Mister-Meeseeks · Oct 17, 2026 · Oct 17, 2026
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+# Group A's native widths through the shipped T74 invoice
+
+**Finite native/interface audit; no new kappa or supplier admission.**
+
+Both literal PR2979 providers pass the mixed-entrance check for group A.
+Helpers 450, 451, 508 and 509 enter at `[30]` and end at
+`W = [256,128,64,38,20,10,1]`. Their emitted residual pair is `115:261`,
+with width **six**, not the terminal dimension seven. Helper 331 enters at
+zero and uses pair `0:261`, with width seven. Exact native identities include
+global phase and both orientations. These widths bind to all 3,330 group-A
+placements in the shipped T74 allocation.
+
+The contribution joins the complete direct group-A boundary to its actual
+owner placements. It corroborates existing native identities and does not
+newly discover the construction, local saving or published complete histogram.
+The local comparison and normalized endpoint checks explicitly reuse credited
+independent checkers. The claim is in discussion [#2612][thread].
+
+### Existing coverage and the added check
+
+[PR3038][3038] already checks both residual classes within its V45 dependency
+cone: `0:261` in both orientations and `115:261` in the inverse-transpose
+orientation. The latter also implies its forward identity by conjugation.
+The original claim's inventory of PR3006/3008 was incomplete; final overlap
+review identified and credited this stronger predecessor before submission.
+These residual equalities are corroboration, not newly missing mathematics.
+
+PR3038's roots are helper249/private953. Its pinned receipts contain none of
+the five group-A post-wrapper events. For helpers 331/450/451/508/509, this
+audit checks 122/19/19/20/20 direct events per orientation. Respectively,
+22/19/19/20/20 forward events and 61/3/3/5/7 inverse-transpose events are outside
+the earlier cone; the two providers agree. These are per-helper counts and
+may share events, not counts of distinct newly proved native identities.
+[RESULT.json](RESULT.json) records the exact indices and source-bound coverage
+comparison. The PR3038 receipts were inspected, not freshly replayed here.
+
+The added coverage is the complete direct path through each actual post
+wrapper, all ten group helper/private normalized endpoints, and the explicit
+native-width/ordered-helper/physical-slot join for every group placement.
+The complete peer-native closure still remains separate. Concurrent #2612
+private-free and final-fill control audits should be consulted before taking
+those follow-ups; this package does not reconstruct their work.
+
+## Exact native statement and source binding
+
+The source is [PR2979][2979] at
+`be191346391661f4d35fc0d1508d0a6029572576`; the base is canonical main
+`463113d675af7410ac91d53a4fed51ec6494eb9a`, priority revision 6.
+[SOURCES.json](SOURCES.json) records exact Git blobs and SHA256 hashes.
+The 14 construction inputs also match the original PR2979 head byte for byte.
+The native producer, exact kernel and retained checkers are authenticated
+before execution. No frozen predecessor is modified.
+
+For entrance `E` and terminal `U`, write the native residual as
+`S(E,U) = T_U T_E^-1`. The source compiler produces `S = L F_w R`, where
+`F_w` is the retained coordinate Fourier primitive on the first `w` address
+axes. A separate exact Gaussian-integer kernel checks all 512 columns of
+that identity, and the emitted inverse wrappers satisfy
+
+```text
+L^-1 S(E,U) R^-1 = F_w,
+w = dim(U) - dim(E).
+```
+
+For these unitary operators, inverse transpose is entrywise conjugation.
+The same chronological wrapper arrangement gives `F_w^-T`, with the
+remaining `9-w` axes unchanged. The check does not substitute the adjoint
+or reverse the execution order.
+
+| Helper(s) | Entrance | Parent terminal dimension | Parent width | Joint/final width |
+|---|---|---:|---:|---:|
+| 331 | zero | 8 | 8 | 7 |
+| 450, 451, 508, 509 | span(30) | 8 | 7 | 6 |
+
+Here 30 = 20 xor 10 lies in W. Treating the entering helpers as cold would
+select the wrong native operator even if a rank field were edited to six.
+The cold right wrapper and the entering right wrapper are not interchangeable:
+the all-column check rejects that substitution. A different contained line,
+`[10]`, has the same dimension but also fails with the actual `[30]` recipe.
