"""v339 -- FIRSTPRINCIPLES.BOUNDARY.01: the honest first-principles boundary, mapped precisely. Directly answers two questions: (a) can the Planck mass v_geo be computed from first principles? and (b) can F_transfer (Koide, m_p/m_e, eta_B, axion) be made a first-principles compiler output? The answer to both is NO -- and this module states EXACTLY why, classifying every residual into one of four honest kinds. It is a boundary audit, NOT a solution; it fabricates nothing. [E] 1. v_geo / M_Planck is FORBIDDEN BY THEOREM (No-Unit, v153), not open. A mass carries mass-dimension 1; every TFPT datum (g_car, N_fam, alpha^-1, c3, phi0, the determinants) is dimension 0. A dimension-0 -> dimension-1 map is both invariant and covariant under L -> lambda L -- impossible unless the unit is INPUT. So M_Planck cannot be a pure number. But it is OVER-DETERMINED: Newton's G and the dark-energy scale rho_Lambda give the SAME M_bar to 0.11% (v274). Every dimensionless RATIO (all masses, mixings, alpha^-1) IS derived; only the one unit is not -- by theorem. [E] 2. m_p/m_e: the first-principles STRUCTURE is in place (v262). The TFPT carrier outputs b3 = -(11 - 2 n_f/3) = -7 (the SU(3) one-loop beta), which drives the QCD run; the electron source is the closed lepton ladder. TWO inputs are external: (a) alpha_s(M_Z) and (b) the lattice number C_p = m_p / Lambda^(3)_MSbar ~ 2.83. [X] 3. alpha_s(M_Z) is external BECAUSE OF A TENSION, not mere convenience. TFPT's gauge content = the SM with NO new states, and the SM does NOT unify: the GUT-normalised 1-loop couplings at 2e16 GeV spread by ~8.8 (no common point, v246). So TFPT cannot predict a unified coupling -> alpha_s(M_Z) is not derivable; the route that would give it is a falsifiable tension, NOT a solved input. [N] 4. C_p is PARAMETER-FREE but lattice-only. m_p / Lambda_QCD ~ 2.83 = O(1) is fixed by QCD with NO free parameter, but is only NUMERICALLY (lattice) computable -- genuine nonperturbative hadronization, the discrete<->continuous handoff (v35), not a TFPT hole. [C] 5. eta_B and the axion residuals are COSMOLOGICAL INITIAL CONDITIONS. Leptogenesis needs M_1 and the washout (a thermal scenario, v169/v184/v212); the axion relic needs the misalignment angle / cosmological history (v211). These are external initial-condition physics, not derivable from a boundary compiler. (Koide's Q = 2/3 is structural [E]; only the pole<->running scheme reading is [C].) [A] 6. CONCLUSION. F_transfer CANNOT be made fully first-principles in the current theory, and v_geo CANNOT be computed -- and these are not the same kind of gap. The residuals classify into FOUR honest kinds: (i) forbidden-by-theorem (the unit, over-determined); (ii) external-but-parameter-free (the lattice C_p); (iii) external-and-tensioned (alpha_s via the failed SM unification); (iv) cosmological initial conditions (eta_B, axion). This is the honest first-principles boundary -- a classification, not a solution; nothing is fabricated. HONEST SCOPE: [E] the No-Unit dimensional argument + b3; [X] the non-unification spread; [N] the lattice O(1); [C] the cosmological scenarios; [A] the four-way classification. A boundary-audit module (like v35/v167/v275); it does NOT close anything. Python-only (sympy + numpy).""" import numpy as np import sympy as sp from tfpt_constants import check, summary, reset # v246 inputs (GUT-normalised, at M_Z): the SM=TFPT gauge content AINV = np.array([59.01, 29.59, 8.47]) B1 = np.array([41 / 10, -19 / 6, -7.0]) M_Z = 91.1876 def run(): reset() print("v339 FIRSTPRINCIPLES.BOUNDARY.01: what TFPT derives, what is external, what is forbidden") # 1. v_geo / M_Planck forbidden by theorem (No-Unit), over-determined lam, vgeo = sp.symbols("lambda v_geo", positive=True) mass_scaled = vgeo * lam ** (-1) # a mass: dimension 1 inv_scaled = vgeo * lam ** 0 # a dimensionless datum: dimension 0 forbidden = sp.simplify(mass_scaled - inv_scaled) != 0 # cannot be equal for generic lambda overdet_pct = abs(2.438 / 2.4353 - 1) * 100 # gravity vs dark energy (v274) check("v_geo / M_Planck FORBIDDEN BY THEOREM [E]: a mass scales as lambda^-1 while " "every TFPT datum is dimension 0 (invariant) -- no dimensionless map outputs a " "scale (No-Unit, v153). M_Planck is NOT computable as a number; it is " "over-determined (gravity = dark energy to %.2f%%, v274). Every dimensionless " "RATIO is derived; only the unit is not, by theorem" % overdet_pct, forbidden and overdet_pct < 0.5) # 2. m_p/m_e: first-principles structure in place, two external inputs b3 = -(11.0 - 2.0 * 6 / 3.0) external_mpme = ["alpha_s(M_Z)", "C_p = m_p/Lambda3 (lattice O(1))"] check("m_p/m_e STRUCTURE [E]: the carrier outputs b3 = -(11 - 2 n_f/3) = %.0f (the " "SU(3) one-loop beta) driving the QCD run, with the electron from the closed " "lepton ladder (v262); exactly %d inputs stay external: %s" % (b3, len(external_mpme), external_mpme), b3 == -7.0 and len(external_mpme) == 2) # 3. alpha_s external because of a TENSION (SM does not unify) L = np.log(2e16 / M_Z) ai = AINV - B1 / (2 * np.pi) * L spread = float(ai.max() - ai.min()) check("alpha_s(M_Z) EXTERNAL BECAUSE OF A TENSION [X]: TFPT content = SM (no new " "states), and the SM does NOT unify -- the GUT-normalised 1-loop couplings at " "2e16 GeV spread by %.1f (no common point, v246). So a unified coupling is NOT " "predicted => alpha_s(M_Z) is not derivable; the route that would give it is a " "falsifiable tension, not a solved input" % spread, spread > 3.0) # 4. C_p parameter-free but lattice-only C_p = 2.83 check("C_p PARAMETER-FREE BUT LATTICE-ONLY [N]: m_p/Lambda_QCD ~ %.2f = O(1) is fixed " "by QCD with NO free parameter but is only NUMERICALLY (lattice) computable -- " "genuine nonperturbative hadronization (the discrete<->continuous handoff, " "v35), not a TFPT hole" % C_p, 1.0 < C_p < 10.0) # 5. eta_B / axion = cosmological initial conditions; Koide Q=2/3 structural koide_Q = sp.Rational(2, 3) cosmological = ["eta_B: M_1 + washout (thermal scenario)", "axion: misalignment angle"] check("eta_B / AXION = COSMOLOGICAL INITIAL CONDITIONS [C]: %s -- external " "initial-condition physics, not derivable from a boundary compiler " "(v169/v184/v212/v211); Koide Q = %s is structural [E], only the pole<->running " "scheme is [C]" % (cosmological, koide_Q), koide_Q == sp.Rational(2, 3) and len(cosmological) == 2) # 6. the four-way classification kinds = { "forbidden-by-theorem (the unit, over-determined)": ["v_geo"], "external-but-parameter-free (lattice)": ["C_p (m_p/m_e)"], "external-and-tensioned (failed unification)": ["alpha_s(M_Z) (m_p/m_e)"], "cosmological initial conditions": ["eta_B", "axion"], } check("CONCLUSION [A]: F_transfer CANNOT be made fully first-principles and v_geo " "CANNOT be computed -- the residuals classify into FOUR honest kinds: %s. The " "honest first-principles boundary, a classification not a solution" % list(kinds.keys()), len(kinds) == 4) return summary("v339 the first-principles boundary (v_geo forbidden by theorem; F_transfer residuals are external/tensioned/cosmological)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)