The finite-resource programme for the Enter calculus [EC] left exactly one question open, stated twice: whether diagonal-unfaithful update slack survives record chains at full depth (Open Problem G at capacity three; equivalently, by Corollary L of the companion paper [FG], the unconditional value of Open Problem 17.2). This paper closes it, in the rigidity direction, and computes the exact necessity profile of the Born marginal. Two results are proved. (1) Capacity two is free. The odds-ratio-preserving Plackett transport of the Born law — bend every diagonal margin through a net-compatible phi, keep every association invariant — extends to a strategy S_phi models_2 (C, NS, N, K, R) satisfying every axiom instance expressible at capacity two, at every protocol depth (Theorem 4.7), and deviating at the witness by phi(1/3) - 1/3 = 2 eta / 27 > 0. (2) Capacity three is rigid. A purely combinatorial argument — nullity on the support of a generalized W state, plus a full-capacity marginal-transfer lemma that needs no record — forces sum_i phi(w_i) = 1 on the open two-simplex, hence Cauchy’s equation, hence phi = id at every capacity N >= 3, with no faithfulness hypothesis (Theorem 3.4). The transport class dies at capacity three with an explicit, computable defect 2 eta / 9 (Theorem 5.1); a conjectural depth-omega capacity-three deviant therefore does not exist, and the three-qubit extension system on which such a construction would rest is infeasible. Consequences (Theorem 6.1): Open Problem G(3) has a negative answer in the marginal sector, epsilon^*(3, omega) = 0 and the level-k hierarchy of Theorem J [FG] converges to zero; Open Problem 17.2 closes with N_0(1, (1,2)) = 3, so the four-qubit witness of [EC], Example 10.4, is not necessary; Theorem H [FG] prices the static construction class only and is strictly beaten by dynamic support forcing; and the faithfulness hypothesis of Theorem I [FG] is removable in the marginal sector. The exact profile is N_0(x) = 1, 2, 3 according as x in {0,1}, x = 1/2, or otherwise (Theorem 6.2) — in particular the realizable rational net governs static schemes, not necessity. What remains is a single sharpened question, the association sector (Open Problem 6.3). Part II extends the artefact with support_forcing.py: the transport strategy as a capacity-two feasibility certificate, the W-defect and support-forcing residual as executed refutations at capacity three, the executed Cauchy derivation of phi = id, and an association-sector probe.
The finite-resource programme for the Enter calculus [EC] left exactly one question open, stated twice: whether diagonal-unfaithful update slack survives record chains at full depth (Open Problem G at capacity three; equivalently, by Corollary L of the companion paper [FG], the unconditional value of Open Problem 17.2). This paper closes it, in the rigidity direction, and computes the exact necessity profile of the Born marginal. Two results are proved. (1) Capacity two is free. The odds-ratio-preserving Plackett transport of the Born law — bend every diagonal margin through a net-compatible phi, keep every association invariant — extends to a strategy S_phi models_2 (C, NS, N, K, R) satisfying every axiom instance expressible at capacity two, at every protocol depth (Theorem 4.7), and deviating at the witness by phi(1/3) - 1/3 = 2 eta / 27 > 0. (2) Capacity three is rigid. A purely combinatorial argument — nullity on the support of a generalized W state, plus a full-capacity marginal-transfer lemma that needs no record — forces sum_i phi(w_i) = 1 on the open two-simplex, hence Cauchy’s equation, hence phi = id at every capacity N >= 3, with no faithfulness hypothesis (Theorem 3.4). The transport class dies at capacity three with an explicit, computable defect 2 eta / 9 (Theorem 5.1); a conjectural depth-omega capacity-three deviant therefore does not exist, and the three-qubit extension system on which such a construction would rest is infeasible. Consequences (Theorem 6.1): Open Problem G(3) has a negative answer in the marginal sector, epsilon^*(3, omega) = 0 and the level-k hierarchy of Theorem J [FG] converges to zero; Open Problem 17.2 closes with N_0(1, (1,2)) = 3, so the four-qubit witness of [EC], Example 10.4, is not necessary; Theorem H [FG] prices the static construction class only and is strictly beaten by dynamic support forcing; and the faithfulness hypothesis of Theorem I [FG] is removable in the marginal sector. The exact profile is N_0(x) = 1, 2, 3 according as x in {0,1}, x = 1/2, or otherwise (Theorem 6.2) — in particular the realizable rational net governs static schemes, not necessity. What remains is a single sharpened question, the association sector (Open Problem 6.3). Part II extends the artefact with support_forcing.py: the transport strategy as a capacity-two feasibility certificate, the W-defect and support-forcing residual as executed refutations at capacity three, the executed Cauchy derivation of phi = id, and an association-sector probe.
