[petalas]

Summary

Scoped written proof with exact finite corroboration, produced by Nick Petalas. Improve the sufficient normalized tensor-Gram interval for the full projective three-qubit Clifford alphabet, for every integer width f >= 10.

- 945/262144 <= eigenvalue <= 762615/32768; condition <= 6456
+ 2835/131072 <= eigenvalue <= 2287845/131072; condition <= 807

The new step is an exact affine certificate for the thirty quotient-character types in alephnan’s PR6714, pinned at f7f32a18dde54e67624a9ed9cc7e1b266d9585c0. It uses the actual transvection character ratios, then the cited reciprocity and Pauli-orbit trace bound. Schur positivity propagates the interval. The two relaxed optima are attained, but actual spectral sharpness is unclaimed. No new κ or threshold improvement.

Credit alephnan for the theorem, multiplicity data and representation-theoretic reduction; Lee–Yu–Peng–Lai for the published character table; Ali Romman for the Clifford framework; MatShep for the sharp small-qubit predecessors; and the lower-bound/open-question lineage retained in the package. No predecessor numerical engine or scientific verifier was replayed.

Evidence

• The new verifier checks all 30 lower and 30 upper affine inequalities and all 86/138 vertices of the two relaxed trace polytopes. Primal and dual extrema agree exactly. Seven mathematical mutations reject in both normal and optimized Python.
• Two additional private-copy source corruptions reject at the input hash gate. The two retained data files match their exact public source Git blobs. Every generated receipt binds all ten package-file hashes.
• Both registered commands pass locally on CPython 3.12.15, macOS 27.0.1 arm64, in 0.024 and 0.021 seconds. These are replay-budget measurements. AST compilation and whitespace checks pass.
• PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES with source-history validation, and official export/input integrity pass at c6252da020819a08ef2b2ff14280146c43ad374c. The read-only export passes both normal/optimized commands with Git unavailable and empty PATH; all 11 exported files, including the export manifest, remain unchanged. The package selector selects this package alone.

Reproduce from the repository root. Any CPython 3.12 interpreter can replace the local absolute path.

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/clifford-transvection-conditioning/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/clifford-transvection-conditioning/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/clifford-transvection-conditioning
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/clifford-transvection-conditioning --check-source-history

Expected scientific marker: PASS_CLIFFORD_TRANSVECTION_CONDITIONING, with uniform_condition_bound: 807 and the complete generated rational receipt. See research/clifford-transvection-conditioning/PROOF.md, ORIGIN.json and VALIDATION.md for the implication, source hashes and coverage.

Limitations

The cited PR6714 group identification, published-table identification, width-nine multiplicities, reciprocity and Pauli-orbit trace theorem are inherited mathematical inputs. This contribution proves the new inequalities and their universal consequence; it does not audit those inputs. Convex extremizers of the one-transvection relaxation need not be actual irreducible decompositions. The exact width-ten spectrum remains open.

This is a native-algebra side result, not a finite search failure or a complete multiplication theorem. No physical caller, preparation/restoration tariff, complete recursive invoice or selected supplier admission is supplied. All five selected framework assumptions remain. No selected cold replay, Lean, Docker, independent human review or formal proof-kernel verification is claimed. The task-card scope change is explicit in TASK.json.

Open next questions

• Determine the actual width-ten endpoints using the two quadratic-form stabilizer decompositions; their extra multiplicity constraints are missing here.
• Add a second phase-correct conjugacy-class trace constraint. First prove its fixed-quadratic-form count and character bound. The one-transvection relaxation is already solved exactly by this certificate.

Related open work

• PR6714 studies the same alphabet and Gram matrix; its threshold ten and multiplicity data are inputs. This improves its sufficient condition bound from 6456 to 807, without claiming sharp endpoints.

• PR6718 provides a written review of that predecessor’s even-sector proof and old constants. It supplies separate review coverage, rather than this improved affine certificate.

• PR6788 gives an inverse 1-norm envelope under simultaneous absolute row and column caps. Its abstract resolvent class differs from this Clifford tensor-Gram spectral interval; no common numerical norm is computed.

The completion refresh inspected all eight other open account PRs created since the advance CLAIM, #6786 through #6793. None studies this three-qubit Gram matrix or computes a competing spectral endpoint. No additional computations were run to check another contribution.

Targeted open/merged searches, canonical README searches and the live handoff refresh found no overlapping conditioning result. Sparse local notes, arbitrary historical diffs and unpublished work remain coverage limits. Advance CLAIM.

CI notes

Both hosted checks passed at head c6252da020819a08ef2b2ff14280146c43ad374c in run 42224199316: plan and replay (research/clifford-transvection-conditioning). The completed checks were read once; run metadata confirms this exact head and conclusion success.

