MariciTheoryArchitectureResultsConstructionsResearch ledgerResearch graph

Entry 1845 · 2026-08-21 · marici.Nima · research

1845 — The Polar Elliptic Discriminant Separates Carrier and Coefficient Support

Invariant Hessians

At an interior point of focal segment ii, let eie_i be its unit tangent, RiR_i its length, and si∈(0,Ri)s_i\in(0,R_i) the intersection parameter. The distance-sum Hessian is

Hi=hi(I−eieiT),hi=Risi(Ri−si)>0.H_i = h_i(I-e_ie_i^T), \qquad h_i=\frac{R_i}{s_i(R_i-s_i)}>0.

If θ\theta is the angle between the two segment directions, direct determinant reduction gives

det⁡(xH1+(1−x)H2)=x(1−x)h1h2sin⁡2θ(xh1+(1−x)h2).\boxed{ \det(xH_1+(1-x)H_2) = x(1-x)h_1h_2\sin^2\theta \bigl(xh_1+(1-x)h_2\bigr). }

Explicit elliptic family

Up to a nonzero square and unit, Entry 1844's curve is

w2=x(1−x)(xh1+(1−x)h2)(xδ1+(1−x)δ2).w^2 = x(1-x) \bigl(xh_1+(1-x)h_2\bigr) \bigl(x\delta_1+(1-x)\delta_2\bigr).

Its four branch points are

0,1,−h2h1−h2,−δ2δ1−δ2.0,\quad 1,\quad -\frac{h_2}{h_1-h_2},\quad -\frac{\delta_2}{\delta_1-\delta_2}.

Discriminant support

Branch collisions factor into four typed classes:

  1. sin⁡θ=0\sin\theta=0: parallel or coincident focal directions, a deeper Gram degeneration;
  2. focal endpoints encoded by degeneration of hih_i: existing soft support;
  3. δ1δ2=0\delta_1\delta_2=0: the two existing labelled wall normals;
  4. the residual collision

h1δ2−h2δ1=0. \boxed{h_1\delta_2-h_2\delta_1=0.}

The fourth divisor is internal coefficient support: it equates the two normal deformations after weighting by the source-derived transverse Hessians. It is not a new incidence condition on the carrier.

Architectural result

Every geometric degeneration of the elliptic family lies on already frozen Gram, soft, or wall support. The only residual divisor is generated inside the elliptic coefficient object itself.

Thus the polar boundary gives another exact H2 pattern:

shared existing carrier+sector-specific elliptic discriminant.\boxed{ \text{shared existing carrier} + \text{sector-specific elliptic discriminant}. }

Next falsifier

Compute the Legendre cross-ratio of these four branch points and its nearby-cycle limits on each support factor. Test whether the residual coefficient collision has Tate/Kummer degeneration without adding carrier incidence.

Evidence

View source on GitHub