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Relational Emergence of Born's Rule and a Golden-Ratio Active Surface in PDL

Cédric Laubscher

Published Feb 16, 2026• 0 citations
Publisher

Abstract

We analyze how two fundamental quantum-theoretic ingredients—Born’s rule for spin-$1/2$ measurements and the fine-structure constant governing proton–electron interactions—can be reformulated within the Projective Dynamic Logo (PDL) framework as emergent properties of finite active surfaces embedded in a relational hierarchy. In PDL, physical entities are represented as finite signed graphs selected by coherence and minimality principles, with the minimal $(4,6)$ closure serving as a logical prototype for the electron, and a two-level hierarchy comprising valence cores and a relational sea modelling the proton. Building on earlier work, in which the proton’s active surface $R_{\mathrm{surf}}=\varphi\,r_{\mathrm{val}}/3$ yields a structural relation $\alpha^{-1}\simeq\gamma^2/R_{\mathrm{surf}}^2$ that reproduces the empirical fine-structure constant without continuous fit parameters, we generalize the concept of active surface to a Stern–Gerlach-type spin measurement apparatus. In this construction, spin eigenstates correspond to two stationary sign configurations on the $(4,6)$ block, while quantum superpositions are encoded as weights on mixed triangles that couple the $(4,6)$ structure to two disjoint interface branches representing the apparatus. A weighted coherence cost assigned to these mixed triangles, together with an exponential selection over global graph configurations, induces effective probabilities for the two outcome basins that coincide with Born’s rule for spin-$1/2$ measurements. We additionally introduce an instrumental coupling parameter $\alpha_{\mathrm{spin}}$ that links the apparatus’ internal coherence budget to the extent of its active surface, in direct analogy with the protonic interpretation of $\alpha$. This unified perspective interprets both proton structure and spin measurement as instances of coherence allocation between internal relational hierarchies and finite active surfaces, and it motivates a potential role for the golden ratio as a relational optimization factor. The status of these results is explicitly propositional: they demonstrate that the PDL framework can reproduce both the functional dependence of Born’s rule and the numerical value of $\alpha$, while leaving open the questions of uniqueness and of rigorously deriving the emergence of the golden ratio from the underlying axioms.

