The Lorenz butterfly never leaves its ball

Static snapshot of Lorenz.lean · source revision dated 2026-10-19T10:47:08.000Z · source SHA-256 bdade6cf9bdae7dfa8356862451d32b44c0c702a3bcc56c9f1f840ed584e30b7. Saved results, shown as recorded. Each result is shown as its check last saved it, compared only with this saved source and not with the reader's Lean environment. Nothing was checked or run to show it. This static page does not check or update results.

import Mathlib
L1–1
import Gimle.Forseti.Syntax.Certificate
L2–2
import Gimle.Forseti.Examples.Lorenz
L3–3

The Lorenz system is the most famous picture in dynamics: three coupled equations whose solutions wander forever over a two-winged surface, never repeating and never settling, so sensitive to their start that no simulation of them can be trusted for long. This notebook proves one thing about it that no simulation could: from the start (1,1,1)(1, 1, 1), the trajectory stays inside an explicit ball for all time. It also shows, honestly, what the same machinery cannot settle.

The equations, with the parameters Lorenz used in 1963, are

x′=σ(y−x),y′=x(ρ−z)−y,z′=xy−βz,x' = \sigma (y - x), \qquad y' = x(\rho - z) - y, \qquad z' = xy - \beta z,

with σ=10\sigma = 10, ρ=28\rho = 28, β=8/3\beta = 8/3, started from (x,y,z)=(1,1,1)(x, y, z) = (1, 1, 1) at t=0t = 0. We watch

V=x2+y2+(z−38)2,V = x^2 + y^2 + (z - 38)^2,

the squared distance from the point (0,0,σ+ρ)(0, 0, \sigma + \rho), which starts at V(0)=1+1+372=1371V(0) = 1 + 1 + 37^2 = 1371.

Lorenz derived these equations by truncating Saltzman's expansion of the Boussinesq equations for convection, the Navier–Stokes equations coupled to heat, to three modes. A bound on this system is therefore a bound on a three-mode truncation of Boussinesq convection. The pure Navier–Stokes truncations, whose smallest member is a single triad rather than the Lorenz system, have their own notebooks, T3 (NS_T3.lean), K5 (NS_K5.lean) and B2 (NS_B2.lean); this one stands beside them as the convective case.

Proved. A trajectory from (1,1,1)(1, 1, 1) exists for all t≥0t \ge 0, is unique, and satisfies 0≤V≤16000 \le V \le 1600 throughout. The bound is decided by the trajectory decider from a named Lean contract and its certificate replays without the solver.

Refuted. V≤1300V \le 1300 for all t≥0t \ge 0: false at the start, where V=1371V = 1371, and Lean checks that witness.

Unknown. V≤1400V \le 1400 for all t≥0t \ge 0. A simulation says it is true (the largest value it sees is about 13871387), but the contract proves only 16001600, and the start does not refute it. The decider says so rather than guess; that is the point of it.

Not claimed. Anything about chaos, the attractor, or sensitivity to the start: the simulation below is evidence, and no status reads it. Nothing about other starts or other parameters.

The system, compiled

The equations are declared once, in Asgard's language, and compiled to a circuit whose exact field Lean draws and reads:

namespace Notebook.Lorenz
L58–58
open Gimle.Asgard Gimle.Forseti.Examples.Lorenz
L59–59

model diagram Gimle.Forseti.Examples.Lorenz.compiled L61–61

Continuous modelt · start 0Vector fieldIntegrate · tdx/dt = (10 × (y + (-x)))∫ → xx(0) = 1xxdy/dt = ((x × (28 + (-z))) + (-y))∫ → yy(0) = 1yydz/dt = ((x × y) + (-(8/3 × z)))∫ → zz(0) = 1zz
Observations x y z V
Evidence · not proofsimulate Gimle.Forseti.Examples.Lorenz.compiled · forsetiNever runL62–62

Not run yet.

What a simulation shows

The run above integrates the compiled field with RK4 in floating point from t=0t = 0 to t=30t = 30, and shows the butterfly: xx and yy swinging between the two wings, zz oscillating between about 55 and 4848. The picture is right and also the reason the picture cannot be a proof: two runs with different step sizes diverge from each other after a few wing changes. What follows does not depend on it.

