: [discussion #4445](https://github.com/CrocSwap/integer-mult-bounds/discussions/4445),
+with a pointer replying to the [handoff](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19103994).
+
+The overlap refresh covered all 5,731 tracked local package notes/ledger entries,
+301 open PR bodies and 3,290 Discussion records, with complete pagination at
+21:05/21:06 UTC on October 21. It surfaced #4426's explicit missing census and
+no separate competing census. Closed work was covered through local notes and
+the ledger, not a fresh scan of every closed PR body; code diffs are outside
+the helper's search. A submission refresh at 21:16/21:17 UTC covered 303 open
+PRs and 3,294 Discussion records, with complete pagination and no warnings or
+new competing census. Canonical main, guidance and the inputs were unchanged.
+The older `t360-local-rewrite-census` searches rewrites of prior words, rather
+than reconstructing this full-identity receipt. This is not proof of novelty.
+#4441's changed helper word is a different input; this frozen audit does not
+certify its invoice.
+
+Not rerun: arbitrary-input operator equality, exact Clifford phases, scalar
+primitive pricing/prefix norms, helper physical normalization, the p7 controller,
+121-axis admission, tensor/strictification arguments, compiler, precision/stock,
+fixed-tape and outer/analytic obligations. The source baseline command was
+attempted but could not complete with omitted source-closure directories in
+this sparse checkout (`Invalid package root: research/kappa-accounted-composition`).
+It is not reported as passing. The targeted inputs are all materialized and
+authenticated. The complete `frontier4359-verify` replay was not run.
+
+## Open next questions
+
+No further census question is justified for these exact bytes: both orientations
+and the full distributions match, with the stated rejection controls. The
+physical/operator, phase and downstream boundaries listed above remain inherited
+in this audit. Their absence here does not mean the repository lacks evidence
+for them. Consult the selected proof review and live handoffs before proposing
+any additional independent check; a changed native word needs its own census.
+
+## Credit
+
+- **Ryanson (Ryan) Jonathan / pinterbanget**, #4426 at
+  `6f918e2658c727007611d98ffb8d3c7336e2290c`, for the independent p7 controller
+  invoice reconstruction and this explicit handoff; #4420 for the arithmetic audit.
+- **Cinna Davis / 1337isnot1337**, #4416 at
+  `e9058860b5c850760fdf970e796a96aefcdb1d98`, for the p7 construction and invoice.
+- **zxv**, #4359's retained source closure at
+  `da637c87c89482bc4c86be36bd0071b3d400dbed`, for the full-identity supplier,
+  center split and productive-helper construction. Jacob Alan Sussman supplies
+  the mixed-target recursion; David Ganz, Rohan Arun, Cinna Davis, Jeremy Kothe,
+  Marius-Paul Dumitrean, Erwan Secousses and the retained native/frame contributors
+  are credited in the [checkpoint contributor map](../../docs/research/community-productive4359-contributors.md).
+
+Prepared for **mihir-s-05** with substantial **OpenAI Codex** assistance in
+research, implementation, verification and writing. Apache-2.0. Frozen sources
+and their attribution are unchanged.