This paper derives the Lorentzian structure and dimensionality of spacetime from irreversible record formation on a compact Kähler configuration space. Building on a programme in which physical law emerges from the contraction of possibility under irreversible measurement events, the framework begins with the configuration manifold M = Fl₁,₂(ℂ³) × ℂP¹ and a bridge law governing the depletion of accessible states. The bridge law relates the contraction rate of the possibility measure Ω to the local rate of irreversible record formation Γ through d/dτ ln Ω = −κΓ where κ = π arises as the Fubini–Study volume of CP¹. Physical time is identified with the contraction of possibility, dt = −kₜ d ln Ω. From this relation the paper derives several structural results: • The temporal one-form θ = −d ln Ω defines a canonical causal ordering on the event manifold.• Finite record-propagation speed produces a cone field whose compatible quadratic form has Lorentzian signature.• Quantisation of the entropy screen measure in Planck area units implies that spatial dimension must be d = 3.• Among simple Lie groups, SU(3) uniquely matches the resulting spatial dimensionality through the dimension of its flag manifold. The resulting (3+1)-dimensional Lorentzian spacetime therefore appears not as a fundamental arena but as the structural residue of irreversible contraction on configuration space. The manifold furnishes structured possibility, the Fubini sphere furnishes recordability, and spacetime is the ordered geometry of irreversible configuration selection Within the broader research programme, the same geometric framework has previously produced quantitative predictions for fundamental constants, particle masses, the Born rule, and semiclassical gravitational dynamics without adjustable parameters. This paper closes the remaining gap by deriving the causal structure and dimensionality of spacetime itself from the same mechanism.
This paper derives the Lorentzian structure and dimensionality of spacetime from irreversible record formation on a compact Kähler configuration space. Building on a programme in which physical law emerges from the contraction of possibility under irreversible measurement events, the framework begins with the configuration manifold M = Fl₁,₂(ℂ³) × ℂP¹ and a bridge law governing the depletion of accessible states. The bridge law relates the contraction rate of the possibility measure Ω to the local rate of irreversible record formation Γ through d/dτ ln Ω = −κΓ where κ = π arises as the Fubini–Study volume of CP¹. Physical time is identified with the contraction of possibility, dt = −kₜ d ln Ω. From this relation the paper derives several structural results: • The temporal one-form θ = −d ln Ω defines a canonical causal ordering on the event manifold.• Finite record-propagation speed produces a cone field whose compatible quadratic form has Lorentzian signature.• Quantisation of the entropy screen measure in Planck area units implies that spatial dimension must be d = 3.• Among simple Lie groups, SU(3) uniquely matches the resulting spatial dimensionality through the dimension of its flag manifold. The resulting (3+1)-dimensional Lorentzian spacetime therefore appears not as a fundamental arena but as the structural residue of irreversible contraction on configuration space. The manifold furnishes structured possibility, the Fubini sphere furnishes recordability, and spacetime is the ordered geometry of irreversible configuration selection Within the broader research programme, the same geometric framework has previously produced quantitative predictions for fundamental constants, particle masses, the Born rule, and semiclassical gravitational dynamics without adjustable parameters. This paper closes the remaining gap by deriving the causal structure and dimensionality of spacetime itself from the same mechanism.
