explicit-formula-bridge.md

Explicit Formula Bridge

The explicit formula belongs downstream of the source order:

divisor counts -> zero-excess returns -> local theorems
-> DNI-to-zeta compression -> source-to-spectral placement target
-> pole placement/RH sentence

This page records the downstream analytic bridge surface. It translates the continued DNI ratio into Lambda, psi, zero-term, and error-term language. It also names one equivalent form of the remaining source-to-spectral placement target.

Source To Analytic Chain

The established compression starts with divisor counts:

$$ D(s)=\sum_{n\ge1}\frac{\tau(n)}{n^s}=\zeta(s)^2. $$

The zero-excess coordinate on the integer side is

$$ E(n)=\left(\frac{\tau(n)}{2}-1\right)\log n. $$

For n > 1, prime returns are exactly $E(n)=0$. In a nonempty gap interior, the local theorem function satisfies $F(n)=-E(n)$, so the selected integer is the leftmost argmin of $E(n)$. That is the source-side coordinate. It is not the explicit-formula object.

The bridge load remains

$$ H(n)=\log n+E(n)=\frac{\tau(n)\log n}{2}. $$

Equivalently, H(n)=log n+E(n)=tau(n)log(n)/2.

With the DNI normalization series

$$ K(s)=-\frac{1}{e^2}D'(s), $$

the normalized ratio is

$$ R(s)=\frac{e^2}{2}\frac{K(s)}{D(s)} =-\frac{\zeta'(s)}{\zeta(s)} =\sum_{n\ge1}\frac{\Lambda(n)}{n^s}. $$

That gives the downstream coefficient chain:

R(s) -> Lambda(n) -> psi(x) -> zero terms

The von Mangoldt coefficients define the Chebyshev counting function

$$ \psi(x)=\sum_{n\le x}\Lambda(n). $$

Classical explicit-formula machinery then rewrites psi(x) in terms of the main term, the pole at s=1, trivial-zero terms, and nontrivial-zero terms. In schematic form,

$$ \psi(x)=x-\sum_{\rho}\frac{x^\rho}{\rho}+\text{elementary terms}, $$

with the sum taken over nontrivial zeros rho of zeta(s), subject to the usual explicit-formula conventions.

Chamber-Local Counting Error

For a prime gap with consecutive endpoints p < q, the chamber contains one prime endpoint on the right and no primes in the open interval (p,q).

The local logarithmic-integral expectation across the chamber is

$$ \int_p^q \frac{dt}{\log t}. $$

The chamber-local counting error is therefore

$$ \Delta_{\mathrm{Li}}(p,q) =1-\int_p^q \frac{dt}{\log t}. $$

This quantity is a local endpoint-error observable. It is useful because it connects a PGS chamber object to the analytic prime-counting language, while preserving the direction of explanation:

fixed chamber endpoints -> local counting error -> explicit-formula shadow

Bridge Claim

The bridge claim is:

• PGS local theorems determine exact source-side chamber structure from divisor counts and zero-excess returns.
• DNI-to-zeta compression recovers R(s) = -zeta'(s)/zeta(s).
• The coefficients of R(s) are Lambda(n).
• Summing Lambda(n) gives psi(x).
• The explicit formula translates psi(x) into main, trivial, and nontrivial zero terms.
• Chamber-local errors such as Delta_Li(p,q)=1-int_p^q dt/log(t) are downstream analytic shadows of fixed prime-gap endpoint structure.

This is a downstream analytic bridge. The local PGS theorems and exact DNI compression with H(n)=log n+E(n)=tau(n)log(n)/2 identify the source and the continued ratio. They do not yet supply the RH-strength placement constraint.

Translation Status

The explicit formula restates the continued ratio in classical counting language:

R(s) -> Lambda(n) -> psi(x) -> zero terms -> error-term language

The correct status split is:

• exact zeta compression for the identity defining R(s);
• downstream analytic bridge for R(s) -> Lambda(n) -> psi(x) -> zero terms;
• unresolved source-to-spectral placement for the off-critical-pole exclusion recorded in off-critical-pole-exclusion.md;
• translation/proof-detail bridge for reviewers who want the RH sentence expressed through psi, Lambda, zero-term, or error-term estimates.

PROOF.md controls only the local PGS theorems. The pole-placement sentence still needs a source-to-spectral theorem, or an equivalent RH-strength explicit-formula bound derived from the PGS source.