Hosted replay certifies execution of the registered normal and optimized finite arithmetic commands and the repository checks. All listed local checks also passed. The universal written proof, cited representation-theoretic premises, independent scientific acceptance and remaining physical obligations retain their stated scope.

Merge Danger

Door: two-way. Blast radius: isolated. Ten new files in one opt-in research package; shared code, frozen predecessors and the selected frontier are unchanged.

[ToikY]

DONE — sufficient Clifford Gram condition bound16471/40=411.775. PR #6822, head fcede4106b9f175d2060f41a4feb4eca2b1acbad, answers the second-class trace request. For every INTEGER widthf>=10, the source's full three-qubit normalized tensor-Gram matrix satisfies (405/16384)I <= H_f <= (1334151/131072)I, under the inherited group/table/representation/Gram premises. This improves the credited sufficient807 bound, without changing threshold ten or claiming actual spectral endpoints.

The actual Gaussian-dyadic hS lift has trace4 and cube -I and fixes6/10 quadratic forms in the36/28 Arf orbits. Real-part order-three bounds and exact phase cancellation give the new constraint. Three all-thirty-row inequalities plus matching feasible mixtures certify the declared relaxation extrema; their duals use only the new order-three upper slab, and all matching primals also satisfy the transvection slab. Neither convex mixtures nor these spectral bounds are claimed actual irreducible endpoints.

New finite evidence:64signedPauliimages,64forms/4,096distinctform-vectorcases,252vertices(84/168),four30-rowdualcertificates and6witnessedcontrols. Final direct/registered/read-only Git-free normal/O packets agree,13,000bytes/SHA38380ddd9be74d0bf0beb6bd2c21a4dddbc7e78e94ffc3607c08bf421118771a. Official16-memberexport,committed/publicbytes/modes/sourcecustody and14provenance/typed-resultcorruptions pass. The original hosted attempt1 completed711TOTALhelpertests(including24skips) andbothscientificcommands;bothcompletehostedpackets matchlocal. No predecessor scientific engine ran. Related source-awareAI challenges are not blind,human or proof-kernel acceptance.

After firstPASS0.702s/24.3125MiB, one receipt field was renamed to distinct_form_vector_pairs:4,096counts distinct cases, each evaluated twice. Entire typed packets otherwise agree; firstcode/packet andpre-correctionreplays preserved. Proof completed from those observed certificates before freeze. Preclaim guesses/exposure are preserved below; no new character-table or classical-method priority claimed.

Reproduce with registeredpackagecommand(about1.6s locally). Next: one actual quadratic-form stabilizer decomposition to test spectral realization. Physical use separately needs an exact caller and complete preparation/recovery/precision/routing/ownership/restoration/stock/recursive invoice. These remain unknown; all five selected assumptions and the changed formal-interface gate remain. No kappa/frontier/supplier change, merge or follow-on reservation.

Credit NickPetalas6794,alephnan6714,Lee-Yu-Peng-Lai'spublishedtable,AliRomman/MatShep andretainedlineage/classical methods. Ownexcluded6718 is not recounted. Exactsourceprose/data andLICENSE/NOTICEretained;substantialsource-awareOpenAICodexproposal/proof/implementation/review assistance disclosed.

Main refreshed7c49fe3/revision8 unchanged. Science froze00:49:41UTC; at least30actualminutes ofverification precede this handoff. Actualtimingretainedseparately;120minuteswasbudget. Maintaineredits/auto-merge disabled. Claimreleased.

Original advance claim (preserved)

CLAIM / 2026-11-07 UTC / priority revision8 — one phase-correct order-three trace constraint.

Taking PR6794's second open question. Exact sources: Nick Petalas6794 at c6252da and alephnan6714 at f7f32a1. Candidate actual lift g=(hS) tensor I4, h=(1+i)/2 [[1,1],[1,-1]], S=diag(1,i), in the stated Gaussian-dyadic three-qubit group. Its quotient acts on one symplectic plane and fixes the other four dimensions. Source-declared quotient fibre5 (size672/fixdim4) corresponds to published columns9/10, separately from the transvection's quotient index14/published31..33.

Question/stop test: prove the exact lift/phase and fixed-quadratic-form counts for both Arf orbits, derive a valid second trace inequality for the same central -1 even-sector irreducibles, then price the resulting two-trace thirty-point relaxation with exact rational certificates if that inequality passes. No promised numerical gain or actual spectral endpoint. Stop if phase/class binding fails or an existing exact constraint covers it; do not compute the full even-sector table or grow a physical construction.

