#!/usr/bin/env python3 """ PROVENANCE: NUMERICAL Computation 143 -- Born compatibility and uniqueness of the threshold-crossing rule ==================================================================================== Backs the open item in sec:open-frontiers: "how weak a coupling still forces a threshold crossing". The collapse rule of sec:collapse-rule sets tau_i(C) = tau_0(C) + kappa (lambda o_i)^2 (eq:tau-i-explicit) C crosses outcome i iff w_i := |Pi^{-1}(P_i psi)|^2 > tau_i(C) and clause (U) of the threshold-crossing-uniqueness theorem asserts the crossing set is a singleton (or empty). (U) is proved only for kappa lambda^2 >> tau_0. WHERE THE RANDOMNESS LIVES -- the point that decides the whole analysis ----------------------------------------------------------------------- For a GIVEN state psi the weights w_i are fixed numbers, not random variables. What varies from run to run is the substrate configuration C, and with it tau_0(C). So the object to study is P(select i) = P_C ( w_i > tau_0(C) + kappa (lambda o_i)^2 ), a statement about the DISTRIBUTION OF tau_0 OVER CONFIGURATIONS. The proof's own bound sum_i w_i <= ||psi||^2 = 1 is used throughout; s = kappa lambda^2 is carried in the same units, and no other normalisation is imposed. CHECKS ------ (1) BORN IS RECOVERED EXACTLY, UNDER AN UNSTATED CONDITION. At zero coupling, P_C(tau_0 < w_i) = w_i for every i precisely when tau_0(C) is UNIFORM on [0,1]. Verified to sampling precision, and shown to fail for a non-uniform tau_0 (Beta(2,2)) on the same states. This is a real constraint on the law of tau_0 over substrate configurations, and the book does not state it. (2) UNIQUENESS FAILS AT WEAK COUPLING. A single low tau_0 draw sits below several w_i at once, so the crossing set is not a singleton. Measured. (3) THE STRONG-COUPLING CONDITION IS THE RIGHT HYPOTHESIS. Raising s raises every threshold and monotonically drives P(>=2 crossings) to zero. The proof's stated hypothesis does exactly the work the proof claims for it. (4) BUT (U) AND BORN PULL APART. Conditioned on the crossing set being a singleton, the selected outcome is the maximum-margin one, which for fixed w is DETERMINISTIC -- the same outcome every run. Where (U) holds, Born does not; where Born holds, (U) does not. Checked across three states, including one with the largest weight on the largest |o_i| to rule out a correlation artefact. (5) SYMMETRIC SPECTRA GET NO SEPARATION FROM THE COUPLING. For o = +-1 the quadratic form is outcome-independent, so eq:tau-i-explicit contributes nothing; the book's own degenerate-case clause routes this to einselection. Recorded for completeness, not as a defect. """ import numpy as np rng = np.random.default_rng(20260926) N = 2_000_000 U = rng.uniform(0.0, 1.0, N) # tau_0(C) ~ Uniform[0,1] B22 = rng.beta(2.0, 2.0, N) # a non-uniform control on the same support def cross_sets(w, O, s, tau0): """Crossing iff tau_0 < w_i - s o_i^2.""" margin = np.asarray(w)[None, :] - s * np.asarray(O)[None, :] ** 2 cr = tau0[:, None] < margin return cr, cr.sum(axis=1) print("=" * 78) print("Computation 143 -- Born compatibility and uniqueness of the collapse rule") print("=" * 78) STATES = [ ("A weight aligned with small |o|", np.array([0.5, 0.3, 0.2]), np.array([0.5, 1.0, 1.5])), ("B weight aligned with large |o|", np.array([0.2, 0.3, 0.5]), np.array([0.5, 1.0, 1.5])), ("C near-degenerate weights ", np.array([0.34, 0.33, 0.33]), np.array([0.5, 1.0, 1.5])), ] # ---- CHECK 1 ------------------------------------------------------------- print("\n(1) BORN RECOVERY AT ZERO COUPLING, AND WHAT IT REQUIRES") print(f" {'state':>34} {'P(cross i), tau_0 uniform':>30} Born") for name, w, O in STATES: p = np.array([(U < w[i]).mean() for i in range(len(w))]) print(f" {name:>34} {str(np.round(p,4)):>30} {w}") assert np.abs(p - w).max() < 2e-3 print(" -> P(tau_0 < w_i) = w_i EXACTLY. The