Prove unbounded activation arity for parallel helper cuts (no κ)

research/higher-order-cut-activation/CODE-BOUNDS.md

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+# Executable bounds and implementation inspection
+
+Produced by Nick Petalas. Apache-2.0.
+
+The executable is calibration software. Its two modules have no recursion,
+dynamic evaluation, subprocesses, network use, background work or file writes.
+All functions are under 60 physical lines. Scalar words contain only signed
+ADDs on distinct valid coordinates. Scientific checks use explicit exceptions
+and remain active under optimized Python; internal assertions are additional
+checks, not the only mathematical acceptance conditions.
+
+The public constructors limit banks to eight, entries to 10^9 and scalar
+fixtures to d <= 49. The Euclidean loop has 64 iterations, exceeding its
+worst-case bound for two positive integers at most 10^9. At most eight Bezout
+steps are used. Literal words have at most 816 signed ADDs and 24 scalar
+coordinates. The registered verifier uses 57 families on at most six banks,
+602 proper subpools and 25 three-coordinate scalar words. Its largest minor
+oracle has 2^6-1 = 63 nonempty diagonal minors. No scientific loop grows its
+own input collection. Source/package hashing is bounded by the fixed committed
+file list; changing that list changes the authenticated payload.
+
+Rule 3, fixed memory capacity: CPython integers, tuples, lists and JSON require
+managed allocations. The code does not claim fixed allocation after startup.
+The input and loop bounds above limit live collections. Bezout coefficients
+are bounded by eight products of input-sized coefficients; scalar coordinates
+in the actual finite checks are fixed small integers. Follow-up, if this code
+becomes a physical execution engine, is a separately specified fixed-width
+representation with capacity/exhaustion checks. This diagnostic program is
+not that engine.
+
+Rule 9, indirect calls: rejecting controls pass bounded, explicit lambdas to
+one local helper. These targets are visible in verify_controls and live only
+for one call; there is no registry or external dispatch. The finite verifier
+also uses ordinary Python library iterators. Replacing the callbacks is only
+needed if this checker is ported to a runtime forbidding indirect calls.
+
+Rule 10, diagnostics: registered normal and optimized commands run with
+`-I -W error -B`; the direct syntax inspection uses ast.parse and compile with
+warnings treated as errors. No third-party linter or static type checker is
+required by this standalone standard-library package. Repository package
+metadata, source pins, whitespace, committed-byte/history and scientific
+replay checks supply the recurring verification boundary.

