Literal caption construction: higher-step bicentric normal trace
================================================================

This is a bounded analytic extension of the published bicentric pole method,
not a statement that skipping vertices of the original billiard gives another
confocal billiard. It closes the two-step NORMAL trace used by the literal
vertex-inversion formula, provided the matched bicentric Jacobi parametrization
and ordinary primitive finite Poncelet hypotheses below are used. Literature
priority has not been established; no new-invariant claim is made.

Constructions, circle pair, symbols C,r,Q_i,D_i,I_i, and their finite domains
are defined in dual_side_derivation.txt. In particular

    I_i=(D_i+D_{i+1})/2,
    sum_i |I_i-I_{i-1}|^2=(N-T_2)/(2r^2),
    T_2=sum_i n_{i-1}.n_{i+1}.

Borrowed input
--------------
Use the Jacobi parametrization of a fixed bicentric Poncelet circle pair as
in Roitman-Garcia-Reznik2021, Section2 and the proof of Theorem1. Translate
the fixed circumcircle center to the origin and rotate its center-line axis.
For real u write

    Q_i=R(cos(2phi_i),sin(2phi_i)),
    phi_i=am(u+i*sigma,k),
    sigma=2*tau*K/N, gcd(tau,N)=1, 0<tau<N.

The actual geometric period of Q(u) is2K, not4K. Thus the primitive-period
condition is N*sigma=2*tau*K. Phi is lifted to a strictly increasing real
amplitude, not reduced modulo pi individually. Strict nested nonconcentric
circles give0<k<1. The concentric k=0 limit is treated separately below.
This is a fixed bicentric parametrization; the Jacobi modulus/step are not
changed by an unsupported two-step confocal-billiard reinterpretation.

The published parametrization/angle convention is already matched in
problems/AMR-050-0010/proof.md, especially Sections4-5. The real-period
normalization correction there must be retained for primitive even N.

1. Tangent normals in this parametrization
------------------------------------------
The oriented edge Q_i-Q_{i-1} equals

    2R*sin(phi_i-phi_{i-1})*J*(cos(phi_i+phi_{i-1}),
                              sin(phi_i+phi_{i-1})).

Since0<sigma<2K and am is strictly increasing with increment pi over2K,
0<phi_i-phi_{i-1}<pi. Its sine is positive. The incircle stays to the left
of each directed edge; therefore its outward unit normal is

    n_i=(cos(phi_{i-1}+phi_i),sin(phi_{i-1}+phi_i)).

No convexity of a globally self-intersecting star polygon is used. This is
the same normal in ell_i:(U-C).n_i=r, with its sign fixed by r>0.

2. A common elliptic function and all possible singularities
----------------------------------------------------------
Define complex meromorphic functions

    E(v)=cn(v,k)+i*sn(v,k), H(v)=cn(v,k)-i*sn(v,k).

The identity cn^2+sn^2=1 gives E*H=1, meromorphically. Both sn and cn
have common simple poles, with real spacing2K. At a common pole z their
odd-reflection identities are

    sn(z+t)=-sn(z-t), cn(z+t)=-cn(z-t),
    E(z+t)=-E(z-t), H(z+t)=-H(z-t).

These follow, for example, from the imaginary-quarter-period formulas
sn(iK'+t)=1/(k*sn(t)) and cn(iK'+t)=-i*dn(t)/(k*sn(t)), then their
quasiperiods at all other common poles. Exactly one of E,H has a simple
pole at z and the other a simple zero, because their product is1. Away
from those common poles neither can vanish: an analytic finite H cannot
have E*H=1 at a zero of E. Consequently these are all E/H singularities
or zeros.

With E_j=E(u+j*sigma), the complex representation of n_i is E_{i-1}E_i.
Hence its scalar product with n_{i+2} is the real cosine expression given
meromorphically by the following reciprocal sum. After indexing its center,

    R_j(u)=E_{j-2}E_{j-1}/(E_j E_{j+1}),
    T_2(u)=1/2*sum_{j=0}^{N-1}(R_j+1/R_j).

