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.
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.
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.
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 Enter Calculus derives the Born rule and the Lüders update from five operational requirements on evaluation strategies. This manuscript studies the quantitative regime in which those requirements hold only approximately. The analysis identifies the local variables that may legitimately control a per-measurement tolerance, proves a linear residual-state estimate, and shows that outcome maps inherit an almost-Lipschitz modulus without requiring an independent Hölder assumption.The record axiom is separated into three inequivalent scopes: pair, protocol, and controlled. This distinction determines the accumulation law for adaptive protocols. On the Lüders-exact residual class, the optimal adaptive deviation is exactly 1 minus (1 minus epsilon) raised to the mth power. A rotated-residual construction proves that this class is maximal.A noisy Born witness gives the crossover profile 1 minus (1 minus epsilon divided by 2) raised to the mth power. The reflected direct/suspended gap has the closed form: 1 minus epsilon, multiplied by the quantity 1 minus (1 minus epsilon divided by 2) raised to the mth power. This expression is independent of coherent copy multiplicity.A multi-axiom tolerance geometry yields a cube-root transfer into derived anchoring, while credit constraints produce a scaling collapse onto the function F of x, equal to 1 minus e raised to the negative x divided by 2.The accompanying born-rates package verifies the closed forms and falsifies invalid scope promotions, but it does not replace proof. The resulting geometry is governed by three features: the local budget, the scope of the record axiom, and the residual class.
Quantum mechanics supplies outcome probabilities through the Born rule, while measurement physics describes how interactions, decoherence, amplification, and environmental encoding produce records. We reformulate quantum admissibility around the distinction between quantum support, Born probability, operational record support, and realized record. The reformulation is motivated by a no-go result for a legacy strong ansatz in which Born probabilities are multiplied by an independently specified scalar admissibility factor: independence from the Born distribution together with arbitrary coarse-graining consistency forces that factor to be uniform, while the corresponding positive-operator construction reduces to an ordinary quantum state update. Rather than replacing this failed weighting by an unstructured support predicate, we connect record support explicitly to existing quantum-to-classical formalisms. Quantum-Darwinist redundancy, classically accessible information, quantum discord, and distance to spectrum-broadcast structure provide a context-dependent operational record profile from which prospectively specified support criteria can be constructed without modifying the Born rule. We also show why Bell correlations cannot supply the missing content of a binary support relation: a non-product joint support set is shared by classically correlated systems, while generic Bell-violating distributions may have full Cartesian support and differ only in their probabilityweights. Bell and no-signaling therefore constrain any future probability-modifying extension but do not establish relational realization as a new physical quantity. The resulting framework has three levels: a conceptual grammar, an operational record-support layer entirely within standard quantum theory, and a distinct strong route requiring independently predicted deviations from the best standard quantum-plus-apparatus model. The principal claim is consequently a boundary result: objective record support can be operationalized without a second probability law, whereas the natural scalar attempt to convert such support into probability reweighting is stronglyconstrained.
The Admissibility of Natural Physics, Part I begins where an admissibility-first reconstruction of single-system quantum mechanics ends. Its central question is not whether the familiar tensor-product formalism can be assumed, but whether any physical composition law can be derived without importing composite systems, global state spaces, entanglement, locality, or signalling as primitives. The book develops a disciplined reconstruction of binary quantum composition from operational and structural requirements. It first records the exact content and limitations of the inherited single-system closure, establishes dependency and non-circularity statutes, and introduces the possible object of natural physics only as a provisional target ledger rather than as an assumed ontology. It then examines joint preparation, product states and effects, marginal consistency, operational no-signalling, minimal and maximal probabilistic composites, local tomography, and the classification of admissible tensor-product representations. Particular attention is given to the distinction between a mathematical representation theorem and the uniqueness of a physical composition rule. Real, complex, quaternionic, Jordan-algebraic, minimal, maximal, restricted, and non-tomographic alternatives are compared under explicitly declared admissibility conditions. The complex Hilbert-space tensor product is recovered only on branches satisfying the required factorization, coherence, scalar, and operational closure conditions. Where several inequivalent composites survive, the result is a classified family rather than a falsely asserted uniqueness theorem. Entanglement is introduced only after composition has been commissioned. It is defined as the failure of an admitted joint state to belong to the convex closure of admitted product preparations. The analysis proves, within the resulting predictive grammar, that entanglement is a property of composite-state geometry, not an additional force, substance, interaction mechanism, or signalling channel. Throughout, the argument enforces quotient primacy, refinement stability, finite diagnostic closure, regeneration stability, explicit dependency receipts, local termination, and the possibility of Undefined as a legitimate scientific outcome. Part I therefore supplies neither a speculative ontology nor a presupposed many-world formalism. It provides a conditional classification theorem for binary physical composition and identifies precisely what remains to be established before interaction, repeated composition, particle identity, locality, fields, symmetries, and particle content may be addressed.
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.