AMR-050-0027: quotient-torus / character audit, 1 October 2026

Verdict: the proposed nome and exact phase are correct, conditional on the
earlier established pole/product facts. There is no hidden winding-parity
restriction. This does not itself establish nonvanishing of the pedal area.

Notation and prerequisites
--------------------------
n>=3 is odd and primitive, gcd(tau,n)=1, 0<tau<n/2; h=4tau K/n,
Delta=h/2; P=(-a sn,b cn), alpha=a^2>beta=b^2>lambda>0.
q=[alpha beta-lambda(alpha+beta)]/(2alpha beta) is assumed nonzero.
Its exclusion via the four-periodic caustic is a separate obligation.

Required previously established inputs, not numerical extrapolations:
(i) A*S=L !=0, S=sum dn/W, W=alpha-(alpha-beta)sn^2;
(ii) A has at-most-simple poles only at shifted Jacobi vertex poles;
(iii) S vanishes at those vertex poles and has simple nonzero-residue
      poles at shifted W roots, with no other poles;
(iv) X=sum dn sn/W^2=(1/(2beta))sum dn sn/W, and
     Y=sum dn cn/W^2=(1/(2alpha))sum dn cn/W;
(v) Cx=-a(beta-lambda)X/(qS), Cy=b(alpha-lambda)Y/(qS).
The velocity_telescope_reduction.txt proves (iv). The earlier center-pedal
proof establishes (i)-(iii); a final proof must incorporate or cite them.

1. Small quasiperiod rectangle and all coprime windings

Set ell=2K/n and eta=2iK'. Characters under original shifts are:
              2K      eta
 A             +       -
 S             +       -
 Cx            -       +
 Cy            -       -

Indeed eta sends sn->sn, cn->-cn, dn->-dn, W->W, hence reflects the
point polygon across the x-axis. Shift2K negates P, preserving A.
Choose integers r,s with2tau*r+n*s=1. They exist because gcd(2tau,n)=1,
and s is odd. Since ell=r*h+s*(2K), h-invariance gives the SAME characters
under ell as under2K. This works for both even and odd tau.
All functions are elliptic on the doubled rectangle2ell Z+2eta Z.

2. Complete divisor check

Every shifted vertex pole is iK'+2mK+jh+2riK', hence the single class
p=iK' modulo ell Z+eta Z. Every W root is Z+/-Delta, Z=iK'+K.
Since Delta=tau*ell and K=n*ell/2 with n odd, all W-root traces reduce to
p+ell/2, disjoint from p. In the original2K rectangle exactly two summands
contribute at each W pole: the branches differ byh, while each branch has
order n underh. Their residues of dn/W agree by oddness around Z, so
the S pole is genuinely simple, with nonzero residue.

At p, S is analytic and zero by the paired-term reflection argument.
A has at most a simple pole and A*S=L!=0, forcing a simple S zero and a
genuine simple A pole. No extra S zero is possible, as it would force
another A pole. Similarly, A has only the zeros supplied by S poles.

The reduced numerators sum dn sn/W and sum dn cn/W are analytic at p:
numerator and denominator both have order2 at their individual Jacobi
poles, while other terms are regular by the disjointness above.
Division by the simple S zero gives at-most-simple centroid poles at p.
At p+ell/2 both numerators have at-most-simple poles and S has a simple
pole with nonzero residue, so their ratios are holomorphic.
This covers all possible C poles, including denominator zeros.

3. Nome and exact phase

Use capital Q, distinct from the scalar q:
 Q=exp(-pi K'/K), Q_n=Q^n,
 K_n=K(k_n), K_n'/K_n=nK'/K,
 v=(nK_n/K)u.

The usual real modulus satisfies0<k_n<1. The map sends ell->2K_n,
eta->2iK_n', and p=iK'->iK_n' EXACTLY. Thus Q^n, not Q^(1/n), is correct.
No tau-dependent phase shift occurs. The functions dn(v,k_n),sn(v,k_n),
cn(v,k_n) have respectively characters(+,-),(-,+),(-,-) and only simple
poles in class p on the small character rectangle.

Primary conventions checked on1 October2026:
DLMF22.2 Eq1 (nome), Eq4-6 (theta definitions) and22.4 (periods/poles):
https://dlmf.nist.gov/22.2
https://dlmf.nist.gov/22.4

4. Principal-part uniqueness, without an additive ambiguity

Let F have one of those three nontrivial character pairs and at-most-simple
poles only in p+ell Z+eta Z. Let g be its corresponding scaled Jacobi
function. Match the residue of F at p by t*g; res(g,p)!=0. Characters
propagate this principal-part cancellation to all lattice translates.
F-tg is entire and periodic under2ell and2eta, so is bounded and constant.
At least one character is negative, so that constant is zero.
This proves the required one-dimensional span directly.

Consequently:
 A(u)=A0 dn(v,k_n), Cx(u)=cx sn(v,k_n), Cy(u)=cy cn(v,k_n),
 S(u)=sigma/dn(v,k_n), sigma=L/A0.

A0=A(0)>0 and sigma=S(0)>0. The constants cx=Cx(K/n), cy=Cy(0) are real.
Phase also agrees with parity: A is even, Cx odd, Cy even under u->-u,
using index reversal and reflection of P. The abstract lemma ALLOWS cx=0
or cy=0, so their nonvanishing should not be assumed from uniqueness alone.

5. Exact axis values remove that possible degeneracy, not all gaps

At a W root r, the already-checked residue calculation gives
 Cx(r)=-a(beta-lambda)sn(r,k)/(2beta q),
 Cy(r)= b(alpha-lambda)cn(r,k)/(2alpha q),
 sn(r,k)^2=alpha/(alpha-beta), cn(r,k)^2=-beta/(alpha-beta).

Its reduced phase is K_n+iK_n', where
sn(v,k_n)^2=1/k_n^2, cn(v,k_n)^2=-k_n'^2/k_n^2.
Equivalent translates can change signs but not these squares. Thus

 cx^2=k_n^2 alpha^2(beta-lambda)^2/[4 beta^2 q^2(alpha-beta)],
 cy^2=k_n^2 beta^2(alpha-lambda)^2/
                     [4 alpha^2 q^2(alpha-beta) k_n'^2].

Every factor is nonzero in the stated domain, so the real centroid locus
is a genuine ellipse. This does NOT automatically order its semiaxes or
prove |C|<=a.

C(r).C(r)=c/q becomes cx^2-k_n'^2 cy^2=k_n^2*c/q. Hence
 c-q(C.C)=(c-q*cy^2)dn(v,k_n)^2.
The remaining possibility A_C identically0 is equivalent to
cx^2=cy^2=c/q. Excluding this circular locus, or proving a sufficient
strict bound, remains separate from the character/isogeny audit.

