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-$\tfrac12$ and spin-$\tfrac32$ representations of ${2I}$ induce the same Cayley-graph Laplacian eigenvalue; it establishes no Casimir/isotropy normalisation, Born rule, Tsirelson-type bound, or ${\mathrm{SU}}(2)$ 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 ${\mathrm{SU}}(2)$ 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.
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.
Duplicate record. The canonical deposit for this paper is APF Paper 5 — Quantum Structure from Finite Enforceability (Technical Supplement) (DOI 10.5281/zenodo.19908706). This record is retained because published Zenodo DOIs cannot be removed; please cite the canonical record. Technical Supplement to Paper 5 of the Admissibility Physics Framework (APF), Quantum Structure as Least-Cost Admissible Bookkeeping. Paper 5 derives the complete quantum formalism — linearity, complex Hilbert space, tensor products, Born's rule, unitary and CPTP dynamics, measurement as class change, spectral-action internalisation, and UV-completion closure — from PLEC (A1, MD, A2, BW) plus the capacity quantization Ctot = 61, deff = 102 imported from Papers 1–4. The supplement provides self-contained proofs of the theorems compressed to sketch form in the main paper. Linearity from coherent-distinction closure. Tlinearity: if sector states |ψ1⟩, |ψ2⟩ are admissible and coherent-distinction-closed, then their normalised linear combination α|ψ1⟩ + β|ψ2⟩ with |α|2 + |β|2 = 1 is admissible. Complex Hilbert space. THilbert: the state space is a complex Hilbert space ℋ with inner product induced by the cost functional, and the field selection 𝔽 = ℂ is inherited from Paper 1's field-selection tier. Tensor product. Ttensor: composite systems factorise as ℋAB = ℋA ⊗ ℋB under admissibility of joint distinction. Born's rule. TBorn: measurement outcome probabilities are Pr(k | ρ) = Tr(Pk ρ) with {Pk} the projective decomposition operators corresponding to admissible partition classes — derived, not postulated. Unitary and CPTP dynamics. THermitian and TCPTP: admissible state evolution is Hermitian-generated for closed systems and CPTP for open systems, both from minimum-cost bookkeeping. Measurement as class change. Tdecoherence: measurement is the transition from an admissible coherent superposition to an admissible classical register, formalised as a class change on the equivalence structure of admissible states. Spectral-action internalisation. Lspectral_action_internal: the Connes spectral action coincides with the APF partition function under the Boltzmann cutoff. Full proof in Paper 7 Technical Supplement §S.4. UV-completion closure. LQG_P1_closure: the admissibility framework remains consistent under Paper 1's structural extension to UV scales. Explicit countermodels exclude the main published quantum-reconstruction alternatives (Hardy 2001, Masanes–Müller 2011, d'Ariano–Chiribella–Perinotti 2010, Rovelli relational). Red-team section analyses three paper-specific hypotheses (H1 coherent-distinction closure, H2 tensor-product admissibility, H3 spectral-action compatibility). Theorem index, dependency diagram, and changelog close the document. Readable without prior exposure to the APF series. Code and reproducibility. GitHub repository Colab walkthrough notebook (one-click) Interactive dependency DAG About the series. Admissibility Physics derives the Standard Model gauge group, fermion content, quantum formalism, Lorentzian spacetime, the Einstein field equations, the cosmological constant, and the neutrino, strong-CP, and dark sectors from a single assumption — that enforcement is finite at every causally connected region — with zero free parameters. The core derivation runs Papers 0–11; the extension papers carry the individual sectors. Each paper's main text and Technical Supplement