Glossary

Epistemic tag: derivation. Result derived from the PT chain, testable against experiment. Stronger than [PRED], weaker than [THM].

Epistemic tag: algebraic identity. True with no physical assumption (e.g. GFT: log m = D_KL + H). Maximum epistemic strength, equivalent to [THM].

Epistemic tag: falsifiable prediction. Numerical statement not yet verified but imposed by PT. E.g. δ_CP(PMNS) = 197.4° (DUNE ~2032).

Epistemic tag: PT theorem. Result proved unconditionally inside the mathematical PT framework. Highest epistemic level in PT.

Prime p with γ_p > s = 1/2 at the fixed point μ*. Three active primes: {3, 5, 7}. All primes ≥ 11 are inactive, but may appear as echoes. Determines N_gen = N_c = 3.

Logarithmic sensitivity exponent γ_p = −d ln(sin²θ_p)/d ln μ. It measures how fast the cyclic phase of a prime p changes as the sieve level μ varies. The criterion γ_p > s selects active primes.

Fifth bridge axiom. Derived theorem: α_EM is the Pontryagin product of sin²(θ_p) over active primes. Not a postulate — a consequence.

At μ* = 15, the system splits into two sectors: q⁺ (couplings) and q⁻ (geometry). Separates leptons/vertex from quarks/propagator, PMNS from CKM.

Chinese Remainder Theorem. Z/(p·q·r)Z ≅ Z/pZ × Z/qZ × Z/rZ for distinct primes. Source of the orthogonality of the three circles {3, 5, 7} and 3D dimensionality.

Kullback–Leibler divergence. In PT, measures the persistence of a distribution P relative to uniform U_m. Measured in bits. This is what literally persists.

Inactive prime (γ_p < s). It does not contribute as a dynamical primary, but may leave an echo polarization that dresses couplings. {11, 13} are the dominant echo primes.

Binary informational leakage. F(2) ≈ 0.7583. Dresses α_bare = 1/136.28 to α_EM = 1/137.036 (leading order). Originates from the p = 2 channel (info/anti-info boundary).

Gap Fundamental Theorem. Algebraic identity log₂(m) = D_KL(P ‖ U_m) + H(P). It is the fundamental principle of persistence: informational capacity is conserved and decomposes exactly into persistence and entropy.

Mathematical name for phase transport on the circle Z/pZ. In the monograph nomenclature, the usual PT term is “cyclic phase”: the Cyclic Phase Identity defines θ_p by cos θ_p = 1 − δ_p, hence sin²(θ_p) = δ_p(2 − δ_p).

Maximum entropy lemma. The geometric distribution is the unique memoryless max-entropy distribution on ℕ. No choice: forced.

Number of fermion generations. In PT: N_gen = |{active primes}| = |{3, 5, 7}| = 3. Equal to N_c (quark colors) — not a coincidence.

Structured part of a distribution, measured in bits by D_KL. Conserved along the PT cascade. Per-channel cap = 1 bit (sin²θ_p ≤ 1).

Coupling branch: q⁺ = 1 − 2/μ. Gives sin²(θ_p, q⁺) → vertex, leptons, α_EM. Bifurcates from μ* = 15.

Geometry branch: q⁻ = e^(−1/μ). Gives sin²(θ_p, q⁻) → propagator, metric, quark masses, CKM.

Derived fundamental symmetry of the sieve. Every other dimensionless constant descends from it together with the arithmetic structure. It is forced by mod-3 forbidden transitions (T1), rather than chosen as a free parameter.

Algebraic holonomy identity (T6): sin²(θ_p) = δ_p (2 − δ_p). Trigonometry emerges naturally from transport around the circle Z/pZ.

Dynamical Field Theorem. Under conditions U1–U4, the gap sequence {g_n} is the unique admissible dynamical field of the sieve; BA0 is therefore forced.

Forbidden Transitions theorem. At mod-3 sieve level, transitions 1→1 and 2→2 are forbidden; this zero diagonal forces the symmetry s = 1/2.

Arithmetic torus T³ = Z/(3·5·7)Z = Z/105Z. Unified gauge space of the three active channels. Physical amplitudes factor on it via CRT.

Fixed point theorem. μ* = 15 is the unique finite solution of Σ p_active = μ with γ_p(μ) > s. Verified exhaustively by exact rational arithmetic.

Holonomy theorem. sin²(θ_p) = δ_p(2 − δ_p) is exact (not approximate). Identified to 3.5 × 10⁻⁷ bits with the Gibbs measure on all tested moduli.

Fine-structure constant. In PT: α_EM = ∏ sin²(θ_p, q⁺) over {3, 5, 7} = 1/136.28, dressed by F(2) to 1/137.036 (CODATA). Zero fitted parameters.

Anomalous dimension of prime p at the fixed point: γ_p = −d ln(sin²θ_p) / d ln μ. A prime is active if γ_p > s = 1/2. At μ* = 15: γ_3 = 0.808, γ_5 = 0.696, γ_7 = 0.595, γ_11 = 0.426 (inactive).

Gap fraction for prime p: δ_p = (1 − qᵖ) / p. Elementary building block of holonomy (T6).

Unique sieve fixed point: μ* = 3 + 5 + 7 = 15. Sum of the three active primes (T5). No other finite prime subset closes the self-consistency equation.