HoTT-Riemann Hypothesis
Volume IV
HoTTCubical AgdaCondensed MathematicsFormal Verification

Volume IV
From Formulation to Formal Proof

The HoTT-Riemann Hypothesis programme advances from informal formulation to machine-checked formal proof. Volume IV targets OQ1 (coinductive ζ(2k)), OQ3 (directed univalence), and OQ5 (analytic Langlands topos), alongside three cross-cutting infrastructure papers.

Read SynthesisDownload PDF
7
Papers
3
Primary Targets
3
Infrastructure
157+
Total Pages

Primary Research Targets

OQ1, OQ3, and OQ5 — the three open questions from Volume III now advanced toward formal proof

Full Cubical Agda Formalisation of Coinductive ζ(2k)
Part I
math.LOcode28p

Full Cubical Agda Formalisation of Coinductive ζ(2k)

We give a full Cubical Agda formalisation of the coinductive digit-stream witness for ζ(2k)/2, with a constructive Bernoulli library using only the generating-function recurrence and exact rationals. We prove the central contractibility theorem and provide a Lean 4 shadow using Mathlib.

Directed Univalence for Non-Discrete Types: A No-Go Theorem and a Half-Reduction
Part II
math.CTcode26p

Directed Univalence for Non-Discrete Types: A No-Go Theorem and a Half-Reduction

Voevodsky's univalence axiom is symmetric: for any types A, B in a univalent universe, the identity type equals the equivalence type. We prove a no-go theorem for a directed analogue of univalence for non-discrete types, and establish a half-reduction result.

Identifying the (∞,1)-Topos for Analytic Langlands
Part III
math.NT25p

Identifying the (∞,1)-Topos for Analytic Langlands

OQ5 asks: which elementary (∞,1)-topos most naturally hosts automorphic representations of GL(n, A_Q)? We identify condensed pyknotic spaces over extremally disconnected sets as the correct ambient topos and verify the six Rasekh elementarity axioms.

Cross-Cutting Infrastructure

Three infrastructure papers enabling the primary research targets

Pre-Warmed Mathlib Build Cache
Part IV
cs.LOcode22p

Pre-Warmed Mathlib Build Cache

Volume III produced 29 incomplete theorems, of which roughly 25% are infrastructure gaps. We provide a pre-warmed Mathlib build cache with verified stubs for the key analytic facts used across OQ1, OQ3, and OQ5, reducing downstream compilation times significantly.

Completing Cubical.HITs.CauchyReals
Part V
math.LOcode24p

Completing Cubical.HITs.CauchyReals

Volume III produced a verdict matrix of 3 valid, 0 invalid, 29 incomplete theorems. A substantial share of the incompletes are blocked on the unmerged Cubical.HITs.CauchyReals module. We complete the missing proofs and prepare the PR for the cubical-agda library.

The Comparison Lemma Library (Theorem A)
Part VI
math.LOcode20p

The Comparison Lemma Library (Theorem A)

Volume IV has six parallel research workstreams. The infrastructure papers enable the primary research targets. We provide a comparison lemma library (Theorem A) connecting the classical analytic theory with the HoTT-native formulation across all three OQ targets.

Volume Synthesis

From Formulation to Formal Proof — Volume IV Synthesis
Synthesis32 pages

From Formulation to Formal Proof — Volume IV Synthesis

The HoTT-Riemann Hypothesis programme is a multi-volume effort to formulate, then prove, the Riemann Hypothesis as a proposition in homotopy type theory. This synthesis paper integrates the results of Parts I–VI and charts the path from formulation to formal proof.