We extend the operational uniqueness result of the ℓ2-norm sample-and-query rule [1] to the full quantum reconstruction programme. Starting from the five operational axioms A1–A5 of the companion note, we generalise them to the framework of generalised probabilistic theories (GPTs). We impose two additional structural postulates: the strict non-commutative UEIR ordering R ◦ I ◦ E ◦ U and the conductance geometry Φ = k/n derived from the Boolean hypercube Qn.We prove (Theorem 5.1) that the only GPT satisfying A1–A5 together with the UEIR ordering and the conductance geometry is complex Hilbert-space quantum mechanics (d ≥ 2). As corollaries we recover the standard Born rule p(k|ρ) = Tr(ρEk) on density operators and the emergence of Tang’s ℓ2-norm SQ access model as the unique R-layer classical realisation.We further prove (Theorem 7.1) that under Born-rule ℓ2-norm sampling, the expected conductance of Haar-random quantum states satisfies EHaar[Φeff ] = (n + 1)/(2n) → 1/2 as n → ∞. This makes the quantum-classical threshold pc = 1/2 a proved theorem derivable from quantum mechanics, not an external postulate. We identify a structural parallel between this convergence and the Riemann critical line Re(s) = 1/2, arising from the shared involution structure of L2-norm squaring, and state a precise open conjecture.
We extend the operational uniqueness result of the ℓ2-norm sample-and-query rule [1] to the full quantum reconstruction programme. Starting from the five operational axioms A1–A5 of the companion note, we generalise them to the framework of generalised probabilistic theories (GPTs). We impose two additional structural postulates: the strict non-commutative UEIR ordering R ◦ I ◦ E ◦ U and the conductance geometry Φ = k/n derived from the Boolean hypercube Qn.We prove (Theorem 5.1) that the only GPT satisfying A1–A5 together with the UEIR ordering and the conductance geometry is complex Hilbert-space quantum mechanics (d ≥ 2). As corollaries we recover the standard Born rule p(k|ρ) = Tr(ρEk) on density operators and the emergence of Tang’s ℓ2-norm SQ access model as the unique R-layer classical realisation.We further prove (Theorem 7.1) that under Born-rule ℓ2-norm sampling, the expected conductance of Haar-random quantum states satisfies EHaar[Φeff ] = (n + 1)/(2n) → 1/2 as n → ∞. This makes the quantum-classical threshold pc = 1/2 a proved theorem derivable from quantum mechanics, not an external postulate. We identify a structural parallel between this convergence and the Riemann critical line Re(s) = 1/2, arising from the shared involution structure of L2-norm squaring, and state a precise open conjecture.
Tang’s dequantisation programme established that several quantum machine learning algorithms admit efficient classical simulation when given ℓ2-norm sample-and-query (SQ) access to their input data. The specific choice of ℓ2-norm sampling—rather than uniform, ℓ1, or any other sampling rule—has hitherto been justified operationally by its polynomial-time implementability, but not axiomatically. We show that ℓ2-norm SQ sampling is the unique sampling functional satisfying five natural operational axioms (normalisation and positivity, additivity under coarse-graining, noncontextuality, unitary covariance, and continuity). The proof is direct and does not invoke Gleason’s theorem.This note operates top-down: it begins from the standard quantum formalism (Hilbert spaces, unitaries, density operators) and works downward to show that the five axioms uniquely force the ℓ2-norm sampling rule at the Instrumental layer of the Unified Emergence of Information and Reality (UEIR) operational pipeline [15]. It therefore provides a strong contribution to Criterion #2 (axiomatic justification for the Born rule) and Criterion #3 (the classical shadow / SQ access rule is the unique classical implementation of Born-rule measurement) of the quantum reconstruction programme. It does not derive Hilbert-spacestructure from first principles (Criterion #1), which remains an open direction stated explicitly as Open Problem 2.The same five axioms underlie the operational derivations of the Born rule in quantum mechanics (Hardy [7]; Chiribella–D’Ariano–Perinotti [8]; Masanes–M¨uller [9]). Tang’s SQ access is therefore not a design choice but the unique classical implementation of Born-rulemeasurement.
The postulates of quantum mechanics — complex Hilbert space structure, unitary evolution, the Born rule, definite measurement outcomes, the canonical commutation relations, and the Pauli exclusion principle — have traditionally been taken as axioms. In this work we show that, within the framework of Modular Entropic Gravity (MEG), these structures admit a unified derivation from four geometric axioms governing the informational structure of the quantum vacuum.The framework is non-circular: its inputs are operator-algebraic structures of quantum field theory — including modular Hamiltonians, the JLMS identity, and the Bisognano–Wichmann theorem — which are defined independently of any probability interpretation. The probabilistic and dynamical rules of textbook quantum mechanics then arise as an infrared effective description of this underlying structure.The six results are obtained in logical order: (Q1) the emergence of complex structure from the entropion mass gap; (Q2) the L2 Hilbert space and Schrödinger equation from the long-wavelength limit of the vacuum propagator; (Q3) the Born rule from a relative entropy identity combined with the PLES–Petz correspondence; (Q4) definite measurement outcomes from PLES boundary conditions imposed by classical apparatus; (Q5) the canonical commutation relations from modular flow via the Bisognano–Wichmann and Araki reconstruction; and (Q6) the Pauli exclusion principle from the Z3 topological structure of the MEG vacuum.Three of these results (Q1, Q2, Q5) are established as complete theorems within the framework. The remaining three (Q3, Q4, Q6) are structurally complete derivations in which the underlying mechanism is fully identified, and any remaining conditions concern either formal infinite-dimensional extension or quantitative refinement.The paper therefore presents a coherent and non-circular programme in which the postulates of quantum mechanics are reconstructed from a smaller set of underlying principles, together with an explicit account of the remaining technical steps required for complete formalisation.
