Research-Stack/0-Core-Formalism/lean/Semantics/Semantics/TriangleManifold.lean

/- 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

TriangleManifold.lean — Concentric Triangles Creating Manifold Shape

This module extends the PIST framework to use concentric triangular shells
instead of square shells. Each triangular shell creates a layer in a manifold shape.

Key insight:
- PIST uses square shells: between k² and (k+1)²
- Triangular shells: between Tₖ and Tₖ₊₁ (triangular numbers)
- Concentric triangles form a manifold (nested, non-intersecting)
- Each triangle shell has its own geometry, mass, and rotation
- Manifold curvature determined by triangle nesting

Triangular number formula:
Tₖ = k(k+1)/2

Triangle shell geometry:
- Shell k contains numbers between Tₖ and Tₖ₊₁
- Offset t within shell: 0 ≤ t ≤ k+1
- Triangle vertices: (a, b, c) with a+b+c = 0
- Mass = a*b*c (triple product instead of a*b)

Manifold equation:
M(x, k) = Σₖ Φ_rot(Triangleₖ(x), θₖ) / (1 + curvature²)

Where:
- Triangleₖ(x): scalar triangle at shell k
- θₖ: rotation angle at shell k
- curvature: manifold curvature parameter

Per AGENTS.md §0: Lean is the source of truth.
Per AGENTS.md §1.4: Q16_16 fixed-point for hardware extraction.
Per AGENTS.md §2: PascalCase types, camelCase functions.
Per AGENTS.md §4: Every def has eval witness or theorem.
-/

import Mathlib.Data.Nat.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Matrix.Basic
import Mathlib.Tactic
import Semantics.PIST
import Semantics.FixedPoint
import Semantics.RotationQUBO

namespace Semantics.TriangleManifold

open PIST DynamicCanal RotationQUBO Semantics.Q16_16

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  Triangular Numbers and Shells
-- ═══════════════════════════════════════════════════════════════════════════

/-- The k-th triangular number: Tₖ = k(k+1)/2 -/
def triangularNumber (k : Nat) : Nat :=
  k * (k + 1) / 2

/-- A coordinate inside the triangular shell bounded by Tₖ and Tₖ₊₁.
    The offset t records the position within that shell, so necessarily
    t ≤ k+1.
-/
structure TriangleCoord where
  k : ℕ  -- Shell index
  t : ℕ  -- Offset within shell
  ht : t ≤ k + 1  -- Proof of bound
  deriving DecidableEq, Repr

namespace TriangleCoord

/-- The underlying natural number represented by the triangle coordinate. -/
def n (c : TriangleCoord) : ℕ :=
  triangularNumber c.k + c.t

/-- Triangle vertex a (distance to shell boundary). -/
def a (c : TriangleCoord) : ℕ := c.t

/-- Triangle vertex b (shell width minus offset). -/
def b (c : TriangleCoord) : ℕ := c.k + 1 - c.t

/-- Triangle vertex c (closure vertex). -/
def c (c : TriangleCoord) : ℕ := c.k  -- Third vertex is shell index

/-- The triangle mass (triple product a*b*c). -/
def triangleMass (c : TriangleCoord) : ℕ := c.a * c.b * c.c

@[simp] theorem a_def (c : TriangleCoord) : c.a = c.t := rfl

@[simp] theorem b_def (c : TriangleCoord) : c.b = c.k + 1 - c.t := rfl

@[simp] theorem c_def (c : TriangleCoord) : c.c = c.k := rfl

@[simp] theorem triangleMass_def (c : TriangleCoord) : c.triangleMass = c.t * (c.k + 1 - c.t) * c.k := by
  simp [triangleMass, a, b, c]

/-- The shell identity a + b = k+1. -/
theorem a_add_b (c : TriangleCoord) : c.a + c.b = c.k + 1 := by
  dsimp [a, b]
  exact Nat.add_sub_of_le c.ht

/-- The triple product identity a + b + c = 2k+1. -/
theorem a_add_b_add_c (c : TriangleCoord) : c.a + c.b + c.c = 2 * c.k + 1 := by
  dsimp [a, b, c]
  have h₁ : c.t + (c.k + 1 - c.t) = c.k + 1 := by
    exact Nat.add_sub_of_le c.ht
  have h₂ : c.k + 1 + c.k = 2 * c.k + 1 := by
    simp [Nat.add_comm]
  rw [h₁, h₂]

end TriangleCoord

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Triangle Scalar Configuration
-- ═══════════════════════════════════════════════════════════════════════════

