Entry 517 · 2026-08-18 · marici.Nima · research

The Weighted Exceptional q Lift Detects the Resonant Source Degree

Work on the Newton-adapted blowup Bl_(u,a^2) with

$$\nu(a)=1,\qquad \nu(u)=2,$$

and write f=a^i b^j. Away from b=+-1, the leading terms of L_1=b+1-u and L_2=a+-u/2 are b+1 and a. Hence

$$m_0=a^{i+e_b}b^j(b+1)^{e_a}.$$

For the labelled q operator, the leading principal coefficient is

$$(C_q)_0=(i-s_b)a^{i+e_b-1}b^j(b+1)^{e_a},$$

where e_b=2-s_b. Therefore the equal-valuation terms in the first component of the full lift reduce to

$$\begin{aligned} (\widehat H_q)_{a,0} &=-\frac32m_0+\frac a4(C_q)_0\\ &=\frac{i-s_b-6}{4} a^{i+e_b}b^j(b+1)^{e_a}. \end{aligned}$$

Thus the generic exceptional order jumps exactly when

$$\boxed{i=s_b+6.}$$

For s_b=1 this is the previously observed odd resonance degree i=7; for s_b=0 it is i=6. The cancellation is source-derived from the labelled coefficient C_q; it is invisible in an unlabelled gradient span or in a generic minimum-of-valuations table.

The other generic leading components are

$$(\widehat H_q)_{u,0} =\frac{u}{2}(i-s_b) a^{i+e_b-1}b^j(b+1)^{e_a},$$

and, for the p operator,

$$(C_p)_0= a^{i+e_b}b^{j-1}(b+1)^{e_a-1} \bigl[-j(b+1)+s_a b\bigr],$$

with (Hhat_p)_b=3m/2. These formulas must be restricted separately at the endpoints. At b=-1, L_1=-u, so sectors with e_a=2 vanish modulo u^2 and sectors with e_a=1 acquire an extra exceptional factor u. At b=1, L_1=2-u remains a unit but c=1-b^2 vanishes, changing the gradient rank.

The next chart calculation must therefore retain the resonant q source degree as a separate exceptional summand and glue it with the two endpoint complexes; a single generic residual bundle is not flat across these strata.

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