EXACT ZERO-STEINER-PEDAL-AREA CONSTRUCTION FOR k503
Research algebra lane, 1 October 2026. No canonical acceptance is made here.

1. Scope and logical distinction

The construction concerns the inner polygon of caustic contact points and
its OWN Steiner curvature centroid, using its OWN internal angles. It uses
a genuine convex least-period-five orbit, so star-angle conventions and
repeated-walk conventions are irrelevant to the obstruction.

At the constructed parameter the Steiner weight denominator is nonzero,
but the inner polygon's own-centroid signed pedal area vanishes while its
signed area is strictly positive. Consequently A''/A''_K is undefined.
This refutes an unqualified assertion that this ratio is defined for every
admitted family. It does NOT exhibit two unequal finite ratios in one
family, and does NOT refute a constancy assertion restricted to the domain
where the denominator is nonzero. The source's intended domain must be
disclosed explicitly in any claim. Do not label numerical phase agreement
as a proof of the restricted-domain invariant.

2. Algebraic one-parameter family

Let x be in I=[101/100,11/10]. Set

  D=(x^2-1)^2+4,
  y=x*(x^2-1+sqrt(D))/2,
  U=x^2/y=y-x^3+x,
  B=U*(y-1)^2*(U*(1+y)^2-4*y)/(16*y^2),
  lambda=4*B*y/(y-1)^2,
  alpha=U+lambda, beta=B+lambda.

The positive algebraic branch y satisfies

  y^2-(x^3-x)*y-x^2=0.

Here B=b^2, and physical coordinates below are (X,bY). Thus dot products
in coefficient coordinates use diag(1,B); physical signed areas and
turn-weight sums are b times their normalized versions.

Inner vertices, in positive cyclic order, are

  Q0=(0,1),
  Q1=(-2U/(1+U),(1-U)/(1+U)),
  Q2=(-2x/(1+y),(1-y)/(1+y)),
  Q3=( 2x/(1+y),(1-y)/(1+y)),
  Q4=( 2U/(1+U),(1-U)/(1+U)).

They lie on the inner physical ellipse X^2/U+(bY)^2/B=1.
The consecutive tangent intersections are

  P0=(-U,1),
  P1=(-(U+x)/(1+x),(1-x)/(1+x)),
  P2=(0,(1+y)/(1-y)),
  P3=((U+x)/(1+x),(1-x)/(1+x)),
  P4=(U,1).

They lie on X^2/alpha+(bY)^2/beta=1. The ellipses are confocal because
alpha-U=beta-B=lambda. The chord P_i P_(i+1) is tangent at Q_(i+1).

Derivation: take inner-ellipse tangent half-angles 0,u,v,-v,-u,
where u^2=U, v^2=y, uv=x. Consecutive tangent intersections of
(-2a t/(1+t^2), b(1-t^2)/(1+t^2)) have coordinates
(-a(s+t)/(1+st), b(1-st)/(1+st)). Requiring P0,P1,P2 to lie on
one confocal outer ellipse reduces the final boundary equation to

  v^2+uv=u^2+u^3*v^3,

equivalently the displayed quadratic for y. This derivation contains no
elliptic integral, numerical root shooting or unproved billiard transfer.

3. Nondegenerate geometry throughout the interval

The positive y branch is continuous, x^2<y<3/2, and 0<U<1. Indeed,
the quadratic evaluated at y=x^2 is -x^2*(x-1)^2*(x+1)<0, while
at y=3/2 it is bounded below by
9/4-(3/2)*((11/10)^3-11/10)-(11/10)^2>0.

Let N denote conjugation norm for y -> (x^3-x)-y. Direct exact field
arithmetic gives

  N(B)=(x-1)^5*(x+1)^5*(x^2-5)/256.

This is nonzero throughout I. The explicit formula for B has no pole
on the positive branch. Exact endpoint evaluation gives B>0; continuity
and the nonzero norm imply B>0 throughout I. Moreover

  B/U=(y-1)^2*(U*(1+y)^2-4y)/(16*y^2)<25/256<1.

Thus U>B>0, lambda>0, alpha>beta>lambda and the caustic is genuinely
elliptic, not degenerate. Because 0<u<1<v, the five inner tangency
parameters are distinct, cyclically ordered, with consecutive angular
gaps below pi. Consecutive tangent intersections form a strictly convex
five-gon, with each contact strictly inside its side segment. A polygon
inscribed in an ellipse and tangent to its confocal inner ellipse obeys
the specular reflection law. Alternatively every reflection equation can
be checked directly with metric diag(1,B), as the exact fixture checker
does at six rational parameters. No shorter period is possible for these
five distinct vertices.

