ORIGINAL-POLYGON ANTIPEDAL CENTROIDS: AN ELEMENTARY PERIMETER REDUCTION
AMR-050-0023, k405. Algebra lane, 2 October 2026 local date.

Scope and conventions
---------------------
This note concerns antipedals of the ORIGINAL billiard polygon P, not
the outer tangent polygon. All antipedal sides are complete supporting
lines through P_i perpendicular to P_i-M. The vertex centroid is their
consecutive-intersection arithmetic mean; no area denominator occurs.
The source's period convention still needs explicit treatment. The full
argument below uses an even primitive period, allowing repetitions of
that orbit. It does NOT cover an odd primitive orbit merely listed twice.
An exact warning about that distinction is included below.

Put O=0, E: x^2/a^2+y^2/b^2=1, a>b>0, c^2=a^2-b^2, F=(c,0),
and caustic parameter 0<lambda<b^2. Define
  D=a^2 b^2, B=b^2-lambda,
  K=(D-lambda c^2)/B.
For any effective-even billiard with N displayed vertices and perimeter L,
the proposed exact result is
  C_0^*(O)=O,
  C_0^*(F)=kappa F,
  kappa=-1+K/c^2*(1-b^2 L/(2N sqrt(lambda D))).                 (A)
The opposite focus has the same coefficient kappa. In translated and
rotated coordinates the formula is O+kappa(F-O). Uniform scaling leaves
kappa unchanged. The ratio L/N is unchanged by repeated traversal.

1. Finiteness
-------------
Both the ellipse center and either focus lie strictly inside the
confocal elliptic caustic: its squared major semiaxis is a^2-lambda>c^2,
and b^2-lambda>0. A chord sideline tangent to that caustic cannot pass
through any of these three points. Thus consecutive vectors P_i-M,
P_(i+1)-M are linearly independent. The antipedal intersection is finite.
No assumption of nonzero signed area is needed, and focal intersections
do not degenerate as long as the caustic remains elliptic/nondegenerate.

2. One chord and its opposite
----------------------------
Choose unit normal n=(nx,ny) to a chord, its positive support h>0, and
t=(-ny,nx). Write the endpoints as
  P=h n+(s-ell)t, Q=h n+(s+ell)t,
where ell>0 is the half chord length. Set
  g=a^2 nx^2+b^2 ny^2=b^2+c^2 nx^2=h^2+lambda.
Restricting the ellipse equation to h n+u t gives exactly
  (g/D)*[(u-s)^2-ell^2]=0,
  s=-h c^2 nx ny/g, ell^2=lambda D/g^2.                       (B)
In particular the chord length is 2sqrt(lambda D)/g.

For F=(c,0), put f=F.n=c nx and v=F.t=-c ny. Then
  (h-f)(h+f)=b^2-lambda=B>0.
Let W_F(P,Q) denote the consecutive antipedal intersection. Subtracting
its two defining line equations, then substituting into either one, gives
  W_F(P,Q).t = 2s-v,
  W_F(P,Q).n = h+[ell^2-(s-v)^2]/(h-f).                       (C)
The opposite chord has endpoints -P,-Q. In the SAME frame n,t,
its corresponding components are
  W_F(-P,-Q).t = -2s-v,
  W_F(-P,-Q).n = -h-[ell^2-(s+v)^2]/(h+f).
Therefore (B), c^2=a^2-b^2 and nx^2+ny^2=1 reduce the pair to
  W_F(P,Q)+W_F(-P,-Q) = -2F+2K*(f/g)n.                      (D)
The exact checker verifies this rational identity by polynomial reduction,
as well as the chord-section identity in (B); it does not substitute a
finite numeric grid for these symbolic identities.

3. The off-diagonal trace from billiard stationarity
--------------------------------------------------
Every nondegenerate periodic billiard is stationary for total positive
perimeter under independent variations of its vertices tangent to E.
Indeed the coefficient of delta P_i in the first variation is the
incoming unit velocity minus the outgoing unit velocity, hence a normal
to E by reflection. This argument also applies to self-intersecting
orbits; no signed edge lengths or convex-polygon restriction is used.

