"""v425 -- DYN.TRANSFER.UNIVERSAL.01: the single-flow reduction of F_transfer. The four frontier transfers (F_pole, F_Boltzmann, F_relic, F_QCD) do NOT each need a separate external solver: their DYNAMICS is the ONE native flow of the theory -- the seam's modular/recovery semigroup (v238/v240, gap 6 ln(3/2)) -- restricted to each sector. Native flow + EXTERNAL ANCHORS, firewall-clean: the observable stays [C] until the dimensionful anchor is supplied; only the anchors (v_geo, C_p, M_R, the cosmological IC theta_i) are external, NOT the dynamics. This extends v383 (the STATIC universal gap) to a DYNAMIC universal flow. [E] the flow identities; [C] the reduction verdict; the anchors stay [O]. [E] 1. THE ONE NATIVE FLOW. The recovery transfer has spectrum {1,(2/3)^6,(1/3)^6}; the OS Hamiltonian H_OS=-log T has gap Delta=6 ln(3/2)>0 with e^{-Delta}=(2/3)^6 EXACTLY (v240); an explicit 2-state column-stochastic recovery T realises it and L=log T is a genuine Markov generator (off-diagonals >=0, columns sum to 0) -- the native continuous flow (v238). [E] 2. F_pole IS THIS FLOW (lepton sector). The v99 Koide generator dq/dt=(Delta/N_fam)(q-2)(q-5) linearised at the attractor q=2 has rate g'(2)=-Delta, so the time-1 contraction is e^{-Delta}=(2/3)^6 -- the SAME recovery eigenvalue; q=5 is the unstable root (g'(5)=+Delta). The flow RATE is native; the external piece is only the IR pole scale (v_geo). [E] 3. F_Boltzmann washout IS THIS CONTRACTION. The washout kappa_f in (0,1) is bounded by the native sub-unit eigenvalue (2/3)^6, and the CP phase delta_PMNS=4pi/3 (240 deg) is native (v320); the external piece is only the absolute seesaw scale M_R (v_geo-class). [E] 4. F_QCD rate IS NATIVE. b3=-7 carrier (v159); b0(n_f=6)=11-2*6/3=7=-b3, so the dimensional-transmutation exponent -2pi/(b0 alpha_s) is native; the external pieces are alpha_s(M_Z) and the lattice O(1) C_p (QCD.LAMBDA.01). [C] 5. VERDICT. The transfer DYNAMICS is one native object (the v238 recovery semigroup, v383 universal); only the ANCHORS stay external. F_relic is the one genuine wall: its free input is a cosmological initial condition theta_i, not theory-derivable (best case: mu4-symmetry-reduced). A single-flow reduction, NOT internalisation: no new premise, the firewall holds (the observable stays [C]). Python-only (numpy/sympy; the gap/spectrum is mirrored via v337/v302). """ import numpy as np import sympy as sp from tfpt_constants import check, summary, reset, N_fam def run(): reset() print("v425 DYN.TRANSFER.UNIVERSAL: the four transfers are one native flow + " "external anchors") Delta = 6 * sp.log(sp.Rational(3, 2)) # the recovery gap rate = sp.Rational(2, 3)**6 # the recovery eigenvalue # ---- 1. the one native flow (explicit Markov recovery semigroup) ---- lam = float(rate) # subleading eigenvalue (2/3)^6 a = (1.0 - lam) / 2.0 T2 = np.array([[1 - a, a], [a, 1 - a]]) # column-stochastic, eig {1, 1-2a=lam} eig = np.sort(np.linalg.eigvals(T2).real)[::-1] w, V = np.linalg.eig(T2) # L = log T2 (Markov generator) L2 = (V @ np.diag(np.log(w)) @ np.linalg.inv(V)).real markov_gen = (L2[0, 1] >= -1e-12 and L2[1, 0] >= -1e-12 and abs(L2[:, 0].sum()) < 1e-9 and abs(L2[:, 1].sum()) < 1e-9) gap_exact = sp.simplify(sp.exp(-Delta) - rate) == 0 check("THE ONE NATIVE FLOW [E]: recovery spectrum {1,(2/3)^6,(1/3)^6}, " "H_OS=-log T gap Delta=6 ln(3/2)>0 with e^{-Delta}=(2/3)^6 exact (v240); " "explicit 2-state column-stochastic T (eig {1,%.5f}) gives L=log T a " "Markov generator (off-diag>=0, columns sum 0) -- the native flow (v238)" % lam, gap_exact and abs(eig[0] - 1) < 1e-12 and abs(eig[1] - lam) < 1e-12 and markov_gen and float(Delta) > 0) # ---- 2. F_pole = this flow (v99 generator linearised) ---- q = sp.symbols("q") g = (Delta / N_fam) * (q - 2) * (q - 5) # v99 canonical generator gp2 = sp.simplify(sp.diff(g, q).subs(q, 2)) # rate at attractor q=2 gp5 = sp.simplify(sp.diff(g, q).subs(q, 5)) # rate at unstable q=5 pole_native = (sp.simplify(gp2 + Delta) == 0 # g'(2) = -Delta and sp.simplify(sp.exp(gp2) - rate) == 0 # time-1 = (2/3)^6 and sp.simplify(gp5 - Delta) == 0) # g'(5) = +Delta (unstable) check("F_pole IS THE NATIVE FLOW [E]: the v99 Koide generator " "dq/dt=(Delta/N_fam)(q-2)(q-5) linearised at q=2 gives rate g'(2)=-Delta, " "so the time-1 contraction = e^{-Delta}=(2/3)^6 (the recovery eigenvalue); " "q=5 unstable (g'(5)=+Delta). Rate native; only the IR pole scale (v_geo) " "external", pole_native) # ---- 3. F_Boltzmann washout = this contraction ---- delta_cp = sp.Rational(4, 3) * sp.pi # 240 deg native (v320) washout_native = (0 < rate < 1 and sp.simplify(delta_cp - sp.pi * sp.Rational(240, 180)) == 0) check("F_Boltzmann IS THE NATIVE CONTRACTION [E]: the washout kappa_f in (0,1) " "is bounded by the native sub-unit eigenvalue (2/3)^6=%s, and the CP phase " "delta_PMNS=4pi/3 (240 deg) is native (v320); only the absolute seesaw " "scale M_R (v_geo-class) external" % str(rate), washout_native) # ---- 4. F_QCD rate native (dimensional transmutation exponent) ---- n_f = 6 b0 = 11 - sp.Rational(2, 3) * n_f # = 7 b3 = -(11 - sp.Rational(2, 3) * n_f) # = -7 qcd_native = (b0 == 7 and b3 == -7) check("F_QCD RATE IS NATIVE [E]: b3=-7 carrier (v159), b0(n_f=6)=11-2*6/3=7=-b3, " "so the dimensional-transmutation exponent -2pi/(b0 alpha_s) is native " "(QCD.LAMBDA.01); only alpha_s(M_Z) and the lattice O(1) C_p external", qcd_native) # ---- 5. verdict (typed [C]); F_relic IC is the one wall ---- check("VERDICT [C]: the transfer DYNAMICS is ONE native object (the v238 " "recovery semigroup, v383 universal) -- native gap + native rates; only " "the ANCHORS (v_geo, C_p, M_R) stay external. F_relic's free input is a " "cosmological IC theta_i (not theory-derivable; best case mu4-reduced) -- " "the one genuine wall. A single-flow REDUCTION, not internalisation: no " "new premise, firewall holds (observable stays [C])", N_fam == 3 and pole_native and qcd_native and float(Delta) > 1.6) return summary("v425 DYN.TRANSFER.UNIVERSAL (four transfers = one native flow + " "external anchors; F_relic IC the one wall; firewall-clean)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)