import Mathlib.Data.Real.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Tactic

noncomputable section

namespace QFD.Empirical

open Real

/-!
# Gate C-2: The Core Compression Law (CCL)

This file formalizes the "Backbone" of the nuclear chart derived in the uploaded
"Universal Nuclear Scaling" papers [McSheery 2025].

Unlike the Semi-Empirical Mass Formula (which sums volume/surface/Coulomb terms),
QFD models the nucleus as an elastic soliton minimizing geometric stress.

## The Physical Model
1. **Backbone**: The ideal charge Q* for mass A is defined by competing geometric capacities:
   Q*(A) = c₁ A^(2/3) + c₂ A
   (c₁ ≈ Surface Flux, c₂ ≈ Volume Compression)

2. **Strain Energy**: Isotopes off this backbone experience an elastic restoring force:
   E(Q) = ½ k (Q - Q*)^2

3. **Stability**: Decay (Beta/Alpha) is the process of minimizing this strain.
-/

variable (A : ℝ) (hA : 0 < A)

/-! ## 1. The Geometric Backbone -/

/--
The "Zero Stress" charge trajectory.
Matches the functional form Q = c₁A^(2/3) + c₂A from the NuBase 2020 analysis.
-/
def backbone_charge (c₁ c₂ : ℝ) : ℝ :=
  c₁ * (A ^ (2 / 3 : ℝ)) + c₂ * A

/-! ## 2. The Elastic Energy Functional -/

/--
The Effective Potential Energy of a nucleus with mass A and charge Q.
This corresponds to the "Charge Stress" in the provided papers.
-/
def deformation_energy (c₁ c₂ k : ℝ) (Q : ℝ) : ℝ :=
  0.5 * k * (Q - backbone_charge A c₁ c₂)^2

/-! ## 3. Stability Theorems -/

/--
**Theorem CCL-1a**: The backbone minimizes deformation energy.
-/
theorem backbone_minimizes_energy
    (c₁ c₂ k : ℝ) (hk : 0 < k) (Q : ℝ) :
    deformation_energy A c₁ c₂ k (backbone_charge A c₁ c₂) ≤ deformation_energy A c₁ c₂ k Q := by
  unfold deformation_energy
  have h_zero : 0.5 * k * (backbone_charge A c₁ c₂ - backbone_charge A c₁ c₂)^2 = 0 := by ring
  rw [h_zero]
  apply mul_nonneg
  · linarith
  · exact pow_two_nonneg _

/--
**Theorem CCL-1b**: The backbone is the unique minimizer.
-/
theorem backbone_unique_minimizer
    (c₁ c₂ k : ℝ) (hk : 0 < k) (Q : ℝ)
    (h_min : deformation_energy A c₁ c₂ k Q = 0) :
    Q = backbone_charge A c₁ c₂ := by
  unfold deformation_energy at h_min
  have : (Q - backbone_charge A c₁ c₂)^2 = 0 := by nlinarith
  have : Q - backbone_charge A c₁ c₂ = 0 := eq_zero_of_pow_eq_zero this
  linarith

/-! ## 4. Decay Dynamics (The Gradient) -/

/--
**Theorem CCL-2**: Beta Decay Gradient ("Rolling Down the Hill").
If a nucleus is "Overcharged" (Q > Q*), reducing charge (Q - δ) strictly reduces Energy.
This formally predicts the β+ / Electron Capture decay channel.
-/
theorem beta_decay_favorable
    (c₁ c₂ k : ℝ) (hk : 0 < k)
    (Q : ℝ) (_h_excess : Q > backbone_charge A c₁ c₂) -- "Overcharged" condition
    (delta : ℝ) (h_delta_pos : 0 < delta) -- Small decay step
    (h_small_step : delta < Q - backbone_charge A c₁ c₂) : -- Step doesn't overshoot
    deformation_energy A c₁ c₂ k (Q - delta) < deformation_energy A c₁ c₂ k Q := by
  unfold deformation_energy
  let target := backbone_charge A c₁ c₂
  -- We need to show: 0.5*k*(Q - δ - target)^2 < 0.5*k*(Q - target)^2
  -- Since 0.5*k > 0, we just need to compare the squares.
  have h_factor_pos : 0 < 0.5 * k := by linarith
  apply mul_lt_mul_of_pos_left _ h_factor_pos
  -- We compare x_new^2 < x_old^2
  -- Let x_old = Q - target
  -- Let x_new = Q - delta - target
  apply sq_lt_sq'
  · -- Need: -(Q - target) < Q - delta - target
    linarith
  · -- Need: Q - delta - target < Q - target
    linarith

end QFD.Empirical
