The paper studies whether four natural axioms—sequential composability, nullity, continuity, and records—force exact no-signaling in operational strategies. It proves that, for strategies with Lüders residual states, composability and nullity alone imply strong structural properties, so signaling can arise only through a remote unitary; moreover, any uniform signaling bound smaller than one already guarantees exact no-signaling. A first-order analysis around the Born rule identifies a constrained family of perturbations governed by a cocycle equation. Within guarded quadratic purity–overlap functionals, the record axiom removes overlap and marginal-distance terms, leaving six exact invariants, although three could signal; the cocycle eliminates all six. Finally, an analytic order-by-order argument extends this rigidity, reducing the problem to a single precisely formulated infinitesimal question.
The paper studies whether four natural axioms—sequential composability, nullity, continuity, and records—force exact no-signaling in operational strategies. It proves that, for strategies with Lüders residual states, composability and nullity alone imply strong structural properties, so signaling can arise only through a remote unitary; moreover, any uniform signaling bound smaller than one already guarantees exact no-signaling. A first-order analysis around the Born rule identifies a constrained family of perturbations governed by a cocycle equation. Within guarded quadratic purity–overlap functionals, the record axiom removes overlap and marginal-distance terms, leaving six exact invariants, although three could signal; the cocycle eliminates all six. Finally, an analytic order-by-order argument extends this rigidity, reducing the problem to a single precisely formulated infinitesimal question.
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.
We study operational strategies, consisting of outcome laws and residual states, assigned to measurements of subsystems of a pure state and required to satisfy five exact axioms: sequential composability, nullity, continuity, equivariance, and records. We then examine what changes when no-signaling is weakened to a quantitative tolerance delta. There are two inequivalent versions: a one-shot tolerance imposed separately for each configuration, and a uniform tolerance required across adaptive procedures of every size. We prove five main results. First, composition obeys a sharp modulus: the error is delta for coarse bipartitions and 2 delta for pairwise compositions, and both bounds are tight. Second, for strategies that factor across tensor products, any uniform delta-no-signaling condition with delta less than 1 already implies exact no-signaling, by Hellinger amplification. Third, constraints based on causality, reliability, or cryptography can impose only an upper limit on delta, never a positive lower limit, because the Born strategy satisfies all of them exactly when delta is zero. Fourth, the exact axioms together with one-shot delta-no-signaling imply quantitative closeness to the Born rule: the total-variation distance is bounded by a function eta sub d of delta that scales as the cube root of Lambda squared d times delta; this yields a signaling bound of at most twice eta sub d and Hausdorff convergence of the admissible strategies to the Born strategy. Finally, we construct an explicit purity-signal strategy satisfying nullity, continuity, equivariance, and records exactly while changing a remote outcome law by precisely epsilon divided by four. Thus, all substantive no-signaling content is carried by sequential composability, leaving a single precisely formulated open problem.
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.
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
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.
No-signalling constraints on isolated outcome probabilities do not determine whether an apparatus preserves the joint law of its accessible records under repeated readout, network composition, or reuse. We study these extensions through a conditional-record criterion and four explicit models. For a wrapped-Cauchy retrocausal source, protection under two conditionally independent continuous readouts forces an angle-independent response; three readouts suffice without continuity. A finite angular-bandwidth assumption gives a quantitative leakage bound, whereas continuity alone gives no uniform stability. At a specified two-source Bell junction, the required correction matrix has full nonnegative rank on every even uniform angular grid. A Gaussian controller can cancel branch-mass dependence while leaving a distinguishable joint record, for which we derive exact noise and law-change tradeoffs. Finally, a fixed singlet with shared classical noise reproduces the variable-width isolated outcome family, but retained and refreshed controllers generate different temporal statistics. We prove a finite-resource obstruction and construct positive finite-controller approximations. Numerical studies comprising 8.3 million independent simulated histories support the model-specific predictions. Together, the results identify conditions that must accompany marginal no-signalling when extending an apparatus to richer records and interventions.