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arxiv: 2603.11900 · v2 · submitted 2026-03-12 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Existence as Distinguishability: Quantum Mechanics from Finite Graded Equality

Julian G. Zilly
Authors on Pith no claims yet

Pith reviewed 2026-05-15 12:20 UTC · model grok-4.3

classification 🪐 quant-ph
keywords distinguishabilityquantum mechanicsBorn rulefinite dimensionalself-referential consistencycomplex projective spaceindeterminismunitary dynamics
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The pith

Existence as distinguishability derives quantum mechanics on complex projective space for each finite capacity N at least 3.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives finite-dimensional quantum mechanics from the principle that existence is constituted by distinguishability together with finite capacity N and self-referential consistency. Self-referential consistency expands into eight structural conditions whose final member, structural unambiguity, forces the graded kernel K equal to one minus the squared overlap. This kernel is realized uniquely on complex projective space of dimension N minus one, from which complex coefficients, the Born rule, unitary dynamics and tensor-product composition are obtained as direct consequences. Indeterminism appears automatically once the number of outcomes exceeds the finite capacity. A sympathetic reader cares because the derivation starts from an operational distinction principle rather than from Hilbert-space axioms or probability postulates.

Core claim

For each N greater than or equal to 3 the unique distinguishability space is the complex projective space of dimension N minus one equipped with the kernel K of psi and phi equal to one minus the squared modulus of their inner product. A state is identified with the profile of its distinguishability values against every other state. Complex coefficients supply the coordinates that realize this kernel, the Born rule supplies probabilities as squared moduli, dynamics must be unitary to preserve the kernel, and composite systems are formed by tensor product so that the distinguishability grading is preserved. Finite capacity forces indeterminism through overflow, while the usual quantum theory

What carries the argument

The graded distinguishability kernel K taking values in the unit interval, together with the self-referential consistency closure schema that enforces structural unambiguity and selects the geometry.

If this is right

  • Complex coefficients appear as the natural coordinates realizing the kernel on projective space.
  • Probabilities must be given by the Born rule as squared moduli to close the structure consistently.
  • Time evolution is restricted to unitary maps that leave the distinguishability kernel invariant.
  • Composite systems compose via the tensor product to keep the grading of distinguishability intact.
  • Indeterminism is required for any finite N because capacity overflow prevents deterministic assignment of outcomes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Indeterminism receives a direct information-capacity explanation rather than being introduced by a separate measurement postulate.
  • Hidden-variable theories can be classified according to whether they satisfy or violate the self-referential consistency conditions.
  • Finite-N models may be tested in resource-limited quantum devices where the effective number of distinguishable states is bounded.
  • The same distinguishability principle could be applied to continuous or field-theoretic settings to constrain possible extensions.

Load-bearing premise

Self-referential consistency is the correct closure needed to turn the distinguishability principle into a structurally unambiguous theory.

What would settle it

An explicit construction, for some N at least 3, of a different finite geometry or kernel that satisfies all eight conditions derived from self-referential consistency would falsify uniqueness.

Figures

Figures reproduced from arXiv: 2603.11900 by Julian G. Zilly.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
read the original abstract

We derive finite-dimensional quantum mechanics from a single ontological principle, that \emph{existence is constituted by distinguishability}, together with two structural commitments: finite capacity $N$ (parametric input) and self-referential consistency (SRC, a closure schema with two equivalent forms, operational and information-theoretic). SRC unpacks into eight derived structural conditions; structural unambiguity (S5) completes the hierarchy, uniquely selecting the Born rule as the geometric/probabilistic closure. The graded distinguishability kernel $K(x,y) \in [0,1]$ realises both axioms, with a state constituted by its $K$-profile against all others. For each $N \geq 3$, the unique distinguishability space is $(\mathbb{C} P^{N-1}, K)$ with $K(\psi,\phi) = 1 - |\langle\psi|\phi\rangle|^2$, from which complex coefficients, the Born rule $p_k = |c_k|^2$, unitary dynamics, and tensor-product composition all follow. Indeterminism is forced by capacity overflow; alternatives (e.g. Bohmian mechanics) are classified rather than refuted. Standard QM is the $N \to \infty$ limit; finite $N$ is the only free parameter. The algebraic spine is machine-checked in Lean 4 modulo five imported classical theorems and the existence direction of Stone's theorem; the Appendix states the verification scope.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper derives finite-dimensional quantum mechanics from the single ontological principle that existence is constituted by distinguishability, together with finite capacity N (a free parameter) and self-referential consistency (SRC), a closure schema presented in operational and information-theoretic forms. SRC unpacks into eight structural conditions; structural unambiguity (S5) selects the Born rule as geometric closure. For each N ≥ 3 the unique distinguishability space is (ℂP^{N-1}, K) with K(ψ,ϕ) = 1 − |⟨ψ|ϕ⟩|^2, from which complex coefficients, p_k = |c_k|^2, unitary dynamics, and tensor-product composition follow. Indeterminism arises from capacity overflow; the algebraic spine is machine-checked in Lean 4 modulo five classical theorems and the existence direction of Stone's theorem.

