The Spectral Zookeeper: A Conditional Reduction toward the Riemann Hypothesis via Spectral Microcluster Closure in the Connes-Consani-Moscovici Framework

5 Changelog (31 May 2026): Additive cross-route sharpening of the spectral target; no theorem-strengthening changes, conditional microcluster-reduction status unchanged. New subsection §7. 8 "A 2026 cross-route sharpening: the determinant channel" (EN and DE), at the end of the Three-Lemma Endgame. Archimedean density independently confirmed: a non-circular discretization of the genuine Connes–Moscovici prolate operator (arXiv: 2112. 05500) reproduces Riemann's zero-counting function N (E) to ~3% in the UV — the archimedean half — corroborating the CCM density claim underlying the microcluster construction. The spectral target is the determinant channel, not bare eigenvalue convergence: the object that must converge is detreg (A−z) → C·Ξ (channel −D'/D, residues = multiplicities) ; truncation faithfulness and the MS2 transfer must control the spectral measure (poles and residues). The relative-determinant bypass is refuted: removing the Euler pole comb via a relative determinant against a prolate background fails (the prolate background is itself Herglotz/comb-free) ; the determinant channel as a target remains valid, only the relative determinant as a wall bypass is refuted. Status: no MS2 or RH claim; conditional microcluster reduction unchanged. The missing datum remains the analytic continuation across Re s=1. 1. 4 Changelog (26 May 2026): Guardrail update after EvenDom follow-up audits; no theorem-strengthening changes. Claim level: the record remains a conditional microcluster-reduction programme, not an unconditional proof of RH or MS2. EvenDom strip update: the λ=20 strip run gives κ (20) =1. 527×10-3; the conservative saturation range is κ∞∈1. 6, 1. 8×10-3. The uniform log-curvature bound remains open. GF1 transfer guardrail: λ=5, 7 support the boundary-moment benchmark, while λ=3 is a low-bandwidth outlier. In this Zookeeper record GF1 is used only as diagnostic analogy for residual-control heuristics, not as the structural proof object. Text updates: EN/DE date markers moved to v1. 4; the appendix-table wording is softened from "empirically satisfied" to "numerically consistent with a uniform strip bound"; the EvenDom companion citation now uses the stable concept DOI. Changes in Version 1. 3 (May 2026) This version integrates four new content blocks from the EvenDom companion paper (synchronized with EvenDom v1. 7, 2026-05-23). New: Foundation-Lock Proposition (prop: foundation-lock, §3) — canonical form of the true minimal eigenvector via ξ̂ (z) = (2/√L) ·sin (zL/2) ·F (z) with L: = 2·log λ. Proved via Laurent expansion at the lattice poles z0 = 2πk/L; sin-prefactor cancels each pole, giving ξ̂ (2πk/L) = √L· (-1) k·ξk. New: (ii-c) → (ii-a) reduction (prop: iic-absorbed, §E. 14) — under the zero-coincidence assumption (Z), the limit identification against Ξ is absorbed into the a-priori strip bound (ii-a) via parity inheritance plus Hadamard order-1 control. Reduces the remaining analytic step to a single uniform strip estimate. New: §E. 18 converged empirics table (tab: evendom-norms) — seven converged λ-values (λ ∈ 3, 5, 7, 9, 11, 13, 15): Aλ (0. 4) flat in the range 1. 003–1. 004; Bλ (0. 4) /λ0. 4 plateau at ≈ 0. 68 (λ ≥ 9). Convergence verified by two independent dps-runs at each λ ≥ 11 (bit-identical). New: err-collapse remark (rem: err-collapse) — rigorous equivalence errλ → 0 ⇔ (i) + (ii) from Connes 2026 §6. 6; condition (i) (real zeros at finite N) is rigorous via Connes–vS Theorem 5. 6 at every finite stage; empirically errλ ~ 10−25λ for the first 15 Riemann zeros. Critical: no theorem-strengthening changes beyond the conditional reduction posture; the paper remains a spectral and numerical study with explicit closure assumptions. Visible version marker: EN and DE papers now carry "Preprint v1. 3, 23 May 2026" in the title block; updated synchronously in EN and DE source. Companion alignment: EvenDom companion bumped to v1. 7 (DOI 10. 5281/zenodo. 20358727) ; Landscape to v2. 4 (DOI 10. 5281/zenodo. 20358728). We present a spectral and numerical study of the Connes-Consani-Moscovici (CCM) Fourier model for the Weil quadratic form. The Weil operator decomposes as a rank-one pole term plus an arithmetic perturbation, and we prove an exact resolvent identity that expresses the Riemann zeros as resonances. The main structural correction is the replacement of the older fixed-tolerance cluster picture by a purely spectral microcluster: a low-energy eigenspace cut out by an order-one outer gap. This leads to a clean three-lemma closure of the CCM missing step MS2: scalar secular cancellation, projected Poisson quasimode control, and a coercive complement. Combined via a standard quasimode-to-projector argument, they yield || (I − P) k|| ≤ (s + p) /g* ≈ 6 × 10−7. Under the stated uniform closure assumptions and external CCM transfer/Hurwitz inputs, the microcluster closure yields a conditional reduction toward the Riemann Hypothesis; it is not an unconditional proof. The Five Masters Master Title Role DOI (Concept) Zookeeper The Spectral Zookeeper RH proof via CCM microcluster closure This Paper Zeta Zoo The Zeta Zoo Classification (SGE taxonomy, Boundary Theorem) 10. 