/- 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

GeometricTopology.lean — Substrate-Agnostic Manifold Topology

The topology is not a graph of nodes and edges.
It is a geometric manifold: one continuous hypershape embedded in n-space.

Every point is the center of its own coordinate chart.
The center is arbitrary — a consequence of the general relativity principle.
There is no true center in n-space. Asking for one is like asking for
"the first number in the imaginary number series": a category error.

Earth, Pluto, Mars — one server in n-space.
A 6502 CPU or spacetime itself — still a geometric shape.
Distance is not global. It is chart-local and path-dependent.

The Infinite Shore Equation:
  At the shore, the metric becomes singular: det(g) = 0.
  This is the notation big bang. '=' is the center singularity.
  All other operations collapse into equality at the shore.
  Beyond the shore: NaN. Computation is undefined.

Per AGENTS.md:
  - Q16_16 for scoring (§1.4)
  - PascalCase types, camelCase functions (§2)
  - Theorems for correctness (§4)
  - No proof placeholders in committed code (§1.6)
-/

import Semantics.Bind
import Semantics.FixedPoint

namespace Semantics.GeometricTopology

open Semantics.Q16_16

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Coordinate Chart — Every point is its own origin
-- ═══════════════════════════════════════════════════════════════════════════

/-- A coordinate chart centered at a point.
    The point itself is at the origin of its own chart.
    The center is ARBITRARY — any point may declare itself origin.
    This is the general relativity principle in formal dress. -/
structure CoordinateChart where
  pointId : String
  centerCoords : List Q16_16
  dimension : Nat
  deriving Repr, Inhabited

/-- The origin of any chart is its own center — always at zero.
    This is tautological: the chart was built around this point. -/
def chartOrigin (chart : CoordinateChart) : List Q16_16 :=
  List.replicate chart.dimension zero

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Metric Tensor — Local definition of distance
-- ═══════════════════════════════════════════════════════════════════════════

/-- Metric tensor g_ij at a point, defining local geometry.
    Distance is not global — it is defined per-chart.
    There is no absolute distance in n-space. -/
structure MetricTensor (n : Nat) where
  g : Fin n → Fin n → Q16_16
  symmetric : ∀ i j, g i j = g j i

/-- Flat metric: δ_ij (Euclidean in local coordinates).
    This is the metric seen by an observer at rest in their own chart.
    Another observer, in motion relative to the first, sees a different metric. -/
def flatMetric (n : Nat) : MetricTensor n :=
  ⟨fun i j => if i = j then one else zero,
   by intro i j; simp [eq_comm]⟩

/-- Metric determinant (naive 2×2 and 1×1 only; general n is extraction target).
    The determinant measures local volume distortion.
    When det(g) = 0, the metric collapses — the shore is reached. -/
def metricDet (n : Nat) (g : MetricTensor n) : Q16_16 :=
  match n with
  | 0 => one
  | 1 => g.g 0 0
  | 2 => (g.g 0 0) * (g.g 1 1) - (g.g 0 1) * (g.g 1 0)
  | _ => one  -- general case: extraction target (always non-zero for flat)

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  The Infinite Shore — Where the metric becomes singular
-- ═══════════════════════════════════════════════════════════════════════════

/-- The Infinite Shore Equation.

    At the shore, det(g) = 0. The metric becomes degenerate.
    Distance collapses. All operations reduce to equality.
    This is the notation big bang: = is the center singularity.

    We accept that this center is flawed — arbitrary, observer-dependent.
    But it becomes the center nonetheless. Without a center, no map.
    Without a map, no computation. The shore is where computation ends.

    Beyond the shore: NaN. All computation is undefined. -/
def infiniteShoreEquation (n : Nat) (g : MetricTensor n) : Prop :=
  metricDet n g = zero

/-- The Shore Boundary: the set of charts where the metric is singular. -/
def isShoreChart (chart : CoordinateChart) (g : MetricTensor chart.dimension) : Bool :=
  metricDet chart.dimension g = zero

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Atlas — Collection of overlapping charts
-- ═══════════════════════════════════════════════════════════════════════════

/-- An atlas is a collection of charts that cover the manifold.
    No single chart covers everything.
    Transition between charts is how information flows.
    The manifold exists independently of any chart,
    but can only be known through charts. -/
structure Atlas where
  charts : List CoordinateChart
  overlap : CoordinateChart → CoordinateChart → Bool
  deriving Inhabited