We investigate the extent to which thermodynamic reasonableness, formalized as two operational postulates on entropy and reversible dynamics within the framework of Generalized Probabilistic Theories (GPTs), uniquely determines the complex Hilbert space formalism of standard quantum mechanics. Building on Krumm, Müller, and Masanes (2017), we provide an extended classification of state spaces satisfying thermodynamic consistency for systems of (N \ge 3) distinguishable states, proving that self-duality of the state cone, spectral entropy compatibility, and continuous reversible dynamics jointly eliminate real and quaternionic quantum theories as viable candidates. We show that augmenting the thermodynamic postulates with local tomographic closure and a purification principle is sufficient to recover full complex Hilbert space structure, deriving analogues of the Born rule via entropy maximization arguments. A central novel result is a no-go theorem establishing that non-complex state spaces cannot support a continuous, reversible, and entropy-preserving dynamics compatible with thermodynamic equilibration in composite systems. We further propose experimentally falsifiable signatures of post-quantum deviations using higher-order interference, generalized Landauer erasure, and multi-slit protocols. Also available on Academia.edu: https://www.academia.edu/attachments/132550729/download_file?s=portfolio
Quantum Theory as the Operational Shadow of the M--A Channel: a Substratum Reconstruction in Absolute Frame Theory
The reconstruction program of quantum foundations asks why nature is described by complex Hilbert spaces with the Born rule, and why its correlations stop at the Tsirelson bound rather than saturating the no-signaling limit. Absolute Frame Theory (AFT) embeds the observable four-dimensional manifold M in a continuous Euclidean substratum A carrying a pilot wave Ψ, with an information-bottleneck channel that projects A onto M and traces out an inaccessible fiber. We propose that finite-dimensional quantum theory is the operational shadow of that channel---the channel viewed from within M, restricted to its M-accessible, identifiable part---and we carry out the reconstruction in that reading. This is a program, not a completed theorem: we mark every claim as derived, as a structural correspondence, or as an open frontier, and we state plainly that AFT is not yet formulated as an operational-probabilistic theory in full rigor and that the complex Hilbert structure is here a mechanism (the analytic-signal representation of the channel's sampling), not a posit. Within those limits the reconstruction reduces the why-of-quantum-theory to a single physical fact---the M--A signature transition, the Euclidean-to-Lorentzian Wick rotation---of which the complex structure and the Born rule are the two faces. Concretely, a linear map carries both the complex generator and the conserved ∣Ψ∣2 density if and only if it is unitary. The same signature transition is, in AFT, the root of parity violation and of the arrow of time; quantum theory, parity violation, and time's arrow thus share one origin. The operator-level complex structure that makes the theory locally tomographic is supplied by the Osterwalder--Schrader reconstruction about that single Euclidean time---reflection positivity being a theorem for the free embedding sector and preserved, by a local Feynman--Kac argument, for static configurations---hence complex quantum theory is derived, not posited, in the static sector. The named open frontier is then the time-dependent and interacting reconstruction, while the Born measure ∣ψ∣2 is shown to be the equivariant density of the reconstructed unitary and generic macroscopic matter is derived to be classical, leaving only the strict Dürr--Goldstein--Zanghì uniqueness open. That the embedding yields the Lorentzian signature, and with it emergent local Lorentz invariance, is established in a companion paper on emergent Lorentz invariance.
We construct an arithmetic model space ℓ²(Z>0) carrying additive shifts and multiplicative scalings coupled by the distributive law, and show that close analogues of all five standard quantum axioms arise from the interplay of additive and multiplicative structures, together with explicitly stated probabilistic and dynamical assumptions. The Beckner-Babenko theorem singles out ℓ² as the unique Fourier-isometric norm; Gleason's theorem yields Born's rule; Wigner-Stone gives unitary dynamics; unique factorization gives the tensor product. The Riemann Hypothesis is equivalent to positive semi-definiteness of a spectral quadratic form analogous to Born-type inner-product positivity.
This record contains the manuscript Quantum Mechanics as the Unique Lorentzian Continuation of an Osterwalder–Schrader Positive Theory by Chast K. Wolfe. The paper proves that the full mathematical structure of quantum mechanics—complex Hilbert space, linear and unitary time evolution, the Born rule, canonical (anti-)commutation relations, Poincaré invariance, the spin–statistics correspondence, and the existence of a unique vacuum—arises uniquely from the standard Osterwalder–Schrader (OS) axioms: reflection positivity, Euclidean covariance, locality, regularity, and the Markov semigroup property. Starting from a reflection-positive Euclidean field theory, the work establishes a monotone chain of implications showing that OS positivity forces:(1) a unique complex structure on the reconstructed Hilbert space,(2) linear unitary Lorentzian time evolution generated by a self-adjoint Hamiltonian,(3) the Born rule via Gleason’s theorem,(4) CCR/CAR relations in the Gaussian case, and(5) full OS → Wightman reconstruction, including Poincaré covariance, TCP analyticity, spin–statistics, and vacuum uniqueness. The paper also proves that nonlinear, real, or quaternionic variants of quantum theory are incompatible with OS positivity together with Euclidean covariance and locality. Thus, within the class of OS-positive models, standard complex quantum mechanics is the unique Lorentzian continuation. This Zenodo deposit includes:• the full manuscript (PDF),• a license file (CC BY-NC-ND 4.0),• a bibliography file, and• a README describing the archive contents.