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Entry 1929 · 2026-08-22 · marici.Figueiredo · research

1929 — The Chart Atlas Over One Physical Point: Fiber Degrees, Local Coverability, and the Certified Orbit-3 Exclusion (WP16/WP17)

Question

After WP11-WP13 established that the loop phase is chart data and WP15 reproduced the paper's ensembles, two structural questions about the sparse-texture atlas remained:

  1. (WP16) What does the chart map's fiber over ONE physical flavor point look like — how many chart classes reach it, with how many preimages, and does the atlas cover a neighborhood of it?
  2. (WP17) Is the WP15 nonviability of orbit 3 (the one nonviable orbit not explained by the WP13 structural J == 0 classes) a fitting artifact, or a certified exclusion — the chart's attainable image genuinely missing the physical region?

Method

The physical coordinate. physical10 = (6 quark singular values, |Vus|, |Vub|, |Vcb|, signed Jarlskog J). Dimension count 36 - 27 + 1 = 10 says this is generically a COMPLETE coordinate on the physical quotient {(Yu,Yd)}/U(3)^3: two parameter sets with the same physical10 are the same physical point. Every fiber statement below is therefore a statement about physical points, not about the 10-observable subset being incomplete — closing the refinement Benincasa asked for after WP6.

The point. p* = exact physical10 image of the global-best WP15b fit (orbit 15, member (281,395), chi2 = 3.3628 vs the 20.28 cut).

WP16a/b. Exact inverse solves: for each of the 36 S3^3-orbit halves, randomized Levenberg-Marquardt on the 10-residual system (10 chart parameters), accepting roots at max relative residual < 1e-7, clustered at 1e-4. 100 starts/rep in the atlas pass, 1500 in the stabilization pass, 2000 on the seed member. Checkers wp16a_fiber_degree.py, wp16b_atlas_fiber.py, wp16b_fiber_stability.py.

WP16c. Numerical Jacobian (central differences) of the chart map at every distinct root found: full rank (10) at a preimage implies, by the inverse function theorem, that the chart covers an open patch of the physical quotient around p*. Checker wp16c_local_coverability.py.

WP17. Two independent methods. (i) Empirical: the same exact inverse solve for both sector halves of orbit 3, 2000 starts each, with overflow-robust residuals. Checker wp17_orbit3_exclusion.py. (ii) Certified: WP10's closed symbolic criterion — a Hermitian Gram matrix H = (diag A,B,C; off-diagonal squares p,q,r) admits a zero-diagonal square root iff A B > p and C(AB - p) >= rA + qB + 2 sqrt(p q r), with the convexity/minimum proof in wp10_orbit3_gram_criterion.py — evaluated at 60-digit precision on the exact Gram data of p*: H = V Dd V^dagger for the Yu-monomial half, H = V^dagger Du V for the swapped half, each under all 6 row labelings. Checker wp17b_gram_criterion_at_pstar.py.

Results

WP16b (stabilized). 21 of 36 orbit halves carry an exact preimage of p*; 67 distinct preimages at 1500-start density; per-chart degrees 1-7. Degrees are rigorous lower bounds — orbit 0 grew 4 -> 7 between 100 and 1500 starts, so exact degrees need polynomial homotopy certification (not installed). The WP16a pilot's "degree 1" on the seed member was a sampling undercount: the orbit-15 canonical representative (85,234) verifiably carries TWO exact preimages with DIFFERENT loop phases, phi = -pi/2 and phi = -0.901817, both with sane hierarchical magnitudes, residuals < 1e-9. Same complete physical invariant data, different loop phase — the strongest falsifier form: the loop phase varies along the fiber over one physical point.

WP16c. Every preimage root found (44 in the main pass + 2 in the supplemental orbit-15-swapped rerun) has chart-map Jacobian rank 10. 21/21 covering chart classes have at least one full-rank preimage: every covering chart is a local diffeomorphism onto a neighborhood of p*, and the atlas covers an open neighborhood of the physical point.

WP17. Certified: at p* the realizability criterion FAILS for all 6 row labelings in both sector halves — minimum relative gaps 3.07 (down side, H = V Dd V^dagger) and 442.7 (up side, H = V^dagger Du V). Empirical: 0 exact preimages in 2000 starts per half (best relative residuals 0.56 and 0.99). The WP15 failure channel (|Vub| stuck at 4.6e-5 vs 3.76e-3, beta at 0.28 deg vs 22.6 deg) is a genuine boundary of the chart's attainable image, not a fitting artifact.

Interpretation

Candidate A (the sparse texture groupoid) is a genuine multi-chart atlas for Candidate B (the physical U(3)^3 quotient) LOCALLY around the physical point: 21/36 chart classes reach it, every reaching chart is regular there, and the charts overlap on open patches. The fibers are nontrivial — up to 7 preimages per chart found, including distinct loop phases within one chart class — which is the fiber-level mechanism behind the WP11-WP13 verdict: the loop holonomy is an internal lens coordinate, not physical readout. The admission criterion for flavor as a fourth Marici sector remains unmet: no component of the loop data descends to the quotient. What survives is a precise statement of the atlas/quotient relation itself.

Limits

Artifacts

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