[petalas]

Result

Answers Open next question 1 of DreamingOfClouds' #6345 (research/p104-frontier-symmetric-ladders) in that package's own register-tax model. Claim type: scoped written proof with exact finite corroboration. No κ change.

The question: is the set of optimal frame assignments exactly the union of the three freedoms (one-step helper re-spans, stop reorders with fixed endpoints, a stop move in one register cancelled by one in another)? In the tax model, no:

• Theorem 1. Two one-stop window chains with different window lengths never cancel each other's tax change unless neither changes. Pairwise cancellation needs equal windows.
• Corollary. In a configuration of one-stop main chains with pairwise distinct windows, every configuration reachable by any sequence of the three freedoms keeps each register's step multiset.
• Theorem 2. For every k ≥ 3 there are k-register price-null changes with no price-null nonempty proper subset: register i has window u_i + u_{i+1} + c and moves its stop from u_i to u_i + c, with all u_j and u_j + c distinct. With u_j = 2^(j−1), c = 2^k the windows are distinct, so the freedoms cannot reach these changes even through intermediate states.
• Smallest case. An exhaustive search finds none with largest window below 6. At 6: window 4 {1,3}→{2,2}, window 5 {2,3}→{1,4}, window 6 {2,4}→{3,3}.

A classifier or local search over the fibre therefore needs multi-register cyclic exchanges.

Evidence and reproduction

From the repository root, standard library only (interpreter used: CPython 3.12 at ~/git/integer-mult-bounds/.venv/bin/python3.12):

python3.12 -I -W error -B research/price-null-cycle-exchanges/verify.py
python3.12 -I -W error -O -B research/price-null-cycle-exchanges/verify.py
python3.12 -B scripts/research_package.py --package research/price-null-cycle-exchanges
python3.12 -B scripts/check_research_commit.py --package research/price-null-cycle-exchanges --check-source-history

Local results: both verifier runs print PASS_PRICE_NULL_CYCLE_EXCHANGES in about 0.6 s. The package check prints PASS_REGISTERED_RESEARCH_PACKAGE; the commit check prints PASS_RESEARCH_COMMIT_BYTES. The verifier checks Theorem 1 over all 216,062 combinations of two one-stop chain changes with distinct windows up to 12, Theorem 2 for k = 3..8, the exhaustive smallest three-register example with a pinned witness, and three rejecting controls (dropped register, equal-window exchange, non-generic split).

Limitations

• This is a statement about the tax model and the freedoms as No new κ: a price-null fibre of PR6167's word (per-rank helper−paid invariant) #6345 defines them. It does not show that the PR6167 word admits a legal cycle: frames must exist at the right dimensions and events, and the frame table, inventories and replay fields must stay fixed.
• Conditional κ = 0.003128645 (+2.59e-7 over #6144): retain an operand to remove a root stop #6167 and No new κ: a price-null fibre of PR6167's word (per-rank helper−paid invariant) #6345 are cited, not replayed or audited.
• No supplier, caller, native, compiler, Lean or Docker obligation is discharged. Independent human and proof-kernel review are unclaimed.

Open next questions

1. Does the PR6167 frame table admit a legal three-register cycle on its actual events?
2. Is every price-null change a composition of irreducible cycles once chains have several stops?
3. Can an irreducible cycle enable a stop drop that no single freedom reaches?

Related open work

• No open PR studies the No new κ: a price-null fibre of PR6167's word (per-rank helper−paid invariant) #6345 price-null fibre. Prove a fixed-schedule obstruction for PR6345 helper respanning #6381 (open) studies No new κ: a price-null fibre of PR6167's word (per-rank helper−paid invariant) #6345's second question, helper re-spanning; it computes no number this package computes. No other open PR studies the No new κ: a price-null fibre of PR6167's word (per-rank helper−paid invariant) #6345 tax-model fibre.

CI notes

All hosted checks passed at head 7369dbc043dfdf680b01946064fce0bd966b46f7: plan and replay (research/price-null-cycle-exchanges) in run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/41670652772.

Credit: the tax model, freedoms and question are DreamingOfClouds' (#6345), over Ali Romman's #6167 word. The new theorems, checker and examples were produced by Nick Petalas.