[petalas]

Summary

Finite exact certificate in the three-register positive F2[Dic3] model. All 890 diagonal monomial targets with separate positive-generator bill at least seven have no word of at most six calls. Consequently, all 360 targets whose separate bill is seven have exact minimum seven.

#2069 open six-call layer
  -> complete three-call inventory x complete three-call inventory
  -> 890 targets, 84,307,030 exact lookups, zero endpoints
  -> 360 matching seven-call coordinate constructions

This answers Cinna Davis's #2069 explicit six-call question. Its depth-four/five exclusions remain cited predecessors, rather than this contribution's deliverable. The new verifier consumes no predecessor executable or table. The exact alphabet is positive u,v, all 28 binary rank-one idempotents, three arbitrary dirty registers and coefficientwise full formal endpoints. Free scalar GL3 mixing is covered by the inherited augmentation/conjugation normalization. No free address maps, extra registers, clean inputs or free inverse labels are permitted.

Produced by Nick Petalas. Credit Cinna Davis / 1337isnot1337 #2069, David Govea / davidgovea #720, eumemic #288 and the retained register-model lineage. Sources are bibliographic pins at canonical base ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c; no pending Git object is needed. Advance CLAIM.

Evidence

Before: #2069 explicitly left the six-call layer open. After: complete exact exclusion, with matching seven-call constructions for the entire direct-bill-seven family.

The replay covers 94,727 exact prefix states; all 175,616 raw ordered three-call words; all retained matrix inverses; 3,136 gate-pair update comparisons; eight positive joins; 360 independent coordinate witnesses; 2,520 omitted-call rejections; and three disposable semantic mutations. A separate literal-permutation model agrees with all 144 native group products and every endpoint distance. Hash buckets use full coefficient equality, not digest acceptance.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/dicyclic-six-call-boundary/verify.py --controls
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/dicyclic-six-call-boundary/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/dicyclic-six-call-boundary
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/dicyclic-six-call-boundary --check-source-history

CPython 3.12.15, macOS 27.0.1 arm64, Apple Clang 21, C++17; no Python dependencies. Substitute another CPython 3.12 path and set CXX to a C++17 compiler executable as needed. All four commands passed locally. Each registered check was timed separately: 3.78 seconds with mutations and 1.80 seconds optimized. Compilation enables warnings as errors. Generated JSON receipts bind every package source by SHA256.

Committed-byte/source-history preflight and official export integrity pass at 16e8665a2043c9842b9dda37bf7157a582abfea9. Both direct modes also pass from a read-only Git-free export; all seven exported files, including its integrity manifest, remain unchanged. Source-bound selected-baseline metadata orientation passed. No selected scientific replay, Lean or Docker run was needed or run.

Limitations

This is a finite certificate with exact finite minima, not an arbitrary-group/register theorem, physical supplier or complete multiplication theorem. Targets with separate bill eight through twelve receive only the lower bound seven. Physical realization, compiler/routing, ownership, precision, stock and recursive costs remain unresolved. Independent human and formal proof-kernel review are unclaimed. No kappa or selected-frontier change.

Open next questions

• Does a seven-call word share a target with separate bill at least eight? Start with a concrete priced target and a structural restriction before considering a larger inventory. No deeper search is reserved.
• Can an algebraic invariant replace the finite join while charging both positive labels and preserving arbitrary dirt?

Related open work

No open PR studying the Dic3 positive-generator diagonal tariff was found in the submission-time live search. Merged #2069 is the direct predecessor; neighboring nonabelian-group claims use different groups, alphabets or workspace boundaries and supply no numeric comparison for this object. Coverage includes complete remote overlap snapshots, distinctive main-tree/merged searches and live handoffs; sparse local notes, arbitrary diffs and unpublished work limit it.

The completion refresh inspected all nine other same-account open PRs created since the CLAIM. None studies this Dic3 tariff or computes a competing value. The neighboring formal group-algebra contributions are:

• #5199 proves an A4 workspace minimum; its address group, endpoint and workspace domain differ. No common target value is computed.
• #5202 proves a normal-C7 tariff in the order-21 group; its paid alphabet and target family differ. No Dic3 value is computed.
• #5203 prices even S4 pairs with every odd label and arbitrary dirty workspace. This package fixes positive u,v in Dic3 and exactly three roles, so it supplies no common numerical comparison.

CI notes

Both hosted checks passed at exact contribution head 16e8665a2043c9842b9dda37bf7157a582abfea9 in Registered research packages run 39954163808: plan in 4m5s and the focused registered package replay in 1m30s. Run metadata confirms completion, conclusion success and the exact head SHA. The replay executes both registered normal and optimized commands, including native compilation and the declared controls. All registered checks also passed locally; no local obligation was deferred to CI. Execution success remains separate from independent mathematical acceptance.

Merge Danger

Two-way door. Blast radius: one isolated research package. Frozen predecessor files and the selected frontier are unchanged.