Add visual primitive witness formalism

0-Core-Formalism/otom/tools/lean/Semantics/Semantics/VisualPrimitive.lean

@@ -0,0 +1,142 @@
+/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Research Stack Team
+
+VisualPrimitive.lean — Coordinate-bearing visual witnesses
+
+This module records the theorem-safe core of the "thinking with visual
+primitives" bridge: a visual reference is treated as a bounded spatial
+witness, not merely as an annotation or natural-language phrase.
+
+Design boundary:
+  - This file formalizes the minimal coordinate witness substrate.
+  - It does not claim benchmark performance or model capability.
+  - Empirical claims belong in docs/evidence receipts, not this module.
+
+Connection to the stack:
+  visual primitive → bounded region → goxel witness → replayable trace atom
+-/
+
+import Mathlib.Data.Fin.Basic
+import Mathlib.Data.Nat.Basic
+
+namespace Semantics.VisualPrimitive
+
+/-- Minimal kind taxonomy for coordinate-bearing visual thought units. -/
+inductive PrimitiveKind where
+  | point
+  | box
+  | contour
+  | mask
+  | vector
+  | spectral
+  deriving Repr, BEq, DecidableEq
+
+/-- A bounded integer image/manifold region.
+
+`width > 0 ∧ height > 0` is the theorem-level nonempty-region gate.  A
+point-like primitive may be represented as a `1 × 1` region. -/
+structure BoundedRegion where
+  x      : Nat
+  y      : Nat
+  width  : Nat
+  height : Nat
+  deriving Repr, BEq, DecidableEq
+
+namespace BoundedRegion
+
+/-- Nonempty coordinate support: the primitive actually binds somewhere. -/
+def Nonempty (r : BoundedRegion) : Prop :=
+  r.width > 0 ∧ r.height > 0
+
+/-- A `1 × 1` point region is nonempty. -/
+theorem point_nonempty (x y : Nat) :
+    Nonempty { x := x, y := y, width := 1, height := 1 } := by
+  exact ⟨Nat.succ_pos 0, Nat.succ_pos 0⟩
+
+/-- The area cost of a bounded region, using integer arithmetic only. -/
+def areaCost (r : BoundedRegion) : Nat :=
+  r.width * r.height
+
+/-- Nonempty regions have positive area. -/
+theorem area_positive_of_nonempty (r : BoundedRegion) (h : r.Nonempty) :
+    r.areaCost > 0 := by
+  unfold areaCost
+  exact Nat.mul_pos h.1 h.2
+
+end BoundedRegion
+
+/-- Coordinate-bearing visual primitive.
+
+`confidenceQ16` is intentionally an integer field.  Interpret values through the
+project Q16.16 convention at system boundaries; theorem-critical code should not
+use floating-point coordinates or confidences. -/
+structure VisualPrimitive where
+  kind          : PrimitiveKind
+  region        : BoundedRegion
+  confidenceQ16 : Nat
+  deriving Repr, BEq, DecidableEq
+
+namespace VisualPrimitive
+
+/-- A visual primitive binds when its region is nonempty. -/
+def Binds (p : VisualPrimitive) : Prop :=
+  p.region.Nonempty
+
+/-- Bound primitives have positive spatial cost. -/
+theorem area_positive_of_binds (p : VisualPrimitive) (h : p.Binds) :
+    p.region.areaCost > 0 := by
+  exact BoundedRegion.area_positive_of_nonempty p.region h
+
+/-- Canonical point primitive witness. -/
+def point (x y confidenceQ16 : Nat) : VisualPrimitive :=
+  { kind := PrimitiveKind.point,
+    region := { x := x, y := y, width := 1, height := 1 },
+    confidenceQ16 := confidenceQ16 }
+
+/-- Canonical point primitives bind to a nonempty region. -/
+theorem point_binds (x y confidenceQ16 : Nat) :
+    (point x y confidenceQ16).Binds := by
+  exact BoundedRegion.point_nonempty x y
+
+end VisualPrimitive
+
+/-- A Goxel witness binds a semantic claim to a bounded visual/manifold region.
+
+This is the repo-facing bridge from external visual-primitives work into the
+Research Stack vocabulary:
+  - primitive: coordinate witness payload
+  - semanticTag: language-side binding label
+  - timestep: reasoning/replay index
+  - register: finite 4-bit state register compatible with TSM-style traces
+-/
+structure GoxelWitness where
+  primitive   : VisualPrimitive
+  semanticTag : String
+  timestep    : Nat
+  register    : Fin 16
+  deriving Repr, BEq, DecidableEq
+
+namespace GoxelWitness
+
+/-- A Goxel witness is grounded when its primitive has nonempty coordinate support. -/
+def Grounded (w : GoxelWitness) : Prop :=
+  w.primitive.Binds
+
+/-- Constructing a witness from a point primitive always gives a grounded witness. -/
+theorem point_grounded (x y confidenceQ16 timestep : Nat) (tag : String) (register : Fin 16) :
+    Grounded
+      { primitive := VisualPrimitive.point x y confidenceQ16,
+        semanticTag := tag,
+        timestep := timestep,
+        register := register } := by
+  exact VisualPrimitive.point_binds x y confidenceQ16
+
+/-- Grounded witnesses have a positive spatial receipt cost. -/
+theorem area_positive_of_grounded (w : GoxelWitness) (h : w.Grounded) :
+    w.primitive.region.areaCost > 0 := by
+  exact VisualPrimitive.area_positive_of_binds w.primitive h
+
+end GoxelWitness
+
+end Semantics.VisualPrimitive