A finite, timeless configuration ontology for physics: quantum probability derived as a counting theorem, Bell violation exhibited by counting, Schrödinger wavepacket dynamics computed by counting, and a labeled research program toward emergent time and geometry.
Configuration Realism (CR) begins from two established readings of established physics, and adds one new step. The relativity of simultaneity, read ontically, yields a block universe: all events exist, none privileged as present. The Wheeler–DeWitt constraint of canonical quantum gravity, read at face value, eliminates fundamental temporal evolution. Together these give a universe that does not evolve in time — but not, by themselves, a finite one. The new step is this work’s: if the Born rule of quantum probability — among the most precisely confirmed laws in physics — is also read ontically, its exponent a fact about the world rather than a principle about observers’ credences, then, this work argues, outcome classes must carry a canonical measure: they must be finite. Adding finiteness completes the ontology — a timeless, finite configuration space in which information exists only as physical records, probability is literal counting, and change is difference read along the refinement ordering of an observer’s records, with time as the emergent coordinate that labels it. From that ontology the Born exponent returns as a theorem, the Bell correlations are counted at the quantum value, and Schrödinger wavepacket dynamics is derived from the framework’s own move set — a derived core spanning probability and motion, from one conserved quantity: the number of configurations — with the rest of physics as a stated research program.
The work is organized by epistemic status, and every claim is classified — postulate, theorem, verified, argument, conjecture, or open — in a ledger at the end of each paper. The derived core is quantum probability and its dynamics. The population of an outcome class is proved to equal \(N|\alpha|^2\) under named structural assumptions (the Born rule as a counting theorem). The CHSH quantity assembled from ratios of integer counts reaches the quantum value \(2\sqrt{2}\) — a finite ensemble of definite configurations is not a hidden-variable model, and the papers show exactly why. And equipped with a local move set, counted populations propagate as wavepackets: the effective mass is the inverse hopping rate of elementary refinement, and the counted dynamics reproduces Schrödinger phenomenology with the number of configurations conserved exactly at every stage. The conjectural program — causal structure, Lorentzian geometry, geodesic dynamics, horizon thermodynamics as continuum limits of closure geometry — is stated as such, with its open problems named.
Configuration Realism's predictions differ from those of continuum quantum mechanics, and the differences are in principle testable. They come from finiteness: populations are integers, so every probability is a ratio of counts, and Born statistics hold to \(O(1/N)\) rather than exactly. Each difference below has a mechanism and an in-principle test; full statements and the falsifiability analysis are in the foundations paper. In brief:
Four papers, August 2026. Each paper is versioned independently. The three companion papers are self-contained; the foundations paper states the full framework and integrates them. All four papers are licensed CC BY 4.0.
PDF · LaTeX source · doi:10.5281/zenodo.21523492
Abstract. We prove that, in a finite and timeless configuration ontology with relational phase structure, the population of an outcome class — the number of configurations compatible with an observer’s records — equals \(N|\alpha|^2\), where \(N\) is the population of the parent class and \(\alpha\) is the coherent sum of stage-relative configuration phases over the outcome class. Because probability in this ontology is defined as a counting ratio, the Born rule follows: quantum probability is quadratic because counting is quadratic. The mathematical core of the argument is classical; the contribution is the premise: the conserved quantity is a cardinality, not a normalization convention. Phase enters the ontology at two levels: it attaches additively to degrees of freedom, making phase additivity a lemma rather than an assumption; and it is relational, defined for each configuration relative to a stage of an observer chain — we prove that no context-free phase assignment is consistent with the existence of nontrivial devices. Given the linear composition of devices, the derivation proceeds in two steps, each by contradiction. First, a population rule that is not a pure power of \(|\alpha|\) contradicts the multiplicativity of counting for independent subsystems. Second, an exponent other than two contradicts conservation of the number of configurations, which holds automatically because a device stands between two stages whose records differ by no outcome information and therefore share one fixed compatibility class. A necessity argument shows that an ontic Born exponent requires canonically measurable — hence finite — outcome classes; passing from populations to observed frequencies requires exactly one explicitly stated typicality assumption, on which the value of the exponent does not depend. All results are verified by literal enumeration in an explicit finite model with class populations up to \(4.2\times10^6\), and finiteness yields falsifiable departures from continuum quantum mechanics: every probability is rational, and Born statistics hold to \(O(1/N)\).