4. Steiner denominator stays nonzero on I

For inner coefficient edges e_i=Q_(i+1)-Q_i and metric dot_B, define

  w_i=-2*det(e_(i-1),e_i)*dot_B(e_(i-1),e_i)
       /(dot_B(e_(i-1),e_(i-1))*dot_B(e_i,e_i)),
  S=sum w_i, C=sum w_i Q_i/S.

These w_i are the physical sin(2 theta_i) weights divided by b>0.
All edge norm denominators are positive throughout I. Direct exact
quadratic-function-field calculation yields

  N(S)=16384*(x-1)*(x+1)*(x^2-5)^2*F8/(G8*H9^2),

  F8=x^8-2x^7-2x^6+22x^5-38x^3+66x^2+18x-1,
  G8=x^8+2x^7-6x^6-14x^5+4x^4+22x^3+38x^2-10x-101,
  H9=x^9-x^8-8x^7+8x^6+34x^5-34x^4-80x^3+80x^2+53x-181.

The whole-interval rational Bernstein certificate verifies F8>0, G8<0,
H9<0 on I. It checks all coefficients, not a mesh. Hence N(S) is finite
and nonzero, S cannot vanish, and the own Steiner centroid is finite and
continuous throughout I. Exact endpoint values have S<0.

5. An exact algebraic zero of the signed pedal area

Compute the perpendicular feet to the INNER chord supporting lines with
metric diag(1,B). Write A=area(Q)>0 and BC for their normalized signed
area at C. Reflection symmetry gives C_x=0. The general pedal quadratic
gives BC=B0+B*T_y^2/(8S), where T_y=sum w_i*Q_i.y.

Independent exact quadratic-field fixtures establish

  BC/A < 0 at x=101/100,
  BC/A > 0 at x=11/10.

The checker verifies each fixture's original boundary, five specular
reflections, five strict interior contacts, centroid and direct pedal
area; signs use rational comparisons in Q(sqrt(D)), with no floats.
Continuity and S nonzero imply some x* in the open interval has BC=0.
This already proves existence of a nondegenerate algebraic-family
zero-area example; an explicit defining polynomial follows.

Exact field arithmetic gives

  N(BC/A)=-(x-1)^2*R18/[16*x^2*(x+1)^2*(x^2-5)*F8*G8],

  R18=x^18-58x^17-383x^16-288x^15+2948x^14+6840x^13
      -2508x^12-21376x^11-10946x^10+27892x^9+17134x^8
      -84768x^7-189388x^6-121896x^5+127044x^4+217280x^3
      +64265x^2+950x+25.

The Bernstein coefficients of R18 on I have exactly one sign variation
and opposite endpoint signs. By the Bernstein/Descartes transformation,
there is exactly one real root in the open interval. Define x* to be that
root and use the positive y branch above. Since the physical BC changes
sign, this unique norm zero is a zero of the physical branch, not merely
its conjugate. This supplies an exact algebraic specification of every
ellipse and polygon coordinate in the counterexample.

6. Retained actual computational evidence

All paths here are relative to this attacks/algebra directory.

* check_exact_pentagons.py: exact metric geometry and signs.
  executions/run-005/stdout.json SHA256
  00a4aa342fdb80f43065134d8b3089e6fd9c2d4cdb244ffdec3d4c8dee08a94b.
* derive_pentagon_formula.py: exact rational-function field arithmetic
  with the already-present SymPy installation; symbolic norm identities.
  symbolic-executions/run-001/stdout.txt SHA256
  d5ee1960d323535b7eb608371b7d5b6b8216bb42c096a1181d761ab76e2b60ba.
* certify_pentagon_interval.py: Python standard-library Fraction-only
  exact whole-interval Bernstein certificates, conditional on the shown
  norm identities; also isolates exactly one R18 root in I.
  executions/run-006/stdout.json SHA256
  4b37556b75f73f098cad1620454e51da5920cb1764e26cd0abf373280af9d6a1.

The earlier numerical runs remain evidence of discovery, never proof.
The exact geometric continuity argument and interpretation of the source
are separate mathematical responsibilities; no acceptance, novelty,
independent human review, formal verification or publication is recorded
by these attack files.
