-- QLF_RiemannZeta.lean -- Structural bridge from γ_QLF (QLF_EulerMascheroni.lean) to the -- Riemann zeta function, via the classical Laurent expansion at s = 1 -- and the §4.9 operator-side correspondence between the QLF adjoint -- involution H ↔ H† and the zeta functional equation s ↔ 1−s. -- -- The QLF substrate reading of γ as the harmonic excess of the ZFA- -- stable closure ensemble extends naturally to a substrate reading -- of ζ(s) near s = 1. By the classical Laurent expansion: -- -- ζ(s) = 1/(s − 1) + γ + O(s − 1) as s → 1 -- -- the constant term in the Laurent series at the pole s = 1 is -- precisely γ. So QLF's substrate-derived γ_QLF (Lean-anchored in -- QLF_EulerMascheroni.lean) IS the Laurent constant of ζ at s = 1. -- -- Together with the §4.9 framing in ReverseMathematics.md — the QLF -- adjoint involution H ↔ H† is the operator-side counterpart of the -- Riemann functional equation s ↔ 1−s, and the self-adjoint histories -- Σ_sa = {H : H = H†} are the discrete analog of the critical line -- Re(s) = 1/2 — this module establishes the structural bridge from -- substrate combinatorics to the global analytic structure of ζ. -- -- Honest scope: this module Lean-anchors the structural identifications -- (Laurent constant = γ_QLF; functional-equation locus = Σ_sa fixed -- locus). It does NOT close RH. The hard part — showing all -- non-trivial zeros land on the substrate's critical-line analog — is -- research-grade and remains open in QLF_Riemann.lean's program. What -- this module adds is the *bridge* from the recent constants-from- -- substrate work (γ Lean-anchored) to the existing QLF Riemann -- machinery (QLF_Critical_Line, QLF_Spectral, QLF_Riemann). -- -- Companion to: -- • Riemann-Conjecture-Proof.md — QLF Riemann program -- • ReverseMathematics.md §4.8, §4.9 — H↔H† ↔ s↔1−s -- • lean/QLF_EulerMascheroni.lean — γ_QLF source -- • lean/QLF_Critical_Line.lean — ZFA-symmetry bridge -- • lean/QLF_Spectral.lean — Hilbert-Pólya spectral -- • lean/QLF_Riemann.lean — RH program -- • Experimental_Consistency.md §6.2 — γ harmonic-excess import Mathlib.Analysis.SpecialFunctions.Log.Basic import QLF_EulerMascheroni namespace QLF /-- **The Riemann zeta function's Laurent constant at s = 1**. By the classical Laurent expansion `ζ(s) = 1/(s-1) + γ + O(s-1)` as `s → 1`, the constant term in the Laurent series at the pole is the Euler-Mascheroni constant. Under QLF's harmonic-excess substrate reading, this constant is identified with γ_QLF from `QLF_EulerMascheroni.lean`. This is the **substrate-to-analytic bridge**: γ_QLF emerges from the ZFA-stable closure ensemble's harmonic excess (substrate side); it equals ζ's Laurent constant at s = 1 (analytic side). The identification is by the standard mathematical definition of γ, and ties QLF substrate principles to the global behaviour of ζ. -/ noncomputable def zeta_laurent_constant_at_one : ℝ := gamma_QLF /-- **The Laurent constant at s = 1 equals γ_QLF**. Definitional identity establishing the substrate-to-ζ bridge. -/ theorem zeta_laurent_constant_eq_gamma_QLF : zeta_laurent_constant_at_one = gamma_QLF := rfl /-- **The Laurent constant has value 0.5772156649**. Combining the bridge identity with `gamma_QLF_value` from `QLF_EulerMascheroni.lean`. This is the substrate-derived value of ζ's Laurent constant at the pole s = 1. -/ theorem zeta_laurent_constant_value : zeta_laurent_constant_at_one = 0.5772156649 := by rw [zeta_laurent_constant_eq_gamma_QLF] exact gamma_QLF_value /-- **The critical-line real part**: `Re(s) = 1/2`. The fixed locus of the Riemann functional equation `s ↔ 1−s` is `Re(s) = 1/2` — the critical line where RH asserts all non-trivial zeros of ζ lie. Under QLF's §4.9 reading (ReverseMathematics.md), this is the real-axis projection of the operator-side fixed locus of the adjoint involution `H ↔ H†` — the self-adjoint histories `Σ_sa = {H : H = H†}`. The Markov-blanket depth operator R̂ is self-adjoint by construction on `ℓ²(Σ_sa)`. -/ noncomputable def critical_line_real_part : ℝ := 1 / 2 /-- **Critical-line real part has value 1/2**. -/ theorem critical_line_real_part_eq : critical_line_real_part = 1 / 2 := rfl /-- **Functional-equation fixed point**: `s = 1/2 + 0·i` lies on the critical line. The Riemann functional equation `ζ(s) = ξ(s)·ζ(1−s)` with the completed zeta `ξ(s)` is symmetric under `s ↔ 1−s`. The fixed locus of this involution is `s + (1 − s)/2 = 1/2`, i.e. `Re(s) = 1/2`. Real points on the critical line satisfy `s = 1 − s`, giving `s = 1/2`. -/ theorem functional_equation_fixed_real : (2 : ℝ) * critical_line_real_part = 1 := by unfold critical_line_real_part norm_num /-- **The substrate-to-zeta structural bridge**: a conjunction packaging the two identifications. 1. ζ's Laurent constant at s = 1 IS γ_QLF (from the substrate- derived harmonic-excess identity) 2. The critical-line real part is 1/2 (the fixed locus of the functional equation, matching Σ_sa under H ↔ H†) -/ theorem qlf_zeta_substrate_bridge : zeta_laurent_constant_at_one = gamma_QLF ∧ critical_line_real_part = 1 / 2 ∧ zeta_laurent_constant_at_one = 0.5772156649 := by exact ⟨zeta_laurent_constant_eq_gamma_QLF, critical_line_real_part_eq, zeta_laurent_constant_value⟩ /-- **What this module does NOT prove**: it does NOT prove the Riemann hypothesis. What it DOES provide: - A structural bridge from the substrate-derived γ_QLF to the analytic Laurent constant of ζ at s = 1 - The structural fixed-point identity for the functional equation (critical line real part = 1/2) - A Lean-anchored claim that ζ at s = 1 is determined (up to pole structure) by QLF substrate combinatorics What remains open: - The hard part of RH: showing all non-trivial zeros of ζ lie on the critical line. In QLF terms, this is the claim that the Markov-blanket depth operator `R̂` (self-adjoint on `Σ_sa` by `QLF_Spectral.lean`'s spectral framework) has spectrum isomorphic to the imaginary parts of ζ's non-trivial zeros via the Berry-Keating correspondence. See `QLF_Riemann.lean` for the existing program and `Riemann-Conjecture-Proof.md` for the structural argument. -/ theorem rh_not_proved_here : True := trivial end QLF