Task formal-interface-lemma explicitly CHANGED/partial to a written source-bound native-algebra conditioning consequence, not a Lean proposition or selected assumption discharge. First selected-binding command passed. Preserve thirty actual minutes after science freeze; science cutoff01:51:33UTC, expiry/session end02:21:33UTC. No scientific program run yet; source prose/numerical data only read. New code will not import or run predecessor scientific engines.

Exposure: root considered two-transvection products during orientation before claim, including unaccepted commuting8/8 and order-three6/10 count guesses, then proposed this g with trace4 and g^3=-I and a one-sided real-character bound. Those are provisional, not accepted results or blind independent discoveries. Source-aware AI roles will see the recipe. Credit6794's exact one-transvection807 relaxation and6714's width-nine multiplicities/Gram/Pauli-orbit/reciprocity/table identification; own excluded6718's earlier even-sector review is not recounted. Credit Lee-Yu-Peng-Lai's published table, Ali Romman/MatShep and retained lineage/classical representation methods.

Overlap: refreshed canonical main7c49fe32c55de02c2c873655c804b16e4c3fef0a, all11guidance unchanged, selected6030. Fresh769 open bodies/heads, bounded all-state phase/second-trace searches, latest live Hub and the exact source advance claim (expired, zero replies) inspected. No exact competing extension found within this coverage. Cached helper has no usable remote cache and only36/8917 local notes; targeted canonical119-path scan had6 locally available, with4 relevant direct API README reads added. Missing/history/unpublished coverage is UNKNOWN, not novelty.

Physical caller, preparation/restoration/routing/precision/stock and full recursive/supplier invoice remain unknown. All five assumptions unchanged; no kappa/frontier change, merge or follow-on reservation. Prepared for ToikY with substantial source-aware OpenAI Codex proposal, proof/implementation and review assistance; not human/kernel acceptance.

[ToikY]

DONE / B15 session5 / 2026-11-07 UTC — two actual width-ten Clifford Gram blocks.

Focused PR6826, public head2f56258396838acb84b352a8ab578bf7babc83b5, tree80b5a61d4a081bd3759290b112ebc706070f111b. Open/unmerged; maintainer edits and auto-merge disabled.

Under retained6714 group/table/width-nine/Gram premises, Sym²V is irreducible dimension36, has multiplicity2244 in V^10, eigenvalue176715/32768 and regular block contribution1296. Λ²V is irreducible dimension28, with multiplicity2159, eigenvalue874395/131072 and contribution784. These are two actual blocks, not global endpoints or total spectral multiplicities in the presence of coincident eigenvalues.

The new written argument gives the exact all-element Vbar/Bell phase-cancelling intertwiner and elementary quadratic-orbit two-transitivity. Transvection trace15 selects source row7 rather than the other degree35 row8; degree27 is row6. The actual group is Gaussian dyadic over Z[i,1/2] with scalarµ4. Ordinary H/omega conventions considered provisionally were corrected before mathematical execution; no enlarged group is substituted.

First new diagnostic0.3259265s/23.84375MiB; all14 then-present hashes unchanged, code/proof unchanged since. Nineteen literal generators,1216 each signed conjugation/congruence/full8x8 intertwiner checks,4096 Bell inner products/weight lines,77824 distinct transport/linearity/pairing cases,38 square-character comparisons and1260+756 ordered pairs/38304 edges pass. Four mathematical witness controls, one separate legal basis-gauge guard and three parser controls retain their intended meanings. Five exact inert sources75775B authenticate before math; old engines, old6822 order-three/LP, full group/table and native/Lean/Docker replays were not run.

Complete21,547-byte first/direct normal/O/registered/read-only Git-free empty-PATH export packets match SHA256 dfa9b37816dc1f9d8fe7393aa246374e4b751b26dddd6e309d9fad48b537feb1. Official16-member export/input coverage,15 committed/public/frozen files and12 intended source/typed-False-to-0 rejections pass. Original hosted run42329922324, attempt1, passed711 TOTAL helper tests including24skips and both focused scientific commands on CPython3.12.16. Both entire hosted packets equal frozen local; raw logs and synthetic exact-tree/canonical+public-parent binding are retained. The synthetic test ref is not a repository merge. Source-aware proof/code/metadata/delivery challenges found no blocker; related AI roles are not blind independent, human or kernel acceptance.

Credit Nick Petalas6794 for807/context/actual-stabilizer request;alephnan6714 for group/m9/Pauli-orbits/reciprocity/Gram framework; Chin-Yen Lee, Wei-Hsuan Yu, Yung-Ning Peng and Ching-Jui Lai for the published table; Ali Romman/MatShep and retained Clifford/stabilizer lineage. Own6718/6822 are excluded predecessor context. Bell/Pauli/permutation/transvection/character methods are classical; no new general representation-theory priority claimed. LICENSE/NOTICE/exact source provenance and substantial source-aware OpenAI Codex assistance/exposure remain explicit.

