[petalas]

Summary

Resolve the maximality question left open in Matarener's #6217. The pinned 104 P104 ports admit exactly 33 disjoint A2 triangles, so at least five partners need another scheme in that routing architecture. Claim type: scoped written proof with a finite attainment certificate. The original contribution is the parity obstruction to 34; the source's reported 33-triangle construction retains its credit.

- 33 triangles reported; maximality open
+ 33 triangles optimal; minimum uncovered port count is 5

For integral scaled roots, 3r(q) mod 2 = 511 XOR q. Every A2 triangle consequently has label XOR 511. All 104 labels XOR to zero. Thirty-four triangles would leave two distinct labels whose XOR is zero, a contradiction. A separately found, solver-free-checkable 33-triangle certificate attains the bound. No predecessor triangle catalogue or routing result was audited.

Only research/p104-a2-packing-maximum/ is added. The contribution was produced by Nick Petalas. Credit Matarener's #6217 question, #6166 port/root interpretation, Ali Romman, DreamingOfClouds and the retained P104/native/caller/compiler/BIT lineage; binary parity and root-system ingredients are classical.

Evidence

The written upper-bound proof covers every A2 triangle without a solver verdict or catalogue enumeration. Local standard-library checks pass on CPython 3.12.15, macOS 27.0.1 arm64:

• All 104 root/label parity identities and the source's total label XOR; all 33 certificate triangles and 99 distinct covered ports.
• All 487,200 ordered disjoint-triple pairs in zero-XOR eight-subsets of four-bit labels; fifteen rejecting geometry, overlap, type, index, leave and parity-premise controls.
• Warning-strict normal and optimized direct and registered replays. The two registered commands take 0.123 and 0.124 seconds locally, each with a 30-second timeout.
• Committed-byte/mode and full source-history preflight at f9f096be89878e4fcbf75c4be98bc9b52f0e6c56 passes for seven package/input files. Canonical main was refreshed before committing and remains a5bff3bf133e482541c09eba4e773884bdb2b1b7.
• Official committed export and declared-input coverage pass. Normal and optimized replays pass in a read-only export with empty PATH and no Git. All eight export-file hashes and modes remain unchanged. Receipts are generated from source hashes, not handwritten.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/p104-a2-packing-maximum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/p104-a2-packing-maximum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/research_package.py --package research/p104-a2-packing-maximum
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/check_research_commit.py --package research/p104-a2-packing-maximum --check-source-history

Elsewhere substitute CPython 3.12 for the absolute interpreter path. Expected markers are PASS_P104_A2_PACKING_MAXIMUM, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations

This is a packing optimum for this exact port set and disjoint A2 triples. It supplies no physical wire/helper program, dirty restoration, complete signed invoice, supplier admission, new kappa or complete multiplication theorem. Overlapping and multistage routing remain outside the obstruction. The selected frontier and all five framework assumptions are unchanged. Independent human and formal proof-kernel review are unclaimed. No selected scientific cold replay, Lean, Docker or predecessor theorem audit was run.

The advance claim asked whether 34 triangles exist. The progress update records the exact negative resolution and the change from a possible larger construction to an optimality proof.

Open next questions

• Construct a legal wire-plus-helper program for one certified A2 triangle and price every entry, exit, sign correction and restoration. A packing is not an implementation.
• Specify an overlapping or multistage contract for the five exceptional partners and its complete cost allowance. No favorable bill is supplied here.

Related open work

• Prove sharp 68-point E8 plane unions and isolate the P104 puncture step (no κ) #6221 proves a full-E8 two-plane union bound and isolates a puncture-specific P104 step. It shares the binary root-label setting but computes a different geometric quantity; it cites this packing optimum without extending or challenging it.
• Construct optimal 39/8-bit P104 labels with length-preserving updates #6224 constructs prefix codes for rank-two P104 signatures. It does not study disjoint A2 triangles or the exceptional-port minimum.
• Prove unbounded activation arity for parallel helper cuts (no κ) #6222 studies algebraic activation across parallel helper banks. It supplies no A2 packing result or routing construction on the fixed 104 ports.
• Exclude arbitrary-plane temporal rails on the fixed PR1094 word (no κ) #6223 studies temporal-rail closure on the historical PR1094 word. Its injected-plane accounting quantity differs from this root-triangle packing problem.
• Prove the exact fixed-cubic coframe rank frontier (no new κ) #6220 studies fixed cubic Clifford coframes and complete boundary rank costs. It supplies no competing P104 root-packing optimum.
• research: P104 wire routing obstructions (scoped negative, no κ) #6217 poses the exact maximality question and reports 33 triangles. This result proves 33 is maximal and supplies a separate attaining witness. It strengthens the necessary number of exceptional partners to five, without auditing its routing lemmas.
• research: P104 B-form floor follow-up: RootFact/SpanFact proved (hand + Lean), Cases II/III without the census (no κ claim) #6218 studies B-form decoder support/span facts and compatible label planes on P104. It does not determine this disjoint A2-triangle packing optimum; its finite and formal obligations are separate.
• Prove E8 span inputs over every odd-characteristic field (no new κ) #6198 proves odd-characteristic E8 span facts on the full root-line system. This contribution uses the fixed 104-port subset and a binary parity obstruction, without extending or auditing those span facts.
• Exhibit complex rank-seven P104 approximations below the real 1/4 floor (no κ) #6211 studies complex rank-seven approximation of the P104 Gram matrix. This packing result supplies no approximation bound or correction program.
• Prove sharp 1/4 error for every real rank-seven P104 approximation (no κ) #6191 studies the real rank-seven P104 approximation error floor. Its approximation quantity is separate from disjoint-triangle packing.

Live open PRs and handoffs, main key-quantity/general-form searches, merged PR searches, recent account work and local reservations were refreshed before committing. The overlap helper covered 8,623 main notes and 337 open PRs; its Discussion refresh timed out, so separate live GraphQL reads covered the latest handoffs and active Discussions. This is bounded coverage, not proof of global novelty. No additional computations checked another PR's numbers.

CI notes

Both hosted checks, plan and replay (research/p104-a2-packing-maximum), passed at exact head f9f096be89878e4fcbf75c4be98bc9b52f0e6c56 in run 40963748313. The completed check list was read once, and the run's head and successful conclusion were confirmed. Both registered checks also passed locally; no required local command was left unrun.

The completion refresh inspected all five other open contribution PRs created after the advance claim, #6220 through #6224, and added their scope relationships above. None supplies a competing value for the disjoint A2-triangle packing maximum. No same-instance numerical disagreement was identified and no extra computations checked another contribution.

Merge Danger

Two-way door. The change adds one isolated research package. Shared code, frozen evidence and the selected frontier are unchanged.