In a finite-dimensional causal convex operational theory with closed physical effect sets and essentially unique purification, we derive pure products, general purity preservation and effect non-restriction. Exact binary steering follows from an informative noisy pointer; support order intervals, pure dilations and probabilistic returns then force unrestricted effects without local tomography, ideal compression, product domination or a bound on purifying partners. Finite conditioned control is also derived. Spectrality, pure support projections and symmetric transitions yield a Jordan representation. For an essentially small catalogue, we then classify compositions without local tomography. Exact pure-ancilla faces restrict simple factors to a common real or complex matrix field. Sector translations embed in a locally finite abelian group. Complete event representations and simultaneous phase normalizations identify the full sequential and symmetric monoidal structure. At fixed field, equivalence classes are determined by the effective charge group, the actual product-closed object catalogue and a parity character. The proof retains all rectangular event spaces and does not add charge objects or compressions to an incomplete catalogue. Local tomography selects the ordinary complex quantum case on the available multiplicative catalogue. Exact one-way teleportation supplies an operational comparison. In the specified protocol class, one state is shared on two copies of the input system, without an additional correlated record. Exact preservation of all reference correlations forces a pure balanced resource and reversible corrections on nonzero outcomes. A system with three rank-one sectors and Z3 fusion has exact one-copy outcome minima twelve over the reals and nine over the complexes, with corresponding message entropies, although both alphabets fit in four binary digits. Joint protocols have the same regularized rate. Alternative geometric proofs, restricted-model obstructions, optimal physical returns, a quantum-pointer application and an open-effect counter-theory delimit the assumptions and conclusions. Preparation note: This preprint builds on Yoann Dumont’s research programme, earlier manuscript, and research notes. ChatGPT assisted with proof development and checking, comparison with these prior materials, bibliographic searches, English drafting, and typesetting. AI-assisted checks do not constitute independent peer review. As author, Yoann Dumont takes responsibility for the claims, references, and any necessary corrections to the published text.
In a finite-dimensional causal convex operational theory with closed physical effect sets and essentially unique purification, we derive pure products, general purity preservation and effect non-restriction. Exact binary steering follows from an informative noisy pointer; support order intervals, pure dilations and probabilistic returns then force unrestricted effects without local tomography, ideal compression, product domination or a bound on purifying partners. Finite conditioned control is also derived. Spectrality, pure support projections and symmetric transitions yield a Jordan representation. For an essentially small catalogue, we then classify compositions without local tomography. Exact pure-ancilla faces restrict simple factors to a common real or complex matrix field. Sector translations embed in a locally finite abelian group. Complete event representations and simultaneous phase normalizations identify the full sequential and symmetric monoidal structure. At fixed field, equivalence classes are determined by the effective charge group, the actual product-closed object catalogue and a parity character. The proof retains all rectangular event spaces and does not add charge objects or compressions to an incomplete catalogue. Local tomography selects the ordinary complex quantum case on the available multiplicative catalogue. Exact one-way teleportation supplies an operational comparison. In the specified protocol class, one state is shared on two copies of the input system, without an additional correlated record. Exact preservation of all reference correlations forces a pure balanced resource and reversible corrections on nonzero outcomes. A system with three rank-one sectors and Z3 fusion has exact one-copy outcome minima twelve over the reals and nine over the complexes, with corresponding message entropies, although both alphabets fit in four binary digits. Joint protocols have the same regularized rate. Alternative geometric proofs, restricted-model obstructions, optimal physical returns, a quantum-pointer application and an open-effect counter-theory delimit the assumptions and conclusions. Preparation note: This preprint builds on Yoann Dumont’s research programme, earlier manuscript, and research notes. ChatGPT assisted with proof development and checking, comparison with these prior materials, bibliographic searches, English drafting, and typesetting. AI-assisted checks do not constitute independent peer review. As author, Yoann Dumont takes responsibility for the claims, references, and any necessary corrections to the published text.