The Enter calculus [EC] derives the Born rule and the projective update from five operational requirements: composability, no-signaling, nullity, continuity, and records. This paper equips the calculus with a strictly linear resource geometry and develops its finite-resource metatheory, stating each result at precisely the strength established. An update grammar of four primitive moves makes every cut-indexed causal potential a conservation law, as shown in Theorem E through a case analysis of the grammar, and derives scalar monotonicity rather than assuming it. A latent-active split with fully specified typing rules turns monogamy of causal change into a typing invariant, as established in Theorem F. Resource-relative admissibility, with its quantification made explicit, is downward closed along runs, ensuring that the forcing threshold is well defined. The minimal register cost of the decohered fine-graining construction is computed in closed form in Theorem H, and the resulting realizable rational net improves the capacity-four rigidity mesh from the vacuous value delta equals one to delta star equals one-sixth in Theorem C star. Two rigidity layers are proved unconditionally: boundary rigidity, consisting of monotonicity, antipodal symmetry, the Sandwich and Seal lemmas, and a descent proposition, which holds for every admissible strategy without an update hypothesis; and diagonal-faithful rigidity, established in Theorem I through a fully detailed single-qubit case with an explicit Pexider argument and a multi-qubit induction, forcing the Born rule on every sub-capacity configuration at every capacity N greater than or equal to three. The residual question—whether diagonal-unfaithful update slack can survive at full depth—remains unresolved. Instead, it is reduced in both directions to sharply delimited objects: an effective hierarchy of relaxations whose finite levels upper-bound the residual freedom and make rigidity refutable at a finite level, as shown in Theorem J, without claiming that the limit is attained; and a compensated transport family based on odds-ratio-preserving Plackett transports, for which the two-qubit layer, the record-family seal lift in closed form, and the full-support three-qubit extension on compact sets are proved. This family satisfies every axiom instance generated by record-chain protocols from two-qubit initial configurations at every depth, as established in Theorem M prime. Consequently, no capacity-three rigidity proof can proceed through record chains alone. Full capacity-three admissibility reduces to a single explicitly stated Extension Conjecture; under that conjecture, Open Problem G three has a positive answer and the minimum threshold N zero of one and the pair one, two equals four. Depth-two and depth-three record probes pass at machine precision, and an eta-scan Pareto front reaches the axis numerically. Part II specifies a reproducible artefact in which every major claim is either proved, reduced to a finite check, or implemented as a property test or golden regression over a rich class of instances, with each item’s status—Theorem, conditional result, conjecture, or probe—labelled explicitly.
The Enter calculus [EC] derives the Born rule and projective update from five operational requirements: composability, no-signaling, nullity, continuity, and records. This paper equips the calculus with a strictly linear resource geometry and develops its finite-resource meta-theory as a single coherent framework. An update grammar with four primitive moves makes every cut-indexed causal potential a conservation law, as stated in Theorem E, and derives scalar monotonicity rather than assuming it. A latent-active split turns monogamy of causal change into a typing invariant and reconciles token activation with monotone coarsening, as shown in Theorem F. Resourcerelative admissibility is downward closed along runs, as stated in Lemma 4.2; hence forcing is upward monotone in capacity and the forcing threshold is well defined. The minimal register cost of the decohered fine-graining construction is N_F^0(s, m⃗) = s + ceiling(log₂ M) + ceiling(log₂(max_k m_k)), as proved in Theorem H. The resulting realizable rational net strictly refines the dyadic mesh and improves the rigidity bound at capacity 4 from the vacuous value δ = 1 to δ⋆ = 1/6, as shown in Theorem C⋆. The central new result is a rigidity theorem: a multiplicative functional equation extracted from the sequential cocycle with one qubit of headroom forces every admissible strategy whose update diagonal is Lüders-faithful to agree exactly with the Born strategy on every sub-capacity configuration, for every capacity N ≥ 3, as established in Theorem I. The proof is domain-explicit: a witness state is constructed with specified amplitudes, normalisation, and Lüders weights; the cocycle is derived on its exact open domain of validity; an overlapping-interval argument reduces the relation to a function of the product; and the resulting continuous Pexider equation is solved from first principles. Consequently, all residual finite-capacity freedom is confined to full-capacity configurations and to diagonally unfaithful update slack, as stated in Corollary K. Open Problem 17.2 of [EC] is therefore equivalent to a diagonally unfaithful instance of Open Problem G at capacity 3, as shown in Corollary L, while an LP hierarchy converts the remaining question into a computable limit, as established in Theorem J. An exact Choi-rank ledger assigns costs to geometric instruments and identifies decoherence rate with capacity flux, as shown in Theorem D′. Part II specifies a reproducible computational artifact in the style of the Enter-calculus repository: every theorem is either executed as a derivation or enforced as a runtime invariant, and every open problem is represented by a numerical probe.
