"""v986 -- FLAV.NUSCALE.05: the SCALARON-TRACE RUNG M_R = N_fam M_scal = 3 c3^{7/2} Mbar, a structured point INSIDE the v481 window, 1.8% from the 1-loop requirement. Candidate [C]/[N] -- NO seesaw closure, mechanism [O]. THE POINT (external master-object review, wave 2, 2026-08-28). v481 left the honest endpoint 'M_R free inside the PS window; required M_R = 9.3e13 GeV under the carrier normalisation y_nu = y_t [P]'; v482 declined the integer rung c3^3 Mbar (m_3 comes out 40% low) and v488 killed the 3/5 group-theory Clebsch rescue. The review proposes a DIFFERENT mechanism class for a structured point: the Majorana operator rides the scalaron (the common trace mode, M_scal = c3^{7/2} Mbar, v253/v272 anchors) times a generation-space insertion whose third-generation slot is the family count N_fam = 3: M_R = N_fam * M_scal = 3 c3^{7/2} Mbar. HONEST MOTIVATION (wave-3 retype, 2026-08-28): the original trace argument Tr_A3 I = 3 is LOOSE -- Spec(I) = {1,1,1}, and a trace of 3 does not make 3 an eigenvalue. The operator reading is the Q_+ eigenvalue route Spec{1,2,3} (v10/v50/v69); the mixed insertion texture M_R = M_scal diag(eps, 2eps, 3) is typed DATA_CONSTRAINS_TEXTURE in the nu-scalaron v2 project (untextured Y prop I is killed x10^4; aligned diagonal Y incompatible with v270 PMNS). The NUMBER below is unchanged: M3 = 3 M_scal remains the candidate rung. The mechanism stays [O]. [N] 1. THE NUMBER: 3 M_scal = 9.180e13 GeV vs the v481-required M_R = 9.346e13 GeV (same 1-loop SM inversion, y_nu = y_t) -- deviation -1.8%; m_3 at the candidate rung = 0.0512 eV vs the observed 0.0503 eV (+1.9%, well inside the >50% RG scheme envelope named in v482). NOTE: the review text claimed 'about one percent'; the honest number at the frozen v481 inputs is 1.8. [E] 2. EXACT LADDER POSITION: 3 c3^{7/2} = (3 sqrt(c3)) c3^3, i.e. the candidate sits a factor 3 sqrt(c3) = 3/sqrt(8 pi) = 0.598413 below the declined integer rung -- numerically 0.26% from the 3/5 = 0.6 of the v482/v488 story. ADJACENCY TYPED HONESTLY: the v488 negative killed 3/5 as an SO(10)/PS CLEBSCH; the scalaron-trace reading is a different mechanism class (an absolute scale from the gravity/scalaron sector times a generation-space insertion whose third slot is N_fam -- the Q_+ Spec{1,2,3} eigenvalue route, not Tr I = 3; texture DATA_CONSTRAINS_TEXTURE), and it is NOT derived here: the mechanism stays [O]. [N] 3. SHARPER THAN THE DECLINED RUNG: |m_3(cand)/obs - 1| = 0.019 vs 0.40 for c3^3 Mbar -- a 20x closer structured point. [E] 4. CONSISTENCY ACROSS THE SCALARON SECTOR: the same M_scal already carries f_a = M_scal/128 (v185) and the inflation scale (v253); the candidate adds no new scale atom -- it reuses the frozen c3^{7/2} rung times the frozen N_fam. [X] 5. KILL TEST (inherited and sharpened): Sigma m_nu (NO, m1 ~ 0) at the candidate rung = 0.059 eV; a robust cosmological bound below 0.059 eV, or a measured m_3 pulling the required M_R more than ~5% away from 9.18e13 GeV under the same premise chain, kills the candidate. HONEST SCOPE (firewall): [P] y_nu = y_t stays a NAMED premise (v481); the 1-loop SM running is the frozen v481 tool (scheme envelope >50% per v482); the scalaron-trace mechanism is a typed hypothesis, not a derivation -- this row does NOT close FLAV.NUSCALE, does not move the window candidate, and makes no PMNS claim. It registers a structured, falsifiable point inside the window. """ import math from tfpt_constants import check, summary, reset, c3, Mbar, N_fam MZ = 91.1876 V_EW = 246.22 YT_MZ = math.sqrt(2) * 162.5 / V_EW LAM_MZ = 0.130 A_INV_MZ = (59.01, 29.59, 8.44) M3_OBS = 0.0503 # eV (NuFIT NO, v481 frozen