We investigate a broad class of physical theories in which fundamentally localizable entities possess ontic positions that can be represented uniquely by numbers. This includes any framework in which spatial position is a fundamental physical attribute, independent of measurement, and can in principle be represented exactly by numerical coordinates. Starting from a small set of physically motivated structural axioms, we formulate a measurement framework that distinguishes ontic positions from their numerical encoding. We show that every exact position measurement necessarily yields a finite numerical representation, while making no assumption that the underlying ontic position space is discrete. The measurement framework provides a canonical arithmetic representation of all admissible exact position measurements. Once this representation has been established, the remainder of the proof proceeds entirely within elementary number theory and requires no further physical assumptions. Under the stated axioms, we derive a contradiction showing that no ontic position-based theory admitting such a unique numerical representation can exist. The key idea is that the set of admissible numerical values for a measurement is invariant under unit conversion. But this invariance will be violated once we do the math. The contradiction is independent of dynamical laws, quantum-mechanical postulates, relativistic structure, probabilistic assumptions, and the particular realization of the measurement process. It therefore establishes a general number-theoretic no-go theorem for ontic position-based theories satisfying the stated axioms. Publication update (October 7, 2026) This work has subsequently been published in Quantum Speculations, Volume 8, Issue 4 (2026), pp. 61–97, under the title “On the Existence of Fundamental Ontic Positions: A Number-Theoretic No-Go Theorem.” Published version: https://ijqf.org/archives/8657
We investigate a broad class of physical theories in which fundamentally localizable entities possess ontic positions that can be represented uniquely by numbers. This includes any framework in which spatial position is a fundamental physical attribute, independent of measurement, and can in principle be represented exactly by numerical coordinates. Starting from a small set of physically motivated structural axioms, we formulate a measurement framework that distinguishes ontic positions from their numerical encoding. We show that every exact position measurement necessarily yields a finite numerical representation, while making no assumption that the underlying ontic position space is discrete. The measurement framework provides a canonical arithmetic representation of all admissible exact position measurements. Once this representation has been established, the remainder of the proof proceeds entirely within elementary number theory and requires no further physical assumptions. Under the stated axioms, we derive a contradiction showing that no ontic position-based theory admitting such a unique numerical representation can exist. The key idea is that the set of admissible numerical values for a measurement is invariant under unit conversion. But this invariance will be violated once we do the math. The contradiction is independent of dynamical laws, quantum-mechanical postulates, relativistic structure, probabilistic assumptions, and the particular realization of the measurement process. It therefore establishes a general number-theoretic no-go theorem for ontic position-based theories satisfying the stated axioms. Publication update (October 7, 2026) This work has subsequently been published in Quantum Speculations, Volume 8, Issue 4 (2026), pp. 61–97, under the title “On the Existence of Fundamental Ontic Positions: A Number-Theoretic No-Go Theorem.” Published version: https://ijqf.org/archives/8657
This paper represents one approach to making explicit some of the assumptions and conditions implied in the widespread representation of numbers by composite quantum systems. Any nonempty set and associated operations is a set of natural numbers or a model of arithmetic if the set and operations satisfy the axioms of number theory or arithmetic. This paper is limited to $k\ensuremath{-}\mathrm{ary}$ representations of length L and to the axioms for arithmetic modulo kL. A model of the axioms is described based on an abstract L-fold tensor product Hilbert space Harith. Unitary maps of this space onto a physical parameter based product space Hphy are then described. Each of these maps makes states in Hphy, and the induced operators, a model of the axioms. Consequences of the existence of many of these maps are discussed along with the dependence of Grover's and Shor's algorithms on these maps. The importance of the main physical requirement, that the basic arithmetic operations are efficiently implementable, is discussed. This condition states that there exist physically realizable Hamiltonians that can implement the basic arithmetic operations and that the space-time and thermodynamic resources required are polynomial in L.