The storage function, and why the usual argument fails

Along any solution, the chain rule and the equations give

V′=−20x2−2y2−163z2+6083z.V' = -20x^2 - 2y^2 - \tfrac{16}{3} z^2 + \tfrac{608}{3} z.

The cubic terms cancel, and so do the xyxy terms, because the centre was placed at z=σ+ρz = \sigma + \rho; that choice is the whole trick. But V′V' is not always negative: near the origin, where zz is small and positive, VV grows. So VV is not monotone along the trajectory, and the simple argument used for a damped system, "the energy only ever decreases", is unavailable. Lean checks both facts about the compiled field below.

Theorem energy_at_start Kernel checkedAs recorded L88–90

Namespace Notebook.Lorenz

The compiled observation at the start is 13711371.

Statement
energy  ![1,  1,  1]  =  1371\text{energy}\;!\text{[}1,\;1,\;1\text{]}\;=\;1371

Assumptions: propext, Classical.choice, Quot.sound

Lean source & proof · 3 lines
/-- The compiled observation at the start is $1371$. -/
theorem energy_at_start : energy ![1, 1, 1] = 1371 := by
  norm_num [energy, Matrix.cons_val_two, Matrix.tail_cons]

Theorem rate_positive_at_origin_side Kernel checkedAs recorded L92–94

Namespace Notebook.Lorenz

VV is not monotone: at (0,0,1)(0, 0, 1) its rate along the field is positive.

Statement
0  <  rate  ![0,  0,  1]0\;<\;\text{rate}\;!\text{[}0,\;0,\;1\text{]}

Assumptions: propext, Classical.choice, Quot.sound

Lean source & proof · 3 lines
/-- $V$ is not monotone: at $(0, 0, 1)$ its rate along the field is positive. -/
theorem rate_positive_at_origin_side : 0 < rate ![0, 0, 1] := by
  norm_num [rate, Matrix.cons_val_two, Matrix.tail_cons]

What does hold is a differential inequality: at every state,

V′+2V≤2⋅1541.V' + 2V \le 2 \cdot 1541.

The least constant for which this is true is 23104/15≈1540.2723104/15 \approx 1540.27; Lean checks it with 15411541. Then e2t(V−1600)e^{2t}(V - 1600) is non-increasing along any solution, and since it starts negative, V≤1600V \le 1600 forever. The margin between 15411541 and 16001600 is not decoration: it is what makes the trajectory exist for all time, as the next section explains.

Theorem storage_inequality Kernel checkedAs recorded L108–110

Namespace Notebook.Lorenz

The differential inequality at every state, checked for the closed-form rate.

Statement
(x  :  Point  3)  :  rate  x  ≤  2  ⋅  1541  −  2  ⋅  energy  x(\text{x}\;:\;\text{Point}\;3)\;:\;\text{rate}\;\text{x}\;\le \;2\;\cdot \;1541\;-\;2\;\cdot \;\text{energy}\;\text{x}

Assumptions: propext, Classical.choice, Quot.sound

Lean source & proof · 3 lines
/-- The differential inequality at every state, checked for the closed-form rate. -/
theorem storage_inequality (x : Point 3) : rate x ≤ 2 * 1541 - 2 * energy x := by
  exact storage_bound x

Existence is the hard part

For a linear system, a solution for all time is written down. For Lorenz it is not, and Mathlib's Picard–Lindelöf theorem only gives a solution on a short interval. The library's proof, in Gimle.Forseti.Examples.Lorenz, builds the global solution by hand: clamp the field to a box containing the ball V≤1600V \le 1600, so that it is bounded and globally Lipschitz; get a solution on every [0,n][0, n]; glue them by uniqueness; and show the glued solution never leaves the ball, because at any point where V=1600V = 1600 its rate is at most 2⋅1541−2⋅1600=−118<02 \cdot 1541 - 2 \cdot 1600 = -118 < 0. Inside the ball the clamp is the identity, so the glued curve solves the true equations. Uniqueness is Lipschitz continuity on the same compact ball.

That proof is what the decider's contract names. Every claim below is decided against the compiled circuit, not against these words.

