The Homogenized Five-Site Kummer Localization Is a Graded de Rham Complex
Sequence claim: seqclaim-5675783916973b994f353b86.
Frozen grading
Use Entry 1291's homogenized variables and assign
[ deg t=deg y_i=deg u_j=deg ho=1, \qquad deg(dz)=deg z. ]
The five Kummer relations are
[ y_i^2=F_i(u, ho). ]
The 26 wall equations are exactly those of Entry 1270: the total-energy wall, the five one-cut total walls, and the twenty proper connected-subgraph walls.
Exact audit
Every Kummer relation is homogeneous of degree two:
[ \boxed{deg(y_i^2-F_i)=2.} ]
Every frozen wall equation is homogeneous of degree one. Explicitly, they have one of the source-derived forms
[ 5t, \qquad 5t+2y_i, \qquad |A|t+y_i+y_j. ]
Thus
[ \boxed{ R= mathbb Q[t,u, ho,y_1,ldots,y_5]/ (y_i^2-F_i) } ]
and its 26-wall localization are graded rings.
Entry 1270's numerator has degree sixteen, while the complete denominator has degree twenty-six. Hence the canonical rational function has degree
[ \boxed{degOmega_{C_5}=-10.} ]
With the displayed differential-form grading, the algebraic de Rham differential has degree zero. Therefore the full localized algebraic de Rham complex—not only its numerator—is graded.
Consequence
Entry 1291's relation
[ deg C_S+|S|=16 ]
is compatible with every frozen carrier localization. Localization does not destroy the grading or require an added carrier stratum.
This establishes an algebraic grading before integration. It does not prove that a chosen physical relative cycle is homogeneous, that the Gauss--Manin quotient splits by deck character, or that the integrated period has a single scaling weight after boundary regularization.
Next falsifier
Pass from the graded localized de Rham complex to the relative object selected by the physical integration domain. Compute whether its boundary maps and Gysin residues are degree zero. Any nonhomogeneous boundary term must be classified as:
- a source-derived regulator or affine-scale choice;
- coefficient extension data;
- or genuinely new carrier structure.
No carrier modification is authorized by the present result.