[petalas]

Summary

Prove the exact second generalized support weight of the specified P104 root code:

d_2(C over F3) = d_2(C over Q) = 72
704 minimizing ternary two-dimensional subcodes

This is a finite certificate with a scoped proof. Two independent channels need a union of at least 72 ports, with an explicit attaining pair. Combined with Matarener's cited PR6166 hyperplane capacity 60, a common acquisition frame must be FULL; a common landing frame for independent receiver columns satisfying the same code hypothesis must be ZERO. Asynchronous relays and larger codes remain open. No new kappa, physical supplier or selected-frontier change is claimed.

The contribution was produced by Nick Petalas. Credit Matarener PR6166 for the code identification, coordinate fixture and cited capacity; Ali Romman PR5827 and DreamingOfClouds PR5907/6030 for the retained P104 lineage, with their native/helper/controller/compiler predecessors. Generalized Hamming weights and the minor argument are standard.

Evidence

• Before: PR6166 supplies the single-channel support result. It does not compute this joint-support invariant.
• After: all 3,280 projective codewords and 5,377,560 distinct projective pairs are enumerated exactly. The minimum is 72. Grouping the 4,224 minimizing pairs gives 704 planes, each represented by exactly six pairs.
• The eight-row generator is systematic on the declared clique. Its 832 entries agree with separate explicit rational-root and intersection formulas. The rational witness has weights 49 and 71, its ternary reduction has weights 49 and 69, and both unions are the same 72-coordinate set. A written rank-reduction proof supplies the rational lower bound.
• Registered warning-strict normal and optimized checks, direct replay, metadata/pins, committed bytes/modes and source-history preflight pass. The two registered checks took 0.553 s and 0.548 s in individual local timed runs, below their 120 s limits. No runtime improvement is claimed.
• Official export integrity and both Git-free read-only replays pass. All nine exported files, including the export receipt, retain their bytes and modes.
• The generated RESULT.json binds all seven proof/input/metadata files and records the complete histogram and witness. Changed source bytes, zero/dependent projective inputs and incorrect receipts reject under optimized Python too.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/p104-two-channel-support/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/p104-two-channel-support/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-two-channel-support
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-two-channel-support --check-source-history

Expected markers are PASS_P104_TWO_CHANNEL_SUPPORT d2=72 pairs=5377560 minimizing_subcodes=704, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Any CPython 3.12 installation can replace the local path; no third-party dependencies are required.

Base and publication ancestry pin canonical main 3f0152a32eeebac0df11946dbd50cabe5a7f7839, priority revision 8. The external PR6166 coordinate source is pinned at 096938513138868f0ed27639061f1a5054900f96 by URL and byte hash. That open head is outside publication ancestry. The generic source-history pass covers the canonical catalogue and selected record, not the external head. The retained coordinate bytes make direct replay self-contained.

Limitations and unrun obligations

PR6166's uniqueness and capacity results are cited, not re-verified. The common-frame implications require the specified code reduction and genuinely common acquisition or landing frames. No cost for independently scheduled channels is inferred. Arbitrary nine-dimensional codes, overcomplete canceling carriers and public relays are outside the result. No native, complete caller, restoration, stock/precision, compiler, outer, selected-word or Lean replay is performed. This is not a complete multiplication theorem. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Can two asynchronously acquired/delivered channels escape the common-frame constraint with a favorable complete invoice? Start with explicit independent rows, chronological cuts and all frame/restoration charges. The actual physical caller is missing.
• What joint-support bound holds for a specified nine-dimensional decoding code outside the minimal-code reduction? First provide the code and a legal pair of delivered channels. No larger-code census is supplied here.

Related open work

• P104 carrier code: the length-8 decoding code is unique (E8 root code, min weight 49) #6166 identifies this ternary P104 code and supplies the cited single-channel/capacity results. This adds its second support weight and rational lift; it does not audit the predecessor.

• Conditional κ .003128386 (+2.582e-6 over #6030): owned registers 770, 771 and 772 descend through the frames of helpers 705, 706 and 731, which no longer relay them; helper 732 reads 771 at frame 117 #6144 changes owned-register relay descent on the P104 word. This changes no word or invoice and does not exclude its separately scheduled relays.

• Conditional κ = 0.003128645 (+2.59e-7 over #6144): retain an operand to remove a root stop #6167 retains an operand to remove a P104 root stop. This is a code-support obstruction under common-frame hypotheses, without a competing price computation.

• No new κ: exact multi-gate resynthesis removes 117 P104 events #6169 resynthesizes P104 scalar events while retaining the invoice. This leaves every scalar program unchanged.

• Add P104 designer toolkit v2: exact pricing and legality, #6030 rebuild with fold/hoist/retain, generators, sub-circuit tools (no kappa claim) #6170 supplies P104 design/pricing tools. This standalone exact support certificate invokes none of those tools or word optimizers.

• Prove an all-width tensor-frame reduction under fixed boundaries (no new κ) #6171 reduces tensor frames with fixed productive boundaries. This studies independent supports within a specified code and makes no tensor-moment optimality claim.

• Prove sharp signed tensor gap with variable target boundaries (no new κ) #6172 studies a signed tensor gap with variable target boundaries. This fixes the code space and computes no signed-moment optimum.

• Conditional κ 0.00309871 from a prepared P104 split helper #5606 prepares a split helper and changes its paid/target bill. This supplies no preparation word or invoice change.

• Two successor hand-offs on the P104 word: conditional κ 0.0030428242 #5553 changes two helper handoffs on an earlier P104 word. This does not change or audit those lifetimes.

• Add exact P104 ADD-coefficient fault localization (no new κ) #6175 adds exact ADD-coefficient fault localization on a P104 scalar word. This computes a fixed root code's support invariant and supplies no fault diagnosis or scalar edit.

• Realize unbounded bipartite rank metrics in four-dimensional shears (no κ) #6174 realizes bipartite rank-distance metrics in rational shear operators. This fixes the P104 code and supplies no new shear geometry or distortion value.

The completion refresh checked this account's open PRs created since the advance claim. The newly published scopes above contain no competing computation of this code's second support weight. No additional computations were run to check those contributions.

Live main README, merged-PR, open-PR and Discussion searches found no same-question result within their coverage. The overlap helper reports complete remote pagination and incomplete sparse local-note coverage; separate Git searches cover main's README inventory. Unpublished work and literature-wide novelty are not certified. Advance claim.

CI notes

Both hosted checks passed at exact head 6479ec01b2afb0056e9ee3bb5f053b7d5931bece in run 40879613312: plan and replay (research/p104-two-channel-support). The completed check list was read once; run metadata confirms success and the same head.

Hosted replay certifies the registered normal and optimized package checks at that head. All recorded local checks, committed-byte/source-history checks and read-only Git-free export checks also passed. The cited predecessor results, scoped frame hypotheses and unrun physical/full-supplier obligations retain their stated limits. Independent human and formal proof-kernel acceptance remain unclaimed.

Merge Danger

Door: two-way. Blast radius: isolated. Eight new files in one opt-in research package. No shared code, frozen predecessor, upstream archive or selected-result record changes.