| // ═══════════════════════════════════════════════════════════════════════════ | |
| // 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 | |