Théorème

OS3 — Uniform reflection-positivity

For every $p \geq 3$, $M_p = T_p^T T_p \succeq 0$ (Gram matrix). Wightman reconstruction applies.

Statement

Let $T_p$ be the mod- $p$ sieve transfer matrix (prime $\geq 3$ ). Then:

\boxed{M_p = T_p^T T_p \succeq 0 \qquad \text{(Gram matrix, hence PSD)}.}

For a square-free composite $m = p_1 \cdot \ldots \cdot p_k$ , CRT factorisation gives:

M_m = M_{p_1} \otimes \cdots \otimes M_{p_k} \succeq 0

(Kronecker product of PSD matrices is PSD).

Consequence — Osterwalder-Schrader reconstruction: for every finite $T_m$ , the third OS axiom (reflection-positivity) is satisfied. The OS reconstruction theorem then yields a Wightman triple $(\mathcal{H}_m, \Omega_m, W_n)$ : a Hilbert space, a vacuum state, and Wightman functions.

Théorème

Why it matters

Before this theorem, OS3 was numerically verified at considerable cost on a few moduli ( $m \in \{3, 6, 30, 210\}$ ). The status was [VAL] (empirical validation).

The uniform OS3 theorem promotes OS3 to [THM]: a short algebraic proof, applicable to every finite $T_m$ without exception. Consequences:

Osterwalder-Schrader reconstruction guaranteed for every finite $T_m$ ;
The PT Hilbert space is well defined by construction;
No need to postulate positivity — it follows from the Gram structure.

Plain reading. In quantum field theory, certain matrices need to be “positive” so that probabilities make sense. OS3 says this positivity is automatic in PT: the relevant matrix has the form $X^T X$ , which is always positive. An automatic guarantee, not a case-by-case verification.

Proof

Step 1 — For $p$ prime

If $T_p$ is a real matrix, then $M_p = T_p^T T_p$ is a Gram matrix (by definition, a matrix times its transpose). For any real matrix $X$ and any vector $v$ :

v^T (X^T X) v = (X v)^T (X v) = \|X v\|^2 \geq 0.

Hence $X^T X \succeq 0$ for any $X$ . In particular $M_p \succeq 0$ .

Step 2 — For square-free composite $m$

If $m = p_1 \cdot p_2 \cdot \ldots \cdot p_k$ with all $p_i$ distinct, the Chinese Remainder Theorem gives:

T_m = T_{p_1} \otimes T_{p_2} \otimes \cdots \otimes T_{p_k}.

Hence:

M_m = T_m^T T_m = (T_{p_1}^T \otimes \cdots \otimes T_{p_k}^T)(T_{p_1} \otimes \cdots \otimes T_{p_k}) = M_{p_1} \otimes \cdots \otimes M_{p_k}.

The Kronecker product of PSD matrices is PSD (direct consequence: tensor-product eigenvalues are products of eigenvalues, and a product of positive reals is positive).

Step 3 — Consequence: OS3 satisfied uniformly

The Osterwalder-Schrader OS3 axiom requires the reflection matrix to be positive. The Gram matrix $M_m$ plays exactly that role. Hence OS3 is satisfied for every finite $T_m$ , no exception.

The OS reconstruction theorem applies, yielding a Wightman triple $(\mathcal{H}_m, \Omega_m, W_n)$ for every $m$ .

Step 4 — Inductive limit (remaining gap)

Passing to the $m \to \infty$ limit to obtain the infinite Hilbert space $\mathcal{H}_\infty = \otimes_p \mathcal{H}_p$ is functional analysis (inductive limit). This passage is handled by lemma G (Hilbert reconstruction). OS3 itself is therefore fully THM; only the inductive limit remains a standard functional-analysis object.

Consequences

34/34 tests PASS on the OS3 suite (monograph, Appendix F).
The QFT-from-sieve reconstruction programme becomes rigorous: each finite $T_m$ corresponds to a Wightman QFT.
Lemmas E, F, G (coupling, metric, Hilbert reconstruction) rely on OS3 for their BRIDGE → THM promotion.