One Clifford child under arbitrary reciprocal basis changes: uniform-direction obstruction and opposite-direction exception

[Arham-Tahir64]

DONE — PR1920, head 057b214: #1920

Written all-size characteristic-zero proof: under an arbitrary invertible reciprocal basis, q same-direction width-f Clifford lanes plus fully restored dirty padding can equal a phased width-h tensor child only when padding=0 and h=f. The exact opposite-direction pair classification changes with f mod 4: odd f permits h=f+1 (phase -i or 1); f divisible by four permits only h=f; f congruent to two permits none. Full rational encodings are supplied. For sixteen same-direction rank-one lanes on 32 coordinates, every reciprocal basis plus a left correction F needs rank(F-I)>=6; no sharpness or recursive-cost claim.

The odd-width identity still loses under the inherited joined-volume upper-bound tariff, even with basis conversion free. This compares available budget expressions, not optimal runtime lower bounds. No physical decoder, 5% construction-gate success or new kappa. Independent nonreciprocal maps, clean invariant-subspace injection and changed tariffs remain outside the obstruction.

Evidence: final registered CPython3.12.10/Windows11 replay passes both optimized verifiers and six tests (7.890s). 28032 spectral cases through f=64, independent cyclic multiplicities, 2728 native Gaussian matrix entries, three complete rational basis witnesses, independent Gaussian elimination/correction-rank probes, semantic and actual source/inventory/receipt/type corruption controls, no-Git optimized read-only export and deterministic regeneration. Source/provenance metadata, selected-record integrity and staged whitespace checks pass. Receipt hashes and exact commands are in VALIDATION.json. Full selected cold replay, new physical/compiler/hosting/precision/tape, proof kernel, independent human review and hosted Linux CI are not claimed.

Final snapshots 13:45–13:46 UTC:567 open PRs and539 Discussion records, complete pagination, limited body/metadata coverage. Main and PR1779/1817 heads rechecked unchanged. Credit Greer Guthrie/PR679, Ross Richardson 8a3900c, Jose Manuel/Cinna Davis PR1779, Josie Elliston PR1817, the kernel and selected construction lineage, PR1012 and PR1605. ToikY's clean-subspace claim below is complementary, explicitly outside our full-space model and credited as unfinished parallel work.

Next: a legal inexpensive nonreciprocal adapter or a physical opposite-direction boundary with a proved changed volume tariff. Both need exact full-space contracts and complete costs. Prepared by Arham Tahir with substantial OpenAI Codex assistance; no subagents. Advisory claim released.

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Original claim:

CLAIM — arbitrary reciprocal linear basis changes for one Clifford tensor child.

Base main 0b1cccf86e2f5132ed286a26ef0a681a495740ea; chooser nonce 5539eb9e61a07a24d746ce2018c34aba, complex-mechanisms / complex-rank-shape-decoder, priority revision 3. Scope changed from physical decoder synthesis to a precise contract obstruction; no construction-gate success or new kappa.

Question: can arbitrary invertible linear encoding followed by its inverse turn one higher-rank native tensor child into q equal-width, same-direction independent children, with all dirty padding restored? The prior Ross Richardson/PR679 audit covers monomial pre/post maps. This changes that premise to unrestricted reciprocal maps, while leaving independent nonreciprocal adapters outside scope.

Method/domain: all-size trace-square and trace-norm proof for C=((1+i)I+(1-i)X)/2; exact fourth-root multiplicities and full Gaussian operator fixtures through total address dimension 5. Also test the sharp direction boundary: two opposite-direction odd-width lanes may be similar to a phase times the next-rank child. Charge joined volume under the inherited tariff, even when encodings are free. Expected artifact: proof + standalone stdlib checker, exact receipt, pins and meaningful controls.

Credits: Greer Guthrie/PR679, Ross Richardson commit 8a3900c84213fd7603b99beaceaff48f501b4232, Jose Manuel/Cinna Davis PR1779, Josie Elliston PR1817, and the selected construction/kernel lineage. Their results are retained; no batch recount or monomial support replay is intended. Fresh entry search covered 543 open PRs and 529 Discussion records with full pagination, plus local notes. Limited text coverage does not certify novelty.

Prepared by Arham Tahir with substantial OpenAI Codex assistance; no subagents. Claim expires: 2026-10-14T15:32:50.224464+00:00

[ToikY]

DONE / HANDOFF — PR1924, head 0a7c54d. Advisory claim released. No merge or frontier change.

Established for the literal clean tensor-host contract: q=2^s same-direction C_n lanes have least host width N+1 with any fourth-root global phase, N=n+s. At exact phase, n>=3 always uses N+1; n=1 passes exactly at selector residues s mod8 in {0,6,7}, and n=2 at {0,1,5,6,7}; other small exact-phase cases need N+2. The all-size Pascal/character proof, explicit dyadic Walsh injection and exact left inverse are in research/clean-selector-embedding/. Sixteen width-one lanes need C7, or -C6 with the allowed sign.

The complement must start at zero. The spectator extension is E tensor I with the same factor: reassigning spectator axes is outside both this minimum-width theorem and its cost formula. The proof gives explicit changed-spectator examples, including an unchanged-volume width-one budget tie. Under the stated literal full-volume contract, the inherited child-budget ratio exceeds 2 even with encoder/decoder/phase/preparation costs waived. This is an upper-expression comparison, not a runtime lower bound or minimum physical recursive-rank theorem. The card’s physical 5% construction gate is not met; no supplier or new kappa.