This record contains a set of documents presenting a foundational framework in quantum foundations organized around a single postulate and the constraints that follow from it. The central proposal is that observation may constitute an emergent descriptive coordinate structurally distinct from space and time. This coordinate, designated Type III, is introduced on an extended five dimensional descriptive manifold. The proposal is stated as a postulate rather than a derived result. Its purpose is not to solve the measurement problem by mechanism, but to reframe it structurally and to constrain what explanations at that address can coherently take. Taking the postulate seriously yields a strict no go boundary. No physical or probabilistic facts may be generated by evolution along the observational coordinate. In particular, any deterministic, stochastic, or geometric dynamics that attempt to recover definite outcomes or Born rule statistics along the observer dimension are formally excluded. The collection documents both the postulate and the consequences of enforcing this boundary. The package includes: An orienting note that situates the postulate within the landscape of quantum measurement paradoxes and clarifies the explanatory demands it explicitly refuses. The Reid Postulate (Rev 1), which introduces a dimensional taxonomy classifying observation as a Type III coordinate, establishes a composability constraint grounding partial trace operations geometrically, and formulates a small set of precisely stated open problems, including a structural parallel to the Wheeler DeWitt problem of time. A companion no go theorem, Definiteness Is Not Kinematic, which proves that diagonal dynamics along the observational coordinate cannot reproduce Born rule statistics for any functional form within this construction class. The proof is closed against all known escape routes of this type. The author response to the IJQF referee report associated with an earlier submission, clarifying scope, limitations, and points of agreement. This framework does not propose new collapse dynamics and does not commit to an ontological proliferation of outcomes. It identifies a geometric address for the measurement problem and demonstrates what dynamics at that address cannot be if observational homogeneity is to be preserved. Probability is treated as an invariant measure over admissible descriptions, while definiteness is treated as an indexical fact of location rather than a physical event. Falsification conditions and open problems are explicitly stated. The work was developed in collaboration with Claude (Anthropic), which served as a mathematical and expository partner. The AI collaborative nature of the work is disclosed throughout. Archived predecessor: Reid and Claude (2026), Observation as Emergent Dimension: Original Conjecture Package
We formulate physically motivated axioms for a physical theory which for systems with a finite number of degrees of freedom uniquely lead to quantum mechanics as the only nontrivial consistent theory. Complex numbers and the existence of the Planck constant common to all systems arise naturally in this approach. The axioms are divided into two groups covering kinematics and basic measurement theory, respectively. We show that even if the second group of axioms is dropped, there are no deformations of quantum mechanics which preserve the kinematic axioms. Thus, any theory going beyond quantum mechanics must represent a radical departure from the usual a priori assumptions about the laws of nature.
AbstractStandard quantum mechanics is formulated using complex Hilbert spaces, real-valued probabilities, continuous one-parameter dynamics, and—where particles and fields have continuous degrees of freedom —function spaces constructed over continua such as R3 . In that precise mathematical sense, the real and complex continua are deeply embedded in the standard theory. It does not follow, however, that experiments have demonstrated the physical existence of an uncountable set of exact real numbers, still less that individual quantum systems physically “store” infinitely many digits. Every preparation, control operation, and measurement has finite precision, and many quantum predictions can be approximated arbitrarily closely using countable or finite descriptions.This distinction matters because the continuum is sometimes blamed for quantum theory's conceptual strangeness. That diagnosis is mostly unsupported. Superposition, interference, uncertainty, noncommutativity, entanglement, contextuality, Bell-inequality violations, and no-cloning already occur in finite-dimensional quantum systems and do not depend in any straightforward way on continuous position space or arbitrary irrational numbers. Restricting a Hilbert space to rational, algebraic, computable, or finitely specified states therefore does not automatically produce classical physics. Indeed, dense restricted subsets can approximate ordinary quantum predictions to arbitrary finite accuracy, although turning such a subset into a complete physical theory requires much more: closed dynamics, a composition rule, a measurement theory, robustness, and new empirical predictions.A particularly developed contemporary alternative is Tim Palmer's Invariant Set Theory and its recent formulation as Rational Quantum Mechanics (RaQM). RaQM is not accurately described as simply “quantum mechanics with irrational numbers banned.” It restricts the bases in which states are physically defined through number-theoretic rationality conditions, represents admissible states using finite bit strings, invokes a fractal invariant state space equipped with a non-Euclidean, p-adic-like metric, and violates Bell measurement independence at the level of exact counterfactual settings. Palmer's 2026 proposal further predicts a finite qubit-information capacity and an eventual saturation of quantum computational advantage. These are genuine, potentially falsifiable proposals, but they remain speculative and have not displaced standard quantum theory.The strongest conclusion is consequently asymmetric. There is a serious foundational question about whether the continuum is an ontological feature of nature or an extraordinarily efficient mathematical completion. Finite precision prevents existing experiments from proving a literal continuum. But there is presently little evidence that continuum mathematics causes quantum phenomena. Any discrete replacement must still account quantitatively for interference, contextuality, Bell correlations, entanglement, spectroscopy, atomic structure, and quantum computation. Discreteness can change quantum physics only when it is accompanied by additional physical postulates—and those postulates, rather than discreteness alone, must ultimately face experiment.
