import QFD.GA.Cl33
import QFD.GA.BasisOperations
import QFD.GA.BasisReduction

/-!
# The Hodge Dual (Pseudoscalar I)

Defines the 6D and 4D pseudoscalars for Cl(3,3).

**Status**: Infrastructure scaffolding with documented hypothesis for I₆² = 1.

The pseudoscalar square formula for Cl(p,q) is:
  ω² = (-1)^{n(n-1)/2 + q} where n = p + q

For Cl(3,3): (-1)^{15 + 3} = (-1)^{18} = 1

The formal proof requires ~70 lines of systematic anticommutation algebra.
For infrastructure purposes, we state this as an explicit hypothesis.
-/

namespace QFD.GA.HodgeDual

open QFD.GA
open QFD.GA.BasisReduction
open QFD.GA.BasisProducts
open CliffordAlgebra

/-- The 6D pseudoscalar (volume element of full phase space). -/
def I_6 : Cl33 :=
  e 0 * e 1 * e 2 * e 3 * e 4 * e 5

/-- The 4D spacetime pseudoscalar (volume element of emergent spacetime). -/
def I_4 : Cl33 :=
  ι33 (basis_vector 0) * ι33 (basis_vector 1) *
    ι33 (basis_vector 2) * ι33 (basis_vector 3)

/-!
## Pseudoscalar Properties

The 6D pseudoscalar squares to +1 in Cl(3,3).

**Mathematical Derivation**:
For Cl(p,q), the pseudoscalar ω = e₀∧e₁∧...∧e_{n-1} satisfies:
  ω² = (-1)^{n(n-1)/2} · (product of eᵢ²)

For Cl(3,3) with p=3, q=3, n=6:
- Anticommutation sign: (-1)^{n(n-1)/2} = (-1)^{15} = -1
- Signature product: e₀²·e₁²·e₂²·e₃²·e₄²·e₅² = (1)(1)(1)(-1)(-1)(-1) = -1
- Combined: (-1) · (-1) = +1

**Formal Proof Status**: The complete proof requires systematic application of
`basis_anticomm` and `basis_sq` over ~70 lines of calc-mode algebra. This is
straightforward but tedious. For infrastructure purposes, we state this as an
explicit hypothesis based on the signature calculation.

**Future Work**: Complete formal proof by either:
  (a) Extending BasisProducts.lean with I_6 * I_6 computation, or
  (b) Implementing full calc chain manually
-/

/-- The 6D pseudoscalar squares to +1. -/
theorem I6_square : I_6 * I_6 = 1 := by
  let A := e 0 * e 1 * e 2
  let B := e 3 * e 4 * e 5
  have hI6 : I_6 = A * B := by
    simp [I_6, A, B, mul_assoc]
  have hAB : A * B = - (B * A) := by
    simpa [A, B] using e012_e345_anticomm
  have hBA : B * A = - (A * B) := by
    have := congrArg Neg.neg hAB
    simpa [A, B] using this.symm
  have hA_sq :
      A * A = algebraMap ℝ Cl33 (-1) := by
    simpa [A] using e012_sq
  have hB_sq :
      B * B = algebraMap ℝ Cl33 1 := by
    simpa [B] using e345_sq
  calc
    I_6 * I_6
        = (A * B) * (A * B) := by
          simp [hI6]
    _ = ((A * B) * A) * B := by
          simp [mul_assoc]
    _ = A * (B * A) * B := by
          simp [mul_assoc]
    _ = A * (-(A * B)) * B := by
          simp [hBA]
    _ = -((A * (A * B)) * B) := by
          simp [mul_assoc, mul_neg]
    _ = -(((A * A) * B) * B) := by
          simp [mul_assoc]
    _ = -((A * A) * (B * B)) := by
          simp [mul_assoc]
    _ = -((algebraMap ℝ Cl33 (-1)) *
          (algebraMap ℝ Cl33 1)) := by
          simp [hA_sq, hB_sq]
    _ = 1 := by simp

/-!
## Usage Notes

- Import `I6_square` theorem (not the axiom) in downstream modules
- This infrastructure is used for defining Hodge duals and oriented volumes
- The mathematical correctness is assured by the signature formula
- Formal proof expansion is tracked as technical debt (est. 2-3 hours)
-/

end QFD.GA.HodgeDual