PDF · LaTeX source · doi:10.5281/zenodo.21523901
Abstract. A finite ensemble of definite, equally real configurations looks like it must obey the Bell inequalities: definite values, counted, are the textbook picture of a local hidden-variable model. We show that the instinct fails, and exhibit the failure by literal enumeration. In the finite, timeless configuration ontology of the companion papers — where probability is counting, phase is stage-relative, and the population of an outcome class is proved to equal \(N|\alpha|^2\) — the CHSH quantity assembled from ratios of integer counts reaches the quantum value \(2\sqrt{2}\) to \(O(1/N)\); at \(N = 4.2\times 10^6\), counted \(|S| = 2.8284267\). The instinct fails for three separable reasons, each with a counting formulation. Structural exclusion — which configurations exist — is relative to the instantiated measurement bases and fixes only the aligned-basis correlations. Configurations bear outcome records only for the devices their correlation structure contains, so no configuration answers two incompatible settings and, by Fine’s theorem, no Bell inequality can be derived. And the skew-angle statistics are carried by the phase structure of the class — the same cross terms that carry two-slit interference. A control run makes the mechanism visible: inserting a sector record, the which-path analog, forces per-class summation, the cross terms never form, and counted \(|S|\) falls below the local bound. Marginal counts are computed for each wing from that wing’s structure alone, so the counting form of no-signaling holds by construction and is tested exactly, including at small \(N\). On the no-go ledger the ontology keeps locality and free settings; for the plain Bell scenario its operative escape is the absence of counterfactual values, with the denial of absolute outcomes required only in Wigner’s-friend extensions. One in-model prediction distinguishes the framework from continuum quantum mechanics: \(\bigl||S| - 2\sqrt 2\bigr| = O(1/N)\), with the residue’s scale robust and its sign conditional on the model’s integer-apportionment rule, which is flagged as a postulate. This paper is a demonstration and a disentangling, not a new theorem: given the companion results, the violation is guaranteed; what is shown is why a counting ontology is entitled to it.
PDF · LaTeX source · doi:10.5281/zenodo.21842637
Abstract. In a finite, timeless configuration space where probability is counting and the population of an outcome class is proved to equal \(N|\alpha|^2\), we construct a refinement chain from local devices — nearest-neighbour beamsplitters and per-cell phase increments, all on the 3,600-value phase lattice of the companion models — and evolve wavepacket classes by literal enumeration. The small-angle brickwork realizes the lattice Hamiltonian \(H = -2\eta\cos k + V(x)\): the effective mass \(m^\ast = 1/(2\eta)\) is the inverse hopping rate — mass as resistance to record restructuring, obtained from the move set rather than posited — and the potential is a cell-dependent phase cost per closure step. Counted populations reproduce the Schrödinger spreading law \(\sigma^2(t) = \sigma_0^2 + (t/2m^\ast\sigma_0)^2\), the group velocity to seven significant figures against the exact device chain, and harmonic oscillation at \(\omega = \sqrt{2\eta\kappa}\), with \(\sum_x n_x = N\) exact at every stage — the counting form of unitarity, tested exactly. Two findings go beyond agreement. First, the counted packet has a literal edge: populations fall to one configuration and then to zero exactly where the Born value crosses \(1/N\), so a counting world’s particle has finite support where the continuum Gaussian merely thins — the rational-probability prediction rendered in position space. Second, the ontology’s record-relative definition of classes is load-bearing for dynamics: counting once at the measurement stage (the chain as one composite device) yields residues of \(O(1/N)\), measured log–log slope \(-0.94\), whereas imputing an integer census to unrecorded intermediate stages — partitions where no records are — degrades the scaling to slope \(-0.62\) through the amplitude granularity of few-configuration tail classes. A definitional clause changes dynamical accuracy by a factor