The Born rule — the statement that the probability of obtaining outcome |ϕi⟩ from quantum state |ψ⟩ equals |⟨ϕi|ψ⟩|² — is normally taken as a fundamental postulate of quantum mechanics. This paper derives the Born rule from the bridge law connecting the the Fubini–Study geometry of quantum state space to physical time through irreversible contraction of the accessible future state space Ω. The bridge law, d(ln Ω)/dτ = −κ Γ(τ), specifies that each recording event depletes generative capacity at a rate set by the structural efficiency constant κ = π — the Fubini–Study volume of ℂP1. This identifies the Fubini–Study metric as the natural measure of state-space contraction. Three physical requirements — unitary invariance, projective consistency, and smooth locality — uniquely select this metric, leaving no freedom in the probability assignment. The probability of each outcome is the fraction of the local Fubini–Study contraction associated with that outcome, yielding pi = |⟨ϕi|ψ⟩|². The framework further predicts finite-time corrections to the Born rule. Because records form through irreversible thermodynamic processes with characteristic timescale τdet, the effective probability of registering independent events at short delay τ is modulated by the record-completion function R(τ) = 1 − e−|τ|/τdet. This produces a distinctive suppression of photon coincidences in the second-order correlation function g(2)(τ) at picosecond timescales in Hanbury Brown–Twiss measurements using superconducting nanowire single-photon detectors. The effect depends on detector material properties through the detector transition time τdet and is experimentally testable with existing instrumentation. The result therefore provides both a geometric derivation of the Born rule from the bridge law of irreversible record formation and a concrete experimental prediction linking quantum probability to the thermodynamics of measurement. This work forms part of the author's research programme Emergence as Regime Formation, investigating the origin of time, probability, and physical constants through irreversible record formation.
Quantum mechanics, gauge theory, and general relativity have recently been derived as regime-dependent limits of gradient flow on a compact Kähler manifold. In that framework, six structural conditions identified the product geometry M = Fl₁,₂(ℂ³) × ℂP¹ as the unique candidate consistent with physical observations. This paper shows that those six conditions reduce to two axioms: (1) Physical dynamics is the negative gradient flow of a moment map component on a compact Kähler manifold.(2) Each irreversible recording event is a single, irreducible binary act on ℂP¹. From these axioms, three results follow. First, the admissible geometry is uniquely determined. The irreducibility requirement forces the tangent weight lattice to be non-decomposable, eliminating all compact Kähler threefolds except the flag manifold Fl₁,₂(ℂ³). Combined with the ℂP¹ recording sector, this yields the unique solution M = Fl₁,₂(ℂ³) × ℂP¹. Second, quantum entanglement is identified as a topological property rather than a dynamical one. The cohomology ring of Fl₁,₂(ℂ³) satisfies the Weyl relation x² + xy + y² = 0, which prevents factorization of internal quantum numbers. Entanglement therefore originates from geometric irreducibility of the configuration space, not from state preparation. Third, the recording sphere ℂP¹_rec and the canonical fiber ℂP¹_fib of the flag manifold are shown to be isomorphic as Hamiltonian U(1) spaces. Measurement is thus internal to the geometry: the recorder is realized as a fiber of the configuration space itself. The observer–system distinction is topologically obstructed. The framework eliminates the landscape of possible geometries and introduces no free parameters. The Standard Model geometry emerges as the unique fixed point of a self-consistency condition in which the recording structure determines the manifold and the manifold returns the recording structure.
AbstractStandard quantum theory presupposes an external temporal parameter, while general relativityadmits no preferred global time, producing the problem of time in quantum gravity. We developa framework in which physical time is not fundamental but is locally generated by irreversibleclassical record formation in open quantum systems.We introduce the Future Contraction Postulate (FCP), defining the local physical time incrementas proportional to the logarithmic contraction of the record-compatible future state-spacevolume Ω.Δt = −kₜ ΔlnΩ.where Ω denotes the coarse-grained set of decoherence-stable macroscopic futures compatiblewith existing classical records. In the continuum limit of Markovian decoherence, we derive theunique local time-generation lawdt = kₜ Γ(τ ) dτ,relating temporal generation to the irreversible record-formation (decoherence) rate Γ(τ ). The framework predicts the existence of atemporal regimes in the limit Γ → 0 and establishesa sharp thermodynamic falsifiability criterion: any system exhibiting stable classical recordswithout monotonic contraction of Ω would contradict the postulate. In macroscopic thermodynamicregimes the generated time coincides with standard proper time measured by physicalclocks. We further discuss speculative cosmological implications in which classical spacetimegeometry is interpreted as an emergent historical residue of irreversible record formation.
