Outer-tangent focal-distance products: exact factor and cyclic norm
AMR-050-0007, literal invariant k115, raw ID 5100007.

1. Domain and genuine period

Let the caustic semiaxes be A>B>0, c^2=A^2-B^2, k=c/A,
k'=B/A, and K,K' the real complete quarter periods. Let 0<delta<K,
a=A dn(delta)/cn(delta), b=B/cn(delta). The original boundary ellipse
has these semiaxes and the same foci F_epsilon=(epsilon*c,0).
Write h=2delta=4tau K/n, gcd(tau,n)=1, 0<tau<n/2.
The theorem below requires genuine least period n divisible by four.
Any repeated traversal of such an orbit also has an invariant product.
An arbitrary listed period divisible by four is not enough: Section 6
gives an exact padded-odd obstruction. Hyperbolic and degenerate
caustics are not introduced into this elliptic-caustic statement.

All Jacobi functions here have modulus k (parameter m=k^2). Put
P(v)=(-a sn(v),b cn(v)). The intersection of boundary tangents at
P(v-delta), P(v+delta) is

  R(v)=(-a dn(delta) sn(v)/cn(delta), b cn(v)/cn(delta)).       (1)

To check the actual outer construction, substitute (1) into either
tangent equation -sn(v +/- delta)*X/a + cn(v +/- delta)*Y/b=1.
The two Jacobi addition numerators have cancelling mixed terms;
the remaining numerator is cn(delta)*(dn(delta)^2 sn(v)^2+cn(v)^2),
equal to cn(delta)*(1-m sn(delta)^2 sn(v)^2). This is exactly the
addition denominator. Thus (1) is not an inverted/polar/inner polygon.
It is finite because cn(delta)>0. No adjacent original vertices are
antipodal since h<2K. All real focal distances below are positive.

2. Squared distance factor

Set x=sn(delta)^2, q=1-x=cn(delta)^2, D=1-mx=dn(delta)^2,
J=1-2x+mx^2, U=1-2mx+mx^2, V=1-mx^2. Then

  U+J=2Dq,  cn(h)=J/V,  dn(h)=U/V, V>0.

Using B^2=A^2(1-m), (1) is
R=(-A D sn(v)/q, B cn(v)/q). Direct Euclidean expansion gives

  |R-F_epsilon|^2
   = A^2/q^2 * [U+mJ sn(v)^2+2epsilon*k*Dq*sn(v)]
   = A^2 V/q^2 * (1+epsilon*k*sn(v))
                        * (dn(h)+epsilon*k*cn(h)*sn(v)).    (2)

The coefficientwise polynomial identities are checked in the portable
script. Since dn(h)^2-m cn(h)^2=1-m>0 and dn(h)>0, the last factor
is strictly positive on the real axis, even when cn(h)<0. The first
factor is at least 1-k>0. Thus a focus is never an outer vertex and
no absolute-value/square-root sign ambiguity has been suppressed.

3. Two cyclic products

For v_j=v+jh, let

  H_epsilon(v)=product_j (1+epsilon*k*sn(v_j)),
  G_epsilon(v)=product_j (dn(h)+epsilon*k*cn(h)*sn(v_j)),
  Q(v)=product_j (1-m*sn(v_j)^2*sn(h)^2), j=0,...,n-1.

Since n=4r is primitive, tau is odd. Translation by n/2 indices
adds 2tau K and reverses sn. Pairing these indices gives

  H_epsilon=product_{j=0}^{n/2-1} dn(v_j)^2.

Translation by n/4 indices adds tau K, congruent to K or -K modulo
2K. The standard identity dn(u)dn(u+K)=k' therefore pairs this
half-grid into n/4 pairs and gives H_epsilon=(k')^(n/2), both signs.

