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PGS to RH

Exact divisor-count structure is the source; RH is the downstream pole-placement sentence after that structure is compressed into zeta language.

The required reading order is:

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

What This Folder Does

docs/rh is the affirmative narrative spine for the PGS-to-RH documentation bundle. It starts from arithmetic objects the reader can inspect directly: divisor counts, consecutive prime-gap interiors, and the selected integer inside each nonempty gap. It then moves through the local theorem authority, the exact DNI-to-zeta compression, the source-to-spectral placement target, and the pole-placement language that gives the RH-facing sentence.

This folder is the bundle index and source-order guide. The FAQ remains the objection-handling surface. Note (post-archival): The detailed research/12-rh-bridge workbench material (classical completion strategy, loop infrastructure, etc.) has been moved outside the repository to /Users/velocityworks/prime-gap-structure-archives/2026-05-classical-rh-bridge-completion-route/ due to classical drift and prompt injection concerns. See the pointer at research/12-rh-bridge/README.md and the ARCHIVAL_HANDOFF.md there.

The public narrative spine in docs/rh/ remains, but deep technical workbench content is no longer on the live surface.

For Zero-Excess DNI Phase 1, zero-excess is an exact coordinate reformulation. The zero-excess floor is integer-side; the critical line is zeta-side.

Table Of Contents

Page Role Status
Source Order Establishes the direction from integer objects to RH language. explanatory consequence
DNI-to-Zeta Compression Shows the coefficient bridge from divisor counts to $R(s)=-\zeta'(s)/\zeta(s)$. exact zeta compression
Pole Placement Records how zeros of $\zeta$ become poles of the continued DNI ratio. exact zeta compression
Off-Critical Pole Exclusion Records the residual-closure route and the current obstruction to using it as a proof of pole placement. unresolved proof target
Critical Line And Zero Geometry Names the critical strip and critical line as downstream coordinate language. explanatory consequence
Explicit Formula Bridge Connects $R(s)$ to $\Lambda(n)$, $\psi(x)$, and zero terms. downstream translation bridge
Status Ledger Separates proved theorems, exact compression, source-side closure, and downstream translation. reviewer control
Reviewer Map Gives the checking order for the bundle. reviewer control

Primary Sources

Source Role Status
Root Proof Authority Proves the local next-prime rule and the prime-gap interior maximizer theorem. proved theorem
DNI-to-Zeta Bridge Records the full bridge workbench for the DNI ratio. exact zeta compression
FAQ Handles recurring objections and category errors. objection handling
The Riemann Hypothesis Is Obsolete States the public-facing consequence of starting from the arithmetic source. explanatory consequence

Status Spine

Source Layer Object Status Label
Divisor counts Each integer carries tau(n), with primes exactly at tau(n)=2. arithmetic source
Zero-Excess DNI The zero-excess floor is the integer-side coordinate $E(n)=0$ under the $n>1$ prime guard. exact coordinate reformulation
PGS local theorems PROOF.md controls the next-prime theorem and the interior maximizer theorem. proved theorem
DNI-to-zeta compression The native DNI series gives (e^2/2)K(s)/D(s) = -zeta'(s)/zeta(s); in zero-excess notation the bridge load is $H(n)=\log n+E(n)$, not $E(n)$ alone. exact zeta compression
Source-to-spectral placement The residual test closes failed identities and coefficient-bookkeeping errors, but the no-extra-carrier argument does not yet rule out zeros carried by the same global zeta object. unresolved proof target
Pole placement/RH sentence Zeros of zeta become poles of the continued DNI ratio. Placing every nontrivial pole on the critical line still requires a source-to-spectral placement theorem. unresolved proof target
Explicit-formula bridge The R(s) -> Lambda(n) -> psi(x) path restates the result in classical zero-term and error-term language. downstream translation bridge

Bundle Boundary

The bundle begins with ordered arithmetic and ends with RH vocabulary. Zeta zeros enter after the divisor-count source and exact compression have been stated. The FAQ carries objection handling; this folder carries the main claim. The spine is constructive:

  1. Count divisors on the integer line.
  2. Read the finite interior between consecutive primes.
  3. Apply the local PGS theorems controlled by PROOF.md.
  4. Compress the same divisor-count source into the DNI zeta ratio.
  5. Apply the source-side residual test and its current obstruction for off-critical poles.
  6. Translate the continued ratio into pole-placement/RH language.
  7. Use the explicit-formula bridge only as downstream translation into Lambda, psi, zero-term, and error-term language.

PROOF.md controls local PGS theorem status. It does not itself prove RH. The remaining bridge is source-to-spectral placement: chamber geometry and the $Z=1$ / $E=0$ return law must still be shown to force the nontrivial poles of the continued DNI ratio onto the critical line.