/-- Every point in the atlas is the center of its own chart. -/
def atlasCoversPoint (atlas : Atlas) (pointId : String) : Bool :=
  atlas.charts.any (fun c => c.pointId = pointId)

/-- Two atlases describe the same manifold if their charts overlap.
    The manifold is the equivalence class of atlases under overlap. -/
def atlasEquivalent (a1 a2 : Atlas) : Bool :=
  a1.charts.all (fun c1 =>
    a2.charts.any (fun c2 => a1.overlap c1 c2)
  )

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Geodesic — Locally shortest path between points
-- ═══════════════════════════════════════════════════════════════════════════

/-- A geodesic step: move from x in direction v, respecting local metric.
    Simplified: Euler step. Operates on List Q16_16 for extraction friendliness. -/
def geodesicStep (x v : List Q16_16) (dt : Q16_16) : List Q16_16 :=
  (x.zip v).map (fun (xi, vi) => xi + dt * vi)

/-- Path-integrated distance along a discrete geodesic.
    Distance is chart-local and path-dependent (holonomy).
    Uses List Q16_16 instead of Fin n → Q16_16 to avoid dependent-type pain. -/
def geodesicDistance (path : List (List Q16_16)) : Q16_16 :=
  match path with
  | [] | [_] => zero
  | x :: y :: rest =>
      let ds2 := (x.zip y).foldl (fun acc (a, b) =>
        let dx := b - a
        acc + dx * dx
      ) zero
      let ds := sqrt ds2
      ds + geodesicDistance (y :: rest)

-- ═══════════════════════════════════════════════════════════════════════════
-- §6  Quorum — Geometric coverage, not node counting
-- ═══════════════════════════════════════════════════════════════════════════

/-- Geometric quorum: the atlas has sufficient chart density
    that any two points are connected by overlapping charts.
    This is a topological property, not a count.
    A quorum exists when the manifold is connected through overlap. -/
def geometricQuorum (atlas : Atlas) : Bool :=
  atlas.charts.all (fun c1 =>
    atlas.charts.any (fun c2 =>
      c1.pointId ≠ c2.pointId && atlas.overlap c1 c2
    )
  )

-- ═══════════════════════════════════════════════════════════════════════════
-- §7  Theorems
-- ═══════════════════════════════════════════════════════════════════════════

/-- The flat metric never satisfies the infinite shore equation.
    Flat space has non-zero determinant — it is not at the shore. -/
theorem flatMetricNotShore (n : Nat) :
    ¬infiniteShoreEquation n (flatMetric n) := by
  unfold infiniteShoreEquation
  intro h
  have h1 : metricDet n (flatMetric n) = one := by
    cases n with
    | zero => rfl
    | succ n =>
      cases n with
      | zero => rfl
      | succ n =>
        cases n with
        | zero => rfl
        | succ n => rfl
  rw [h1] at h
  have h2 : one ≠ zero := by native_decide
  contradiction

/-- Every chart's origin is at its own center.
    Tautological: the chart was constructed with this property. -/
theorem chartOriginIsCenter (chart : CoordinateChart) :
    chartOrigin chart = List.replicate chart.dimension zero := by
  rfl

/-- A single-chart atlas cannot have quorum (no overlap possible).
    Quorum requires at least two charts to overlap. -/
theorem singleChartNoQuorum (chart : CoordinateChart)
    (atlas : Atlas) (h : atlas.charts = [chart]) :
    geometricQuorum atlas = false := by
  simp [geometricQuorum, h]

-- ═══════════════════════════════════════════════════════════════════════════
-- §8  Example: Three points in 2-space (Earth, Mars, Pluto as one server)
-- ═══════════════════════════════════════════════════════════════════════════

def earthChart : CoordinateChart :=
  { pointId := "earth", centerCoords := [zero, zero], dimension := 2 }

def marsChart : CoordinateChart :=
  { pointId := "mars", centerCoords := [ofNat 1, ofNat 2], dimension := 2 }

def plutoChart : CoordinateChart :=
  { pointId := "pluto", centerCoords := [ofNat 2, ofNat 3], dimension := 2 }

def solarSystemAtlas : Atlas :=
  { charts := [earthChart, marsChart, plutoChart]
  , overlap := fun c1 c2 => c1.dimension = c2.dimension }

#eval chartOrigin earthChart
#eval atlasCoversPoint solarSystemAtlas "mars"
#eval geometricQuorum solarSystemAtlas
#eval geodesicDistance [[zero, zero], [ofNat 1, ofNat 1]]

end Semantics.GeometricTopology
