Exact convex five-periodic circle/ellipse pair from literal vertex inversion
===========================================================================

This note verifies the projective construction and an explicit algebraic
primitive pentagon. The general trace theorem is an application of published
bicentric Jacobi parametrization and the Khare-Lakshminarayan-Sukhatme2003
cyclic master identity, not an empirical extrapolation. The note does not
classify canonical AMR-050-0053 or assert an absolute novelty claim. Comparison
with a differently scoped conjecture must retain that conjecture's exact text.

1. General projectivity, horizon and transformed circle/conic
------------------------------------------------------------
Translate a bicentric pair so that its inner circle has center C=(0,0),
radius r>0, and outer circle center O=(d,0),radius R, with

    d>0, R>d+r, q=R^2-d^2>0.

Set T(x,y)=(x,y)/(2d*x+q). On the outer-circle boundary
|X|^2=2d*x+q, so T(X) is exactly inversion in the unit circle centered C.
Away from that boundary T is a projectivity, not pointwise inversion.
In homogeneous coordinates it is the matrix

    [1  0  0]
    [0  1  0]
    [2d 0  q],

whose determinant is q>0. Its affine horizon is2d*x+q=0. Throughout the
entire closed outer disk x>=d-R,

    2d*x+q >=2d(d-R)+R^2-d^2=(R-d)^2>0.

Thus it is a smooth orientation-preserving projectivity on a neighborhood
of the whole disk, including every polygon edge, disk and caustic contact.
Its inverse is X=q*U/(1-2d*u). It maps the outer disk homeomorphically
to the disk bounded by the circle

    center O'=(-d/q,0), radius R'=R/q.

The inner circle maps to

    q^2*(u^2+v^2)=r^2*(1-2d*u)^2.

Put D0=q^2-4r^2d^2. Strict nesting yields q>2dr+r^2>2dr, so D0>0.
Completing the square gives a genuine noncircular ellipse with

    center E0=(-2r^2d/D0,0),
    horizontal semiaxis A=rq/D0,
    vertical semiaxis B=r/sqrt(D0),
    foci (0,0) and F2=(-4r^2d/D0,0).

Indeed A^2-B^2=4r^4d^2/D0^2. The circle strictly contains the ellipse,
since they are images of strictly nested compact disks under this common
projectivity. This is not merely an algebraic conic with unverified nesting.

Lines and tangencies are preserved projectively. More explicitly, a tangent
line n.X=r with |n|=1 maps to

    (q*n_x+2dr)*u+q*n_y*v=r.

The image of its original contact r*n lies on this line and the ellipse.
Since the map is nonsingular there, it is the ellipse tangency point.
It remains inside the actual edge segment: for an original segment with
positive endpoint denominators g_P,g_Q, its parameter s maps to
s'=s*g_Q/[(1-s)*g_P+s*g_Q], which is strictly between0 and1 when0<s<1.
Thus supporting tangency to actual segments, not merely their extensions,
is preserved in every convex orbit.

For d>0, O' cannot equal the first focus0. Direct calculation shows

    O'=E0 iff D0=2r^2q,
    O'=F2 iff D0=4r^2q.

These algebraic conditions match the usual nonconcentric bicentric Fuss
quadrilateral and Euler triangle conditions, respectively:
q^2=2r^2(R^2+d^2), and d^2=R(R-2r). No such theorem is required to
exclude the conditions for the explicit pentagon below; direct bounds do so.

2. Exact algebraic primitive pentagon
-------------------------------------
Take R=1,d=1/5, hence q=24/25. Let t be the unique root in(1/10,1/5) of

    g(t)=25t^3-50t^2-40t+8=0,
    L=sqrt(1-t^2)>0, r=4/5-2t^2.

Existence and uniqueness use g(1/10)=141/40>0,g(1/5)=-9/5<0 and
g'(t)=75t^2-100t-40<-37 throughout the interval. Mod7 the cubic takes
the values1,6,5,1,4,3,1 at0,...,6, so it has no root and is irreducible
over Q. The initially obtained quartic is not assumed irreducible:

    (5t+2)*g(t)=125t^4-200t^3-300t^2-40t+16.

In particular18/25<r<39/50<4/5=R-d, establishing strict nesting.
Introduce real angles theta,psi in(0,pi/2) by

    cos(theta)=5r/6, sin(theta)=5rL/[4(1+t)],
    cos(psi)=t, sin(psi)=L.

The quartic/cubic relation gives

    cos^2(theta)+sin^2(theta)=1,
    (4/5)*sin(theta)*L=r*(1-t).

All signs are positive, so no extraneous root of a squared tangency
equation is included. Since cos(theta)>3/5>t, theta<psi. Thus
0<2theta<2psi<pi<2pi-2psi<2pi-2theta<2pi. Define

    P0=(6/5,0),
    P1=(-4/5+25r^2/18, 25r^2L/[12(1+t)]),
    P2=(-r,2tL),
    P3=(-r,-2tL),
    P4=(P1_x,-P1_y).

Relative to the outer center(1/5,0), their circle arguments are respectively
0,2theta,2psi,2pi-2psi,2pi-2theta. This proves all five are distinct,
the polygon is strictly convex, its winding is1, and its least period is5.
It is not a star, padded triangle/quadrilateral, or degenerate periodic list.

The line P0P1 has outward unit normal
n0=(cos(theta),sin(theta)); its support from0 is(6/5)cos(theta)=r.
The line P1P2 has unit normal

    n1=(cos(theta)*t-sin(theta)*L,
        sin(theta)*t+cos(theta)*L).

Its support is

    (1+d)cos(theta)*t+(1-d)sin(theta)*L
      =r*t+(4/5)sin(theta)*L=r.

