Light as Causal Coherence Structure in Compact S³ Geometry - Null Relational Propagation and Compact Causal Accessibility

Light as Causal Coherence Structure in Compact S³ Geometry treats light not as a material substance and not as a mystical entity, but as the maximal causal-coherence structure allowed by compact, boundaryless S³ geometry. In this model, space has a finite causal diameter Dc = πR, so null propagation defines a finite timescale for complete causal accessibility: tcoh = πR/c. This does not imply simultaneity, superluminal communication, or that a photon “experiences” the universe. The claim is strictly geometric: in compact S³, every spatial point becomes causally accessible from every other after a finite time, while the local light-cone structure remains fully respected. The key diagnostic object is the Causal Accessibility Ratio Γ(T): the fraction of compact space that can be causally reached from a given point after time T. For 0 < T < tcoh, it is given by Γ(T) = [cT/R − 1/2 sin(2cT/R)]/π, and for T ≥ πR/c it reaches Γ(T) = 1. Compact topology therefore turns causality from an indefinitely open process, as in non-compact ℝ³, into a structure with finite global closure. In this sense, light functions as maximal relational coherence: not as an echo returning through space, but as the boundary of causal accessibility permitted by geometry. The role of the speed of light c is treated with particular care. The paper does not derive its numerical value and does not modify Maxwell’s equations, special relativity, or general relativity. Instead, c is interpreted as a causal self-consistency parameter of compact geometry, expressed through C(c, R) = 0 and tcoh · c = πR. This is an identity-level structural relation between propagation speed, S³ radius, and global accessibility time — not a new formula for calculating c. The framework therefore sharpens the language: c is not “produced by topology”, but in compact space it acquires an additional global meaning as the parameter translating the spatial invariant πR into the temporal scale tcoh. The spectral part of the work shows that compactness does not imply a simple cyclic repetition of the whole universe. The causal coherence operator Kc(t) = eit√−∆gR is almost periodic but has no exact period: Kc(T0) ≠ id for every T0 > 0. The reason is the irrational structure of eigenfrequency ratios associated with the Laplacian on S³, where λℓ = ℓ(ℓ + 2)/R². Thus compact causal coherence allows local quasi-periodic structures, but prevents a global resonant collapse into exact recurrence. At the observational level, the paper proposes a research programme rather than a completed prediction. It asks whether finite causal accessibility, compact spectral structure, and the suppression of modes larger than the compact scale could leave traces in the CMB, especially in the low-ℓ sector. A crucial caution remains: low quadrupole power alone does not prove S³ topology, because similar effects may arise from other mechanisms. The value of the framework is its disciplined test structure: S³ geometry → Dc = πR → tcoh = πR/c → Γ(T) → compact causal observables. The central takeaway is that, if the universe is compact in an S³-like sense, light is not merely a local signal speed; it becomes the geometric measure of global causal reachability. Null propagation then describes how a compact universe organizes itself into a finite causal network. This makes the proposal both conceptually powerful and scientifically constrained: it does not attribute consciousness to light, does not derive c from numerology, and does not claim immediate observational proof. It formulates a testable geometric hypothesis about how compact topology may structure causality, coherence, and large-scale cosmological signatures.