is deposited separately, and every quantitative claim traces to a named, executable check in the public theorem bank (over 3,800 bank-registered theorems across roughly 450 modules, 48 quantitative predictions). Website: admissibilityphysics.com. Engine and theorem bank: https://github.com/Ethan-Brooke/APF-Engine. Full series and latest deposits: Zenodo community admissibility_physics. The Engine — Admissibility Physics: An Executable Constraint Framework and Constants Map — DOI 10.5281/zenodo.18529115. Paper 0 — What Physics Permits: A Constraint-First Framework for Physics — DOI 10.5281/zenodo.18439523 Paper 1 — The Enforceability of Distinction — DOI 10.5281/zenodo.18439200 Paper 2 — Finite Admissibility and the Failure of Global Description — DOI 10.5281/zenodo.18439274 Paper 3 — Entropy, Time, and Accumulated Cost — DOI 10.5281/zenodo.18439363 Paper 4 — Admissibility Constraints and Structural Saturation — DOI 10.5281/zenodo.18439397 Paper 5 — Quantum Structure from Finite Enforceability — DOI 10.5281/zenodo.18439433 Paper 6 — Dynamics and Geometry as Optimal Admissible Reallocation — DOI 10.5281/zenodo.18439445 Paper 7 — A Minimal Quantum of Action from Finite Admissibility — DOI 10.5281/zenodo.18439513 Paper 8 — The Admissibility-Capacity Ledger — DOI 10.5281/zenodo.19721384 Paper 9 — The Geometric Substrate as Cost Structure of Comparison Continuations — DOI 10.5281/zenodo.20041675 Paper 10 — The Calculus of Finite Continuability — DOI 10.5281/zenodo.20041680 Paper 11 — Forced Universality from Capacity-Bounded Admissibility — DOI 10.5281/zenodo.20684198 Paper 13 — The Minimal Admissibility Core — DOI 10.5281/zenodo.18361446 Paper 16 — Markov Breakdown and the Hard Problems — DOI 10.5281/zenodo.20684207 Paper 18 — The Electroweak Sector as a Capacity Equilibrium — DOI 10.5281/zenodo.20684209 Paper 20 — The Enforcement Crystal — DOI 10.5281/zenodo.18531732 Paper 24 — The Recruitment-Radius Extension — Foundations — DOI 10.5281/zenodo.20684211 Paper 28 — Absolute Mass Scales from Electroweak Capacity Saturation — DOI 10.5281/zenodo.20684215 Paper 29 — Plaquette Representation Dominance and Confinement — DOI 10.5281/zenodo.20684218 Paper 30 — A Tube Mechanism for the Lattice Mass Gap — DOI 10.5281/zenodo.20684220 Paper 31 — Osterwalder–Schrader Structure of Lattice Yang–Mills — DOI 10.5281/zenodo.20684222 Paper 33 — Trace-to-Scheme Export Architecture — DOI 10.5281/zenodo.20684224 Paper 35 — The Dark Sector as a Two-Role Capacity Decomposition — DOI 10.5281/zenodo.20684228 Paper 40 — The Thermodynamics of Finite Distinction (Between Symmetry and the Void) — DOI 10.5281/zenodo.20684235 Paper 41 — The Horizon as a Continuation Ledger — DOI 10.5281/zenodo.20684241 Paper 42 — The Weak Mixing Angle Is Not Free — DOI 10.5281/zenodo.20684245 Paper 43 — The Neutrino Sector — DOI 10.5281/zenodo.21194191 Paper 44 — What a Finite Theory Owes the Continuum — DOI 10.5281/zenodo.21195874 Paper 45 — What the Vacuum Can Keep — DOI 10.5281/zenodo.21194195 Paper 46 — The Strong CP Angle Is Not Free — DOI 10.5281/zenodo.21194197 Paper 47 — The Confinement Antecedent Is Not Forced — DOI 10.5281/zenodo.21194199 Technical Supplements cross-link to their main paper via isSupplementTo and to the engine via isDocumentedBy. Concept DOIs above resolve to the latest version. Author. Ethan Brooke, Independent Researcher, San Anselmo, California, USA. ORCID: 0009-0001-2261-4682 LinkedIn: linkedin.com/in/ethanbrooke GitHub: github.com/Ethan-Brooke Contact: brooke.ethan@gmail.com