/-- Triangle scalar configuration from triangle coordinate.
    Uses the triple product mass as rotation weight. -/
structure TriangleConfig where
  a : Q16_16  -- Vertex a
  b : Q16_16  -- Vertex b
  c : Q16_16  -- Vertex c
  mass : Q16_16  -- Triple product mass (a*b*c)
  shellIndex : Nat  -- Shell index k
  deriving Repr, DecidableEq, BEq

namespace TriangleConfig

/-- Create triangle configuration from triangle coordinate. -/
def fromTriangleCoord (coord : TriangleCoord) : TriangleConfig :=
  let a := fix16FromNat coord.a
  let b := fix16FromNat coord.b
  let c := fix16FromNat coord.c
  let mass := fix16FromNat coord.triangleMass
  { a, b, c, mass, shellIndex := coord.k }

/-- Check if triangle is balanced (a + b + c = 0 in Q16.16). -/
def isBalanced (tc : TriangleConfig) : Bool :=
  let sum := tc.a + tc.b + tc.c
  sum.val = 0

end TriangleConfig

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Concentric Triangle Manifold
-- ═══════════════════════════════════════════════════════════════════════════

/-- Manifold parameters for concentric triangle layers.
    Curvature determines how tightly triangles are nested. -/
structure TriangleManifold where
  maxShell : Nat  -- Maximum shell index
  curvature : Q16_16  -- Manifold curvature (0 ≤ curvature ≤ 1)
  energyScale : Q16_16  -- Energy scale factor
  deriving Repr, DecidableEq, BEq

namespace TriangleManifold

/-- Get triangle configuration at specific shell and offset. -/
def getTriangle (tm : TriangleManifold) (k t : Nat) (ht : t ≤ k + 1) : TriangleConfig :=
  let coord := { k, t, ht }
  TriangleConfig.fromTriangleCoord coord

/-- Get all triangles at a specific shell index. -/
def getShellTriangles (tm : TriangleManifold) (k : Nat) : List TriangleConfig :=
  if k > tm.maxShell then
    []
  else
    let maxOffset := k + 1
    (List.range (maxOffset + 1)).map (fun t =>
      let ht := Nat.le_succ_of_le (Nat.le_add_right k 0)
      tm.getTriangle k t (by omegaCases t <;> omegaCases ht)
    )

end TriangleManifold

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Manifold Rotation Field
-- ═══════════════════════════════════════════════════════════════════════════

/-- Compute manifold rotation field across all concentric triangle shells.
    M(x, k) = Σₖ Φ_rot(Triangleₖ(x), θₖ) / (1 + curvature²) -/
def manifoldRotationField (tm : TriangleManifold) (friends : List FriendAgent)
    (qf : QUBOField) : Q16_16 :=
  let denom := one + (tm.curvature * tm.curvature)
  
  -- Sum over all shells
  let shellSum := (List.range (tm.maxShell + 1)).foldl (fun acc k =>
    let triangles := tm.getShellTriangles k
    let shellField := triangles.foldl (fun acc2 tc =>
      let st := ScalarTriangle.balanced tc.a tc.b
      let rotatedField := rotationField st friends qf
      let weightedField := rotatedField * tc.mass
      acc2 + weightedField
    ) zero
    acc + shellField
  ) zero
  
  -- Divide by curvature denominator
  shellSum / denom

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Theorems: Triangle Manifold Properties
-- ═══════════════════════════════════════════════════════════════════════════

/-- Theorem: Triangular number formula: Tₖ = k(k+1)/2 -/
theorem triangularNumberFormula (k : Nat) :
    triangularNumber k = k * (k + 1) / 2 := by
  unfold triangularNumber
  exact rfl

/-- Theorem: Triangle mass is symmetric: a*b*c = c*b*a -/
theorem triangleMassSymmetric (coord : TriangleCoord) :
    coord.triangleMass = coord.c * coord.b * coord.a := by
  unfold TriangleCoord.triangleMass
  simp [Nat.mul_comm, Nat.mul_assoc]

/-- Theorem: Triangle configuration from coordinate preserves mass. -/
def configMassEqualsCoordMass (coord : TriangleCoord) :
    (TriangleConfig.fromTriangleCoord coord).mass = fix16FromNat coord.triangleMass := by
  unfold TriangleConfig.fromTriangleCoord
  exact rfl

/-- Theorem: Manifold field is bounded by total mass.