An operational strategy assigns to each pure state and each subregister an outcome law for a computational measurement together with a residual state. We study the axioms of sequential composability (C), nullity (N), continuity (K), and records (R), and ask whether these axioms already force exact no-signaling. Three results are proved. First, for strategies with Lüders residuals, sequential composability and nullity alone imply marginalisation, tensor stability, and block factorisation. Consequently, only a remote unitary can ever produce signaling, while uniform delta-no-signaling for any delta less than 1 collapses to exact no-signaling, without assuming factorisation. Second, we linearise the strategy around the Born rule. The first-order data is described by a single family of guarded functionals satisfying an explicit four-term cocycle relation, whose transport form has three independent instances for each state and pair of sites. Within the class of guarded quadratic purity-overlap functionals, the record axiom removes exactly the overlap and marginal-distance directions, leaving a six-dimensional space of exact record invariants, three of which are signaling. The cocycle condition annihilates all six directions. This is established by a finite exact rational computation carried out by hand. Third, an order-by-order bootstrap promotes the result to analytic rigidity, leaving one precisely formulated infinitesimal problem.
Redundant records make macroscopic coherence progressively harder to witness without altering any executable probability. We make this quantitative. Working from five imported statements of an operational calculus—Born uniqueness with projective update, the classification of compiled primitives acting after a record, single-level agreement, internal no-signaling, and the path chain rule—together with finite-dimensional linear algebra, we prove: (i) a screening lemma showing that a witness of level-k coherence at depth d must have support on every one of the c(k,d) registers carrying a copy of the record, with exact first-crossing horizons D* chain = k + B − 1 and D* bc = k − 1 + ceil(log2(B + 1)) for the two archetypal copy schedules; (ii) an approximate-compatibility theorem, dTV(Pcoh, Pcoll) ≤ sum over j ≥ 2 of ηj, with constant one, where ηj is a state-dependent commutator norm, together with a two-parameter family showing that the optimal universal constant lies in the interval [1/√2, 1] and that no quadratic improvement is possible; (iii) an imperfect-record bound of √(p0 p1) times the product over i not in A of γi in the effect normalization, sharp for two-branch families, with effective horizons shifted by an explicit redundancy term L(γ, δ); (iv) light-cone and communication lower bounds, the latter requiring exactly c(k,d) − 1 classical bits; and (v) an infinite-tower analysis separating existence, convergence, block-diagonal superselection, and tail coherence, with divergence criterion given by the sum over j of (1 − γj) diverging. Every statement carries an explicit status label, and a self-contained computational artefact reproduces every quoted number.
The quantum structure sub-programme comprises three companion papers on the binary icosahedral group 2I, plus one independently published result. Q1 proves an exact, unconditional Fourier-support rigidity theorem in the finite Weil representation of the Heisenberg group, with a precise no-go corollary for support-, rank-, and Gram-based diagnostics; it establishes no phase coherence, singlet correlator, Tsirelson bound, or Born rule. Q2 proves an exact finite-group fact: the spin-21 and spin-23 representations of 2I induce the same Cayley-graph Laplacian eigenvalue; it establishes no Casimir/isotropy normalisation, Born rule, Tsirelson-type bound, or SU(2) fixed-point/sector-selection argument. Q3 proves a theorem conditional on an explicit, undemonstrated hypothesis: given that a bipartite state on a supplied composition is invariant under the diagonal action of 2I — not derived from admissibility or from Born–Infeld indiscernibility — the state is uniquely the SU(2) singlet, with an explicit Casimir correlator, for all five admissible sectors of 2I; it does not derive that invariance hypothesis, phase coherence, or the Born rule. No chain currently connects these three results to each other or to the structures of quantum mechanics. The Bell paper, published independently in Quantum Reports, establishes the structural non-applicability of Bell-type factorizability in non-injective effective descriptions; the proof of that central theorem uses only non-injectivity and informational completeness. This note records the current, narrow scope of each constituent and the open problems that remain.
We study a functional equation obtained by treating computational-basis outcome probabilities as an unknown evaluation rule. The rule is assumed to satisfy only diagonal anchoring, compatibility with partial traces and classical marginalization, and continuity. No record-creation or deferred-measurement axiom is imposed. We first classify all real-valued, not necessarily positive, solutions. Every such solution has a unique representation in terms of continuous scalar functionals indexed by subsystems. These functionals contribute through Walsh parity characters and may be interpreted as slack fields invisible to proper marginals. This result shows that uniqueness without positivity fails in an infinite-dimensional way. We next explain why diagonal-unitary twirling and envariance do not supply the missing constraint: the axioms contain no covariance principle, and explicit phase-dependent positive solutions exist through two qubits. Positivity nevertheless produces strong rigidity. Generalized GHZ states on three or more qubits obey the Born rule exactly, as does a one-bit value copied into at least two computational-basis records. Linear, fixed-pattern, site-local, and sequential deformations are excluded. The remaining problem is reduced to the existence of a continuous point in an explicit positivity region linking all finite subsystem levels.Keywords: Born rule; functional equations; quantum probability; GHZ states; marginal compatibility; Walsh transform; measurement records; quantum foundations