The claims, decided

Decisionin_ball · trajectory · forsetiNever runL132–132
trajectory lorenz from (1, 1, 1) at 0 observe V always : V >= 0 and V <= 1600

Not run yet.

namespace DecidedShared_72f9d7627148
L134–134

Definition forsetiTime : Gimle.Asgard.Dynamics.TimeDomain L136–137

Namespace Notebook.Lorenz.DecidedShared_72f9d7627148

⟨"time",  (((mkRat  (0)  1)  :  Q)  :  R)⟩\langle \texttt{"time"},\;(((\text{mkRat}\;(0)\;1)\;:\;\mathbb{Q} )\;:\;\mathbb{R} )\rangle
Lean source · 2 lines
def forsetiTime : Gimle.Asgard.Dynamics.TimeDomain :=
  ⟨"time", (((mkRat (0) 1) : ℚ) : ℝ)⟩

Definition forsetiInitial : Gimle.Asgard.Point 3 L139–140

Namespace Notebook.Lorenz.DecidedShared_72f9d7627148

![(((mkRat  (1)  1)  :  Q)  :  R),  (((mkRat  (1)  1)  :  Q)  :  R),  (((mkRat  (1)  1)  :  Q)  :  R)]!\text{[}(((\text{mkRat}\;(1)\;1)\;:\;\mathbb{Q} )\;:\;\mathbb{R} ),\;(((\text{mkRat}\;(1)\;1)\;:\;\mathbb{Q} )\;:\;\mathbb{R} ),\;(((\text{mkRat}\;(1)\;1)\;:\;\mathbb{Q} )\;:\;\mathbb{R} )\text{]}
Lean source · 2 lines
def forsetiInitial : Gimle.Asgard.Point 3 :=
  ![(((mkRat (1) 1) : ℚ) : ℝ), (((mkRat (1) 1) : ℚ) : ℝ), (((mkRat (1) 1) : ℚ) : ℝ)]

Definition forsetiObservation : Fin 4 L142–142

Namespace Notebook.Lorenz.DecidedShared_72f9d7627148

⟨3,  by  decide⟩\langle 3,\;\text{by}\;\text{decide}\rangle
Lean source · 1 line
def forsetiObservation : Fin 4 := ⟨3, by decide⟩

Definition forsetiIdentities : Prop L144–148

Namespace Notebook.Lorenz.DecidedShared_72f9d7627148

Gimle.Forseti.Examples.Lorenz.evolution.states.map  (⋅.initialId)  =  ["initial-x",  "initial-y",  "initial-z"]  ∧\text{Gimle}.\text{Forseti}.\text{Examples}.\text{Lorenz}.\text{evolution}.\text{states}.\text{map}\;(\text{·}.\text{initialId})\;=\;\text{[}\texttt{"initial-x"},\;\texttt{"initial-y"},\;\texttt{"initial-z"}\text{]}\;\land   (Gimle.Forseti.Examples.Lorenz.body.observations.get  forsetiObservation).port.id  =\;(\text{Gimle}.\text{Forseti}.\text{Examples}.\text{Lorenz}.\text{body}.\text{observations}.\text{get}\;\text{forsetiObservation}).\text{port}.\text{id}\;=  "obs-v"  ∧\;\texttt{"obs-v"}\;\land   Gimle.Forseti.Examples.Lorenz.evolution.axis.id  =  "time"\;\text{Gimle}.\text{Forseti}.\text{Examples}.\text{Lorenz}.\text{evolution}.\text{axis}.\text{id}\;=\;\texttt{"time"}
Lean source · 5 lines
def forsetiIdentities : Prop :=
  Gimle.Forseti.Examples.Lorenz.evolution.states.map (·.initialId) = ["initial-x", "initial-y", "initial-z"] ∧
  (Gimle.Forseti.Examples.Lorenz.body.observations.get forsetiObservation).port.id =
    "obs-v" ∧
  Gimle.Forseti.Examples.Lorenz.evolution.axis.id = "time"
end DecidedShared_72f9d7627148
L150–150
namespace Decided_in_ball
L152–152
open DecidedShared_72f9d7627148
L154–154

Example example (line 156) Kernel checkedAs recorded L156–156

Namespace Notebook.Lorenz.Decided_in_ball

Checked statement, as this project’s notation prints it
forsetiTime  =  forsetiTime\text{forsetiTime}\;=\;\text{forsetiTime}
Kernel form, without notation
Eq Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiTime Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiTime

Assumptions: propext, Classical.choice, Quot.sound

Lean source & proof · 1 line
example : @forsetiTime = @DecidedShared_72f9d7627148.forsetiTime := rfl

Example example (line 157) Kernel checkedAs recorded L157–157

Namespace Notebook.Lorenz.Decided_in_ball

Checked statement, as this project’s notation prints it
forsetiInitial  =  forsetiInitial\text{forsetiInitial}\;=\;\text{forsetiInitial}
Kernel form, without notation
Eq Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiInitial Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiInitial

Assumptions: propext, Classical.choice, Quot.sound

Lean source & proof · 1 line
example : @forsetiInitial = @DecidedShared_72f9d7627148.forsetiInitial := rfl

Example example (line 158) Kernel checkedAs recorded L158–158

Namespace Notebook.Lorenz.Decided_in_ball

Checked statement, as this project’s notation prints it
forsetiObservation  =  forsetiObservation\text{forsetiObservation}\;=\;\text{forsetiObservation}
Kernel form, without notation
Eq Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiObservation
  Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiObservation

Assumptions: none

Lean source & proof · 1 line
example : @forsetiObservation = @DecidedShared_72f9d7627148.forsetiObservation := rfl

Example example (line 159) Kernel checkedAs recorded L159–159

Namespace Notebook.Lorenz.Decided_in_ball

Checked statement, as this project’s notation prints it
forsetiIdentities  =  forsetiIdentities\text{forsetiIdentities}\;=\;\text{forsetiIdentities}
Kernel form, without notation
Eq Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiIdentities
  Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiIdentities

Assumptions: propext, Classical.choice, Quot.sound

Lean source & proof · 1 line
example : @forsetiIdentities = @DecidedShared_72f9d7627148.forsetiIdentities := rfl

Definition forsetiPost : Gimle.Forseti.Syntax.Formula 1 L161–162

Namespace Notebook.Lorenz.Decided_in_ball

(.and  (.atom  .ge  (.var  ⟨0,  by  decide⟩))  (.atom  .le  (.add  (.var  ⟨0,  by  decide⟩)  (.neg  (.constant  (mkRat  (1600)  1))))))(.\text{and}\;(.\text{atom}\;.\text{ge}\;(.\text{var}\;\langle 0,\;\text{by}\;\text{decide}\rangle ))\;(.\text{atom}\;.\text{le}\;(.\text{add}\;(.\text{var}\;\langle 0,\;\text{by}\;\text{decide}\rangle )\;(.\text{neg}\;(.\text{constant}\;(\text{mkRat}\;(1600)\;1))))))
Lean source · 2 lines
def forsetiPost : Gimle.Forseti.Syntax.Formula 1 :=
  (.and (.atom .ge (.var ⟨0, by decide⟩)) (.atom .le (.add (.var ⟨0, by decide⟩) (.neg (.constant (mkRat (1600) 1))))))

Definition forsetiLeaf : Gimle.Forseti.Syntax.Goal L164–167

Namespace Notebook.Lorenz.Decided_in_ball

⟨⟨["V"],  by  decide,  by  decide,  by  decide⟩,\langle \langle \text{[}\texttt{"V"}\text{]},\;\text{by}\;\text{decide},\;\text{by}\;\text{decide},\;\text{by}\;\text{decide}\rangle ,  (.and  (.atom  .le  (.neg  (.var  ⟨0,  by  decide⟩)))  (.atom  .le  (.add  (.var  ⟨0,  by  decide⟩)  (.neg  (.constant  (mkRat  (1600)  1)))))),\;(.\text{and}\;(.\text{atom}\;.\text{le}\;(.\text{neg}\;(.\text{var}\;\langle 0,\;\text{by}\;\text{decide}\rangle )))\;(.\text{atom}\;.\text{le}\;(.\text{add}\;(.\text{var}\;\langle 0,\;\text{by}\;\text{decide}\rangle )\;(.\text{neg}\;(.\text{constant}\;(\text{mkRat}\;(1600)\;1)))))),  forsetiPost⟩\;\text{forsetiPost}\rangle
Lean source · 4 lines
def forsetiLeaf : Gimle.Forseti.Syntax.Goal :=
  ⟨⟨["V"], by decide, by decide, by decide⟩,
    (.and (.atom .le (.neg (.var ⟨0, by decide⟩))) (.atom .le (.add (.var ⟨0, by decide⟩) (.neg (.constant (mkRat (1600) 1)))))),
    forsetiPost⟩

