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75 lines
2.2 KiB
Text
75 lines
2.2 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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AVMR.lean — Algebraic Vector Mountain Range (Core)
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This is the reduced core module for the AVMR framework.
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Component definitions are modularized.
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-/
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import Semantics.Spectrum
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import Semantics.GeneticCode
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import Semantics.ShellModel
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import Semantics.SpectralField
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import Semantics.VecState
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import Semantics.FixedPoint
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namespace Semantics.AVMR
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open Semantics
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open Semantics.Spectrum
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open Semantics.GeneticCode
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open Semantics.ShellModel
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open Semantics.VecState
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/-! # Algebraic Vector Mountain Range (AVMR) — Reduced Core -/
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/-- Hyperbola index for a natural number n. -/
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def hyperbolaIndex (n : Nat) : Nat :=
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let k := isqrt n
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let a := n - k*k
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let b := (k+1)*(k+1) - n
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a * b
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/-- Tip Coordinate Mass Resonance (Theorem 122).
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Uses `ShellModel.isqrt` which delegates to `Nat.sqrt`, so `isqrt (m*m) = m`
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by `Nat.sqrt_eq`. -/
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theorem resonanceHubDegeneracy (m : Nat) :
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let n := m*m
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let k := isqrt n
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let a := n - k*k
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let b := (k+1)*(k+1) - n
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a = 0 ∧ b = 2*m + 1 := by
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have h_isqrt : isqrt (m * m) = m := by
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simp [isqrt, Nat.sqrt_eq]
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simp [h_isqrt]
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have h_expand : (m + 1) * (m + 1) = m * m + 2 * m + 1 := by
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simp [Nat.mul_add, Nat.add_mul]
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<;> omega
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omega
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/-- Theorem 19: Axial Generator Exhaustivity.
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NOTE: Original statement used strict inequality for the last conjunct,
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which is false at k=1 (3 < 3 is false). Weakened to ≤. -/
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theorem axialGeneratorExhaustivity (k : Nat) (_hk : k ≥ 1) :
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k*k < k*k + k ∧ k*k + k < k*k + k + 1 ∧ k*k + k + 1 ≤ (k+1)*(k+1) - 1 := by
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refine ⟨?_, ?_, ?_⟩
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· -- k*k < k*k + k for k ≥ 1
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omega
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· -- k*k + k < k*k + k + 1 always
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omega
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· -- k*k + k + 1 ≤ (k+1)*(k+1) - 1 for k ≥ 1
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have h_expand : (k + 1) * (k + 1) = k * k + 2 * k + 1 := by
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simp [Nat.mul_add, Nat.add_mul]
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<;> omega
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rw [h_expand]
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omega
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/-- The Missing Link ODE (Model 131). -/
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def vectorField (a b : Float) (ε : Float) : Float × Float :=
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(1.0 + ε * (b * 0.5 + 0.3), -1.0 + ε * (a * 0.5 - 0.3))
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/-! ## End Core AVMR -/
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end Semantics.AVMR
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