The Starting Point

1.1 — The four axioms

You have four axioms.

AP20 Paper D installed them. The 420 Code has been using them.

Look at them now as the floor of what is about to happen.

S (Symmetry). Two disjoint sectors ℒ and 𝓟 with order-reversing involution σ. Extensive quantities match: Q(ℒ) = Q(𝓟) in the unbroken case.

B (Unique Breaking). One element ε ∈ ℒ with no σ-image. Valuation: ν(ℒ) − ν(𝓟) = ν(ε) = 1. This is the break. Without B, the system is the 1:1 — perfectly symmetric, and nothing exists.

R (Record Monotonicity). Sequential composition (·) forms a monoid, not a group, within each sector. No non-identity element has an inverse. Records cannot be annihilated locally. History is irreversible.

C (Constraint). Finite invariant rate c constraining sequential propagation. Structural, not electromagnetic.

These four axioms are independent (the four removal arguments at AP20 section 4.2; full formalisation fenced at KS-P.2) and jointly consistent — any existing record exhibits all four jointly satisfied (the undeniable premise, AP20 section 6).

Independence means no axiom is derivable from the other three. Removing any one axiom produces a strictly weaker system. Each axiom adds irreducible content to the algebra.

1.2 — What is now proven

Two hypotheses were carried as conditionals in earlier APs. Both are now proven (AP20).

EH (Embedding Hypothesis). The algebraic pre-state structure defined by {S, B, R, C} embeds into physical reality as a smooth manifold M. Proven in AP20.

QRA (Quantum-Record Alignment). Quantum states are pre-state records. Closed at AP20 section 5.5 (KS-P.4). The differentiability and leading-order quadratic structure of the cone boundary invoked here rest on the smooth-manifold structure fenced at KS-P.5.

From the axioms, the record algebra produces a Lorentzian manifold (M, g) with signature (−, +, …, +) and symmetry group SO(1, N).

1.3 — What was left undetermined

The number of + signs — the number of spatial dimensions N — is not fixed by the signature. The signature argument (section 4; formal derivation fenced at KS-D.4) establishes that the manifold is Lorentzian and that the symmetry group is SO(1, N). It does not fix N.

This paper fixes N.

Specifically: N is determined by the number of independent axioms, with a structural consistency check from the multidimensional residual. A primary derivation — the axiom count — fixes N = 3; the residual face structure confirms it. The two are not doubly independent: the residual includes the temporal face, so it is a cross-check, not a second derivation.