[petalas]

Summary

Scoped written proof with exact finite corroboration: merging reader 497's two operands once cannot improve the PR6167 schedule in the stated monotone-cache relaxation. This answers one mixed-source case explicitly left open by CubeJerry's #6248. No new κ.

V at S -> descending cache --+
                            merge at M -> sum at A -> ascending reader -> R
B at U -> descending cache --+                    root H -> F -> L

Every cache MOVE is charged. Capture, waiting, clean storage, the merge and disposal are optimistically free. For H_r=r^t, 0<t<1, the exact minimum is

min(H1 + 2H3 + 2H6, 2H1 + 3H3 + H5)

The unchanged bill is 2H1+2H3+H5. The exact positive excess is min(2H6-H1-H5,H3). The written proof gives the corresponding exact two-profile formula for every field of frame coordinates, every 1<=s<u, every permitted flag and every finite monotone path, under any nonnegative increasing concave tariff. Native availability is granted in this cost relaxation, not proved.

Only research/p104-single-merge-cache-floor/ is added. Base: canonical main 2485805a6d47f36ea83ddde6327d453ac82d44dc, refreshed before commit without relevant changes. Advance claim.

Evidence

Before: #6248 covers separate descending caches and leaves mixed operands outside its model. After: the single-merge class has an exact optimum with two attaining flag recipes, both strictly worse than the unchanged schedule for every power in (0,1).

• All 50 reader-497 dimension recipes have explicit binary flag witnesses. Independent geometric rank extraction agrees with the invoice, and all six capped sums dominate the appropriate endpoint profile.
• The same exact checks cover 20,839 recipes over 91 dimension pairs, 2<=u<=14, 1<=s<u. These finite cases corroborate the written all-dimension proof.
• Four independent public basis columns and 81 rational dirty examples check the relaxed scalar payload. Thirteen controls test incidence, unpaid cache transport, operand/root loss, incoming dirt and cleanup. They are not native operator checks.
• Warning-strict isolated normal and optimized direct checks pass, each timed at 0.28 seconds. Both complete registered checks pass. Generated normal/optimized receipts agree and bind all seven package files and the complete 50-recipe certificate.
• PASS_RESEARCH_COMMIT_BYTES at 06fc8ef80235f02c67295e474d7766f843f1eaa0: all seven files, executable modes and declared source ancestry pass.
• Official committed export integrity and declared-input coverage pass. Both modes pass with empty PATH from the read-only Git-free export. All eight exported files, including the export manifest, retain their exact bytes and read-only modes; complete receipts agree.

Local environment: CPython 3.12.15, macOS 27.0.1 arm64; standard library only. From the repository root:

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

Expected markers are PASS_SINGLE_MERGE_CACHE_FLOOR, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Any CPython 3.12 interpreter can replace the displayed local path. Optional verify.py --receipt /absolute/path/outside/checkout/receipt.json generates the hash-bound certificate; no receipt is hand-written.

Limitations and unrun obligations

This is an exact tariff theorem for one merge and one sum delivery, with descending owned caches, an ascending reader and an uncloned descending root. Extra reads, repeated merges/splits, ascending public carriers, changed productive targets and coordinated deletions are outside scope. Disposal is deliberately free even on nonzero cache values; attainment is only in that favorable relaxation. Additional nonnegative physical costs cannot improve its loss.

PR6248's source geometry and its identification with PR6167 are cited premises, not independently audited. No predecessor verifier or theorem was replayed. Native operators/phases, actual physical delivery or erasure, caller placement, compiler closure, rounding, tape and analytic recovery are not proved here. No selected cold replay, Lean, Docker, supplier admission or full multiplication theorem is claimed. The selected frontier and its five framework assumptions are unchanged. Independent human and formal proof-kernel review are unclaimed.

Related open work

The completion refresh covers all 330 open PRs and all 13 other open PRs from this account created since the advance CLAIM. These scopes are related by the same reader, source word or P104 family. No extra computation checked their reported numbers.

• Prove correlated dirt is necessary at P104 wire meetings (no κ) #6261 proves a dirty-entry condition at P104 wire meetings; this keeps the independent operand-capture contract and computes a monotone cache tariff, without a shared dirt-entry census.
• Prove linear drift in a rounded native identity cycle (no κ) #6258 proves drift in a rounded native identity cycle; this exact abstract rank-tariff proof computes no rounded orbit or error bound.
• Prove the P104 reader-497 two-cache cost floor (no new kappa) #6248 supplies the same reader's separate-cache floor and the mixed-source question; this adds the single-merge representation and its exact tariff. Its source geometry remains an inherited premise.
• Independently audit PR6167 numerical compiler premises #6249 audits numerical compiler premises on PR6167; this proves a local mixed-cache cost obstruction and does not evaluate its compiler quantities.
• Remove one scalar ADD from the selected P104 word (no κ) #6251 removes one scalar ADD from selected PR6030 with unchanged native profiles; this package concerns PR6167 cache transport and computes no scalar ADD reduction or event count.
• Construct an exact dyadic rank-seven P104 projector (no new kappa) #6247 constructs a dyadic rank-seven P104 Gram approximation; this keeps the fixed frame geometry and exact source contract.
• Conditional κ 0.00309871 from a prepared P104 split helper #5606 proposes a prepared split helper in an older P104 word; this changes no productive helper target.
• Two successor hand-offs on the P104 word: conditional κ 0.0030428242 #5553 proposes successor handoffs in an older P104 word; this keeps the source and consumer fixed in the PR6167 geometry.
• Isolate explicit historical receipt adapters from P104 research #4103 concerns historical receipt adapters; this is a self-contained tariff proof without a predecessor runtime dependency.
• Add P104 flag 0 coefficient transport audit #3915 audits a historical P104 channel coefficient; this introduces no coefficient-transport construction.
• Research P104 singular-mixing lifetime floor #3907 studies historical source-lifetime floors; this prices explicitly merged descending operand caches.
• research: price full P104 end-cut correction #3894 prices a historical full-frame correction suffix; this introduces no correction suffix.
• 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 fitting root-code floors; this proves no new root-code bound.
• Draft construction: P104 dirty-restored gather and low-rank channel with COPY56 (no new kappa) #1972 proposes an earlier dirty-restored channel; this supplies no new channel or COPY tariff.

Open next questions

Can a cache representation with splitting and repeated merging change the fully charged signed bill? Start with one chronological diagram specifying source capture, every MOVE, all merges/splits, delivery and physical erasure. The missing obligation is a legal complete consumer/cost contract. This result does not reserve or implement that follow-on.

CI notes

Both hosted checks, plan and replay (research/p104-single-merge-cache-floor), passed at exact head 06fc8ef80235f02c67295e474d7766f843f1eaa0 in run 41048665067. The checks were read once after completion, and the run metadata confirms completed/success with the same head SHA. Both registered finite checks also passed locally. Hosted replay ran those checks successfully; the universal tariff theorem rests on the written proof. Independent human and formal proof-kernel review remain unclaimed.

Merge danger

Door: two-way. Blast radius: isolated research package. No shared code, predecessor package, frozen source or selected record changes.

The new single-merge theorem, exact verifier and write-up were produced by Nick Petalas. Jerry Chen / CubeJerry retains #6248's separate-cache theorem, question and source geometry at 9aaf76f43493cd2392eaae248274c651f29bc28e. Ali Romman retains #6167 at 564a076b113621a93edc4e39968ee4e4a8c9caf0; DreamingOfClouds retains #6144/#6030/#5907/#6170; Matarener retains #6214. Earlier P104/native/caller/compiler/BIT contributors retain their source credit. Apache-2.0.