1985 — The c-Anchor Class Is Exact Too; Erratum to WP38/WP41 (WP42)
Erratum
WP38 reported and WP41 repeated that the c-anchor class "genuinely fluctuates at ~1% in all ratios" and has no exact closed form. This is superseded. The ~1% scatter was an artifact of testing incomplete monomials. With the full monomial the c-class is exact:
q_c = C_c * y_t^2 * |V_ub| * sin(gamma), C_c = 1.0008814042
constant across all 424 c-anchor sheets with maximum deviation 6.3e-6 (181 sheets within 1e-9 of the median). The old q_c/|V_ub| ratio scatters 7.6e-3 because y_t^2*sin(gamma) itself scatters that much across the fitted ensemble; the monomial absorbs exactly that wandering (G3: old bulk dev 7.6e-3, new max dev 6.3e-6).
The complete anchor taxonomy
All three anchor classes of the up-sector singlet/block decomposition now have exact physical closed forms for the weak charge q = Re(H_block,12)*sin(phi_loop):
u-anchor (block {c,t}): q_u = C_u * y_t^2 * |V_cb| * sin(gamma) C_u = 0.9993729706 (dev 2.3e-11) c-anchor (block {u,t}): q_c = C_c * y_t^2 * |V_ub| * sin(gamma) C_c = 1.0008814042 (dev 6.3e-6) t-anchor (block {u,c}): q_t = C_t * y_c^2 * |V_ub| / |V_cb| C_t = 0.9920333163 (dev 1.6e-10)
The scale is always the largest Yukawa of the off-diagonal block, as Re(H_12) = (m_large^2 - m_small^2) sin(theta) cos(theta) requires. The CKM factors are the block mixing angles in physical units.
Consistency: at the [417,482] d12 same-point pair, q_u/q_c direct = 11.15538774 vs monomial prediction 11.15538774 (agreement 2e-11). The earlier "11.155" pair ratio is thus exactly accounted for.
C_c is a true constant
Refit-gradient of ln C_c over the physical 10-space: all partials < 1.8e-3 (G5), matching C_u's certified constancy (WP41). All three constants are pure numbers of the construction: 1 - 6.27e-4, 1 + 8.81e-4, 1 - 7.97e-3.
Process note (honesty)
Two interactive exploratory runs in this package produced a spurious non-integer c-gradient (y_s^0.72 y_b^-0.72 ...). Root cause: a Python loop variable shadowed the phase-edge index, corrupting three of seventeen refits. The published wp40/wp41 checkers use literal/parameter indices and were verified unaffected by code reading; the correct c-gradient (G2) is clean on both tested textures (y_t +2.001, Vub +1.000, gamma = gamma*cot(gamma)).
Open
- Identification of the three pure numbers C_u, C_c, C_t.
- The single principle behind the three CKM factors: Vcbsin(gamma), Vubsin(gamma), Vub/Vcb.
- Whether the 6.3e-6 c-class residual has its own exact structure.
Certificate: checkers/wp42_c_class_closed_form.py -> results/wp42_c_class_closed_form.json (5/5 gates).