The Action Principle

You have read the structural argument. Now read where it lives in formal mathematics.

section 7 carries the derivation of c from ε and the structure of G’s determination in the induced-gravity form named in the literature as Sakharov-Visser.

7.1 — Strategy

The simplest possible action for the pre-state S is constructed satisfying three requirements: (i) ℤ₂ symmetry, (ii) symmetry-breaking vacuum parameterised by a single amplitude ε, and (iii) broken phase yielding two quantities identifiable with c and G, both determined by ε.

The analogue gravity programme (Unruh 1981, Barceló, Liberati & Visser 2005) serves as a structural template, not as a physical identity.

7.2 — The pre-geometric action

Let φ be a real scalar field with ℤ₂ symmetry (φ → −φ). The most general renormalisable potential respecting this symmetry is:

V(φ) = −½μ²φ² + ¼λφ⁴ (7.1)

with μ² > 0 (symmetric point unstable) and λ > 0 (potential bounded below). The minima occur at φ = ±v where v = μ / √λ (7.2). The dimensionless breaking amplitude:

ε ≡ μ² / Λ² (7.3)

where Λ is the fundamental scale. When ε → 0, symmetry is restored. When ε > 0, symmetry is broken. The vacuum displacement is v = Λ√(ε/λ).

The kinetic structure cannot presuppose a Lorentzian metric (which has not yet emerged). The most general quadratic kinetic term compatible with ℤ₂:

ℒ = ½α(∂_τφ)² − ½β(∇φ)² − V(φ) (7.4)

where α and β are pre-geometric substrate parameters (temporal and spatial stiffness respectively) and τ is a pre-geometric time parameter. The full action:

S[φ] = ∫ dτ d³x [ ½α(∂_τφ)² − ½β(∇φ)² − V(φ) ] (7.5)

7.3 — The broken phase

The field settles into φ = +v (without loss of generality). Write φ = v + χ. Expanding to quadratic order:

S[χ] = ∫ dτ d³x [ ½α(∂_τχ)² − ½β(∇χ)² − ½ m²χ² + … ] (7.6)

with effective mass:

m² = V″(v) = 2μ² = 2εΛ² (7.7)

The equation of motion yields propagation speed:

c² = β / α (7.8)

Result A.1 — Emergence of c. Perturbations on the broken vacuum propagate with speed c = √(β/α). c is not an input but an output of the broken phase. Determined by the ratio of spatial to temporal stiffness of the pre-geometric substrate. Structurally identical to the emergence of the speed of sound in fluid mechanics: cₛ = √(B/ρ). Tested by KS-R.5.

You did not put c into the action.

c came out.

The speed of light is not a law. It is a consequence — of the ratio between how hard the substrate resists deformation in space versus time.

The same mathematics that gives you the speed of sound in water gives you the speed of light in vacuum.

You already knew this intuitively. Sound has a speed because the medium has stiffness. Light has a speed because the vacuum has stiffness.

The universe is a medium. The break made it one.

7.4 — What this action derives

Derived. The propagation speed c (equation 7.8).

Derived. The effective mass of perturbations: m² = 2εΛ² (equation 7.7). The mass of excitations above the vacuum is set by ε.

Derived. The condensate energy density: ρ₀ = |V(v)| = μ⁴/(4λ) = ε²Λ⁴/(4λ) (equation 7.9). The energy density of the background condensate — the Mode 0 content. Scales as ε².

7.5 — The form of G’s dependence on ε

The derivation of G was the central open problem at AP03’s publication.

AP20’s forcing proof on Axiom B shows it could not remain open: Axiom B admits exactly one free parameter, so G cannot be independent of ε. The structural argument is in place before any specific functional form is named. This is the load-bearing step — it comes from the axioms, not from any external scaffolding.

What remains is to find the specific form. The structural inputs are two. First, the condensate energy density from section 7.3:

ρ₀ = ε²Λ⁴ / (4λ) (7.9 — restated)

This is a fact about the broken-phase substrate, derived from the {S, B, R, C}-instantiated action. The ε² dependence is structural — it follows from the depth of the well |V(v)| = ¼ μ⁴/λ together with μ² = εΛ² (the breaking-amplitude scaling of the potential).

Second, AP20’s forcing proof: G has ε-dependence, no other parameter available.

Combining these two structural inputs forces the form. Once the broken vacuum has a finite energy density ρ₀ ∝ ε²Λ⁴, the induced gravitational coupling on that vacuum has the dimensional form G ∝ 1/(v²) where v² = µ²/λ = εΛ²/λ. Substituting:

G = κ / (εΛ²) (7.10)

where κ is a dimensionless coefficient set by the structural details of how the broken vacuum responds to perturbation. The Planck mass is mₚ = √(ℏc/G) ∝ √ε, modulo the dependence on Λ that AP24 pins.

Correspondence with the literature. The same form appears in standard physics under the name Sakharov-Visser induced gravity — the parallel calculation in standard QFT terms produces exactly this scaling for a scalar-field condensate of our energy density. This is correspondence with our structural result, not its source. Two routes arrive at the same place: the structural route through {S, B, R, C} + AP20 + the section 7.3 condensate density, and the literature calculation that produces the same form independently. The literature parallel confirms; it does not derive.

Numerical check. With ε ≈ 1/137 (identified with the electromagnetic fine-structure constant via AP06), the dimensionless conjugacy α · α_G(m) = κ m²/(Λ² ℏc) is independent of ε; the hierarchy α/α_G(e) ≈ 10⁴³ at the electron mass tracks ε² · (Λ/m_e)² once Λ is pinned via AP24. (The dimensionless hierarchy α/α_G(e) ≈ 10⁴³ is the unit-independent fact; the SI quantity c/G ≈ 10¹⁹ is a different number and must not be conflated with it. AP28 derives this same hierarchy parameter-free as α_em⁻²⁰/(1 + 1/π) ≈ 4 × 10⁴², dissolving it as a channel count rather than a fine-tuning.)

Status. The ε-dependence of G is theorem under AP20’s forcing proof. The specific functional form is derived in this section from the structural condensate density. The structural coefficient κ is not derived from {S, B, R, C} alone; computing it requires the response calculation of the broken vacuum, which leans on the standard QFT cutoff structure of the literature. This is debt D1, partially closed by AP08 + this paper, not fully closed (see section 0.6).

7.6 — Summary

Established. (1) The ℤ₂-symmetric double-well action is the minimal pre-geometric theory with SSB. (2) In the broken phase, perturbations propagate with speed c = √(β/α). (3) The condensate energy density scales as ε². (4) Via AP20 forcing + the section 7 derivation: G = κ/(εΛ²), c/G ∝ ε, mₚ ∝ √ε.

Open. (1) The substrate parameters α and β are introduced but not derived; c is parameterised but not predicted (debt D-AP03.1). (2) The kinetic term presupposes a distinction between temporal and spatial coordinates (τ vs x) — a structural input, not an output. The deepest open problem in AP03, distinct from and prior to the G derivation. A truly pre-geometric formulation would derive this distinction.