The companion paper on effective composition characterises, for a finite substrate without a supplied subsystem factorisation, the possibilistic support layer completely and the preparation layer for two reference classes, and explicitly leaves open the states-and-effects layer of the resulting composition contract. We resolve that layer's finite, mono-system, and bipartite-composition questions, keeping the general preparation layer and the tensor product exactly as open as the companion paper left them. The primitive family arising from cylindrical conditioning of a strictly positive, rank-one reference measure is not itself convex: it is finite while its own convex hull is a continuum, unconditionally, for any local outcome set of size at least two. An explicit classical-randomisation postulate closes it onto the full classical probability simplex, independently of the reference measure's rank; rank governs composition of the primitive family, not the shape of its closure once randomisation is added. Simplicial geometry fixes the mathematical representation of affine effects as vectors in a cube, but not their physical availability as measurements; a weak coarse-graining axiom already yields the entire Boolean skeleton of sharp effects for free, while the full continuum of effects requires a further, dual randomisation postulate on the measurement side. Under two further, distinct, and logically independent axioms — one for a common classical control resource, one for the physical existence of a composed bipartite effect — the pointwise product of two local effects is proved unique, not postulated, and local tomography holds exactly when the local effect families linearly span their ambient functional spaces, equivalently when they separate their local states and contain the unit effect in their linear span. All results are finite, exact, and reproduced by committed scripts using rational arithmetic. Interpretive reading (structural, not a result): the finite hypotheses examined here force, successively and for free, a classical, simplicial operational skeleton — convex states once mixing is supplied, a Boolean effect algebra once coarse-graining is supplied, and exact local tomography once spanning local families are supplied — under precisely identified imports, and produce neither a quantum state geometry nor the Born rule. Locating exactly how much operational structure a finite admissibility contract yields for free, and where every further richness must instead be imported, is the intended service of this result to the non-injective foundations programme: it marks the boundary this classical skeleton cannot cross on its own, and hands that boundary, precisely typed, to whatever module or phase law is proposed next.
Complementarity can organize a relational starting point for quantum reconstruction, but a completion relation does not yet supply preparations, effects, probabilities, continuous trans- formations, composites, or a scalar field. We formulate the missing passage as an assumption-control problem. Beginning from the typed primitive C= [a⊣b], we separate incidence, operationalization, continuity, scalar coherence, transition probability, effects, instruments, composition, purification, and process identification. Every negative reconstruction bridge is tied to an exact premise package and a typed target; geometric or pre-operational witnesses are not promoted into countermod- els of a larger theory they do not instantiate. At exact complex-closure resolution, let AOPT denote the finite-dimensional causal operational framework of an external reconstruction theo- rem, L local equivalence, and E equivalent-system purification. Finite classical theory witnesses AOPT + L without E, while finite-dimensional real quantum theory witnesses AOPT + E without L; the external theorem closes only AOPT + L+ E. Neither selector is derived from primitive complementarity here. The real-quantum stress test is sharpened by the two-rebit decomposition Sym(4) = Sym(2) ⊗Sym(2) ⊕R(J ⊗J): product effects see a nine-dimensional shadow of a ten- dimensional carrier. The identity setting together with one additional algebraically transverse joint-interaction setting gives complete terminal-product state exposure. This later observability remains distinct from original local tomography, universal process tomography, and complex-field selection. The contribution is a source-generated dependency architecture, exact finite witnesses, and a precise selector-origin problem: complex closure is formally conditional, while the physical origin of local equivalence and equivalent-system purification remains open.
Quantum-reconstruction programmes normally begin after several structures have already stabilised: a system type, a tensor-product composition rule, an operational perspective, and a reference-frame regime. Quantum-reference-frame (QRF) research, by contrast, shows that precisely these structures may depend on physical perspective. We develop Relational-Perspectival Quantum Reconstruction (RPQR), an operational architecture that places perspective formation and subsystem composition upstream of a fixed-perspective quantum-reconstruction core. The primitive data are complete candidate situations, physically implementable protocols, outcome statistics, sequential composition, and admissible multi-fragment gluing. An informational perspective is the quotient induced by the protocols available in that perspective. We prove a quotient-factorisation theorem defining an information order, then separate information perspectives from stronger process perspectives carrying actual statistics-preserving simulator functors. Coherent simulator data yield an indexed operational theory and hence a fibration. Composition is not assumed primitive: a perspective-local tensor product is introduced only when admissible multi-input gluing is representable; preservation of the corresponding universal gluing is the criterion for strong monoidality of a perspective change. Finite reference, clock and control resources are compared only after transport to a common operational comparison theory, allowing ideal backgrounds to be represented as context-stable completion limits. A fixed, resource-stable, representable sector then conservatively recovers the operational signature required by a companion fixed-perspective reconstruction, without deriving its independent quantum-selection principles. The architecture is shown to be nontrivial by an explicit three-qubit Z2 QRF realisation based on the extra-particle framework. Two invariant global states are identical in a naive Alice-relative description but orthogonal in Bob’s relative description. The missing information is identified with the central charge algebra C ⊕ C, providing a finite transition-remainder witness. The chargeaware QRF transformation induces, after an explicit operational-saturation model specification, a typed equivalence of whole process fibres. Each perspective contains a preferred representable compositional sector, while the preferred subsystem decomposition is perspective dependent. The QRF is an exact finite-resource witness, and fixing a charge sector yields an ordinary 𝑀4 (C) quantum subsector in which the fixed-perspective reconstruction signature is recovered. Thus RPQR possesses an explicit finite nontrivial model realising the full T1–T5 architecture. We also state falsifiability diagnostics and delimit the claims: no universal information-to-process lifting theorem, no universal cross-perspective tensor covariance, and no new quantum dynamics are assumed.
