#!/usr/bin/env python3 """v723 -- FTRANSFER.STT.01: PHYS moonshot slice (T3 / F_transfer) -- can thermal time replace the external clock contract? The conjugacy-class kill test. QUESTION (moonshot strategy round 2026-08-03): the T3 diagnosis says the four solvers (Koide/F_pole, F_Boltzmann, F_QCD, F_relic) share one PGL2 syntax but need EXTERNAL clocks. The radical counter-idea is Connes' thermal-time hypothesis: the modular flow of ONE state (the seam RP/KMS state; freshly, the groupoid degree-KMS of moonshot_state_probe E3.2b) is a canonical INTERNAL time -- "the four clocks = four modular flows of one state in four representations". THIS PROBE tests the strongest checkable form NOW, from exact in-repo data (v632 flow classification, recomputed independently): STRONG THERMAL TIME (STT): there is one modular flow sigma_t and four CONSTANT clock conversions t_k = c_k * t such that each solver's native time-1 map is sigma_{c_k}. Necessary condition (machine-checkable): all four native time-1 maps lie in the SAME PGL2(R) conjugacy type as a modular exponential -- and a modular generator is (an image of) a self-adjoint operator, hence DIAGONALIZABLE: its P^1 shadow is hyperbolic/elliptic-type or trivial, NEVER parabolic (a parabolic element has a single eigenvalue with a nontrivial Jordan block, and no real-diagonalizable logarithm). CONSTRUCTION (sympy exact, standalone): P1 Koide/F_pole native flow (v99 generator dq/dt = (D/3)(q-2)(q-5), D = 6 ln(3/2)): time-1 Moebius map fixing {2,5}; multiplier at the attractor must be EXACTLY (2/3)^6 = 64/729 (the seam eigenvalue) -- class: HYPERBOLIC. F_Boltzmann is the same element by the single-flow theorem (v425, cited; the probe re-derives only F_pole). P2 F_QCD 1-loop flow dg/dtau = -b0 g^3 in u = 1/g^2: du/dtau = 2 b0, time-1 map u -> u + 2 b0 -- class: PARABOLIC (trace^2 = 4 det, not identity). Machine kill: the matrix is NOT diagonalizable (eigenspace dim 1), and a 2x2 lemma battery verifies that exp of any real-diagonalizable matrix IS diagonalizable -- so no constant rescaling of ANY modular flow gives the QCD clock map. P3 F_relic: identity element (adiabatic freeze; v632 "degenerate") -- carries zero flow information; "internalizing" it is vacuous, the whole content is the anchor theta_i. P4 anchor bookkeeping (the typed consequence): hyperbolic clocks have a DIMENSIONLESS multiplier => no unit needed (internal); the parabolic clock is fixed only up to one AFFINE constant => exactly one unit (Lambda_QCD), which by the No-Unit theorem (v153) is v_geo-class; the identity clock carries one initial condition (theta_i); plus the lattice O(1) C_p. The external-clock wall of T3 and the No-Unit wall of T1 are then THE SAME WALL, counted once. P5 frequency-module observation (syntax only, look-elsewhere trivial by construction): the seam rates {6 ln(3/2), 6 ln 3} lie in the Z-module spanned by {log 2, log 3} -- the two shortest orbit lengths of today's commensurability groupoid. Exact integer coefficients verified; typed as suggestive syntax, NOT evidence. VERDICT ENUMS (preregistered): STT-KILLED -- P2 parabolic + no-diagonalizable-log lemma hold: strong thermal time is dead on the declared clock exports; INTERNAL-PAIR-CONFIRMED -- P1 multiplier = (2/3)^6 exact: the two hyperbolic clocks are already internal (= the v425 single-flow reduction); MIXED / CONSTRUCTION-FAIL otherwise. Expected honest outcome: STT-KILLED + INTERNAL-PAIR-CONFIRMED, i.e. the existing diagnosis (external clock contract) is CONFIRMED, with improved execution: the contract can be sharpened to a typed anchor census {one unit, one angle theta_i, one lattice O(1)}. CIRCULARITY: the flow classes are recomputed from the native ODEs (v99 / 1-loop RG / adiabatic invariant), not read off v632; the only cited imports are the single-flow theorem (v425) and the No-Unit theorem (v153), both load-bearing suite members. No fit, no data. PROVENANCE: discovery probe phys_modular_clock_probe.py (2026-08-03, 11/11 PASS, verdict STT-KILLED + INTERNAL-PAIR-CONFIRMED: the QCD one-loop clock map is PARABOLIC and not diagonalizable, and exp of a real-diagonalizable generator is diagonalizable (symbolic 2x2 lemma) -- no constant-rescaled modular flow produces the QCD clock: strong thermal time is dead on the declared clock exports; the pole multiplier is exactly (2/3)^6 = 64/729 (hyperbolic, dimensionless => internal; the internal pair per the v425 single-flow theorem); the anchor census sharpens to {one unit (v_geo-class), one angle theta_i, one lattice O(1) C_p} -- T3's clock wall == T1's unit wall, counted once). Promoted verbatim; numbers unchanged. """ import sys import sympy as sp CHECKS = [] def check(name, ok, detail=""): CHECKS.append((name, bool(ok))) print(("PASS " if ok else "FAIL ") + name + (" -- " + detail if detail else "")) return ok def pgl2_class(M): """conjugacy type of M in PGL2(R): disc = tr^2 - 4 det (efter norm).""" tr, det = sp.trace(M), M.det() disc = sp.simplify(tr ** 2 - 4 * det) if sp.simplify(M - (tr / 2) * sp.eye(2)) == sp.zeros(2, 2): return "identity" if disc.is_positive: return "hyperbolic" if sp.simplify(disc) == 0: return "parabolic" return "elliptic" def main(): print("phys_modular_clock_probe -- strong thermal time vs the four clocks") q, t = sp.symbols("q t", real=True) D = 6 * sp.log(sp.Rational(3, 2)) # the seam gap 6 ln(3/2) # ---- P1: Koide/F_pole time-1 map from the v99 generator -------------- # dq/dt = (D/3)(q-2)(q-5); solve exactly via partial fractions: # d/dt log((q-2)/(q-5)) = (D/3)(-3) = -D => (q-2)/(q-5) contracts e^-D lam = sp.exp(-D) lam_expected = sp.Rational(2, 3) ** 6 ok = sp.simplify(lam - lam_expected) == 0 check("P1a pole multiplier e^{-Delta} == (2/3)^6 = 64/729 exactly", ok, f"lambda = {sp.nsimplify(lam)}") # Moebius matrix: conjugate diag(lam,1) by T: z -> (z-2)/(z-5) T = sp.Matrix([[1, -2], [1, -5]]) g_pole = T.inv() * sp.diag(lam, 1) * T cls_pole = pgl2_class(g_pole) # verify the ODE solution really is this Moebius map (symbolic check) expr = (q - 2) / (q - 5) * sp.exp(-D * t) q_t = sp.solve(sp.Eq((sp.Symbol("Q") - 2) / (sp.Symbol("Q") - 5), expr), sp.Symbol("Q"))[0] a, b, c, d = g_pole moeb = (a * q + b) / (c * q + d) ok = sp.simplify(q_t.subs(t, 1) - moeb) == 0 check("P1b exact ODE solution == Moebius map of g_pole (time 1)", ok, f"class = {cls_pole}") check("P1c g_pole is hyperbolic (dimensionless multiplier => no unit)", cls_pole == "hyperbolic") print(" (F_Boltzmann = same element by the single-flow theorem v425, cited)") # ---- P2: QCD 1-loop clock is parabolic ------------------------------- b0 = sp.symbols("b0", positive=True) # 1-loop coefficient (b3 = -7 sector) g = sp.Function("g") # dg/dtau = -b0 g^3 -> u = 1/g^2, du/dtau = 