[petalas]

Summary

Prove the exact directed-cycle tariff for lee-phys's #6210 independent radical demand (e,h,e+h+z) over R16=F2[e,h,z]/(e,h,z)^2. Claim type: scoped written proof with exact finite corroboration, produced by Nick Petalas. No new κ.

cycle ADD prices c > 0; m = min_i(insertion price mu_i)
m >= 6c; every reverse ADD price >= 6c
exact complete price = 2m + 6c

At insertion/reverse prices 100 and cycle price 1, the exact optimum is 206. Every two-insertion cycle word needs at least six ADDs. An eight-gate attaining word, listed chronologically, is

P_0(1+e), E_10(1+h), E_21(e+h+z), E_10(1),
P_0(1),   E_10(1+h), E_21(e+h+z), E_10(1).

The word preserves the source and restores all three independent dirt columns at every positive data width. Cyclic relabeling gives an attaining word at any minimum-price insertion root. The all-word proof separates the unit/nonunit cases of a possible five-ADD chain; the full price cone also prices additional insertions and reverse edges.

This answers one independently directed-price family of #6210 Open next question 2. Its demand, arbitrary-dirty contract, incoming-row return lemma and two-unit insertion argument remain credited to lee-phys. The new contribution is the six-cycle-ADD lower bound, word and sharp price cone. The first-order subspace method follows Nick Petalas's older #6693; the new checker has standalone bit packing and imports no predecessor executable. Retain #5259/#5163, Ali Romman #5310/#5266, Krzysztof Wojt #5045, Ross Richardson #3063/#3121, Nick Petalas #4916/#5208 and the earlier fanout lineage. The older #5933 and #6663 have different demands. No general priority is claimed for square-zero expansion or dirty echoes.

Evidence

Before, the destination-row relaxation does not determine this directed price. After, a written proof and literal attaining word establish 206 on the displayed instance and 2m+6c on the cone.

At head fd2952b2b759ef5a5506920469ab2abccac6ab93, local CPython 3.12.15 on macOS arm64 passes:

• 504 full dirt/source matrix identities and inverse checks, covering all 168 independent radical triples and three insertion roots.
• 393,216 literal forward/inverse scalar evaluations of the requested demand, ten price fixtures and sixteen rejection controls.
• An exact first-order exclusion of every skeleton with two unit insertions and at most five cycle ADDs. Layer sizes [1,9,57,283,1186,4201,11842,18061], 95,562 transitions and 2,481 closing transitions. Counts refer to merged skeleton states/transitions, not distinct literal words. No symmetry or heuristic pruning is used; zero-residue coefficients are overapproximated harmlessly for exclusion.
• Normal and optimized warning-strict standalone commands, each timed locally at about 0.56 seconds, and both registered package commands. Both are below their 120-second limits.
• Metadata/source-pin validation, AST compilation, function-length inspection, staged whitespace and exact committed-byte/source-history checks. All six files and the canonical base's ancestry pass. Each scientific invocation prints a generated receipt bound to all six package hashes.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/r16-independent-cycle-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/r16-independent-cycle-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/r16-independent-cycle-tariff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/r16-independent-cycle-tariff --check-source-history

Other machines may substitute their CPython 3.12 path. Standard library only. Expected markers are PASS_R16_INDEPENDENT_CYCLE_TARIFF, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Scientific base remains canonical main e54523517d0f03e58352e647c403a5af3e525a57, re-fetched before commit. The selected binding baseline passes at κ 0.00363005817; the historical priority-revision-8 card is explicitly marked changed to this algebraic interface side result. No selected scientific replay is claimed. Advance CLAIM.

Entry overlap reported complete pagination for 477 open PRs and 7,122 Discussion records, supplemented by all-main README and merged/all-state object/formula searches. A later complete 479-open-PR source-reference scan found no #6210 overlap; live claims, the open list and local reservations were refreshed before commit. Sparse local note coverage omitted 9,973 HEAD paths. Unpublished work and arbitrary historical diffs remain coverage limits.

Limitations and unrun obligations

The theorem uses exactly the stated three-row, preserved-source R16 contract and price cone. No arbitrary-price classification, larger-ring theorem, extra-workspace optimum, physical coefficient preparation, routing, precision/tape, stock, supplier admission, selected-frontier change or complete multiplication theorem follows. Independent human and formal proof-kernel review are unclaimed. The universal statement rests on the written proof; bounded enumeration corroborates it. No native suite, selected cold replay, Lean or Docker run was required or run. Every registered command ran locally.

Open next questions

• Determine the exact tariff outside this cone. Start with the same demand and one cheap reverse edge, recording each directed edge separately. An arbitrary positive-price lower bound is missing.
• Find a physical caller with these independent radical demands and complete coefficient, source, routing, precision and restoration prices. No such caller or full supplier invoice is supplied.

Related open work

No same-object open PR was found in the refreshed source-reference scan, targeted R16/cycle searches and live open list. Merged #6210 supplies the receiving-row theorem and question; merged #6693 supplies the receiving-row frontier. Their claims are cited and not replayed. No other contributor's numerical result was recomputed for comparison.

The completion refresh read the titles and bodies of this account's nine other open PRs created since the CLAIM: #7178, #7179, #7180, #7181, #7182, #7184, #7185, #7186 and #7187. They study other objects and quantities; none computes this independent-radical three-row fanout tariff. No extra computations were run to check them.

CI notes

CI did not certify this package. At exact head fd2952b2b759ef5a5506920469ab2abccac6ab93, run 42992841885 finished with plan failed and replay skipped. I read the completed checks once and inspected plan job 130429636564.

The only failed step was Onboarding helper tests. Its summary is Ran 760 tests and FAILED (failures=1, skipped=24). The exact failing test is test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record. Its assertion says that 'make frontier7000-verify' was not found in the current CI verification document.

The cause is in unchanged canonical-main files at e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. The test reads those files and compares their current replay command. This PR adds only its six package files and changes neither input. Source inspection and the job log confirm the mismatch; no extra test or job rerun was needed.

I also inspected the plan logs of all eight comparison runs below. Each has the same single failing test, the same missing-command assertion, and the same 760-test / 24-skipped summary:

• 42905094314
• 42903313790
• 42992088329
• 42992340683
• 42992579989
• 42992679183
• 42992720936
• 42993026387

The package evidence remains local: both registered scientific commands, direct normal and optimized checks, metadata/source pins and committed-byte/source-history preflights pass. No fix, new commit, push, retrigger or CI rerun was made for this repository-level mismatch. The written mathematical proof still requires independent review.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Only six files in research/r16-independent-cycle-tariff/ are added.