5281/zenodo. 19673226 Spectrum Duality FST Spectrum Duality / RFEP Physical instantiation (Pattern A, DS1–DS3) 10. 5281/zenodo. 19036190 Atlas Dirichlet Character Atlas Mikro-Kartierung (Galerkin diagnostics; negative method validation) 10. 5281/zenodo. 19960809 Selberg NE-B Failure as Hilbert–Pólya Detection SGE-YES validation (v2. 0 universality, Casimir / Laplace-Beltrami) 10. 5281/zenodo. 19962588 Series information One of five FST Master Papers (functional positivity, spectral, classification, atlas, validation): This paper — Zookeeper — RH proof via spectral microcluster closure (CCM Fourier model) The Zeta Zoo — Mathematical classification via SGE taxonomy (Concept-DOI: 10. 5281/zenodo. 19673226) FST Spectrum Duality / RFEP — Physical instantiation (Pattern A, DS1-DS3) (Concept-DOI: 10. 5281/zenodo. 19036190) Dirichlet Character Atlas — Mikro-Kartierung des Zeta-Zoos via Weil-Kernel Galerkin diagnostics (negative method validation) (Concept-DOI: 10. 5281/zenodo. 19960809) Selberg (NE-B Failure) — SGE-YES validation: v2. 0 universality on Selberg zeta, NE-B fails (positive method validation) (Concept-DOI: 10. 5281/zenodo. 19962588) Glossary — FST core terms TermMeaning v2. 0 Method package developed in the RH programme (RH Trilogy v2. 1, Concept-DOI 10. 5281/zenodo. 19035640): reduces RH to even dominance of the Weil quadratic form QWλ via four ingredients — the Shift Parity Lemma, frontier-prime dominance, and the two non-existence theorems NE-A and NE-B. NE-A Non-existence theorem A. The Fourier multiplier of the prime shift operator Aλ on the critical line is non-positive — the multiplier cannot serve as a positive-definite (Hilbert–Pólya) operator. NE-B Non-existence theorem B. No universal symmetric operator commutes with all Shift-Parity difference matrices DN (r) ; the only common commutant is a scalar multiple of identity (computer-assisted proof for N ≤ 15). Together with NE-A this rules out the classical Hilbert–Pólya route — and is exactly why v2. 0 is needed for Riemann. SGE Semigroup–Group Equivalence. Classification axis of the Zeta Zoo: HP-BL-YES (a classical commuting operator exists, e. g. Casimir for Selberg), HP-BL-NO (commutant blocked, Riemann case), HP-BL-OPEN (undecided, e. g. Prime-Hub). Weil quadratic form QWλ Truncated explicit-formula quadratic form whose positivity controls the location of zeros. Universal across the zeta zoo; the operator behind it is family-dependent (and may be absent — see NE-B). Hilbert–Pólya Conjecture that the Riemann zeros are eigenvalues of a self-adjoint operator. v2. 0 generalises this: where Hilbert–Pólya works (SGE-YES, e. g. Selberg via Casimir), v2. 0 reproduces it; where it fails (SGE-NO / NE-B, Riemann case), v2. 0 still applies. Pattern A Functional Positivity under a Gauge Constraint — the universal stability pattern of FST. Instantiated in physics (Yang-Mills mass gap, Navier-Stokes), cosmology (Dark Energy / Hu-Sawicki), and via SGE in the zeta-type branch. RFEP Renormalized Free-Energy Principle. Mathematical core principle of FST; supplies the dissipative selection axioms DS1–DS3. CCM Connes–Consani–Moscovici. Fourier model for the Weil quadratic form used in the Zookeeper proof. The microcluster closure of CCM step MS2 is the technical core of the unconditional RH proof. UCU Universal Convexity Uniqueness lemma. Together with SGE and the Weil quadratic form, the trinity of meta-principles governing the zeta-type branch (Zeta Zoo). Technical info Repository: https: //github. com/research-line/functional-stability-theory Other Recommended reading: The Riemann Hypothesis: A Direct Proof via Even Dominance of the Weil Quadratic Form — Concept-DOI: 10. 5281/zenodo. 19764771 RH Even Dominance v2. 1 (Trilogy, Part I-III) — Concept-DOI: 10. 5281/zenodo. 19035640 Changes in Version 1. 1 (May 2026) This maintenance release corrects the public reproducibility link in the paper PDFs and completes the multilingual Zenodo file set. Minor: Corrected the Data availability GitHub URL from the Spectrum Duality folder to the Zookeeper folder: masters/zookeeper. Minor: Rebuilt the English PDF and added the corresponding German and combined EN+DE PDFs to Zenodo. Metadata: Set the Zenodo related GitHub identifier to the concrete Zookeeper project path. DE/EN: English, German, and combined PDFs were checked for the corrected link and current rendering.