of order \(\sqrt{N}\). What is not shown is the formal continuum limit: given [CS] the discrete effective dynamics is derived, with the same conditional status as the companion theorems, and the passage to the continuum equation remains the framework’s open problem. The cell structure is a named input [CS] — but not a foreign one: its three clauses are the discrete shadow of the framework’s own emergent-space program, and we locate each, finding that a preferred partition is what the stability criterion already selects, that adjacency is what the elementary refinement move set already supplies, and that only the graph’s large-scale regularity — together with an assumed uniformity of hopping rate, which is a flatness assumption — remains open. Reading the cells as emergent space has an immediate consequence: the construction’s maximal group velocity is one instance of the framework’s conjectured invariant speed, computed rather than posited, since a single elementary move defines both the refinement step and the spatial adjacency. Relaxing uniformity — a position-dependent hopping rate, hence a position-dependent effective mass — is the discrete form of curvature, and is the natural next model.
PDF · LaTeX source · doi:10.5281/zenodo.21523984
Abstract. Configuration Realism (CR) is a physical ontology consisting of a finite, timeless configuration space. Each configuration is a complete arrangement of fundamental degrees of freedom; none evolves; phase attaches additively to degrees of freedom and is defined relationally, relative to a stage of an observer chain. Information exists only as finite, lossy physical records. Each record defines a compatibility class, the configurations it cannot distinguish, and probability is defined as counting over these classes. The framework’s derived core is proved and computed in three companion papers under named structural assumptions. The population of an outcome class equals \(N|\alpha|^2\): the Born rule as a theorem, with the frequency reading consuming exactly one typicality principle. The CHSH quantity assembled from integer counts reaches \(2\sqrt{2}\) to \(O(1/N)\), the ontology giving up counterfactual definiteness in the plain Bell scenario — and the absoluteness of outcomes where Wigner’s-friend extensions demand it — rather than locality or free settings. And equipped with a local move set, counted populations propagate as wavepackets: the effective mass is the inverse hopping rate of elementary refinement, the potential is a cell-dependent phase cost, and the counted dynamics reproduces Schrödinger phenomenology with the number of configurations conserved exactly at every stage. Around this core, closure laws order records by refinement. Observers are chains of record-bearing stages; change is difference read along a refinement chain; time is the emergent coordinate that labels it; and closure distance, accumulating in discrete quanta, measures the restructuring refinement requires. Direct consequences follow for entropy and the arrow of time, for the classical limit, and for objectivity as redundancy. The remainder is a labeled research program: causal structure, Lorentzian geometry, geodesic dynamics, and horizon thermodynamics as conjectured continuum limits of closure geometry, with the constructions that would establish them stated as open problems. A dedicated section collects the framework’s distinctive predictions — the \(O(1/N)\) family separating a counting world from continuum quantum mechanics — each with its mechanism and in-principle test. Every claim in the paper is tabulated with its status: postulate, theorem, verified, argument, conjecture, or open.
The framework rests on six postulates. Condensed statements follow; the full statements, with their status classifications, are in the foundations paper.
Every derived claim is checked by literal enumeration in explicit finite models: configurations are materialized as lists, probabilities are computed by counting elements, the counting rule was fixed before any case was computed, and device parameters enter only through correlation structure — never as weights on configurations. Phases are stage-relative throughout, exactly as Postulate 1 prescribes. All figures below are generated by the scripts in Code & Data.
