[petalas]

Summary

Prove the sharp owned paid floor eight for a P104 block consisting of two mutually orthogonal four-port cliques, even with arbitrary finite acyclic REL chains, branching, merging, cancellation and public re-ascents, provided every gathering and landing frame stays inside the block span. Four-move and eight-move relayed constructions attain the floor.

Claim type: scoped written proof with exact finite corroboration. No new kappa, physical supplier or selected-frontier change.

local two-dimensional quotient
  -> rational subspace representation
  -> cumulative source-row space
  -> rank(C Hadamard-product Gamma) <= complete owned paid
  -> identity on eight independent coordinates requires paid >= 8

This answers the confined-chain part of DreamingOfClouds PR6170, P104-LEMMAS.md section 11.5. Its isolated-move floor is a cited predecessor, not an audit deliverable. The new invariant handles shared cost: an individual relayed term can have shadow rank two while its own frame drop is only one.

Only research/p104-two-clique-relay-floor/ is added. This contribution was produced by Nick Petalas. Credit DreamingOfClouds for the question and REL contract, Ali Romman PR5599 for the retained port definition, Matarener and the retained P104/native/helper/controller/caller lineage. The linear-algebra ingredients retain their classical status.

Evidence

Before: the source leaves chains unresolved. After: the written proof covers every finite confined relay network, including merged parent content. The exact eight-port fixture uses (73,239,336,502) and (81,247,328,494).

• All 67 frames and 513 nested pairs; 128 local-source and 128 receiver-functional checks.
• Two exact attaining networks. Their complete source endpoint has rank four, while its two diagonal class blocks form the eight-dimensional identity. Both pay eight.
• 512 deterministic sixteen-move networks, including 18,885 nonzero parent edges, 4,429 branching parents and 4,407 multiple-parent moves. Literal coordinates agree with the packet calculation.
• A separate minor oracle on all 512 binary 3-by-3 matrices; five invalid networks reject; a concrete chain refutes independent per-move rank billing.
• Warning-strict normal and optimized direct checks and both registered commands pass locally. Direct final-byte exported replay timings were 3.960 s and 4.066 s; each registered check is bounded at 120 s. These are limit checks, not performance claims.
• Exact committed bytes/modes and declared source-history ancestry pass at 50540b2a7e0d219bc4920e6252e75113b5a9d8d8. Official export integrity and declared-input coverage pass. Normal and optimized read-only Git-free export replays pass; all nine exported files, including the export receipt, retain their bytes and modes.

From the worktree root, CPython 3.12 and the standard library:

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

Expected scientific marker: PASS_P104_TWO_CLIQUE_RELAY_FLOOR. Receipts are generated and bound to package-file hashes. Another CPython 3.12 path can replace the local path. The only live external input is read for its port definition; predecessor programs, invoices and census results are not replayed.

Limitations and unrun obligations

All gather and landing frames must lie in the block span. Same-class pairing functionals can differ outside that span, so the proof does not exclude outside-reaching constructions or the full E8 mechanism. Arbitrary rational coefficients and free public re-ascents make this an optimistic abstract REL relaxation. No physical word, full productive/paid invoice, native realization, restoration schedule, compiler/precision/tape/analytic admission or multiplication theorem is supplied. The five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed. No selected cold replay, Lean, Docker or historical scientific suite was run.

The finite relay fixtures corroborate the proof; they are not an exhaustive network census. The reference checker fixes eight ports and at most sixteen moves. Its Python allocation deviation and explicit bounds are documented in the README. No registered local obligation is left for CI alone.