This uses the algebraic continuation of the real cosine, not complex
conjugation off the real axis. Under v->v+2K, both cn,sn change sign,
so each R_j is2K-periodic. A common imaginary period is4iK'. Therefore
T_2 is an elliptic function, and N*sigma=2*tau*K makes its summation
genuinely cyclic. Individual E factors are not assumed N-periodic: their
possible sign changes cancel in R_j and the normal products.

3. Pole cancellation for N>=4
-----------------------------
At a possible singularity of T_2, cyclically reindex its summands so that
E_0 is singular at u=z. The primitive condition implies that1,2,3 times
sigma are not integer multiples of2K. Thus E_{+/-1},E_{+/-2},E_{+/-3}
are all finite and nonzero there, and the four affected summands have
distinct indices modulo N. This includes N4. More generally exactly one
of the N sampled E_j is singular modulo the real pole spacing.

First suppose E_0 has a simple pole, with residue h. Put e_j=E(z+j*sigma)
for nonzero indices. The reflection formula gives e_{-j}=-e_j. The only
simple-pole terms are R_2,R_1,1/R_0,1/R_{-1}. Their coefficients of
h/(u-z), before the common factor1/2, are respectively

    e_1/(e_2*e_3),
    e_{-1}/(e_1*e_2)=-1/e_2,
    e_1/(e_{-2}*e_{-1})=1/e_2,
    e_{-1}/(e_{-3}*e_{-2})=-e_1/(e_2*e_3).

Their sum is exactly zero. Their reciprocal partners vanish rather than
have a pole; all other summands are analytic. Each affected term has at
most a simple pole because the four shifted factors are distinct. Thus
no higher principal part remains and the possible singularity is removable.

If E_0 instead has a simple zero, replace E everywhere by H=1/E. This
swaps R_j with1/R_j and leaves T_2 unchanged. Now H_0 has a simple pole
and H also obeys the same odd reflection; the preceding cancellation
applies identically. These cases exhaust all possible poles.

T_2 is consequently an entire elliptic function. It is bounded on a
compact fundamental parallelogram and hence constant. This is a full
analytic two-step-normal trace proof, not a finite numerical extrapolation.

4. N3, concentric circles, orientation and repetition
----------------------------------------------------
For primitive N3, the sampled four shifts can coincide. Do NOT apply the
simple-pole calculation without this exception. Instead cyclic indexing
of the genuinely periodic normals gives immediately

    sum_i n_{i-1}.n_{i+1}=sum_i n_i.n_{i+1}.

The latter is the adjacent normal trace, minus the published bicentric
internal-cosine trace, and is constant by Theorem1 (the k120 reduction).

For concentric circles k=0, phi_i=u+i*sigma: normals advance uniformly,
so each n_{i-1}.n_{i+1}=cos(4sigma) is independently constant. Reversal
preserves scalar products/squared sides, and a fixed r-fold traversal of
a smaller primitive period repeats every trace r times. It is not assigned
a false larger primitive period. Finite cycles with primitive period1 or2
are outside the tangent-polygon hypotheses, including parallel-polars and
focal-segment-caustic cases retained in the exact construction control.

Conclusion
-----------
Both explicitly distinguished constructions have all-N square-trace
constancy on the genuine finite bicentric images of elliptic-caustic
billiards, but with DIFFERENT formulas and values:

    reciprocal dual D: (2/r^2)*(N-S_F),
    caption's literal vertex inversion I: (1/(2r^2))*(N-T_2).

The literal circle-center claim is still geometrically false; proving
its trace does not identify I with D or repair that incorrect locus label.
This lane offers a conditional matching theorem for each precise operation,
not authority to erase the source ambiguity or claim publication novelty.

Earlier fallback superseded, not deleted
---------------------------------------
dual_side_derivation.txt Section3 preserves the earlier honest remaining
gap and suggested two-step-original-billiard route. The latter is rejected:
the fixed boundary's sn uniformization modulus depends on its caustic, so
altering its step at the same modulus does not generally preserve the
required confocal billiard constraint. The present proof changes no
caustic and operates solely on the actual bicentric normal trace. That
new derivation supersedes the earlier literal-trace gap, subject to root
scope/source audit; all original exact and numerical controls remain intact.
