2x2 grid, 16:9, AI infers different subjects

2x2 grid, 16:9, AI infers different subjects: UNKNOWNS   T_soot = flame temperature ;  p_f, scale = flame position & size ;  Q̇ = heat release ;  fuel bed
LAWS
  incandescence  luminous zone = soot glowing as a greybody:
                 ~900K dull red · ~1200K orange · ~1500K yellow-white
                 blue is NOT blackbody — it is CH*/C₂* chemiluminescence ⇒ premixed/lean only
                 ⇒ a yellow flame is BY DEFINITION sooty ⇒ it stains, and it smokes when quenched
  buoyancy       plume rises anti-parallel to ĝ ; entrains air ⇒ widens with height (~15° half-angle)
                 flame height H ∝ Q̇^{2/5} ; puffing frequency f ≈ 1.5·D^{−1/2} Hz
                 under a ceiling the plume turns and STRATIFIES into a descending layer
  radiometry     the flame illuminates: E(x) ≈ Q̇_rad·(n̂·r̂)/4πr² , extended-source softening near-field
                 ⇒ every cast shadow RADIATES from p_f ; penumbra widens with occluder→receiver gap
                 ⇒ the light it casts carries the SAME chromaticity as the flame's own luminous zone
  optics of gas  a flame is a translucent emissive VOLUME: no hard silhouette, no self-shadow;
                 objects in front of it read as dark silhouettes, never the reverse

CONSERVATION   fuel → volatiles + soot + CO₂/H₂O + heat. A flame requires an attached fuel bed and an
               air path; sooty yellow luminosity and a perfectly clean smokeless surround cannot coexist.

OVERDETERMINATION   2 unknowns (T_soot, p_f) recovered 7× ⇒ 5 redundant equalities
  T̂_flame (luminous-zone chromaticity)   T̂_cast (colour of light on nearby surfaces)   T̂_height (H vs Q̇, fuel)
  p̂_shadow (every cast direction back-projects to one origin)   p̂_falloff (inverse-square gradient)
  p̂_spec (highlight positions on glossy objects)   p̂_penumbra (softness vs gap)

HONEST      one flame explains its own color, the color and falloff of every lit surface, every shadow
            direction and softness, the plume, and the soot record
FORBIDDEN   a blazing fire whose surroundings are lit by neutral or daylight-balanced ambient ;
            cast shadows that do not radiate from the flame, or that point toward it ;
            firelight rendered cool/neutral while the flame renders warm ;
            a flame with a crisp opaque edge, or casting a shadow of itself ;
            blue premixed flames rising off solid wood ;
            a plume that neither widens nor rises, or that ignores a ceiling instead of stratifying ;
            a flame floating free of any fuel bed.