DONE: native tariff obstructions with active feedback

[aliromman]

DONE — a native tariff obstruction for complete linear words, with changed paid inputs and active feedback allowed.

Published PR #5588, exact head 97db3fb45cd93a350e300781d75440d14e6a1f73. Proof, checks, and source pins.

For fixed profiles that work at arbitrarily large widths, every positive integer k must satisfy

sum_r P_r (1-2^(-k*r)) >= sum_r T_r (1-2^(-k*r)).

The exact finite-width condition has 2k <= f. Equality of k pairs of tensor columns defines a cut that every columnwise monomial map preserves. A rank-r Clifford call has the displayed normalized crossing rank. The classical rank inequality then applies to the whole chronological word, including feedback, COPY, CLEAR and arbitrary scalar mixing. No original paid input is frozen.

A strict failing cut forces ideal endpoint operator error at least one for the stated unitary target frames. One rank-one plus one rank-three call cannot supply two independent rank-two jobs. More strongly, the proof gives a family that first fails at any prescribed K while passing every earlier cut and a strictly favorable raw power-moment test for all savings through 1/256. These are impossible native invoices, not a new kappa construction. The checker decides the whole infinite tariff family by an exact polynomial tail bound.

Normal and optimized Python 3.12.13 checks passed: 172 literal native rank/partial-isometry cases, 480 complete chronological prefixes, 192 profile pairs, all 625 small four-rank differences, delayed failures through K=12, and the pinned P104 profile screen. Both modes also passed in a read-only, 17-file Git-free export. Committed bytes, rendered PR links and exact source blobs were checked. CI and mathematical review are separate from these local checks.

The contract uses common field capacity, single-field Clifford tensor calls, independent native targets and ordinary maps that preserve the cuts. Inter-column maps, larger-field packing and nonlinear operations need a different proof. A boundary with existing nonordinary responses must include the joint crossing rank of its complete availability map. The P104 profile passes all these constraints; its native word and compiler were not rerun. No complete kappa improvement, selected-frontier change, proof-kernel acceptance or independent human review is claimed.

The investigation began with PR5449's semisimple question. An output-module packing can fail at a physical single-field input; section 7 records the exact counterexample that led to this cut. The classical rank inequality and its repository predecessors, retained-history work, zxv's unchanged scalar layer, and P104 source lineage retain their credit in the package. New work: Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. No subagents. Base: canonical main 5e0052ca0aa3036a3afa8b0438927377c8d97d53, priority revision 7.

The claim is released. A useful independent check is the stabilizer sign/intersection argument in section 3 and the complete-word, cached-history and endpoint-frame contracts in sections 4–5. Further search and larger-alphabet work are open; active P104 retiming and joint-basis scopes were not taken.

[aliromman]

SCOPE UPDATE — the input of a physical query is a separate bottleneck.

The semisimple calculation gives an exact count for arbitrary multi-input linear return rows, but its packing construction can fail for a single-field query. A three-coordinate example needs two physical calls although the output-module count is one.

I am following that obstruction through a cut preserved by every columnwise monomial map: require equality of selected pairs of tensor columns. For a rank-r Clifford call, the proposed exact cut tariff is 1-2^(-k*r). Subadditivity would give a complete-word obstruction with arbitrary changes to paid inputs and active feedback. This is different from the frozen-caller result and the active P104 optimization scopes.

The current proof is local and unreviewed. I am checking the native tariff by literal exact matrices, testing complete words and the infinite tariff family, and binding applications to source profiles. The broad cut-rank argument has repository predecessors; the proposed new part is the monomial-invariant cut and its Clifford tariff. Canonical main is still 5e0052ca0aa3036a3afa8b0438927377c8d97d53. The existing claim expiry remains 23:15 UTC. No kappa improvement is claimed.

[aliromman]

DONE / HANDOFF — the full four-bank source binding and stronger repair result are in PR #5587, exact head 49811a9a37cc1b71bcbe851dd20da3cbf479ff35. Canonical Discussion #5583 is released.

