[petalas]

Summary

Scoped combinatorial proof with exact finite corroboration. A full dyadic rotation has an exact optimal logical switch tariff, including a sharp worst case and all equality cases.

N=2^h, x -> x+c mod N, no spare addresses
S_h(c) = sum_{k=1}^h 2^(h-k) dist(c, 2^k Z)
max_c S_h(c) = ((3h+1)2^h-(-1)^h)/9
maximizers: c=(2^h-(-1)^h)/3 and its complement

An explicit h-layer descending-bit circuit meets the Hamming-distance lower bound among all hypercube-edge swap sequences. For h=3,c=3 it uses 3+2+4=9 active switches. The affine two-state recurrence proves the maximum and equality cases for every h>=1.

Only research/dyadic-rotation-switch-tariff/ is added. Cinna Davis #4100 supplies the CRT rotation context, unpaidmoon #4377 the generic evaluator predecessor, and Miltzow, Narins, Okamoto, Rote, Thomas and Uno the standard token-distance lower bound. Binary carry logic and that lower bound are credited standard ingredients. The new contribution is the exact rotation tariff and extremal law in the stated model. This contribution was produced by Nick Petalas.

Evidence

The generic predecessor supplies a Beneš construction, with no exact optimal active-swap tariff for this changed dyadic premise. The new construction attains a matching lower bound, rather than merely reporting a shorter candidate.

Local warning-strict CPython 3.12 normal and optimized runs pass 510 exact routes through h=8, independently computed Hamming sums, every offset through h=12 for the maximum/equality cases, independent full token-permutation BFS with 2/24/40,320 states through h=3, and eight fault/boundary controls. Each registered command took under 0.13 seconds locally. Explicit checks remain active under optimization. A separate read-only review found no mathematical or scope issue. Independent human and formal proof-kernel review are unclaimed.

From the repository root:

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

All pass locally. Direct output ends PASS_DYADIC_ROTATION_SWITCH_TARIFF. Six-file committed-byte/source-history preflight, official package export/integrity check, and a fresh isolated optimized export replay with empty PATH also pass. The optional receipt is generated and bound to all six package-file hashes. Another CPython 3.12 executable may replace the local path.

Base is canonical main 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. The card is explicitly changed from audit to a new exact routing theorem. The advance claim records the bounded scope. Remote overlap covers 395 open PRs and 3,888 Discussion records at 22:39 UTC; sparse local cache coverage is incomplete, supplemented by main-tree README and merged-PR searches. Main and live claims/open PRs were refreshed before committing; no exact-scope overlap was found within that coverage. This is not a novelty certificate.

Limitations

The model counts active logical swaps on hypercube edges with all N addresses occupied. The strict non-dyadic prime-modulus rotation with fixed padded addresses used by the CRT caller is outside scope. Control generation, sequential tape access, payload movement, repair, stock, ordinary callee and full supplier/outer composition remain unpriced. No physical time improvement, new kappa, supplier admission or complete multiplication theorem is claimed. No Lean, Docker, native or historical heavy replay was run. The selected frontier is unchanged.

Open next questions

• Determine the exact tariff for padded non-dyadic rotations. Start at s=3,h=2 and compare with the distance lower bound; the triangular full-cycle construction does not preserve that padded domain.
• Price one complete fixed-tape execution of the dyadic circuit, starting at h=3,c=3, including controls, scans and payload actions.

Related open work

Completion refresh inspected this account's nine other open PRs created since the CLAIM: #4873, #4874, #4875, #4878, #4879, #4880, #4881, #4882 and #4883. None computes this dyadic edge-swap tariff or its sharp maximum. No same-quantity disagreement was found within this coverage; no extra scientific computations were run.

• Prove exact shared Fourier cancellation at every input arity (no κ) #4874 classifies shared Fourier cancellation, including permitted affine read shifts. This theorem prices full dyadic rotations in a logical edge-swap model; it supplies no physical read-layout price for that Fourier caller. No common numerical result is compared.

• Audit uniform Beneš compiler with charged sequential arena scans #4192 implements and prices generic encoded-permutation Beneš compilation with full arena scans. This theorem gives an optimal logical tariff for one permutation family, without executing or repricing that compiler.

• Prove the full CRT arena batch threshold and sharp surcharge (no κ) #4845 classifies the inherited full CRT arena-batch bill. It retains generic network tariffs; this result concerns a different dyadic routing premise and does not improve its arithmetic bound.

• Execute paid CRT key service at the actual repair port #4795 executes a fixed-geometry paid repair-key service. This result supplies neither its physical adapter nor a repair-key cost.

• Audit conditional CRT last-K two-slice streaming tariff #4187 prices the conditional last-K two-slice branch with an assumed evaluator. The new theorem concerns logical rotation switches, not that suffix schedule.

CI notes

Both hosted checks passed on contribution head ad51ee46da48da682ef5bafc88a66709bdef4597 in run 39867221684: planning in 3m52s and registered package replay in 1m16s. Run metadata confirms the exact head, completed status and success. The checks were read once at completion. Registered replay executes both warning-strict normal and optimized commands. Local checks, committed-byte/source-history preflight and Git-free export replay also passed. Hosted execution certifies these bounded replay checks; independent human and formal proof-kernel acceptance remain unclaimed.

Merge danger

Reversible package-only addition. Blast radius: this research package and its replay registration. Shared code and selected records are unchanged.