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.

This is a preprint. Major revisions are not excluded. The record states a conditional reduction programme, not an unconditional proof of RH or MS2.

The Five Masters

Series information

One of five FST Master Papers (functional positivity, spectral, classification, atlas, validation):

• This paper — Zookeeper — conditional RH-reduction programme 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

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

CHANGELOG

1.6 Changelog (15 August 2026):

Additive interim update (instrument status); no theorem-strengthening changes, conditional-reduction status unchanged, no progress on MS2 is claimed.

• New section “Update (v1.6): Instrument Status and Contract-Form Applications, June–August 2026” (EN and DE): (a) state of the cross-route consistency instrumentation (shadow registry, revision V06; the sister abc programme maintains a separate register, canonical branch v118 with active continuation v119); (b) three contract forms from the sister programme recorded as transferable audit instruments — bridge certificate (10.5281/zenodo.21925803), ledger audit (10.5281/zenodo.21951155), rigidity / effective-weight test (10.5281/zenodo.21952486) — with status envisaged, not carried out for the spectral transport step of this paper; (c) five instrument status lines (limit topology: norm route excluded by a rank argument, strong topology named but unverified; contour admissibility: only an upper bound derivable; window/seed: index-free finite contract passed, uniformity and limit open; selection family: not canonical, finite gate passed, limit gate open; inclusion angles: one-sided diagnostics, fixed-anchor quantity not yet computed) — each line names what remains open.
• Companion reference update: the landscape/atlas companion is cited at its v3.2 record (10.5281/zenodo.21952620).
• Status: everything in this update is instrument and validation progress; no MS2 step, no new claims, active analytic fronts are pursued separately and will be reported in a dedicated follow-up paper.

1.5 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 det_reg(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·ξ_k.
• 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}): A_λ(0.4) flat in the range 1.003–1.004; B_λ(0.4)/λ^0.4 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 err_λ → 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 err_λ ~ 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).

Changes in Version 1.2 (May 2026)

This version integrates a healing iteration after external critical review (2026-05-14/15) and adds four new content blocks from the EvenDom companion paper (2026-05-23).

• Claim level: Theorem 8.3 reformulated as a conditional reduction (not an RH-proof claim).
• Theorem 4.4 (η_j > 0) re-proved via secular equation (Bunch–Nielsen–Sorensen identity for rank-1 perturbations), replacing the circular Cauchy-interlacing argument.
• Lemmas 7.3–7.5 reformulated as conjectural closure statements with honest separation between architectural identity and uniform constant control. rem:endgame-status lists three closure statements with honest status.
• Conjecture 8.2 (CCM Hurwitz passage) upgraded to Theorem 8.2 with Slepian–Pollak justification; sourcing caveat documented.
• Abstract updated: "spectral and numerical study within the CCM framework" with conditional reduction framing.
• 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 poles z₀ = 2πk/L.
• New: (ii-c)→(ii-a) reduction (prop:iic-absorbed, §E.14) — limit identification against Ξ absorbed into a-priori strip bound (ii-a). Reduces remaining analytic step to a single strip estimate.
• New: §E.18 empirics table (tab:evendom-norms) — seven converged λ-values (λ ∈ {3,5,7,9,11,13,15}): A_λ(0.4) flat near 1.004, B_λ(0.4)/λ^0.4 plateau near 0.68.
• New: err-collapse remark (rem:err-collapse) — err-collapse equivalence err_λ → 0 iff (i)+(ii) from Connes 2026 §6.6 (rigorous); condition (i) rigorousness at each stage via Connes–vS Theorem 5.6; empirically err_λ ~ 10^−25λ for the first 15 Riemann zeros.

This is a preprint. Major revisions are not excluded.

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.