Ontological Resolution Theory

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Published July 6, 2026 | Version v9
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Paper R5 v2.2 — Spectral-Budget Isomorphism: The von Mangoldt Function from XOR-Annihilation and Closure Depth

Authors/Creators

Description

Paper R5 v2.2 — Spectral-Budget Isomorphism:
The von Mangoldt Function from XOR-Annihilation and Closure Depth

Authors: Fedor Kapitanov
Publication Date: June 2026
Status: Closed — Derived v2.2

Overview

We close open problem OP-R5 of the ORT Riemann programme by proving the Spectral-Budget Isomorphism:

Tr(Bnbm) = C · Λ(m)

where Λ(m) is the von Mangoldt function and C is a carrier normalisation constant.

Two Physical Mechanisms

XOR-Annihilation

Horizontal composition of distinct primitive defects produces ω=0. Such paths annihilate before reaching Branch 2. Eliminates all m with ≥2 distinct prime factors.

Closure Depth

Vertical recursion of a single primitive defect ηr increases closure depth without charge cancellation. Survives, accumulates budget, produces CE event with weight ln p. Preserves all m = pk.

New in v2.2 — Three Structural Closures

  • T1 — Prime Length Theorem: Every primitive defect with ω=1 has prime length. Proven via F₂ decomposition.
  • T2 — Uniqueness via Phase Coherence: For each prime p, MDE filtration collapses all geometric cycles of length p into exactly one executable class via phase coherence requirement.
  • T3 — Dirichlet Trace Bridge: The connection between the combinatorial trace and the Riemann zeta function is established rigorously via the Dirichlet series generating function, avoiding the mismatch between power series (graph zetas) and Dirichlet series (number theoretic zetas).

Complete Derivation Chain

φ → g(r) → r=½ → Ganc → Bnb → Tr(Bnbm) = C·Λ(m) → ZORT(s) = eK ζ(s)C → ΞCE = ξ → Re(s) = ½

Status Summary

XOR-Annihilation Derived
Closure Depth Exact
Prime Length Theorem Derived
Uniqueness via Phase Coherence Derived
Dirichlet Trace Bridge Derived
Spectral-Budget Isomorphism Derived
ΞCE = ξ Derived
OP-R5 Closed

Free parameters: Zero  ·  New axioms: Zero

Prime numbers are the only topologically protected primitive cycles of the executable carrier.
Everything else annihilates.

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Paper_R5_v_2_2_close.pdf

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