The line P2P3 is x=-r and has outward unit normal(-1,0). The remaining
two lines are reflections of the first two. Thus all five edges are tangent
to the same inner circle. Each contact lies in the corresponding actual
segment: it lies strictly inside the outer disk, and on the chord cutting
that disk, hence between its two endpoints. The exact checker additionally
computes and interval-certifies all five segment parameters in(0,1).

3. Transformed exact pair and center/focus exclusion
----------------------------------------------------
Apply T to all five vertices and to their contacts. The transformed pentagon
is strictly convex and primitive with least period5, inscribed in the fixed
circle centered(-5/24,0) of radius25/24, and tangent on actual segments to
the fixed ellipse above. No horizon is crossed: the global lower bound
on its original disk is(R-d)^2=16/25.

The circumcenter is not the ellipse center or either focus. For clarity,
the two potentially vanishing quantities reduce to

    D0-2r^2q=576/625-(52/25)r^2<0,
    D0-4r^2q=576/625-4r^2<0,

using r>18/25. Also O'=-5/24 is not0. Nonzero algebraic field elements
and strict rational interval enclosures independently confirm these facts
in the executable certificate.

The conic center and axes simplify in the isolated real embedding to

    E0_x=-5/18+(125/288)t^2,
    A^2=125/192+(625/1152)t-(625/2304)t^2,
    B^2=25/36-(625/576)t^2,
    F2_x=2*E0_x.

These formulas are exact, not displayed floating estimates.

4. Why the whole family closes and has a constant squared-side sum
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The existence of one genuine five-cycle supplies a Poncelet porism for
the fixed circle pair. More concretely, the published bicentric Jacobi
uniformization realizes its tangent map as u->u+sigma with geometric
period2K. Closure of the explicit cycle gives5sigma=2tau*K; the strict
convex winding1 established above gives tau=1. Every initial phase on the
same pair consequently closes after5 steps, with the same branch. This is
the standard Poncelet/Jacobi input, not an inference that all polygonal
tangencies happen to close from a few numerical examples. Projectivity T
carries that entire family to a fixed nested circle/ellipse Poncelet family.

For each original bicentric family member, T(P_i)=P_i/|P_i|^2 on the outer
circle. These are exactly the literal vertex inverses I of
literal_higher_trace_proof.txt. Hence their square trace is constant by
that proof. Its analytic constancy should be credited as the direct
corollary of the published master identity explained below, rather than
claimed as a newly discovered general Jacobi identity. At the explicit
member the value reduces exactly to

    sum_i |T(P_{i+1})-T(P_i)|^2=175/18-(625/48)t.

The full-family conclusion thus combines the established general trace
argument and a fixed-pair exact construction. Passing the single-cycle
checker by itself does not prove this whole-family statement.

5. Explicit prior-art credit for the analytic load-bearing step
--------------------------------------------------------------
Khare, Lakshminarayan and Sukhatme, Cyclic Identities Involving Jacobi
Elliptic Functions. II, arXiv:math-ph/0207019v2 (24March2003), Journal of
Mathematical Physics44 (2003),1822-1841, DOI10.1063/1.1560856, provides
master identity MI-II in Section2.1.2, equation(26), printed/PDF pp7-8.
The retained full master-identity setup/derivation pp2-8 were read for this
note; source paths and detailed pins are in literature_lane/literal-followup.

For the actual quartet f=(R+1/R)/2, replace denominators by
H=cn-i*sn=1/E: f is a homogeneous degree-four Jacobi polynomial. Under2K
all four factors negate; under2iK', E->-H andH->-E swaps R with1/R.
Thus it has both periods2K and2iK', the MI-II class P=Q=0. For primitive
N>=4 its four real shifts are distinct modulo2K, so f has at most simple
poles on the required real2K/N grid. Its total simple-residue coefficient
is0 over the fundamental parallelogram and there are no higher principal
parts. MI-II(26) therefore gives a cyclic sum independent of base phase.
Coprimality merely reindexes the sigma grid. This is precisely the analytic
constancy used above, with N3 treated separately by the known adjacent trace.
The direct four-residue calculation in the earlier note is a transparent
reproduction of this published master principle, not separate prior art.

The geometric construction's comparison with Murad2026 Section6 Conjecture1
belongs to root's exact source-scope audit. If that conjecture has the stated
unqualified all-n necessity of circumcenter=center/focus for a constant
circle/conic square-side sum, this fixed convex primitive-five family supplies
a counterexample to that necessity. No assertion about its triangle theorem,
its authors' intentions, a stronger omitted genericity assumption, or absolute
publication priority is made here.

6. Actual reproducible certificate and exclusions
------------------------------------------------
check_projective_pentagon.py is portable standard-library Python. The actual
isolated run saved pentagon-exact-run-001.json with
PASS_EXACT_PROJECTIVE_PRIMITIVE_PENTAGON. It uses exact cubic-field arithmetic,
the positive radical L, irreducibility mod7,96 rational root bisections and
integer-sqrt enclosures. It verifies all original circle incidences/tangencies,
all transformed circle/ellipse incidences and actual segment contacts, fixed
circle/conic parameters, noncoincidence with both foci/center, and the exact
single-cycle square trace. No floating/CAS/module dependency is used by it.

The scratch CAS factorization revealed the quartic's linear factor; it was not
used as an irreducibility certificate. The final cubic factorization is checked
by exact polynomial multiplication in the portable certificate. Earlier source
ambiguities and the rejected skip-original-confocal argument remain preserved.
No canonical status/claims, publication records, or external accounts were
modified by this mathematical lane.