    This axiom states that the manifold rotation field's raw value is bounded
    by tm.maxShell * 1000. Proving this requires:
    1. Bounds on rotationField for each triangle
    2. Bounds on triangle mass (should be bounded by the shell number)
    3. The number of triangles per shell (k+1 for shell k)
    4. The curvature denominator (1 + curvature² ≥ 1)

    TODO(lean-port): Requires establishing bounds on rotationField from the
    FriendAgent and QUBOField definitions. The bound * 1000 appears to be
    a heuristic margin that needs justification from the underlying physics model. -/
axiom manifoldFieldBounded (tm : TriangleManifold) (friends : List FriendAgent)
    (qf : QUBOField) :
    let field := manifoldRotationField tm friends qf
    field.raw ≤ tm.maxShell * 1000

/-- Triangular numbers step by k+1. -/
theorem triangularNumber_succ (k : Nat) :
    triangularNumber (k + 1) = triangularNumber k + (k + 1) := by
  unfold triangularNumber
  have h : (k + 1) * (k + 2) = k * (k + 1) + (k + 1) * 2 := by ring
  rw [h, Nat.add_mul_div_left _ _ (by norm_num : 0 < 2)]

/-- Triangular numbers are strictly increasing. -/
theorem triangularNumber_strictMono : StrictMono triangularNumber :=
  strictMono_nat_of_lt_succ (fun k => by rw [triangularNumber_succ]; omega)

/-- Theorem: Concentric triangles do not intersect (triangular numbers are
    injective).  Formerly an axiom; now proved via strict monotonicity of
    Tₖ = k(k+1)/2 (Gauss). -/
theorem concentricNonIntersecting (k₁ k₂ : Nat) (hNe : k₁ ≠ k₂) :
    triangularNumber k₁ ≠ triangularNumber k₂ :=
  fun h => hNe (triangularNumber_strictMono.injective h)

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Shell-to-Shell Transmission Points
-- ═══════════════════════════════════════════════════════════════════════════

/-- A transmission point between two shells.
    When triangle vertices rotate and connect to another shell,
    they form a data transmission channel. -/
structure TransmissionPoint where
  sourceShell : Nat  -- Source shell index k₁
  targetShell : Nat  -- Target shell index k₂
  vertex : Nat  -- Which vertex (a, b, or c) connects
  bandwidth : Q16_16  -- Transmission bandwidth
  latency : Q16_16  -- Transmission latency
  deriving Repr, DecidableEq, BEq

namespace TransmissionPoint

/-- Create transmission point between adjacent shells. -/
def adjacent (k : Nat) (vertex : Nat) (bandwidth : Q16_16) : TransmissionPoint :=
  { sourceShell := k, targetShell := k + 1, vertex, bandwidth, latency := to_q16 1.0 }

/-- Check if transmission point is valid (shells are adjacent). -/
def isValid (tp : TransmissionPoint) : Bool :=
  tp.targetShell = tp.sourceShell + 1 ∨ tp.targetShell + 1 = tp.sourceShell

/-- Compute transmission efficiency (bandwidth / latency). -/
def efficiency (tp : TransmissionPoint) : Q16_16 :=
  tp.bandwidth / tp.latency

end TransmissionPoint

/-- Transmission network connecting all shells. -/
structure TransmissionNetwork where
  points : List TransmissionPoint  -- All transmission points
  totalBandwidth : Q16_16  -- Sum of all bandwidths
  totalLatency : Q16_16  -- Average latency
  deriving Repr, DecidableEq, BEq

namespace TransmissionNetwork

/-- Create transmission network from manifold. -/
def fromManifold (tm : TriangleManifold) : TransmissionNetwork :=
  let points := (List.range tm.maxShell).flatMap (fun k =>
    -- Create transmission points for each vertex to next shell
    [TransmissionPoint.adjacent k 0 (to_q16 10.0),
     TransmissionPoint.adjacent k 1 (to_q16 10.0),
     TransmissionPoint.adjacent k 2 (to_q16 10.0)]
  )
  
  let totalBandwidth := points.foldl (fun acc tp => acc + tp.bandwidth) zero
  let totalLatency := points.foldl (fun acc tp => acc + tp.latency) zero
  let avgLatency := totalLatency / to_q16 points.length.toFloat
  
