Théorème

Lemma F — Metric reconstruction

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

Statement

The spacetime metric of the quantum field theory reconstructed (by Osterwalder–Schrader, cf. Lemma E) is the Fisher information metric $g^F_{ab}$ of the sieve transfer system, evaluated at the fixed point $\mu^* = 15$ .

\boxed{g_{\rm spacetime} = g^F_{\rm sieve}\big|_{\mu^* = 15}.}

Théorème

Plain reading. Spacetime itself is not postulated in PT. It emerges from the statistical geometry of the sieve. Specifically: the Fisher metric (which measures the “distance” between probability distributions) coincides, at the fixed point, with the Riemannian spacetime metric of the reconstructed theory.

Why it matters

Lemma F is the bridge from arithmetic to gravity. The cascade T0 → T6 + T5 + Lemma E gives the coupling $\alpha_{\rm EM}$ (electromagnetic interaction). Lemma F gives the structure of space itself.

This is what justifies the derived identity $G_{\rm SCU} / \alpha_{\rm EM} = 2\pi$ (observable #42): gravity is not a separate fundamental force, it is the geometric envelope of the same cascade.

Proof — outline

Step F.1: Fisher metric = Hessian of the log-partition function.
Step F.2: the partition function is a spectral invariant (by Lemma E.2, spectrum rigidity).
Step F.3: the metric decomposes additively over CRT factors.
Step F.4: Čencov (G3) gives uniqueness of the metric on the probability simplex.
Conclusion: the metric identity spacetime ↔ Fisher is forced.

Detailed proof

Step F.1 — Fisher = Hessian of log-partition

For an exponential family $p(x; \theta) = e^{\theta \cdot T(x) - A(\theta)}$ , the Fisher metric is:

g^F_{ab}(\theta) = \frac{\partial^2 A(\theta)}{\partial \theta^a \, \partial \theta^b},

where $A(\theta)$ is the log-partition. Standard result of information geometry.

For the sieve transfer system, $A = \ln Z$ with $Z = \mathrm{Tr}(T^N)$ the Ruelle partition function, and $\theta = \mu$ the scale parameter.

Step F.2 — Log-partition is a spectral invariant

By step E.2 of Lemma E, the spectrum of $\{T_m\}$ is rigid:

T1 fixes the zero pattern,
T5 fixes the active primes,
T6 fixes the sin² themselves.

So $Z_N = \sum_i \lambda_i^N$ and its Hessian are entirely determined with zero free parameters.

Step F.3 — CRT separability

The Fisher metric decomposes additively over CRT factors:

g_{00} = \sum_p g_{00}^{(p)},

because $T_m = \bigotimes_p T_p$ implies $\ln Z = \sum_p \ln Z_p$ , and the Hessian of a sum is the sum of Hessians.

This allows computing the metric factor by factor: three independent contributions for the three active primes $\{3, 5, 7\}$ .

Step F.4 — Čencov uniqueness (G3)

Čencov’s theorem ensures that $F_{\mu\mu}$ is the unique monotone Riemannian metric on the probability simplex (up to a constant). “Monotone” means it decreases under Markov morphisms — that is the projection-invariance axiom.

Physical invariance of spacetime under information-non-creating transformations imposes the same condition. So the spacetime metric must be the Čencov–Fisher metric.

Step F.5 — Bianchi I decomposition

The reconstructed metric has Bianchi I structure (anisotropic, three independent axes), direct consequence of the CRT decomposition into three factors:

ds^2 = -dt^2 + a_3(t)^2 dx_3^2 + a_5(t)^2 dx_5^2 + a_7(t)^2 dx_7^2,

with each $a_p(t)$ driven by $\gamma_p$ . PT’s geometric signature (cf. essay Why three dimensions?).

For the complete derivation, see chapter 9 of the monograph, section Metric Reconstruction (Lemma F).