Théorème

T5 — Informational attractor μ* = 15

After p = 2 crystallises, the reduced sieve sector admits a unique physical attractor: $\mu^* = 3+5+7 = 15$.

Statement

The sieve self-consistency equation

\mu^* = \sum_{p\,:\,\gamma_p(\mu^*) > s} p, \qquad s = 1/2,

admits a unique stable solution in the reduced information sector, namely:

\boxed{\mu^* = 3 + 5 + 7 = 15, \qquad \text{active set} = \{3, 5, 7\}.}

The proof combines an exact rational exhaustive scan over $\mu \in [3, 100]$ with an analytic monotonicity bound for $p \geq 7$ , excluding any additional fixed point in the reduced sector $\{3, 5, 7\}$ .

Théorème

Plain reading. We have a self-referential equation: ” $\mu$ must equal the sum of primes active at $\mu$ ”. In the physical sector where $p=2$ has crystallised into binary infrastructure, T5 says there is only one stable closure. It is $\mu = 15$ with active primes $\{3, 5, 7\}$ . No other choice of primes closes the equation in that reduced sector. This is why it is better to call it an informational attractor: $\mu^* = 15$ is not merely a static equality, but the stable closure toward which the admissible cascade slides.

Why it matters

T5 is the summit of the chain of theorems. All preceding ones (T0–T6, GFT, L0) are needed to formulate T5, and T5 closes the sieve: at $\mu^* = 15$ , the parity split seeded by $p = 2$ is frozen as the $q_+/q_-$ bifurcation, and the 43 physical observables are then derived.

Three regimes exist in fact:

$\mu = 10$ , set $\{2, 3, 5\}$ — binary/spin sector,
$\mu = 17$ , set $\{2, 3, 5, 7\}$ — transitional sector,
$\mu^* = 15$ , set $\{3, 5, 7\}$ — information sector, that of our physics.

It is $\mu^* = 15$ that drives PT physics, after $p = 2$ has “crystallised” into the binary infrastructure.

Proof — outline

Define $\gamma_p(\mu) = -d\ln(\sin^2\theta_p)/d\ln\mu$ (anomalous dimension).
Compute $\gamma_p$ via exact rational arithmetic at $\mu = 15$ : $\gamma_3, \gamma_5, \gamma_7 > 1/2$ , $\gamma_{11}, \gamma_{13} < 1/2$ .
Verify $3 + 5 + 7 = 15$ — consistency reached.
Analytic bound: $\gamma_p$ strictly decreasing in $p$ for $p \geq 7$ , so no prime $\geq 11$ is active at any reasonable $\mu$ .
Exhaustive scan of finite subsets of $\{3, 5, 7, 11, 13, ...\}$ to confirm uniqueness in the reduced sector.

Detailed proof

Step 1 — Exact rational computation of $\gamma_p$ at $\mu = 15$

At $q = 13/15$ , every quantity is an exact rational number (fractions.Fraction, zero floating-point error). Values are:

$p$	$\gamma_p$ (exact fraction)	$\gamma_p$ (decimal)	$> 1/2$ ?
3	$4\,536\,129 / 5\,616\,704$	0.80761…	✓
5	$486\,792\,684\,365 / 699\,097\,512\,194$	0.69632…	✓
7	$2\,827\,519\,972\,576\,117 / 4\,748\,396\,022\,746\,468$	0.59547…	✓
11	(exact rational, omitted)	0.42573…	✗
13	(exact rational, omitted)	0.35624…	✗

Differences $\gamma_3 - \gamma_5$ , $\gamma_5 - \gamma_7$ , $\gamma_7 - \gamma_{11}$ are strictly positive (verified by exact fraction subtraction).

Step 2 — Numerical consistency

The set $\{3, 5, 7\}$ is active at $\mu = 15$ . Its sum:

3 + 5 + 7 = 15 = \mu.

The self-consistency equation is satisfied. This is a fixed point.

Step 3 — Analytic monotonicity for $p \geq 7$

$\gamma_p$ factors as $\gamma_p = F(p) \cdot G(p)$ with:

$F(p) = 4 q^{p-1}/\mu$ — exponentially decreasing,
$G(p) = (1 - \delta_p)/(\delta_p (2 - \delta_p))$ — sign analysis.

The gap fraction $\delta_p = (1 - q^p)/p$ is strictly decreasing in $p$ (real-variable analysis $x \mapsto (1 - e^{-Lx})/x$ with $L = \ln(15/13) > 0$ ).

Consequence: for every $p \geq 7$ , $\gamma_p < \gamma_7 \approx 0.595$ . Adding a prime $p \geq 11$ to the active set is therefore impossible: $\gamma_p < 1/2$ would violate activity.

Step 4 — Exhaustive scan of finite candidates

Enumerate all finite subsets $S \subset \{3, 5, 7, 11, 13, 17, 19, 23, ...\}$ , compute $\mu_S = \sum_{p \in S} p$ , and test the consistency $\gamma_p(\mu_S) > 1/2$ for $p \in S$ .

Exhaustive scan results (cf. proof_T5_fixed_point.py, 33 PASS):

$S = \{3, 5, 7\}$ : $\mu = 15$ , consistent ✓
$S = \{3, 5\}$ : $\mu = 8$ , but $\gamma_3(8), \gamma_5(8), \gamma_7(8) > 1/2$ imply 7 should be included — contradiction.
$S = \{3, 5, 7, 11\}$ : $\mu = 26$ , but $\gamma_{11}(26) < 1/2$ — inconsistent.
$S = \{2, 3, 5\}$ : $\mu = 10$ , binary/spin sector (distinct fixed point, $p = 2$ included).
$S = \{2, 3, 5, 7\}$ : $\mu = 17$ , transitional sector.

No other subset closes the equation in the information sector (without $p = 2$ ). $\mu^* = 15$ is unique for the info sector.

Step 5 — Linear stability and attractive basin

Stability of the fixed point is checked by computing the Jacobian:

J_{ij} = \frac{\partial \gamma_{p_j}}{\partial \mu_i} \bigg|_{\mu = \mu^*}.

The spectral radius $|\lambda_\max(J)| < 1$ confirms $\mu^* = 15$ is an attractor (cf. ch. 8, Stability Theorem). In the reduced reading, the relevant basin is $\mu_0 \in [12,17]$ : $17$ slides to $15$ after crystallisation of $p=2$ , while $18$ triggers the divergent cascade.

J1–J6: six independent justifications

The monograph lists six independent justifications of $\mu^* = 15$ :

Exhaustive scan (step 4).
Linear stability (step 5).
$N_c = 3$ : unique solution of $(N_c + 1)! / (N_c + 3) = 2^{N_s - 1}$ with $N_s = 3$ .
Prime 7 = integer oracle of $e^2$ : drift $A_p = \ln p / \sqrt{p}$ is maximal at $p = 7$ .
Echo product identity: $\prod_{p \in \{3,5,7\}} q_-^p = 1/e$ exactly.
Cosmogony: the sequence $\mu = 10 \to 17 \to 15$ gives the evolutionary scenario.

For the complete derivation, see chapter 8 of the monograph.