Significance. If the SRC schema can be shown to be the minimal, non-circular closure entailed by distinguishability alone, the result would be a significant foundational derivation of QM structures (including the Born rule and complex Hilbert space geometry) from an ontological starting point, with the formal verification providing additional rigor. The classification of alternatives and the identification of finite N as the sole parameter are also noteworthy strengths. The current presentation, however, leaves open whether SRC is independently required or selected to yield the target space.

major comments (2)
  1. [SRC definition and unpacking (operational and information-theoretic forms)] The central uniqueness claim for (ℂP^{N-1}, K) with K(ψ,ϕ)=1−|⟨ψ|ϕ⟩|^2 rests on SRC unpacking into the eight conditions S1–S8, with S5 (structural unambiguity) enforcing the Born rule. The manuscript presents SRC as required by the distinguishability principle, yet provides no independent derivation showing these exact eight conditions are the minimal consequences of “existence constituted by distinguishability” without reference to the target features (linearity, complex scalars, inner-product geometry). This makes the derivation conditional on the choice of SRC rather than a direct consequence of the ontological axiom.
  2. [Appendix on Lean verification] The paper states that the algebraic spine is machine-checked in Lean 4 modulo five imported classical theorems and the existence direction of Stone's theorem. The verification scope should be stated more precisely: which specific lemmas establishing uniqueness of the distinguishability space (e.g., the step from the eight conditions to the complex projective geometry) are covered by the formalization, and which remain interpretive.
minor comments (2)
  1. [Definition of K] Notation for the graded kernel K(x,y) ∈ [0,1] is introduced without an explicit contrast to the standard overlap |⟨ψ|ϕ⟩|^2; a short comparison table would clarify the geometric meaning.
  2. [N → ∞ limit paragraph] The claim that standard QM is recovered in the N → ∞ limit is stated but not derived in detail; a brief sketch of how the finite-N structures converge would strengthen the presentation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the presentation of our derivation. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [SRC definition and unpacking (operational and information-theoretic forms)] The central uniqueness claim for (ℂP^{N-1}, K) with K(ψ,ϕ)=1−|⟨ψ|ϕ⟩|^2 rests on SRC unpacking into the eight conditions S1–S8, with S5 (structural unambiguity) enforcing the Born rule. The manuscript presents SRC as required by the distinguishability principle, yet provides no independent derivation showing these exact eight conditions are the minimal consequences of “existence constituted by distinguishability” without reference to the target features (linearity, complex scalars, inner-product geometry). This makes the derivation conditional on the choice of SRC rather than a direct consequence of the ontological axiom.

    Authors: We agree that the independence of the SRC conditions from the target structure merits explicit emphasis. In the manuscript, the operational form of SRC is derived directly from the ontological principle that existence is constituted by distinguishability (any two distinct states must admit a distinguishing measurement) together with finite capacity, while the information-theoretic form follows from the requirement that the distinguishability kernel be closed under self-reference. The eight conditions S1–S8 are presented as the minimal unpacking of this closure in Section 3. To address the concern that the choice may appear tailored, we will add a new subsection (3.1) that derives each condition step-by-step from the ontological axiom and finite capacity alone, without presupposing linearity or inner-product geometry. This will make the non-circular character of the derivation clearer while preserving the existing formal structure. revision: partial

  2. Referee: [Appendix on Lean verification] The paper states that the algebraic spine is machine-checked in Lean 4 modulo five imported classical theorems and the existence direction of Stone's theorem. The verification scope should be stated more precisely: which specific lemmas establishing uniqueness of the distinguishability space (e.g., the step from the eight conditions to the complex projective geometry) are covered by the formalization, and which remain interpretive.

    Authors: We accept this recommendation. The current Appendix already notes that the algebraic spine is verified modulo five classical theorems and the existence direction of Stone's theorem, but we will expand it with an explicit table listing each lemma in the uniqueness proof (from the eight SRC conditions through the classification of the distinguishability space as (ℂP^{N-1}, K)) and indicating which steps are formally checked in Lean 4 versus those that rely on the imported theorems or interpretive steps. This will be included in the revised Appendix. revision: yes

Circularity Check

1 steps flagged

SRC closure schema selects Born rule by construction via its own unpacking conditions

specific steps
  1. self definitional [Abstract (SRC definition and unpacking)]
    "together with two structural commitments: finite capacity N (parametric input) and self-referential consistency (SRC, a closure schema with two equivalent forms, operational and information-theoretic). SRC unpacks into eight derived structural conditions; structural unambiguity (S5) completes the hierarchy, uniquely selecting the Born rule as the geometric/probabilistic closure."

    SRC is posited as required by the ontological principle, yet its unpacking is defined to produce exactly the eight conditions whose final step (S5) forces selection of the Born rule and CP^{N-1} geometry. The 'uniqueness' therefore reduces to the schema being constructed to close on the target structure, with no independent derivation shown that these precise conditions are the minimal entailment of distinguishability without reference to the desired Hilbert-space features.

full rationale

The derivation claims uniqueness of (ℂP^{N-1}, K) with K=1−|⟨ψ|ϕ⟩|^2 from the single principle 'existence is constituted by distinguishability' plus finite N and SRC. However, SRC is introduced as a closure schema whose operational unpacking directly yields eight structural conditions (S1–S8), with S5 (structural unambiguity) explicitly completing the hierarchy to select the Born rule as geometric closure. This makes the central uniqueness claim conditional on a schema whose form is defined to enforce consistency with the target probabilistic structure, reducing the step from distinguishability alone to a self-consistent choice of closure rather than an independent derivation. The algebraic spine is machine-checked, but the load-bearing ontological step remains internal to the SRC definition.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The derivation rests on one free parameter N (finite capacity) and two core axioms: the ontological principle that existence is constituted by distinguishability, and the self-referential consistency closure. No new physical entities are postulated beyond the graded kernel K, which is derived rather than invented.

free parameters (1)
  • N
    Finite capacity of the distinguishability space; the sole parametric input, with standard QM recovered only as N → ∞.
axioms (2)
  • domain assumption Existence is constituted by distinguishability
    Single ontological principle stated as the starting point.
  • ad hoc to paper Self-referential consistency (SRC)
    Closure schema with operational and information-theoretic forms that unpacks into the eight structural conditions.

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Reference graph

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