Admissibility Physics — Quantum Structure from Finite Enforceability (Paper 5) This Zenodo community collects research outputs associated with Admissibility Physics, with primary focus on Paper 5 — Quantum Structure from Finite Enforceability and the derivation of quantum mechanics as an admissible representational regime. Paper 5 shows that the mathematical structure of quantum mechanics—Hilbert space, linearity, inner products, tensor products, unitary evolution, CPTP maps, and the Born rule—is not postulated, but forced by admissibility once coherent, jointly enforceable distinctions exist under finite correlation capacity. Quantum structure emerges as the unique bookkeeping required to represent interference, refinement, and recombination while preserving admissibility. The derivation does not assume probability, measurement postulates, dynamics, or spacetime geometry. Probability enters only as a secondary representation, with the Born rule arising uniquely from admissibility invariance and refinement consistency. Measurement is treated as a regime transition associated with saturation and record formation, not as a fundamental collapse mechanism. Quantum mechanics is shown to be conditionally valid: exact in regimes where coherence and non-saturation permit admissible composition, and structurally inapplicable where capacity is exhausted. Monogamy, entanglement limits, and decoherence follow from non-closure under composition and finite interface capacity, rather than from quantum axioms. All subsequent works in this community rely on the quantum-admissible representation established in Paper 5. Later papers show how variational dynamics and spacetime geometry arise as further conditional representations layered on top of quantum structure, and how quantum description itself fails at saturation boundaries, horizons, and measurement-like events. This community serves as a curated, citable archive centered on the derivation and limits of quantum structure within Admissibility Physics. Paper 5 functions as the representational keystone of the program: the point at which finite enforceability yields the quantum formalism without postulation. Core papers are frozen where noted; companion papers, mathematical appendices, figures, and computational artifacts document derivations, regime conditions, and testable consequences.
We prove an exact, unconditional identity in the finite Weil (oscillator) representation of the Heisenberg group Heis3(Z/qZ), q an odd prime: for every complete (inversion-symmetric) BFS shell of the Cayley graph Cay(Heis3(Z/qZ),Sq), the orbit vectors generated by conjugate Weil sectors c and q−c span identical subspaces of Cq, for the special character block and generic character triples alike. The proof reduces every orbit vector to a single pure Fourier mode and shows the reachable frequency set is closed under negation whenever the generating shell is inversion-symmetric — unconditional, coherence-independent, verified independently by 1,548 exact integer-arithmetic checks across six primes with zero failures, plus a negative control on a deliberately truncated shell. We characterise precisely which diagnostics this forecloses: not only Fourier support, rank, projectors, dimensions, and principal angles, but any quantity invariant under independent rephasing and reindexing of the frame — including normalised Bargmann invariants, proved trivial by an explicit frame-orbit argument. A bibliography search confirms ρq−c=ρc is standard; no matching result was found for the subspace-equality identity or its no-go corollaries. This paper corrects an earlier version (v1.1.2): independent review found its central-theorem proof unsound and its separate Born-rule proof independently defective; both are withdrawn, and the singlet correlator and Tsirelson bound survive only as a standard, conditional remark. What remains is the rigidity result above and its precisely scoped no-go corollaries. Read as a methodological instance for the wider programme (interpretive, not a further result): a numerical near-zero-residual observation can be forced by a construction's own structural degeneracy rather than signal an intended physical mechanism.