A physical identification requires a domain of experiments, a realization, and an observation that distinguishes competing mechanisms. We construct these for the δ framework. Within finite causal quantum experiments, computable Choi data are equivalent to finite arithmetic programs producing valid instrument approximations at every requested accuracy. A separate pointer reader preserves the quantum posterior and supports adaptive composition. For the native two-atom census, the actual join update and common-mode retention select golden scaling and a signed contraction. Coherent return selects its minimal conserving completion; an independently calibrated spectral band then selects its constant interaction. A bright-mode construction and a stable short-time calibration procedure supply an explicit realization and uncertainty budget; measured duration and frequency supply the dimensional units. We derive a general endpoint identity between stored record energy and comparison imbalance, and distinguish the monotone fresh-record trajectory from coherent recurrence. The logical-to-physical bridge also has a sharp boundary: an explicit instance of the proposed completion generator has no effective value-reflecting physical reader or numerical comparison receipt. Sequence completion requires precisely the corresponding halting-oracle resource, not an unexplained scalar cost. This obstructs operational completion while leaving purely proof-language transport distinct. Native implementations of identical logical strength nevertheless carry different physical receipts. Joint covariance and the exact endpoint law state the valid relationship between those quantities. Finally, calibrated return and three-outcome tests distinguish the relevant phase and recording alternatives, with quantitative error bounds. First public preprint release, version 1.0. The files include the PDF and the main TeX file. AI-use disclosure: AI assistance was used for source analysis, mathematical exploration, manuscript drafting and revision, diagram preparation, and computational verification support. GPT-6 Astra reviews assessed arguments, scope, readability, and schematics. These were internal model reviews, not external peer review. Jonathan Washburn is the sole listed author and is responsible for the released manuscript.
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.
A partial-transpose image of the positive-semidefinite cone supplies an established alternative ordered presentation on a fixed composite carrier. Transporting both the cone and the processes preserves an abstract equivalence, whereas fixing the process representation can discriminate the two orders. We study the iteration of one specified right-argument transpose rule on supplied finite qubit carriers. With three standard leaves, its two bracketings have different cones, so no linear identification fixing every product operator pointwise can make them agree. Subset-orientation labels give an explicit repair. Relative to a separately supplied tensor-closed standard process category, we construct the raw tensor of arrows, associator, symmetry, asymmetric unit maps, and quasi-inverse tensorator, and prove symmetric monoidal equivalence by faithful transport. Complementary subsets define the same isolated cone, but their identification is not a congruence for the raw tensor product. At fixed rank, a precisely defined two-lift groupoid is a split central Z₂ extension; its odd lift is not an ordinary completely positive endomorphism at fixed representatives. A five-qubit example then separates matched reference transport, a fixed calibrated anchor, and an effect-based quotient. The latter loses a linear subspace rather than only a two-point transpose orbit and is preserved only by histories satisfying the standard kernel-invariance condition. A named real local unitary connects the anchor pair, while a real controlled-NOT (CNOT) gate exposes the limitation of an enlarged single-anchor history class. The contribution is an assumption-controlled analysis of one finite composition rule and its interfaces, not a new general coherence, orientation, or resource theorem, and not a derivation of quantum theory from primitive complementarity. Selected supporting statements and finite witnesses are accompanied by a partial Lean 4.19.0 formalization. A recorded local build and type/axiom audit accepted 117 theorem declarations and 57 definitions or abbreviations. The full symmetric monoidal category and equivalence in Theorems 4.5 and 4.6, and other explicitly listed manuscript links, remain incompletely formalized. The public source supplement also includes three preserved exact Python implementations, expected results, pinned dependencies and claim-to-code maps. Packaging and source/log audits are not another Lean compiler run or external peer review. The public evidence selection preserves the successful type/axiom output and the failure/success chronology. Machine-local account paths are omitted from derived run summaries, and one failed-build diagnostic has an explicitly normalized home-directory prefix. Accepted Lean proof sources remain unchanged; private review archives and original machine-local receipts are not public upload files.