Volume V●Work in Progress

A HoTT/Yoneda No-Phantom Route
Toward the Riemann Hypothesis

The current volume of an ongoing programme that uses homotopy type theory and condensed mathematics to mount a machine-checked, dual-tracked attack on the Riemann Hypothesis. Thirteen specialised AI agents worked in parallel to attempt the proof along two independent routes — the operator-theoretic Hilbert–Pólya programme and the cohomological Connes–Deligne programme — with every theorem verified by Lean and independently re-checked by Codex. The work continues.

The result: a precise, machine-checked reduction of RH to two named open problems, and a formal proof that those two problems are the same mathematical barrier seen from different angles. 239 valid Lean theorems, zero invalid.

What this volume has accomplished so far:
  • RH formalised inside an elementary (∞,1)-topos. Both proof routes — Hilbert–Pólya (operator-theoretic) and Connes–Deligne (cohomological) — written as Lean statements inside the condensed topos ℰcond.
  • Cross-track equivalence theorem proven. The two formulations are formally equivalent inside ℰcond: the operator-theoretic and cohomological views are not two attacks but one, viewed from two angles.
  • Two precise obstruction theorems. Each route reduces RH to a single named gap: a polarisation existence problem on Track A, and a condensed Frobenius-purity transfer on Track B. These are the most precise statements to date of what blocks RH from inside a condensed-topos formulation.
  • 239 valid Lean theorems, zero invalid. Every theorem machine-checked. Independent Codex verification caught and forced fixes for six AI cheating attempts during the run, so the ledger is honest.
  • Four open questions from earlier volumes were closed — including a constructive Cubical Agda proof of ζ(2)/2 = π²/12 via the boundary postulates B1–B4.
  • Post-pipeline addendum. A sharper formal reduction of Track A’s residual obstruction: finite-rank RKHS / Yoneda detector semantics → no-phantom Blaschke defect → internal Blaschke triviality → RHclassical. The RH-level burden is relocated from a positive density obligation in L2(0,1) to a generation-and-descent obligation in the condensed topos. 178 additional public Lean theorems, zero sorries. RH is not proved; the attack surface is sharpened.

Work in progress. The two named gaps — and the cross-track equivalence between them — are concrete targets the programme continues to push on, alongside the wider mathematical community. The post-pipeline addendum sharpens the Track A target into a categorical/topos-side obligation rather than a classical density one, and further refinements are expected as the work continues. RH is not proved.

239Valid Lean Theorems
0Invalid
22Incomplete
13Agent Teams
2Obstruction Theorems

Synthesis

Synthesis
Volume V Synthesis — Dual-Track Obstruction Reductions (Work in Progress)
Synthesis of the HoTT-Riemann programme to date. Thirteen concurrent agent teams across five months produced 239 valid / 0 invalid Lean theorems. The current state: theorem ObstructionTrackA reduces RH to wedge-polarization convergence; theorem ObstructionTrackB reduces RH to CondensedPurityTransfer SpecZCond inside the condensed topos. The dual obstructions are honest logical equivalences characterising the same barrier from operator-theoretic and cohomological angles. The post-pipeline Yoneda–Nyman addendum sharpens Track A further. Work in progress — RH remains open, precisely quantified, and the programme continues.
239 valid22 incompletemath.NT
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Synthesis cover

Post-Pipeline Addendum — Yoneda–Nyman Frontier

Addendum
Yoneda–Nyman Frontier — Decomposed Proof-Run and No-Phantom Blaschke Defect
Post-pipeline addendum to Volume V's Track A. Records a sharper formal reduction inside the condensed topos: finite-rank RKHS / Yoneda detector semantics → no-phantom Blaschke defect payload → zero…
178 valid38 incompletemath.NT
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Proof Teams — The Obstruction Theorems

V6a
Track A Proof Attempt — Operator-Theoretic Obstruction in the Hilbert–Pólya Formulation
Track A proof team. Applies the Hilbert–Pólya operator-theoretic formulation of RH within the condensed topos. Produces theorem ObstructionTrackA: RH_classical ↔ ∃ (J : Polarization), Compatible J ω —…
14 valid1 incompletemath.NT
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V6b
Track B Proof Attempt — Cohomological Obstruction via Condensed Purity Transfer
Track B proof team. Works inside the condensed étale cohomology framework. Produces theorem ObstructionTrackB: RH_classical ↔ CondensedPurityTransfer SpecZCond, decomposed into sub-obstructions O1…
7 valid1 incompletemath.NT
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Formulation Teams — Dual-Track Setup

