Paper R5 v 4.1 The Riemann Sector of Ontological Resolution Theory

This paper presents the unified Riemann programme of Ontological Resolution Theory (ORT v2.1). It combines prior results into a single document with a complete chain from the discrete FCC carrier to the critical line of the Riemann zeta function.

Three-layer structure.

• Ontological layer. The (G, R, U, D) foundation and the tag-decoder framework. The Riemann zeta function is identified as the mathematical structure of the decoder D, not postulated but constructed from Carry Exit dynamics.
• Algebraic layer. The Spectral-Budget Isomorphism Tr(B_nb^m) = Λ(m), derived from XOR-Annihilation and Closure Depth via the budget-weighted trace. Independent of any external number theory.
• Spectral layer. The connection between the critical line Re(s) = 1/2 and the executable-positivity property of the FCC carrier. The functional equation (symmetry axis) is unconditional; the placement of zeros on that axis is conditional, pending an executable-positivity criterion.

Key structural advances over v4.0.

1. The state-update map Δ(γ), silently used in the proof of XOR-Annihilation, is now formally defined and its equivalence with the topological charge ω is proved, closing the last unformalised inference in the algebraic layer.
2. The previous appeal to the Ramanujan property of FCC as the source of the critical line is withdrawn: it was logically disconnected from the Dirichlet Trace Bridge (established purely combinatorially) and could not, in principle, select a single symmetry axis, only bound a region.
3. The functional equation ξ_ORT(s) = ξ_ORT(1−s) is shown to be an unconditional consequence of the Dirichlet Trace Bridge, giving Re(s) = 1/2 as the architecturally forced encoding/decoding symmetry axis between the graph's address space and the observer's address space — without any conjecture.
4. The genuinely open step (placement of zeros on that axis, not merely symmetry about it) is correctly relocated to an executable-positivity conjecture, the ORT-native analogue of the classical Weil/Li positivity criteria for the Riemann Hypothesis.

Complete chain.

D = 5 → FCC(k = 12) → K_cell = 133 → CE events → Λ(m) → ζ(s) ⟶^uncond. symmetry axis ½ ⟶^Positivity* Re(s) = ½

Unconditional results.

• Spectral-Budget Isomorphism: Tr(B_nb^m) = Λ(m) for all m ≥ 1.
• ORT zeta function: ζ_ORT(s) = e^Kζ(s).
• Functional equation: ξ_ORT(s) = ξ_ORT(1−s).
• Symmetry axis Re(s) = 1/2 as the fixed line of the backend/frontend encoding involution.
• Decoder D constructively identified with analytic continuation of ζ(s).

Conditional result. If the executable-positivity conjecture (ORT-native Weil/Li criterion) holds, then all non-trivial zeros of ζ(s) lie on Re(s) = 1/2. ORT does not claim to have reduced the difficulty of the Riemann Hypothesis; it claims to have placed it in its correct location within the architecture.

Honest separation of layers.

• If positivity is verified: the Riemann Hypothesis becomes a theorem within ORT.
• If positivity fails: only zero placement is falsified. The Spectral-Budget Isomorphism, the decoder construction, and the unconditional functional equation / symmetry axis all survive.

Free parameters: zero.
New axioms: zero.
External number theory imported: the classical (unconditional) functional equation of ζ(s), and the classical (equivalent-to-RH) Weil and Li positivity criteria, used only to correctly state — not to solve — the remaining open step.

Two physical mechanisms (no external number theory).

• 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.
• 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 CE event with information weight ln p. This preserves all m = p^k.

Three structural theorems.

• T1 (Prime Length): Every primitive defect has prime length. Proved via explicit block-decomposition lemma on FCC carriers, using the extended definition of ω on open walk segments.
• T2 (Uniqueness via Phase Coherence): For every prime p, the MDE filtration collapses all geometric cycles of length p into exactly one executable class.
• T3 (Dirichlet Trace Bridge): ζ_ORT(s) = e^Kζ(s), established rigorously via the Dirichlet series generating function with carrier normalisation constant C = 1.