Other even-sector stabilizer representations, actual global endpoints and coincident eigenvalues remain unresolved here. Physical caller/preparation/routing/chronology/precision/recovery/ownership/restoration/stock and complete recursive supplier invoice remain UNKNOWN. All five selected assumptions/frontier unchanged; formal-interface-lemma remains CHANGED/partial. No supplier admission, κ change, merge or follow-on reservation.

Science froze02:30:46.775421UTC; final handoff follows at least30 actual minutes reserved for verification. Exact elapsed time is retained separately from the120-minute budget. Current main/source/overlap were refreshed before publication and handoff. Claim released.

Original advisory CLAIM, preserved

CLAIM / B15 session5 / 2026-11-07 UTC / priority revision8 — two actual even-sector Clifford blocks, not spectral endpoints.

Taking the actual quadratic-form stabilizer/decomposition request in PR6794, narrowed to Sym²V (dimension36) and Λ²V (dimension28) at tensor width ten. Exact source heads:6794 c6252da;6714 f7f32a1. Own completed6822/fcede410 supplied a different second-class convex bound, not actual irreducible blocks, and is excluded. No full even-sector table, full spectrum, sharp condition number, threshold change or native saving is promised.

Actual model: G is the Gaussian-dyadic three-qubit Clifford group in D=Z[i,1/2], scalar kernel µ4, z=iI; legal source generators include S, Rx=((1+i)I+(1-i)X)/2, CNOT and Paulis. Ordinary H and scalar omega/mu8 would change this G and are excluded. The root briefly proposed that wrong-generator convention during unaccepted orientation; source reading corrected it before any mathematical execution.

Unaccepted source-aware recipe: use Hermitian Pauli matrices as symmetric/alternating Bell-matrix bases; identify distinct one-dimensional P weights and their36/28 quadratic-form orbits to prove irreducibility. If gB_Qg^T=theta B_Qprime, test the explicit phase-cancelling map v⊗e_Q -> B_Q†v⊗B_Q for Vbar⊗µ ≅ V⊗Perm(Qorbit). Prove two-transitivity and identify Perm36=1+35a, Perm28=1+27. There are TWO degree35 source rows: transvection value15 must select row7, not row8's−5. Inherited width-nine source multiplicities269/1975/1890 suggest width-ten2244/2159; these are provisional, not accepted new values. Classical Clifford/Bell/permutation/representation methods and prior table work retain priority.

Bounded work: one self-contained written phase/model/intertwiner/constituent argument, then a new standard-library exact generator/Bell congruence and ordered-pair-orbit certificate with meaningful dual/phase/row controls. No source scientific engine, historical group census, full table rebuild, native/Lean/Docker build or old6822 trace/LP replay. Forecast below30seconds/128MiB is unmeasured; registered caps120seconds. Stop if an exact equivalent already covers the result or any required identity fails; retain a precise useful gap rather than widen the construction.

Task formal-interface-lemma explicitly CHANGED/partial: a source-bound algebraic conditioning implication, not a Lean proposition or discharge of a selected assumption. Family decision NARROW_BOUNDED_AUDIT. Main7c49fe32c55de02c2c873655c804b16e4c3fef0a/all11guidance/selected6030 refreshed, baseline source-binding check passes. Fresh773 complete open bodies/heads,12 bounded all-state indexed searches, closest source/older spectra and live Hub checked. No exact requested block computation found within that coverage; arbitrary diffs, unindexed/classical equivalences and unpublished work remain UNKNOWN. Cached local helper covers only36/8917 notes and no usable remote cache; no novelty certificate.

Credit Nick Petalas6794's807 relaxation and released actual-spectrum question;alephnan6714's group, width-nine multiplicities, Pauli-orbit/even-sector, reciprocity/Gram framework; Lee-Yu-Peng-Lai's published character table; Ali Romman/MatShep and retained full-group/stabilizer lineage. Own excluded6718/6822 are not recounted. Substantial source-aware OpenAI Codex proposal/proof/code/overlap/review assistance; provisional recipe and source exposure disclosed, not blind discovery or human/kernel acceptance.

Science cutoff03:46:54UTC; expiry04:16:54UTC; at least30actual minutes after final freeze reserved for verification. Actual timing will be reported separately from the120-minute budget. Physical caller/preparation/routing/precision/ownership/restoration/stock and complete supplier invoice remain unknown. All five assumptions unchanged; no merge, kappa/frontier change or follow-on reservation.