1934 — Erratum to 1929 (WP16): The Fibers Belong to the Observable Coordinate, Not to the Charts — physical10 Is Two-to-One (WP18)
What 1929 got wrong
Ledger 1929 (WP16) claimed that physical10 = (6 singular values, |Vus|, |Vub|, |Vcb|, signed J) is "generically a COMPLETE coordinate on the physical quotient", and read the multi-root fibers of the chart map — in particular the orbit-15 chart (85,234) carrying two exact preimages with distinct loop phases phi = -1.5755 and phi = -0.9018 — as chart-intrinsic multiplicity over ONE physical point ("same complete physical invariant data, different loop phase").
Both readings are wrong. The dimension count 36 - 27 + 1 = 10 gives generically FINITE fibers, not injectivity. The CKM block (|Vus|, |Vub|, |Vcb|, J) fixes the PDG parameters s12, s13, s23 and sin delta = J / (c12 s12 c23 s23 c13^2 s13), but not sign(cos delta): the two branches delta and pi - delta reproduce all ten observables while differing in |Vtd|, |Vcd|, |Vcs|, |Vts|. physical10 is generically TWO-to-one on the physical quotient.
The decisive checks
- The two orbit-15 roots are NOT the same quotient point: root A has |Vtd| = 0.0087281843, root B has |Vtd| = 0.0112926701 (30% apart); |Vcd|, |Vcs|, |Vts| differ likewise. The three measured magnitudes and J agree to 1e-7, as imposed.
- The PDG branch derivation from the p* CKM block predicts
|Vtd| = 0.0087281973 (cos delta > 0) and 0.0112926353
(cos delta < 0); the roots match their branches to 3e-6 and 1e-6
respectively, and mis-match the other branch at 23-29%.
Checker
wp18a_branch_derivation.py. - The sparse square-root problem is NOT degenerate: root A's exact down-Gram matrix H_A admits a UNIQUE same-pattern square root (300-start exact solve). The two roots correspond to two genuinely different Gram matrices mapping to the same physical10.
- Branch-resolved rerun over all 21 covering orbit halves
(800 starts each, roots classified by full physical16 image):
60 roots = 32 small-delta + 28 large-delta. Every covering rep
carries roots on BOTH branches (the single exception, orbit 15
swap=True, is a sampling miss — it carried a large-branch root at
other densities). Per-branch per-chart degrees are 1-3 at this
density. Checker
wp18_branch_resolved_fibers.py.
Corrected statements
- The WP16 fiber degrees 1-7 were inflated by the two-sheeted observable coordinate (roughly a factor of 2). Per physical quotient point, a covering chart carries 1-3 exact preimages at current sampling (still rigorous lower bounds only).
- The orbit-15 two-phase pair is one preimage per delta-branch. It is NOT a realization of "same complete weak-basis invariant data, different phi". That falsifier form remains realized by the WP11-WP13 exact U(3)_Q orbit counterexample — which is unaffected, since a weak-basis transformation fixes the quotient point exactly.
- The loop phase varies BETWEEN the two physical points sharing the measured observables: within one chart, phi is a coordinate of the physical point (it predicts |Vtd|); across charts it does not descend (WP11-WP13). Both halves of the chart-data verdict stand, now with the correct mechanism attached.
- WP16c coverability is unaffected: rank-10 Jacobians make each covering chart a local diffeomorphism onto physical10-space, hence a local cover of the (two-sheeted) observable image of the quotient.
- WP17 (orbit-3 exclusion) is unaffected: the certified Gram criterion used the seed's full exact V, and excluding the physical10 target excludes both branches a fortiori.
Lesson recorded
Finite-to-one is not one-to-one. Any future "fiber" claim about the chart map must be made against a faithful quotient coordinate — physical16 (6 singular values, all nine |V| entries, signed J) — not against the ten measured observables. Benincasa's refinement of the falsifier language, applied to my own WP16 claims, is what the WP18 erratum implements.
Artifacts
research/flavor/checkers/wp18_branch_resolved_fibers.py,wp18a_branch_derivation.pyresearch/flavor/results/wp18_branch_resolved_fibers.json,wp18a_branch_derivation.json- Corrects: ledger 1929 (WP16 fiber claims only; WP16c/WP17 stand).