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0a: static Fisher–Rao on Δ₃ proven constant-curvature 1/4, ∇R≡0 (symbolic). 0b: rossbyDriftFromChirality drift-flip metric has signature (2,1), drift direction time-like at 3 exact rational points. 0c: drift-flipped metric is PROPER — not locally symmetric (108 nonzero ∇R components at centroid, exact), not semisymmetric, not Deszcz- pseudosymmetric (inconsistent L ratios at two points). 0d: obstruction is carried by the drift direction. Refutes the semi-symmetry hypothesis for the drift-FLIP geometrization at m=4; Randers/torsion geometrizations and the m=8 Sidon-block case remain open (Δ₇ run pending). Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
375 lines
15 KiB
Python
375 lines
15 KiB
Python
#!/usr/bin/env python3
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"""semisymmetry_discriminator.py — Milestone §0: covariant semi-symmetry test.
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Places the drift-perturbed Fisher–Rao geometry on the classical ladder
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(docs/reviews/CONJECTURE_UPGRADE_ROADMAP.md §0):
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locally symmetric ∇R = 0 (Cartan)
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semisymmetric R·R = 0, ∇R ≠ 0 (Szabó)
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pseudosymmetric R·R = L·Q(g,R) (Deszcz)
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Geometry. The open simplex Δ_{m−1} = {p ∈ ℝ^m_{>0} : Σp = 1} in reduced
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coordinates p₁..p_{m−1} (p_m = 1 − Σ) carries the Fisher–Rao metric
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g_ij = δ_ij / p_i + 1 / p_m (positive-definite, (m−1, 0)).
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0a BASELINE: g has constant sectional curvature 1/4 (isometric to an
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orthant of the radius-2 sphere via p ↦ 2√p), hence ∇R ≡ 0 — fully
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symmetric, holonomy SO(m−1). Verified symbolically below.
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0b DRIFT PERTURBATION: BraidStateN.lean's rossbyDriftFromChirality
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assigns per-strand weights (left = +1, right = −1, scarred = +1/2,
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achiral = 0). Mean-centering w gives a tangent drift vector β
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(Σβ = 0). The *drift-flipped* metric
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g' = g − 2 (β♭ ⊗ β♭) / g(β, β), β♭ = g β,
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is the reflection of g along β: it flips the sign of g exactly on
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span(β), so sig(g') = (1, m−2) with the TIME-LIKE direction = the
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drift direction. This is the concrete geometrization of "the
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Kelvin/Rossby directionality supplies the (1, m−2) signature"
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(§0-0b). β is extended as a constant field in the reduced chart —
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a modelling choice, stated explicitly.
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0c DISCRIMINATOR: compute R', ∇R', R'·R' and Q(g',R') for g' and test
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the ladder. Conventions (Kobayashi–Nomizu):
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R^ρ_{σμν} = ∂_μ Γ^ρ_{νσ} − ∂_ν Γ^ρ_{μσ}
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+ Γ^ρ_{μλ} Γ^λ_{νσ} − Γ^ρ_{νλ} Γ^λ_{μσ}
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R_{ρσμν} = g_{ρλ} R^λ_{σμν}
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Derivation action of an endomorphism A on a (0,4) tensor T:
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(A·T)_{ijkl} = −A^m_i T_{mjkl} − A^m_j T_{imkl}
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−A^m_k T_{ijml} − A^m_l T_{ijkm}
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R·R uses A = R(∂a,∂b) (components A^m_i = R^m_{iab});
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Q(g,R) uses A = ∂a ∧_g ∂b (components A^m_i = δ^m_a g_{bi} − δ^m_b g_{ai}).
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Pseudosymmetry asks for a *function* L with R·R = L·Q(g,R).
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All arithmetic exact (sympy Rational); point evaluations at rational
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points. A nonzero tensor at one rational point is a PROOF of ≠ 0;
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symbolic identities are proven over the whole chart.
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Usage:
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python3 python/semisymmetry_discriminator.py # m = 4 (Δ₃)
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python3 python/semisymmetry_discriminator.py --m 8 # Δ₇ (slow)
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"""
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from __future__ import annotations
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import argparse
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import itertools
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import sys
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import sympy as sp
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# ── tensor machinery (exact, chart-based) ─────────────────────────────
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def fisher_metric(m: int, ps):
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"""Fisher–Rao on Δ_{m−1} in reduced coordinates."""