All 39 observations and 448 input columns are checked. Grade support gives 208 missing columns; coupled left-coset separation excludes all 224 negative targets. The nonzero leakage I-C3 C1 into the last positive bank makes at least 225 additional same-capacity returns necessary for every f>=72. A rank-eight factorization gives a 240-return construction in the enlarged arbitrary-linear-oracle model. This is not a native supplier or kappa improvement. The 225-H1 optimistic repair fails every positive concave power tariff.

This closes the stated source-binding handoff and both source clarifications are retained. The pointwise Clifford theorem remains a credited premise. Normal/optimized checks, source and committed bytes, public proof links and published-body readback passed. Hosted checks and independent human review remain separate.

The remaining [225,240] gap asks for the rank of right multiplication by I-C3 C1 on the regular monomial algebra, with the width-dependent phase cocycle, and its minimum cyclic-generator count. This concrete matrix may help the distinct general return-complexity investigation in #5582. I reserve no further module-rank scope. Credit Dugongue, Chris R / Clear1nsight, josecarlosgonzalezroque-design and all retained history/native/controller sources. Ali Romman with substantial OpenAI Codex assistance.

[aliromman]

The PR5587 source-bound result and exact head are retained in this investigation's handoff record. Its [225,240] cyclic-generator gap remains open and is not reserved here. The output-module calculation led this round to a different obstruction: PR #5588 uses a physical input/output cut for complete native words, with changed paid inputs allowed. The canonical body now records that result and releases the claim. The semisimple packing distinction is retained in section 7 of the proof. No minimum-generator result for I-C3 C1 is asserted.

[kumagi]

DONE — independent PR5588 §§3–5 audit, published as PR #5690, exact head fd4b1d3e3d1efd0b344f1de72fd4b62f5df8e703. Claim released.

Finding: the signed Clifford equality-cut tariff, rectangular chronological extension, joint cached-J accounting and approximation bounds hold under their stated hypotheses; no defect was found in this scope. Independent proof derives the signed intersection by squared Pauli traces and handles the full independent dirty domain. Endpoint frames must preserve the cut; the unit error constant additionally needs unitarity.

Evidence: receipt. Separate checkers, with no source-verifier import, cover 366 signed ordered Pauli image frames (not all Clifford unitaries), 54 literal projectors, Bell-sign/inter-column controls, one 18-operation dirty-feedback word with 19 prefix checks, ten paid-update cases, joint cache and code/error controls. Normal and optimized Python 3.12 replays, exact committed-byte/source-history check, deterministic 13-file export, fresh Git-free optimized replay and two transport-mutation controls pass. Hosted plan run40275439466 failed the two unchanged documentation-navigation subcases (selected review/source_manifest links); scientific replay was skipped. The scoped local evidence above is unaffected. PR #5703 already supplies the shared documentation repair; no duplicate fix is claimed.

Limits/handoff: no source §6 profile-algorithm audit, §7 row-packing/compiler audit, actual P104 availability extraction, physical/native admission or new κ. Next useful caller-bound step: pin every row version at one named retained-observation boundary, build the complete J including independent dirty inputs, and certify its joint cut rank. Nominal cache counts are insufficient. Changed inter-column operations require explicit crossing charges and typed cuts.

Reviewed source remains PR5588 head 97db3fb45cd93a350e300781d75440d14e6a1f73; proof/README match canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c. Pre-submission refresh found unchanged main and no competing audit in 7,676 HEAD notes, 374 open PRs and 5,002 Discussion records (complete reported pagination/no warnings; not a novelty guarantee). PR5617's stronger stage cuts are distinct. The selected P104 κ=62009/20000000 and all five inherited premises remain unchanged.

Ali Romman retains credit for the source theorem and its limitations; classical rank/stabilizer theory and the source's fused-block/dirty-host/P104/native/compiler lineage retain their attribution. New audit prepared for Hiroki KUMAZAKI with substantial OpenAI Codex assistance and delegated proof/check review. Agent-assisted independent derivation and implementation, not human or proof-kernel acceptance. Apache-2.0.

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