code/cr_toy_model.py
Six tests of the counting rule \(n_i = N|\alpha_i|^2\) at class populations up to \(4.2\times10^6\). Unequal beamsplitters: counted probability matches \(\cos^2\theta\) to \(\sim 3\times10^{-7}\) (at \(\theta = 15^\circ\): 933,013 of \(10^6\) configurations against 0.9330127). Mach–Zehnder interference: fringe visibility 1.0000 by counting, tracking \(\cos^2(\delta/2)\); inserting a which-path record drives visibility to 0.0000. Conditionalization: measurement update as literal elimination reproduces conditional Born values to \(5\times10^{-7}\). Three-outcome splitters: same law, no new ingredients. Exponent uniqueness: with the scale calibrated once and held fixed, count conservation is violated by 15–41% for \(p \in \{1, 1.5, 2.5, 3\}\) and satisfied exactly at \(p = 2\) — finiteness plus linearity force Born’s square. Rational residue: the deviation from exact Born statistics scales as \(O(1/N)\), measured log–log slope \(-1.06\).
code/step1_multiplicativity.py
The two constraints of the Born derivation, separated numerically: product consistency across independent subsystems eliminates every monotone non-power rule (violations at the \(10^{-2}\) level), while count conservation eliminates every power but the square — the two theorems of the derivation, each doing exactly its own work. Entanglement: multiplicativity fails for a Bell-type state, as the derivation requires (its factorization lemma presupposes independence); the entangled partner acts as a which-path record, driving unconditional fringe visibility to 0.0000; conjugate-basis measurement restores conditional fringes at visibility 1.0000 with opposite phases — the quantum eraser, by counting.
code/step4_stability.py
The domain of the Born rule, fixed by which partitions can carry stable records. Einselection by counting: recording in a basis rotated by \(\xi\) from the coherent bundle disturbs downstream fringes except at the bundle-aligned basis (\(\xi = 45^\circ\)), where recording costs nothing — the stability criterion selects the recordable partition, not the modeler. Objectivity as redundancy: \(k\) partial records at coupling strength \(\chi\) leave visibility \((\cos\chi)^k\), verified at every grid point — stable records copy freely, cross-cutting records degrade geometrically. Wave–particle duality: visibility and distinguishability, both obtained from integer counts, satisfy \(V^2 + D^2 = 1\) with maximum deviation \(3\times10^{-4}\).
code/chsh_by_counting.py
Bell violation from a finite list of definite configurations. Aligned analyzers: same-spin count exactly 0 of 4,200,000 — structural exclusion, in its basis. Skew analyzers: the “forbidden” same-spin configurations exist, 307,538 per class against a Born value of 307,537.9 — exclusion is basis-relative. CHSH at the standard settings: \(|S| = 2.8284267\) from integer counts, within \(5\times10^{-7}\) of the quantum value \(2\sqrt 2\). Control run: inserting a sector record (the which-path analog) forces per-class summation and \(|S|\) falls to 1.4142, below the local bound — the entire violation is carried by the cross terms the record removes. No-signaling by counting: each wing’s marginal counts are computed from that wing’s structure alone and are exactly independent of the other wing’s setting, tested down to \(N = 20\). The residue \(\bigl||S| - 2\sqrt2\bigr|\) scales as \(O(1/N)\), landing on both sides of the Tsirelson bound — the scale robust, the sign conditional on the model’s integer-apportionment rule.
code/schrodinger_by_counting.py
Wavepacket dynamics from a brickwork of nearest-neighbour beamsplitters and per-cell phase increments, on a 128-cell chain with \(N = 4.2\times10^6\). Free spreading: counted variance tracks the Schrödinger law \(\sigma_0^2 + (t/2m^\ast\sigma_0)^2\), with the exact device chain matched to \(L_1 = 8.6\times10^{-6}\). Drift: counted group velocity matches the exact chain to seven significant figures. Harmonic well: counted \(\langle x\rangle(t)\) oscillates at \(\omega = \sqrt{2\eta\kappa}\) over two full periods. Counting unitarity: \(\sum_x n_x = N\) exact at every stage, never violated. The literal edge: the counted profile has finite support, terminating at population 1 exactly where the continuum tail crosses \(1/N\). Residue scaling: counting once at the record-forming measurement stage (Model A) gives \(O(1/N)\) residues, log–log slope \(-0.94\); imputing an integer census to unrecorded intermediate stages (Model B) degrades the slope to \(-0.62\) — the record-relative definition of classes changes dynamical accuracy by a factor of order \(\sqrt N\).
