The Theory of Persistence

Theorems

Theorems

The central results of Persistence Theory, in two tiers: the fundamentals (GFT, then the T₀ → T₆ chain and L0) closing the derivation at μ* = 15, and the secondaries (naturality, bridges, identifications) widening and tightening the edifice.

Fundamental theorems

Nine results. GFT gives the fundamental principle: capacity = persistence + entropy. T0 then closes the BA0 axiomatic basis, L0 gives the uniqueness lemma, and T1–T6 close the arithmetic chain at $\mu^* = 15$.

GFT — Gap Fundamental Theorem

$\log_2 m = D_\mathrm{KL} + H$ — fundamental principle of persistence.

T0 — BA0 Closure

The gap sequence is the unique dynamical output of the sieve.

L0 — Unique geometric distribution

The unique memoryless maximum-entropy distribution on even gaps.

T1 — Forbidden auto-transitions mod 3

The sieve forbids two consecutive identical residues mod 3.

T2 — Spectral conservation

Exact spectral identity $|\lambda_2(T_{30})| = s^2 = 1/4$.

T3 — Antidiagonal transfer

$T_3 = \mathrm{antidiag}(1,1)$ — the mod-3 matrix is purely off-diagonal.

T4 — Spectral convergence

$\alpha_k \to 1/2$ as the sieve depth $k \to \infty$.

T5 — Informational attractor μ* = 15

After p = 2 crystallises, the reduced sieve sector admits a unique physical attractor: $\mu^* = 3+5+7 = 15$.

T6 — Holonomy

$\sin^2 \theta_p = \delta_p (2 - \delta_p)$ — trigonometry emerges from the sieve.

Secondary theorems

Sieve naturality (N1–N4), derived bridge axioms (BA5), identification lemmas (E, F, G), and associated structural results.

N1 — Algebraic uniqueness of primes

The primes are the unique atoms of the multiplicative monoid $(\mathbb{N}_{\geq 1}, \times)$.

N2 — Sieve self-consistency

The Eratosthenes sieve is the unique self-consistent multiplicative sieve on $\mathbb{N}_{\geq 2}$.

N3 — Structural minimality of ℕ

$(\mathbb{N}_{\geq 1}, \times)$ is the unique free commutative cancellative UFD monoid with countable atoms.

N4 — First cascade level

In the canonical sieve ordering, $p = 3$ is the first dynamical level. $p = 2$ has a structurally distinct role (info / anti-info).

BA5 — Pontryagin product

At the reduced attractor $\mu^* = 15$, the sieve coupling is the product $\prod_{p \in \{3,5,7\}} \sin^2\theta_p(q_+)$.

Lemma E — Coupling reconstruction

The reconstructed QFT's coupling is $g^2 = \prod_{p \in \{3,5,7\}} \sin^2\theta_p(q_+)$ — a spectral invariant, not an identification.

Lemma F — Metric reconstruction

The reconstructed QFT's spacetime metric is the sieve's Fisher metric evaluated at $\mu^* = 15$.

Lemma G — Hilbert reconstruction

The reconstructed QFT's Hilbert space is $\mathcal{H}_\infty = \varinjlim \bigotimes_{p \mid m_K} \mathcal{H}_p$.

Fisher-Koide identity

$C_K = G_{\rm Fisher}/\sin^2\theta_3 + (1 + 5\delta_3^2/18)/21$ — exact derivation of the Koide coefficient at 0.04 ppm.

Fourier-Koide lemma

$Q_{\rm Koide} = 2/3 \iff |a_1|/|a_0| = 1/\sqrt{2} = \sqrt{s}$ — Parseval equivalence on $\mathbb{Z}/3\mathbb{Z}$.

OS3 — Uniform reflection-positivity

For every $p \geq 3$, $M_p = T_p^T T_p \succeq 0$ (Gram matrix). Wightman reconstruction applies.

G1 — Uniqueness of $D_{KL}$ (Shore–Johnson)

On the CRT-structured sieve simplex, $D_{KL}$ is the unique consistent divergence.

G3 — Fisher metric uniqueness (Čencov)

Fisher is the unique Markov-monotone Riemannian metric on the simplex.

Mertens — Compactness of $M(x)$

The function $M(x) - \log\log x$ is bounded — classical, imported into PT.

Active prime criterion

A prime $p$ is active iff $\gamma_p > s = 1/2$ — the active set is exactly $\{3,5,7\}$.

$N_c = 3$ — Colour from the sieve

$N_c = N_{\text{spatial}} = 3$: unique integer solution of $(N_c+1)!/(N_c+3) = 2^{N_{\text{spatial}}-1}$.

CRT — Sieve decoupling and causal invariance

The sieve factors additively: $\mathbb{Z}/P \cong \bigoplus_p \mathbb{Z}/p$. Operations at different primes commute.

Thermodynamics — GFT = first law

The GFT identity $\log_2 m = D_{KL} + H$ is the first law of sieve thermodynamics.

Depth