The Formalisation: From Axiom to Conjecture

7.1 — Assumptions

Assumption A1. Working model: Einstein–Cartan gravity with fermionic matter and spin–torsion coupling.

Assumption A2. There exists a non-singular transition hypersurface Σ, defined by an invariant criterion (e.g. a curvature-scalar threshold or an effective quantum-gravity completion condition reaching the Planck regime), replacing the classical singularity.

Assumption A3. There exists an approximate ℤ₂ mapping across Σ in the chosen completion.

Assumption A4. The expanding side is approximately FRW after Popławski’s torsion-driven isotropisation mechanism.

Assumption A5. Deviations between the collapsing and expanding geometries are captured by a perturbation of the form ε·h, where ε is small and h is a symmetric 2-tensor on M.

7.2 — Interpreting 1:1

Interpretation I1. 1:1 ≡ ℤ₂ symmetry on Σ (Assumption A3).

7.3 — Interpreting + 1×ε

Interpretation I2. + 1×ε ≡ ℤ₂ symmetry breaking on Σ.

7.4 — The conjecture

Conjecture L7 (The Loop Hypothesis, guarded by KS-L0.1, KS-L0.2, KS-L0.8, KS-L0.9). g⁺ = (φ⁻¹)*(g⁻) + ε·h, where φ: M⁻ → M⁺ is the transition diffeomorphism (φ|Σ = id, normal-reversing), (φ⁻¹)* denotes the pullback of φ⁻¹ (mapping tensors on M⁻ to tensors on M⁺), ε is the symmetry-breaking parameter, and h is the perturbation tensor encoding the anisotropy difference, the information content, and the initial conditions of expansion. [Layer 2 + Layer 3]

The collapsing geometry and the expanding geometry are the same structure, mapped across Σ. The difference between them is ε·h — a small perturbation carrying everything that makes a universe a universe. If the equation is wrong, nine kill switches will find it.

7.5 — What h encodes

Anisotropy difference. M⁻ is Kantowski-Sachs (anisotropic). M⁺ is nearly FRW (isotropic). h carries the difference. [Layer 2]

Information content. Information is deviation from symmetry. If ε = 0, perfect ℤ₂, no records. If ε > 0, h carries the records compressed into the transition. [Layer 3]

Initial conditions. The trace of h on Σ determines the initial perturbation spectrum. [Layer 2 + 3]

7.6 — The electron and α

The electron is read as the minimal low-energy carrier of the break (section 5.3), and α as the coupling strength (section 5.4). This is an interpretive identification, physically motivated but not derived from the conjecture. The value of α is not derived (D3); its underivability is consistent with AP41’s structural argument.

7.7 — Limits

ε → 0. g⁺ → (φ⁻¹)*(g⁻). Perfect ℤ₂. No records. No distinguishable time direction. Sterile symmetry.

ε large. g⁺ departs from (φ⁻¹)*(g⁻). Rich structure. But the axiom constrains ε to be small (1×ε). The perturbation is gentle.

7.8 — The full chain

Step 1. Axiom: 1:1 + 1×ε. Step 2. 1:1 ≡ ℤ₂ on Σ (Assumptions A2, A3). Step 3. + 1×ε ≡ ℤ₂ breaking; ∃ φ: M⁻ → M⁺. Step 4. Conjecture: g⁺ = (φ⁻¹)*(g⁻) + ε·h (Assumption A5). Step 5. h carries records; ε·h induces Pₘ(k) on the expanding side. Step 6 (interpretive identification, not derived from Steps 1–5). Electron = minimal low-energy carrier of the break (section 5.3); α = coupling strength (section 5.4) — physically motivated but not derived from the conjecture. Step 7 (layer shift: entirely Layer 3). Awareness ≡ ε > 0 (naming choice). Structural result: all actualisation shares a common origin. Perspectival result: a conscious entity recognising this unity can identify cruelty as incoherent with it. The obligation is in the recognition, not in the structure. [Layer 3]