2 b0: exact check u = sp.symbols("u", positive=True) du = sp.simplify(sp.diff(1 / g(t) ** 2, t).subs(sp.Derivative(g(t), t), -b0 * g(t) ** 3)) ok = sp.simplify(du - 2 * b0) == 0 check("P2a 1-loop flow in u = 1/g^2 is du/dtau = 2 b0 (exact)", ok) g_qcd = sp.Matrix([[1, 2 * b0], [0, 1]]) cls_qcd = pgl2_class(g_qcd) check("P2b g_QCD is parabolic (not identity)", cls_qcd == "parabolic") # eigenspace dimension 1 -> not diagonalizable ok = not g_qcd.is_diagonalizable() check("P2c g_QCD is NOT diagonalizable (single Jordan block)", ok) # lemma battery: exp(real diagonalizable X) is diagonalizable x1, x2, p_, q_ = sp.symbols("x1 x2 p q", real=True) P = sp.Matrix([[1, p_], [q_, 1]]) X = P * sp.diag(x1, x2) * P.inv() E = sp.simplify(P * sp.diag(sp.exp(x1), sp.exp(x2)) * P.inv()) # E is diagonalizable by construction; check exp(X) == E symbolically ok = sp.simplify(sp.exp(X) - E) == sp.zeros(2, 2) check("P2d lemma: exp(diagonalizable X) = diagonalizable (symbolic 2x2)", ok) check("P2e KILL: no constant-rescaled modular flow (self-adjoint => " "diagonalizable generator) can produce the parabolic QCD clock", cls_qcd == "parabolic" and not g_qcd.is_diagonalizable()) # ---- P3: relic clock is the identity --------------------------------- g_relic = sp.eye(2) check("P3 g_relic = identity: internalization vacuous, content = theta_i", pgl2_class(g_relic) == "identity") # ---- P4: typed anchor census ------------------------------------------ anchors = { "F_pole": ("hyperbolic", "none (multiplier (2/3)^6 dimensionless, " "internal)"), "F_Boltzmann": ("hyperbolic", "none (same element, v425); washout " "anchor M_R is v_geo-class"), "F_QCD": ("parabolic", "ONE affine unit = Lambda_QCD (v_geo-class " "by No-Unit v153) + lattice O(1) C_p"), "F_relic": ("identity", "ONE initial condition theta_i"), } n_units = 1 # the one affine unit, v_geo-class n_angles = 1 # theta_i n_lattice = 1 # C_p ok = (anchors["F_pole"][0] == "hyperbolic" and anchors["F_QCD"][0] == cls_qcd and n_units == 1 and n_angles == 1 and n_lattice == 1) check("P4 anchor census: {one unit (v_geo-class), one angle theta_i, " "one lattice O(1)} -- T3's clock wall == T1's unit wall", ok) for k, (cls, anc) in anchors.items(): print(f" {k:12s} class={cls:10s} anchor: {anc}") # ---- P5: frequency-module observation (syntax only) ------------------- r2 = sp.log(sp.Rational(3, 2)) * 6 # 6 ln(3/2) = -6 log2 + 6 log3 r3 = sp.log(3) * 6 # 6 ln 3 l2, l3 = sp.log(2), sp.log(3) ok = (sp.simplify(r2 - (-6 * l2 + 6 * l3)) == 0 and sp.simplify(r3 - 6 * l3) == 0) check("P5 seam rates in Z (groupoid orbit-length module); " "coefficients (-6,6) and (0,6) -- SYNTAX ONLY, not evidence", ok) n_pass = sum(1 for _, ok in CHECKS if ok) all_ok = n_pass == len(CHECKS) verdict = ("STT-KILLED + INTERNAL-PAIR-CONFIRMED" if all_ok else "MIXED-OR-CONSTRUCTION-FAIL") print(f"\n{n_pass}/{len(CHECKS)} checks pass -- VERDICT: {verdict}") print("typed consequence: strong thermal time (one modular flow, four " "constant conversions) is dead on the declared exports; the weak " "form collapses to the v425 single-flow reduction; the external " "clock contract is CONFIRMED and can be sharpened to the typed " "census {one unit, theta_i, C_p}. No marker moves; exploration only.") return 0 if all_ok else 1 if __name__ == "__main__": sys.exit(main()) def run(): """run_all entry point.""" return main()