/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Research Stack Team

AVMR.lean — Algebraic Vector Mountain Range (Core)

This is the reduced core module for the AVMR framework.
Component definitions are modularized.
-/

import Semantics.Spectrum
import Semantics.GeneticCode
import Semantics.ShellModel
import Semantics.SpectralField
import Semantics.VecState
import Semantics.FixedPoint

namespace Semantics.AVMR

open Semantics
open Semantics.Spectrum
open Semantics.GeneticCode
open Semantics.ShellModel
open Semantics.VecState

/-! # Algebraic Vector Mountain Range (AVMR) — Reduced Core -/

/-- Hyperbola index for a natural number n. -/
def hyperbolaIndex (n : Nat) : Nat :=
  let k := isqrt n
  let a := n - k*k
  let b := (k+1)*(k+1) - n
  a * b

/-- Tip Coordinate Mass Resonance (Theorem 122).
    Uses `ShellModel.isqrt` which delegates to `Nat.sqrt`, so `isqrt (m*m) = m`
    by `Nat.sqrt_eq`. -/
theorem resonanceHubDegeneracy (m : Nat) :
  let n := m*m
  let k := isqrt n
  let a := n - k*k
  let b := (k+1)*(k+1) - n
  a = 0 ∧ b = 2*m + 1 := by
  have h_isqrt : isqrt (m * m) = m := by
    simp [isqrt, Nat.sqrt_eq]
  simp [h_isqrt]
  have h_expand : (m + 1) * (m + 1) = m * m + 2 * m + 1 := by
    simp [Nat.mul_add, Nat.add_mul]; omega
  omega

/-- Theorem 19: Axial Generator Exhaustivity.
    NOTE: Original statement used strict inequality for the last conjunct,
    which is false at k=1 (3 < 3 is false). Weakened to ≤. -/
theorem axialGeneratorExhaustivity (k : Nat) (_hk : k ≥ 1) :
  k*k < k*k + k ∧ k*k + k < k*k + k + 1 ∧ k*k + k + 1 ≤ (k+1)*(k+1) - 1 := by
  refine ⟨?_, ?_, ?_⟩
  · -- k*k < k*k + k for k ≥ 1
    omega
  · -- k*k + k < k*k + k + 1 always
    omega
  · -- k*k + k + 1 ≤ (k+1)*(k+1) - 1 for k ≥ 1
    have h_expand : (k + 1) * (k + 1) = k * k + 2 * k + 1 := by
      simp [Nat.mul_add, Nat.add_mul]; omega
    rw [h_expand]
    omega

/-- The Missing Link ODE (Model 131). -/
def vectorField (a b : Float) (ε : Float) : Float × Float :=
  (1.0 + ε * (b * 0.5 + 0.3), -1.0 + ε * (a * 0.5 - 0.3))

/-! ## End Core AVMR -/

end Semantics.AVMR
