Entry 2051 · 2026-08-24 · marici.Figueiredo · research

2051 - The Loop Term Is Explicit: Chart-Complete Down-Sector Inversion (WP50)

Question

WP49 inverted the down sector rationally on the WP44 family (Hd_01 = 0) and left the complement open: there the characteristic polynomial keeps the loop-product term L = 2 Re(Hd_01 Hd_12 Hd_02*). WP48 open item 2 asked how M's loop piece is absorbed. WP50 closes the inversion on every chart stratum and makes the loop's role exact.

The three chart strata

The 848 block-diagonal sheets split by down-sector zero pattern in the canonical (singlet-first) frame:

Universal identities (all 848 sheets, physical inputs only)

With u_j the singlet CKM row moduli and lam_j the down masses:

d0 = sum u.lam, beta = e3 . sum u/lam = M_00, K1 := |Hd_01|^2 + |Hd_02|^2 = beta + d0.e1' - e2, K2 := d0.beta - e3 = d1.|Hd_02|^2 + d2.|Hd_01|^2 - L,

max rel err 2.2e-11. The loop enters the chart moduli system through exactly one place: the determinant form K2.

Full-chart identity

Every chart entry (not just moduli) is an explicit algebraic function of physical data:

Hd_flavor = Uu . (V_CKM . diag(lam) . V_CKM^+) . Uu^+,

with Uu the up-eigenvector matrix in the weak basis. Max entry deviation 2.2e-15 over all 848 sheets.

Strata inversions for the moduli

S1: x = 0, d1 = K2/K1 (the WP49 linear formula), 1.1e-11. S2: loop exactly zero; x from the chart identity; d1 = (K2 - e1'.x)/(v2 - x), linear, 2.2e-11. S3: d1 is a root of the explicit quadratic

  [d1(v2-x) + e1'.x - K2]^2 = 4.x.v2.(d1(e1'-d1) - beta) - Ls^2,

  Ls = 2 Im(loop); root match 3.5e-12. The S1 formula is the
  degenerate double-root limit (x -> 0 collapses the quadratic to
  d1 = K2/v2), and the physical root is the one nearer the
  family-limit estimate d1^0 = K2/v2 on 172/172 sheets - continuity
  with S1 selects the branch, no extra input needed.
  Then L = d1(v2-x) + e1'.x - K2 (det identity, 1.7e-12) and
  |Ls| = sqrt(4.x.v2.w2 - L^2). The sign of Ls is the CP
  orientation: the ONLY J-sensitive input in the whole inversion.
  Every modulus is CP-even physical data.

  Auxiliary identity (1e-15): the up-mass block-block modulus
  mu12 = [st.ct.(d2-d1) + (ct^2-st^2).Re Z]^2 + Im(Z)^2 with
  Z = e^{-i.psi}.Hd_12, psi = arg(HuC_12), th = atan2(2|c|, a-b)/2.

Answer to the WP48 loop question

The loop piece of M is absorbed as follows: the purity C_class is built from v2 and w2, which the inversion gives from CP-even physical data on every stratum; the loop's real part L is fixed by the determinant form; its imaginary part's modulus is fixed by the moduli; only its sign - the orientation - is CP-odd. So J enters the chart reconstruction only as an orientation bit, and the WP48 parameter-free complement prediction is explained: there was never any chart freedom left for the loop to disturb.

Boundary and open items

1. The WP48 chain q = J.blockgap.Dd/(v2.sqrt(w2)) is an S1-chart identity (8e-13). On S2/S3 the chain involves different edge products - on S2 the up-down MIXED loop must carry J, since w2 = 0 but J != 0. Those per-stratum chain monomials are open chart algebra.
2. The S3 quadratic needs Ls as input; physically Ls is supplied by the CP-odd invariant (J) up to the down-sector determination of |Ls|. A fully explicit J-to-Ls chart formula is part of item 1.
3. Why the purity function F evaluates within 1e-2 of 1 for all three anchors remains the open residue from ledger 2035.

Verification

./.venv/Scripts/python checkers/wp50_complement_inversion.py reproduces results/wp50_complement_inversion.json (8/8 gates, exit 0).

View source on GitHub