[petalas]

Summary

Finite certificate: the resonant partial target coefficient of #4797 has exact complex rank 100 at tensor width four, in dimension 4,096. This answers the first new-width case of its explicit rank question. Produced by Nick Petalas.

frozen 256-term coefficient
  -> enumerate all 4,096 columns
  -> exact rational upper rank 100
  -> 100 x 100 integer minor with determinant -2^24

The predecessor's partial-channel identity and norm theorem are cited inputs, not audit deliverables. Only research/resonant-channel-rank/ is added. A committed byte export of its source receipt authenticates the coefficient, so the unmerged predecessor commit is not needed for replay.

Evidence

• Before: Construct a resonant partial target channel at every even width #4797 determines rank at widths one and two and leaves general even-width rank open. After: rank(A4) = rank(A4 K4) = 100, with 6,440 nonzero entries, 496 nonzero rows and 446 nonzero columns.
• Exact rational reduction and separate integer Bareiss elimination; six rejection controls; generated result and SHA256 file receipt. Independent read-only review reproduced every matrix entry and the new rank certificate with no mathematical flags.
• Registered normal and optimized warning-strict Python 3.12 checks pass locally, each under one second. Metadata/replay, committed bytes/source ancestry, committed export integrity and read-only Git-free normal/optimized export replay pass.

Exact reproduction from the repository root:

~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/resonant-channel-rank/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/resonant-channel-rank/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/resonant-channel-rank
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/resonant-channel-rank --check-source-history

Expected output: PASS_RESONANT_CHANNEL_RANK, rank 100, dimension 4096. Local environment: CPython 3.12.15, macOS arm64, standard library only.

Limitations

One finite width only. No general rank formula, complete target, paid saving, new kappa or supplier admission. Original inputs, fresh-owner complement and intermediates remain retained as in #4797. Physical preparation, missing coordinates, routing/storage, terminal frames, dirty restoration and recursive supplier costs remain unpriced. Independent human/proof-kernel review, Lean, Docker and selected scientific cold replay are unrun.

The historical card control reaches KeyError: 'complex_profile' against the current selected schema; the current selected source-binding baseline passes. Neither is used as evidence for the new rank result.

Open next questions

• Find an analytic image/kernel factorization giving the general even-width rank. The norm formula and this boundary value alone do not provide it.
• Write and price a complete consumer contract that uses this image while recovering missing coordinates and restoring dirty state.

Related open work

• Construct a resonant partial target channel at every even width #4797 supplies the exact partial channel and frozen coefficient, and explicitly asks for its even-width rank. This contribution adds only the new width-four rank; it inherits the channel theorem and leaves its full construction obligations open.

• Prove sharp conditioning of the complete complex port (no new κ) #4828 studies the sharp norm, inverse norm and singular spectrum of complete dirty two-bank ports in the same Add reversible native ports and swap-sector completion for the complex prefix #4743/Construct a target-aligned phase repair of the block-3 two-call prefix (no new kappa) #4710/Construct an invertible two-call prefix tight on eight block-3 hook cuts #4688 lineage. It treats a complete port rather than the resonant partial coefficient A4; this certificate computes none of its conditioning values, so no numerical agreement or disagreement is asserted.

This account's open PRs created since the CLAIM at 2026-10-22 21:08:38 UTC were refreshed before DONE. The other new submissions concern Toeplitz cuts, projector tariffs, integer splitting or prime-field connectors and do not determine this frozen coefficient's rank.

CI notes

Both hosted checks passed on c396428eacde0de53e2fe29330308b4a87d3d99b in run 39855849435: plan in 3m7s and the registered package replay in 1m10s. The replay executes both warning-strict normal and optimized commands. Run metadata confirms the exact head and successful conclusion. Hosted CI certifies execution of these bounded checks; independent human and proof-kernel acceptance remain unclaimed.

Merge Danger

Door: two-way. Blast radius: isolated research package and its opt-in replay. The selected frontier and shared implementation are unchanged.