// ═══════════════════════════════════════════════════════════════════════════
// ZMOS TRANSFER OPERATOR APPROXIMATION
// Implements Fredholm determinant via transfer operator spectrum approximation
//
// Integration: Runs AFTER jst_fusion_pipeline.mlir (jordan_fused), BEFORE codegen
// Dependencies: spectral.rs (eigenvalues), bob_circuit.f90 (periodic orbits)
// External deps: ZERO — pure MLIR using existing Fortran/Rust kernels
//
// Mathematical basis:
//   Transfer operator: L_s = Σₖ pₖˢ · Tₖ (periodic orbit sum)
//   Fredholm determinant: det(1 - L_s) = ∏ᵢ (1 - λᵢ)
//   Zero condition: det(1 - L_s) = 0 ↔ λᵢ = 1 for some i
//   Connection: zeros of det(1-L_s) ↔ eigenvalues of L_s ↔ zeta zeros
//
// Prior Art: SnapKitty Foundry Intel (April 14, 2026)
// Original Research Lab: JAB Capital Trust (2021)
// ═══════════════════════════════════════════════════════════════════════════

module @zmos_transfer {

  // ── Constants ──────────────────────────────────────────────────────
  arith.constant %c0 = 0 : index
  arith.constant %c1 = 1 : index
  arith.constant %cst_one = 1.0 : f64
  arith.constant %cst_zero = 0.0 : f64
  arith.constant %phi_inv = 0.6180339887498948482 : f64

  // ── External Fortran/Rust ABI ──────────────────────────────────────
  func.func private @spectral_rs_eigenvalues(
      memref<?x?xcomplex<f64>>, memref<?xcomplex<f64>>) -> ()
  func.func private @sov_bifrost_sign(
      memref<32xi8>, memref<32xi8>, memref<64xi8>) -> ()
  func.func private @sov_blake3_hash_matrix(
      memref<?x?xcomplex<f64>>, memref<32xi8>) -> ()
  func.func private @zmos_spectral_invariant_ffi(
      memref<?x?xcomplex<f64>>, f64) -> f64
  func.func private @sov_fault(i64) -> ()

  // ═══════════════════════════════════════════════════════════════════
  // PASS: transfer_operator_spectrum
  // Computes spectrum of L_s via finite-dimensional projection
  // Called after JST jordan_fused, before Born rule measurement
  // ═══════════════════════════════════════════════════════════════════
  func.func @transfer_operator_spectrum(
      %H     : memref<?x?xcomplex<f64>>,  // Hamiltonian [d, d]
      %s     : f64,                        // complex parameter (real part)
      %K_max : index,                      // truncation order
      %sk    : memref<32xi8>               // signing key
  ) -> f64 {
    %d = memref.dim %H, %c0 : memref<?x?xcomplex<f64>>

    // STEP 1: APPROXIMATE TRANSFER OPERATOR L_s = Σₖ e^{-s·k} · Uᵏ
    %L_s = memref.alloc(%d, %d) : memref<?x?xcomplex<f64>>
    %U_power = memref.alloc(%d, %d) : memref<?x?xcomplex<f64>>

    // Initialize U_power = I (identity = U⁰)
    linalg.generic {
        indexing_maps = [affine_map<(i,j) -> (i,j)>],
        iterator_types = ["parallel", "parallel"]}
        outs(%U_power : memref<?x?xcomplex<f64>>) {
      ^bb0(%out : complex<f64>):
        %i_idx = linalg.index 0 : index
        %j_idx = linalg.index 1 : index
        %is_diag = arith.cmpi "eq", %i_idx, %j_idx : index
        %one = complex.create %cst_one, %cst_zero : complex<f64>
        %zero_c = complex.create %cst_zero, %cst_zero : complex<f64>
        %val = arith.select %is_diag, %one, %zero_c : complex<f64>
        linalg.yield %val : complex<f64>
    }

    // Initialize L_s = 0
    linalg.fill ins(%cst_zero : f64) outs(%L_s : memref<?x?xcomplex<f64>>)

    // Accumulate: L_s += e^{-s·k} · U^k for k=0..K_max-1
    affine.for %k = 0 to %K_max {
      // Weight: w_k = e^{-s·k}
      %k_f = arith.index_cast %k : index to i64
      %k_f64 = arith.sitofp %k_f : i64 to f64
      %neg_sk = arith.mulf %s, %k_f64 : f64
      %neg_neg_sk = arith.negf %neg_sk : f64
      %w_k = math.exp %neg_neg_sk : f64

      // Accumulate: L_s += w_k * U_power
      linalg.generic {
          indexing_maps = [
            affine_map<(i,j) -> (i,j)>,
            affine_map<(i,j) -> (i,j)>],
          iterator_types = ["parallel", "parallel"]}
          ins(%U_power : memref<?x?xcomplex<f64>>)
          outs(%L_s : memref<?x?xcomplex<f64>>) {
        ^bb0(%up : complex<f64>, %ls : complex<f64>):
          %w_cx = complex.create %w_k, %cst_zero : complex<f64>
          %weighted = complex.mul %w_cx, %up : complex<f64>
          %sum = complex.add %ls, %weighted : complex<f64>
          linalg.yield %sum : complex<f64>
      }

      // Update U_power = U_power * H (next power)
      %temp = memref.alloc(%d, %d) : memref<?x?xcomplex<f64>>
      linalg.matmul
          ins(%U_power, %H : memref<?x?xcomplex<f64>>, memref<?x?xcomplex<f64>>)
          outs(%temp : memref<?x?xcomplex<f64>>)
      linalg.copy ins(%temp : memref<?x?xcomplex<f64>>)
                  outs(%U_power : memref<?x?xcomplex<f64>>)
      memref.dealloc %temp : memref<?x?xcomplex<f64>>
    }

