Théorème

BA5 — Pontryagin product

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

Statement

Under bridge axioms BA0–BA4, the multiplicative functional on the CRT-decomposed sieve algebra at the fixed point $\mu^* = 15$ has product form:

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

evaluated on the vertex branch ( $q_+ = 13/15$ ). The product structure is forced by Pontryagin duality on the additive direct sum

\mathbb{Z}/3\mathbb{Z} \oplus \mathbb{Z}/5\mathbb{Z} \oplus \mathbb{Z}/7\mathbb{Z}.

Théorème

Plain reading. Why is $\alpha_{\rm EM}$ a product of three numbers, not a sum or an integral? Because the three channels $p \in \{3, 5, 7\}$ are independent (orthogonal in the torus $\mathbb{T}^3$ ). When independent channels combine, their amplitudes multiply — that is Pontryagin duality applied to the sieve. Not an aesthetic choice, a mathematical necessity.

Why it matters

BA5 is the essential bridge between sieve arithmetic and the value of the fine-structure constant. It was historically stated as an axiom (hence the “bridge axiom” name), but its structural part is promoted to theorem in the monograph: the product form follows from BA0–BA4 + Pontryagin duality + T5 + T6.

The distinction matters: the arithmetic identity $\alpha_{\rm sieve}=\prod \sin^2\theta_p$ is THM-level; the physical identification with the measured electromagnetic coupling and its radiative dressing is a later reconstruction/physical-derivation step.

Without BA5, the computation $\alpha_{\rm EM} = 1/136.28$ (then $1/137.036$ after dressing) would have no structural justification. With BA5, it has one that leaves no choice.

Proof — outline

CRT (Chinese Remainder Theorem): $\mathbb{Z}/(3 \cdot 5 \cdot 7)\mathbb{Z} \cong \mathbb{Z}/3 \oplus \mathbb{Z}/5 \oplus \mathbb{Z}/7$ .
Additive characters: on each factor, the Pontryagin dual group is cyclic.
Multiplicative functional: on the direct sum, any multiplicative functional factors into a product over factors.
Identification: the relevant per-factor functional is $\sin^2\theta_p$ (by T6).
Evaluation at the fixed point: at $\mu^* = 15$ , $q_+$ branch, the product gives $1/136.28$ .

Detailed proof

Step 1 — CRT decomposition

The Chinese Remainder Theorem gives, for distinct primes:

\mathbb{Z}/(p_1 p_2 p_3)\mathbb{Z} \cong \mathbb{Z}/p_1\mathbb{Z} \oplus \mathbb{Z}/p_2\mathbb{Z} \oplus \mathbb{Z}/p_3\mathbb{Z}.

For $\{p_1, p_2, p_3\} = \{3, 5, 7\}$ , the arithmetic torus $\mathbb{T}^3 = \mathbb{Z}/105\mathbb{Z}$ factors into three independent components.

Step 2 — Additive characters and Pontryagin duality

For each factor $\mathbb{Z}/p\mathbb{Z}$ , the Pontryagin dual group is the set of characters $\chi_k : r \mapsto e^{2\pi i k r/p}$ for $k = 0, 1, \ldots, p-1$ . This dual is itself cyclic of order $p$ (self-duality of finite cyclic groups).

On the direct sum $\bigoplus_p \mathbb{Z}/p\mathbb{Z}$ , the Pontryagin dual is the direct sum of duals. Any multiplicative functional $f : \bigoplus_p \mathbb{Z}/p\mathbb{Z} \to \mathbb{C}$ therefore necessarily factors:

f(r_3, r_5, r_7) = f_3(r_3) \cdot f_5(r_5) \cdot f_7(r_7).

Step 3 — Functional identification

The relevant per-factor functional, by BA3 (holonomy axiom) and T6, is the squared transition amplitude:

f_p \equiv \sin^2\theta_p.

Justification: $\sin^2\theta_p$ is the orthogonality measure of the fundamental character $\chi_1$ relative to the stationary measure (cf. T6, route 2: $1 - |\widehat{T}_p(\chi_1)|^2 = \sin^2\theta_p$ ).

Step 4 — Product over active primes

At the fixed point $\mu^* = 15$ , by T5, active primes are $\{3, 5, 7\}$ . The global multiplicative functional therefore evaluates to:

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

Step 5 — Numerical value

At $q_+ = 13/15$ :

$\sin^2\theta_3 = 0.21916$
$\sin^2\theta_5 = 0.19397$
$\sin^2\theta_7 = 0.17261$

Product: $\alpha_{\rm sieve} = 1/136.28$ .

This value is then dressed by the binary channel $F(2)$ (cf. observable 1/α_EM) to give the observed value $1/137.036$ ; this part is a physical derivation, not the pure BA5 identity.

Why not a sum

Pontryagin duality forbids the additive form: on an abelian group, multiplicative functionals are necessarily multiplicative in the strict mathematical sense (homomorphisms to $\mathbb{C}^*$ ). The product form is unique, by duality.

For the complete derivation, see chapter 9 of the monograph.