[petalas]

Summary

Scoped proof with exact finite corroboration: S distinct-bank monomial ADD scans and unary gates of total row-support exponent R can produce two banks of row support D only if 2^R F_(S+1)>=D. This tracks the two largest bank support sizes and strengthens the inherited single-maximum estimate. Arbitrary finite restored dirty helpers and arbitrary per-bank monomial maps are allowed.

before: one support maximum gives 2^(R+S) >= D
new:    two bank maxima give 2^R F_(S+1) >= D
L=3, R=0, two full chirps: necessary ADD count 3 -> 5

This takes the explicit constant-scan next question in Ross Richardson's merged #4023. His #4082 and #4057 retain the earlier support and paid-endpoint method credit. The new contribution is the two-bank invariant and its tradeoff, not a new general support-union or Fibonacci method. This contribution was produced by Nick Petalas. Only research/fibonacci-scan-floor/ is added; no selected-frontier change or new kappa is claimed.

Evidence

PASS_FIBONACCI_SCAN_FLOOR: 315884 enumerated support-relaxation layer states, 2304 exact F5 matrix prefixes, dense chirp tensor rows through seven bits, and ten rejection controls. Two algorithms corroborate the written induction; no predecessor executable is replayed. A generated complete receipt binds all mathematical fields to proof, code and metadata hashes. Boolean receipt changes, proof-byte drift and occupied receipt output reject; Git-free optimized replay passes.

Run from the repository root with CPython 3.12, standard library only:

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

Other machines may substitute their CPython 3.12 path. Both registered commands pass locally in 0.530 and 0.527 seconds on CPython 3.12.15, macOS arm64. Metadata/provenance, exact committed bytes/source history and official export checks pass at 1c98c7261ccb03e089c61f006c29ba0a47d3f42f.

Limitations

The count is a serial full-bank ADD count with distinct source and destination banks. General free bank GL, same-bank buffered additions charged as ordinary scans, multi-bank native gates, partial volumes and fused traversals require different models. An explicit invertible F5 same-bank example demonstrates that boundary. R is a sum of supplied row-support exponents; actual native recipes must establish their own support laws. No matching scan optimum, complete physical supplier, precision/tape price, selected cold replay, formal proof-kernel or independent human review is claimed. All registered obligations ran locally; full supplier and formal proof obligations remain outside scope.

Open next questions

• Determine the exact inter-bank ADD optimum between this bound and Prove sharp fixed-support unary-query bounds with restored dirty banks #4023's cited six-per-bit construction. Start at one bit with explicit coefficients and two banks; no matching construction is supplied here.
• Derive a weighted support bound for a declared partial-volume or fused traversal model. Its row-update and movement tariff must be specified first.

Related open work

• Prove six calls for the open three-value shear case (no new κ) #4875 counts same-phase native C/C^-1 calls for three dirty source-to-target shears. This package counts monomial ADD scans for two complete chirps under a different alphabet; it does not recompute that PR's optimum.
• Prove optimal weighted borrowed-source pair tours (no new κ) #4899 optimizes weighted borrowed-source tours for prescribed pair-parity targets. Its scalar row ADD count concerns a different endpoint and alphabet; no shared numerical quantity is computed here.

Live overlap covered 411 open PRs plus the refreshed recent-open list, latest handoff claims, all-main README searches and merged searches for the object, formula and Fibonacci terms. Sparse local note coverage and arbitrary diffs remain incomplete. No other open PR studying this exact two-bank chirp scan bound was found within that coverage. Canonical claim.

CI notes

Hosted Registered research packages run 39870058467 passed at 1c98c7261ccb03e089c61f006c29ba0a47d3f42f: plan and replay (research/fibonacci-scan-floor) / replay (research/fibonacci-scan-floor) both completed successfully. All registered checks also passed locally; none was deferred to CI. This certifies the registered replay, not independent mathematical acceptance.

Before DONE, the account's open PRs created since the advance CLAIM were refreshed and their titles and bodies read. None computes this same chirp scan bound; #4899 is recorded above as nearby ADD-cost work. No shared computed number disagrees.

Merge danger

Two-way door. The change is confined to one isolated research package. Shared code, frozen sources, predecessor credit and selected frontier files are unchanged.