[petalas]

Summary

Finite exact certificate with a scoped rewrite proof: the selected PR6030 word needs one fewer logical scalar ADD, with every native transition, owned lifetime and boundary retained.

- event 231: 210 += 403/4
+ event 231: 210 += 403/2
  ... 160 commuting scalar events ...
- event 392: 210 += 403/4

The word has 22,802 events instead of 22,803, and 18,403 ADDs instead of 18,404. The recursive paid and productive profiles are unchanged. No new κ, selected-frontier change or complete multiplication theorem is claimed. The first improvement ended the construction search.

Only research/selected-p104-scalar-fusion/ is added. Base: canonical main 2485805a6d47f36ea83ddde6327d453ac82d44dc. Advance claim.

Evidence

• Before: both retained source events add one quarter of role 403 into role 210.
• After: one half-coefficient update has exactly the same complete 133-role region operator. All 17,689 rational entries agree; separate scalar execution passes all 133 basis columns and eight dirty vectors. All 160 intervening commutation guards pass.
• Eleven unsafe-gate, wrong-coefficient and source-corruption controls reject. Warning-strict normal and optimized direct and registered replays pass. Each registered command was timed locally below one second.
• Committed bytes, executable modes and complete declared source ancestry pass. The official export passes integrity and declared-input coverage. Both normal and optimized replays pass from a read-only, Git-free export with empty PATH; all seven exported files, including the export receipt, retain their bytes and modes. Two actual exported-source corruption executions reject.
• The complete emitted word agrees with generated SHA256 03788d8092813118feaad115e26521f11c888bb1c4346652bc57b1092a98da38. Every receipt is generated and source-bound.

Local environment: CPython 3.12.15, Darwin 27.0.0, arm64; standard library only.

From the repository root, using CPython 3.12 and its standard library:

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

Expected markers are PASS_SELECTED_P104_SCALAR_FUSION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The verifier writes nothing; --emit-word prints the complete verified word JSON for redirection to an external scratch path. Any CPython 3.12 installation can replace the local interpreter path.

Limitations

This is a scalar-event improvement. Rounded intermediate errors, physical runtime, the full native/caller/BIT/outer suites, Lean and Docker were not run. The unchanged native stream and exact region identity preserve the predecessor's endpoint conditional on its existing correctness; this is not a new audit of that predecessor. Native all-column implementation, owned erasure, compiler closure, precision/tape and analytic recovery remain inherited assumptions. Independent human and formal proof-kernel review are unclaimed.

The helper-split card is explicitly changed to this bounded scalar side result; its invoice-improvement gate remains unmet. No larger search or optimality claim is made.

Related open work

These P104-family scopes were read for overlap; no extra computation checked their reported numbers.

• Draft construction: P104 dirty-restored gather and low-rank channel with COPY56 (no new kappa) #1972 constructs the historical dirty-restored channel; this edit uses the later selected PR6030 word.
• Tooling: 3.09× faster production frame-pair enumeration via eight-bit neighborhoods with identical receipts #2375 changes frame-pair enumeration tooling; this contribution changes a scalar word rather than its enumeration.
• F-star residue class (ii): non-3-dichromatic zero digraphs - matched-pivot floor, fixture dichromatic structure, smallest-first falsifier search (no kappa) #2774 studies root-decoder zero-digraph obstructions; no decoder lower bound changes here.
• 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 reduces root-decoder floor questions; this exact word edit supplies no new floor.
• 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-pairing root-code floors; this contribution keeps the decoder and native stream.
• research: price full P104 end-cut correction #3894 constructs a historical full-frame correction suffix; this edit is inside the selected word's initial scalar region.
• Research P104 singular-mixing lifetime floor #3907 reviews historical source-lifetime floors; this edit preserves all lifetimes.
• Add P104 flag 0 coefficient transport audit #3915 audits a historical channel coefficient; this contribution constructs an exact selected-word replacement.
• Two successor hand-offs on the P104 word: conditional κ 0.0030428242 #5553 changes productive helper handoffs in an older P104 word; this edit changes no productive target.
• Conditional κ 0.00309871 from a prepared P104 split helper #5606 replaces a prepared helper's queries in an older P104 invoice; this edit changes no query or recursive profile.
• Keep selected invoice stdout bytes consistent with its receipt #5973 repairs selected-invoice stdout binding; this package adds no exporter or invoice changes.
• Construct an exact dyadic rank-seven P104 projector (no new kappa) #6247 constructs a dyadic rank-seven Gram approximation; this edit retains the exact selected native operators.
• Prove the P104 reader-497 two-cache cost floor (no new kappa) #6248 proves a reader-497 cache floor on PR6167; this edit is on PR6030 and changes no cache or transport.
• Independently audit PR6167 numerical compiler premises #6249 audits PR6167 compiler arithmetic; this scalar certificate changes no compiler recipe.
• Prove the exact reader-497 single-merge cache floor (no κ) #6257 proves the reader-497 single-merge cache floor on PR6167; this edit is on selected PR6030, preserves transports and does not compute its cache tariff. It separately identifies this same one-ADD reduction without a conflicting scalar count.

Merged PR6169's distinct common-response certificate on PR6144 changes neither of the two 210 <- 403 events retained by its input. Its theorem and verifier were not replayed as this contribution. PR5828 and older commuting-cancellation work are methodological predecessors. Sparse local overlap coverage is incomplete; main and live searches supplement it without guaranteeing global novelty.

Open next questions

Can a scalar rewrite remove an actual paid native transition or legally change a productive target? First give one topology and its full signed invoice; the current profile supplies no κ gain and does not justify another construction round.

What rounded-error and precision costs does this fusion have under an actual caller? First compare both complete region programs within that caller. Exact endpoints alone do not settle the rounded comparison. Neither follow-on is reserved.

CI notes

Both hosted checks, plan and replay (research/selected-p104-scalar-fusion), passed at exact head 892844b6bdb9302bc23a9dbf10ac6ddca35abf2a in run 41048194031. The checks were read once after completion; the run is completed/success and its head SHA matches the publication. All registered checks also passed locally. Hosted replay certifies the registered finite checks; independent human and formal proof-kernel review remain unclaimed.

Merge Danger

Door: two-way. This is an isolated additive package.

Blast Radius: research. Selected records, predecessor credit and frozen inputs are unchanged.

Credit DreamingOfClouds for PR6030/5907, Ali Romman for PR5827/6169/5828 and the retained P104/native/helper/controller/caller/compiler/BIT lineage. Elementary-matrix identities are classical. The literal selected-source reduction and its certificate were produced by Nick Petalas. Apache-2.0.