# WSFC State Crystal 04

**Code:** `WSFC-SC-04-v0.1`  
**Date:** 2026-09-23  
**Canonical Round:** `WSFC-R04-v0.1`

## Closed gates

```text
SCALE DUALITY                    CLOSED
FINITE-SEGMENT CONVEXITY         SOURCE-CLOSED
ENDPOINT-STRIP / DRAPE           SOURCE-CLOSED
ENDPOINT-DISC WIDTH REDUCTION    SOURCE-CLOSED / ADAPTED
EQUAL-ANGLE PARTITION            DERIVED-CLOSED
SUPPORT-CELL DIMINISHING RETURN  DERIVED-CLOSED
EVEN BALANCED ALLOCATION         REDUCED-MODEL THEOREM
```

## Core support-cell cost

$$
g(q,\Delta)
=
2q\tan\frac{\Delta}{2q}.
$$

$$
\frac{\partial g}{\partial q}<0,
\qquad
\frac{\partial^2 g}{\partial q^2}>0.
$$

## Unified reduced functional

$$
\mathcal L_n(d,x;q_L,q_R)
=
2\sqrt{d^2-1}
+d
+
g(q_L,\Delta_L)
+
g(q_R,\Delta_R),
$$

$$
q_L+q_R=n-2.
$$

## Even theorem

For $n=2m$:

$$
\boxed{
x^\star=d/2,
\qquad
q_L^\star=q_R^\star=m-1.
}
$$

## Odd state

Enumeration through $n=20$:

```text
BEST ALLOCATION ALWAYS |qL-qR| = 1
```

Status:

```text
STRONG NUMERICAL EVIDENCE
GENERAL THEOREM OPEN
```

## Top-segment stress

For nominal $n=5,\ldots,12$, explicit positive top segment collapses numerically to zero length.

Status:

```text
STRONG NUMERICAL EVIDENCE
GLOBAL THEOREM OPEN
```

## Main remaining proof gate

$$
\boxed{
\text{finite shortest disc-avoiding polygon}
\Rightarrow
\text{every free side is tangent to the active endpoint disc}.
}
$$

Label:

```text
OBSTACLE-SUPPORT-ACTIVATION
```

This is the next primary target.

===END===