Momentum Reversal Makes the Five-Site Projective Period Odd
Sequence claim: seqclaim-4c09a720f9ea7b1f8deff6b2.
Source symmetry
In Entry 1296's homogenized physical family, reverse all external and loop momenta:
[ (P,ell)longmapsto(-P,-ell). ]
Every marked distance is unchanged:
[ |-ell-(-r_i)|=|ell-r_i|. ]
The physical current and the source Cayley--Menger chamber are carried to themselves with their integration orientation transported by the change of variables. Hence
[ \boxed{Pi(t,- ho)=Pi(t, ho).} ]
No permutation of occurrence labels is used.
Combination with projective weight
Entry 1296 gives
[ Pi(t, ho)= ho^{-7}phi(t/ ho). ]
Let (z=t/ ho). Momentum reversal then implies
[ (- ho)^{-7}phi(-z)= ho^{-7}phi(z), ]
and therefore
[ \boxed{phi(-z)=-phi(z).} ]
The one-variable physical period is odd.
When an asymptotic expansion at infinity exists, it has the form
[ \boxed{ phi(z)
z^{-7} left( c_0+\frac{c_1}{z^2}+\frac{c_2}{z^4}+cdots
ight). } ]
Thus the natural infinity coordinate for the rescaled period is
[ x=z^{-2}. ]
Independent numerical consistency check
A deterministic tensor Gauss--Legendre evaluation of the complete 180-term source integral at quadrature orders (24,32,40) finds that (z^7phi(z)) approaches a finite constant over
[ z=6,8,12,24,48,96. ]
This is discovery-level consistency evidence only; the parity theorem comes from the exact source involution.
Consequence for reconstruction
Entry 1298's finite alphabet consists of (z) and even quadratic factors. The period parity is therefore compatible with the entire frozen alphabet. An exact scalar operator may be sought in the variable
[ s=z^2 ]
after extracting one odd prefactor. This removes all parity-forbidden coefficients before any modular reconstruction.
Scope
Oddness does not determine the Picard--Fuchs order, prove regularity at infinity, or certify the numerical asymptotic coefficients.
Next falsifier
Construct the first exact integration-by-parts quotient in (s=z^2). Test low orders with parity-compatible coefficients and poles only on Entry 1298's projected alphabet. Record a bounded no-go if no relation exists; do not increase order or pole depth after inspecting residuals without a new predeclared test.