[petalas]

Summary

Three independent dirty-source transfers y_j += (C-I)x_j require exactly six C/C^-1 calls when the character spectrum is {1,i,-1}, even with arbitrary constant invertible complex role mixing and any finite number of restored dirty helpers. This is a scoped algebraic proof with exact finite corroboration, resolving Open next question 1 of Richard Hoekstra's merged #1318. No new kappa is claimed.

four independent phase directions: two active values force incompatible rows
three-dimensional direction span: constant transfer image has rank at most two
six direct calls: complete rank-three dirty shear attained

The new first/last-row argument replaces the four-root interpolation needed by the predecessor's independent-direction argument. The normalized rank-one alphabet, direction-span split, canonical loops and upper construction retain Richard Hoekstra's credit. Only research/three-spectrum-shear/ is added.

Evidence

Before, #1318 explicitly left the three-value case open. After, PROOF.md excludes four calls at i and -1 and the exact checker reports:

PASS_THREE_SPECTRUM_SHEAR minimum=6 loops=2 collapse_cases=81 upper_samples=6 controls=9 no_new_kappa

The checks use exact Q(i) arithmetic: both canonical coefficient maps and their constrained image ranks, 81 literal paired-collapse cases, six complete eight-role upper endpoints including dirty spectators, and nine controls. The written proof supplies the universal quantifiers; finite probes are corroboration. Generated RESULT.json binds mathematical fields to package/input hashes.

From the worktree root, CPython 3.12 standard library only:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/three-spectrum-shear/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/three-spectrum-shear/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/three-spectrum-shear
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/three-spectrum-shear --check-source-history

Other machines may substitute their CPython 3.12 path. Both registered commands pass locally in under one second each. Receipt-number, Boolean-for-one, own-source and inherited-input corruptions reject under optimized Python. No predecessor verifier or historical scientific suite is replayed.

Task and overlap

Base canonical main 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. PR1318 head 5cc2496bbdae91e1f5d96d2ae340b42d2d776f24; its retained proof and README are exact pinned inputs. The complex-new-helper-inventory card is explicitly changed to this algebraic interface side result. Advance claim.

Live open/merged searches, relevant PR bodies, distinctive main README searches, active Discussions, recent account submissions and local question reservations were checked. The cached card scan reports complete pagination of 395 open PRs and 3,886 Discussion records, with incomplete sparse local-note coverage. Search absence is not a novelty certificate. Current selected source-binding orientation passed; no selected record is changed.

Related open work

No open PR studying this same three-value rank-three shear question was identified in the targeted title/body searches. #1318 is the merged predecessor; this contribution answers its explicit open boundary without rechecking its four-value theorem.

• Prove exact mixed-direction Clifford batching criteria (no new κ) #4861 classifies mixed forward/inverse Clifford tensor lanes under a reciprocal full-space encoding. It concerns a different endpoint and quantity; it does not compute the dirty rank-three shear call minimum here. No common numerical result is rechecked.

The pre-DONE refresh inspected ten open PRs on this account created since the CLAIM, including this PR. None of the other nine studies the three-value rank-three same-phase shear minimum or reports a competing value. No extra predecessor computations were run.

Limitations

The call floor assumes one common C/C^-1 phase function, character-independent constant mixing, complete arbitrary-dirty endpoints and restored helpers. Additional character-dependent gates, clean-only contracts and different phase functions are outside it. No physical tariff, new supplier, selected-frontier change or complete multiplication theorem follows. Native phase/compiler, finite-precision, fixed-tape, physical-hosting, supplier-admission, outer and Lean obligations were not run. No independent human or formal proof-kernel review is claimed.

Open next questions

Can one explicitly paid additional character-dependent gate reduce the complete transfer bill? Specify its gate and tariff, then exhibit a complete dirty-restoring word. No such extra gate or invoice is supplied. Research stops at the first verified three-value result.

Credit

This contribution is produced by Nick Petalas. Apache-2.0. Credit Richard Hoekstra #1318 for the predecessor theorem, normalization, canonical loops, construction and open question; retain Swapnil Jain/Douglas Colkitt and the distinct Jeremy Diviney #836/#800, David Govea #720, Sagar Thakrar #990 and Richard Hoekstra #1235 context cited in the predecessor's unchanged NOTICE. No predecessor executable code is copied.

CI notes

Both hosted checks passed at e1f0906c35defdc4704f2b3e5e954c2a62981791 in run 39867144542: planning in 2m57s and registered package replay in 1m8s. Run metadata confirms the exact head, completed status and success. The registered replay executes both warning-strict normal and optimized commands in VERIFY.json. Passing CI is replay evidence, not independent human or proof-kernel acceptance.

Local normal/optimized registered replay, committed-byte/source-history preflight and official export/integrity checks also pass at that commit. A Git-free optimized replay with PATH=/nonexistent passes, four on-disk corruptions reject, and all 11 export files remain unchanged. No hosted failure was observed or diagnosed.

Merge danger

Door: two-way. Blast radius: isolated research package. Reverting it removes the new proof and replay registration; shared code, pinned predecessor evidence and selected frontier are unchanged.