Use the specific tangent variation delta P_i=T P_i, where
  T=[[0,-a/b],[b/a,0]].
It is tangent because (x/a^2,y/b^2).T(x,y)=0. For each edge of length
d_i and unit direction u_i, its first variation is d_i u_i.T u_i.
Hence
  0=(b/a-a/b)*sum_i d_i u_ix u_iy.
For a noncircle, its nonzero prefactor gives sum_i d_i u_ix u_iy=0.
Since u_i=+/-t_i, u_ix u_iy=-n_ix n_iy. Using (B),
  sum_i n_ix n_iy/g_i=0.                                   (E)
The unsigned direction choice of each chord does not affect this identity.

4. The diagonal trace and the centroid
-------------------------------------
An even primitive confocal elliptic billiard is centrally symmetric,
as in the classical Jacobi parametrization with coprime winding number.
Pair opposite chords and sum (D). Because f_i=c n_ix,
  C_0^*(F)=-F+(K/N)*sum_i (F.n_i)n_i/g_i.
Its y component vanishes by (E). The elementary diagonal relation is
  nx^2/g=(1-b^2/g)/c^2.
Also sum_i 1/g_i=L/(2sqrt(lambda D)) by the chord length in (B).
These identities give (A). For M=O, direct half-turn equivariance of
the antipedal intersections gives the centered centroid O.

The total perimeter is constant on a connected smooth Poncelet family:
differentiate it along that family, whose vertex derivatives are tangent
to E, and apply the same first-variation stationarity. Thus (A) is a
phase-independent value. The parametrized ellipse billiard family is
smooth in the nondegenerate elliptic-caustic regime. Reversing traversal
does not change positive perimeter or an arithmetic vertex mean.

5. Circle, covariance and repeated lists
----------------------------------------
In a circle the two foci coincide with O. The common antipedal of a
genuine regular star orbit is again a concentric regular star, obtained
by intersecting equal-radius radial-normal lines. Nondegenerate interior
caustic excludes the diameter case, so consecutive intersections are
finite; the arithmetic centroid is O. This handles the circle directly
without dividing by c^2 in (A).

Translations, rotations and positive uniform scalings commute with the
line construction and vertex averaging. They also preserve kappa: its
dimensions cancel in (A). Repeating an effective-even orbit multiplies
L and N by the same integer, while retaining its centroid.

The following exact odd-primitive control forbids silently weakening
the effective-period convention. On a^2=1,b^2=3/8,lambda=9/25, take
  P=((1,0),(-4/5,3sqrt(6)/20),(-4/5,-3sqrt(6)/20)).
It is a genuine primitive three-periodic billiard. Its center-antipedal
vertices have centroid (11/32,0). The opposite phase -P is in the same
ellipse/caustic family and has centroid (-11/32,0). Listing P twice
and -P twice makes two displayed six-lists but leaves those centroids
unequal. This is NOT a counterexample within the effective-even theorem.

6. Exact controls and limitations
--------------------------------
The standalone lane script check_antipedal_centroid.py uses existing
SymPy1.13.1 with its dependency SHA pinned. It checks symbolic (B)-(D),
exact ellipse incidence/reflection/interior caustic contacts and direct
antipedal intersections for two phases of one N4 family, three axial N6
examples, and one circle N6. Each source pole O,F+ and F- is included.
Rational translation, rotation and positive scaling are then checked
on the actual intersection coordinates, not only on formula (A).

N4 a=2,b=1,lambda=4/5: axial and rectangle phases give kappa=0.
Axial N6 a=2,3,1+sqrt(3),b=1,lambda=a^2/(a+1)^2: kappa=-1/3.
The last value remains finite at the outer-antipedal zero from0036;
here the polygon used is original P, not outer P'.

No novelty assertion, whole-source acceptance, status update, manuscript,
publication or external request is made by this algebra lane. Classical
central symmetry, first variation and constant perimeter should be
properly credited in the source/literature and main proof lanes.

Rerun with an actual fresh evidence directory:
  python3 -I -B problems/AMR-050-0023/attacks/algebra/run_check.py
The runner preserves actual stdout/stderr, command, time and input hashes.
