AMR-050-0053 / k811: exact dual-side formulas and source distinction
=================================================================

This lane has not classified or changed the canonical problem. The following
is a general proved reduction for the reciprocal dual, and an exact distinction
from the caption's literal vertex-inversion construction. The latter is NOT
claimed to be nonconstant. Finite controls do not settle source terminology.

Source pins read
----------------
Reznik-Garcia-Koiller, arXiv:2004.12497v11, retained PDF:
problems/AMR-050-0048/attacks/literature/sources/reznik-2004-12497v11.pdf
SHA256 c56bb4ea29734ed04ee153206bb8619286df544714fd3f77fe97bdc945dfe1da.
Section 3.9 on PDF p10 defines w_i as the dual polygon's side length and says
the dual is the inverse of the pedal [7]. Table9 p11 asserts sum(w_i^2)
constant for all N (k811). Figure8 p12 instead says to invert polar vertices
about their incircle center O_p while asserting a circumcircle centered O_p.
Those two constructions differ; the latter assertion of center is false for
literal vertex inversion when a>b and lambda>0. The bibliography [5] calls a
private communication 'Sum of squared sidelengths of focus-polar polygon is
invariant'; it must not be silently treated as a publicly proved theorem.

Published load-bearing theorem for the shortest corrected reduction:
Roitman-Garcia-Reznik, New Invariants of Poncelet-Jacobi Bicentric Polygons,
Arnold Mathematical Journal7 (2021),619-637, DOI10.1007/s40598-021-00188-6,
Theorem1: sum of the bicentric polygon's internal-angle cosines is invariant.
Its exact focal-polar transfer is already preserved in
problems/AMR-050-0010/proof.md,
SHA25672f8997ac6c1419a2e990b140455e9aacf9a4ee7d466e1e110e5dc8b7762ba21.
This is borrowed published mathematics, not a new invariant/priority claim.

1. Coordinates and finite domain
-------------------------------
Outer ellipse E: x^2/a^2+y^2/b^2=1, a>b>0, c^2=a^2-b^2.
Confocal elliptic caustic: semiaxes sqrt(a^2-lambda),sqrt(b^2-lambda),
0<lambda<b^2. Fix F=(c,0) and translate F to the origin. Let X_i=P_i-F,
d_i=|X_i|>0, n_i=X_i/d_i. Unit-circle focal polarity sends P_i to
ell_i: X_i.U=1. Put

    C=(c/b^2,0), r=a/b^2,
    O=(c/(b^2-lambda),0), R=sqrt(a^2-lambda)/(b^2-lambda).

The ell_i are tangent to the circle (C,r), and
Q_i=ell_i intersect ell_{i+1} lies on the fixed outer circle (O,R).
Indeed the ellipse focal-distance identity gives

    d_i=a-c*x_i/a,
    1/d_i-C.n_i=r,
    ell_i: (U-C).n_i=r.

Choose the orientation keeping the confocal caustic left of every directed
chord. The focus is strictly inside this caustic, so every angular increment
delta_i of the n_i is in (0,pi). This remains true for winding star orbits:
it is a local oriented tangent condition, not global polygon convexity.
Consequently det(X_i,X_{i+1})>0, and Q_i is finite. Reversal changes this
orientation convention, not a sum of squared lengths. Repetition only
multiplies every square trace by the number of repetitions.

2. Reciprocal dual: explicit vertices and squared sides
------------------------------------------------------
The pedal of the POLAR polygon about C consists of the feet
H_i=C+r*n_i on ell_i. Inverting these feet in the UNIT circle centered C
produces its reciprocal dual:

    D_i=C+n_i/r.

This is polarity of the polar polygon about C, not pointwise inversion of
its Q_i vertices. It has vertices on the circle centered C of radius1/r.
For an inversion circle of fixed radius rho, replace 1/r by rho^2/r.
The explicit original-coordinate formula avoids square roots in each vertex:

    D_i=C + b^2*(P_i-F)/(a^2-c*x_i).

Here C is in focus-centered coordinates and x_i is the original x-coordinate.
Any common translation changes no side length. Adjacent sides satisfy

    w_i^2 = |D_{i+1}-D_i|^2
          = (2/r^2)*(1-n_i.n_{i+1})
          = (2*b^4/a^2)*(1-cos angle(P_i,F,P_{i+1})).

Thus, writing S_F=sum_i cos angle(P_i,F,P_{i+1}),

    sum_i w_i^2 = (2*b^4/a^2)*(N-S_F).

The bicentric internal cosine at Q_i is -n_i.n_{i+1}. To check its sign,
ell_i's oriented side runs parallel to J*n_i: Q_i-Q_{i-1} is a positive
multiple of J*n_i. The two rays at Q_i therefore have unit directions
-J*n_i and J*n_{i+1}. Its ordinary internal cosine is their dot product,
-n_i.n_{i+1}. The published bicentric cosine theorem, with the fixed circle
pair above and its proper cyclic winding convention, makes S_F constant.
This proves all-N constancy for THIS reciprocal-dual definition, including
primitive convex/star families and fixed repeated traversals. A circular
limit follows from regular stars. No claim about all arbitrary polygons is
made; the fixed Poncelet pair is essential.