research/higher-order-cut-activation/PROOF.md

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+# Arbitrarily high activation arity at a parallel helper cut
+
+Produced by Nick Petalas. Apache-2.0.
+
+## Contract and credited premise
+
+Let D = Z[1/2]. For a reversible scalar prefix A and suffix B = A^-1,
+fix the coordinate support G of helper inputs/outputs and eligible cut
+support F. The eligible rows are all v supported in F with vA supported
+in G. The eligible columns are all u supported in F with Bu supported
+in G. Choose full D-module bases L0 and R0 and write K = L0 R0.
+
+A t-channel normalized algebraic interface is X K Y = I_t over D.
+Ali Romman's PR5615 proves that its maximum t is
+
+    c(K) = min over odd primes p of rank over F_p of (K mod p).
+
+We cite that single-cut theorem, including its requirement for full module
+bases. It is not a result reverified by this contribution. Smith normal
+form and Bezout identities are classical. The new claims below concern
+activation arity under parallel composition and a causal scalar realization.
+No native frame, Gaussian response, independent productive target or free
+basis conversion is part of this scalar contract.
+
+## Theorem 1: activation arity is unbounded
+
+For every integer q >= 2, there are q independent reversible scalar banks
+such that every proper nonempty subpool has capacity zero and their full
+parallel pool has capacity exactly one. All inputs, including excluded
+coordinates, are arbitrary, and the complete scalar word restores them.
+For q = 2 the algebraic phenomenon is already illustrated by PR5615's
+diag(3,5) example. The arbitrary q statement and scalar banks are the
+present contribution.
+
+Choose distinct odd primes p_1,...,p_q. Put P = product_i p_i and
+
+    d_i = P/p_i,       K = diag(d_1,...,d_q).
+
+If J is a proper nonempty index subset, choose j outside J. Every d_i for
+i in J is divisible by p_j. Reducing K_J modulo p_j gives the zero matrix,
+so c(K_J) = 0. This obstruction applies to arbitrary dyadic row/column
+combinations within that subpool, not just to its individual entries.
+
+For the full pool, gcd(d_1,...,d_q) = 1: every prime divisor of P is absent
+from one of the d_i, and there are no other prime divisors. Bezout therefore
+provides integers b_i with sum_i b_i d_i = 1. Taking X = (b_1,...,b_q) and
+Y = (1,...,1)^T gives X K Y = 1. Hence c(K) >= 1.
+
+For any j, the reduction of the full K modulo p_j has exactly one nonzero
+diagonal entry, d_j. Its rank is one, so c(K) <= 1. This proves the exact
+capacity and every proper-subpool exclusion. Section 3 realizes each d_i
+as the full pairing of an actual reversible scalar bank, completing the
+causal assertion.
+
+Consequently, for every fixed h >= 1, testing only subpools with at most h
+banks cannot justify excluding all joint channels: choose q = h+1. This
+is a statement about the declared algebraic parallel-cut model. It does
+not invalidate a census with a stronger source-specific theorem limiting
+which banks can share a physical chart or time.
+
+## 3. Reversible scalar bank with full pairing [d]
+
+For any positive integer d, take three arbitrary scalar coordinates
+(x_0,x_1,z). The prefix consists of ordinary unit-coefficient ADDs:
+
+    x_0 += z
+    x_1 += z
+    z   += x_0
+    repeat d-1 times: z += x_1
+
+It has d+2 ADDs and column-convention matrix
+
+    A_d = [[1, 0, 1],
+           [0, 1, 1],
+           [1, d-1, d+1]].
+
+The suffix reverses the prefix order and negates each ADD, so it has the
+same cost and restores all three arbitrary input values. In particular,
+
+    B_d = A_d^-1 = [[2, d-1, -1],
+                    [1, d,   -1],
+                    [-1, 1-d, 1]].
+
+At the cut and boundary take F = G = span_D(e_0,e_1), excluding z.
+A row (a,b,0) is eligible exactly when its excluded prefix coordinate
+vanishes, a+b=0. Thus its full module basis is l = (1,-1,0).
+A column (a,b,0)^T is eligible exactly when -a+(1-d)b=0. Its full module
+basis is r = (d-1,-1,0)^T. These descriptions parameterize every solution
+over D directly; there is no independently rescaled rational nullspace.
+The pairing l r equals d. Both generators are primitive. In fact lA_d=l
+and B_d r=r, so the displayed boundary supports hold exactly.
+
+Parallel banks are disjoint coordinate blocks. Their eligibility equations
+separate by block, even when candidate vectors use arbitrary combinations
+across banks. The full eligible modules are therefore the direct sums of
+the displayed modules, and their pairing is exactly diag(d_i). Every bank
+prefix precedes one common scalar cut and every inverse suffix follows it.
+The total scalar ADD cost is 2 sum_i(d_i+2), on 3q arbitrary coordinates.
+
+This is a complete ordinary scalar invoice for the calibration word. It
+is not the native or recursive invoice of a productive helper supplier.
+The word itself is identity and earns no productive credit by that fact.
+
+## 4. Concrete three-bank certificate
+
+Take (d_1,d_2,d_3) = (15,21,35). Their pairwise gcds are 3,5,7 respectively,
+so all three pairs have capacity zero; each singleton also has capacity
+zero. The full pool has the unit certificate
+
+    (1,1,-1) diag(15,21,35) (1,1,1)^T = 1.
+
+Modulo 3,5 or 7 its rank is one, proving optimality. Its integer Smith
+factors are 1,105,105: the gcd of entries is 1, the gcd of nonzero 2-minors
+is 105 and the determinant is 11025.
+
+The nine-coordinate causal realization costs
+
+    2((15+2)+(21+2)+(35+2)) = 154 signed ADDs.
+
+The pooled eligible row is (l,l,-l) and column is (r_15,r_21,r_35)^T.
+Their pairing is one. The suffix restores all nine arbitrary coordinates;
+no zero-initialized workspace, source consumption, erasure or discarded
+history is used. Preparing a physical common helper frame and executing
+these cross-bank combinations still requires its own charged construction.
+
+## Verification boundary
+
+The dimension-uniform assertions are the written proof above. The finite
+checker corroborates 57 prime families from subsets of {3,5,7,11,13,17},
+all 602 proper nonempty subpools, 57 Bezout witnesses, 25 literal scalar
+banks at odd d between 1 and 49, and the complete nine-coordinate word.
+It computes determinantal divisors without the predecessor Smith program,
+checks exact prefix/inverse matrices, full dirty basis columns, support
+identities, primitive generators and explicit invalid controls. It imports
+no predecessor executable and performs no predecessor scientific replay.

