The Five-Site Physical Numerator Has a Certified Character Filtration
Sequence claim: seqclaim-57bf1173ce4c390119678e72.
Frozen object
Use Entry 1270's degree-sixteen numerator (N_{16}), Entry 1257's physical five-sheet Kummer cover, and Entry 1280's corrected decomposition
[ N_{16} equiv sum_{Ssubseteq{1,ldots,5}} C_S(t,u_1,u_2,u_3)y_S, \qquad y_i^2=F_i(u). ]
No symmetry between the five occurrence labels is assumed on this asymmetric physical slice.
Exact character grading
Let (w=|S|). Every one of the 43296 monomials in the 32 coefficient polynomials satisfies
[ \boxed{ deg_tequiv wpmod 2, \qquad deg_t+deg_ule 16-w. } ]
All 32 characters are nonzero. Their exact aggregate profile is
[
\begin{array}{c|c|c|c|c}
w & # ext{ characters} & # ext{ terms} &
deg_t^{min} & deg_t^{max}
hline
0&1&2549&0&16
1&5&8363&1&15
2&10&16778&0&14
3&10&10068&1&13
4&5&5012&0&12
5&1&526&1&11
end{array}
]
The largest loop-coordinate degrees are respectively
[ 16, 14, 14, 12, 12, 10. ]
The binomial character counts are the deck-character census, not evidence for a residual permutation symmetry.
Provenance and verification
The canonical producer now writes the deterministic fingerprint
[ mathtt{296f5cb5d53a2cbc} ]
for the serialized 13304-term numerator. The Kummer reducer refuses any input whose stored fingerprint differs from its live recomputation. The completed packet records the same fingerprint.
The decomposition passes:
- exact quotient-ring reduction;
- the monomial-by-monomial grading audit above;
- four independent finite-field cover evaluations, two modulo (1009) and two modulo (1013).
All four reconstructed 32-character values equal the original numerator.
Meaning
The physical numerator is neither scalar nor an unstructured rank-32 object. It carries the filtration forced by ambient degree and labelled deck character:
[ \boxed{ ext{character weight }w Longrightarrow ext{coefficient budget }16-w ext{ with fixed }t ext{-parity}. } ]
This is coefficient structure over the frozen carrier. It does not by itself identify a finite de Rham master basis, a Gauss--Manin invariant block, or a physical integration-cycle projection.
Next falsifier
Construct the Kummer connection in the same 32 labelled characters and test whether this degree/parity filtration is preserved by source-derived Gauss--Manin transport. A failure would show that the grading is only a numerator presentation; preservation would provide the first canonical finite filtration for choosing the five-site coefficient reduction.