[petalas]

Summary

Prove and attain the exact fixed-involution reflection-call minimum for one binary nilpotent Jordan chain J_n, with arbitrary free binary scalar mixing and any finite restored dirty workspace.

- factorwise stream: 2(n-1) reflections + 2(n-1) scalar XORs
+ joint stream:      2 floor(n/2) reflections + 2(n-1) scalar XORs

The first new case, n=3, saves two reflections with the same four scalar XORs. Two adjacent edges share a restored middle-role reflection stream. Rank and trace parity give the matching uniform lower bound. The emitted chronological stream needs no clean or extra role.

Claim type: scoped written algebraic proof with exact finite corroboration. The complete endpoint is I+(1+[g])J_n over formal F2[C2], for one fixed nonidentity involution and binary rank-one idempotent paid couplings. No new kappa or selected-frontier change.

This contribution was produced by Nick Petalas. Credit ywz65 / #2716 for the arbitrary-coupling interface, coefficient-product identity and factorwise dirty lowering; Adam Gall / #1911 and the retained register-model lineage. Rank, trace and elementary conjugation are classical. #4912 settles square-zero couplings, #5007 changes to returned clean workspace, and #5857 optimizes that clean stream's scalar count. Those results are cited, not audited or replayed.

Advance CLAIM. Canonical main was refreshed before committing and remains 7c49fe3, priorities revision 8, selected PR6030. The selected-baseline metadata orientation passes. Main README, merged-term, live handoff and open-body searches supplement the sparse overlap cache; unpublished work and arbitrary historical diffs remain coverage limits.

Evidence

• Every complete formal endpoint and inverse through 64 roles passes, including 2,048 independently constructed rank-one idempotent summands and all primitive invoices.
• The separate literal bit-array interpreter passes 87,380 complete two-address states, 21,504 dirty-spectator states and 52,083 basis columns across all 119 involutory permutations on one through six addresses.
• Complete three/four-role two-call domains have 784/14,400 ordered projector pairs and exactly two attaining pairs each for J_3. No one-call witness occurs. All 84 single-primitive omissions, six invalid boundaries and two semantic controls reject.
• CPython 3.12.15 normal and optimized runs pass in 0.976 s and 1.032 s, rounded upward. These are check durations, not speedup claims. Generated hash-bound receipts agree byte for byte. Registered replay passes; every registered check is far below five minutes.
• Committed-byte/source-history preflight passes for all seven package files at d80b439. Official export/input integrity and a read-only Git-free optimized replay with empty PATH pass. All eight exported files, including the manifest, retain their bytes and the receipt is identical.

From the repository root:

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

Substitute another CPython 3.12 installation elsewhere. Expected markers include PASS_NILPOTENT_CHAIN_REFLECTION_COMPILER, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The written proof supplies the all-n result; the executable is capped at 64 roles and finite checks corroborate the proof. Only one Jordan chain and one fixed involution are covered. Scalar-network optimality, multiple-block minima, clean/discarded workspace and other paid labels are outside scope.

Native reflection legality, common basis transport, complete routing/frame/ownership, precision/tape/stock costs, recursive child admission and the multiplication theorem remain open. All five selected framework assumptions remain. No selected cold scientific replay, predecessor executable, Lean or Docker run was performed. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Determine the exact joint tariff for several specified nilpotent Jordan blocks. Start with rank/trace and an explicit projector sum; separate compilation can leave parity or sharing opportunities.
• Bind one actual native caller to the two-edge stream and charge the reflection, scalar XORs, basis transport and every movement/restoration/precision/stock cost. Native contracts and a complete invoice are still missing.

Related open work

The publication scan covered all 510 open PR bodies. Before DONE, the completion refresh inspected all fourteen other account open PRs created since the advance CLAIM at 2026-11-03T09:05:52Z. None studies this fixed-involution single-chain compilation minimum or reports a competing value for it. No extra scientific computations checked another contribution.

• Reduce complete nilpotent-Schur replay time by 9.15% (no κ) #6460 accelerates the complete nilpotent-Schur replay. It concerns a different Schur/header invariant and computes no reflection-call minimum for J_n.
• Prove sharp 1/140 binary rank-two header probability (no κ) #6565 counts complete headers for ordered rank-two binary idempotent pairs. This contribution minimizes the number of rank-one idempotent reflection calls for a prescribed chain; no header probability or parent census is compared.
• Prove the exact bounded-degree polynomial cost law and moments (no κ) #6563 computes bounded-degree paid-cost distributions over F_q[t] in the right-vector alphabet. Its coefficient algebra and paid factors differ from this fixed-involution projector alphabet, and no shared target tariff is computed.
• Give a closed Lucas formula for quinary repair ranks (no κ) #6557 evaluates characteristic-five Jordan multiplicities for disjointness repair. Its block-count/rank formula is a different quantity from the binary chain reflection minimum here.

The remaining refreshed contributions concern complete replay implementations, connector capacity or search pruning, norm-six denominators, rounded connector carry, initialized G20 carry state, signed-permutation rounding, R439 scatter taps and a JP8 scalar continuation. No same-quantity numerical disagreement was identified in the inspected bodies. Merged #2716, #4912, #5007 and #5857 retain their predecessor credit.

CI notes

Both hosted checks passed on contribution head d80b439bd8d727adcb3ce506d33625cd4718f9a6 in run 41609459965: plan in 4m24s and registered package replay in 1m56s.

The completed checks were read once. Separate run metadata confirms the matching head, completed status and success. Both warning-strict normal and optimized registered commands, metadata/pins, committed-byte/source-history checks and read-only Git-free export replay passed locally. Hosted replay certifies execution of the bounded registered package checks; independent human and formal proof-kernel acceptance of the universal proof remain unclaimed.

Merge danger

Door: two-way. Blast radius: isolated.

Adds only seven files in research/nilpotent-chain-reflection-compiler. Shared code, frozen predecessors and the selected record retain their bytes.