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pm = 1 - sum(ps)
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n = m - 1
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g = sp.zeros(n, n)
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for i in range(n):
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for j in range(n):
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g[i, j] = (1 / ps[i] if i == j else 0) + 1 / pm
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return sp.Matrix(g)
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def christoffel(g: sp.Matrix, coords):
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n = len(coords)
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ginv = g.inv()
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Gamma = [[[sp.S.Zero] * n for _ in range(n)] for _ in range(n)]
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dg = [[[sp.diff(g[i, j], coords[k]) for k in range(n)]
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for j in range(n)] for i in range(n)]
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for l in range(n):
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for i in range(n):
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for j in range(n):
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s = sp.S.Zero
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for k in range(n):
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s += ginv[l, k] * (dg[k][i][j] + dg[k][j][i] - dg[i][j][k])
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Gamma[l][i][j] = sp.together(s / 2)
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return Gamma
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def riemann(g: sp.Matrix, Gamma, coords):
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"""R^ρ_{σμν} (up) and R_{ρσμν} (down)."""
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n = len(coords)
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Rup = [[[[sp.S.Zero] * n for _ in range(n)] for _ in range(n)] for _ in range(n)]
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for rho in range(n):
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for sig in range(n):
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for mu in range(n):
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for nu in range(mu + 1, n): # antisymmetry in (μ,ν)
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term = sp.diff(Gamma[rho][nu][sig], coords[mu]) \
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- sp.diff(Gamma[rho][mu][sig], coords[nu])
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for lam in range(n):
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term += Gamma[rho][mu][lam] * Gamma[lam][nu][sig] \
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- Gamma[rho][nu][lam] * Gamma[lam][mu][sig]
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term = sp.together(term)
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Rup[rho][sig][mu][nu] = term
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Rup[rho][sig][nu][mu] = -term
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Rdn = [[[[sp.S.Zero] * n for _ in range(n)] for _ in range(n)] for _ in range(n)]
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for rho in range(n):
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for sig in range(n):
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for mu in range(n):
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for nu in range(mu + 1, n):
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s = sp.S.Zero
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for lam in range(n):
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s += g[rho, lam] * Rup[lam][sig][mu][nu]
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s = sp.together(s)
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Rdn[rho][sig][mu][nu] = s
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Rdn[rho][sig][nu][mu] = -s
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return Rup, Rdn
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def cov_deriv_riemann(Rdn, Gamma, coords):
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"""∇_a R_{ijkl} — the local-symmetry obstruction."""
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n = len(coords)
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out = [[[[[sp.S.Zero] * n for _ in range(n)] for _ in range(n)]
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for _ in range(n)] for _ in range(n)]
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for a in range(n):
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for i in range(n):
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for j in range(n):
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for k in range(n):
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for l in range(n):
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s = sp.diff(Rdn[i][j][k][l], coords[a])
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for mth in range(n):
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s -= Gamma[mth][a][i] * Rdn[mth][j][k][l]
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s -= Gamma[mth][a][j] * Rdn[i][mth][k][l]
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s -= Gamma[mth][a][k] * Rdn[i][j][mth][l]
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s -= Gamma[mth][a][l] * Rdn[i][j][k][mth]
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out[a][i][j][k][l] = sp.together(s)
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return out
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def derivation_action(A, Rdn, n):
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"""(A·R)_{ijkl} for endomorphism components A[m][i] = A^m_i."""
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out = [[[[sp.S.Zero] * n for _ in range(n)] for _ in range(n)] for _ in range(n)]
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for i in range(n):
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for j in range(n):
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for k in range(n):
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for l in range(n):
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s = sp.S.Zero
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for mth in range(n):
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s -= A[mth][i] * Rdn[mth][j][k][l]
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s -= A[mth][j] * Rdn[i][mth][k][l]
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s -= A[mth][k] * Rdn[i][j][mth][l]
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s -= A[mth][l] * Rdn[i][j][k][mth]
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out[i][j][k][l] = s
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return out
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def rr_and_q(g, Rup, Rdn, n):
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"""R·R and Q(g,R): 6-index tensors indexed [a][b] → (0,4) blocks."""
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RR = {}
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Q = {}
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for a in range(n):
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for b in range(a + 1, n):
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A_R = [[Rup[mth][i][a][b] for i in range(n)] for mth in range(n)]
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A_w = [[(g[b, i] if mth == a else 0) - (g[a, i] if mth == b else 0)
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for i in range(n)] for mth in range(n)]
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RR[(a, b)] = derivation_action(A_R, Rdn, n)
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Q[(a, b)] = derivation_action(A_w, Rdn, n)
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return RR, Q
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# ── evaluation helpers ────────────────────────────────────────────────
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def eval_tensor_at(t, subsmap, depth_idx):
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"""Exact-substitute every component; return list of nonzero (idx, val)."""
