#!/usr/bin/env python3 """ Computation 77 -- Threshold-induced dynamical generation-symmetry breaking ========================================================================== Continuation of deriving the Yukawa hierarchy from structural PST inputs. This time the mechanism is dynamical: the directed modal threshold (P3) selects configurations based on their angular structure in V_7, and the threshold-crossing rate per quaternionic sub-algebra Q_k gives a hierarchy of generation Yukawa couplings. CONTEXT ------- Comp 76 ruled out the SIMPLEST formulation of Option B (single-direction V_7 constraint with maximal G_2 symmetry preserves the S_3 of the three Q_k). The honest residual: the structural-scope theorem is overcome only by either a second postulate- level input OR a dynamical mechanism induced by the threshold crossing. This computation tests the DYNAMICAL option. Setup: (1) V_7 = R^7 decomposes under the G_2-stabilizer of tau-hat as tau-hat ⊕ tau-hat^⊥, with tau-hat^⊥ = R^6. (2) The three quaternionic sub-algebras Q_k of O containing tau-hat give a decomposition tau-hat^⊥ = (Q_1 ∩ tau-hat^⊥) ⊕ (Q_2 ∩ tau-hat^⊥) ⊕ (Q_3 ∩ tau-hat^⊥) into three mutually-orthogonal 2D subspaces. (3) For a configuration C with directional content T(C) ∈ V_7, its 'residual direction' T(C) - (T(C)·tau-hat) tau-hat lies in tau-hat^⊥ and has projections onto each Q_k ∩ tau-hat^⊥. (4) The threshold-crossing 'rate' for configurations populating generation k is taken to be proportional to a Boltzmann-like factor in the squared projection onto Q_k: rate_k ~ exp(-beta * (1 - p_k)) where p_k = |proj_{Q_k ∩ tau-hat^⊥}(T(C) residual)|^2 / |T(C) residual|^2, and beta is a structural inverse- temperature parameter at the threshold. (5) The effective Yukawa coupling for generation k is then proportional to rate_k (configurations cross more often → more 'mass density' → larger Yukawa coupling). THIS IS SPECULATIVE. The mechanism is structurally motivated (the threshold crossing is the right place to look for symmetry breaking, since it already breaks parity), but the specific form of rate_k is an assumption not yet derived from postulates. Honest goal: see whether ANY value of beta produces a Yukawa-like hierarchy. EXPECTED HIERARCHY (TARGET) --------------------------- Up-type SM Yukawas: y_t = 1.0, y_c ~ 0.0076, y_u ~ 1.3e-5 Hierarchy spans 5 orders of magnitude. For the dynamical mechanism to deliver this, beta * (p_max - p_min) must be ~5*log(10) ~ 11.5. We test the angular structure to see if any natural beta fits. """ from __future__ import annotations import math import numpy as np def setup_V7_and_Qs(): """V_7 = R^7. tau-hat = e_1. Q_k ∩ tau-hat^⊥ are three orthogonal 2D subspaces of tau-hat^⊥ = R^6, corresponding to the three Fano-plane lines through e_1. """ tau_hat = np.array([1, 0, 0, 0, 0, 0, 0], dtype=float) # Q_k ∩ tau-hat^⊥ basis vectors (in V_7 = R^7) Q1_perp = np.array([[0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0]], dtype=float) Q2_perp = np.array([[0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0]], dtype=float) Q3_perp = np.array([[0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 1]], dtype=float) return tau_hat, [Q1_perp, Q2_perp, Q3_perp] def projection_squared(v: np.ndarray, basis: np.ndarray) -> float: """Returns |proj_{span(basis)}(v)|^2 / |v|^2.""" v_norm = np.linalg.norm(v) if v_norm < 1e-12: return 0.0 coeffs = basis @ v proj_norm_sq = (coeffs ** 2).sum() return proj_norm_sq / v_norm ** 2 def boltzmann_yukawas(T_residual: np.ndarray, Q_perps: list[np.ndarray], beta: float) -> np.ndarray: """Returns the three Boltzmann-weighted generation Yukawa couplings. rate_k ~ exp(-beta * (1 - p_k)) Yukawa_k normalised so the largest is 1. """ p_k = np.array([projection_squared(T_residual, Q_perp) for Q_perp in Q_perps]) rates = np.exp(-beta * (1 - p_k)) yukawas = rates / rates.max() return yukawas, p_k def main(): print("=" * 100) print(" Computation 77 -- Threshold-induced dynamical generation symmetry breaking") print("=" * 100) print() tau_hat, Q_perps = setup_V7_and_Qs() print("MODEL: rate_k ~ exp(-beta * (1 - p_k))") print(" where p_k is the squared fraction of T(C) residual lying in Q_k ∩ tau-hat^⊥") print() print("SCANS: vary beta from 1 to 20, see what hierarchy emerges for an EXTREME") print(" residual direction (all weight in Q_1: p_1 = 1, p_2 = p_3 = 0)") print("-" * 100) print() # Extreme case: residual entirely in Q_1 T_extreme = np.array([0, 1, 0, 0, 0, 0, 0], dtype=float) # in Q_1 print(f" Extreme case: T residual = (0, 1, 0, 0, 0, 0, 0) (entirely in Q_1)") print(f" p_1 = 1, p_2 = 0, p_3 = 0") print() print(f" {'beta':>6} {'y_1':>10} {'y_2':>12} {'y_3':>12} {'y_1/y_2':>10} {'y_1/y_3':>10}") for beta in [1, 3, 5, 8, 10, 12, 15, 20]: yukawas, p_k = boltzmann_yukawas(T_extreme, Q_perps, beta) ratio12 = yukawas[0] / yukawas[1] if yukawas[1] > 0 else float('inf') ratio13 = yukawas[0] / yukawas[2] if yukawas[2] > 0 else float('inf') print(f" {beta:>6.1f} {yukawas[0]:>10.4f} {yukawas[1]:>12.4e} " f"{yukawas[2]:>12.4e} {ratio12:>10.2e} {ratio13:>10.2e}") print() print(" Observed SM up-type ratios:") print(f" y_t/y_c = 1.0/0.0076 = {1.0/0.0076:.2e}") print(f" y_t/y_u = 1.0/1.3e-5 = {1.0/1.3e-5:.2e}") print() print(" For y_t/y_c = 130 (extreme case beta needed for log(130) = 4.87, so beta ~ 5)") print(" For y_t/y_u = 77000 (extreme case beta needed for log(77000) = 11.25, so beta ~ 11)") print() print(" The two ratios DIFFER, so a single beta cannot fit both. Either the model") print(" is wrong, or the p_k values for generations 2 and 3 are NOT equal (they're") print(" both 0 in the extreme case, but in reality they would differ by structural") print(" V_7 geometry).") print() print("=" * 100) print(" GENERIC CASE: T residual at a generic angle in tau-hat^⊥") print("=" * 100) print() print(" For T residual aligned at angle (theta, phi) in tau-hat^⊥ (where tau-hat^⊥ = R^6,") print(" decomposed as Q_1 ∩ tau-hat^⊥ + Q_2 ∩ tau-hat^⊥ + Q_3 ∩ tau-hat^⊥),") print(" scan over generic T directions and find best-fit beta to SM hierarchy.") print() print(" Generic parameterisation: T residual = (cos(a), sin(a)*cos(b), sin(a)*sin(b)*cos(c),") print(" sin(a)*sin(b)*sin(c)*cos(d), ...)") print(" where (a, b, c, ...) are angles. For each (a, b, c, d, e), compute") print(" p_1, p_2, p_3 (which depend on the angles), then check if any beta value") print(" gives SM-like hierarchy (1.0, 0.0076, 1.3e-5).") print() target = np.array([1.0, 0.0076, 1.3e-5]) rng = np.random.default_rng(42) best_error = float('inf') best_params = None n_samples = 100_000 for _ in range(n_samples): # Sample a unit vector in R^6 uniformly v = rng.normal(size=6) v = v / np.linalg.norm(v) # Build T residual in V_7 with v in tau-hat^⊥ T_residual = np.array([0, v[0], v[1], v[2], v[3], v[4], v[5]]) p_k = np.array([projection_squared(T_residual, Q_perp) for Q_perp in Q_perps]) if p_k.max() < 1e-9: continue # Sort so p_1 is largest (assign to top generation) order = np.argsort(p_k)[::-1] p_sorted = p_k[order] for beta in np.linspace(1, 30, 100): rates = np.exp(-beta * (1 - p_sorted)) yukawas = rates / rates.max() err = np.sum((np.log(yukawas[1:] + 1e-30) - np.log(target[1:])) ** 2) if err < best_error: best_error = err best_params = (T_residual.copy(), p_sorted.copy(), beta, yukawas.copy()) if best_params is not None: T_best, p_best, beta_best, yukawas_best = best_params print(f" Best-fit over {n_samples} random T directions and beta in [1, 30]:") print(f" p_sorted = ({p_best[0]:.4f}, {p_best[1]:.4f}, {p_best[2]:.4f})") print(f" beta = {beta_best:.2f}") print(f" Yukawas: ({yukawas_best[0]:.4e}, {yukawas_best[1]:.4e}, {yukawas_best[2]:.4e})") print(f" Target: (1.0, {target[1]:.4e}, {target[2]:.4e})") print(f" log-error sum: {best_error:.4f}") print() print("=" * 100) print(" STRUCTURAL ASSESSMENT") print("=" * 100) print() print(" The dynamical Boltzmann mechanism produces a Yukawa-like hierarchy") print(" for specific (beta, T-residual-direction) values, but:") print() print(" (a) beta and T-residual-direction are TWO additional structural inputs that") print(" are themselves not derived from P1-P3.") print() print(" (b) The T-residual-direction is configuration-dependent (it varies with C),") print(" so a SPECIFIC hierarchy emerges only after specifying which configuration") print(" is realised. This is the same 'lives in T(C)' content the structural-scope theorem") print(" flagged -- moved one level deeper but not eliminated.") print() print(" (c) For a fit to SM Yukawas, the best-fit p_k values would themselves need") print(" to be predicted structurally, which they currently are not.") print() print(" CONCLUSION: the dynamical mechanism CAN produce 5-order-of-magnitude") print(" hierarchies for some choices of (beta, T-residual), but it does NOT") print(" overcome the structural-scope theorem in the sense required -- the hierarchy still depends") print(" on contingent input (configuration-specific T-residual direction and") print(" the structural beta whose origin is open).") print() print(" Same status as Comp 76: real reduction, identifies WHAT would have to") print(" be true for closure, but does not close the structural-scope theorem.") print() print(" The next research frontier: derive beta and the residual-direction") print(" structure from a deeper structural principle (e.g., maximal-entropy") print(" threshold-crossing or extremal-action selection). Such a principle") print(" would be a P4 candidate.") if __name__ == "__main__": main()