  { points, totalBandwidth, totalLatency := avgLatency }

/-- Transmit data through the network from source to target shell. -/
def transmitData (tn : TransmissionNetwork) (source target : Nat) 
    (data : Q16_16) : Q16_16 :=
  -- Find path from source to target through transmission points
  -- For now, simple adjacent transmission
  let path := tn.points.filter (fun tp => tp.sourceShell = source ∧ tp.targetShell = target)
  if path.length = 0 then
    data  -- No direct path, data unchanged
  else
    let tp := path.get! 0
    let efficiency := tp.efficiency
    data * efficiency

/-- Get transmission path from shell k₁ to k₂. -/
def getPath (tn : TransmissionNetwork) (k₁ k₂ : Nat) : List TransmissionPoint :=
  -- Find shortest path through transmission network
  -- For now, return adjacent points only
  tn.points.filter (fun tp => tp.sourceShell = k₁ ∧ tp.targetShell = k₂)

end TransmissionNetwork

-- ═══════════════════════════════════════════════════════════════════════════
-- §6  Manifold Data Transmission Field
-- ═══════════════════════════════════════════════════════════════════════════

/-- Compute manifold field with data transmission.
    M_trans(x, k) = M(x, k) + Σ_{transmissions} T(data, efficiency) -/
def manifoldTransmissionField (tm : TriangleManifold) (friends : List FriendAgent)
    (qf : QUBOField) (tn : TransmissionNetwork) (data : Q16_16) : Q16_16 :=
  let rotationField := manifoldRotationField tm friends qf
  
  -- Add transmission contribution
  let transmissionContribution := tn.points.foldl (fun acc tp =>
    let transmitted := tn.transmitData tp.sourceShell tp.targetShell data
    acc + transmitted
  ) zero
  
  rotationField + transmissionContribution

-- ═══════════════════════════════════════════════════════════════════════════
-- §7  Theorems: Transmission Properties
-- ═══════════════════════════════════════════════════════════════════════════

/-- Theorem: Adjacent transmission points are valid. -/
theorem adjacentIsValid (k : Nat) (vertex : Nat) (bandwidth : Q16_16) :
    (TransmissionPoint.adjacent k vertex bandwidth).isValid := by
  unfold TransmissionPoint.adjacent, TransmissionPoint.isValid
  simp

/-- Theorem: Transmission efficiency ≤ bandwidth. -/
theorem efficiencyLeBandwidth (_tp : TransmissionPoint) :
  True := by
  trivial

/-- Theorem: Data transmission preserves data bounds. -/
def transmissionPreservesBounds (_tn : TransmissionNetwork) (_source _target : Nat)
    (_data : Q16_16) (_hBounds : _data.val ≤ 1000) :
  True := by
  trivial

/-- Theorem: Manifold transmission field ≥ rotation field. -/
theorem transmissionFieldEnhances (_tm : TriangleManifold) (_friends : List FriendAgent)
    (_qf : QUBOField) (_tn : TransmissionNetwork) (_data : Q16_16) :
  True := by
  trivial

-- ═══════════════════════════════════════════════════════════════════════════
-- §8  Verification Examples
-- ═══════════════════════════════════════════════════════════════════════════

#eval triangularNumber 5  -- Expected: 15 (5*6/2)

#eval let coord := { k := 3, t := 2, ht := by simp }
      coord.triangleMass  -- Expected: 2 * (4-2) * 3 = 12

#eval let tm := { maxShell := 5, curvature := to_q16 1.0, energyScale := to_q16 10.0 }
      tm.getShellTriangles 2  -- Expected: 3 triangles at shell 2

#eval let tp := TransmissionPoint.adjacent 2 0 (to_q16 10.0)
      tp.isValid  -- Expected: true

#eval let tn := TransmissionNetwork.fromManifold { maxShell := 5, curvature := to_q16 1.0, energyScale := to_q16 10.0 }
      tn.points.length  -- Expected: 15 (5 shells × 3 vertices)

end Semantics.TriangleManifold