DOI
10.5281/zenodo.18661257

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Preliminary sketch extending the Projective Dynamic Logo (PDL) framework toward an effective Schrödinger-type dynamics. The electron is modeled as a minimal (4,6)(4,6)(4,6) signed graph and the proton as a hierarchical structure with valence cores, active surface, and relational sea. A discrete position toy model and a flux ansatz are used to motivate Laplacian-like kinetic and Coulomb-like potential terms, and to show that the structurally defined fine-structure constant αPDL\alpha_{\mathrm{PDL}}αPDL​ is compatible with hydrogenic phenomenology. The work is programmatic and emphasises modelling assumptions and consistency checks rather than a full derivation of quantum dynamics.
Gleason-Type Uniqueness of Born's Rule for Spin-12\tfrac{1}{2}21​ in the PDL Framework
We subsequently reanalyse the mixed-triangle measurement scheme previously formulated within PDL, in which a minimal (4,6)(4,6)(4,6) closure encodes a spin doublet and a Stern–Gerlach-type device is represented by a finite active surface composed of mixed triangles. We prove that, under mild symmetry constraints on the apparatus, the effective probabilities generated by this scheme satisfy our operational axioms and thus fall within the scope of the uniqueness theorem. It follows that Born’s rule for spin-12\tfrac1221​ measurements is not merely implementable in PDL, but is uniquely enforced by the stipulated operational and combinatorial conditions. This result provides an initial link between Gleason-type derivations and the discrete relational substrate underlying PDL, and suggests the feasibility of a more general uniqueness programme for quantum probabilities within this framework.
The Projective Dynamic Logo (PDL) framework derives quantum dynamics from combinatorial axioms on finite signed graphs. This paper addresses the measurement problem at Level 1: it proves that the Born rule P₊ = |⟨+θ|ψ⟩|² emerges from the (A)∧(B) stability criterion without presupposing probability or the Hilbert space inner product. The derivation proceeds in three steps. First, the coherence indicator χ(τ; σ) ∈ {+1, −1} is defined as a purely combinatorial quantity over triangle orientations, with no probabilistic content. Second, the coherence cost εcoh(σ₊; θ, ψ) = |⟨−θ|ψ⟩|² is established from the (A)∧(B) apparatus symmetry conditions, with numerical residual 5.55×10⁻¹⁶ over 20,000 state-angle pairs. Third, P₊ = 1 − εcoh = |⟨+θ|ψ⟩|² follows by normalisation, algebraically. All five Gleason axioms are verified for this assignment with residuals ≤ 1.1×10⁻¹⁵, and the Gleason uniqueness theorem — stated in full as a self-contained result — guarantees that this is the unique admissible probability assignment on the Hilbert space of spin-½ states. A structural diagnostic is provided: uniform triangle weights α_τ = 1/N_T yield the Heaviside function Θ(cosθ), with maximum discrepancy 1.00 (exact), confirming that the Born cos²(θ/2) distribution is not trivial and requires the specific geometric weight structure of K₄⊗G_app. This result closes OP5 of D32 at Level 1. Level 2 — the derivation of triangle weights α_τ from PDL axioms without presupposing the spin Hilbert space inner product — remains open.
The Projective Dynamic Logo (PDL) framework derives physical reality from four axioms on finite signed graphs, without presupposing space, time, or wavefunctions. This paper shows that the Schrödinger equation emerges from the (A)∧(B) stability criterion — the condition characterising stable mixed triangles between K₄ closures and proton vertices — without any additional quantum mechanical postulate. The key result is Proposition 2: the amplitude coefficient b = −iħΔt/(2m_e(Δx)²) is derived from condition (B) together with the Compton pulsation scale, with no free parameter. From this coefficient, the full time-dependent Schrödinger equation iħ∂ψ/∂t = [−ħ²/(2m_e)∇² − α_PDL·ħc/r]ψ follows by the continuum limit, with the fine-structure constant α_PDL entering from the PDL proton quintuplet. The S₄ transitivity established in D31 forces isotropy of the transition amplitudes, ensuring spatial homogeneity of the derived dynamics. The non-relativistic limit is self-consistent, using the same α_PDL and m_e as the full theory. This result closes open problem OP6 of D31 and constitutes the first derivation of a quantum dynamical equation from the PDL closure axioms. Conditions (A)∧(B) are identified as the natural entry point for the relativistic extension to the Dirac equation, addressed in D33