threshold rule reproduces the Born") print(" rule, provided tau_0(C) is uniformly distributed over configurations.") print(f"\n same states, tau_0 ~ Beta(2,2) instead (still supported on [0,1]):") for name, w, O in STATES[:2]: p = np.array([(B22 < w[i]).mean() for i in range(len(w))]) print(f" {name:>34} {str(np.round(p,4)):>30} {w}") print(" -> Born FAILS. The rule reads off the CDF of tau_0, so uniformity is not") print(" cosmetic: it is the condition under which the collapse rule agrees with") print(" the Born rule that sec:born-rule derives independently. UNSTATED.") # ---- CHECK 2, 3 ---------------------------------------------------------- print("\n(2,3) UNIQUENESS AGAINST COUPLING (tau_0 uniform)") for name, w, O in STATES: print(f"\n {name} Born = {w}") print(f" {'s':>6} {'P(0)':>8} {'P(1)':>8} {'P(>=2)':>8}") prev = None for s in [0.0, 0.05, 0.10, 0.20, 0.30, 0.40]: _, k = cross_sets(w, O, s, U) p2 = (k >= 2).mean() print(f" {s:>6.2f} {(k==0).mean():>8.4f} {(k==1).mean():>8.4f} {p2:>8.4f}") if prev is not None: assert p2 <= prev + 1e-6, "P(>=2) should be non-increasing in s" prev = p2 print("\n -> P(>=2 crossings) falls monotonically to zero as the coupling rises.") print(" The strong-coupling condition kappa lambda^2 >> tau_0 is doing exactly") print(" the work the proof of (U) claims for it.") # ---- CHECK 4 ------------------------------------------------------------- print("\n(4) WHERE (U) HOLDS, BORN DOES NOT") print(f" {'state':>34} {'s':>6} {'P(>=2)':>8} P(select i | unique) Born") for name, w, O in STATES: for s in [0.0, 0.30, 0.40]: cr, k = cross_sets(w, O, s, U) one = k == 1 if one.sum() < 1000: continue sel = cr[one].argmax(axis=1) f = np.bincount(sel, minlength=len(w)) / one.sum() print(f" {name:>34} {s:>6.2f} {(k>=2).mean():>8.4f} {str(np.round(f,3)):>20} {w}") print("\n -> conditioned on the crossing set being a singleton, the selected outcome") print(" is the maximum-margin one. For a fixed state that is the SAME outcome") print(" every run, so the conditional distribution is a point mass, not Born.") print(" Case B rules out a correlation artefact: its selected outcome is 3 at") print(" weak coupling and 1 at strong, so the effect tracks the margin, not the") print(" labelling.") # ---- CHECK 5 ------------------------------------------------------------- print("\n(5) SYMMETRIC SPECTRA") SYM = np.array([-1.0, 1.0]) for s in [0.0, 1.0, 100.0]: t = 0.3 + s * SYM ** 2 assert np.ptp(t) == 0.0 print(" for o = +-1 the term kappa(lambda o_i)^2 is outcome-independent at every") print(" coupling, so eq:tau-i-explicit separates nothing. The book's own") print(" degenerate-case clause routes this to einselection; recorded, not a defect.") print("\n" + "=" * 78) print("""SUMMARY 1. The threshold rule REPRODUCES THE BORN RULE exactly, at zero coupling, iff tau_0(C) is uniformly distributed over substrate configurations. This is a positive result about the construction and a genuine constraint on the law of tau_0. The book states neither. 2. The strong-coupling hypothesis of clause (U) is the correct one: raising kappa lambda^2 raises every threshold and drives the probability of multiple simultaneous crossings monotonically to zero. 3. The two pull apart. At the coupling where Born holds the crossing set is not a singleton (about 30% of runs here), so (U) fails. At the coupling where (U) holds the selection is the maximum-margin outcome, which for a fixed state is deterministic, so Born fails. eq:tau-i-explicit cannot deliver both at once. CONSEQUENCE FOR THE OPEN ITEM. "How weak a coupling still forces a crossing" is answerable -- P(cross) is computed above and moves smoothly with s -- but the answer is not the interesting quantity. The interesting quantity is the Born/(U) trade-off along the same axis, which is a consistency question inside the theory rather than a prediction to be measured. """)