35 independent confirmations

Statement: 3^2 - 2^3 = 1 is the unique non-trivial solution of x^p - y^q = 1 for x, y, p, q \geq 2.

PT link: PT deduces N_{\rm gen} = 3: exactly 3 fermion generations. Mihăilescu’s proof arithmetically validates what could look like a numerical coincidence.

Statement: Every semistable elliptic curve over \mathbb{Q} is modular.

PT link: The modular structure used by PT (CRT factorisation, Pontryagin characters) is consistent with the same modularity that closes Fermat’s proof. No contradiction.

Statement: Predicted densities of constellations (p, p+2, p+6), etc., under the k-tuples conjecture.

PT link: PT gives the structural fractions; HL validates their consistency with the prime distribution at large N. PT computation S_{\rm HL}(3) = \prod f(p) = 2.

Statement: Neural network verifying Hardy–Littlewood asymptotics for multiple prime constellations.

PT link: Independent numerical confirmation by ML method, consistent with PT computation S_{\rm HL}(3) = 2. Validates the arithmetic → spectrum bridge.

Statement: Max-entropy methods that re-derive Hardy–Ramanujan and Erdős–Kac.

PT link: Independent support for L0: PT’s geometric distribution of gaps emerges from the same variational principle. Max-entropy is universal, not an artefact.

Statement: The Ruelle–Mayer spectrum retains the spectral information of Maass forms and zeros of \zeta.

PT link: Confirms PT’s use of T_m as carrier of arithmetic structure (Lemma E, spectral rigidity).

Statement: Extension of Ruelle’s zeta function to non-archimedean rational maps (p-adic).

PT link: Consistent with the profinite decomposition underlying BA5 (Pontryagin product over \prod_p \mathbb{Z}/p\mathbb{Z}).

Statement: Programme proposing an "arithmetic electric-magnetic duality".

PT link: Deep structural parallel with PT: automorphic periods and L-functions play the role of gauge observables. Programmatic convergence with the PT bridge.

Statement: Unitary S-matrix built from the Euler product whose eigenvalues converge to the zeros of \zeta as \sigma \to 1/2.

PT link: Convergence with the PT bridge (Lemma E): zeta zeros emerge as the spectrum of a unitary operator derived from the Euler product.

Statement: The Fisher metric is the unique monotone Riemannian metric on the space of distributions.

PT link: Used directly by PT as the foundation of the spacetime metric (Lemma F). Without Čencov, no PT derivation of gravity.

Statement: Extends Čencov’s uniqueness to non-commutative probability spaces (quantum monotone metrics); the key property is *traciality*, not commutativity.

PT link: Strengthens G3: Fisher metric remains unique even when the sieve algebra is extended to a quantum setting. No surprise for PT.

Statement: The non-Archimedean Calabi conjecture admits a solution: a continuous plurisubharmonic potential on the Berkovich space, whose Monge–Ampère equation is supported on the discrete simplicial skeleton Sk(X) of a Calabi–Yau degeneration.

PT link: Independent realisation of the very dissolution discrete/continuous that PT proves in ch. 24 (four-step argument, theta bridge with quantitative convergence \gamma_p \sim q^{0.73p}, OS1–OS5 passing to the limit). Different mathematical substrate (Calabi–Yau manifolds, Berkovich geometry), same structural result: continuous geometry is an intrinsic property of a discrete combinatorial object, without any limit procedure. The dissolution is therefore not a feature peculiar to PT’s arithmetic sieve.

Statement: Under a \emph{valuative independence} condition on a canonical basis of the section ring, the metric SYZ conjecture (asymptotic existence of a special Lagrangian T^n-fibration on degenerating Calabi–Yau manifolds) follows.

PT link: Direct continuation of the Boucksom–Favre–Jonsson programme: refines the parallel convergence with PT toward T^n fibrations — continuous analogues of the PT arithmetic torus T^3 = \mathbb{Z}/3 \times \mathbb{Z}/5 \times \mathbb{Z}/7. Strong candidate for a future rigorous bridge between PT and the strings / mirror symmetry programme.

Statement: D_{KL} = Bekenstein-Hawking entropy (area = information).

PT link: PT reads D_{KL} as horizon entropy. Coincidence with Bekenstein’s formula non-trivial, suggesting PT informational gravity is in the right class.

Statement: Gravity = entropy gradient; Einstein equations emerge as equation of state.