Related open work

No open PR found studies this exact confined two-clique REL paid optimum. Nearby P104 work has different contracts:

• research: P104 wire routing obstructions (scoped negative, no κ) #6217 studies live-wire routing and an A2 cover obstruction; this proves a cost floor for confined two-clique owned networks.
• research: P104 B-form floor follow-up: RootFact/SpanFact proved (hand + Lean), Cases II/III without the census (no κ claim) #6218 studies rank-nine B-form decoder floors and compatible label planes; this bounds chronological relay cost with arbitrary fitting cross-class blocks.
• Independent audit of #6144 and #6167: replay, deltas, invoice, full dense native, caller and funding all pass (κ 0.003128645 corroborated, no new κ) #6214 audits the complete 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/Conditional κ = 0.003128645 (+2.59e-7 over #6144): retain an operand to remove a root stop #6167 candidate words; no such word or kappa is replayed here.
• Exhibit complex rank-seven P104 approximations below the real 1/4 floor (no κ) #6211 studies complex approximate rank-seven P104 matrices; this theorem requires an exact rational pattern.
• Prove sharp 1/4 error for every real rank-seven P104 approximation (no κ) #6191 studies real approximate rank-seven P104 matrices; no approximation error is computed here.
• Conditional κ 0.00309871 from a prepared P104 split helper #5606 changes an owned prefix helper and productive credit in a complete P104 word; this supplies an abstract local obstruction, with no new word.
• Two successor hand-offs on the P104 word: conditional κ 0.0030428242 #5553 changes successor helper handoffs; this permits general confined handoffs but does not price its complete word.
• Research P104 singular-mixing lifetime floor #3907 concerns a frozen full-matrix source-lifetime floor; the present quantity is owned descent cost on one eight-port block.
• research: price full P104 end-cut correction #3894 constructs a full-frame end-cut correction; the present attainers are abstract REL maps.
• Add P104 flag 0 coefficient transport audit #3915 follows one native flag coefficient; no native coefficient is audited here.
• Draft construction: P104 dirty-restored gather and low-rank channel with COPY56 (no new kappa) #1972 constructs the full P104 dirty-restored low-rank channel; this does not constrain its whole-bank frames or claim the same paid quantity.
• Non-dense fitting B: every E8 term with x·y ≠ 0 has rank ≥ 9; square root-code floors E8 ≥ 72, P112 ≥ 64, P104 ≥ 56, D4 ≥ 12 for any fitting B (no κ claim) #3045 studies static fitting-matrix decoder floors; this bound charges complete chronological frame descent.
• Decoder floors: universality, projection transfer and batching reductions; C(P112) ≥ 64 reduced to named conjectures, C(P120) ≥ 72 restated on N₀ (no κ claim) #2832 gives static decoder universality and reductions; the cumulative relay invariant adds a different temporal constraint.
• Exclude birth-lowered P104 markets without paid deletions (no κ) #6197 bounds market-birth preparation with retained paid moves; this instead permits arbitrary confined REL resynthesis.
• Prove the P104 disjoint A2 packing maximum is 33 (no κ) #6219 determines disjoint A2 packing; this computes no triangle count or packing.
• Prove sharp 68-point E8 plane unions and isolate the P104 puncture step (no κ) #6221 proves a quadratic-plane union bound; this computes no plane-union optimum.
• Construct optimal 39/8-bit P104 labels with length-preserving updates #6224 constructs length-preserving variable-length P104 labels; this computes no label-storage tariff.

No competing numerical claim about this exact paid-eight optimum was identified. No extra predecessor computations were run for comparison.

Open next questions

Can an outside-reaching REL network lower the eight-unit bill? Start with one explicit gather/landing graph and exact source and receiving maps, then test the rational-lift reader condition. The blocker is loss of classwise functional agreement outside the block span. Any physical successor must also pay public helper lifetimes, restoration, native access and the complete caller costs.

CI notes

CI pending at head 50540b2a7e0d219bc4920e6252e75113b5a9d8d8. All registered checks ran locally. Hosted CI is a separate pending step and no hosted pass is claimed.

Merge Danger

Door: two-way.

Blast Radius: isolated. Seven new package files, one unchanged declared input, no shared checker or selected-frontier edits.