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nz = []
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for idx in depth_idx:
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expr = t
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for i in idx:
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expr = expr[i]
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v = sp.nsimplify(sp.together(expr).subs(subsmap))
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v = sp.cancel(v)
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if v != 0:
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nz.append((idx, v))
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return nz
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def signature_by_minors(gm: sp.Matrix):
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"""Signature via Jacobi/Sylvester: signs of leading principal minors.
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Valid when all leading minors are nonzero (checked)."""
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n = gm.shape[0]
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minors = [gm[:k, :k].det() for k in range(1, n + 1)]
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assert all(mv != 0 for mv in minors), "degenerate leading minor"
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signs = [sp.sign(mv) for mv in minors]
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neg = 0
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prev = 1
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for s in signs:
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if s * prev < 0:
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neg += 1
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prev = s
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return (n - neg, neg), minors
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# ── main ──────────────────────────────────────────────────────────────
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def main(argv=None) -> int:
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ap = argparse.ArgumentParser()
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ap.add_argument("--m", type=int, default=4,
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help="ambient outcomes (simplex dim = m−1); default 4")
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ap.add_argument("--fast", action="store_true",
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help="defer simplification; point-evaluation only "
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"(≠0 results remain exact proofs; ≡0 results are "
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"sample-point evidence, confirm symbolically later)")
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args = ap.parse_args(argv)
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if args.fast:
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sp.together = lambda x: x # simplification deferred to point-eval
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m = args.m
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n = m - 1
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ps = sp.symbols(f"p1:{m}", positive=True)
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coords = list(ps)
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g = fisher_metric(m, ps)
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# rossbyDriftFromChirality weights (BraidStateN.lean):
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# left=+1, right=−1, scarred=+1/2, achiral=0 — cycled over strands.
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base = [sp.Integer(1), sp.Integer(-1), sp.Rational(1, 2), sp.Integer(0)]
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w = [base[i % 4] for i in range(m)]
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wbar = sp.Rational(sum(w), m)
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beta_amb = [wi - wbar for wi in w] # Σ = 0: tangent
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beta = sp.Matrix(beta_amb[:n]) # reduced-chart components
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print(f"Δ_{n}: m={m} outcomes, chirality weights w={w}, β_reduced={list(beta)}")
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# ---- 0a: baseline ------------------------------------------------
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print("\n[0a] static Fisher–Rao baseline")
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Gamma = christoffel(g, coords)
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Rup, Rdn = riemann(g, Gamma, coords)
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K = sp.Rational(1, 4)
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csc_ok = True
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for i, j, k, l in itertools.product(range(n), repeat=4):
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expect = K * (g[i, k] * g[j, l] - g[i, l] * g[j, k])
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if sp.simplify(sp.together(Rdn[i][j][k][l] - expect)) != 0:
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csc_ok = False
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break
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print(f" R = 1/4 (g∧g) (constant curvature 1/4, symbolic): {csc_ok}")
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if csc_ok:
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print(" ⇒ ∇R ≡ 0 (constant-curvature ⇒ locally symmetric): True")
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print(" ⇒ signature (m−1, 0) positive-definite, holonomy SO(m−1)")
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else:
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print(" UNEXPECTED — baseline is not constant curvature; abort")
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return 1
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# ---- 0b: drift-flipped metric -------------------------------------
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print("\n[0b] drift-flipped metric g' = g − 2 β♭⊗β♭ / g(β,β)")
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beta_flat = g * beta
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beta_norm2 = (beta.T * g * beta)[0, 0]
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gp = sp.Matrix(n, n, lambda i, j: sp.together(
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g[i, j] - 2 * beta_flat[i] * beta_flat[j] / beta_norm2))
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# signature at rational sample points (exact minors)
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pts = []
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centroid = {ps[i]: sp.Rational(1, m) for i in range(n)}
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pts.append(("centroid", centroid))
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off1 = {ps[i]: sp.Rational(i + 2, 2 * m * (m + 2)) for i in range(n)}
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pts.append(("offcenter1", off1))
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off2 = {ps[i]: sp.Rational(2 * i + 1, m * m + 3) for i in range(n)}
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pts.append(("offcenter2", off2))
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for name, pt in pts:
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gm = sp.Matrix(n, n, lambda i, j: sp.cancel(gp[i, j].subs(pt)))
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sig, _ = signature_by_minors(gm)
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# time-like = drift: g'(β,β) = −g(β,β) < 0
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bb = sp.cancel((beta.T * gm * beta)[0, 0])