Evidence: registered CPython 3.12.14 replay passes, with 131 independent multiplicity counts, 4096 width cases, 12 full Gaussian-dyadic fixtures (9440 intertwiner entries, 3360 Gram entries), seven semantic/domain controls and five actual no-Git subprocess tests. Normal/optimized exports and the final archive replay pass; source, numeric and boolean-alias receipt corruptions reject; occupied output is preserved. 18 onboarding tests, exact source/Git pins and diff checks pass. Separate bounded Codex reviews and a code-disjoint 4096-case recurrence agree. All ten final public blobs match the local tree.

Final 14:14 UTC overlap covered 580 open PR bodies including drafts and 873 local notes; relevant live Discussions were read directly. Numbered pages are non-atomic and there is no complete Discussion census or novelty certificate. Main and PR1920 heads were unchanged. At 14:13 UTC, 61 final-head hosted checks were queued; hosted success and independent human review are unverified. No selected cold replay, native/physical/hosting/production precision, all-size tape compiler, analytic-transfer or Lean multiplication claim.

Next: identify one actual caller’s tensor/spectator split and prove its clean invariant entry state, then price the complete encoder, decoder, workspace preparation or dirty repair, phases, routing, frames, hosting and precision. A changed spectator or recursion cost model needs a fresh invoice.

Credit Douglas R. Colkitt/PR665 for the existing spectral-capacity/Walsh method; Greer Guthrie/PR679 for kernel, dirty padding and volume; Arham Tahir/PR1920 and 1605; Ross Richardson’s 8a3900c audit; Jose Manuel/jmbarrancoml with publishing account Cinna Davis/1337isnot1337 (PR1779), Josie Elliston/PR1817, icekylinx and the Rohan Arun/Jeremy Kothe/Hao Yan retained lineage. New specialization/checker by ToikY with substantial OpenAI Codex assistance and bounded proof/review agents; Apache-2.0.

Original advisory claim

CLAIM — complementary clean invariant-subspace encoding; one pre-implementation overlap pivot.

Selector: seed 9a396c07a6d107acdb9a76567c1010d0, complex-mechanisms / complex-rank-shape-decoder, priority revision 3; base 0b1cccf; selected-record SHA256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32. Scope changed from physical decoder synthesis to an explicit algebraic interface and cost obstruction.

I read Arham Tahir’s fresh claim above and leave its full-space reciprocal basis / opposite-direction result to that work. Question here: for n,s>=1, q=2^s independent same-direction C_n lanes, when does an injective clean encoding E satisfy C_d E = E (I_q tensor C_n)? The changed premise is a proved reachable invariant subspace with a zero complement, rather than arbitrary dirty ambient padding. Determine the sharp least d by exact fourth-root multiplicities and construct a rational Walsh-eigenvector injection. Price the ambient volume even if the encoding is granted free; no physical or kappa claim.

Bounded evidence: integer multiplicity oracle for n,s=1..64, explicit full Gaussian-dyadic intertwiners on small complete fixtures and the q=16,n=1 motivating batch, plus phase, injectivity, wrong-width and dirty-complement controls. Written all-size proof is separate from finite checks. Expected package: research/clean-selector-embedding/, or a source-pinned handoff if overlap changes. Stop after this model; do not search hardware or alter the selected frontier.

Nearest credit: Greer Guthrie/PR679, Ross Richardson’s 8a3900c audit, Jose Manuel/Cinna Davis PR1779, Josie Elliston PR1817, Arham Tahir/this Discussion and PR1605, selected kernel authors. Entry overlap:554 open PR bodies across7 REST pages with empty terminal page, local notes/helper report, current coordination/handoff/active Discussions. No complete Discussion census or novelty certificate. First baseline and assigned ceiling control pass.

Prepared by @ToikY with substantial OpenAI Codex assistance and bounded proof/review agents. Claim expires 2026-10-14 15:46 UTC; session ends around14:33 UTC.

[thakrarsagar]

CLAIM — Reciprocal single-child merges of unequal-volume Clifford lane multisets: exact criterion, h >= f_min, width-lowering merges and their free-conversion budget

Base: 20ecf2891d80b5acf18d980b2164fb44ea7faa90. Predecessors: PR1920 / PR1605 (Arham-Tahir64), PR1946 (adamgall), PR1924 (ToikY), PR2516 (ywz65).
Milestone: local module (scoped theorem + verified constructions + exact budget). No adapter, supplier, recursive invoice or kappa; no frontier change.

Question: can one reciprocal native child E^{-1}[λ C_h^η ⊗ I_t]E beat the inherited separate-lane bill under free conversion? PR1920/PR1946 say no for one width or equal volumes. Hypothesis: yes for unequal volumes (e.g. 8·C_2 ⊕ C_6^{-1} ~ -i C_4 ⊗ I_6), which turns the blocker into an exact adapter-cost budget under A·V·width^σ.

Method: written proof of the Z4-multiplicity criterion (d = 0 required) and h >= f_min; exhaustive census (k <= 3 lane types, widths <= 8, h <= 10); independent Gaussian-integer trace check plus an exact dense rational witness E; PR1920/PR1946 controls and one mutation sweep.
Deliverable: one focused PR research/clifford-unequal-volume-reciprocal-merge/, with a DONE reply here.
Expiry: 2026-10-19T05:33:13+00:00.

AI-assisted session (Claude Opus 5.5) by @thakrarsagar; advisory claim.

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  "scope": "Reciprocal single-child merges of unequal-volume Clifford lane multisets: criterion, h>=f_min, width-lowering merges, free-conversion budget",
  "base": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
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