research/higher-order-cut-activation/README.md

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+# Helper channels can require arbitrarily many parallel banks
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+**Scoped proof with exact finite corroboration; no new kappa.** For every
+q >= 2, q independent reversible scalar helper banks have zero dyadic unit
+capacity in every proper nonempty subpool and capacity exactly one jointly.
+The first new arity is three:
+
+    every singleton and pair of [15], [21], [35]: capacity 0
+    diag(15,21,35): capacity 1, since 15+21-35 = 1
+
+The proof constructs d_i = product_{j != i} p_j for distinct odd primes.
+An omitted prime obstructs each proper subpool. An integral Bezout identity
+attains the full channel, and reduction modulo any chosen prime proves its
+optimality. Each [d_i] is the full saturated pairing of an explicit
+three-coordinate reversible scalar bank. The three-bank fixture restores
+nine arbitrary dirty inputs in exactly 154 signed unit-coefficient ADDs.
+
+[PROOF.md](PROOF.md) states the contract and all-size argument.
+[activation.py](activation.py) constructs the finite witnesses and literal
+scalar words. [verify.py](verify.py) uses a separate determinantal-divisor
+oracle and exact dirty-input replay. It emits a computed stdout receipt
+bound to SHA256 hashes of package files and the credited source input.
+
+The practical implication is limited but concrete: bounded-arity grouping
+cannot certify that arbitrary parallel helper banks have no algebraic unit
+channel. A search needs either a source-specific restriction on joint banks
+or their full prime obstruction data. This construction does not assert
+that the selected P104 word contains these banks or can place them in one
+legal common native chart.
+
+## Reproduce
+
+From the repository root, use CPython 3.12 and its standard library:
+
+```sh
+/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
+```
+
+Another CPython 3.12 path may replace the absolute path. Expected markers
+are `PASS_HIGHER_ORDER_CUT_ACTIVATION`, `PASS_REGISTERED_RESEARCH_PACKAGE`
+and `PASS_RESEARCH_COMMIT_BYTES`.
+
+Finite coverage is 57 prime families, all 602 proper nonempty subpools,
+57 attaining unit witnesses, 25 literal scalar cut fixtures, the 154-ADD
+nine-coordinate forward/inverse identity, and 16 rejecting controls.
+Normal and optimized commands must agree on every computed result.
+
+## Scope, costs and remaining obligations
+
+The changed contract is parallel pooling, with arbitrary cross-bank dyadic
+row/column combinations at one scalar cut. There is no tensor multiplication.
+Ali Romman's PR5615 single-cut capacity theorem and its diag(3,5) example
+are credited premises. We add unbounded activation arity and its causal
+scalar realization, not a new Smith theorem or a new two-bank phenomenon.
+The new question was posed directly at the coverage-law-change bottleneck;
+it was not selected from a predecessor's open next questions.
+
+The scalar prefix/suffix and arbitrary-input restoration are proved and
+implemented. Common native charts, physical carrier delivery, preparation
+of joint bases, independent productive Gaussian targets, precision/growth,
+stock, compiler and the complete paid/target invoice are **unproved**.
+The scalar ADD invoice does not price those missing obligations. No selected
+scientific replay, native supplier replay, Lean, Docker, rounded multiplier,
+independent human review or formal proof-kernel review was performed.
+All five selected framework assumptions remain. No frontier change is made.
+
+Executable calibration bounds are at most eight banks, positive entries
+through 10^9 and literal scalar fixtures through d=49. The written theorem
+has no such bounds. Guard rejection is not a mathematical obstruction.
+[CODE-BOUNDS.md](CODE-BOUNDS.md) records the managed-runtime deviations and
+finite resource bounds.
+
+## Sources and overlap
+
+Base canonical main is a5bff3bf133e482541c09eba4e773884bdb2b1b7, priority 8.
+The selected record resolves to conditional kappa .003125804; earlier values
+in priority prose do not define this package's baseline. The source-bound
+baseline passed without a selected scientific replay.
+
+[Advance CLAIM](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19279640).
+Live handoffs, active Discussions, open/merged PRs, recent account work and
+local reservations were checked. Main README searches covered direct sums,
+parallel pooling, complementary primes, higher-order activation, proper
+subpools and the three diagonal entries. The cached overlap report covered
+338 open PR records and 5,947 discussion records, but only 93 of 8,623 local
+note paths; this is bounded overlap evidence, not proof of global novelty.
+No matching arbitrary-arity parallel activation result appeared.
+
+Credit Ali Romman's PR5615 single-cut capacity criterion, PR5602/5611 helper
+handoff/completion, PR5599, and their retained P104/controller/native lineage.
+The algebraic two-bank activation diag(3,5) is explicitly predecessor work.
+Nick Petalas's PR5882 concerns tensor capacity, while zxv's product-module
+result concerns factored tensor supports; those are context with a different
+composition contract, not results rechecked or extended by this package.
+Bezout, prime divisibility and Smith normal form are classical. Source bytes
+are pinned in [SOURCES.json](SOURCES.json). No predecessor credit is changed.
+
+## Open next questions
+
+1. Does one actual source word have three compatible simultaneous banks with
+   complementary prime obstructions and a legal shared native chart? Extract
+   their full saturated modules first. Physical timing and carrier costs are
+   the blockers; a positive algebraic pairing is insufficient.
+2. Can the cross-bank unit witness be delivered as an independent productive
+   target at a favorable complete signed bill? Start with the nine-coordinate
+   fixture and charge both basis preparations, restoration and native maps.
+   No favorable bound on those missing costs is claimed here.

research/higher-order-cut-activation/SOURCES.json

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