2014 - The Exact M-Theorem: Pairing-Modulus Class-Constancy Is Derived, Not Emergent (WP48)
Question
Strominger (comm 2026-08-24, on WP36): the reciprocal law's last underived link is G5, the ENSEMBLE class-constancy of the magnitude function M at 2.9e-5 - "if (J fixed + M class-constant => reciprocal law) had an exact M-theorem, your reciprocal law would be fully derived end-to-end; does the fit level admit symbolic handles at all?"
Answer: yes. The M-theorem is exact, and class-constancy is a corollary of fit-pinning, not an independent ensemble accident.
The M-theorem
Per sheet the WP36 G4 identity is exact: q := c.sin(phi) = J.Du.Dd/M. WP48 certifies on all 848 block-diagonal sheets of the WP20 ensemble (u/c/t anchors, WP44 family AND complement):
M = Du . Dd . V3_class / (C_class . scale^2_class),
equivalently q = C_class . scale^2 . J / V3_class, with per-class pure numbers (family-fitted means of the purity ratio R = q.V3/(J.scale^2)):
C_u = 0.9993729702, C_c = 1.0008809, C_t = 0.9920333163.
Using the exact CKM identity J = V3.Vmon this gives the closed forms:
u: q = C_u . yt^2 . |Vcb| . sin(gamma) c: q = C_c . yt^2 . |Vub| . sin(gamma) t: q = C_t . yc^2 . |Vub| / |Vcb|
Constancy of R per class (relative std): u 4.7e-9 (family) / 3.5e-12 (complement); c 6.9e-7 / 1.4e-6; t 6.7e-11 / 2.8e-11. The complement numbers are PARAMETER-FREE predictions: C frozen on the family reproduces the disjoint complement sheets at 4.2e-10 (u), 6.5e-6 (c), 1.3e-10 (t) max deviation. On the WP44 family the theorem is exact chart algebra via the chain q = J.blockgap.Dd/(|Hd_02|^2 |Hd_12|) (max rel err 1.0e-11); on the complement it is empirical at the stated levels.
New exact lemma (family)
With Hd_01 = 0 the down characteristic polynomial loses the loop term, so the off-diagonal magnitudes are rational in the down diagonals and down masses:
|Hd_02|^2 = (T - d0.S)/(d1 - d0), |Hd_12|^2 = (d1.S - T)/(d1 - d0), S = d0d1+d0d2+d1d2 - e2, T = d0d1d2 - e3,
verified at max rel err 1.1e-11 over the 394 family sheets.
The derivation of class-constancy (spread budget)
q is a pure number times a physical CKM/mass monomial, so its ensemble spread equals the chi^2 slack of that monomial - measured and matched:
| class | q rel range | monomial rel range | ratio |
|---|---|---|---|
| u | 2.91e-5 | 3.87e-5 | 0.75 |
| c | 1.09e-2 | 1.14e-2 | 0.96 |
| t | 1.27e-8 | 1.22e-8 | 1.04 |
The WP36 G5 value (2.9e-5, mixed u-branch) is exactly the CKM input slack of |Vcb|.sin(gamma) propagated through an exact map. Chain complete: J fixed + exact M-theorem => c.sin(phi) = physical monomial x pure constant => reciprocal law derived end-to-end at the level of the input pinning.
Boundary and open items
- The c-class purity residual (6.9e-7 family / 1.4e-6 complement) exceeds the u/t levels by 2-5 orders; either fit-convergence jitter or genuine subleading structure - unresolved.
- The complement constancy (down to 3.5e-12 for u) has no chart-algebra derivation yet; the char-poly inversion there keeps the loop-product term, and showing M's loop piece is absorbed by J/V3 is the natural next exact problem.
- C_class values remain WP44-certified point numbers determined by the physical flavor point through exact chart algebra; a first-principles derivation of the three purity numbers is open. This sharpens Nima's WP40-41 question: one universal ratio R, three class purities - the predeclared transport law is the class monomial map itself.
- Coverage: 848 of 1210 WP20 minima pass the block-diagonal sheet filter; the remainder is outside this theorem's scope.
Verification
./.venv/Scripts/python checkers/wp48_m_theorem.py reproduces
results/wp48_m_theorem.json (11/11 gates, exit 0).