2035 - The Purity Numbers Are Physical-Data Functions: Exact Down-Sector Inversion (WP49)
Question
WP48 closed the M-theorem M = Du.Dd.V3/(C_class.scale^2) but left the three purity numbers C_u = 0.9993729702, C_c = 1.0008809, C_t = 0.9920333163 as unidentified ensemble constants (WP48 open item 3). Nima's WP40-41 instruction was explicit: cherish the unidentified constants until they are derived. WP49 derives them.
The inversion: down sector from physical inputs only
On the WP44 family (Hd_01 = 0), the down Hermitian sector inverts EXACTLY and RATIONALLY from physical data alone - the down masses lam_j and the singlet CKM row moduli u_j = |V_ckm[k,j]|^2 at the eigen-sorted singlet position k (u -> 0, c -> 1, t -> 2):
d0 = sum_j u_j lam_j (spectral diagonal) beta = e3 . sum_j u_j / lam_j ((0,0) minor / resolvent) d1 = (beta.d0 - e3)/(e1'.d0 - e2 + beta) (linear in d1: the cubic terms cancel exactly) d2 = e1' - d1 v^2 = p(d0)/(d1 - d0), w^2 = p(d1)/(d0 - d1), p(x) = prod_j (x - lam_j), e1' = e1 - d0
where (d0, d1, d2, v^2, w^2) are the chart entries Hd = [[d0, 0, *], [0, d1, ], [, *, d2]], v^2 = |Hd_02|^2, w^2 = |Hd_12|^2 of the family presentation. The formulas are rational: no square roots, no fitting, no chart data beyond the physical invariants on the right-hand side.
Consequence: C_class is an explicit function of physical data
Substituting into the WP48 purity definition,
C_class = blockgap . Dd . V3 / (scale^2 . v^2 . sqrt(w^2)),
every factor is now physical: blockgap and scale^2 are up-sector mass data, Dd and the v^2, w^2 above are down masses plus CKM row moduli, V3 is a CKM modulus product. The purities are no longer unidentified numbers - they are evaluations of an explicit algebraic function F of the physical flavor point.
Certification (all 394 family sheets, machine precision):
| class | max rel err (d0,d1,d2,v2,w2) | max rel err C |
|---|---|---|
| u | 7.3e-13 | 1.4e-14 |
| c | 1.2e-11 | 1.7e-13 |
| t | 4.9e-13 | 2.3e-13 |
Cross-frame consistency: the exact-frame singlet row (first basis vector e_0) and the eigh eigen-sorted singlet row give identical d0 to 2.4e-13. The one implementation trap is the row index: the singlet CKM row sits at the eigen-SORTED position k, not at its flavor index; a flavor-index lookup silently reads the wrong row on permuted sheets.
Measured-point evaluation
F evaluated at the measured flavor point (OBS17 central values) reproduces the WP48 fitted constants within fit slack:
| class | C (measured point) | C (WP48 fitted) | deviation |
|---|---|---|---|
| u | 0.9993900742 | 0.9993729702 | 1.7e-5 |
| c | 1.0008845110 | 1.0008809 | 3.3e-6 |
| t | 0.9914853279 | 0.9920333163 | 5.5e-4 |
The deviations track the chi^2 slack of the physical inputs (the t-class monomial |Vub|/|Vcb| is the slack-sensitive one). The c-class deviation 3.3e-6 sits at the scale of the WP48 c-purity residual (1.4e-6), indicating that residual is fit offset, not structure - this tentatively settles WP48 open item 1.
Boundary and open items
- The linear d1 formula assumes the family (Hd_01 = 0). The complement-family inversion retains the loop-product term; deriving it is the natural continuation and would close WP48 open item 2.
- The purities are NOT simple CKM monomials - the WP44 negative control stands. F is an honest algebraic function (spectral sums and resolvent minors), not a monomial.
- Why F evaluates within 1e-2 of 1 for all three anchors is NOT explained by this derivation. That near-unity question is the new residue, stated openly.
- The measured-point evaluation computes the FAMILY-presentation value of F at the measured point; family presentations of the physical point exist (the fitted sheets), so this is legitimate, but a complement-only point would need item 1 first.
Verification
./.venv/Scripts/python checkers/wp49_purity_derivation.py reproduces
results/wp49_purity_derivation.json (10/10 gates, exit 0).