[petalas]

Summary

Scoped proof with exact finite corroboration: parallel dyadic helper-cut activation can require arbitrarily many banks. For every q >= 2, there is a reversible scalar construction with capacity zero on every proper nonempty subpool and capacity exactly one on the full pool. The new arbitrary-arity construction includes a nine-coordinate, 154-ADD dirty-input fixture. No new kappa or selected-frontier change.

[15]     [21]     [35]          each capacity 0
any two of these banks          capacity 0
all three, paired jointly       capacity 1: 15 + 21 - 35 = 1

The proof chooses d_i = product of all distinct odd primes except p_i. Every omitted-bank prime obstructs the corresponding proper pool; Bezout attains the full channel and a rank-one reduction proves optimality. Explicit reversible three-coordinate banks realize the full saturated pairing [d_i]. This shows why bounded-arity grouping needs an additional source-specific restriction before it can exclude all algebraic joint channels.

Only research/higher-order-cut-activation/ is added. This contribution was produced by Nick Petalas. Credit Ali Romman's PR5615 single-cut theorem and diag(3,5) example, PR5602/5611/5599 and the retained helper/native/controller lineage. Bezout, prime divisibility and Smith normal form are classical. The two-bank phenomenon is predecessor work; the new result is unbounded activation arity with a causal scalar realization.

Advance CLAIM. Base canonical main a5bff3bf133e482541c09eba4e773884bdb2b1b7, priority 8, selected record .003125804. Live overlap, merged/main searches and reservations were refreshed before the commit; no matching higher-order parallel-pooling result appeared in that bounded coverage.

Evidence

• Before: the credited single-cut result already exhibited two-bank algebraic activation. After: the written proof gives every finite activation arity, a full-module scalar construction and the limitation of bounded-arity grouping.
• The new checker passes 57 prime families, all 602 proper nonempty subpools, 57 integral attaining witnesses, 25 literal scalar banks, the complete nine-coordinate identity in both orders, and 16 rejecting controls. Its determinantal-divisor oracle imports no predecessor executable and performs no predecessor scientific replay.
• Warning-strict normal and optimized direct outputs are identical. Each registered command took approximately 0.024 seconds locally on CPython 3.12.15/macOS. Metadata and source pins pass. These durations establish a bounded replay, not a performance comparison.
• Committed bytes, modes and source history pass at dd40722fba14ba7446a1b1d58b9c78170c739721. Official export and declared-input coverage pass. Both direct modes pass with empty PATH in a read-only Git-free export; all ten exported files, including its manifest, retain their bytes and modes. A changed-source export rejects before mathematical execution. Receipts are computed on stdout and bound to file hashes.

Exact reproduction from the repository root:

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

Another CPython 3.12 path may replace the absolute interpreter path. Expected markers are PASS_HIGHER_ORDER_CUT_ACTIVATION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

This is a scalar algebraic parallel-cut contract. It allows arbitrary joint dyadic row/column combinations in one common scalar cut, while tensor multiplication is absent. It is not a counterpart obtained by changing the group or target of a tensor-cut construction. The fixture itself is an identity word; algebraic unit capacity supplies no productive Gaussian credit by itself.

Common legal native charts, physical timing/carriers, both basis preparations, independent productive targets, precision/growth, storage, compiler and the complete paid/target invoice remain unproved. The 154 scalar ADDs do not price those missing obligations. No actual selected-word activation, physical gain, native supplier replay, selected scientific replay, Lean, Docker or complete rounded multiplier is claimed. The written theorem is universal; the executable calibration domains are explicitly finite. All five framework assumptions remain. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Find three compatible simultaneous banks in one actual source word. First extract their full saturated modules and legal common native chart/time. Physical delivery and routing costs block a productive claim.
2. Price both joint basis preparations, native maps and restoration for the nine-coordinate fixture. A favorable complete signed invoice is required before expanding this into a supplier construction; none is supplied here.

Related open work

No inspected open PR computes the same higher-order parallel scalar-cut capacity. The completion refresh found no open account PR created since the advance CLAIM at 2026-10-29T22:10:48Z. No competing common value was found or recomputed. Adjacent helper work has different contracts:

• Conditional κ 0.00309871 from a prepared P104 split helper #5606 prepares a particular P104 helper and debits its productive target. This contribution neither prepares that helper nor computes its paid moment; it supplies an abstract simultaneous-bank capacity construction.
  Merged adjacent work Independent audit of #6144 and #6167: replay, deltas, invoice, full dense native, caller and funding all pass (κ 0.003128645 corroborated, no new κ) #6214 independently audits the PR6144/6167 P104 words and their full bills. This contribution cites no new activation in those words and performs no comparison or replay of their reported kappas.

Merged context PR5882 concerns tensor capacity, and the retained product-module theorem concerns factored tensor supports. The present composition is a direct sum with joint cross-bank combinations. Their results and open questions are not this deliverable.

CI notes

Both hosted checks passed at exact contribution head dd40722fba14ba7446a1b1d58b9c78170c739721 in run 40964508904: plan in 4m34s and replay (research/higher-order-cut-activation) in 2m48s. The completed checks were read once, and run metadata confirms completed success at that same head. Hosted replay passed the registered bounded package checks. Local normal/optimized, metadata, committed-byte/mode/source-history, official export and read-only Git-free checks also passed. The universal written theorem and unrun physical/supplier obligations retain their stated scope; independent human and formal proof-kernel review remain unclaimed. A maintainer has merged this PR.

Merge Danger

Door: two-way.

Blast radius: package. Eight isolated new files and opt-in replay only; shared code, pinned predecessors and the selected record are unchanged.