"""v99 -- Koide flow time: canonical generator, m_tau data-fragility, and the conditional discrete-step prediction. [I] generator + [N] sensitivity + [P] conditional m_tau values. Corrects the ROBUSTNESS of one v93 negative. v93 (FR.KOIDE.05) recorded the honest negative "flow time t = 2.838 is not an integer => literal discrete iteration pole = F^n(source) EXCLUDED; the missing P2 object is a continuous-time generator". This script does three things to that statement: [I] 1. THE CONTINUOUS GENERATOR HAS A CANONICAL FORM. The v82 flow rho -> mu^t rho is generated by the autonomous ODE dq/dt = (Delta/N_fam) * (q-2)(q-5) = (Delta/N_fam) * det B(q), with Delta = 6 log(3/2) the established transfer gap and (q-2)(q-5) = det B(K+xQ) in the branch coordinate q = -3x. The time-1 map of this flow IS the v82 Moebius map F (machine- checked). So the missing P2 object is precisely a QFT mechanism producing "gap x anchor-block quadric / family count" -- the form is canonical, only its physical derivation is open. [N] 2. THE NON-INTEGRALITY OF t IS DATA-FRAGILE (correction of v93's robustness, not of its arithmetic). rho_pole = -2.2e-6 sits so close to the branch point that t is hypersensitive to m_tau (PDG 1776.93 +- 0.09 MeV): t(-1 sigma) = 2.35, t(central) = 2.84, and Q crosses 2/3 INSIDE the error band (at about +0.46 sigma), where t -> infinity. Within +0.5 sigma, t passes through EVERY integer >= 3. The v93 exclusion of pole = F^n(source) therefore holds only at the central value -- it is NOT robust. [P] 3. CONDITIONAL DISCRETE-STEP PREDICTIONS (registry untouched; these are typed conditional, NOT predictions of record): pole = F^2(source): m_tau = 1776.6690 MeV (-2.9 sigma, EXCLUDED by current data [N]); pole = F^3(source): m_tau = 1776.9427 MeV (+0.14 sigma) -- n = 3 = N_fam steps, one per family; pole = F^4(source): m_tau = 1776.9667 MeV (+0.41 sigma); Q = 2/3 exactly: m_tau = 1776.9690 MeV (+0.43 sigma). Current data cannot separate n = 3 / n = 4 / exact-branch; the decision needs sigma(m_tau) ~ 0.01 MeV (Belle II). The n = 3 value is stable under the source variant (exact ladder vs the -phi0/24 conjecture: shift 0.0002 MeV). [I] 4. LOOK-ELSEWHERE FIREWALL (documented destroyed candidate): a scale reading t = ln(M_scal/m_tau)/ln(W) matched W = 46080 = |W(D5)|x |W(A3)| to 1e-4 -- but the SAME hypersensitivity makes the self-consistent m_tau land within 0.1 sigma for ANY W in a wide range (negative controls W = 40320, 51840 both give pulls < 0.04 sigma). The t-scale reading carries no information at current m_tau precision; the candidate is REJECTED and recorded so it is not re-fished. STATUS: P2 unchanged in type ([P], one identification) but sharper on both ends: the generator form is canonical [I], and the discrete/continuous dichotomy is now an EXPERIMENTAL question (m_tau at 0.01 MeV) rather than a settled negative. """ import mpmath as mp from tfpt_constants import check, summary, reset, phi0, N_fam, c3, Mbar # PDG 2024 pole masses (MeV) M_E, M_MU = mp.mpf('0.51099895069'), mp.mpf('105.6583755') MTAU, SIG_MTAU = mp.mpf('1776.93'), mp.mpf('0.09') MU_RATE = (mp.mpf(2) / 3)**6 DELTA = 6 * mp.log(mp.mpf(3) / 2) def koide(a, b, c): return (a + b + c) / (mp.sqrt(a) + mp.sqrt(b) + mp.sqrt(c))**2 def rho(q): return (q - 2) / (5 - q) def F(q): return 2 * (569 * q - 3325) / (665 * q - 3517) def q_of_rho(r): return (2 + 5 * r) / (1 + r) def source_rho(): me = mp.mpf(16) / 7 * phi0**5 mmu = mp.mpf(4) / 3 * phi0**3 mtau = mp.mpf(7) / 6 * phi0**2 return rho(N_fam * koide(me, mmu, mtau)) def t_flow(mtau): return mp.log(rho(N_fam * koide(M_E, M_MU, mtau)) / source_rho()) \ / mp.log(MU_RATE) def mtau_for_rho(r_target): q_target = q_of_rho(r_target) / N_fam return mp.findroot(lambda m: koide(M_E, M_MU, m) - q_target, mp.mpf('1776.94')) def run(): reset() mp.mp.dps = 40 print("v99 Koide flow time (generator, m_tau fragility, conditional steps)") # 1. canonical generator: dq/dt = (Delta/N_fam)(q-2)(q-5); time-1 map = F def flow_time1(q0): f = lambda t_, q: [(DELTA / N_fam) * (q[0] - 2) * (q[0] - 5)] return mp.odefun(f, 0, [q0])(1)[0] probes = [mp.mpf('1.2'), mp.mpf('1.9939'), mp.mpf('2.7')] check("CANONICAL GENERATOR [I]: the time-1 map of dq/dt = " "(Delta/N_fam)(q-2)(q-5) = (Delta/N_fam) det B(q) IS the v82 " "Moebius map F (3 probe points, 1e-12)", all(abs(flow_time1(q0) - F(q0)) < mp.mpf('1e-12') for q0 in probes)) check("(equivalently: rho(t) = e^{-Delta t} rho(0); e^{-Delta} = " "(2/3)^6 = mu exactly)", abs(mp.e**(-DELTA) - MU_RATE) < mp.mpf('1e-35')) # 2. data fragility of the flow time check("flow time at the PDG central value: t = 2.8385 (v93 import)", t_flow(MTAU), mp.mpf('2.838456'), tol=mp.mpf('1e-5')) check("DATA-FRAGILE [N]: t(-1 sigma) = 2.347, t(-0.5 sigma) = 2.523 " "-- t moves by ~0.5 steps within one sigma of m_tau", abs(t_flow(MTAU - SIG_MTAU) - mp.mpf('2.34693')) < mp.mpf('1e-4') and abs(t_flow(MTAU - SIG_MTAU / 2) - mp.mpf('2.52323')) < mp.mpf('1e-4')) m_branch = mtau_for_rho(mp.mpf(0)) pull_branch = (m_branch - MTAU) / SIG_MTAU check("Q crosses 2/3 INSIDE the error band: at m_tau = 1776.9690 MeV " "= +0.43 sigma (there t -> infinity; within +0.5 sigma t passes " "every integer >= 3) => the v93 exclusion of pole = F^n(source) " "is NOT robust", pull_branch, mp.mpf('0.4337'), tol=mp.mpf('1e-3')) # 3. conditional discrete-step values (typed [P]; registry untouched) r_src = source_rho() preds = {n: mtau_for_rho(r_src * MU_RATE**n) for n in (2, 3, 4)} pulls = {n: (preds[n] - MTAU) / SIG_MTAU for n in preds} check("pole = F^2(source) EXCLUDED [N]: m_tau = 1776.6690 MeV = " "-2.9 sigma", pulls[2], mp.mpf('-2.900'), tol=mp.mpf('1e-2')) check("CONDITIONAL [P]: pole = F^3(source) (n = 3 = N_fam steps, one " "per family) predicts m_tau = 1776.9427 MeV (+0.14 sigma)", preds[3], mp.mpf('1776.94268'), tol=mp.mpf('1e-7')) check("neighbours not separable yet: n = 4 gives 1776.9667 (+0.41 " "sigma), exact branch 1776.9690 (+0.43 sigma); decision needs " "sigma(m_tau) ~ 0.01 MeV (Belle II) -- the kill criterion", abs(pulls[4] - mp.mpf('0.408')) < mp.mpf('1e-2') and abs(preds[4] - m_branch) < mp.mpf('0.01') * 3) me_alt = mtau_for_rho(-phi0 / 24 * MU_RATE**3) check("n = 3 value is source-variant stable: -phi0/24 source shifts it " "by only 0.0002 MeV", abs(me_alt - preds[3]) < mp.mpf('0.001')) # 4. look-elsewhere firewall Mscal = c3**mp.mpf('3.5') * Mbar # GeV def mtau_scale_reading(W): f = lambda m: t_flow(m) - mp.log(Mscal / (m / 1000)) / mp.log(W) return mp.findroot(f, mp.mpf('1776.93')) pulls_W = {W: (mtau_scale_reading(mp.mpf(W)) - MTAU) / SIG_MTAU for W in (46080, 40320, 51840)} check("FIREWALL [I/N]: the scale reading t = ln(M_scal/m_tau)/ln(W) is " "destroyed by negative controls -- W = 46080, 40320, 51840 ALL " "give |pull| < 0.1 sigma (the t-reading carries no information " "at current m_tau precision; candidate REJECTED, not re-fished)", all(abs(mp.re(p)) < mp.mpf('0.1') for p in pulls_W.values())) check("STATUS: P2 type unchanged ([P], one identification); generator " "form canonical [I]; discrete-vs-continuous now an EXPERIMENTAL " "question (m_tau at 0.01 MeV), not a settled negative; freeze " "registry untouched", True) return summary("v99 Koide flow time") if __name__ == "__main__": raise SystemExit(1 if run() else 0)