Papers I–III derived specific physical structures from three operational axioms: an action gap, finite accessibility, and intersubjective stability. This paper inverts the question: what structures are forbidden? Seven exclusion theorems are proved, showing that large classes of alternative physics are inadmissible under the operational axioms. Each theorem includes explicit assumptions and rigorous mathematical proof. The excluded structures are: 1. **Arbitrarily precise extended synchronisation** — no protocol can certify synchronisation across a domain exceeding the coherence radius R_coh 2. **Higher-than-second-order local dynamics** — via Ostrogradsky instability and certification cost arguments 3. **Non-Born probability assignments** — via the Cauchy functional equation; the Born rule is the unique continuous, non-contextual invariant measure 4. **Non-three-dimensional geometry** — via flux invariance; only d = 3 permits radius-independent total distinguishability 5. **Non-quartic high-frequency dispersion** — via complex analysis of the EFT derivative expansion 6. **Stable relativistic gravitationally-coupled matter** above the accessibility ceiling 7. **Protocol-dependent physical constants** — universality is forced by intersubjective stability The exclusions collectively define an admissibility corridor: the narrow set of physical structures compatible with bounded intersubjective certification. The paper shows that most logically possible alternatives violate the axioms, and that observed physics lies on structural boundaries of the corridor. **This paper is part of the Finite Accessibility series, which reconstructs gravitational geometry, quantum dynamics, gauge structure, cosmological perturbation theory, and the dark sector from a single operational primitive: a finite certification ceiling ω* governing the maximum rate at which physical relations can be maintained across extended domains.** **Introductory papers** - Geometric Limits to Coherent Timekeeping in Extended Reference Frames — https://doi.org/10.5281/zenodo.18824556 - Accessibility Selection Law (ASL) — https://doi.org/10.5281/zenodo.18824649 - Phase Boundary Gravity — https://doi.org/10.5281/zenodo.18824707 **Pillars (numbered I–IV)** - Paper I: Deriving Gravitational Geometry and Schrödinger Dynamics from Finite Accessibility — https://doi.org/10.5281/zenodo.18601600 - Paper II: Deriving Inertia, Motion, and Wave Propagation from Finite Accessibility — https://doi.org/10.5281/zenodo.18625174 - Paper III: Accessible State Spaces, Record Stability, and the Emergence of Quantum Statistics — https://doi.org/10.5281/zenodo.18625192 - Paper IV: Frame Dissolution, the Certification Ceiling, and Observable Deformations — https://doi.org/10.5281/zenodo.18654458 **Extension papers** - Dark Matter from Finite Accessibility — https://doi.org/10.5281/zenodo.18654433 - Dark Energy from Finite Accessibility — https://doi.org/10.5281/zenodo.18654523 - No-Go Theorems and the Admissibility Corridor — https://doi.org/10.5281/zenodo.18654493 - Gauge Structure from Finite Accessibility — https://doi.org/10.5281/zenodo.18928152 - Perturbation Theory from Finite Accessibility — https://doi.org/10.5281/zenodo.18928213 **Experimental paper** - Experimental Probes of Finite Accessibility: GW Dispersion and Vacuum Path Independence — https://doi.org/10.5281/zenodo.18931126 **Keywords:** no-go theorems, admissibility corridor, finite accessibility, exclusion theorems, Born rule uniqueness, spatial dimensionality, Ostrogradsky instability, higher-order dynamics, extended synchronisation, intersubjective stability, certification cost, dispersion constraints, protocol invariance, operational axioms, action gap, quantum gravity constraints
Measurement is an important scientific activity. In most of science, including classical physics, is may be understood as a way of finding out about the physical world and representing the results numerically. No-go theorems show that measurement of quantum observables is not like that: the recorded outcome is typically created rather than revealed in a quantum measurement, in which case there is no objective fact about the observable's prior value. Other no-go theorems show that unitary quantum theory can generally neither explain nor even represent a unique recorded outcome, thereby threatening that outcome's objectivity. Methodological norms inherent in quantum physical practice nevertheless institute the objectivity, not only of unique recorded outcomes of quantum measurements, but also of non-quantum features of the world that physicists and other scientists take their models to represent.