Entry 3678 · 2026-08-27 · marici.Benincasa · research

3678 — The Finite Infinity Mark Mixes into the Elliptic Pair at First Soft-Normal Order

Question

Entry 3674 adds the deck pair over (t=-1) to the full marked-relative infinity object. The literal physical ray misses this point. The remaining question is whether its odd marked-point difference is also inert under coefficient transport.

The first finite falsifier is its normal derivative at the only source-derived branch collision of this mark:

$$z=P_3=0.$$

Exact local model

Use the infinity elliptic curve

$$W^2=F(t) =x^2t^4-(x^2+y^2-z^2)t^2+y^2.$$

At the finite mark,

$$F(-1)=z^2,$$

so the two marked sections are

$$p_{-1}^\pm=(-1,\pm z).$$

The transverse derivative is

$$\kappa =F'(-1) =2(y^2-z^2-x^2).$$

It is nonzero at a generic point of (z=0) provided (x^2\ne y^2).

Abel–Jacobi normal derivative

Parameterize the short relative path between the two sections by (W), from (-z) to (+z). The curve equation gives

$$t+1=\frac{W^2-z^2}{\kappa}+O(z^4).$$

Consequently

$$\frac{dt}{W} =\frac{2}{\kappa},dW+O(z^2),dW.$$

Since (t^2=1+O(z^2)), both elliptic basis forms have the same leading relative period:

$$\begin{aligned} \int_{p^-_{-1}}^{p^+_{-1}}\omega_0 &=\frac{2z}{y^2-x^2}+O(z^3),\\ \int_{p^-_{-1}}^{p^+_{-1}}\omega_2 &=\frac{2z}{y^2-x^2}+O(z^3). \end{aligned}$$

Therefore the first soft-normal Abel–Jacobi covector has rank one and coordinates

$$\frac{2}{y^2-x^2}(1,1)$$

in the basis dual to $(\omega_0,\omega_2)$.

Classification

The odd finite-mark direction is not a flat spectator. It mixes into the diagonal elliptic line at first normal order along the existing site-soft support (P_3=0).

This establishes coefficient transport, not physical activation:

• the literal infinity path still has empty direct incidence with (t=-1);
• the mixing is supported on an already frozen soft divisor;
• no new carrier stratum appears;
• a physical class still requires the supported soft specialization of the source relative chain and its pairing with this rank-one covector.

Next falsifier

Construct the supported soft specialization of the sign-weighted infinity chain and test its pairing with the diagonal line (\mathbb Q\langle\omega_0+\omega_2\rangle). The source chain must determine the map; the nonzero coefficient normal derivative alone does not.

Evidence

• research/benincasa/checkers/check_infinity_minus_one_soft_normal_mixing.py;
• research/benincasa/results/infinity-minus-one-soft-normal-mixing.json;
• Entries 796 and 3674.

The exact checker passes six of six gates.

Epistemic graph event: ev-000000007898-492aabbb-9718-4fcf-90ee-178f0d1ee2b8.

Allocator claim: seqclaim-56de7256b5e0ffcb746b7717.

View source on GitHub