Dependency and Scope

0.1 — What this paper does

It proposes a structural identity between black-hole interiors at the quantum-gravity scale and the origin of expanding universes, and formalises the axiom 1:1 + 1×ε as a metric equation across a transition hypersurface: Conjecture L7, g⁺ = (φ⁻¹)*(g⁻) + ε·h, under five explicit assumptions. It introduces a three-layer certainty system (established / speculative-but-published / philosophical) and tags every claim. It identifies the two fundamental constraints (c and ρₚ), reads the symmetry break through the electron and α, identifies awareness with ε > 0, and lays out nine falsification conditions. It is a conjecture, not a derivation.

0.2 — Dependencies

Established relativity: the Schwarzschild interior as Kantowski-Sachs (Kantowski & Sachs 1966), the Kruskal–Szekeres maximal extension, the Bekenstein–Hawking entropy bound, the Page curve and island formula (Almheiri et al. 2021). Published speculative physics: Popławski’s Einstein–Cartan torsion mechanism (2010, 2016, 2023). Within the 420 Code: AP03 (the conjugacy of c and ρₚ). The Embedding Hypothesis (EH) and Quantum-Record Alignment (QRA) are proved in AP20; the conjecture itself is held at Layer 2/3 and is not claimed as proved.

0.3 — Axiom mapping

Axiom S (Symmetry — the involution σ). 1:1 is interpreted as a ℤ₂ symmetry on the transition hypersurface Σ (Interpretation I1): the transition diffeomorphism φ maps the collapsing geometry to the expanding one, φ|Σ = id, normal-reversing.

Axiom B (Break — the unique break ε). +1×ε is interpreted as the ℤ₂ symmetry breaking on Σ (Interpretation I2). The perturbation ε·h is the single break carried across the transition — the records, the anisotropy difference, the initial conditions of expansion.

Axiom R (Record — monotone accumulation). Information is deviation from symmetry: if ε = 0 there are no records; if ε > 0, h carries the records compressed into the transition. The transition hypersurface is the point of maximum coarse-grained entropy.

Axiom C (Constraint — locality; finite propagation speed c). The propagation constraint c defines the null-cone structure; the compression constraint ρₚ is the Planck density where quantum gravity dominates. These are the two constraints between which the break is held.

0.4 — Epistemic status per section

1–3 (Method, axiom, layers). FOUNDATIONAL.

4–5 (Constraints, the break). LAYER 1 + LAYER 3 — established physics with architectural interpretation.

6 (Awareness). LAYER 3 — entirely philosophical; the awareness identification.

7 (Formalisation). LAYER 2 + 3 — Conjecture L7 under five explicit assumptions.

8 (The structural identity). LAYER 1 + 2 — established mathematics (the diffeomorphism, the entropy resolution) plus the speculative isotropisation mechanism.

9–12 (LQC, predictions, scale, existing physics). LAYER 1–3 — the relation to other programmes and the unique predictions.

0.5 — Kill switch summary

Nine kill switches, KS-L0.1 through KS-L0.9 (canonical registry numbering). KS-L0.1 (structural identity disproved) and KS-L0.2 (no singularity resolution) are LIVE — HARD and target the conjecture; KS-L0.8 (anisotropy irreconcilable) is the sharpest obstacle, LIVE — HARD; KS-L0.7 (fermions unnecessary) is LIVE — HARD; KS-L0.3 (information loss), KS-L0.4 (proton decay), KS-L0.5 (GSL violated), KS-L0.6 (actualisation without awareness / GRW), and KS-L0.9 (CMB mismatch) are LIVE — EMPIRICAL.

0.6 — Structural debts owed

D1 (Isotropisation). A rigorous mechanism for Kantowski-Sachs → FRW isotropisation — the sharpest obstacle (KS-L0.8). Popławski’s torsion mechanism is proposed but not established. Open.

D2 (CMB spectrum). A derivation of the curvature perturbation spectrum Pₘ(k) from ε·h — required for KS-L0.9. Open.

D3 (Fine-structure constant). A derivation of α’s value from the axiom — acknowledged as open. Its underivability is consistent with the structural argument in AP41 (α cannot be read from within the splinter; structurally undetermined at sub-leakage precision). Open.

0.7 — Items held open without claiming debt status

The awareness identification (section 6) is a Layer 3 naming choice with philosophical consequences; the physics is identical under either name, and it is falsifiable only via KS-L0.6 (GRW). The scale question (section 11) — whether the identity is cosmic, astrophysical, or nested-fractal — is a Layer 3 reading. The interpretive identification of the electron and α (Step 6 of the chain) is physically motivated but not derived from the conjecture.

0.8 — Structural relationships

AP03 (The Ratio). The conjugacy of the compression constraint ρₚ and the propagation constraint c. If AP03 holds, AP04’s two constraints are not independent.

AP41 (The Loop). Inherits AP04’s conjecture at Stage 6 (the door): AP41’s KS-41.3 inherits KS-L0.1. AP41’s structural argument for the underivability of α is the body of work-level statement of AP04’s D3.

AP20 (The Proof). EH and QRA proved; the substrate-as-manifold identity that the structural identity rests on at Layer 3.

Established relativity (Kantowski–Sachs, Kruskal–Szekeres). The Layer 1 mathematics: the black-hole interior as a collapsing cosmology, the maximal extension as an expanding one.

Popławski (Einstein–Cartan torsion). The Layer 2 mechanism for singularity avoidance and isotropisation — published, not confirmed; the load-bearing speculative input.