Definition forsetiContract : Prop L169–174

Namespace Notebook.Lorenz.Decided_in_ball

Gimle.Forseti.Trajectory.Contract  Gimle.Forseti.Examples.Lorenz.observed  forsetiTime\text{Gimle}.\text{Forseti}.\text{Trajectory}.\text{Contract}\;\text{Gimle}.\text{Forseti}.\text{Examples}.\text{Lorenz}.\text{observed}\;\text{forsetiTime}  (Gimle.Forseti.Trajectory.Initialized  forsetiTime  (fun  _  =>  True)\;(\text{Gimle}.\text{Forseti}.\text{Trajectory}.\text{Initialized}\;\text{forsetiTime}\;(\text{fun}\;\text{\_}\;=>\;\text{True})  (⋅  =  forsetiInitial))\;(\text{·}\;=\;\text{forsetiInitial}))  (Gimle.Forseti.Trajectory.Always  forsetiTime\;(\text{Gimle}.\text{Forseti}.\text{Trajectory}.\text{Always}\;\text{forsetiTime}  fun  o  =>  forsetiPost.holds  (fun  _  =>  o  forsetiObservation))\;\text{fun}\;\text{o}\;=>\;\text{forsetiPost}.\text{holds}\;(\text{fun}\;\text{\_}\;=>\;\text{o}\;\text{forsetiObservation}))
Lean source · 6 lines
def forsetiContract : Prop :=
  Gimle.Forseti.Trajectory.Contract Gimle.Forseti.Examples.Lorenz.observed forsetiTime
    (Gimle.Forseti.Trajectory.Initialized forsetiTime (fun _ => True)
      (· = forsetiInitial))
    (Gimle.Forseti.Trajectory.Always forsetiTime
      fun o => forsetiPost.holds (fun _ => o forsetiObservation))

Definition forsetiRoot : Prop L176–176

Namespace Notebook.Lorenz.Decided_in_ball

forsetiIdentities  ∧  forsetiContract\text{forsetiIdentities}\;\land \;\text{forsetiContract}
Lean source · 1 line
def forsetiRoot : Prop := forsetiIdentities ∧ forsetiContract

Theorem in_ball Kernel checkedAs recorded L178–219

Namespace Notebook.Lorenz.Decided_in_ball

V is output 3 of the compiled circuit, the port obs-v; the input's initial wires (initial-x, initial-y, initial-z) start at (1, 1, 1) at time = 0. The trajectory decider proved it. Goal (trajectory): trajectory lorenz interpretation continuous-real obligation contract start time = 0 initial initial-x = 1, initial-y = 1, initial-z = 1 observe obs-v as V always : V >= 0 and V <= 1600