3. Literal vertex inversion from the Figure8 caption
---------------------------------------------------
If one instead follows the caption literally, put

    I_i=C+(Q_i-C)/|Q_i-C|^2.

Solving the two tangent equations gives, since 1+n_i.n_{i+1}>0,

    Q_i-C=r*(n_i+n_{i+1})/(1+n_i.n_{i+1}),
    I_i=C+(n_i+n_{i+1})/(2*r)=(D_i+D_{i+1})/2.

Thus I is the MIDPOINT polygon of D. It is not D (except for degeneracies
outside a genuine polygon). Its squared side expression is

    |I_i-I_{i-1}|^2=(1/(2*r^2))*(1-n_{i-1}.n_{i+1}),
    sum_i |I_i-I_{i-1}|^2
      =(1/(2*r^2))*(N-sum_i n_{i-1}.n_{i+1}).

This uses a two-step focal cosine trace rather than S_F. The adjacent-trace
theorem alone does not establish its general constancy. A possible further
route is a valid two-step billiard/Poncelet transfer, but its new caustic,
step collisions, parity, and hyperbolic/degenerate caustic cases need actual
proof. No such all-source argument is asserted by this lane.

Inversion of Q's circle proves the caption's center problem exactly. Let
d=O-C, q=R^2-|d|^2>0. The I_i lie on a circle with center

    C-d/q and radius R/q,

not C unless d=0. For a>b and0<lambda<b^2, d is nonzero. Both reciprocals
are finite, since |Q_i-C|>=r>0. This is a construction correction, not a
counterexample to constancy of the literal square trace.

4. Actual exact controls
------------------------
check_dual_squares.py is self-contained standard-library Python. The actual
isolated-interpreter run saved exact-run-001.json with
PASS_EXACT_DUAL_CONSTRUCTION_CONTROLS. It independently verifies ellipse
incidence, unit-velocity reflection (exact rational length ratios), identical
confocal tangency and internal contacts, original focal-polar intersections,
both circle equations, the reciprocal as inverse of polar pedal, the literal
midpoint identity, both side-square formulas, repetition, and reversal.

Two non-equivalent primitive-three phases of one family:
ellipse squared semiaxes21,16; lambda336/25; foci +/-sqrt5.
H=((sqrt21,0),(-3sqrt21/5,16/5),(-3sqrt21/5,-16/5));
V=((0,4),(-21/5,-8/5),(21/5,-8/5)). For either focus,

    S_F=-19/15,
    reciprocal dual square trace=32768/315,
    literal vertex-inverse square trace=8192/315.

Primitive-four phases of one different family: a5,b3,lambda225/34;
diamond=(5,0),(0,3),(-5,0),(0,-3);
rectangle=(25,9),(-25,9),(-25,-9),(25,-9) all divided bysqrt34.
For either focus,

    S_F=0, two-step focal trace=-36/25,
    reciprocal dual square trace=648/25,
    literal vertex-inverse square trace=5508/625.

The squared side sum of the ORIGINAL FOCAL POLAR Q is instead4400/243
for the diamond and170000/6561 for the rectangle. Those unequal numbers
are an exact counter-control to conflating Q with the reciprocal dual D;
they do not refute k811 under its D interpretation.

A unit-circle regular pentagon and genuine winding-two pentagram are also
checked exactly in Q(sqrt5). Their caustic squared radii are(3+sqrt5)/8
and(3-sqrt5)/8 respectively. Reciprocal traces are
(25-5sqrt5)/2 and(25+5sqrt5)/2; literal traces are
(25+5sqrt5)/8 and(25-5sqrt5)/8. These are additional exact star construction
controls, not noncircular all-star proofs.

Boundary/domain warning: a focal-segment caustic has lambda=b^2 and is
excluded. Its focal-axis diameter produces parallel incompatible polar
equations; the checker records determinant0 for X=(1,0),Y=(-9,0) in
the a5,b3 ellipse at F=(4,0). It is not a genuine elliptic-caustic example.
N2/diametric cycles likewise cannot be silently included as finite Q.

5. Failures and exploratory evidence retained faithfully
--------------------------------------------------------
An initial scratch N4 test accidentally used the center as a focus in an
a5,b3 ellipse. This failed the focus-ellipse identification, so its computed
numbers were discarded before the complete checker was authored. It was an
invalid proposed control, not a failure of either invariant. The executable
now explicitly asserts metric_x*focus^2=a^2-b^2 for every fixture.

A separate65-digit mpmath scratch diagnostic gave apparently constant literal
I traces for N3tau1,N4tau1,N5tau1,N5tau2,N6tau1,N7tau2 at m=.7 and
phases0,K/7,5K/13; differences ranged9.496e-66 to5.697e-65.
Those samples are ordinary floating exploration, not proof, not a certified
counterexample, and not an exact algebraic all-N conclusion. The reproducible
diagnostic is retained separately from the standard-library exact checker.