WF-S Paper V: The Statistical Manifold of Irreversible Records — Quantum Mechanics, Spacetime, and Forces as Geometry of a Record-Bearing Substrate. This paper reformulates the entire WF-S programme in its natural mathematical language: the information geometry of statistical manifolds. Without assuming a Lagrangian, spacetime, or field content, quantum mechanics, Einstein gravity, three fermion generations, the Koide relation, and the LTrion Lagrangian are derived from three conditions on a record-capable substrate (distinguishability, permanence, repeatability) and the Fisher-Rao metric. Key results include: Born rule from entropy-uniform geodesics; canonical commutation relation [x̂,p̂]=iℏ derived via CLT and coherent state overlap without geometric quantisation; trilinear coupling μ = ms/s₀ from Berry curvature; instanton prefactor ε = e⁻⁴(1−e⁻⁴)³ = 0.017327 within 0.035% of the measured value; cosmological constant sign resolved; and the Planck length derived simultaneously from three independent internal routes. The mathematical constant e appears twice in the programme — as the base of the Gaussian distribution arising from the Central Limit Theorem applied to irreversible record formation, and as the base of the complex phase rotation in the Z3 ground state; both are shown to be the same exponential map on the Fisher manifold, unifying the thermodynamic and geometric faces of the theory. The logical spine of WFS is complete within stated regimes. Remaining open problems — the lepton phase angle and full nonlinear Einstein equations — are extensions of derived results, not gaps.
The Quantum Blueprint Formalism (QBF) describes the emergence of coherent structure through projection from a pre-coherent space M_s to a coherence manifold M_Θ. Previous papers have established the topological structure of M_s (Schmieke, 2026l), the emergence of time from circulation (Schmieke, 2026m), and the dissolution of the quantum measurement problem (Schmieke, 2026j). This paper addresses a fundamental question: What is the origin of spacetime? Is spacetime a feature of M_s, of M_Θ, or of the projection itself? We establish three central results. First, M_s is pre-spatial: it has topological structure but no metric. Distance, angle, and dimension are not defined on M_s. Second, spacetime emerges through projection: the metric structure of M_Θ arises from the geometry of the projection map π_Θ and the structure of the tension functional Φ. The dimension of spacetime (3+1) is not arbitrary but determined by the admissibility conditions. Third, the metric signature (−,+,+,+) emerges from the distinction between temporal circulation (Schmieke, 2026m) and spatial distinction: time is the direction of Ω_s, space comprises the directions orthogonal to Ω_s. These results connect the QBF to approaches in quantum gravity. The emergent metric is compatible with general relativity at large scales but predicts deviations at the Planck scale where the projection structure becomes visible. We derive a minimum length scale l_min \~ √(ℏG/c³) below which spatial distinctions cannot be coherently projected---this is the Planck length, now understood as a projection limit rather than a fundamental discreteness.
For over a century, physics has treated time as a continuous background parameter while quantizing everything else—fields, energy, and angular momentum. This paper argues that the assignment of discreteness has been reversed: Light is not quantized; energy is not quantized. The record is quantized—and the record is what generates time. Within the Emergence as Regime Formation framework, temporal advance arises through irreversible record formation governed by the Future Contraction Postulate: Δt = −kₜ Δ ln Ω where Ω represents generative capacity—the number of realizable configurations remaining to a system. Each irreversible recording event reduces Ω and generates a finite increment of physical time. The paper demonstrates that experimental phenomena commonly interpreted as evidence for quantized light—including the photoelectric effect, Compton scattering, blackbody radiation, photon anti-bunching, and Hong–Ou–Mandel interference—are fully consistent with a framework in which: • The electromagnetic field remains continuous• Sources and detectors undergo discrete irreversible recording events• Apparent field quantization arises from projecting continuous processes onto a discrete lattice of records Explicit derivations of photon anti-bunching statistics and the Hong–Ou–Mandel interference dip are provided without invoking Fock space or field quantization. The results reproduce the standard quantitative predictions of quantum optics while relocating the source of discreteness from the field to the recording process. This relocation dissolves several foundational puzzles in physics, including: • Wave–particle duality• The quantum measurement problem• The emergence of classicality• The arrow of time• The “problem of time” in quantum gravity Within the broader Emergence as Regime Formation program, the same recording dynamics generate quantum mechanics, gauge theory, and gravitational structure as regime limits of a single continuity equation with irreversible recording.