AMR-050-0027: exact first-moment reduction from closed velocity jumps
1 October 2026. This is an algebraic reduction, not a proof that A_K is nonzero.

Use P(u)=(-a sn u,b cn u), alpha=a^2, beta=b^2,
W=alpha-(alpha-beta)sn^2 u, and the original nondegenerate elliptic caustic
parameter 0<lambda<beta. Write s=sn u,c=cn u,dn=d.
The outward unit normal is n=(-b s,a c)/sqrt(W).

At a positive-winding reflection the incoming and outgoing unit velocities
satisfy

  v_in-v_out=2sqrt(lambda)(-b s,a c)/W.

Summing around ANY closed traversal cancels every incoming velocity with the
preceding outgoing velocity. Consequently, identically as the starting phase
varies in a Poncelet family,

  sum_i s_i/W_i=0,      sum_i c_i/W_i=0.             (1)

No least-period assumption is used in this cancellation. It is valid for
admissible coprime stars and repetitions under the same physical reflection
convention. The chosen positive winding only fixes a common jump sign.

Differentiate (1) with respect to the common Jacobi phase u. Since
s'=c d, c'=-s d, W'=-2(alpha-beta)s c d,

  (c/W)' = d s/W - 2 beta d s/W^2,
  (s/W)' = 2 alpha d c/W^2 - d c/W.

All sums are finite, so differentiation commutes with summation. Thus

  Tx := sum d_i s_i/W_i^2 = (1/(2beta)) sum d_i s_i/W_i,
  Ty := sum d_i c_i/W_i^2 = (1/(2alpha)) sum d_i c_i/W_i.    (2)

This is stronger than a local residue calculation. The right-hand sides have
only simple W poles individually, whereas the left-hand summands have double
W poles. Hence the trace double-pole cancellation follows directly from (2),
without needing to track adjacent root pairs again.

Let S_J=sum d_i/W_i and q=[alpha beta-lambda(alpha+beta)]/(2alpha beta).
The root's stationary-point formula simplifies globally to

 Kx=-a(beta-lambda)/(2beta q) * (sum d_i s_i/W_i)/S_J,
 Ky= b(alpha-lambda)/(2alpha q) * (sum d_i c_i/W_i)/S_J.    (3)

In physical coordinates, define the positive real weighted vertex mean

 R = [sum (d_i/W_i) P_i] / S_J.

Then

 Kx=alpha(beta-lambda)/(alpha beta-lambda(alpha+beta)) R_x,
 Ky= beta(alpha-lambda)/(alpha beta-lambda(alpha+beta)) R_y. (4)

For q>0 both diagonal factors in (4) exceed1. Therefore positivity of the
weights and R lying in the convex hull do NOT by themselves imply K lies in
the polygon or caustic. Using that invalid inference would leave a real gap.

Possible further route, NOT established in this note:
For genuine odd n, step h=4K tau/n with gcd(tau,n)=1, the traces S_J and
D=sum dn are invariant under phase2K/n, while the numerators in (3) change
sign under2K/n and have period4K/n. This follows from shift-invariance underh,
period/antiperiod2K, and Bezout with n odd. Their imaginary-period characters
also match the standard Jacobi dn,sn,cn pattern. Together with the known
simple-pole divisors, this suggests an odd isogeny/Landen parametrization of
the Steiner centroid locus as an ellipse. A proof and a strict noncircularity
or positivity estimate would still be needed to rule out A_K identically0
in the q>0 regime. No such endpoint conclusion is claimed here.
