[petalas]

Summary

Scoped proof: support-dependent prices of rank-one paid projectors collapse to their conjugacy-orbit minima when scalar mixing is free. This answers the literal free-mixing case of ywz65's #2544 Open next question 2.

wanted D_N(g) = C D_P(g) C^-1
P is a cheapest conjugate of N; both scalar gauges are free
A = min_N c_a(N), B = min_N c_b(N)
exact t-SWAP price = t (A + B + min(A,B))

The compilation lemma is the new contribution. The fixed-price SWAP theorem is inherited from #2544, with credit to ywz65, Ross Richardson #1989/#2415, Aaron Wang's handoff and their retained predecessors. Projector conjugacy is standard linear algebra. This contribution is produced by Nick Petalas.

Evidence

Before, #2544 explicitly left projector-support prices open. The new proof preserves each complete operator through two free scalar gauges. A two-role control shows that removing free mixing changes attainability at price zero.

Local CPython 3.12 checks pass: 1,702 full-formal conjugation checks, 36 tariff endpoints, the six-state restricted-gauge closure and an omitted-inverse rejection. Normal and optimized registered commands took 0.17 and 0.14 seconds locally. Metadata/source pins, committed bytes, declared source history and whitespace checks pass. The verifier prints a generated receipt bound to proof/code/source hashes.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -X dev -W error -B research/projector-price-orbit-collapse/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -B research/projector-price-orbit-collapse/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/projector-price-orbit-collapse
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/projector-price-orbit-collapse --check-source-history

Limitations

This is a formal pricing theorem with all roles restored coefficientwise. It assumes fixed nonnegative displayed-projector prices and free scalar mixing. Charged or restricted gauges remain separate. The general proof uses the inherited fixed-price theorem; no predecessor result is independently audited here. No native supplier, recursive bill, new kappa or complete multiplication theorem is claimed. Lean, Docker, predecessor suites and the selected cold replay were not run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Price both scalar gauges in the explicit two-role example using a named native implementation.
• Classify projector orbits under a proper free subgroup and solve one nonconstant orbit-price instance.

Related open work

• Exact-search tooling for noncommuting words, and a complete six-distinct-value screen of the KB2 thirteen-call candidate #2444 studies the same coordinate-SWAP group-algebra model with uniform prices and supplies noncommuting search tooling. This result addresses support-dependent prices by compilation and does not replay that search.
• Add S3 three-register rank lower bound #3893 studies representation-rank lower bounds for a different three-role S3 target. This contribution concerns projector-dependent prices for coordinate SWAPs, without rechecking that target.

This account's open PRs created since the CLAIM at 2026-10-22 21:12:54 UTC were refreshed before DONE. None of the other new submissions studies the same support-dependent coordinate-SWAP tariff. They concern GL3 address kernels, connectors, integer splitting, complex ports or channels, and Toeplitz norms. No additional computation was run to compare predecessor results.

CI notes

Both hosted checks passed on publication head e965cef1c6b9c6c38ee29afe51c599a312976636 in run 39856235212: plan in 2m37s and the registered package replay in 1m11s. Run metadata confirms the exact head and successful completion. The replay covers both registered normal and optimized commands. Hosted execution does not claim independent human or formal proof-kernel acceptance.

Canonical main was refreshed and remains 624de9d, priority revision 7. Final overlap covered 352 open PRs and 3,793 Discussion records with complete remote pagination, alongside distinctive main README and merged-PR searches. Sparse local notes, arbitrary code diffs and unpublished work are incomplete. The only changed directory is research/projector-price-orbit-collapse.

Claim: #1166 (comment)

Merge danger

Door: reversible. Blast radius: research-only. Adds one opt-in package; no selected-frontier or predecessor-source change.