[petalas]

Summary

A full-support merge of three fixed PR6030 carriers with source-normal retention and a shared tail of paid rank two costs at least 22 paid rank units. Separate delivery costs 21 by the cited predecessor. This is a scoped proof with a finite certificate, answering the rank-two-tail case of Matarener's PR6315 open question under the explicit retention hypothesis identified by CubeJerry's PR6323. No new kappa is claimed.

quotient demand >= 39 > 28 = capacity of two low-rank paths
at least two paths pay >= 7; the third pays >= 6
complete paid rank >= 7 + 7 + 6 + 2 = 22

The only changes are six files in research/p104-triple-merge-tail2/. The proof grants receiver access throughout each path union and permits arbitrary nonzero rational merge coefficients. No predecessor executable or scientific replay is used.

Evidence

• Before: the cited package leaves merges of three or more carriers with longer tails open. After: the explicit source-retaining rank-two-tail model has the proved lower bound 22. The bound is not claimed attainable.
• All 56 triples and 168 triple/pair quotient sets are enumerated. Their cardinalities have histogram {39: 72, 46: 72, 47: 24}. A separate local binary-elimination implementation agrees on all 168 counts.
• All 651 four-spaces of F2^6 and 423801 ordered pairs corroborate the abstract cover bound 28. Exact coset fixtures, strict-threshold checks, four invalid-input rejections, two invalid-geometry rejections and 10648 integer-cost tuples pass.
• Local CPython 3.12.15 normal and optimized direct runs pass in 0.05 and 0.06 seconds, each below the registered 120-second limit. Both registered commands and source metadata pass. The generated normal and optimized receipts agree and bind package/input hashes.
• Exact committed bytes, executable modes and declared source ancestry pass at 1cdfc71b94a2a5efbffac6479f1e712651ad9297. Official export integrity and all four declared inputs pass. Normal and optimized read-only, Git-free export replays preserve all eleven exported files, including the export receipt. Eight actual input-corruption executions reject.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/p104-triple-merge-tail2/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/p104-triple-merge-tail2/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-triple-merge-tail2
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-triple-merge-tail2 --check-source-history --check-modes

On another machine, substitute its CPython 3.12 interpreter. The standard library is sufficient. Expected markers include PASS_P104_TRIPLE_MERGE_TAIL2, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Optional --output /absolute/new/external/receipt.json generates a hash-bound receipt outside the checkout.

Limitations

The fixed carrier code, independent contents, decoder zero supports and walk bound are cited premises. All three source normals must lie in the merge dual; this is not inferred from a common frame or from public-relay rules. Receivers use pure pre-merge contents and one full-support merged scalar. Extra mixtures, feedback and arbitrary dirty-state cancellation are outside the model.

This is a paid-rank statement. Rank mass does not order concave power tariffs, so no recursive-moment, supplier-admission or kappa exclusion follows. Longer tails, merge trees, nine-or-more-carrier codes, physical purity, dirty restoration, chronological realization and complete signed billing remain open. No selected cold replay or Lean build was run. The five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Can a source-normal-retaining triple merge with tail rank at least three save paid rank? First derive its path dimension limits and quotient cover constraints, then price an explicit decoder.
• Can a pure arbitrary-input physical relay relax source-normal retention without repaying the gain in dirty restoration? Specify one complete input/output and restoration contract before using its frame walk.

Related open work

No open PR found in the refreshed search directly targets this fixed-code triple-merge model. Adjacent work:

• Conditional κ = 0.00315120 (+3.3e-7 over #6372): three shared-carrier input lifts #6375 changes P104 helper inputs through shared-carrier lifts on a successor word; those transports and its conditional kappa are not covered by this fixed-code rank bound.

• Conditional κ = 0.00315087 (+3.0e-7 over #6353) #6372 contracts two helper outputs with joint boundary bases; it changes the word rather than using this single-merge model.

• Contract nine native helper outputs at conditional κ 0.00315057 #6353 contracts nine native helper outputs; its decoder and paid/target invoice are separate from the present rank-two-tail obstruction.

• Certify PR6375 matched profiles and unique relative-tariff threshold #6378 studies the complete formal power-tariff comparison for Conditional κ = 0.00315120 (+3.3e-7 over #6372): three shared-carrier input lifts #6375; the present result makes no power-tariff claim and computes no common numerical instance.

• Prove exact tariffs for carriers visiting distinct entrance lines (no κ) #6388 gives exact tariffs for one carrier visiting several entrance lines; it does not study three pure carriers merging into one rank-two tail. No common numerical instance is computed.

• Prove a fixed-schedule obstruction for PR6345 helper respanning #6381 excludes fixed-schedule stop deletions after helper respanning on a different word; its contact obstruction does not cover this changed merge architecture.

• Reduce the concrete nonunit handoff allowance to 232 (no κ) #6392 reduces scalar completion allowance for a historical nonunit handoff while retaining recursive profiles; this result instead bounds paid frame rank in a fixed-code merge model.

• Prove the exact nineteen-leaf query-tree topology optimum (no κ) #6380 optimizes a fixed nineteen-leaf query-tree power invoice; this result concerns a local carrier transport model and makes no power-tariff comparison.

• Construct transported-query substitution for noncommuting coded jobs (no κ) #6382 constructs a noncommuting coded-query merge with paid complements; that operator substitution is outside this pure-scalar single-merge model.

• Prove exact signed idle completion for overlapping stage outputs (no κ) #6391 proves signed-idle completion for overlapping stage demands; its caller inventory differs from the fixed PR6030 carrier code used here.

Merged predecessors #6315 and #6323 retain their distinct results and attribution. No additional computation was run to audit another contribution.

CI notes

Both hosted checks, plan and replay (research/p104-triple-merge-tail2), passed at exact head 1cdfc71b94a2a5efbffac6479f1e712651ad9297 in run 41511559755. The completed checks were read once with gh pr checks 6386 --repo CrocSwap/integer-mult-bounds; run metadata separately confirms successful completion at the matching head. Local registered checks also passed. Hosted replay certifies the registered finite certificate, controls and source bindings; the universal scoped statement also uses the written proof and cited premises. Independent scientific or formal-kernel acceptance is not implied.

The completion refresh listed fourteen open account PRs created since the CLAIM, including this one. Related open work now includes the adjacent new carrier/P104 contracts above. No conflicting computation of this same merge bound was identified, and no additional computations audited another contribution.

Credit

This result was produced by Nick Petalas. Credit Matarener for the question and carrier model, CubeJerry for the corrected source-normal premise, DreamingOfClouds for PR6030, and the fragmentation, owned-boundary and carrier-uniqueness lineage retained in those packages. Apache-2.0.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Only a new opt-in research package is added. Shared code, frozen predecessors and the selected frontier are unchanged.