The Five-Site Character Filtration Is the Shadow of an Exact Homogeneous Grading
Sequence claim: seqclaim-e15fbe68211a934e86a3f4fc.
Question
Entry 1286 found on the physical slice that a character (S) of weight (w=|S|) obeys
[ deg_tequiv wpmod 2, \qquad deg_t+deg_ule16-w. ]
This could have been an accidental filtration caused by setting the external kinematic scale to one.
Frozen homogenization
Restore one scale ( ho) in the five physical radicands:
[ F_1=2u_1^2+2u_2^2+u_3^2-2u_1u_2-2u_2u_3, ]
[ F_2=F_1-2 ho u_1+ ho^2, \qquad F_3=F_1-2 ho u_2+2 ho^2, ]
[ F_4=F_1-2 ho u_3+3 ho^2, ]
[ F_5=F_1+2 ho u_1+2 ho u_2-8 ho u_3+29 ho^2. ]
Each (F_i) is homogeneous of degree two in ((u_1,u_2,u_3, ho)). No coefficient or normalization is fitted.
Exact result
Reduce Entry 1270's degree-sixteen numerator by (y_i^2=F_i):
[ N_{16}
sum_{Ssubseteq{1,ldots,5}} C_S(t,u, ho)y_S. ]
Every one of the 32 coefficients is nonzero, and every monomial of (C_S) has exactly the same total degree:
[ \boxed{ deg_{t,u, ho} C_S=16-|S|. } ]
The degree table is therefore
[
\begin{array}{c|c|c}
|S|&# ext{ characters}° C_S
hline
0&1&16
1&5&15
2&10&14
3&10&13
4&5&12
5&1&11
end{array}
]
The exact term count remains 43296. All 32 rows pass both the homogeneous degree audit and Entry 1286's parity/filtration audit.
Interpretation
The physical ( ho=1) filtration is the specialization of the bigrading
[ \boxed{ deg(C_S)+|S|=16. } ]
Thus deck-character weight consumes one unit of the ambient numerator degree. This grading comes from the frozen Kummer algebra and source numerator; it is not inferred from a residual permutation symmetry.
It is still a grading of the algebraic numerator. It does not yet prove that the full localized canonical form, its de Rham quotient, or Gauss--Manin transport splits by this grading.
Next falsifier
Construct the source-localized differential in the homogenized character basis. Test whether it is homogeneous after assigning
[ deg y_i=1,\qquad deg F_i=2. ]
If localization necessarily mixes total grades beyond the shifts dictated by the 26 frozen walls, then this is only a numerator grading. If the differential is graded, it supplies a canonical filtration for the five-site coefficient complex.