[petalas]

Summary

Scoped written proof with exact finite corroboration, produced by Nick Petalas. For H_N on m independent columns, a one-head linear tape machine carrying one field register has these layout-dependent travel floors:

address-major H_N tensor I_m:  (Nm)^2/2 unit moves
column-major I_m tensor H_N:   m N^2/2 + 2(m-1) unit moves
N=8, m=4:                     512 versus 134 lower bounds

The package proves every spatial cut rank and a closed-form bound for every positive register capacity s, over any field of characteristic other than two. The extra inter-column travel is charged even where communication rank is zero. These are lower bounds, not optimal schedules. With fixed N and s, address-major storage cannot have linear-in-m travel in the stated machine.

The communication-rank method is classical. The new result is the exact pair of layout profiles and their one-head travel conversion. This is an original side result for precision_tape, with an explicitly changed card scope. It neither audits a predecessor nor discharges the selected multitape assumption. No new κ or selected-frontier change. Credit zxv #5441 for the representation-versus-physical-cost distinction, ToikY #4192 for the separate sequential compiler, and the classical linear-communication/Walsh background cited in the README. Advance CLAIM.

Evidence

• Before: algebraic column repetition alone supplies no travel invoice in this restricted machine. After: an all-size linear-span proof gives both exact layout profiles and their register-dependent movement floors.
• Exact Fraction elimination checks both directional ranks at 568 cuts in 32 literal matrices; direct summation checks 224 register-dependent closed forms.
• Eighteen local butterfly programs execute 612 gates and 2,358 unit moves, restore an arbitrary dirty register, satisfy both directional demands at 150 cuts, and pass full inverse-product checks. Eleven controls cover malformed dimensions, layout and travel omissions, dirty restoration and gate changes.
• Both warning-strict normal and optimized direct commands pass, locally timed at 0.254 and 0.307 seconds. These are check durations, not a speed comparison. Both registered commands pass; committed-byte and source-history checks pass on f7ecabe8e04738ee4fd6ac934363ca2c307d85b2. The worktree is clean.

Local environment: CPython 3.12.15, macOS arm64, standard library only. From the repository root:

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

Another CPython 3.12 executable may replace the local path. Expected markers are PASS_WALSH_TAPE_CUT_AREA, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES. Receipts are generated on stdout and bind the five package files by SHA256; default replay writes nothing.

Limitations and unrun obligations

One sequential head, s carried field registers, fixed input-independent linear control, in-place outputs and independent restored scratch are hypotheses. Multiple heads/tapes, changed placement, nonlinear encoding or source-dependent side information can escape the result. No production layout, bit-cost encoding, rounded implementation, complete native caller, optimal schedule, selected cold replay or complete multiplication theorem is supplied. The written proof supplies universal quantifiers; finite checks corroborate it. Independent human and formal proof-kernel review are unclaimed. No required local package check was deferred to CI.

Related open work

• Audit uniform Beneš compiler with charged sequential arena scans #4192 lowers a Beneš compiler with explicit sequential scans; this package proves a different lower bound for repeated Walsh data transforms.
• Add all-width native certificate for the selected #5599 word (native_all_columns; no new kappa) #5974 establishes selected native identities and all-width lifting; it does not price the one-head layout studied here. Its result is cited as context, not rechecked.
• Reduce signed Hadamard lowering from 222 to 208 primitives (no new κ) #5945 constructs a cheaper signed Hadamard connector in a recursive-query alphabet; its primitive-call price is distinct from physical cell movement.
• Prove the exact input-dimension price of unequal Fourier reads #5946 classifies restricted second-input Fourier reads; this package keeps all columns arbitrary and changes the physical tape layout.
• Prove the full n-qubit Clifford tensor threshold is at least 2n+1 (no new κ) #5967 bounds Clifford tensor-independence width; its Gram threshold is distinct from spatial tape cut ranks.
• Prove three Gaussian blocks need five internally restoring stages (no κ) #5981 counts internally restoring moving Gaussian stages; its block-permutation endpoint and stage count differ from local linear tape travel.
• Lower the G20 carry leaf to 1115V/3 transitions and 24V cells (no new κ) #5985 improves a G20 carry leaf's physical transition and storage bounds; it does not implement or bound an in-place Walsh transform.
• Prove sharp 7TP table travel for skew-radix cached chirps (no new κ) #5987 prices unit head movement for a skew-radix cached chirp table; it studies read-only table traversal rather than the Walsh data-transform cut profile.

The live open list and Discussion claims were refreshed before committing, alongside merged-PR and all-main README searches. No PR studying the same spatial-cut formula was found within that coverage. Unpublished work and arbitrary diffs are outside it; the sparse local note cache is incomplete. Completion refresh inspected all 11 account open PRs created since the CLAIM, including this PR. The ten other submissions report no matching spatial-cut formula; no extra computations were run to compare their results.

Open next questions

1. Construct matching-order restored head schedules for these layouts, beginning with complete consecutive column programs. The present literal programs establish feasibility, not optimal growth.
2. Bind one actual caller's layout, carried-register budget and extra head/tape channels to decide whether this restricted obstruction applies.

CI notes

Both hosted checks passed for f7ecabe8e04738ee4fd6ac934363ca2c307d85b2 in CI run 40570458516: plan and replay (research/walsh-tape-cut-area). The completed run has conclusion success and its head SHA matches the PR head. Both registered normal/optimized checks and committed-byte/source-history validation also passed locally. No required local check was deferred to CI.

Merge Danger

Door: two-way.

Blast Radius: research.

Adds one isolated five-file package and its replay declaration. Shared code and frozen evidence are unchanged.