Coalgebraic ζ(2k): Bisimulation-closed predicates for ζ(2k) ∈ ℚ·π^{2k}

We initiate Volume III of the Toward RH\,:\,HoTT-Prop programme by extending the Part I coinductive characterisation of $$ to the even special values $(2k)$ for $k 1$. The substantive contribution is a construction: for each $k 1$, an explicit, coinductively defined element $$ of the digit-stream final coalgebra $ F_b/$, built from the BBP digit stream of $$ by online multiplication and rational scaling, whose underlying real value (in the Cauchy reals $$ of Part II) is $(2k)/2$. The construction is uniform in $k$, parameterised by the Bernoulli stream and the explicit rational coefficient $q_…

Lean 4 Proof Verdict

# Lean proof verdict — oq1-coalgebraic-zeta-2k

_Generated by codex gpt-5.5 at xhigh reasoning effort, with workspace-write._

## Pre-verification `lake build` output

```
info: oq1-coalgebraic-zeta-2k: no previous manifest, creating one from scratch
trace: ././.lake/packages/mathlib> git fetch --tags --force origin
error: ././.lake/packages/mathlib/lakefile.lean:35:40: error: unexpected token ']'; expected term
error: ././.lake/packages/mathlib/lakefile.lean:45:38: error: unexpected token ']'; expected term
error: ././.lake/packages/mathlib/lakefile.lean:64:17: error: unknown identifier 'mathlibLeanOptions'
error: ././.lake/packages/mathlib/lakefile.lean:75:17: error: unknown identifier 'mathlibLeanOptions'
error: ././.lake/packages/mathlib/lakefile.lean:78:17: error: unknown identifier 'mathlibLeanOptions'
error: ././.lake/packages/mathlib/lakefile.lean:143:5: error: invalid field 'baseName', the environment does not contain 'Lake.Package.baseName'
  rootPkg
has type
  Package
error: ././.lake/packages/mathlib/lakefile.lean:143:24: error: invalid field 'baseName', the environment does not contain 'Lake.NPackage.baseName'
  pkg
has type
  NPackage _package.name
error: ././.lake/packages/mathlib/lakefile.lean:149:34: error: invalid field 'trimAscii', the environment does not contain 'String.trimAscii'
  toolchainContent
has type
  String
error: ././.lake/packages/mathlib/lakefile.lean:170:16: error: invalid field 'baseName', the environment does not contain 'Lake.NPackage.baseName'
  pkg
has type
  NPackage _package.name
error: ././.lake/packages/mathlib/lakefile.lean:151:13: error: invalid field 'trimAscii', the environment does not contain 'String.trimAscii'
  toolchainContent
has type
  String
error: ././.lake/packages/mathlib/lakefile.lean:153:29: error: invalid field notation, type is not of the form (C ...) where C is a constant
  toolchainVersion
has type
  ?m.3835 pkg rootPkg y✝ __do_lift✝ toolchainContent
error: ././.lake/packages/mathlib/lakefile.lean: pac

… (truncated)