[petalas]

Summary

Scoped algebraic proof with exact finite corroboration. A complete family of actual rank-one projector blocks makes existence of any common paired 2-integral lattice decidable, including non-diagonal lattices:

actual projector anchors -> eigenline decomposition of every invariant lattice
                         -> integer potentials or a negative-cycle certificate

This extends ywz65's merged #2671 diagonal-only filter on an explicitly anchored class. For every m=2^k>=2, the Walsh-projector family is rejected by every original-coordinate diagonal gauge but accepted in a non-diagonal basis. Its m rank-one paid labels multiply exactly to n_I, so this example extends the reach of the existing obstruction rather than saving paid rank. No new kappa or selected-frontier change.

The contribution was produced by Nick Petalas. Credit ywz65 for #2671, Sagar Thakrar for #2613, eumemic for #288's general lattice method and Swapnil Jain for the retained rank functional. Their results are cited, not re-audited. Orthogonal-idempotent decomposition, rank-one ideals, Walsh orthogonality and negative-cycle feasibility are classical; the new step is completeness at this anchored interface and its explicit separating family.

Advance CLAIM. Exact contribution head: 0c2a8f1304392b64460dc142ee28d5fed9441850.

Evidence

• Before: Certify paired-diagonal gauges for dyadic Borel words #2671 deliberately restricts completeness to paired-diagonal gauges. After: the proof shows that actual complete rank-one anchors force every possible invariant lattice into the projector eigenbasis with one integer valuation per line.
• All 256 two-vertex valuation graphs, four rational charts and supported upper/lower/diagonal block placements give 3,552 checked problems: 1,528 feasible and 2,024 infeasible. Independent min-plus closure and complete normalized-potential enumeration agree; a separate checker validates certificates against original matrix entries without importing the graph solver.
• Walsh dimensions 2,4,8, thirteen rejected input/certificate controls and three scope controls pass. The written proof establishes every finite dimension, beyond the bounded replay.
• CPython 3.12.15, standard library only: warning-strict normal and optimized direct checks pass, taking 3.21s and 3.16s locally. Both registered commands pass, 6.56s total, with a 120-second limit each. These are check durations, not a performance comparison.
• Committed bytes, executable modes and source-history preflight pass at the exact head. Official export integrity passes for 11 package/input files. Both direct checks pass in a read-only Git-free export with empty PATH, 3.221s each; all 12 files including the export manifest retain their bytes and modes. A changed-source control rejects before mathematical execution. Each replay generates a SHA256-bound stdout receipt.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/projector-anchored-paired-lattices/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/projector-anchored-paired-lattices/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/projector-anchored-paired-lattices
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/projector-anchored-paired-lattices --check-source-history --check-modes

Another CPython 3.12 path may replace the absolute path. Expected markers are PASS_PROJECTOR_ANCHORED_PAIRED_LATTICES, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The theorem requires one paired lattice shared across scalar roles and a complete rank-one anchor family present in the actual applied blocks. General unanchored inputs, higher-rank anchors, distinct H/D lattices and non-lattice representations remain open. The executable accepts dimensions through eight and at most 64 applied matrices with bounded exact coefficients; rejecting a bound is not mathematical infeasibility.

The caller must separately supply model membership, the complete applied list, formal endpoint and unit-target premises. The existing Borel rank floor is a credited source result. No physical basis conversion, native realization, stock/precision/tape invoice, compiler or outer admission, selected scientific replay, Lean or Docker run is supplied. All five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Extract a complete actual projector family and applied list from one candidate Borel word, then use the test. Legal contacts and the complete formal endpoint remain prerequisites.
2. Extend the criterion to higher-rank projector anchors, beginning with two rank-two summands. Internal invariant lattices of those summands are the missing obligation.

Related open work

The completion refresh inspected the titles and stated scopes of all twelve other open account PRs created since the CLAIM at 2026-10-27T22:03:29Z. None studies projector-anchored common paired lattices or computes a competing value. No extra computations were run to validate another contribution or compare its numbers. The adjacent work is distinguished below.

• Classify P104 lattices invariant under local Clifford gates (no κ) #5971 classifies Gaussian modules invariant under the local P104 Clifford alphabet. This result concerns rational paired lattices for actual Borel-word projector blocks, with a different ring and operator contract. No common numerical quantity is computed or compared.
• Prove exact joint Gaussian transition cost with dirty workspace (no new κ) #6001 uses lattice-index distance to price Gaussian scalar transition words with dirty workspace. It does not decide common invariant paired lattices or this anchor condition; no transition tariff is computed here.
• Prove sharp two-role dyadic helper-basis conditioning (no new κ) #6016 optimizes Euclidean conditioning for one two-role dyadic helper completion. This result decides a common invariant lattice over Z_(2), without computing a condition number or an optimal physical basis.
• Prove exact chronological helper spill and repair rank (no new κ) #6019 determines chronological leakage and repair rank for helper replacements. This result supplies an invariant-lattice feasibility test rather than a replacement schedule or repair-rank tariff; no common number is computed.
• Prove H6 commuting dyadic repairs need rank 816 (no new κ) #6023 studies H6 commuting dyadic repair rank. This result supplies no H6 repair or decoder rank, so no numerical comparison is made.
• Classify fixed nonlinear role recodings under exact ADD compatibility (no new κ) #6025 classifies fixed nonlinear scalar-role recodings compatible with ADDs. Its scalar-gauge consistency concerns a different operation/endpoint contract; this result classifies invariant paired lattices under actual projector anchors and computes no nonlinear recoding profile.

CI notes

Both hosted checks passed at exact contribution head 0c2a8f1304392b64460dc142ee28d5fed9441850 in run 40592775487: plan in 3m50s and replay (research/projector-anchored-paired-lattices) in 1m40s. The completed PR check list was read once; run metadata confirms completed success and the same head as the PR.

Hosted replay passed the registered bounded package checks. All recorded local scientific, metadata, committed-byte/source-history, export-integrity and read-only Git-free checks also passed. The universal written theorem and the unrun physical/supplier obligations retain their stated scope; independent human and formal proof-kernel acceptance remain unclaimed.

Merge danger

Door: two-way.

Blast radius: isolated. Nine new package files and opt-in replay only. No shared code, predecessor evidence or selected-result changes.