A registered result in its source context

FR.DYNAMICS.01

Original ledger classification: Dynamical-systems unification [E/C] (one shape / one seam-rate)

Original claim text

THE DISCRETE->DYNAMIC LENS ON F_transfer (2026-06-19): asks whether the main-branch mechanism -- a gapped, positivity-preserving relaxation to a UNIQUE attractor (the local-averaging update m<-(A+2I)/4 m -> the E8 marks, gap 6 log(3/2)) -- is also the shape of the four F_transfer instances, or an unrelated bolt-on. Answer: ALL FOUR share the SAME shape (a gapped contraction/flow to a unique attractor: Perron-Frobenius / Boltzmann H-theorem / RG fixed point), but only ONE has the SEAM rate. [E] F_pole (Koide) IS the seam dynamics: the source->pole/branch relaxation is the Moebius map with fixed-point multiplier F'=(2/3)^6 = lambda_2 = the seam-clock subleading eigenvalue (v82/v54); |F'|<1 => a unique attractor at the SEAM rate (gap 6 log(3/2)), two starts converge at rate lambda_2^n. [C] F_Boltzmann (eta_B): the baryon asymmetry is the slow mode of a positivity-preserving Boltzmann operator; washout kappa_f in (0,1) converges to the unique balance (homog. part decays, gap=W>0); compiler fixes A_Lambda=10 (in M_1 and m~_1); the RATE is thermal, NOT the seam gap. [C] F_relic (axion): the misalignment field's comoving number density relaxes to a CONSTANT once H<~m_a (adiabatic invariant, a fixed point); seed theta_i=3pi/5=(N_fam/g_car)pi; the RATE is cosmological. [I] F_QCD (m_p/m_e): the 1-loop RG flow with the carrier coefficient b3=-7 (b0=11-2 nf/3=7, nf=2 N_fam; asymptotic freedom) has the Gaussian UV fixed point as attractor (alpha_s->0 toward the UV), Lambda_QCD the dynamically generated IR scale; b3 from the carrier, the flow standard RG. [O] VERDICT: one shape (gapped relaxation/flow to a unique attractor) underlies all four = the main-branch discrete->dynamic principle; F_pole has the seam rate exactly ((2/3)^6), the other three share only the shape (external rates: thermal/cosmological/RG). So F_transfer is the downstream readout end of the one principle, NOT a bolt-on; the theory's simplicity is preserved precisely because the external rates are honestly fenced (v187), not because they reduce to the seam.

Original classification detail

[E] the F_pole seam-rate identity (Koide multiplier = (2/3)^6 = seam eigenvalue); [C] the Boltzmann/relic relaxations (shared shape, external rate); [I] the QCD RG UV-attractor (carrier coefficient b3=-7); [O] the unifying verdict. Demonstrates the SHARED discrete->dynamic shape across F_transfer; does NOT reduce the three external rates to the seam, does NOT upgrade any frontier number to [E].

Dependencies as recorded

CONTRACT.F.01; FR.KOIDE.04; SEAM.ATTRACTOR.01; SEAM.EQUIV.GAP.01