nx_imgorient.nx

nx_imgorient.nx -- DIHEDRAL (D4) ORIENTATION-INVARIANT perceptual matching. MEASURED MOTIVATION (nx_imgbench, 2026-07-23): the live reverse-image client scores mirror-horizontal recall@1 = 0 permille rotate-90 recall@1 = 31 permille rotate-180 recall@1 = 31 permille and det@10 = ZERO for all three -- i.e. a horizontally mirrored copy of an indexed image is, to the production engine, a completely different picture. Mirroring is not exotic: it is what every meme repost, every "flipped to dodge copyright detection" upload, and every front-camera selfie does. THE FIX, and why it is nearly free: the eight rigid symmetries of the square (identity, three rotations, and those four composed with a mirror) form the dihedral group D4. Because D4 is a GROUP, the orbit {g.A : g in D4} of an image A is IDENTICAL to the orbit of any dihedral variant of A -- applying h to a rotated copy just re-labels which element of the same eight you land on. Therefore: * the MULTISET of the eight orientation fingerprints is a dihedral INVARIANT of the image, and * min-of-eight is a CANONICAL FORM: canonical(A) == canonical(g.A) for every g, exactly, always. No training, no floats, no index change, no extra storage -- pure integer algebra over the existing nx_phash dHash. Eight extra 9x8 downscales per image is nothing. TWO SERVING MODES, deliberately both offered (they trade different things): canonical -- store min-of-8, one probe per query. Cheapest; loses WHICH orientation matched and very slightly raises collision odds (eight images now share one bucket key). query-side -- store the plain dHash unchanged, compute the query's eight and probe eight times. Zero information lost, zero index migration, reports the matching orientation. Costs eight index probes, which a sublinear index (nx_phash_index BK-tree) absorbs. INDEPENDENCE NOTE (rigor, deliberate): the geometric transforms here are implemented from scratch rather than imported from nx_imgxform. nx_imgxform is the ADVERSARY that the ruler attacks us with; this file is the DEFENCE. Sharing one implementation would let a single indexing bug cancel itself out and score as a pass -- the defence must be able to disagree with the attack. genealogy_id: dihedral_group_d4 (classical) over krawetz_2011 dHash license_tier: ORIGINAL