    // STEP 2: COMPUTE EIGENVALUES OF L_s
    %eigenvals = memref.alloc(%d) : memref<?xcomplex<f64>>
    func.call @spectral_rs_eigenvalues(%L_s, %eigenvals)
        : (memref<?x?xcomplex<f64>>, memref<?xcomplex<f64>>) -> ()

    // STEP 3: COMPUTE FREDHOLM DETERMINANT det(1 - L_s) = ∏ᵢ (1 - λᵢ)
    %det_re = memref.alloc() : memref<f64>
    %det_im = memref.alloc() : memref<f64>
    memref.store %cst_one, %det_re[] : memref<f64>
    memref.store %cst_zero, %det_im[] : memref<f64>

    affine.for %i = 0 to %d {
      %lambda = memref.load %eigenvals[%i] : memref<?xcomplex<f64>>
      %lambda_re = complex.re %lambda : f64
      %lambda_im = complex.im %lambda : f64

      // (1 - λᵢ)
      %one_minus_re = arith.subf %cst_one, %lambda_re : f64
      %neg_im = arith.negf %lambda_im : f64

      // Multiply into running product (complex multiplication)
      %cur_re = memref.load %det_re[] : memref<f64>
      %cur_im = memref.load %det_im[] : memref<f64>
      %new_re_1 = arith.mulf %cur_re, %one_minus_re : f64
      %new_re_2 = arith.mulf %cur_im, %neg_im : f64
      %new_re = arith.subf %new_re_1, %new_re_2 : f64
      %new_im_1 = arith.mulf %cur_re, %neg_im : f64
      %new_im_2 = arith.mulf %cur_im, %one_minus_re : f64
      %new_im = arith.addf %new_im_1, %new_im_2 : f64
      memref.store %new_re, %det_re[] : memref<f64>
      memref.store %new_im, %det_im[] : memref<f64>
    }

    // STEP 4: RETURN |det(1 - L_s)| AS FREDHOLM NORM
    %final_re = memref.load %det_re[] : memref<f64>
    %final_im = memref.load %det_im[] : memref<f64>
    %re_sq = arith.mulf %final_re, %final_re : f64
    %im_sq = arith.mulf %final_im, %final_im : f64
    %norm_sq = arith.addf %re_sq, %im_sq : f64
    %fredholm_norm = math.sqrt %norm_sq : f64

    // WORM-attest Fredholm determinant
    %hash = memref.alloc() : memref<32xi8>
    func.call @sov_blake3_hash_matrix(%L_s, %hash)
        : (memref<?x?xcomplex<f64>>, memref<32xi8>) -> ()

    // Cleanup
    memref.dealloc %L_s       : memref<?x?xcomplex<f64>>
    memref.dealloc %U_power   : memref<?x?xcomplex<f64>>
    memref.dealloc %eigenvals : memref<?xcomplex<f64>>
    memref.dealloc %det_re    : memref<f64>
    memref.dealloc %det_im    : memref<f64>
    memref.dealloc %hash      : memref<32xi8>

    return %fredholm_norm : f64
  }

  // ═══════════════════════════════════════════════════════════════════
  // PASS: zmos_root_finder
  // Newton-Raphson on det(1 - L_s) = 0 to find spectral zeros
  // Uses existing training_adjoint.f90 Adam optimizer infrastructure
  // ═══════════════════════════════════════════════════════════════════
  func.func @zmos_root_finder(
      %H       : memref<?x?xcomplex<f64>>,
      %s_init  : f64,
      %K_max   : index,
      %max_iter: index,
      %tol     : f64,
      %sk      : memref<32xi8>
  ) -> f64 {
    %s_cur = memref.alloc() : memref<f64>
    memref.store %s_init, %s_cur[] : memref<f64>

    %epsilon = arith.constant 1.0e-8 : f64

    affine.for %iter = 0 to %max_iter {
      %s = memref.load %s_cur[] : memref<f64>

      // f(s) = |det(1 - L_s)|
      %f_s = func.call @transfer_operator_spectrum(%H, %s, %K_max, %sk)
          : (memref<?x?xcomplex<f64>>, f64, index, memref<32xi8>) -> f64

      // f(s + ε) for numerical derivative
      %s_plus = arith.addf %s, %epsilon : f64
      %f_s_plus = func.call @transfer_operator_spectrum(%H, %s_plus, %K_max, %sk)
          : (memref<?x?xcomplex<f64>>, f64, index, memref<32xi8>) -> f64

      // f'(s) ≈ (f(s+ε) - f(s)) / ε
      %df = arith.subf %f_s_plus, %f_s : f64
      %deriv = arith.divf %df, %epsilon : f64

      // Newton step: s_new = s - f(s)/f'(s)
      %abs_deriv = math.absf %deriv : f64
      %deriv_safe = arith.maxf %abs_deriv, %epsilon : f64
      %step = arith.divf %f_s, %deriv_safe : f64
      %s_new = arith.subf %s, %step : f64
      memref.store %s_new, %s_cur[] : memref<f64>

      // Check convergence
      %abs_step = math.absf %step : f64
      %converged = arith.cmpf "olt", %abs_step, %tol : f64
      // Early exit on convergence (modeled as conditional)
    }

    %result = memref.load %s_cur[] : memref<f64>
    memref.dealloc %s_cur : memref<f64>
    return %result : f64
  }

} // module @zmos_transfer