Claim Summary

Section 0 (Dependency and Scope). Contract with the reader. Dependencies, axiom mapping ({S, B, R, C} ↔ 1:1 + 1×ε @ AS, with empirical premise P sitting separately), epistemic status per section, kill switch summary, structural debts owed, items held open, structural relationships.

Section 1 (The Axiom). DECLARATION + STRUCTURAL ARGUMENT. The axiom 1:1 + 1×ε @ AS stated in canonical form (AP20’s keystone correction making AS explicit). The terms (1:1 = perfect balance; ε = minimal break; 1×ε = one count not a parameter; @ AS = the actualizing prior). The geometric picture (rotational ring + pin = ring acquires two readings). Scope Claim B1 (all structure follows from axiom + B1–B4 + declared empirical premise P). Four bridges B1–B4 declared. Empirical premise P (observer plurality) declared alongside the bridges, separately classed because it is empirical rather than postulational. Universal kill switch KS-B.0 (axiom is generator iff initial conditions on one side derivable from terminal conditions on the other; theoretical-universal consistency check now, empirical operationalization once B-JP solved).

Section 2 (The Chain). DERIVATION CHAIN. Seven steps, each with declared dependency. B-0 (From axiom): before records, AS = 0 at the floor by boundary definition. B-1 (From B-0 + B1): the crack, ε > 0, two record sectors, irreversible via Axiom R / AP01 Theorem T2. B-2 (From B-1 + B3): selection forced by definite outcomes plus multiple sectors. B-3 (From B-2 + B2): agency as ℳ(x) = μ(Reach(x) ∩ Viab(R))/μ(Viab(R)), equation (2.1). B-4 (From B-3 + Empirical Premise P): the load-bearing move from representational duality (two readings of one structure) to interactive plurality (two simultaneous agents); coupling via Geometric Exclusion Principle, AP01 Paper D D1.3. B-5 (From B-4): hierarchy (from coupling alone) / cooperation (requires P’s simultaneity) / deterrence (requires P’s simultaneity) as geometric consequences (D4.3 / D4.4a / D4.4b). B-6 (From B-5 + B4): structural ground (destroying viability contracts the shared kernel; geometry constrains joint domain, not necessarily individual reach) plus perspectival recognition (destroyer’s standing presupposes the substrate that equally grounds the destroyed).

Section 3 (The Boundary). DERIVATION + STRUCTURAL ALTERNATIVE + OPEN PROBLEM. Why not bounce: dynamical bounce cosmologies (Popławski; Lucat and Prokopec) and the structural alternative. The junction conditions on Σ: J1 [γ_ab] = 0 (induced metric continuity; the 1:1 as geometry) and J2 [K_ab] = (ε/ℓ_Σ) · h_ab (extrinsic curvature mismatch with completion-surface length scale ℓ_Σ supplying dimensions, h_ab dimensionless as pullback of L7’s perturbation tensor). Relation to AP04: junction is boundary-data version of AP04 Conjecture L7 (g⁺ = (φ⁻¹)∗(g⁻) + ε · h, global pullback). Observational consequences distinguishable from bounce in principle. Open Problem B-JP: existence of ECSK completion on Kantowski-Sachs interior with J1, J2.

Section 4 (Kill Switches). FORMAL FALSIFICATION REGISTER. KS-B.0 (Universal generator, LIVE — HARD pending B-JP, migrates to EMPIRICAL once B-JP solved). KS-B.1 (Bounce superior, LIVE — EMPIRICAL). KS-B.2 (Spectrum mismatch, LIVE — STAGED pending B-JP and spectrum mapping). KS-B.3 (No junction solution, LIVE — HARD). KS-B.4 (Chain break: any step uses input not in {axiom, B1–B4, P}, LIVE — HARD).

Section 5 (What This Establishes and What Remains Open). SYNTHESIS. Seven structural closures named (axiom in canonical form; four bridges declared at door; empirical premise P declared alongside; seven-step chain walked; Claim B1 stated; junction formulation reformulates AP04 identity with dimensional consistency restored; junction formulation distinguishable from bounce; five kill switches engaged). Two structural debts (Open Problem B-JP partially closed via formal statement of the boundary-value problem; Bridge B4 declared as cost). One item held open (spectrum mapping from [K] to P_ ℛ(k)). Placement within The 420 Code: second chapter of Notebook II (Spacetime). Closing on the same door at the beginning and the end.