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Theorem atlas
41 theorems from the sieve to the Standard Model, each clickable to its full proof.
Interactive replica of the monograph's full-page figure: axioms U1–U4 → T1 → T3 → s = ½ → T2 → L0 → T6 → GFT → G1/G3 → T4 → Mertens → T5 (μ* = 15) → T0 → W7-1 → BA5 → Lemmas E/F/G → α_EM, sin²θ_W, α_s, G, m_e → 43 observables. Each box is colour-coded by epistemic class (THM blue, ID green, DER orange, BRIDGE purple, VAL grey, COND hatched, CONJ crosshatched) and carries its Lean formalisation mark.
Coverage : 39/41 cards link to a dedicated fiche on the site; the rest defer to the monograph via the footer link.
Hover a card to see its immediate neighbours: what it depends on directly (just before) and what it directly enables (just after). On mobile: tap a card to toggle the highlight.
Column 1
I. Arithmetic core
generative primitives {1, +, ×, ≠ 0}
Forbidden self-transitions: $T_3$ has $T_{11}=T_{22}=0$
$T_3$ antidiagonal, unique stochastic form
stationary measure of $T_3$ (the theorem)
Conservation $s^2 = 1/4$
Max-entropy lock on residue classes
Eratosthenes uniqueness (field + axioms + Čencov)
$\log_2 m = D_{KL} + H$
$D_{KL}$ unique divergence (Shore–Johnson)
Fisher metric unique (Čencov 1982, ext.)
$\sin^2\theta_p = \delta_p(2-\delta_p)$
Column 2
II. Convergence & fixed point
Spectral convergence: $\alpha_k \to 1/2$
Compactness: $\|M(x) - \log\log x\| < \infty$
criterion $\gamma_p > s \Leftrightarrow p \in \{3,5,7\}$
Master fixed point: $\mu^* = \sum_{\text{active}} p = 15$
$N_c = N_{\text{gen}} = N_{\text{spatial}} = 3$
Unique gap sequence (lifting U1–U4)
Spiral identity: $\sigma^2 = \pi(k+1)$
chain $s \to \alpha_{\text{bare}}$
subdominant bound (under T5Like axiom)
prime theorem (active spectrum ⇒ $N_{\text{gen}}$)
Column 3
III. Bridge
Pontryagin product: $\widehat{\mathbb{Z}/P} = \prod \widehat{\mathbb{Z}/p}$
physical identification: $\alpha_{\text{sieve}} = \alpha_{\text{EM}}$
OS reconstruction (positivity)
metric reconstruction (Fisher → Riemann)
inductive limit (Wightman W1–W4)
theorem (4 uniqueness: T6 + G1 + G3 + T5)
$\mathbb{Z}/P \cong \bigoplus \mathbb{Z}/p$ — causal gauge invariance
$C^*$-dynamical: $Z(\beta) \to \zeta(\beta)$, $\beta_c = 1$
step-error bound (Lee–Yang tail)
dependency DAG
Column 4
IV. Reconstruction & output
vN1–vN6 axioms; Born rule from $|w_p|^2$
Bianchi I + Lorentz sig.; Friedmann from Fisher
GFT = first law; $Z_F = Z_R = Z_P$
$\alpha_{\text{bare}} = \prod \sin^2\theta_p$ + dressing → 0.004 ppb
$\sin^2\theta_W = \gamma_7^2 / \sum \gamma_p^2$, 0.01–0.05 %
$G/\alpha = 2\pi$ (Friedmann); $m_e = s$ (SCU calibration)
mean 0.30 % (0 fitted parameter)
predictions P1–P28 (falsifiable)
geometric ($F_{\text{inactive}}$); DM/DE programme
via spiral W7 — not reduced
Legend
Epistemic classes
- [THM] proved unconditional
- [ID] algebraic identity
- [DER] derivation
- [BRIDGE] physical identification
- [VAL] empirical validation
- [COND] conditional (hatched)
- [CONJ] conjectural (crosshatched)
- axiom U1–U4 primitives
Lean status
- ✓L kernel-verified, 0 sorry
- ∼L partial (documented sorrys)
- ⋆L external (Mathlib)
- ◦L not formalised
Going further
This map is the web rendering of a full-page figure from the monograph. For the complete dependency arrows (intra-column, inter-column, and the short-circuit feeding the headline α_EM result) and the full context of each result, consult the monograph. For the formalised modules, see the Lean formalisation page; for fine epistemic status, see Result status and Audit.