[petalas]

Summary

Prove a scoped algebraic theorem: w nonidentity normal-C7 targets in the full formal F2[C7 semidirect C3] register model cost exactly w paid rank-one idempotent calls, with any finite restored dirty workspace and arbitrary free binary scalar mixing. This settles Cinna Davis's four-register question in #2011. No new kappa or supplier admission is claimed.

F8 multiplication + Frobenius representation
  target difference rank = 3w
  paid difference ranks = 3 inside C7, 2 outside C7
  rank telescoping: 3 c7 + 2 c3 >= 3w
  w coordinate calls attain the full formal endpoint

Every optimal w-call word therefore uses only normal-C7 labels. The new package contains the written proof, bounded exact verifier and historical provenance pins. Produced by Nick Petalas. The group, alphabet and open question retain Cinna Davis's #2011 credit; #3893 supplies the S3 example of the classical rank method, with the retained eumemic/PR288 register-model lineage.

Evidence

• Before: research: bound order-21 three-register orbit word #2011 settles its fixed three-register target by a finite two-call census and explicitly leaves four outputs open. That census is cited and is not replayed here.
• After: the written F8 proof covers every finite target count and restored workspace size. The verifier checks 441 representation products, all 12,320 paid factors on four/five roles, 1,302 full formal attaining endpoints, one 105-dimensional dirty-mixed regular-action endpoint and ten rejection controls.
• Local CPython 3.12.15 checks pass in normal and optimized warning-strict modes, taking 0.6273 and 0.6247 seconds. Registered replay passes in 1.4844 seconds. Each registered check has a 60-second timeout.
• Committed-byte/source-history preflight passes for 99ed6d2fc13101cb5ec0d3b1cdc826b75bc27fc7, all six files and the pinned base ancestor. Official export and integrity checks pass; Git-free direct replay produces the identical generated source-hash-bound receipt. No receipt is handwritten.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -B -W error research/order21-normal-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -B -W error research/order21-normal-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/order21-normal-tariff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/order21-normal-tariff --check-source-history

Any CPython 3.12 installation may replace the local path. Expected markers are PASS_ORDER21_NORMAL_TARIFF, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

This is a written scoped proof with exact finite corroboration. The finite representation is used only for a lower bound; attainment is checked in the full formal algebra. The proof does not depend on a bounded workspace census.

Mixed order-three target profiles, nonuniform prices, native realization, routing, frames, precision, stock and recursive costs remain outside scope. No physical caller or complete multiplication theorem is supplied. Lean, Docker and the selected operator replay were not run. Independent human and formal proof-kernel review are unclaimed. The selected frontier and predecessor files are unchanged.

Open next questions

1. Determine the exact tariff for a mixed pair with one normal-C7 target and one outside order-three target. Combine the represented-rank budget with the quotient-C3 character; a stronger lower bound or cheaper full formal word is missing.
2. Specify a literal native producer/consumer and price its address actions, ownership, routing and restoration. The physical contract and full cost are unknown.

Related open work

• Add S3 three-register rank lower bound #3893 proves an S3 representation-rank bound at a different group and endpoint. The rank method is a cited predecessor; this contribution proves the order-21 normal-subgroup tariff and does not recheck that PR.
• Prove the A4 eight-call floor with arbitrary dirty workspace (no new κ) #5199 extends an A4 endpoint minimum to arbitrary restored dirty workspace. Its group, target and positive-label alphabet differ; this package computes no A4 minimum.
• Prove four-call optimum for all even S4 pairs with every odd label #5203 proves a four-call minimum for even S4 target pairs with every odd label paid. This package allows all nonidentity order-21 labels and normal-C7 targets; it computes no S4 endpoint cost.
• Prove exact seven-call minima for 360 Dic3 targets (no new κ) #5205 proves seven-call minima for specified Dic3 targets in a positive two-generator alphabet. It studies a different group, alphabet and target family; no common endpoint cost is computed here.
• Prove exact integer-transvection length λ(B^k)=2k (no new κ) #5209 proves lengths of integer-transvection powers over Z. Its faithful matrix alphabet and target powers differ from this binary formal group-algebra tariff; no common length is computed here.
• The refreshed list of this account's open PRs created since the advance CLAIM contains no other order-21 normal-subgroup tariff. research: bound order-21 three-register orbit word #2011 is the merged source of the explicit open question. No additional computations were run to compare predecessor or concurrent results.

CI notes

Both hosted checks passed for 99ed6d2fc13101cb5ec0d3b1cdc826b75bc27fc7 in run 39954092090: plan passed in 6m1s, and the registered package replay passed in 1m11s. Run metadata confirms this exact head. Both registered normal/optimized commands and the source-history preflight also passed locally. These CI checks certify the registered replay, not independent human or proof-kernel acceptance of the mathematical theorem.

Merge Danger

Door: two-way. Six new files in one isolated research package.

Blast Radius: research. No shared code, selected result, frozen source or predecessor package changes.