[petalas]

Summary

Scoped written proof with exact finite corroboration. For any finite paired-root graph, the three-shear coordinate cycle

M = (I+t E_ab)(I+s E_bc)(I+u E_ca), distinct a,b,c
P = R M, Q = R M^(-T)

tsu outside {1,-1}
  => each term's binary COPY rank >= its coordinate degree
  => complete common-pairing COPY bill >= 2|E| = live root count

This extends JoyboyBrian's #5214 two-shear boundary into its explicitly open three-shear question. The common-support degree lemma is credited to that predecessor and #5176. The new contribution is the cycle calculation and its universal graph application, produced by Nick Petalas. The coordinate basis attains the bound, so the joint minimum over this allowed family is the root count. No optimum for each fixed nonzero basis is claimed.

Only research/dn-three-cycle-floor/ is added. No new kappa, supplier admission or selected-frontier change.

Evidence

Before, #5214 excludes two interacting shears and leaves three open. This proof excludes the stated coordinate cycle away from tsu=+/-1. Two explicit exceptional examples lose the particular degree minor, without claiming a complete saving.

Local CPython 3.12.15 checks pass:

• 2,009 exact factorizations and 35,588 complete common directed-pairing bills over every nonempty three-coordinate graph and the stated seven-value rational alphabet outside the exceptional loci.
• 309,960 forced-minor entries and 106,764 independent set-of-span rank comparisons.
• 24 larger factorizations, 168 complete directed-pairing bills, eight rejecting controls and two exceptional-locus witnesses.
• Both registered normal/optimized commands, metadata/source pins, exact committed-byte/mode/source-history checks, AST compilation and whitespace checks.
• Git-free normal/optimized export runs with an empty executable PATH pass and print identical generated hash-bound receipts. No predecessor scientific verifier ran.

Each registered command was timed directly locally at about two seconds, well below its 120-second cap. The complete registered run also passed.

From the repository root, reproduce with the exact local interpreter path:

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

Another CPython 3.12 path can replace this one. Expected markers are PASS_DN_THREE_CYCLE_FLOOR, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations

The all-graph quantifier rests on the written proof, corroborated by finite execution. Rational paired roots, this ordered cycle, square Gram factors and a common directed fitting binary pairing are the exact scope. Neither exceptional locus, other three-shear index patterns, single-sign deletions nor general bases are settled. Physical coefficients, producer/consumer, dirty restoration, frames, native realization, storage, precision, tape and the recursive invoice remain unpriced. No selected scientific cold replay, Lean or Docker check ran. Independent human and formal proof-kernel review are unclaimed.

Credit also ywz65 #2536/#2719, Aaron Wang / navinor #2272/#2518, DreamingOfClouds's Gram/COPY lineage and the retained contributors in #5214. Exact predecessor proof bytes are pinned as historical provenance.

Open next questions

• Can tsu=1 or tsu=-1 lower a complete common-pairing bill? Start with the lost-minor fixtures, then price every term on an actual paired D5 graph.
• Can another three-shear pattern give a fully paid decoder improvement? Supply full Gram equality, one common pairing and the coefficient/restoration/frame/recursive bill.

Related open work

• Non-dense fitting B: every E8 term with x·y ≠ 0 has rank ≥ 9; square root-code floors E8 ≥ 72, P112 ≥ 64, P104 ≥ 56, D4 ≥ 12 for any fitting B (no κ claim) #3045 studies arbitrary fitting pairings and square root-code floors, including full D4; this result addresses one cycle family on arbitrary paired graphs.
• Decoder floors: universality, projection transfer and batching reductions; C(P112) ≥ 64 reduced to named conjectures, C(P120) ≥ 72 restated on N₀ (no κ claim) #2832 gives broader decoder and batching reductions, retaining the full-Dn dense-pairing lineage; this result permits arbitrary directed fitting pairings in its stated basis family.
• C(P112) >= 64: complete reduction to stated conjectures #3154 studies the P112 decoder floor through named conjectures; no P112 floor or conjecture is resolved here.
• Two-sided deficiency budget: Theorem BX for b_Q <= 4 (conditional frame, exact residue, no κ claim) #3029 studies two-sided P112 deficiency budgets; this result keeps both factor families in the paired-root code.
• Checkpoint eight masked-decoder basis searches (exploratory; no new K) #5064 records masked E8 basis searches; this is an all-pairs paired-root obstruction with explicit rational quantifiers.
• Checkpoint symmetric two-star P112 supports and inconclusive COPY63 search #5088 studies P112 two-star support dictionaries and COPY63; no such dictionary or coefficient synthesis is used here.
• Exhibit a non-coordinate COPY71 completion at the first-failure boundary #7169 exhibits a non-coordinate masked E8 completion; its changed geometry and mask remain outside this theorem.
• Bound scalar COPY savings at one 93-line E8 puncture (no κ) #7170 bounds scalar COPY perturbations at one E8 puncture; this result instead changes a paired-root Gram basis by a cycle.
• Bind P112 tetrad decoder to selected ports; no kappa change #3566 prices a P112 tetrad decoder; its geometry, factor count and physical route differ from this paired-root cycle.
• Add exact complex rank-16 block decoder screen #3895 screens a tensor block decoder through rank 16; this result does not cover that product-root model or its tariff.
• Draft construction: P104 dirty-restored gather and low-rank channel with COPY56 (no new kappa) #1972 supplies a physical P104 gather/channel proposal; this abstract paired-root obstruction does not price or reject that channel.
• Research: paid-copy E8 exact-RAM realization with an attributed DFT corollary #647 supplies a paid-copy E8 RAM realization; this result gives no physical implementation or complete supplier comparison.

This account's open PRs created since the advance CLAIM were checked again before DONE. None adds another paired-root coordinate-cycle or common-pairing COPY-bill result, so no additional related-work line or numerical comparison is required.

No additional computations were run to compare another contribution's numbers.

CI notes

CI did not certify this package. At unchanged head b3ad43d2f5c4181cf79abf8dea382d4d2fbcf547, run 43001222481, plan job 130457550993, failed in "Onboarding helper tests". Its log reports 760 tests, one failure and 24 skipped. The exact failing test is test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record, with an AssertionError because `make frontier7000-verify` was not found in the CI verification document.

This is an existing repository-level mismatch. At canonical main e54523517d0f03e58352e647c403a5af3e525a57, certificates/selected-result.json requires make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. The document's Git blob is identical at this PR's head and main. This PR changes only its six package files.

I also read the plan logs of run 42905094314, run 42903313790, and PR7191 run 43000676071. Each reports the same sole failing test and missing-command assertion, with 760 tests and 24 skipped. These are the comparison runs verified here.

The package replay job was skipped because plan failed. The local checks above remain local evidence; no hosted scientific replay or independent acceptance is claimed. No fix, retrigger commit or job rerun was made.

Merge Danger

Door: two-way. Blast Radius: isolated. Only a new research package is added; shared implementation, selected records and frozen predecessor evidence are unchanged.