Focal inverted area product: direct elliptic pole cancellation
AMR-050-0044 / literal k804,a. Algebra lane, 2 October 2026.

This is a mathematical derivation, not canonical acceptance, a publication,
a novelty claim, or a proof-assistant certificate. The root researcher
independently proposed the direct product argument; this note checks its
complete singularity list, signs, and the N=4 adjacency case.

DOMAIN AND PARAMETERS
Let a>b>0, alpha=a^2, beta=b^2, c^2=alpha-beta, 0<lambda<beta.
The confocal caustic has squared axes alpha-lambda,beta-lambda. Put
k^2=c^2/(alpha-lambda), K=K(k), K'=K(sqrt(1-k^2)),
delta=2 tau K/n, h=2delta, gcd(tau,n)=1, 0<tau<n/2.
For a genuine n=4r orbit tau is odd. Its canonical vertices are
P(u+jh)=(-a sn(u+jh),b cn(u+jh)). These are the actual confocal
billiard parameters used in the existing 0007/0014/0021 dossiers, not
equally spaced eccentric anomaly. Star windings are included.
sn^2 delta=lambda/beta, cn^2 delta=(beta-lambda)/beta,
dn^2 delta=alpha(beta-lambda)/(beta(alpha-lambda)).

All areas are ordered signed shoelace areas. Translation does not change
them. For F_e=(e c,0), e=+/-1, define I_e(P)=(P-F_e)/|P-F_e|^2;
the physical inversion adds F_e back, which does not change its area.
Write A(u)=area(P(u+jh)), B_e(u)=area(I_e(P(u+jh))). Complex extensions
use bilinear squares, never Hermitian norms. A common period lattice is
4K Z + 4iK' Z. sn(u+2iK')=sn u and cn(u+2iK')=-cn u;
therefore A and B_e change sign under 2iK', rather than being periodic
under that half-period. This fixes the imaginary-period convention.

REAL REGULARITY
For real u, a>|c sn u|, and
|P(u)-F_e|^2=(a+e c sn u)^2>0.
Thus every inverted vertex is finite. A genuine elliptic-caustic chord
has its caustic, hence O, strictly to one side and is not diametral.
There is no lost finite-real configuration or need to divide by B_e.

ORIGINAL AREA POLES
The common sn/cn poles are p=2jK+(2l+1)iK'. They are simple.
At most one endpoint of any adjacent original edge has such a pole,
because h is not 0 or 2K modulo4K for n>=4. Thus A has at most simple
poles, at p-jh. Nonadjacent simultaneous poles can add residues but not
raise their orders. About every p, P(p+w)=-P(p-w).
At Z=p+K, P(Z+w)=P(Z-w). These follow, for p=iK', from
sn(p+w)=1/(k sn w), cn(p+w)=-i dn w/(k sn w), and
sn(K+iK'+w)=dn w/(k cn w),
cn(K+iK'+w)=-i k'/(k cn w). Translations multiply coordinates by
signs and preserve oddness/evenness.

COMPLETE INVERSE VERTEX POLES
For e=-1, a+e c sn=0 is sn=a/c; its roots on the common torus are
K+iK'+/-delta and K+3iK'+/-delta.
For e=+1 the roots are the same with K replaced by3K (sn changes sign
under2K). At all roots,
cn^2=-beta/c^2, dn^2=-lambda/(alpha-lambda), so sn'=cn dn is nonzero.
Thus the scalar q=a+e c sn has a simple zero and I_e(P) a double pole.
The sn^2 degree-two divisor on2K Z+2iK' Z shows that these exhaust
the roots; the sign choice and4K,4iK' lifts give exactly those listed.
At a Jacobi pole, P-F_e is O(w^-1), whereas q^2 is O(w^-2), so
I_e(P)=O(w) is holomorphic with a zero. There are no other poles.

The inverse-area pole traces have base phase Z-delta-jh, with the
corresponding Z for the chosen focus and imaginary row. The other root
Z+delta is included by one cyclic-index shift since2delta=h.
At any one such phase exactly two adjacent inverse vertices are singular.
A third would require another real shift identifying a root modulo4K;
the only two sn=a/c (or -a/c) roots on that real row differ by h.
The two roots are not congruent modulo4K since0<h<2K. The opposite
vertices, shifted by2K, belong to the OTHER focus and are regular.
In particular n=4,h=K has precisely the same two-adjacent pattern;
there is no additional antipodal pair of poles for the same inversion.

