mirror of
https://github.com/allaunthefox/Research-Stack.git
synced 2026-07-31 03:05:21 +00:00
230 lines
11 KiB
Text
230 lines
11 KiB
Text
/- 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
|