Statement
have  circuit\text{have}\;\text{circuit}

Assumptions: propext, Classical.choice, Quot.sound

Lean source & proof · 42 lines
/-- V is output 3 of the compiled circuit, the port obs-v; the input's initial wires (initial-x, initial-y, initial-z) start at (1, 1, 1) at time = 0. The trajectory decider proved it. Goal (trajectory): trajectory lorenz interpretation continuous-real obligation contract start time = 0 initial initial-x = 1, initial-y = 1, initial-z = 1 observe obs-v as V always : V >= 0 and V <= 1600 -/
theorem in_ball :
    have circuit := Gimle.Forseti.Examples.Lorenz.observed
    have time : Gimle.Asgard.Dynamics.TimeDomain := ⟨"time", 0⟩
    ∀ u : Gimle.Asgard.Dynamics.Signal (0 + 3),
      Gimle.Asgard.Dynamics.signalRight u 0 = ![1, 1, 1] →
      (∃ y : Gimle.Asgard.Dynamics.Signal 4, circuit.Rel time u y) ∧
      (∀ y y' : Gimle.Asgard.Dynamics.Signal 4, circuit.Rel time u y → circuit.Rel time u y' →
        ∀ t ≥ 0, y t = y' t) ∧
      (∀ y : Gimle.Asgard.Dynamics.Signal 4, circuit.Rel time u y →
        ∀ t ≥ 0, have V := y t 3; V ≥ 0 ∧ V ≤ 1600) := by
  have root : forsetiRoot := by
    unfold forsetiRoot forsetiIdentities forsetiContract
    refine ⟨⟨by decide, by decide, by decide⟩, ?_⟩
    have time : forsetiTime = Gimle.Forseti.Examples.Lorenz.evolution.time := by simp [forsetiTime, Gimle.Asgard.Model.Evolution.time, Gimle.Forseti.Examples.Lorenz.evolution] <;> norm_num [Rat.mkRat_eq_div]
    have init : forsetiInitial = Gimle.Forseti.Examples.Lorenz.compiled.initial := by rw [Gimle.Forseti.Examples.Lorenz.initial_eq]; funext i; fin_cases i <;> norm_num [forsetiInitial, Rat.mkRat_eq_div]
    have obs : forsetiObservation = Gimle.Forseti.Examples.Lorenz.energyIndex := rfl
    rw [time, init, obs]
    refine Gimle.Forseti.Trajectory.Contract.consequence Gimle.Forseti.Examples.Lorenz.energy_contract (fun _ admitted => admitted) ?_
    intro o band t within
    have leaf := Gimle.Forseti.Syntax.EntailmentCertificate.sound forsetiLeaf ⟨[⟨⟨[]⟩, [⟨[((mkRat (1) 1), (.constant (mkRat (1) 1)))]⟩, ⟨[]⟩]⟩, ⟨⟨[]⟩, [⟨[]⟩, ⟨[((mkRat (1) 1), (.constant (mkRat (1) 1)))]⟩]⟩]⟩ (by decide +kernel)
    apply leaf
    simp only [forsetiLeaf, Gimle.Forseti.Syntax.Formula.holds, Gimle.Forseti.Syntax.Relation.holds, Gimle.Asgard.Polynomial.Expr.eval]
    obtain ⟨low, high⟩ := band t within
    refine ⟨?_, ?_⟩ <;> norm_num [Rat.mkRat_eq_div] <;> linarith
  have forsetiCast0 : (((mkRat (1600) 1) : ℚ) : ℝ) = 1600 := by norm_num [Rat.mkRat_eq_div]
  have forsetiCast1 : (((mkRat (0) 1) : ℚ) : ℝ) = 0 := by norm_num [Rat.mkRat_eq_div]
  have forsetiCast2 : (((mkRat (1) 1) : ℚ) : ℝ) = 1 := by norm_num [Rat.mkRat_eq_div]
  refine (fun (bridge : _ ↔ forsetiContract) => bridge.mpr (show forsetiIdentities ∧ forsetiContract from root).2) ?_
  constructor
  · intro claim
    unfold forsetiContract
    set_option linter.unusedSimpArgs false in simp only [Gimle.Forseti.Trajectory.Realizable, Gimle.Forseti.Trajectory.Unique, Gimle.Forseti.Trajectory.Holds, Gimle.Forseti.Trajectory.Initialized, Gimle.Forseti.Trajectory.Always, Set.EqOn, Gimle.Asgard.Dynamics.TimeDomain.domain, Set.mem_Ici, forsetiTime, forsetiInitial, forsetiPost, forsetiObservation, Gimle.Forseti.Syntax.Formula.holds, Gimle.Forseti.Syntax.Relation.holds, Gimle.Asgard.Polynomial.Expr.eval, true_and, forsetiCast0, forsetiCast1, forsetiCast2, ← sub_eq_add_neg, sub_zero, sub_nonpos, sub_neg, sub_eq_zero, sub_nonneg, sub_pos, ge_iff_le, gt_iff_lt, and_assoc, or_assoc] at claim
    refine ⟨?_, ?_, ?_⟩ <;> set_option linter.unusedSimpArgs false in simp only [Gimle.Forseti.Trajectory.Realizable, Gimle.Forseti.Trajectory.Unique, Gimle.Forseti.Trajectory.Holds, Gimle.Forseti.Trajectory.Initialized, Gimle.Forseti.Trajectory.Always, Set.EqOn, Gimle.Asgard.Dynamics.TimeDomain.domain, Set.mem_Ici, forsetiTime, forsetiInitial, forsetiPost, forsetiObservation, Gimle.Forseti.Syntax.Formula.holds, Gimle.Forseti.Syntax.Relation.holds, Gimle.Asgard.Polynomial.Expr.eval, true_and, forsetiCast0, forsetiCast1, forsetiCast2, ← sub_eq_add_neg, sub_zero, sub_nonpos, sub_neg, sub_eq_zero, sub_nonneg, sub_pos, ge_iff_le, gt_iff_lt, and_assoc, or_assoc]
    · exact fun u start => (claim u start).1
    · exact fun u start => (claim u start).2.1
    · exact fun u start => (claim u start).2.2
  · intro lane
    unfold forsetiContract at lane
    obtain ⟨realizable, unique, holds⟩ := lane
    set_option linter.unusedSimpArgs false in simp only [Gimle.Forseti.Trajectory.Realizable, Gimle.Forseti.Trajectory.Unique, Gimle.Forseti.Trajectory.Holds, Gimle.Forseti.Trajectory.Initialized, Gimle.Forseti.Trajectory.Always, Set.EqOn, Gimle.Asgard.Dynamics.TimeDomain.domain, Set.mem_Ici, forsetiTime, forsetiInitial, forsetiPost, forsetiObservation, Gimle.Forseti.Syntax.Formula.holds, Gimle.Forseti.Syntax.Relation.holds, Gimle.Asgard.Polynomial.Expr.eval, true_and, forsetiCast0, forsetiCast1, forsetiCast2, ← sub_eq_add_neg, sub_zero, sub_nonpos, sub_neg, sub_eq_zero, sub_nonneg, sub_pos, ge_iff_le, gt_iff_lt, and_assoc, or_assoc] at realizable unique holds ⊢
    exact fun u start => ⟨realizable u start, unique u start, holds u start⟩
end Decided_in_ball
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Decisionbelow_thirteen_hundred · trajectory · forsetiNever runL224–224
trajectory lorenz from (1, 1, 1) at 0 observe V always : V <= 1300

