preference
The preference functional assigns to each wallpaper group the negation of its J-cost on the symmetry ratio, so that higher symmetry yields higher preference. Researchers modeling cross-cultural pattern preference would cite this definition to derive rankings directly from the Recognition Science cost functional. It is a one-line wrapper that negates the output of wallpaperJ.
claimFor a wallpaper group $g$, the preference is defined by $P(g) = -J(r(g))$, where $r(g)$ is the ratio of the canonical orbit count of $g$ to the maximum orbit count (equal to 12) and $J$ denotes the J-cost function.
background
This definition sits inside the module on visual symmetry-group preference, which replaces an earlier placeholder that used raw orbit counts. WallpaperGroup is the inductive enumeration of the seventeen crystallographic wallpaper groups in IUC notation. The symmetry ratio of a group is its orbit count divided by the maximum (the orbit count of p6m). wallpaperJ applies Cost.Jcost to this ratio, producing a non-negative real number that vanishes exactly when the ratio equals one.
proof idea
One-line wrapper that applies negation to the value of wallpaperJ g.
why it matters in Recognition Science
The definition supplies the preference functional used to prove that p6m is the unique maximum and that preference is anti-monotone in orbit count. It directly feeds the downstream theorems preference_p6m_max, preference_anti_mono_in_orbits, and the master certificate SymmetryGroupPreferenceCert. Within the Recognition framework it realizes the J-cost minimum at maximal symmetry (T5 J-uniqueness), furnishing a concrete, hypothesis-free prediction for cross-cultural aesthetic preference.
scope and limits
- Does not compute explicit numerical values for any wallpaper group.
- Does not establish monotonicity or extremal properties.
- Does not invoke physical constants such as phi or the fine-structure constant.
- Does not extend the construction to three-dimensional space groups.
formal statement (Lean)
260def preference (g : WallpaperGroup) : ℝ := - wallpaperJ g
proof body
Definition body.
261
used by (17)
-
Jcost_anti_mono_on_unit_interval -
preference_anti_mono_in_orbits -
preference_p6m_eq_zero -
preference_p6m_max -
symmetryGroupPreferenceCert -
SymmetryGroupPreferenceCert -
wallpaperJ_mono_in_orbits -
proportionCost_nonneg -
abileneCollectiveJCost_succ -
perAgentJCost_pos -
predicts -
Pref -
Composable -
CostModel -
prefer_by_cq -
prefer_trans -
ideal_iff_good