DISJOINTNESS FOR n DIVISIBLE BY FOUR
An original pole and inverse pole could coincide only if
K +/- delta == jh (mod2K), hence
n(1-2l)=2tau(2j +/- 1)
for integers j,l. The left side is divisible by4; the right is2mod4
since tau is odd. Impossible. Consequently A is holomorphic at all
inverse-area poles, and B_e holomorphic at all original-area poles.
This exclusion fails for some n=2mod4; it must not be inferred merely
from the parity of an arbitrary repeated orbit's displayed list length.

LAURENT ORDER AT AN INVERSE-AREA POLE
Put u0=Z-delta and w=u-u0. The two singular vertices have arguments
Z-delta+w and Z+delta+w. At w=0 their original P values coincide by
the evenness about Z. Let v=P-F_e=v0+v1 w+... and q=q1 w+q2 w^2+... .
Because v.v=q^2, v0.v0=0 and v0.v1=0. Here v0 is nonzero: explicitly
its coordinates are (e beta/c,b cn(u*)) and cn(u*)^2=-beta/c^2.
In two-dimensional complex bilinear geometry the orthogonal complement
of a nonzero isotropic vector is its own span. Therefore v1 is parallel
to v0. Both negative Laurent coefficients of v/q^2 are parallel to v0:
w^-2 v0/q1^2 and w^-1(v1/q1^2-2q2 v0/q1^3).
The two singular vertices share that direction. Their mutual determinant
has no order4 or3 pole. Determinants with the remaining finite adjacent
vertices have order at most2. Hence B_e has order at most2 at u0.

The index permutation j -> 1-j reverses the cyclic order, and evenness
P(Z+w)=P(Z-w) gives B_e(u0+w)=-B_e(u0-w). Its Laurent expansion is
odd, so the order2 coefficient is zero. Therefore B_e has at most a
simple pole. The same index reversal gives A(u0+w)=-A(u0-w).
Since A is holomorphic here by disjointness, A(u0)=0. Thus A B_e
has no pole at any inverse-area pole, including any removable one.

ZERO OF INVERSE AREA AT EACH ORIGINAL POLE
Central symmetry P(u+(n/2)h)=-P(u), since tau odd, implies
B_+(u)=B_-(u): inversion at the opposite focus is the negative vector
polygon after cyclic relabeling, and a global sign does not change area.
At an original pole p, oddness P(p+w)=-P(p-w), combined with j -> -j,
gives B_e(p+w)=-B_-e(p-w)=-B_e(p-w). B_e is holomorphic at p, so
B_e(p)=0. It cancels the at-most-simple pole of A. Reindexing covers
all shifted original poles.

CONCLUSION
A B_e is holomorphic on the compact common torus and hence constant.
This proves k804,a for every genuine n divisible by4 and either focus,
including all coprime star windings. No published outer-area theorem is
needed for this direct argument. Repeating such an orbit d times scales
each signed area by d and the product by d^2, so those repetitions are
also covered. Reversal changes both signs, not the product.
For concentric circles (outside the noncircular source convention), a
regular star has inverse vertices P/a^2, hence B=A/a^4; each area is
already phase-independent. The elliptic proof does not take a torus limit.

ATTACKS NOT PROMOTED TO PROOF
At40-digit exploratory precision A^dagger/A' was phase-constant for
sample n4,8,12 families, including n8 tau3 and n12 tau5; it failed at
n3,n5,n6. A naive all-period bridge is therefore false. Proportionality
for4r could also be proved with residues on a reduced trace torus, but
the direct product argument above is shorter and avoids calculating a
constant or importing the Chavez-Caliz area-product theorem.
These numerical samples are regression/failed-route evidence only.

LISTED-PERIOD CAUTION: EXACT SOURCE-VALID ODD CONTROL
The previously certified primitive triangle with a^2=21,b^2=16,
lambda=336/25, vertices (sqrt21,0),(-3sqrt21/5,+/-16/5), has area
128sqrt21/25. Its negation is in the same fixed-caustic family. At the
fixed positive focus(sqrt5,0), their A B values are respectively
(294-2sqrt105)/225 and(294+2sqrt105)/225. Traversing each four times
gives listed N12 and unequal products16 times these values. This is
a counterexample only to the unrestricted listed-length extension,
not to the genuine4r theorem. Reflection/common-caustic certification
is retained in0023/attacks/geometry/check_odd_repeat_centroid.py; the
new exact checker verifies the inversion/area arithmetic independently.
No whole-source-record closure or novelty follows from these observations.