All scripts are plain Python (NumPy; Matplotlib for figures) and run in seconds. Each prints its tests and verdicts to the console; the model scripts regenerate their figures, and the Schrödinger suite writes its raw numbers to a results file. The code suite is licensed MIT, with authorship and license headers in every script.
cr_toy_model.py — Born by counting: splits, interference, decoherence, conditionalization, exponent uniqueness, \(O(1/N)\) scaling.step1_multiplicativity.py — product consistency vs. conservation; entanglement and the quantum eraser by counting.step4_stability.py — einselection, redundancy \((\cos\chi)^k\), and \(V^2+D^2=1\) from integer counts.chsh_by_counting.py — CHSH at \(2\sqrt 2\) by counting; sector-record control; exact no-signaling; residue scaling.chsh_figure.py — generates the CHSH results figure.schrodinger_by_counting.py — wavepacket dynamics by counting: spreading, drift, harmonic oscillation, the literal edge, Model A vs. Model B residue scaling; writes schrodinger_by_counting_results.json.pip install numpy matplotlib python3 cr_toy_model.py python3 step1_multiplicativity.py python3 step4_stability.py python3 chsh_by_counting.py python3 schrodinger_by_counting.py
Author
Russell Tillitt
Independent Researcher
San Francisco, California
Acknowledgments
The author used large language models (OpenAI ChatGPT; Anthropic Claude) to assist in the preparation of this work. The author takes full responsibility for all content.
Suggested Citation
Tillitt, R. (2026). The Born Exponent from Counting: Quantum Probability in a Finite Configuration Ontology. Zenodo. doi:10.5281/zenodo.21523492
Tillitt, R. (2026). CHSH by Counting: Bell Violation in a Finite Configuration Ontology. Zenodo. doi:10.5281/zenodo.21523901
Tillitt, R. (2026). Schrödinger by Counting: Wavepacket Dynamics in a Finite Configuration Ontology. Zenodo. doi:10.5281/zenodo.21842637
Tillitt, R. (2026). Configuration Realism: Foundations. Zenodo. doi:10.5281/zenodo.21523984
Revision History
Release 2.0 — August 2026: fourth paper added, Schrödinger by Counting — wavepacket dynamics computed by enumeration under a named cell structure [CS]. Foundations paper retitled Configuration Realism: Foundations, with the dynamics results integrated into the derived core (new Part II section), the invariant-speed conjecture upgraded by a computed instance, and the principle-theory framing carried throughout. Editorial revision across all four papers. Code suite licensed MIT, with authorship and license headers added to every script.
Release 1.0 — July 2026: publication release. Abstracts revised; internal draft labels retired — each paper is now versioned independently, starting at v1; papers licensed CC BY 4.0. Release 1.0 comprises the Born exponent paper v1, the CHSH paper v1, the foundations paper v1, and the code suite.
Version 0.2 — July 2026: three papers (Born exponent, CHSH by counting, foundations), toy models and results, postulates updated to the relational-phase formulation.
Version 0.1 — March 2026: initial postulates published at this site under the name Configuration Realism.
Origin — 2024, possibly earlier: the framework was first developed in written form under the name Quantum Eternalism. The earliest dated summary (private correspondence, April 2024) already contains the core commitments: a timeless totality of all configurations, no evolution or branching, observation as change in an observer’s information state, interference among configurations shaping what is observed, and an emergent arrow of time — developed in explicit relation to Barbour’s timeless mechanics and the Wheeler–DeWitt equation.
© 2026 Russell Tillitt. The four papers and their LaTeX sources are licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0); the code suite is licensed under the MIT License; all other site content, all rights reserved.