AMR-050-0062 / k903,a: exact finite controls, 2 October 2026
These computations are not a proof of the all-odd-period invariant, a source
closure, a novelty claim, canonical acceptance or independent-human review.

DEFINITION AND SCALE
For the original ordered vertices P_i and original ellipse focus F_e,
Q_(e,i)=(P_i-F_e)/|P_i-F_e|^2. The final translation by F_e has no effect
on shoelace area. B_e=(1/2)sum_i det(Q_(e,i),Q_(e,i+1)). No outer polygon,
other locus focus, absolute-face area or normalized inversion radius is used.

Logical coordinates (x,y) with metric(mx,my) denote physical coordinates
(sqrt(mx)x,sqrt(my)y). A physical signed area is sqrt(mx my) times its
logical area. Thus B_+B_-=mx my times the product of two logical inverse
areas. F_e in logical coordinates is(e*coefficient*sqrt(d),0), and
mx*coefficient^2*d must equal the physical focal square a^2-b^2.

TWO NONTRIVIALLY DIFFERENT PRIMITIVE TRIANGLES
Boundary: x^2/21+y^2/16=1. Foci: (+/-sqrt(5),0).
Confocal caustic: x^2/(189/25)+y^2/(64/25)=1; lambda=336/25.

Horizontal-apex physical triangle:
    H=((sqrt(21),0),(-3sqrt(21)/5,16/5),(-3sqrt(21)/5,-16/5)).
Use metric(21,1), d=105, coefficient=1/21 and rational logical coordinates.
The physical side lengths are8,32/5,8. Contact parameters in cyclic order
are2/5,1/2,3/5. Contact points are
    (9sqrt(21)/25,32/25),(-3sqrt(21)/5,0),(9sqrt(21)/25,-32/25).
The exact original area is128sqrt(21)/25, while
    B_+ =sqrt(21)(7/576-sqrt(105)/12096),
    B_- =sqrt(21)(7/576+sqrt(105)/12096).
Both are positive, unequal, and their product is1/324.

Vertical-apex physical triangle:
    V=((0,4),(-21/5,-8/5),(21/5,-8/5)).
Use metric(1,1), d=5, coefficient=1. Side lengths are7,42/5,7.
Contact parameters are3/5,1/2,2/5. Contact points are
    (-63/25,16/25),(0,-8/5),(63/25,16/25).
The exact original area is588/25. Both inverted signed areas equal1/18,
so their product is also1/324.

Each contact lies inside its segment and on the same confocal caustic.
At every vertex the checker tests boundary incidence, distinctness and
    vin -2(vin dot n)n/(n dot n)=vout,
with exact unit-velocity length ratios. All focal denominators are positive.
No horizontal vertex has x=0, whereas V has such a vertex. Therefore V
is neither a cyclic/reversed copy of H nor a central negative of H.

For alpha=21,beta=16, delta=sqrt(alpha^2-alpha beta+beta^2)=19, the
isosceles coordinates u=(beta-delta)/(alpha-beta)=-3/5 and
v=(alpha-delta)/(beta-alpha)=-2/5 produce these two fixtures. Exact
reflection and tangency, not this construction alone, establish validity.
They belong to the same connected convex primitive-three Poncelet family
by the confocal Jacobi/porism parametrization (Stachel arXiv:2105.03624v2,
Theorem2; source matching is owned by the root/literature lane). The checker
does not claim to infer porism or connectedness merely from two samples.

ODD REPETITIONS AND ORIENTATION
Repeating an ordered polygon r times repeats every edge determinant r
times, so B_e is multiplied by r and the product by r^2. Exact repeats
r=3,5,7 of BOTH triangle phases give listed lengths9,15,21 and common
products1/36,25/324,49/324, respectively. They remain primitive-three
orbits, not new primitive periods. Reversing orientation negates both
areas and preserves the product. Cyclic reindexing and central negation
are separately checked but are not the principal two-phase evidence.

EVEN-PERIOD NEGATIVE CONTROL
Boundary x^2/4+y^2=1, lambda=4/5 and foci(+/-sqrt(3),0).
The axis diamond(2,0),(0,1),(-2,0),(0,-1) and rectangle
(4/sqrt(5),1/sqrt(5)),(-4/sqrt(5),1/sqrt(5)),
(-4/sqrt(5),-1/sqrt(5)),(4/sqrt(5),-1/sqrt(5)) are genuine primitive
four-periodic members of one fixed elliptic-caustic family. The exact
checker certifies their geometry. Their two-focus products are1 and25/16.
This rules out a universal all-period extension, NOT the odd target.

SEPARATE CIRCLE CONTROL
For radius2 the coincident focus inversion divides every coordinate by4,
and hence every area by16. A primitive equilateral triangle has original
area3sqrt(3), inverted area3sqrt(3)/16, and product27/256. The checker
also verifies exact reflection and caustic tangency for this circle.

EXECUTIONS AND LIMITATIONS
check_focal_pair.py is self-contained standard-library Python, adapted from
the read-only0044/0046 exact-field method. Actual exact-run-001.json PASS
and isolated-Python exact-isolated-run-002.json PASS have identical semantic
fields after omitting their truthful execution timestamps. The first
implementation-only mixed-field comparison error is preserved separately.

scout_focal_pair.py uses the already available optional mpmath85-digit
runtime. Actual numerical-run-001.json covers42fixed elliptic families:
m=0.2,0.7,0.95; odd periods3,5,7,9,11; every coprime forward winding
0<2tau<n;5phases each. There are27genuine star families. Measured closure,
reflection, caustic tangency, boundary incidence and winding residuals are
at most2.0590230357872e-82. Maximum scaled product spread is
3.937758002592e-85. Individual focal areas generally DO vary. These are
diagnostics, NOT exact higher-period checks, interval certificates or a
proof of constancy between sampled phases. No noncircular exact primitive
five fixture was asserted. The first inappropriate floating equality test
was preserved and replaced by the declared residual control.

The general meromorphic proof is the root's separate lane. This auxiliary
lane touches no canonical claims/status, publication records or external
systems and introduces no narrower target as a substitute for k903,a.
