1995 - The Valley-Binding Mechanism: the Pi-8 Arc Closed at Constraint Level (WP46)
WP20 (1937) reduced the pi/8 question to one exact open residual: why do the chi2 minima of the nine-link textures select magnitude sets whose a1/(D_u D_d) = |J|/sin(window)? Since sin(phi) = |J| D_u D_d / a1 is an identity at every point (re-verified: G4, max rel err 3.8e-9 over all 1210 viable minima), the phase spectrum IS the a1 spectrum, so the residual is: what pins the valley phases? WP46 answers it at the constraint level with the stored WP20 audit data plus absorbed (magnitude-refit) chi2 profiles. Checker checkers/wp46_valley_binding.py, results results/wp46_valley_binding.json, 5/5 gates.
The valley-binding dictionary (G1, G3)
Re-clustering all 1210 minima at 0.02 deg resolution (the exact fiber phases are delta-peaks; WP20's 0.15 deg bins merged neighbors) gives 23 dense sub-valleys. Every one is bound by exactly ONE physical observable, uniform across every orbit that realizes it (G3: 23/23 shared):
bound by gamma: 68.02, 68.01, 68.00, 46.86, 68.05 (n = 75+58+19+115+12) bound by beta: 23.86, 23.89, 22.82, 22.87, 21.67, 23.67 (n = 48+34+32+26+33+13) bound by |Vcd|: 89.73, 89.69, 68.92, 89.51, 69.72, 68.53 (n = 64+32+84+43+29+12) bound by |Vtd|: 46.94 (n = 127) bound by |Vts|: 89.61, 89.20 (n = 25+33) bound by |Vus|: 44.03 (n = 12) bound by ys/yud: 42.83 (orbit 0, n = 30), 69.22 (orbit 2, n = 12)
The WP20 46.8/46.95 "two-face" bin resolves at fine resolution into two distinct sub-valleys: 46.86 (gamma-bound, the orbit-4 exact fiber phase) and 46.94 (|Vtd|-bound). Every multi-orbit sub-valley carries its binding observable in ALL of its orbits - the active constraint is physical data, identical across presentations, which is exactly why different orbits minimize at the same phase. The sharing mechanism WP20 observed ("the extra valleys are the same valleys") is thereby explained: one constraint equation, one root, many charts.
The active constraint after absorption (G2)
LM magnitude re-fits at phi shifted by +-1,2,4 deg on the 8 largest valleys: the binding observable contributes the largest share of the chi2 growth (top shares 0.35..0.61). The raw chart-facing sensitivity (which CKM element the chart's loop edge feeds: Vcd, Vtd, Vts vary by orbit) and the absorbed landscape constraint differ informatively: after the fit re-absorbs what it can, gamma is the dominant active constraint in every CKM valley outside the beta band (with alpha = 180 - beta - gamma its dependent shadow at 0.23..0.27 share). beta pins the beta band (share 0.61). So at landscape level the valley spectrum is the root set of a small system of scalar physical-data equations:
gamma(phi; refit) = gamma_measured (the 46-89 deg valleys) beta(phi; refit) = beta_measured (the 21-24 deg valleys) ys/yud(phi; refit) = measured ratio (42.83 and 69.22)
The two mass-bound valleys (G5)
Orbit 0's 42.83 valley and orbit 2's 69.22 valley are bound by the MASS ratio ys/yud (uniformity 1.0 in both), not by any CKM readout. Neither is CKM-inherited (min distance to any CKM angle or combination: 2.34 deg and 1.24 deg) and neither sits on the pi/8 lattice (2.17 deg and 1.72 deg off). They are the only valleys whose active constraint lives in the mass sector - and the constraint is the SAME ratio in both, one more instance of the sharing mechanism.
The closure chain (WP20 residual closed)
- A viable minimum must fit the 17 observables; the active (least re-absorbable) constraint at each minimum is a single physical datum - gamma, beta, a CKM modulus, or ys/yud (G1, G2).
- The valley phase is the root of that datum's fit equation; the datum and the equation class are orbit-independent, so the root is shared across orbits (G3) - discreteness and cross-orbit sharing explained.
- The a1 spectrum follows identically: a1 = |J| D_u D_d / sin(phi_valley) (G4) - this answers WP20's question: the minima select those magnitude sets because the fit equation forces phi, and the identity forces a1.
- The pi/8 proximity is not an attractor: it factors as (CKM-readout binding) x (the measured CKM angles' own lattice proximity, WP14b T4), and the two mass-bound valleys are not even CKM-inherited yet still off-lattice. Nowhere does the lattice itself select anything.
What remains genuinely open is only the question WP14b relocated to the physical point itself: why the MEASURED CKM angles sit near pi/8 multiples. That is not a question about sparse presentations and is out of the texture map's reach by its own typing (WP45): it belongs to the UV or to a cross-sector parent-object statement about constant values.
Honesty notes
- G2 top shares are 0.35..0.61, not 1.0: the fits are collective; the binding observable is the dominant active constraint, not the sole one.
- Raw sensitivities are computed at fixed magnitudes (WP20 convention); the absorbed profiles (G2) are the landscape statement.
- Fringe records (sub-clusters n < 10) are reported, not gated.
Artifacts: checkers/wp46_valley_binding.py, results/wp46_valley_binding.json. Depends on: WP14b (1911), WP19 (1936), WP20 (1937), WP45 (1988).