This paper resolves Open Problem OP4 of the Projective Dynamic Logo (PDL) programme by deriving the U(1) phase freedom of quantum mechanics from the combinatorial structure of the complete signed graph K₄, without presupposing complex amplitudes, Hilbert spaces, or gauge symmetry. Document D34 of the PDL corpus established Born's rule at Level 1: the modulus identity P = |ψ|² follows from the Gleason uniqueness theorem applied to the coherence-violation cost χ(τ;σ) ∈ {+1,−1} of the K₄ structure. Level 1 left the origin of complex phases as an open problem. The present document closes this gap. Three propositions are established, each with complete proof. Proposition 1 shows that the global sign equivalence s ∼ −s of K₄ coherent configurations, combined with the half-cycle identity T² = −I₂ proved in D33, generates a canonical U(1) action on the amplitude space H_cyc ≅ ℂ²; the orbits are the fibres of the Hopf fibration S¹ ↪ S³ → S². Proposition 2 proves, by explicit algebraic computation verified over all 8 coherent configurations of K₄, that the (A)∧(B) stability criterion is U(1)-invariant: the cross-products P₁, P₂ ∈ {+1,−1} are independent of any complex phase factor applied to ψ. Proposition 3 proves that the unique U(1)-equivariant probability measure on S³ whose marginal under the Hopf projection satisfies the Level 1 Gleason axioms is the round measure on S³, yielding Born's rule P = |⟨θ|ψ⟩|² on the base ℙ¹(ℂ). Together, these three propositions establish that the U(1) phase freedom of quantum mechanics is not a postulate but a structural consequence of the K₄ pulsation: a state ψ and a phase-rotated state e^{iα}ψ are physically equivalent because they correspond to the same physical K₄ configuration related by the global sign flip s ↦ −s. A connection to the Ginzburg–Landau order parameter of superconductivity is identified: the phase φ in Ψ = |Ψ|e^{iφ} is the fibre coordinate of the Hopf bundle generated by the K₄ pulsation, providing a combinatorial foundation for long-range phase coherence. This result completes the quantum dynamics layer of the PDL programme. The Schrödinger equation (D32), the Dirac equation (D33), Born's rule including the origin of complex phases (D34 + D46), the spin-½ double cover (D33), and the U(1) gauge freedom (D46) all follow from four combinatorial axioms on finite signed graphs, without free parameters and without additional postulates. The paper is self-contained: all necessary results from D29, D32, D33, and D34 are reproduced in full in Section 2.
This preliminary theoretical note proposes a difference-mobility reconstruction of the Born rule, P_i = |psi_i|^2, within the broader Difference-Loop Constraint Hypothesis. The central aim is to reinterpret the Born rule not merely as an empirical postulate linking quantum amplitudes to measurement probabilities, but as a measure-reconstruction principle arising from admissible relational mobility. The framework begins by replacing a particle-first or space-first ontology with a difference-first configuration space. A physical state is represented as a configuration gamma in a relational space Gamma, composed of primitive differences such as spatial separation, delay, phase difference, informational distinguishability, and causal accessibility. For each possible measurement outcome i, the theory defines a corresponding domain Gamma_i of difference configurations capable of realizing that outcome. The paper then introduces triple-difference loops, Omega_ABC = Delta_AB + Delta_BC + Delta_CA, as minimal structures for testing relational closure. The non-closure of such loops is quantified by a triple-difference frustration functional F_Omega. In addition, a light-speed admissibility function chi_c(gamma) excludes configurations that would allow controllable superluminal outcome determination, thereby preserving the distinction between nonlocal quantum correlation and superluminal control. Using these ingredients, the outcome mobility measure M_i is defined as an integral over Gamma_i, weighted by light-speed admissibility and suppressed by triple-difference frustration. Probability is then reconstructed as the normalized mobility occupation ratio, P_i = M_i / sum_j M_j. Quantum amplitude is introduced as a phase-lifted square root of this normalized mobility measure, psi_i = sqrt(P_i) exp(i theta_i). Under this representation, the Born rule follows as the recovery of the original mobility measure through the squared modulus: P_i = |psi_i|^2. The paper also explains why the square-root structure is necessary. A positive measure alone can describe additive classical alternatives, but it cannot generate interference. By lifting the mobility measure into a complex phase-bearing amplitude, coherent alternatives can combine before returning to observable probabilities through the squared modulus. Thus, the Born rule is interpreted as the rule by which admissible difference mobility is represented, interferes, and is reconstructed as measurement probability. This work is not presented as a complete derivation of standard quantum mechanics. It does not derive the full Hilbert-space formalism, canonical commutation relations, Planck's constant, or the measurement postulate. Rather, it offers a preliminary reconstruction and interpretation of the Born rule from the perspective of relational difference geometry, triple-difference frustration, light-speed admissibility, and mobility measure. The discussion situates the proposal alongside Gleason-type arguments, envariance-based approaches, relational quantum mechanics, information-theoretic interpretations, and decoherence-based accounts.