[petalas]

Summary

Intersection counters reduce the complete current E7 classification replay from median 0.370508 s to 0.290648 s, a 21.55% local reduction against Ali Romman's PR5729 engine. All ten alternating fresh-process outputs are byte-identical, with zero errors.

Claim type: measured verifier improvement with a scoped equivalence proof and exact finite conformance. No new support theorem, kappa or selected-frontier change.

fixed lane weights, computed once
  + offset weight
  - twice the lane/offset intersection weight
  -> exact translated weights
  -> unchanged complete classification checks

The implementation and proof were produced by Nick Petalas. Credit Ali Romman / PR5729 for the lane representation, reused population/threshold helpers and complete consumer; PR5669 for the classification; and Nick Petalas / PR5238 for retained geometry. The weight identity and Boolean arithmetic are classical. Predecessor scientific results are cited, not audited as this deliverable. Frozen source files remain unchanged.

Evidence

• Before: the current optimized engine counts every translated coordinate column per offset. After: the adapter reuses fixed weights and counts only the intersection columns, then performs exact modular bit-plane arithmetic. The written proof covers wraparound, all assignments and iterator order.
• Five alternating fresh processes per engine. Original range 0.366887..0.372666 s; intersection range 0.287772..0.293084 s. BENCHMARK.json is a generated receipt bound to implementation/source-manifest hashes. CPython 3.12.15, macOS arm64, eighteen logical CPUs. Shared-host timings are descriptive, not a cross-host guarantee.
• All 34,525 small-code comparisons on 1,381 independent generator sets, 108 wide-code cases, 87,381 scalar word pairs, sixteen rejected inputs and three changed-input histories pass. Both normal and optimized registered commands execute both complete fresh scientific callers and compare exact output. Every original classification check and complete RESULT.json comparison still runs.
• Local registered-command times were 1.237 s normal and 1.244 s optimized, below the 120-second command caps. Metadata, sixteen committed package/input files and source ancestry pass. Official export/input coverage, both Git-free modes, unchanged hashes of all seventeen exported files and actual changed-live-source rejection pass.

Reproduction

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/e7-intersection-counters/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/e7-intersection-counters/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/e7-intersection-counters
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/e7-intersection-counters --check-source-history

A different CPython 3.12 executable can replace that absolute path. Direct marker: PASS_E7_INTERSECTION_COUNTERS. Standard library only. The opt-in complete caller is research/e7-intersection-counters/replay.py, with --engine original selecting the comparator. benchmark.py --output /tmp/e7-intersection-benchmark.json regenerates ten complete timing samples. Timing includes startup, source authentication/imports, all enumeration, downstream checks and output; the extra conformance suite is outside the timing comparison.

Limitations

The adapter has explicit capacities of width 0..127, at most 27 independent generators, batch sizes 1..20 and at most 65,536 high-offset batches. The complete E7 domain fits these limits; no scientific coverage is dropped. This is a private, standalone consumer substitution. Default production callers, concurrent shared-process use and other code installations remain separate.

No physical/native supplier, complete cost invoice, compiler, precision/tape, BIT or outer admission is supplied. Lean, Docker and the selected cold replay were not run. The five selected framework assumptions retain their status. Independent human and formal proof-kernel review are unclaimed.

Open next questions

Can a named production caller adopt the new engine with all affected source pins regenerated? Begin with one explicit installation and source-bound compatibility check. This package supplies the opt-in full comparison; production integration is outside its scope. No further optimization is reserved.

Related open work

• No new κ: covers of E7 by hyperplane flats are exactly the D4-frame triples; near-covers are the A3s (checked) #897 classifies flat covers of the same E7 root-line geometry. This package accelerates a separate minimum-support classification and does not recompute its cover counts.
• 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-rank floors and E7 counterexamples in root-code decoders. This package changes counting arithmetic in the retained classifier and does not re-evaluate those bounds.

The completion refresh listed all sixteen open account PRs created since the CLAIM, including this PR. None of the other fifteen studies this E7 iterator or its complete timing quantity. The refreshed E7 search retains #897 and #3045 above. No shared-number disagreement was identified within that coverage, and no additional computations compared predecessor numbers.

CI notes

Both hosted checks, plan and replay (research/e7-intersection-counters), passed at head a341471127ab570dcca1fa3ee1deaded3c5ca496 in run 41068859265. The completed checks were read once; run metadata separately confirms the matching head and successful conclusion. All local checks described above also passed. Hosted replay certifies the registered finite computation, not a cross-host timing guarantee or independent scientific acceptance.

Merge Danger

Door: two-way

One isolated opt-in research package with six pinned live inputs; no shared verifier, frozen predecessor, upstream archive or selected record changes.

Blast Radius: research