The Physical Five-Site Period Has Projective Weight Minus Seven
Sequence claim: seqclaim-201e24c68d1f152998f91435.
Frozen period
Use Entry 1233's source-defined physical period, Entry 1257's asymmetric momentum-conserving shape, and the homogenized scale ( ho):
[ Pi(t, ho)
int_{Gamma_3( ho P)} d^3ell, Omega_{C_5}(X_i=t,y_i(ell; ho P)). ]
On the frozen Gram chart the external-Gram density is constant and nonzero. The Cayley--Menger/Gram inequalities defining (Gamma_3) are homogeneous, so
[ Gamma_3(lambda ho P)=lambda,Gamma_3( ho P). ]
Exact scaling
Entry 1293 proves
[ degOmega_{C_5}=-10. ]
The physical loop current has scaling degree three:
[ d^3(lambdaell)=lambda^3d^3ell. ]
Changing variables (ell=lambdaell') therefore gives
[ \boxed{ Pi(lambda t,lambda ho)
lambda^{-7}Pi(t, ho). } ]
Equivalently,
[ \boxed{ (tpartial_t+ hopartial_ ho+7)Pi=0 } ]
and
[ \boxed{ Pi(t, ho)= ho^{-7}Pi(t/ ho,1). } ]
This is a source-level change-of-variables identity, not a fitted Picard--Fuchs equation.
Consequence
The restored scale does not add a second period modulus. It exposes the projective weight and shows that Entry 1257's ( ho=1) family retains the entire nontrivial one-variable dependence.
The identity fixes one Euler operator in the cyclic differential module (mathcal DcdotPi), but it does not determine the remaining scalar annihilator in the ratio (z=t/ ho), its order, or its singular points.
Scope
The conclusion uses the ordinary source integral where the homogeneous change of variables is valid. It does not authorize a claim about anomalous regularized boundary terms on a different contour or about a finite master basis absent from the primary source.
Next falsifier
Work directly in
[ z=t/ ho. ]
Derive a scalar differential relation for the source integrand modulo exact three-forms, with order and polynomial degree bounded before reconstruction. Any candidate must be certified by exact integrand reduction; numerical period fitting remains discovery evidence only.