[petalas]

Summary

Scoped algebraic proof with an exhaustive finite certificate: the characteristic-zero two-column action of GL_3(F2) has image dimension 150 and an exact 18-dimensional kernel generated by

E = 6[1] - 2 sum_(order g=2)[g] + 2 sum_(order g=4)[g] - sum_(order g=7)[g]
E^2 = 56E
ker(rho_2) = Q[GL3]E

A literal restored two-register shear is formally nonidentity but passes every input at widths one and two. A third-column regular orbit exposes it and recovers every formal address-algebra coefficient. This contribution was produced by Nick Petalas. Only research/gl3-rational-two-column-kernel/ is added.

The new result is the complete characteristic-zero kernel and restored witness. Ywz65's #2599 supplies the distinct binary-payload boundary and standard regular-orbit method; its group norm does not vanish over Q. Credit woklerx #1440 for finite-address certification, Jacob Alan Sussman #4628 for the complex small-unit context, and the predecessor lineage recorded in the package. No predecessor verifier or census was replayed.

Evidence

Before: a two-column complex address test had no kernel description in the cited stream package. After: all 4,096 entries of rho_2(E) vanish exactly; integer convolution proves E^2=56E; a 150 by 150 Gram minor has determinant 15033 modulo the checked prime 65537. The idempotent trace gives the matching kernel dimension 18. No modular upper-rank inference is used.

The verifier constructs the 168-element group in two ways, checks centrality against every group element, executes all 144 dirty input columns at widths one and two, and executes the width-three witness with 112 nonzero entries. All four omitted-order-class controls are rejected, and the characteristic-zero group norm remains visible. Generated receipts bind all six package files by SHA256.

Both warning-strict registered commands pass locally, normal and optimized, each measured at about 0.24 seconds on CPython 3.12.15, macOS arm64. The registered package replay, exact committed-byte/source-history preflight, committed export integrity and read-only Git-free optimized replay pass. Normal and optimized receipts agree. An existing receipt path is rejected without overwriting it.

From the repository root, reproduce with the exact available interpreter:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/gl3-rational-two-column-kernel/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/gl3-rational-two-column-kernel/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/gl3-rational-two-column-kernel
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/gl3-rational-two-column-kernel --check-source-history

Expected first direct output is PASS_GL3_RATIONAL_TWO_COLUMN_KERNEL group=168 image=150 kernel=18 E_squared=56E support=112 dirty_columns=144 no_new_kappa.

Limitations

This certifies the Q or C address permutation algebra. It supplies no paid-call optimum, phase-letter theorem, complete complex unit, supplier admission or new kappa. Physical movement, precision, restoration price, compiler, stock and outer costs remain unknown. Independent human and formal proof-kernel review, Lean builds and the complete selected scientific replay are unrun.

The selected source-binding orientation command was attempted and stopped at Invalid selected file: research/fresh-helper-thinned/PROOF.md, which is omitted from this sparse checkout. It is not evidence against the selected construction, and this self-contained package does not depend on that file. The selected record, source manifest and linked review were read.

Base and pre-submission canonical main are 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. The changed card scope and dated overlap coverage are explicit in TASK.json. Sparse notes, arbitrary diffs and unpublished work remain outside that coverage.

Open next questions

• Determine the characteristic-zero kernel for AGL3(F2) at its last unfaithful width, starting from exact orbit and Gram data.
• For one pinned candidate with paid calls or phases, identify a sufficient finite tensor width or a formal algebra certificate. The address-only result does not supply that extension.

Related open work

• Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628 studies complex small units with binary address maps, mixing, phases and paid calls. This contribution identifies only the exact address-algebra test kernel; it neither audits nor improves the predecessor's call bounds.
• feat(research): exact 5-array 14-call block-3 unit schedule via 14-neighbor symplectic cascade (κ <= 0.0628) #4813 proposes a five-array block-three complex schedule. This result concerns finite testing of its address-only algebra, supplies no check of that proposal and has no overlapping call-count computation.
• Prove sharp conditioning of the complete complex port (no new κ) #4828 proves conditioning of a different complete two-bank complex port. Its operator-norm quantity differs from this address-algebra image rank.
• Certify resonant partial-channel rank 100 at width four (no new κ) #4825 computes rank 100 for a particular width-four partial channel. That coefficient and 4,096-dimensional action differ from the full two-column GL3 group-algebra map of rank 150 here, so these numbers are not competing counts.

This account's open PRs created since the CLAIM at 2026-10-22 21:15:18 UTC were refreshed before DONE. Related #4825 and #4828 are already listed above and concern different operators or quantities. The other new contributions study Toeplitz cuts, scalar-projector prices, integer splitting or signed prime-field connectors. No competing count for this exact characteristic-zero GL3 kernel was found in that snapshot; no extra predecessor computation was run.

CI notes

Both hosted checks passed on head 46f011c4dc6986de533d3e51c1f7ea1474e31dc2 in run 39856678583: plan in 2m57s and the registered package replay in 1m31s. Run metadata confirms that exact head and successful completed conclusion. The replay executes both registered warning-strict normal and optimized commands. All registered commands and publication checks above also passed locally. These checks certify execution of the bounded verifier; independent human and formal proof-kernel review remain unclaimed. No CI failure or unrun registered local obligation remains.

Claim: #1166 (comment)

Merge danger

Door: reversible. Blast radius: one isolated research package and its opt-in replay. No selected-frontier or shared-code change.