V5a
Track A Formulation — Hilbert–Pólya RH Inside the Condensed Topos
Formalises the Hilbert–Pólya operator-theoretic formulation of the Riemann Hypothesis inside the condensed topos framework, providing the Lean 4 foundations for Track A proof attempt. 3 valid theorems…
3 validmath.NT
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V5b
Track B Formulation — Frobenius RH and the Cross-Track Equivalence Theorem
Formalises the Frobenius-eigenvalue formulation of RH inside the condensed topos and proves theorem RH_HilbertPolya_iff_RH_Frob — the cross-track equivalence establishing that Track A and Track B…
4 valid3 incompletemath.NT
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Axiom-Discharge Teams — Closing Vol IV Gaps

V1
Cosmos-Internal Yoneda — Lifting the Half-Reduction Axiom
Discharges the Vol IV axiom half_reduction (from Oq3DirectedUnivalence.lean) by proving it as a theorem via cosmos-internal Yoneda lemma techniques. Also supersedes axiom DU_Seg_iff_LaxPullbackClosure…
10 validmath.CT
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V2
UCQ at ℵ_ω — Universe Coherence Question Resolved
Resolves the Universe Coherence Question flagged in Vol IV by proving theorem ucq_aleph_omega : UCQ (aleph Ordinal.omega0). 24 valid theorems at 4.8× target, 1 incomplete.
24 valid1 incompletemath.LO
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V3
Cubical Agda Merge — Discharging the Boundary Postulates B1–B4
Discharges boundary postulates B1–B4 from Vol IV in Cubical Agda with --safe --cubical flags. Machine-checks contract-k1 (ζ(2)/2 = π²/12) and contract-k2 (ζ(4)/2 = π⁴/180) as isContr proofs. 35 valid…
35 valid1 incompletemath.LO
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V4
ConjectureC_reverse Discharged — Obstruction Companion Theorem
Discharges the Vol IV axiom ConjectureC_reverse from InfraComparisonLemmas.lean, producing theorem_ConjectureC_reverse_proved plus companion theorem_ObstructionConjectureC. 11 valid theorems, 0…
11 validmath.NT
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Infrastructure Teams

Infra-1
Analytic Langlands Infrastructure — Foundations for Spectral Methods
Provides Lean 4 infrastructure for analytic Langlands correspondence, underpinning the spectral methods used by Track A. 15 valid theorems at exact target, 2 incomplete.
15 valid2 incompletemath.RT
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Infra-2
Langlands GL₁ — Local-Global Compatibility Infrastructure
Develops Lean 4 infrastructure for GL₁ Langlands correspondence and local-global compatibility lemmas needed by the condensed cohomology teams. 22 valid theorems at 2.75× target, 1 incomplete.
22 valid1 incompletemath.RT
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Infra-3
Cubical Analytic Continuation — HoTT Foundations for Complex Analysis
Provides cubical type-theoretic foundations for analytic continuation, enabling the Track B cohomological arguments to access complex-analytic results. 27 valid theorems at 2.7× target, 9 incomplete.
27 valid9 incompletemath.CV
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Infra-4
Categorical Spectral Theorem — Internal Hilbert Space Formalism
Develops the categorical spectral theorem in the internal language of the condensed topos, providing the Hilbert space formalism used by Track A. 20 valid theorems at 3.3× target, 0 incomplete.
20 validmath.FA
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Infra-5
Condensed Étale Cohomology — The ℰ_cond Topos Infrastructure
The largest infrastructure paper: builds the condensed étale cohomology framework inside the condensed topos ℰ_cond, including the motivicGaloisDescent axiom (a shared crux of Track B sub-obstruction…
47 valid3 incompletemath.AG
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Verdict Matrix

#TeamValidInvalidIncompleteMet?
1V6a — Track A Proof Attempt1401✅
2V6b — Track B Proof Attempt701✅
3V5a — Track A Formulation300✅
4V5b — Track B Formulation403✅
5V1 — Cosmos-Internal Yoneda1000✅
6V2 — UCQ at ℵ_ω2401✅
7V3 — Cubical Agda Merge3501✅
8V4 — ConjectureC_reverse Discharged1100✅
9Infra-1 — Analytic Langlands Infrastructure1502✅
10Infra-2 — Langlands GL₁2201✅
11Infra-3 — Cubical Analytic Continuation2709✅
12Infra-4 — Categorical Spectral Theorem2000✅
13Infra-5 — Condensed Étale Cohomology4703✅
Total239022✅