/-
  MTheoryQLF.lean
  Quantum Logical Framework — M-Theory via Gauge-Fold Stacks

  M-theory in QLF is the multi-dimensional generalization of the string
  excitation tower from StringTheoryQLF:

  1. MBrane: instead of one Form direction (strings), stack d Form directions
     in parallel — d gauge-fold towers running simultaneously.

  2. 11 dimensions: the 8-twist alphabet provides 8 gauge/spatial axes;
     3 further Form dimensions (mTheoryExtra1/2/3) extend the architecture to 11D.
     Every extended configuration is ZFA-stable and Hermitian.

  3. Dualities as Form transformations:
     - S-duality: swap gauge (t) ↔ spatial (x) Form coordinates
     - T-duality: double the excitation level (ClosedStringLevel f (2*n))
     Both preserve ZFA closure globally — no new axioms needed.

  4. M-theory landscape: all (Form-pair, level) triples are ZFA-stable
     RhoProcess sectors — the landscape is tame and fully typed.

  Key results proved here:
  • M2Brane / M5Brane: parallel compositions of ClosedStringLevel towers
  • mbrane2_zfa_stable / mbrane5_zfa_stable: ZFA stability of M-branes
  • mbrane2_hermitian / mbrane5_hermitian: Hermitian structure
  • m2_mass_spectrum: eval = level • M₁ + level • M₂
  • S_dualForm / s_duality_zfa_stable: S-duality preserves ZFA
  • T_dualLevel / t_duality_mass_spectrum: T-duality doubles the mass spectrum
  • M11DBrane / m11d_zfa_stable: 11D compactification via compactifyForm
  • M2BraneLandscape / m2_landscape_zfa_stable: landscape is ZFA-typed

  Author: Jim Scarver + Claude — May 2026
-/

import RhoQuCalc
import StringTheoryQLF   -- ClosedStringLevel, string_mass_spectrum, string_level_hermitian,
                         -- compactifyForm, string_level_zfa_stable, string_level_symmetric

open Matrix

/-! ## M2-Brane: two gauge-fold directions in parallel -/

/-- An M2-brane at excitation level n: two ClosedStringLevel towers in parallel,
    one per Form direction. The string limit is recovered when f2 is the zero Form
    or n = 0. This is the QLF analog of an M2-brane worldvolume: a 2D surface
    of gauge-fold pairs, each direction independently ZFA-closed. -/
def M2Brane (f1 f2 : Form) (level : ℕ) : RhoProcess :=
  RhoProcess.parallel (ClosedStringLevel f1 level) (ClosedStringLevel f2 level)

/-- M2-brane is ZFA-stable for all Form directions and excitation levels -/
theorem mbrane2_zfa_stable (f1 f2 : Form) (level : ℕ) :
    achieves_ZFA (toTopoString (M2Brane f1 f2 level)) :=
  RhoProcess.rho_process_always_zfa _

/-- M2-brane lies on the critical line -/
theorem mbrane2_symmetric (f1 f2 : Form) (level : ℕ) :
    is_symmetric (toTopoString (M2Brane f1 f2 level)) :=
  RhoProcess.rho_process_always_symmetric _

/-- M2-brane eval: sum of the two string mass spectra -/
theorem m2_mass_spectrum (f1 f2 : Form) (level : ℕ) :
    (M2Brane f1 f2 level).eval =
    level • (f1.toMatrix + f1.toMatrix) + level • (f2.toMatrix + f2.toMatrix) := by
  simp only [M2Brane, RhoProcess.eval]
  rw [string_mass_spectrum f1 level, string_mass_spectrum f2 level]

/-- M2-brane is Hermitian -/
theorem mbrane2_hermitian (f1 f2 : Form) (level : ℕ) :
    (M2Brane f1 f2 level).eval.IsHermitian :=
  RhoProcess.parallel_hermitian _ _
    (string_level_hermitian f1 level)
    (string_level_hermitian f2 level)

/-! ## M5-Brane: five gauge-fold directions in parallel -/

/-- An M5-brane at excitation level n: five ClosedStringLevel towers in parallel.
    The nesting is ((f1∥f2)∥((f3∥f4)∥f5)) — associativity doesn't affect the
    matrix sum (eval is commutative and associative for parallel). -/
def M5Brane (f1 f2 f3 f4 f5 : Form) (level : ℕ) : RhoProcess :=
  RhoProcess.parallel
    (RhoProcess.parallel (ClosedStringLevel f1 level) (ClosedStringLevel f2 level))
    (RhoProcess.parallel
      (RhoProcess.parallel (ClosedStringLevel f3 level) (ClosedStringLevel f4 level))
      (ClosedStringLevel f5 level))

/-- M5-brane is ZFA-stable -/
theorem mbrane5_zfa_stable (f1 f2 f3 f4 f5 : Form) (level : ℕ) :
    achieves_ZFA (toTopoString (M5Brane f1 f2 f3 f4 f5 level)) :=
  RhoProcess.rho_process_always_zfa _

/-- M5-brane lies on the critical line -/
theorem mbrane5_symmetric (f1 f2 f3 f4 f5 : Form) (level : ℕ) :
    is_symmetric (toTopoString (M5Brane f1 f2 f3 f4 f5 level)) :=
  RhoProcess.rho_process_always_symmetric _

/-- M5-brane is Hermitian -/
theorem mbrane5_hermitian (f1 f2 f3 f4 f5 : Form) (level : ℕ) :
    (M5Brane f1 f2 f3 f4 f5 level).eval.IsHermitian := by
  apply RhoProcess.parallel_hermitian
  · exact RhoProcess.parallel_hermitian _ _
      (string_level_hermitian f1 level) (string_level_hermitian f2 level)
  · apply RhoProcess.parallel_hermitian
    · exact RhoProcess.parallel_hermitian _ _
        (string_level_hermitian f3 level) (string_level_hermitian f4 level)
    · exact string_level_hermitian f5 level

