[petalas]

Summary

Scoped mathematical proof with a finite explicit construction. The two-source
GF(8) scalar-update dirty connector, requesting all 63 nonzero coefficient
pairs, has exact unit complete-call optimum 140, even when arbitrary extra
readouts are allowed.

- two coordinate-axis Gray cycles: 16 source calls + 126 readouts = 142
+ grouped fourteen-prefix walk:   14 source calls + 126 readouts = 140

The new step realizes Aaron Wang's #2128 walk using two GF(8)-linear source
maps and charged scalar letters. The field grouping makes each original DUO
step a single scaled application of one source map. The existing all-readout
counting argument of Tairitsu-Aya #2156 and Cinna Davis #2249 transfers to this
contract; unpaidmoon's #4651 difference-basis exclusion is a cited dependency.
Those predecessor results and the binary walk are credited, not new claims
or audit deliverables. Cinna Davis #2266 supplies the preceding GF(4)
scalar-source interface. No new kappa or selected-frontier change.

Only research/gf8-two-source-optimum/ is added. The changed
connector-alphabet-change scope, source pins, proof, complete scaled-event
price expression and Open next questions are in the package.

Evidence

Before, the explicit axis comparator needs 142 calls. After, the new
field-linear realization needs 140 and preserves both immutable sources,
arbitrary connector dirt and all arbitrary incoming outputs for every allowed
GF(8)-linear map. The lower bound covers all finite words in the stated
alphabet, subject to the cited difference-basis theorem.

Local CPython 3.12.15 checks pass:

PASS_GF8_TWO_SOURCE_OPTIMUM calls=140 sources=14 readouts=126 dense=399 controls=283 cited_lower_bound=PR4651
PASS_REGISTERED_RESEARCH_PACKAGE research/gf8-two-source-optimum outcome=completed
PASS_RESEARCH_COMMIT_BYTES
PASS_RESEARCH_EXPORT

The verifier checks 64 products using two field implementations, 1,024 field-law
instances, seven inverse counts, free noncommutative forward/inverse endpoints,
396 dense basis columns and three mixed dirty inputs. It rejects all 140 call
omissions, fourteen changed source coefficients, 126 changed readout
destinations and three malformed source words. The generated stdout receipt
contains all boundary pairs and hashes of the five package files. It is not
handwritten evidence.

Both registered commands pass locally. Direct warning-strict normal and
optimized replays take 0.67 and 0.66 seconds; the final combined registered
replay takes 1.56 seconds. These are check durations, not performance gains.
Committed bytes, executable modes and declared source ancestry pass at
b81898d08f2c2b88b6276902dfe819394d448584. The official nine-file export passes
integrity and four-input coverage checks. Its read-only package files also
pass the optimized standalone replay without Git and with an empty PATH.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/gf8-two-source-optimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/gf8-two-source-optimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/gf8-two-source-optimum
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/gf8-two-source-optimum --check-source-history --check-modes

Another CPython 3.12 executable can replace this path. Standard library only.
The current selected metadata baseline and historical 945-pairing card control
passed separately as orientation; neither is the new result. No predecessor
scientific suite was replayed as the deliverable.

Limitations and unrun obligations

Exactly one dirty connector, immutable sources, separate unit-coefficient
readouts and GF(8)-linear maps are material. Unit complete-call optimality does
not imply optimality at unequal scaled-event prices. Every scaled call must
include its scalar preparation, application and restoration charges. The
package records the literal event bill and 952 descriptor bits, versus 962 for
the axis comparator, before interpreter/framing costs.

The set-level exclusion and existing binary counting proof are cited inputs.
Finite replay corroborates the new realization and does not independently
verify those predecessors. No real supplier consumer, native scalar service,
physical hosting, frame/stock/tape integration or full signed supplier invoice
is supplied. Selected #7000 and its five assumptions remain unchanged. No
selected cold replay, Lean, Docker, independent human review, formal-kernel
acceptance or complete multiplication theorem is claimed.

