AMR-050-0027 / k501: original chord-pedal algebra

This note proves only the stated general-polygon algebra and reduces the
billiard invariant to another identity. The finite controls do not establish
the all-odd-period invariant.

1. Universal polygon identity.

Let P_i be consecutive vertices, e_i=P_(i+1)-P_i, t_i=e_i/|e_i|,
n_i=J t_i, with J the positive quarter turn. Let delta_i be the oriented
turn from t_(i-1) to t_i. Set

  w_i = -sin(2 delta_i)
      = -2 det(e_(i-1),e_i) (e_(i-1).e_i) / (|e_(i-1)|^2 |e_i|^2),
  S=sum w_i,   T=sum w_i P_i.

For the ordinary internal angle theta_i=pi-delta_i, w_i=sin(2 theta_i).
For signed stars, this is the explicit oriented-turn continuation; source
scope/conventions must not be silently assumed if its angles differ.

Let B(M) be the signed shoelace area of orthogonal projections of M to the
ORIGINAL SIDELINES, indexed in edge order. Then, for any nonzero edges,

  B(M)=B(0)+(T.M)/4-S|M|^2/8.                    (1)

Thus when S is nonzero, K=T/S is the unique stationary point and

  B(M)=B(K)-S|M-K|^2/8,
  B(K)=B(0)+S|K|^2/8.                          (2)

Proof. Translation by M leaves the pedal area unchanged. The two adjacent
translated feet at vertex P_i are
 ((P_i-M).n_(i-1)) n_(i-1) and ((P_i-M).n_i) n_i.
Their contribution is one half sin(delta_i) times the two scalar factors.
Write H_i=2 n_i n_i^T-I. Direct two-dimensional multiplication gives

 sin(delta_i) Sym(n_(i-1) n_i^T)
   = sin(2delta_i) I/4 + (J H_(i-1)-J H_i)/4.

Summing the quadratic part telescopes the H terms. For the linear part,
the remaining sum is sum J H_i(P_(i+1)-P_i); since H_i e_i=-e_i, it is
-J sum e_i=0. The isotropic and linear coefficients are consequently
-S/8 and T/4, establishing (1). No billiard or convexity assumption was used.

2. Reflection-coordinate specialization.

Let alpha=a^2, beta=b^2, D=alpha beta, lambda the nondegenerate elliptic
caustic parameter, and W_i=beta (x_i/a)^2+alpha (y_i/b)^2.
The positive-winding reflection law gives

 sin(delta_i)=2 sqrt(lambda) sqrt(W_i-lambda)/W_i,
 cos(delta_i)=1-2lambda/W_i,
 w_i=-4 sqrt(lambda) sqrt(W_i-lambda)(W_i-2lambda)/W_i^2.

Put R1=sum sqrt(W_i-lambda)/W_i and R2=sum sqrt(W_i-lambda)/W_i^2.
The prior centered-pedal telescoping identity 2D R2=(alpha+beta)R1 gives

 S=-4 sqrt(lambda) [1-lambda(alpha+beta)/D] R1.  (3)

For a real nondegenerate elliptic caustic R1>0. The only zero of (3) is
lambda=D/(alpha+beta), the four-periodic confocal caustic. Excluding this
value for odd closed traversals requires the rotation/least-period argument;
it is not a conclusion of numerical samples.

3. Root Jacobi convention and remaining invariant.

With P=(-a sn u,b cn u), W=alpha-(alpha-beta)sn^2 u, and
 sqrt(W-lambda)=sqrt(alpha-lambda)dn u, define
 S_J=sum dn/W, Tx=sum dn sn/W^2, Ty=sum dn cn/W^2,
 C=sqrt(lambda)*sqrt(alpha-lambda),
 q=[D-lambda(alpha+beta)]/(2D),
 c0=[(alpha+beta)D-lambda(alpha^2+beta^2)]/(2D).

The asserted coefficient formula to check is

 B(M)=C[c0 S_J + 2a(beta-lambda)Mx Tx
                    -2b(alpha-lambda)My Ty +q S_J |M|^2].

It implies S=-8Cq S_J and
 K=(-a(beta-lambda)Tx/(q S_J), b(alpha-lambda)Ty/(q S_J)).
Hence B(K)=C S_J(c0-q|K|^2). Since the prior odd-period theorem supplies
H=A S_J constant, the exact target A/B(K) is constant precisely when

              A^2 / (c0-q|K|^2)

is constant on its nonzero-denominator domain. This last all-period identity
is still a mathematical gap in this algebra lane; it has not been proved
by the finite numerical controls.

4. Current experimental boundary.

The actual run-001 binary64 scout accepts 26 families, each at seven phases:
N=3,5,7,9, aspect ratios1.1,1.6,2, including admitted coprime star windings.
Max relative A/B(K) variation is3.525e-10. One N=9,winding4,a/b2 family
failed the strict geometry tolerance and is retained as a failure. This is
not a certified counterexample search or a general proof. Exact N=3 tests
only verify the classical triangle circumcenter/medial-pedal ratio4.