The addition identity

 (1+epsilon*k*sn(v+h))(1+epsilon*k*sn(v-h))
       = (dn(h)+epsilon*k*cn(h)*sn(v))^2
                         / (1-m*sn(v)^2*sn(h)^2)            (3)

follows directly from the sn addition numerators and cn^2=1-sn^2,
dn^2=1-m sn^2. Its denominator is positive for real v. Multiplying
(3) around the closed grid reindexes the two left products, giving

  G_epsilon(v)^2 = H_epsilon(v)^2 Q(v).                     (4)

Real positivity chooses G_epsilon=H_epsilon sqrt(Q). It remains to
prove Q constant, not merely claim a norm has an invariant divisor.

4. Complete divisor cancellation for Q

Work on the compact torus with lattice Lambda=2K Z+2iK' Z, on which
sn^2 is elliptic. It has exactly one double-pole class p=iK' and
degree two. For 0<h<2K, sn(h)>0. The reciprocal identity

  sn(iK'+z)=1/(k sn(z))

shows that D_h(v)=1-m sn(v)^2 sn(h)^2 has zero classes p+h and p-h.
They are simple unless h=K modulo 2K. In that case they coincide,
D_h=dn^2, and the zero has order two. There are no additional zeros:
the total zero order of this nonconstant elliptic function is two,
equal to its sole double pole. Thus, WITH MULTIPLICITIES,

  div D_h = [p+h]+[p-h]-2[p].

Consequently

  div Q = sum_j ([p+h-jh]+[p-h-jh]-2[p-jh]) = 0,

because nh=4tau K belongs to Lambda and both index shifts permute
the same cyclic multiset. This covers coincident poles, repeated
Lambda-grid points, and the h=K double-zero case; it does not require
the neighbouring factors to be distinct. Q is Lambda-elliptic, and
its zero divisor means every apparent pole is removable. It is
holomorphic on the compact torus and therefore constant. Finally
D_h(v)>0 on the real axis, so that constant is not zero. In particular

  Q=product_{j=0}^{n-1} [1-m sn(jh)^2 sn(h)^2] > 0.          (5)

5. Result and circle/repetition boundary

Taking the positive square root of the product of (2) and using
(4)-(5), for either original ellipse focus,

 product_j |R(v_j)-F_epsilon|
   = (A^2 V/q^2)^(n/2) (k')^(n/2) Q^(1/4),                (6)

independent of phase. All admitted coprime star windings are included.
For d repetitions of this same primitive orbit the actual N=dn-term
product is the d-th power of (6). The circular case k=0 is direct:
both foci are the center and every outer vertex has radius a/cos(delta),
so the product is [a/cos(delta)]^N. No singular complex-period limit
is needed. The nondegenerate caustic excludes a primitive diameter.

6. Unrestricted listed-period obstruction

Reuse the exact source-valid primitive triangle verified in
AMR-050-0023/attacks/geometry/check_odd_repeat_centroid.py:
a=sqrt(21), b=4, c=sqrt(5), lambda=336/25, with original vertices
(a,0),(-3a/5,16/5),(-3a/5,-16/5). The actual outer tangent vertices
are (a,8),(-5a/3,0),(a,-8). For the fixed positive focus the squared
product of the three distances is 413600+5600sqrt(105). The centrally
negated triangle, in the same fixed-ellipse fixed-caustic family,
gives 413600-5600sqrt(105). Both are positive and unequal. Repeating
each triangle four times gives listed N=12 but least period three;
the N-term products are unequal positive powers. This is NOT a
counterexample to the genuine-4r theorem, but it forbids replacing
the genuine-period hypothesis by unrestricted displayed-list length.
The checker verifies this outer construction and exact quadratic
arithmetic; the reused earlier script verifies the primitive triangle's
reflection and common caustic. No original source convention is
silently decided by this control.

7. Evidentiary limits

The portable Fraction checker verifies coefficient identities, a genuine
primitive eight-periodic reflected orbit with lambda=24/7, and the exact
padding calculation. The N8 ellipse has a^2=40,b^2=15,c=5 and focal
outer product 2000000/3. These finite checks support the algebra and
scope, not the all-period analytic proof. Complex divisor facts and
compactness are stated explicitly above. No novelty, independent human
review, formal verification, manuscript readiness, canonical acceptance,
external publication or whole-source closure is performed by this lane.