Open next questions

• Identify one source-pinned recursive caller with this GF(8)-linear boundary
  and price every scaled event, readout, admission, preparation, dispatch and
  dirty return. No such physical caller is supplied here.
• Determine the unequal scaled-event tariff at a concrete caller, beginning
  with the fourteen-step and sixteen-step words. The unit-call theorem does
  not settle that comparison.

Related open work

• Research: high-arity connector toggle cost floor #2249 supplies the existing binary all-readout counting argument and
  high-arity bill. This contribution transports that argument to the two-map
  GF(8) contract and credits its attaining walk.
• Add five-source arity-four connector cost bound #3909 treats a five-source arity-four operation and its dispatch/addition
  ledger. Here a scalar letter acts on one field-linear source.
• research: audit minimal joint-output connector charge #3897 treats fused output readouts. The present contract keeps readouts
  separate and changes the source coefficient alphabet.
• Research: seven-source weight-at-most-four connector boundary (L<=18 excluded, 216-call floor) (no new kappa) #4079 studies a seven-source binary weight-at-most-four demand family. This
  package requests all nonzero pairs on two GF(8) sources.
• Record the length-28 flat-step budget forcing a rank-7 arc (no κ) #7088 studies a length-28 eight-source singleton walk. It does not supply
  this field-linear realization or tariff.
• Prove the exact four-source uniform clean-row tariff (no κ) #7204 studies four binary sources with a paid clean row and uniform prices.
  The present connector starts arbitrarily dirty and has a different demand
  alphabet; no competing numerical value is computed.

Live open PR bodies, recent handoffs, active Discussions, local claim locks,
main README searches and merged object/formula searches were checked before
claiming and committing. The complete overlap snapshots included 502 open PRs,
6,175 closed PRs and 7,176 Discussion records; subsequent live refreshes were
also inspected. Source-reference queries found no recent account follow-up to
#2266/#2128/#4651. Sparse notes, arbitrary historical diffs and unpublished work
remain coverage limits. Advance claim.

The completion refresh inspected this account's ten open PRs created since the CLAIM: #7203, #7204, #7205, #7206, #7207, #7208, #7210, #7211, #7212 and #7213. Only the already-listed #7204 studies the connector-alphabet neighborhood. It uses four binary sources and a paid clean row; no competing value for this GF(8) dirty-connector instance is reported. No extra mathematical computation was run to check other contributions.

CI notes

CI did not certify this package. At head b81898d08f2c2b88b6276902dfe819394d448584, run 43012862401 failed its plan job 130496541977 in Onboarding helper tests. The completed checks were read once and the job log inspected. It reports exactly one failing test among 760, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found in the CI verification document

The package replay job 130498667686 was skipped because plan failed. No hosted package check ran.

This is the unchanged main-base mismatch at e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, while the current section of docs/ci-verification.md still describes #6030 and make frontier6030-verify. Both files were inspected directly from that commit. This PR adds only its isolated package and modifies neither file.

The same sole failing test, missing-command assertion and 760-test/one-failure/24-skipped summary were verified directly in all six comparison run logs:

• Run 42905094314.
• Run 42903313790.
• PR #7203, run 43012497455.
• PR #7204, run 43012524632.
• PR #7205, run 43012595145.
• PR #7207, run 43012902583.

No fix, retrigger commit, push or job rerun was made. The normal/optimized local registered checks, direct verifier, committed-byte/mode/source-history preflight and exported replays remain the verification evidence. No hosted certification or independent scientific acceptance is claimed. The repository-level repair is left to the maintainers.

Credit

This contribution was produced by Nick Petalas. Credit Aaron Wang / navinor
#2128 for the walk; Cinna Davis / 1337isnot1337 #2266 for the paid scalar-source
interface and #2249 for the all-readout counting argument; Tairitsu-Aya #2156
for that argument's DUO predecessor; unpaidmoon #4651 and the credited
Volodymyr Gavrylkiv set-level result; Ross Richardson #934/#943 for the dirty
connector and prefix-readout lineage. Predecessor credit lines are unchanged.
Independent human and formal proof-kernel review are unclaimed. Apache-2.0.

Merge Danger

Door: two-way. Blast radius: package. Only one isolated research directory is
added. Shared code, frozen sources and the selected frontier remain unchanged.