Théorème

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.

Statement

Let $(\mathcal{A}, \Omega)$ be the $C^*$ -algebra and vacuum state obtained by Osterwalder–Schrader reconstruction from the inductive limit of the sieve transfer system $\{(H_{m_K}, T_{m_K}, \Omega_{m_K})\}_{K \geq 1}$ .

Then the coupling of the reconstructed quantum field theory is

\boxed{g^2 = \prod_{p \in \{3, 5, 7\}} \sin^2\theta_p(q_+),}

and this is the unique value compatible with (i) the OS axioms and (ii) spectral rigidity of the transfer system.

Théorème

Plain reading. Lemma E is the bridge that says: “the number $\alpha_{\rm EM}$ is not an arbitrary identification between sieve and physics, it is an intrinsic property of the QFT reconstructed from the sieve.” No interpretive gesture — the value emerges necessarily.

Why it matters

Lemma E eliminates the ontological step of PT. Without E, one would say: “the sieve gives these numbers, and we identify them with the electromagnetic coupling.” That identification would be an external postulate.

With E, one says: “the reconstructed QFT has this coupling.” Not an identification, an intrinsic computation.

This is what makes BA5 (Pontryagin) a theorem rather than a bridge axiom.

Proof — outline

OS reconstruction: from $\{T_m\}$ , build the inductive limit $\mathcal{A}$ and vacuum $\Omega$ .
Spectral rigidity: T1 fixes the zero pattern, T5 fixes the active primes, T6 fixes the amplitudes.
Ruelle partition function: $Z_N = \sum_i \lambda_i^N$ is uniquely determined.
Spectral coupling: the coupling is the invariant $\prod \sin^2\theta_p$ forced by Pontryagin (BA5) and T6.

Detailed proof

Step E.1 — OS reconstruction

The inductive-limit theorem (cf. ch. 9 monograph, thm:inductive_limit) constructs algebra $\mathcal{A}$ and vacuum $\Omega$ from the system $\{(H_{m_K}, T_{m_K}, \Omega_{m_K})\}_{K \geq 1}$ by Osterwalder–Schrader reconstruction. The OS axioms (reflection positivity, Euclidean invariance, correlation decay) are uniformly verified.

Step E.2 — Spectrum rigidity

The spectrum $\sigma(T_m)$ is rigid in the following sense: it is entirely determined by three independent structural theorems:

T1 fixes the zero pattern of the matrix (mod-3 forbidden transitions).
T5 fixes the active primes $\{3, 5, 7\}$ by fixed-point self-consistency.
T6 fixes the sin² values by algebraic identity on each factor.

No residual degree of freedom. The Ruelle partition function:

Z_N = \mathrm{Tr}(T_m^N) = \sum_i \lambda_i^N

is entirely determined by this rigid spectrum.

Step E.3 — Coupling as a spectral invariant

The coupling $g$ of the reconstructed QFT is defined as the long-distance limit of 4-point correlation functions (Feynman vertex in asymptotic). By OS theorems, this coupling is a spectral invariant — it depends only on the spectrum $\sigma(T)$ , not on a particular representation.

Step E.4 — Identification with the product

On the $q_+ = 1 - 2/\mu^*$ branch, T6 gives $\sin^2\theta_p$ for each active prime. By BA5 (Pontryagin product), the relevant spectral invariant is:

g^2 = \prod_{p \in \{3,5,7\}} \sin^2\theta_p(q_+) = \alpha_{\rm bare}.

Uniqueness

Uniqueness comes from four combined external theorems:

G1 (uniqueness of $D_{\rm KL}$ , autonomous CRT Shore–Johnson) — ch. 5.
G3 (Fisher metric uniqueness, Čencov) — ch. 5.
T5 (uniqueness of $\mu^*$ ) — ch. 8.
T6 (holonomy identity) — ch. 6.

These four uniqueness results determine the active primes $\{3, 5, 7\}$ , the angles $\sin^2\theta_p$ , and the branch $q_+$ . By Pontryagin, the coupling is their product. No other value is compatible.

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