Claim Summary

Section 0 (Dependency and Scope). Contract with the reader. Dependencies (AP20 for the canonical axioms and the EH and QRA proofs, with the completeness/minimality obligations fenced at KS-P.1/KS-P.2; the Lorentzian signature via AP10 section 4, fenced at KS-D.4; AP10 for N = 3 contingent on KS-D.3; AP06 Theorem 3.1; The Lock, Edition 04; Lovelock 1971/1972 as external), axiom role mapping (S → eye topology and condensate framing; B → The Lock, the coefficient identification, the horizon leakage; R → weak-field linearity consistency, the R-conservation bridge, density non-negativity; C → differential-form locality, horizon formation, density-change propagation bound), epistemic status per section, kill switch summary, structural debts (The Lock identification; the R → conservation bridge), external mathematical dependencies (Lovelock 1971/1972; potential theory; Schwarzschild + Hawking 1975), items held open (cosmological constant value; Constraint 2 weak-field regime definition), structural relationships.

Section 1 (Axioms Inherited). FOUNDATIONAL. The four canonical axioms {S, B, R, C} re-stated from AP20 Paper D. EH and QRA proven (AP20). Lorentzian manifold (M, g) with signature (−, +, +, +) and symmetry group SO(1, N) carried via AP10 section 4 (formal derivation fenced at KS-D.4). Special relativity is a theorem of that signature, not an assumption.

Section 2 (The Condensate). STRUCTURAL FRAMING. The manifold understood as the structure that emerges from accumulated records. Mode 0 (the metric, the geometry, the background condition) and Mode 1 (propagation on the condensate, coupling events). Gravity is Mode 0.

Section 3 (Record Density). DEFINITION + IDENTIFICATION. Record number density n(x) defined (equation 3.1). Identification with matter density ρ(x) = n(x) · ⟨E⟩/c² (equation 3.2) under The Lock. Properties inherited from axioms: ρ ≥ 0 from R, causality from C, electron coupling from B.

Section 4 (Weak-Field Form). DERIVATION + LIMIT RECOVERY. Four axiom-derived constraints (Lorentz invariance, locality as differential form, second-order in the metric, scalar) applied to the weak-field regime (Constraint 2 as the defining condition of the regime) leave the Helmholtz family a·∇²Φ + b·Φ = A·ρ (equation 4.1). Pure Poisson ∇²Φ = A·ρ (equation 4.2) is the b = 0 case, recovered as the small-cosmological-constant limit of section 8’s full identity in which Λ corrections are negligible at laboratory scales.

Section 5 (Coefficient). IDENTIFICATION + HONEST DISCLOSURE. G = 2κ/m_e² (equation 5.1) via The Lock; A = 4πG = 8πκ/m_e² (equation 5.2); ∇²Φ = (8πκ/m_e²)·ρ (equation 5.3). Structural circularity disclosed: κ defined as Gm_e²/2, so relabelling is tautological at the level of numerical computation. κ structurally inaccessible to independent measurement.

Section 6 (Strong-Field Limit). CONSISTENCY CHECK — RECOVERY. Hawking surface gravity κ_H = c⁴/(4GM) (equation 6.1) and Hawking temperature T_H = ℏc³/(8πk_BGM) (equation 6.2) recovered from the Schwarzschild solution of section 8 plus standard QFT in curved spacetime (Hawking 1975). Under The Lock substitution: T_H = ℏc³ m_e² / (16πk_BκM) (equation 6.3). The m_e at the horizon is the same Lock relabelling reappearing, viewed at the horizon scale, not an independent finding.

Section 7 (Full Nonlinear Structure). STRUCTURAL ARGUMENT. The eye topology: two curves (ℒ and 𝓟 from Axiom S) meeting at the common point (the 1:1 pre-state). Closed topology. Newton is the linearisation near the common point; Einstein is the full curve. The Lock identification carries through both ends as a consequence of the same coupling identification, not as an independent finding at either end.

Section 8 (Tensor Form). DERIVATION. Five covariant constraints (general covariance, symmetric rank-2 tensor, divergence-free source via R bridge, second-order in the metric, four-dimensional from AP10 contingent on KS-D.3) plus Lovelock 1971/1972 force G_μν + Λg_μν = (8πG/c⁴) T_μν (equation 8.1) as the unique symmetric, divergence-free, rank-2 tensor identity in 3+1 dimensions. Einstein’s field equations with cosmological constant, with the Λg_μν term retained as the four-dimensional covariant counterpart of the Helmholtz-family b·Φ term that the weak-field expression drops in the small-Λ limit.

Section 9 (Kill Switches). FORMAL FALSIFICATION REGISTER. KS-I.1 (axiom-to-Poisson, LIVE — HARD), KS-I.2 (Lovelock chain, LIVE — HARD), KS-I.3 (coefficient identification, LIVE — HARD), KS-I.4 (κ inaccessibility, LIVE — HARD), KS-I.5 (curvature-record identity, LIVE — EMPIRICAL), KS-I.6 (N = 3, CLOSED by AP10 contingent on KS-D.3), KS-I.7 (cosmological constant value, LIVE — EMPIRICAL), KS-I.8 (R → conservation bridge, LIVE — HARD).

Section 10 (What This Establishes and What Remains Open). SYNTHESIS. Seven closures named (form of Poisson; form of Einstein’s field equations; cosmological constant existence prediction; coefficient identification; κ structural inaccessibility; Hawking temperature consistency check; Newton-Einstein structural relationship). Seven kill switches live. Two items held open without claiming separate debt status. Placement within The 420 Code: fourth and closing chapter of Notebook II.