input) DM2_21 = 7.42e-5 # eV^2 C3 = float(c3) MBARF = float(Mbar) def run_sm_up(mu_hi, n=20000): """the frozen v481 1-loop SM runner (verbatim tool, read-only).""" g1, g2, g3 = [math.sqrt(4 * math.pi / a) for a in A_INV_MZ] yt, lam = YT_MZ, LAM_MZ h = math.log(mu_hi / MZ) / n k = 1 / (16 * math.pi ** 2) b = (41 / 10, -19 / 6, -7) I_alpha = 0.0 for _ in range(n): I_alpha += (-3 * g2 * g2 + 6 * yt * yt + lam) * h dg1 = k * b[0] * g1 ** 3 dg2 = k * b[1] * g2 ** 3 dg3 = k * b[2] * g3 ** 3 dyt = k * yt * (4.5 * yt ** 2 - 8 * g3 ** 2 - 2.25 * g2 ** 2 - (17 / 20) * g1 ** 2) dlam = k * (24 * lam ** 2 - 6 * yt ** 4 + 12 * lam * yt ** 2 - 3 * lam * (3 * g2 ** 2 + 0.6 * g1 ** 2) + 0.375 * (2 * g2 ** 4 + (g2 ** 2 + 0.6 * g1 ** 2) ** 2)) g1 += h * dg1 g2 += h * dg2 g3 += h * dg3 yt += h * dyt lam += h * dlam return yt, math.exp(-I_alpha / (16 * math.pi ** 2)) def m3_pred_eV(MR): yt, R = run_sm_up(MR) return (yt * V_EW / math.sqrt(2)) ** 2 / MR * R * 1e9 def run(): reset() print("v986 FLAV.NUSCALE.05: the scalaron-trace rung " "M_R = N_fam M_scal = 3 c3^{7/2} Mbar (candidate)") M_scal = C3 ** 3.5 * MBARF MR_cand = N_fam * M_scal lo, hi = 1e13, 1e16 for _ in range(60): mid = math.sqrt(lo * hi) if m3_pred_eV(mid) > M3_OBS: lo = mid else: hi = mid MR_req = math.sqrt(lo * hi) dev = MR_cand / MR_req - 1 check("THE NUMBER [N]: M_R = 3 M_scal = %.3e GeV vs the v481-required " "%.3e GeV (same frozen 1-loop inversion, y_nu = y_t [P]) -- " "deviation %.1f%% (the review said 'about 1%%'; honest value " "1.8), inside the PS window and the >50%% v482 RG envelope" % (MR_cand, MR_req, 100 * dev), abs(dev) < 0.03 and 4.2e13 < MR_cand < 2.4e15) m3_cand = m3_pred_eV(MR_cand) check("m_3 AT THE CANDIDATE [N]: %.4f eV vs observed %.4f eV " "(ratio %.4f, +1.9%%)" % (m3_cand, M3_OBS, m3_cand / M3_OBS), abs(m3_cand / M3_OBS - 1) < 0.03) ladder = 3 * math.sqrt(C3) check("EXACT LADDER POSITION [E]: 3 c3^{7/2} = (3 sqrt c3) c3^3 with " "3 sqrt(c3) = 3/sqrt(8 pi) = %.6f -- 0.26%% from the 3/5 of " "the v482/v488 story; v488 killed 3/5 as an SO(10) CLEBSCH, " "this is a DIFFERENT mechanism class (generation-blind " "scalaron trace), itself [O]" % ladder, abs(ladder - 3 / math.sqrt(8 * math.pi)) < 1e-15 and abs(ladder / 0.6 - 1) < 0.003) m3_old = m3_pred_eV(C3 ** 3 * MBARF) check("SHARPER THAN THE DECLINED RUNG [N]: |m3/obs - 1| = %.3f at the " "candidate vs %.3f at c3^3 Mbar (v482 declined) -- a ~20x " "closer structured point" % (abs(m3_cand / M3_OBS - 1), abs(m3_old / M3_OBS - 1)), abs(m3_cand / M3_OBS - 1) < 0.1 * abs(m3_old / M3_OBS - 1)) f_a = M_scal / 128 check("SCALARON-SECTOR CONSISTENCY [E]: the same M_scal = c3^{7/2} " "Mbar = %.3e GeV already carries f_a = M_scal/128 = %.3e GeV " "(v185) -- the candidate reuses the frozen rung times the " "frozen N_fam = %d, no new scale atom" % (M_scal, f_a, N_fam), 2.0e11 < f_a < 2.8e11 and N_fam == 3) summ = M3_OBS + math.sqrt(DM2_21) check("KILL TEST [X]: Sigma m_nu (NO, m1 ~ 0) = %.4f eV at the " "candidate; a robust bound below 0.059 eV or a required-M_R " "shift beyond ~5%% kills it" % summ, 0.055 < summ < 0.062) check("FIREWALL (scope): y_nu = y_t stays a NAMED premise [P]; the " "scalaron-trace mechanism is a typed hypothesis [O], not a " "derivation (the Tr I = 3 motivation is loose -- Spec(I) = " "{1,1,1}; the operator reading is Q_+ Spec{1,2,3}, texture " "DATA_CONSTRAINS_TEXTURE); FLAV.NUSCALE does NOT close; the " "v481 window candidate and the v488 negative are untouched; " "no PMNS claim", True) return summary("v986 scalaron-trace rung M_R = 3 c3^{7/2} Mbar: " "structured candidate 1.8% inside the window") if __name__ == "__main__": raise SystemExit(1 if run() else 0)