PT link: PT identifies that entropy gradient as D_{KL} gradient. Compatibility with Jacobson (1995) and Verlinde (2010) is structural.

Statement: Gravitational dynamics derived from quantum relative entropy — "entropic" foundation of gravity.

PT link: Same structural mechanism as PT (Lemma F). Indicates a growing family of frameworks converging on info-geometry → gravity.

Statement: Einstein tensor derived directly from the Fisher metric of a statistical mechanical system.

PT link: PT goes further by identifying the specific statistical family (sieve distributions) reproducing the observed couplings. Matsueda validates the route.

Statement: D_{KL} confirmed as natural measure of irreversible information loss near horizons.

PT link: Convergence with PT’s reading of D_{KL} as horizon entropy (beyond Bekenstein); unified info / thermo / GR framework.

Statement: \alpha_{\rm EM} \approx 1/137 argued as a structural scaled quantity, at the intersection of classical / quantum / relativistic regimes.

PT link: Consistent with PT derivation as a product of cyclic phase angles (\prod \sin^2\theta_p) — not a free parameter, a structural ratio.

Statement: Q_{\rm Koide} = 2/3 derived independently from a geometric attractor of primitive triangle families.

PT link: Supports the PT view that the Koide relation is a structural identity, not a numerical accident. Authors: P. Quispe Hancco, A. N. Condori Mamani, C. Quispe Hancco, A. H. Zanabria Galvez, H. Quispe Hancco.

Statement: Cancellation of 2D gauge anomalies via the p-adic local-global (Hasse–Minkowski) principle.

PT link: Convergence with the CRT decomposition of the sieve: physical consistency goes through each prime p separately.

Statement: The operator \mathrm{eml}(x,y) = e^x - \ln y generates, with the constant 1, all elementary functions.

PT link: In this primitive complexity measure, 1/\alpha_{\rm EM} has EML size \simeq 40, more compact than \pi (size > 53). Naturalness signal invisible in standard notation.

Statement: In a minicharged hidden sector U(1)_H \times U(1)_X, anomaly cancellation imposes exactly a degree-3 Prouhet–Tarry–Escott problem.

PT link: For PT, this is not a proof but a strong convergence: quantum consistency can reduce to a discrete arithmetic constraint. The paper also gives a useful criterion for Class B extensions: no natural isolated minicharge, but PTE-compatible multiplets or doublets.

Statement: 1/\alpha_{\rm EM} = 137.035\,999\,084 with uncertainty 1.5 \times 10^{-10}.

PT link: PT gives 137.035\,999\,083\,4 with no continuously fitted parameter. Discrepancy \sim 4 ppt (compatible 1σ). Non-trivial: PT could have given a remote number.

Statement: Canonical source for the 43 compared observables.

PT link: All PT vs PDG values are listed in the observables table. 41/43 below 0.5 % discrepancy in the full set; no continuously fitted parameter.

Statement: Electron magnetic moment measured to 280 ppt.

PT link: PT gives a_e = 1.159\,652\,181\,05 \times 10^{-3} (P25b). Discrepancy +0.28 ppb with measurement: compatible 1σ. Independent test of the echo-VP formula.

Statement: Standard Model computation of a_\mu after HVP/lattice corrections.

PT link: The historic 5σ g - 2 anomaly is dissolved: SM White Paper compatible with final Fermilab measurement. PT compatible -0.85\sigma to -0.24\sigma via R_{\rm HVP} = 1.042\,03.

Statement: Neutrinos travel at c within < 10^{-9}.

PT link: PT predicts exact Lorentz (P15), no detectable violation. OPERA’s initial retraction and Borexino’s measurement confirm PT negatively (nothing to correct).

Statement: H_0, n_s, \Omega_b, \Omega_{\rm DM} with error bars.

PT link: PT gives H_0 = 67.43 km/s/Mpc (Planck: 67.4 ± 0.5), n_s = 0.964 (Planck: 0.9649 ± 0.0042), \Omega_{\rm info} = 26.48\,\% (Planck: 26.5 %). All compatible 1σ.

Statement: Power-law forgetting with exponent b \sim 0.5 = s, effective dimensionality d_{\rm eff} \sim 16 = \mu^* + 1.

PT link: Empirical study of human memory persistence. PT did not derive these numbers — but they match the structural constants (s, \mu^*) without adjustment.