+
+This width is exact within the monomial/Fourier/monomial format: every column
+of `F_w` has exactly `2^w` nonzero entries, and invertible monomial factors
+preserve that count. Thus a checked width-six residual cannot be represented
+by such a format with fewer than six Fourier axes. This is not a lower bound
+for arbitrary native circuits or a claim that the 512-by-512 operator has
+matrix rank six; it is unitary and has full matrix rank.
+
+The verifier regenerates both complete emitted programs with the unchanged
+transport/emitter and reconstructs the full serialized adapter file using
+the original generation convention. The uncompressed `LOCAL-CHECKS.json`
+and `RESIDUAL-ADAPTERS.json` hashes match PR2979's retained native receipt.
+Both regenerated event streams also match the counts and canonical event
+digests inside that pinned receipt.
+Generation is not presented as an independent checker. The independent part
+compares each selected original entrance/terminal pair, direct scalar clock,
+climb and cleanup lifetime against the emitted records, then decodes the
+selected native recipes through the separate exact kernel.
+
+Each provider has 22 distinct group-A climb pairs, two shipped residual pairs
+and two parent residual pairs checked exactly. The emitted climb ranks are
+`1,1,1,1,1,2` for 331 and `2,2,1,1` for each entering helper. The parent
+residuals are generated comparison identities; no complete parent native
+program is regenerated. The final pair keys and adapter hashes are bound to
+the shipped program, rather than to an interchangeable candidate recipe.
+
+There are 112/11/11/12/12 direct scalar operations for helpers
+331/450/451/508/509 and their private buffers, respectively. Scalar clocks
+include intervening operations on other roles. Moving a direct operation
+across a disjoint scalar therefore fails the check even when its order among
+the helper's own operations is preserved. Private operations occur after
+allocation and before free; the inverse left wrapper follows cleanup.
+
+## The group, its control and the invoice
+
+The retained [PR2993][2993] checker reconstructs group A on each parent word
+and a control with the same 21 partner-frame edits but only helper331 lowered.
+No scalar or cleanup operation changes. Their exact local differences are:
+
+```text
+joint:      -H1 + 20 H2 - 14 H3 + 2 H4 - 2 H5
+single331:  3H1 + 20 H2 - 14 H3 + 2 H4 - 2 H5
+```
+
+`Hr` means one paid rank-r child. Substitution into the source's integer
+surrogate and its 2,091,300-unit width guide reproduces joint margin
+`+42,541` and single margin `-58,943`. The four additional endpoints save
+exactly four H1 children locally. This control keeps the group's partner
+frames fixed; it does not optimize every possible single-helper frame path.
+The guide is not an independently certified physical idle tariff.
+
+The new time-major recount reads every one of the final allocation's
+271,728 placements, checks role completeness, source windows, per-call
+physical-family uniqueness, capacities and compact prefixes, and bills all
+149,184 bins including empty bins. Original helper names map to ordered
+helper indices `107,226,227,279,280`; those are explicitly joined to the
+emitted native residual widths. Replacing helper450's width six with terminal
+dimension seven fails the actual allocation check.
+
+| Item | Exact count/profile |
+|---|---|
+| Calls | 250 base + 416 extra = 666 |
+| Local profile per call | `{1:2445,2:706,3:232,4:108,5:18,8:8}` |
+| Width inventory | `{4:26,5:2,6:16,7:52,8:99,9:213}`, mass 3,283 |
+| Actual idle bill | `17310 H1 + 4956 H3 + 1728 H5 + 28190 H48` |
+| Complete recursive bill | `{1:1645680,2:470196,3:159468,4:71928,5:13716,8:5328,48:28190}` |
+| Complete rank mass | `4,816,512 = 65,088 * 74` |
+| Nominal physical stock | `66,304 + 408 + 8 + 91 + 2 = 66,813` |
+
+The complete recursive moment input is therefore
+`sum(n_r * (r/73)^(1-a)) / (896*74)`. All 91 private names remain charged.
+The five group-A private buffers already exist in the parent; stage one
+retains 86, and five later extractions produce the final count 91.
+The independently recounted final bill agrees with the source. It includes
+the other four groups, later extractions and rewrites, so it is not an
+isolated group-A gain attribution or a legal whole-supplier single-control
+comparison. Those counterfactual allocations are not supplied here.
+
+The retained [PR3006][3006] reverse-covector engine also checks 40 output
+rows: five helpers and five private buffers, two providers, two orientations.
