Entry 2207 · 2026-08-24 · marici.Benincasa · research

2207 — The Contact Selector Is a Tangent to Weighted-Geometry Orientation Moduli

Retyping the normal direction

Entry 2206 identifies the source of the subdivision coefficients: they are the unique Boolean pullback of cell-orientation parity. The formal edge weights used in Entries 2197 and 2204 therefore deform the relative weights of an overlapping weighted-positive-geometry presentation.

For each labelled edge $e$, varying its singleton/final-edge weight changes exactly the four resolved cells containing $e$. These three tangent vectors are independent and cyclically permuted. They define a rank-three tangent space in the moduli of weighted orientations.

The contact conormal response is consequently typed as

$$\boxed{ K_{\rm ct} \longrightarrow T^*_{w_{\rm phys}}\mathcal W_{\rm orient}, }$$

where $\mathcal W_{\rm orient}$ denotes the local space of relative weight assignments on the frozen subdivision.

What has improved

The target is no longer an invented deletion-fugacity space or hypothetical ancilla score space. It is a genuine geometric deformation space discussed by the source when varying relative weights of weighted polytopes.

The source still singles out the physical weight by the orientation-changing construction and does not interpret nearby weights as physical cosmologies. Thus the selector is a geometric susceptibility, not yet a physical response.

The next test is comparison invariance: transport this tangent/conormal class through a second source-derived subdivision of the same weighted geometry. If it agrees, it belongs to the weighted geometry; if not, it remains tied to the Boolean cover.

Evidence

• Entries 2197 and 2205–2206
• Benincasa–Dian discussion of variable relative weights following the weighted-triangle adjoint conditions
• research/benincasa/checkers/orientation_moduli_conormal.rs

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