This work presents an axiomatic and structural reconstruction of core elements of quantum mechanics. Starting from minimal principles — symmetry under vanishing preference, normalization, consistency of probability assignments, and coherence-preserving dynamics — we derive several central features of the quantum formalism rather than postulating them. Key results include: A geometric derivation of the constants 1/21/\sqrt{2}1/2 and 1/21/21/2 from symmetry and normalization. A theorem-level derivation of the Born rule (including the qubit case) from noncontextuality and additivity. A reconstruction of Schrödinger dynamics from reversible norm-preserving structure. A derivation of the tensor-product structure for composite systems from composition axioms and local tomography, with entanglement emerging as a necessary consequence. Formal treatment of Bell–CHSH nonlocality, Tsirelson’s bound, and no-signalling. A conditional reconstruction of complex Hilbert space structure from intrinsic phase dynamics and compositional consistency. The manuscript is self-contained and framed in standard mathematical-physics language. It does not propose modifications to quantum mechanics, but instead clarifies why many of its defining structures arise inevitably from deeper structural constraints. Prior computational and model-based work by the authors (including diagnostic toy models exploring phase locking, coherence stabilization, and recursive-time dynamics) is acknowledged but intentionally not reproduced here, in order to keep the present work focused on axiomatic structure rather than model-specific architecture. This upload is intended as a foundations and reconstruction contribution, suitable for readers interested in the mathematical and conceptual structure of quantum theory.
The companion entanglement note and the orthogonality paper E1 identify two distinct entanglement entropies of the conjugate Weil pair {c,q−c}, which must not be conflated. The first is the entropy of the minimal admissible pair sector: the rank-3 admissible projection Πq is spectrally atomic, equal to span{e0,eξc,eq−ξc}; removing the zero mode leaves the two-dimensional Born–Infeld parity pair {eξc,eq−ξc}, identified with the spin-21 representation. The proto-state on this sector is the singlet, and its reduced state is maximally mixed, so Sentadm=log2 for general canonical blocks, unconditionally relative to the cited upstream theorems. This is the sector relevant to the singlet correlator and the Tsirelson bound. The second is the full residual fibre-level rank rpair(n)=R∞−R(n) defined on the full Gram–Schmidt basis (rank q at saturation, not the rank-3 admissible projection). E1 proves Sent(n)=log(rpair(n)) unconditionally for matched single-character blocks; for independently sampled canonical blocks it remains conditional on the residual admissibility indiscernibility hypothesis [Hres]. The metaplectic dilation ϕω is excluded as a bridge for [Hres] because it acts between blocks, not within the residual support of one block.
This series presents a structured and logically sequential formulation of a discrete informational framework governed by global coherence constraints. Although it originates from the same research program developed in the earlier exploratory preprint series (P-series), the works presented here constitute an autonomous and self-contained series, organized in a progressively axiomatic form and focused on the explicit derivation of the framework’s effective structures. The goal of the construction is to investigate whether structures commonly associated with quantum mechanics, thermodynamics, and geometry may emerge from a discrete relational substrate without being introduced as fundamental postulates. Each article addresses a specific level of the theoretical derivation and forms part of a progressive logical structure based on results established in the preceding works of the series. This article develops the emergence of Hilbert-space structure and Schrödinger-type effective dynamics from the coherent two-field sector established in Paper 02. The work demonstrates that the global coherence constraints imposed on the discrete informational substrate naturally generate an effective complex representation endowed with linear superposition properties and unitary-preserving evolution. The analysis shows that the two-component coherent sector admits a mathematically stable reformulation equivalent to a Hilbert-space structure, and that the effective evolution of the coherent amplitudes leads to a Schrödinger-type dynamical equation without introducing quantum mechanics as an initial assumption. The article further derives the probabilistic interpretation associated with the effective amplitudes and clarifies how quantum-like statistical behavior emerges from the internal coherence conditions of the underlying discrete system. No quantum postulates are assumed at the foundational level. The purpose of the article is to demonstrate that the formal structures associated with quantum mechanics can emerge as effective consequences of coherence-preserving informational dynamics. The article does not assume quantum mechanics as a foundational theory and does not postulate the Hilbert-space formalism a priori. Instead, the work demonstrates that, within the internal hypotheses and coherence constraints of the proposed discrete informational framework, structures mathematically equivalent to Hilbert-space representations and Schrödinger-type evolution can arise as effective emergent descriptions. The purpose of the derivation is therefore not to reproduce standard quantum mechanics by assumption, but to show that quantum-like formal structures may consistently emerge from deeper coherence-preserving informational dynamics.