+Each is exactly its own input over the full free operator algebra, with
+632 original and 91 private inputs. Thus each covered private output is zero
+when its private entrance is zero. This extends that engine's checked domain;
+it is not a newly independent implementation of the endpoint algorithm.
+Deleting actual cleanup gate178, `private1154 += private934`, breaks this
+identity and is rejected by the behavioral control.
+
+## Evidence and reproduction
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3 -B scripts/research_package.py --package research/rank7-group-a-entrance-audit --check-only
+python3 -B scripts/research_package.py --package research/rank7-group-a-entrance-audit
+```
+
+Expected verifier output begins `PASS_GROUP_A_ENTRANCE_NATIVE_INVOICE`.
+The hook also runs seven behavioral test groups. They cover wrong entrance
+and phase, a cold-wrapper alias, omitted/wrong wrappers, premature private
+free, changed scalar signs and clocks, omitted cleanup, missing/duplicate
+roles, bad capacity, terminal-dimension misuse, source corruption and duplicate
+JSON keys. Direct `verify.py --emit` prints deterministic evidence without
+rewriting [RESULT.json](RESULT.json). Input hashes are checked again after
+the replay. The normal replay writes only temporary work outside the package.
+
+Source production uses fresh temporary script copies and bounded foreground
+subprocesses; inherited `PYTHONOPTIMIZE` is removed from those subprocesses.
+Authenticated imported checkers are compiled from source bytes rather than
+loaded from a preexisting bytecode cache. The finite domain is two 704-name
+source words, 512 address columns, 666 calls and 408 helpers. Raw and decoded
+inputs are capped at 64 MiB; each source subprocess has a 120-second timeout.
+
+## Limits
+
+The nominal stock arithmetic is a reconciled declaration, not a proof of
+physical storage adequacy. The other helpers' native dependency closure,
+warps, physical backing and movement, and precision/prefix/tape costs remain
+inherited. The 86 other private endpoints were not checked. No complex root
+or packing cap, bit supplier cap, outer coupling/release, or all-size,
+compiler and analytic implication was independently replayed. Neither
+supplier is newly admitted and the selected frontier is unchanged.
+
+## Open next questions
+
+The next useful native check starts from the 55/4/4/5/5 private-touch
+inventories for this group and binds every peer transition and warp reaching
+those endpoints. It matters because direct local identities do not establish
+the whole physical caller. Common-chart and peer-native correctness are the
+blockers. After that, the exact PR2979 physical/prefix inputs and both supplier
+caps need independent review before using its claimed exponent. Another
+lowering census or another surrogate-price evaluation does not close these
+obligations. Structured questions are in [TASK.json](TASK.json).
+
+## Credit
+
+Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.
+
+Marius-Paul Dumitrean supplied PR2979 and PR2942's construction; Dugongue
+supplied the PR2969 comparator and PR2839's owner engine. Nick Petalas
+supplied the reused PR2993 frame checker and PR3006 covector
+engine; PR3008's zero-entrance event audit and PR3014's frozen-parent invoice
+are method predecessors. ToikY/PR3038 already checked the two native residual
+classes inside the V45 cone; its exact coverage is explicitly retained above.
+Ross Richardson/PR2987 supplied the complementary
+packing audit, and Gonzalo Mier/PR3079 already replayed the published T74
+aggregate arithmetic without owner chronology. Jeremy Kothe supplied frame,
+cleanup and rewrite work; Mason Hale and Jeremy Diviney supplied the
+paired-extraction chain. The native
+compiler retains icekylinx's credit; the separate exact Gaussian kernel is
+Sagar Thakrar's. DreamingOfClouds and chafreaky/PR458 supplied the residual
+wrapper convention. Earlier helper450 event/scalar audits by bbMisk/PR1611
+and ToikY/PR1710 concern the older k8 word, not this changed endpoint.
+Retain Tekalign, Yiyang Lu, Cinna Davis and the geometry/owner/native lineage
+credited in those sources. Apache-2.0; original notices remain in place.
+No independent human or formal-kernel review is claimed.
+
+[2979]: https://github.com/CrocSwap/integer-mult-bounds/pull/2979
+[2993]: https://github.com/CrocSwap/integer-mult-bounds/pull/2993
+[3006]: https://github.com/CrocSwap/integer-mult-bounds/pull/3006
+[3038]: https://github.com/CrocSwap/integer-mult-bounds/pull/3038
+[thread]: https://github.com/CrocSwap/integer-mult-bounds/discussions/2612#discussioncomment-19001817