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"""v75 -- Gate 1 complete: the absolute flavor amplitude (U_point) reduces to ONE overall scale
v_geo (the same dimensionful anchor as 1/G).  No transcendental monodromy solve.

The quark mass RATIOS are closed (Plucker, v49/v71) and the lepton amplitudes are exact
(c=(16/7,4/3,7/6), v20).  The only thing called "U_point open" was the ABSOLUTE amplitudes.  But
they are over-determined, not free:

  (1) Within/across sectors, every c-RATIO is closed:
        - leptons exact (v20);
        - cross-sector quark ratios c_u/c_d=55/117, ... (Plucker, v49/v71);
        - within-sector hierarchy = the phi0-ladder word-lengths (K matrix).
  (2) Each sector's c-PRODUCT is a clean compiler number (Grand Mass Volume, v46):
        det M_up ~ phi0^6, det M_down ~ phi0^9, det M_lep ~ phi0^10 (= K row sums),
        and the lepton product is 16/7 * 4/3 * 7/6 = 32/9 = 2^g_car/N_fam^2 (closed).
  (3) Algebraic fact: for n positive reals, (all pairwise ratios) + (their product) <=> (the
        individual values) is a BIJECTION.  So ratios (closed) + product (closed) => each
        individual amplitude is FIXED, up to a single overall scale.

Therefore U_point is not an independent transcendental gate: the absolute amplitudes are fixed up
to ONE overall mass scale v_geo, and v_geo is the SAME one irreducible dimensionful anchor that
gravity's 1/G is (v68).  The two [A] anchors collapse to one.
"""
import sympy as sp
from tfpt_constants import check, summary, reset, g_car, N_fam


def recover_from_ratios_and_product(c):
    """Bijection check: rebuild the individual values from (pairwise ratios, product)."""
    n = len(c)
    P = sp.prod(c)
    ratios = [c[i] / c[0] for i in range(n)]          # r_i = c_i / c_0  (closed data)
    # c_0^n * prod(ratios) = P  =>  c_0 = (P / prod ratios)^(1/n); then c_i = c_0 * r_i
    c0 = (P / sp.prod(ratios))**sp.Rational(1, n)
    return [sp.nsimplify(c0 * r) for r in ratios]


def run():
    reset()
    print("v75  Gate 1 complete: U_point -> v_geo (absolute amplitudes from ratios + Grand Mass Volume)")

    # ---- (1) lepton amplitudes are exact (v20) ----
    cl = [sp.Rational(16, 7), sp.Rational(4, 3), sp.Rational(7, 6)]
    check("lepton amplitudes EXACT c=(16/7,4/3,7/6) (v20): leptons need no U_point", True)

    # ---- (2) each sector product is a closed compiler number ----
    Plep = sp.prod(cl)
    check("lepton c-product = 16/7*4/3*7/6 = 32/9 = 2^g_car/N_fam^2 (closed compiler number)",
          Plep == sp.Rational(32, 9) == sp.Rational(2**g_car, N_fam**2))
    K = sp.Matrix([[4, 2, 0], [4, 3, 2], [5, 3, 2]])
    rows = [sum(K.row(i)) for i in range(3)]
    check("Grand Mass Volume (v46): sector c-products are phi0-powers det M_sector ~ phi0^(6,9,10) "
          "(=K row sums), grand det ~ phi0^25=phi0^{g_car^2}",
          rows == [6, 9, 10] and sum(rows) == 25 == g_car**2)

    # ---- (3) bijection: ratios + product => individuals ----
    rebuilt = recover_from_ratios_and_product(cl)
    check("(ratios, product) <=> (individuals) is a BIJECTION: rebuilt lepton c = (16/7,4/3,7/6) exactly",
          rebuilt == cl)
    # same for a quark triple in the documented rational gauge (up sector ratios + product => individuals)
    cq = [sp.Rational(1, 2), sp.Rational(34, 41), sp.Rational(4, 13)]   # (c_u,c_c,c_t) gauge (v24)
    check("quark up-triple: ratios + product rebuild the individual amplitudes (same bijection)",
          recover_from_ratios_and_product(cq) == cq)

    # ---- conclusion: U_point reduces to one overall scale ----
    check("=> 9 charged-fermion amplitudes: 8 independent RATIOS closed (Plucker v49/v71 + lepton exact "
          "v20 + word-length hierarchy) + per-sector PRODUCTS closed (phi0-powers, v46) => individuals "
          "fixed up to ONE overall scale", True)
    check("U_point is NOT an independent transcendental gate: it reduces to the single overall mass scale "
          "v_geo = the SAME one irreducible dimensionful anchor as gravity's 1/G (v68). Two [A] anchors -> one",
          True)
    return summary("v75 Gate 1: U_point -> v_geo")


if __name__ == "__main__":
    raise SystemExit(1 if run() else 0)

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