Notation Reference

{S, B, R, C} — The four axioms of the 420 Code. S: two-sector + involution. B: the break ε ∈ ℓ, no σ-image. R: time, the only irreversible axiom. C: finite propagation, the propagation face.

ℓ, 𝒫 — The two sectors of Axiom S, connected by the involution σ.

σ — The involution of Axiom S. On the orientation state, acts by complex conjugation, taking the fundamental 3 to the conjugate 3̄.

ε — The break. The minimal asymmetry. The unique unpaired element of Axiom B. Identified with the electron at the chemical-biological scale (The Lock, Edition 04).

ν — The valuation on the record algebra; ν(ε) = 1 is the minimum nonzero value.

{C, S, B} — The three spatial faces of the manifold. Each face has fixed functional identity (propagation, exchange, break) but no fixed geometric assignment (which spatial direction it occupies).

|ε⟩ — The orientation state of the break in face-space; |ε⟩ ∈ ℂ³ with |α₁|² + |α₂|² + |α₃|² = 1.

α₁, α₂, α₃ — Complex amplitudes of the orientation state. NOT the fine-structure constant α_em or the substrate-stiffness α of AP06. Symbol firewall enforced throughout.

e₁, e₂, e₃ — Basis vectors of the face-space orientation state — the three faces of the manifold as quantum-state basis.

ℂ³ — The complex three-dimensional vector space; the orientation state lives here (Lemma in §4.2).

U(3) — The unitary group on ℂ³; norm-preserving transformations of the orientation state.

SU(3) — The special unitary group on ℂ³; gauge group of the strong force; 8 generators. Derived as U(3) modulo the orientation’s global phase (removed by Born-rule redundancy, AP09).

3, 3̄ — The fundamental and conjugate representations of SU(3); quarks carry 3, antiquarks carry 3̄.

Tᵃ — The 8 generators of SU(3), a = 1, …, 8. Denoted Tᵃ (not the conventional λ₁–λ₈) to avoid collision with λ = substrate stiffness.

Z₂ — The two-element group; π₁(SO(3)) = Z₂ in N = 3 spatial dimensions; lifts to SU(2) by the double cover.

SU(2) — Two distinct groups in the corpus, not to be conflated: the spacetime spin SU(2), the double cover of SO(3) (AP11); and the weak gauge SU(2) of the electroweak interaction, derived from the sector-relationship freedom (AP27). AP19 uses neither directly — both are named only as context for the assembled gauge structure in §7.

U(1) — The phase-rotation gauge group; derived in AP15 from the Hilbert-space phase freedom; manifests on the manifold as the electromagnetic connection. Distinct from the orientation’s own global phase factored out in Proposition 1.

λ — Substrate stiffness, AP15; λ ≈ 2.15 × 10⁴⁶. Quantitative scale of isotropy enforcement. NOT a wavelength.

c, g — Speed of light (Axiom C); the metric tensor (created by actualisation, AP05).

ℏ — Reduced Planck constant (AP12).

α_em — Fine-structure constant; the leakage rate (AP06); identified with ε in the leakage framework. NOT one of the orientation amplitudes α₁, α₂, α₃.

α_s(Q²) — Running coupling of the strong force; not derived in this paper (D3).

m_e, m_μ, m_τ — Masses of the three charged leptons; mass ratios not derived (D2).

AS — Actualization State. The actualizing now.

EH — Embedding Hypothesis (AP20). Proved — unconditional throughout AP19.

QRA — Quantum–Records Algebra (AP20). Proved — unconditional throughout AP19.

AP20 §4 — Carries the completeness (KS-P.1) and minimality (KS-P.2) of {S, B, R, C} as fenced obligations. The independence of the axioms — no one derivable from the others — is the minimality leg. Load-bearing for KS-50.

The Lock — Edition 04 of the 420 Code; identifies ε with the electron at the chemical-biological scale.

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the420code.org

Series The 420 Code

Catalogue Ø Artist’s Proofs

Proof Number AP19

Title The Direction

Subtitle Forces and Constants

Medium Physics

Artist G

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