Paper R5 v3.0

Author: Fedor Kapitanov
Date: June 2026
Status: Closing OP-R5 of the ORT Riemann programme

Result

We close open problem OP-R5 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.

Method

The derivation uses no external number theory. It uses only two mechanisms already present in Canon v34.0:

1. XOR-Annihilation. Horizontal composition of distinct primitive defects produces a path with F₂-charge ω = 0. Such paths annihilate before reaching the Branch 2 threshold. This eliminates all m with more than one distinct prime factor.
2. Closure Depth. Vertical recursion of a single primitive defect (η^r) increases closure depth without charge cancellation. The path survives, accumulates budget constructively, and produces a Carry Exit event with information weight ln p. This preserves all m = p^k.

Three Structural Closures

1. T1 (Prime Length Theorem): Every primitive defect with ω = 1 has prime length. Proven via F₂ decomposition.
2. T2 (Uniqueness via Phase Coherence): For each prime p, MDE filtration collapses all geometric cycles of length p into exactly one executable class.
3. T3 (Dirichlet Trace Bridge): The combinatorial trace is connected to the Riemann zeta function via the logarithmic derivative of a Dirichlet series, avoiding the mismatch between power series (graph zetas) and Dirichlet series (number-theoretic zetas).

Status Table

Version Notes

Changes from v2.2:

• T1 strengthened: explicit lemnma on block-decomposition of closed paths of composite length on FCC carriers.
• T2 strengthened: phase coherence explicitly defined as consistency of the P-sector winding number along the cycle.
• T3 strengthened: explicit computation of the carrier normalisation constant C = 1, with discussion of integer-C cases.
• Bridge section: clarification that ζ_ORT is not a graph zeta in the Ihara/Selberg sense, but a Dirichlet-series construction introduced specifically to connect the trace to ζ(s).

Parameters

• Free parameters: zero
• New axioms: zero
• External number theory: none imported

Keywords

ORT, von Mangoldt function, XOR-annihilation, closure depth, Dirichlet series, non-backtracking operator, ancestral graph, Carry Exit, prime numbers, Riemann zeta, spectral isomorphism, phase coherence, FCC carrier, P-sector winding.