Zeta Prime-Product Common Boundary

Abstract

The zeta partition, prime activation product, local entropy, and parameter sensitivity all cross their convergence boundary at one.

Theorem 1.1 (Five concrete thresholds meet at s equals one).

$$\begin{array}{r}\forall s\in \mathbb{R},0<s\Rightarrow \\ \forall p\in \mathbb{P},q_{p,s}=p^{-s},{\gamma }_{p,s}=\mathrm{geometricMeasure}\left(1-q_{p,s}\right),\\ {\Gamma }_s=\mathrm{infinitePi}\left(p\mapsto {\gamma }_{p,s}\right),H_p(s)=-\log \left(1-q_{p,s}\right)+s\log \left(p\right)\frac{q_{p,s}}{1-q_{p,s}},\\ J_p(s)=\frac{\log {\left(p\right)}^2{q}_{p,s}}{(1-q_{p,s}{)}^2},\\ (\mathrm{partitionFunction}\left(s\right)\ne \infty \iff 1<s)\wedge \\ (\mathrm{Summable}\left(p\mapsto q_{p,s}\right)\iff 1<s)\wedge \\ (\mathrm{Pr}\left({\Gamma }_s,FiniteSupportProfiles\right)=1\iff 1<s)\wedge \\ (\mathrm{Summable}\left(p\mapsto H_p(s)\right)\iff 1<s)\wedge \\ (\mathrm{Summable}\left(p\mapsto J_p(s)\right)\iff 1<s).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Analytic/Boundary/ZetaPrimeProductCommonBoundary.zeta_prime_product_common_boundary (✓ std3). ∎

Source. Repository-derived.

Commentary.

For a positive real parameter s, the zeta partition is the sum of the logarithmic Gibbs weights, while q at prime p is the source activation probability p to the power minus s. The exponent law is the canonical independent product of the corresponding zero-start geometric coordinate laws.

The integer and prime p-series criteria put partition finiteness and activation summability exactly above one. Below and at one, the accepted product theorem gives finite-support profiles measure zero; above one, the first Borel-Cantelli lemma gives them measure one.

The displayed H term is the source geometric-coordinate entropy and the displayed J term is its Fisher sensitivity summand. Lower comparison with prime activation forces divergence through the boundary, while logarithm-weighted p-series bounds give summability above it.

References

• Truth anchor: D5/S3/Analytic/Boundary/ZetaPrimeProductCommonBoundary.zeta_prime_product_common_boundary
• Dependency: D5/S3/Analytic/PrimeProducts/FiniteMarginalGlobalSupportContrast