The Common Structure: The Leakage Theorem

Derived. Two physically distinct systems — a black hole and a human eye — share six identical structural features. The common structure is a theorem about what finite c forbids.

3.1 — Structural isomorphism

sections 1–2 establish that two physically distinct systems — gravitational (black hole) and electromagnetic (retina) — share six identical structural features. This is not analogy. It is structural isomorphism: both are instances of the class of near-perfect absorbers whose leakage is constrained by the finitude of c.

3.2 — Universal form (dimensionless structural form)

The common structure is not a single dimensionally exact equation across domains. It is a dimensionless structural dependence: leakage is a function of a finite propagation limit, a finite coupling, and system-specific state variables. Write

η = F(ĉ, γ̂, V̂) (3.1)

where ĉ = c/c*, γ̂ = γ/γ*, and V̂ = V/V* are dimensionless variables defined relative to domain-appropriate reference scales. In specific systems, asymptotic or phenomenological scalings may take power-law form, but the cross-domain claim here is structural, not a universal dimensionally exact monomial.

3.3 — The theorem

Theorem 3.1 (Leakage Theorem — scoped). In the analysed near-perfect-absorber channels of sections 1–2 (gravitational Hawking leakage and electromagnetic retinal leakage), the leakage ratio η is strictly positive whenever c is finite and the relevant coupling constant is finite (G finite for the Hawking channel; α finite for the retinal channel). More generally, AP06 advances the hypothesis that finite propagation limit + finite coupling forbids perfect absorption in the broader class of physical absorbers.

Proof. Forward: If c = 0, no photons exist (E = ℏc/λ = 0); η is undefined. If c → ∞, then α = e²/(4πε₀ℏc) → 0; light does not interact with matter; η is undefined. If γ is infinite, absorption is perfect by definition: η = 0. For any finite c > 0 and any finite γ > 0 in the analysed channels: in the gravitational channel, Hawking radiation ensures nonzero thermal emission (section 1); in the electromagnetic channel, finite α bounds absorption cross-sections below the geometric limit (section 2). In both cases η > 0. Reverse: If η > 0, absorption is imperfect, requiring finite coupling (γ < ∞) and finite c > 0. QED.

3.4 — Precise statement

The eye and the black hole are governed by different equations, different coupling constants, and different mechanisms. The claim is not identity. The claim is that both are members of the class of near-perfect absorbers whose leakage is constrained by the finitude of c and the finitude of the relevant coupling constant.

Result C3 (Derived — scoped). In the two analysed channels, perfect absorption (η = 0) is not achieved for finite c and finite coupling. Leakage is structurally necessary. It is a consequence of the constants, not an imperfection.