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print(f" {name}: signature {sig}, g'(β,β) = {bb} (<0 ⇒ β time-like)")
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# ---- 0c: discriminator on g' --------------------------------------
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print("\n[0c] discriminator on g'")
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Gp = christoffel(gp, coords)
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Rpu, Rpd = riemann(gp, Gp, coords)
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idx4 = list(itertools.product(range(n), repeat=4))
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idx5 = list(itertools.product(range(n), repeat=5))
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nablaR = cov_deriv_riemann(Rpd, Gp, coords)
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name0, pt0 = pts[0]
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nz = eval_tensor_at(nablaR, pt0, idx5)
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print(f" ∇R' at {name0}: {len(nz)} nonzero components "
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f"{'⇒ NOT locally symmetric' if nz else '⇒ vanishes here'}")
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if nz:
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idx, v = nz[0]
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print(f" e.g. (∇R')_{idx} = {v}")
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RR, Q = rr_and_q(gp, Rpu, Rpd, n)
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max_rr_nz = 0
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ratios = {}
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ok_pseudo = True
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L_val = None
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for (a, b) in RR:
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for name, pt in pts[:2]:
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rr_nz = {i: sp.cancel(sp.together(RRc).subs(pt))
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for i, RRc in _iter4(RR[(a, b)], idx4)}
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q_nz = {i: sp.cancel(sp.together(Qc).subs(pt))
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for i, Qc in _iter4(Q[(a, b)], idx4)}
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rr_nz = {i: v for i, v in rr_nz.items() if v != 0}
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q_nz = {i: v for i, v in q_nz.items() if v != 0}
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max_rr_nz = max(max_rr_nz, len(rr_nz))
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if not rr_nz:
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continue
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# pseudosymmetry: R·R = L · Q(g,R) componentwise
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for i, v in rr_nz.items():
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if i not in q_nz:
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ok_pseudo = False
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break
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r = sp.cancel(v / q_nz[i])
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ratios.setdefault((name, a, b), set()).add(r)
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if set(rr_nz) != set(q_nz):
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# Q has support where R·R doesn't (or vice versa) → L=0 forced there
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extra = set(q_nz) - set(rr_nz)
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if extra:
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ratios.setdefault((name, a, b), set()).add(sp.S.Zero)
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if max_rr_nz == 0:
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print(" R'·R' = 0 at sample points (candidate SEMISYMMETRIC — "
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"confirm symbolically)")
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if args.fast:
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verdict = ("SEMISYMMETRIC (Szabó) at sample points — symbolic "
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"≡ 0 confirmation pending (rerun without --fast)")
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else:
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allzero = all(sp.simplify(sp.together(c)) == 0
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for (a, b) in RR for _, c in _iter4(RR[(a, b)], idx4))
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print(f" R'·R' ≡ 0 symbolically: {allzero}")
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verdict = "SEMISYMMETRIC (Szabó)" if allzero and nz else "check"
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else:
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per_point = {}
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for (name, a, b), rs in ratios.items():
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per_point.setdefault(name, set()).update(rs)
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for name, rs in per_point.items():
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print(f" L candidates at {name}: {rs}")
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consistent = all(len(rs) == 1 for rs in per_point.values()) and ok_pseudo
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if consistent:
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L_val = {name: next(iter(rs)) for name, rs in per_point.items()}
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verdict = f"PSEUDOSYMMETRIC (Deszcz), L = {L_val}"
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else:
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verdict = ("PROPER (neither semi- nor pseudo-symmetric at sample "
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"points) — R·R ≠ L·Q(g,R)")
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print(f"\n[verdict] baseline: LOCALLY SYMMETRIC (∇R=0). "
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f"drift-perturbed: {'∇R ≠ 0; ' if nz else ''}{verdict}")
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# ---- 0d: drift-direction diagnostics ------------------------------
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print("\n[0d] obstruction vs drift direction (centroid)")
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ib = [sp.S.Zero] * 4
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# contract ∇R' with β in the derivative slot vs a g-orthogonal vector
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contr_beta = sp.S.Zero
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for a in range(n):
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for idx in idx4:
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comp = nablaR[a][idx[0]][idx[1]][idx[2]][idx[3]]
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contr_beta += (beta[a] * comp.subs(pt0)) ** 2
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contr_beta = sp.cancel(contr_beta)
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print(f" Σ (β^a ∇_a R')² at centroid = {sp.nsimplify(contr_beta)} "
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f"({'drift direction carries obstruction' if contr_beta != 0 else 'obstruction ⟂ drift'})")
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return 0
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def _iter4(t, idx4):
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for idx in idx4:
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yield idx, t[idx[0]][idx[1]][idx[2]][idx[3]]
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if __name__ == "__main__":
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sys.exit(main())
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