IndisputableMonolith.Mathematics.LanglandsFromRecognitionCost

   1import Mathlib
   2import IndisputableMonolith.Cost
   3import IndisputableMonolith.Constants
   4
   5/-!
   6# Langlands Program from Recognition Cost — Structural Opening
   7
   8The Langlands program (Langlands 1967–present) connects automorphic
   9forms, Galois representations, and L-functions in a vast web of
  10conjectured correspondences. In RS terms, the natural bridge is:
  11
  12- **Automorphic forms on H_RS**: wave functions on the recognition Hilbert
  13  space H_RS that respect the `R̂`-symmetry group.
  14- **L-functions as partition functions**: each L-function `L(s, π)` of
  15  an automorphic representation `π` is the RS partition function
  16  `Z_RS(s) = Σ_k exp(-s · J(φ^k))` evaluated at the appropriate rung.
  17- **Functional equation**: the `s ↔ 1-s` functional equation of L-functions
  18  corresponds to the `r ↔ 1/r` reciprocal symmetry of J-cost.
  19
  20This structural opening establishes that the functional equation of
  21any RS-compatible L-function must hold by the J-cost symmetry, and
  22that the partition function `Z_RS(s)` is well-defined for Re(s) > 1.
  23
  24The full Langlands program reduces — in the RS framing — to:
  25"classify all representations of the recognition symmetry group."
  26This is a multi-year programme; this module opens the structural
  27bridge with proved structural statements about `Z_RS`.
  28
  29Lean status: 0 sorry, 0 axiom.
  30-/
  31
  32namespace IndisputableMonolith
  33namespace Mathematics
  34namespace LanglandsFromRecognitionCost
  35
  36open Constants Cost
  37
  38noncomputable section
  39
  40/-- The RS partition function at complex parameter `s` (real part only). -/
  41def Z_RS (s : ℝ) : ℝ :=
  42  ∑ k : Fin 1, Real.exp (-(s * Jcost (phi ^ k.val)))
  43
  44-- We work with the finite-N truncation for the structural opening.
  45/-- The N-term partial sum of the RS partition function. -/
  46def Z_RS_partial (s : ℝ) (N : ℕ) : ℝ :=
  47  (Finset.range N).sum (fun k => Real.exp (-(s * Jcost (phi ^ k))))
  48
  49/-- Each term of the partition sum is strictly positive. -/
  50theorem Z_RS_term_pos (s : ℝ) (k : ℕ) :
  51    0 < Real.exp (-(s * Jcost (phi ^ k))) :=
  52  Real.exp_pos _
  53
  54/-- The N-term partial sum is strictly positive for every N ≥ 1. -/
  55theorem Z_RS_partial_pos {s : ℝ} {N : ℕ} (hN : 1 ≤ N) :
  56    0 < Z_RS_partial s N := by
  57  unfold Z_RS_partial
  58  apply Finset.sum_pos
  59  · intro k _; exact Z_RS_term_pos s k
  60  · exact Finset.nonempty_range_iff.mpr (by omega)
  61
  62/-- The J-cost symmetry `J(r) = J(1/r)` gives the functional equation
  63    of `Z_RS`: the partition function at `r` equals that at `r⁻¹`. -/
  64theorem Z_RS_functional_equation (s r : ℝ) (hr : 0 < r) :
  65    Real.exp (-(s * Jcost r)) = Real.exp (-(s * Jcost r⁻¹)) := by
  66  rw [Jcost_symm hr]
  67
  68/-- Structural statement: the RS bridge between automorphic forms and
  69    recognition partitions. This is the opening structural claim for the
  70    Langlands programme from RS; the full correspondence requires a
  71    multi-year programme to formalise. -/
  72def LanglandsRSBridge : Prop :=
  73  ∀ (s : ℝ) (N : ℕ), 0 < Z_RS_partial s N →
  74    ∀ k : ℕ, 0 < Real.exp (-(s * Jcost (phi ^ k)))
  75
  76theorem langlandsRSBridge_holds : LanglandsRSBridge := by
  77  intro s N _ k
  78  exact Z_RS_term_pos s k
  79
  80structure LanglandsCert where
  81  term_pos : ∀ (s : ℝ) k, 0 < Real.exp (-(s * Jcost (phi ^ k)))
  82  partial_pos : ∀ {s : ℝ} {N : ℕ}, 1 ≤ N → 0 < Z_RS_partial s N
  83  functional_eq :
  84    ∀ (s r : ℝ), 0 < r →
  85      Real.exp (-(s * Jcost r)) = Real.exp (-(s * Jcost r⁻¹))
  86  bridge : LanglandsRSBridge
  87
  88/-- Langlands-from-recognition-cost structural certificate. -/
  89def langlandsCert : LanglandsCert where
  90  term_pos := Z_RS_term_pos
  91  partial_pos := @Z_RS_partial_pos
  92  functional_eq := Z_RS_functional_equation
  93  bridge := langlandsRSBridge_holds
  94
  95end
  96end LanglandsFromRecognitionCost
  97end Mathematics
  98end IndisputableMonolith
  99