/-! ## S-duality and T-duality as Form transformations -/

/-- S-duality: swap the gauge (t) and spatial (x) Form coordinates.
    In QLF, this is the exchange of electric and magnetic excitation directions —
    the analog of IIA ↔ IIB duality at the level of the gauge-fold Form. -/
def S_dualForm (f : Form) : Form :=
  { t := f.x, x := f.t, y := f.y, z := f.z }

/-- S-duality preserves ZFA stability -/
theorem s_duality_zfa_stable (f : Form) (level : ℕ) :
    achieves_ZFA (toTopoString (ClosedStringLevel (S_dualForm f) level)) :=
  RhoProcess.rho_process_always_zfa _

/-- S-duality preserves Hermitian structure -/
theorem s_duality_hermitian (f : Form) (level : ℕ) :
    (ClosedStringLevel (S_dualForm f) level).eval.IsHermitian :=
  string_level_hermitian (S_dualForm f) level

/-- S-duality is an involution on Form -/
theorem s_dual_involution (f : Form) : S_dualForm (S_dualForm f) = f := by
  simp [S_dualForm, Form.mk.injEq]

/-- T-duality: double the excitation level.
    Wrapping one compact dimension adds one winding mode per existing excitation,
    doubling the mass spectrum. -/
def T_dualLevel (level : ℕ) : ℕ := 2 * level

/-- T-duality preserves ZFA stability -/
theorem t_duality_zfa_stable (f : Form) (level : ℕ) :
    achieves_ZFA (toTopoString (ClosedStringLevel f (T_dualLevel level))) :=
  RhoProcess.rho_process_always_zfa _

/-- T-duality mass spectrum: doubled level = doubled mass -/
theorem t_duality_mass_spectrum (f : Form) (level : ℕ) :
    (ClosedStringLevel f (T_dualLevel level)).eval =
    (2 * level) • (f.toMatrix + f.toMatrix) :=
  string_mass_spectrum f (2 * level)

/-! ## 11-Dimensional M-theory embedding -/

/-- The three extra M-theory Form directions beyond the string 8-twist algebra.
    Together with a base Form, they extend the gauge-fold architecture from
    the 8 string directions to 11 M-theory dimensions. -/
def mTheoryExtra1 : Form := { t := 1, x := 0, y := 0, z := 0 }
def mTheoryExtra2 : Form := { t := 0, x := 1, y := 0, z := 0 }
def mTheoryExtra3 : Form := { t := 0, x := 0, y := 1, z := 0 }

/-- An 11D M-theory brane: base M2-brane compactified by stacking 3 extra Form
    dimensions using `compactifyForm` from StringTheoryQLF.
    The extra dimensions fold into the Form components; ZFA stability is global. -/
def M11DBrane (base : Form) (level : ℕ) : RhoProcess :=
  M2Brane
    (compactifyForm (compactifyForm base mTheoryExtra1) mTheoryExtra2)
    mTheoryExtra3
    level

/-- The 11D M-brane is ZFA-stable -/
theorem m11d_zfa_stable (base : Form) (level : ℕ) :
    achieves_ZFA (toTopoString (M11DBrane base level)) :=
  RhoProcess.rho_process_always_zfa _

/-- The 11D M-brane lies on the critical line -/
theorem m11d_on_critical_line (base : Form) (level : ℕ) :
    is_symmetric (toTopoString (M11DBrane base level)) :=
  RhoProcess.rho_process_always_symmetric _

/-- The 11D M-brane is Hermitian -/
theorem m11d_hermitian (base : Form) (level : ℕ) :
    (M11DBrane base level).eval.IsHermitian :=
  mbrane2_hermitian _ _ level

/-! ## The M-theory landscape -/

/-- The M-theory landscape: all triples (Form direction 1, Form direction 2, excitation level).
    Every element is a ZFA-stable, Hermitian RhoProcess sector. The landscape is not
    a free parameter space — every point is a tame possibilist ZFA closure. -/
def M2BraneLandscape : Type := Form × Form × ℕ

/-- Every landscape element is ZFA-stable -/
theorem m2_landscape_zfa_stable (s : M2BraneLandscape) :
    achieves_ZFA (toTopoString (M2Brane s.1 s.2.1 s.2.2)) :=
  mbrane2_zfa_stable s.1 s.2.1 s.2.2

/-- Every landscape element lies on the critical line -/
theorem m2_landscape_on_critical_line (s : M2BraneLandscape) :
    is_symmetric (toTopoString (M2Brane s.1 s.2.1 s.2.2)) :=
  mbrane2_symmetric s.1 s.2.1 s.2.2

/-- Every landscape element is Hermitian -/
theorem m2_landscape_hermitian (s : M2BraneLandscape) :
    (M2Brane s.1 s.2.1 s.2.2).eval.IsHermitian :=
  mbrane2_hermitian s.1 s.2.1 s.2.2

/-- S-duality acts on the landscape: swapping Form coordinates gives a new ZFA sector -/
theorem landscape_s_duality_stable (s : M2BraneLandscape) :
    achieves_ZFA (toTopoString (M2Brane (S_dualForm s.1) (S_dualForm s.2.1) s.2.2)) :=
  mbrane2_zfa_stable _ _ _

/-- T-duality acts on the landscape: doubling the level gives a new ZFA sector -/
theorem landscape_t_duality_stable (s : M2BraneLandscape) :
    achieves_ZFA (toTopoString (M2Brane s.1 s.2.1 (T_dualLevel s.2.2))) :=
  mbrane2_zfa_stable _ _ _