[petalas]

Summary

Scoped proof: the exact five-call minimum for diag([i],[w],[w^-1 i w],1,...,1) over F2[SL2(F3)] survives every finite number of restored dirty roles and arbitrary free constant binary mixing.

#2010: three roles
#4083: four roles, two-or-more dirty roles left open
this proof: every finite r >= 3
           at least 2 quaternion calls + 3 order-three calls = 5

This answers Open next question 1 in Wentao Li / VictorLiwentao's #4083. Cinna Davis #2010 retains the endpoint, alphabet and direct word. Sagar Thakrar #4032 retains the twisted radical-rank method and semidirect-product extension question, with Cinna Davis #2116/#2047/#2248 and the retained register-model lineage credited in the package. The new step is an all-role first-order obstruction and the rigidity of at most two order-three quotient calls. No predecessor census is replayed.

Evidence

Before: the published #4083 scope establishes one dirty role and explicitly leaves two or more open. After: PROOF.md forces two quaternion-label calls by first-order rank and three order-three calls by binary projector rigidity. The inherited coordinate word attains five with every additional dirty role restored.

On CPython 3.12.15, both registered warning-strict checks pass normally and under -O. Direct command times were 0.63 and 0.67 seconds, below their 60-second timeouts. The verifier enumerates 576 cocycle pairs, 308 one-call and 60,880 two-call quotient words, 312 deterministic sampled full-algebra words, 384 regular-action basis inputs and eleven named controls. The samples corroborate the separately written universal proof; they do not exhaust arbitrary role counts.

Generated stdout receipts bind all six package files. Metadata/source pins, committed-byte/source-history preflight at fa2f4bc819a9e895eee624ea53b551eb9e1a3216, official export/integrity, and an optimized Git-free exported replay pass. All seven exported files, including the export manifest, remain byte-identical after replay.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -B research/sl2f3-all-dirty-minimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -O -B research/sl2f3-all-dirty-minimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/sl2f3-all-dirty-minimum
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/sl2f3-all-dirty-minimum --check-source-history

Expected marker: PASS_SL2F3_ALL_DIRTY_MINIMUM minimum=5 i_floor=2 w_floor=3. The registered manifest uses {python} so another CPython 3.12 location is supported. All external source rows are historical provenance; no predecessor executable or live input is needed.

Limitations

The theorem requires binary rank-one idempotent role maps, the four stated paid labels, arbitrary inputs and complete dirty restoration. It does not cover clean-only or partly restored helpers, different targets, enlarged coefficient fields, physical native implementations or supplier admission. The first-order map kills the central element -1 and cannot certify pure central defects. No new kappa or selected-frontier change is claimed. Native, precision/tape, outer-theorem and proof-kernel checks were not run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

Is there a fully priced native realization of this alphabet for a changed endpoint that saves paid work? Start with one pinned arbitrary-input D_N(i) or D_N(w) program, including source access, address movement, hosting and restoration. The missing implementation and complete producer/consumer invoice are the blockers. Further dirty-role censuses for this fixed target cannot improve its five-call bill.

Related open work

The pre-DONE refresh examined the ten open PRs from this account created since the CLAIM at 2026-10-22 23:03:26 UTC, including this PR. None of the other nine studies this SL2(F3) endpoint or computes a competing minimum for it. They concern Hurwitz decoding success (#4900), weighted pair tours (#4899), localized Gaussian headers (#4897), cyclic-source root stock (#4896), nonlinear Eisenstein restrictions (#4895), four-label excess (#4894), a z4 error guard (#4893), chronological library widths (#4892) and dense-output scan costs (#4891). No extra computations were run to check those contributions.

The exact-object predecessors #2010 and #4083 and the methodological predecessor #4032 are merged. The earlier live SL2(F3)/tetrahedral search found no other open same-object PR. Search coverage does not certify novelty.

CI notes

Both hosted checks passed at fa2f4bc819a9e895eee624ea53b551eb9e1a3216 in Registered research packages run 39870358645: plan in 3m01s and registered package replay in 1m12s. Run metadata confirms the matching head and successful completion. The registered replay executes the warning-strict normal and optimized commands. All declared checks also ran locally, including package metadata/source pins, committed-byte/source-history preflight, official export integrity and optimized Git-free exported replay. Hosted execution of these scoped checks does not constitute independent human or formal proof-kernel acceptance.

Merge danger

Door: two-way. Blast radius: one isolated six-file research package. No shared implementation, predecessor package, frozen evidence or selected result changes.

This contribution was produced by Nick Petalas. Apache-2.0.