Not run yet.

namespace Decided_below_thirteen_hundred
L226–226
open DecidedShared_72f9d7627148
L228–228

Example example (line 230) Kernel checkedAs recorded L230–230

Namespace Notebook.Lorenz.Decided_below_thirteen_hundred

Checked statement, as this project’s notation prints it
forsetiTime  =  forsetiTime\text{forsetiTime}\;=\;\text{forsetiTime}
Kernel form, without notation
Eq Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiTime Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiTime

Assumptions: propext, Classical.choice, Quot.sound

Lean source & proof · 1 line
example : @forsetiTime = @DecidedShared_72f9d7627148.forsetiTime := rfl

Example example (line 231) Kernel checkedAs recorded L231–231

Namespace Notebook.Lorenz.Decided_below_thirteen_hundred

Checked statement, as this project’s notation prints it
forsetiInitial  =  forsetiInitial\text{forsetiInitial}\;=\;\text{forsetiInitial}
Kernel form, without notation
Eq Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiInitial Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiInitial

Assumptions: propext, Classical.choice, Quot.sound

Lean source & proof · 1 line
example : @forsetiInitial = @DecidedShared_72f9d7627148.forsetiInitial := rfl

Example example (line 232) Kernel checkedAs recorded L232–232

Namespace Notebook.Lorenz.Decided_below_thirteen_hundred

Checked statement, as this project’s notation prints it
forsetiObservation  =  forsetiObservation\text{forsetiObservation}\;=\;\text{forsetiObservation}
Kernel form, without notation
Eq Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiObservation
  Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiObservation

Assumptions: none

Lean source & proof · 1 line
example : @forsetiObservation = @DecidedShared_72f9d7627148.forsetiObservation := rfl

Example example (line 233) Kernel checkedAs recorded L233–233

Namespace Notebook.Lorenz.Decided_below_thirteen_hundred

Checked statement, as this project’s notation prints it
forsetiIdentities  =  forsetiIdentities\text{forsetiIdentities}\;=\;\text{forsetiIdentities}
Kernel form, without notation
Eq Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiIdentities
  Notebook.Lorenz.DecidedShared_72f9d7627148.forsetiIdentities

Assumptions: propext, Classical.choice, Quot.sound

Lean source & proof · 1 line
example : @forsetiIdentities = @DecidedShared_72f9d7627148.forsetiIdentities := rfl

Definition forsetiPost : Gimle.Forseti.Syntax.Formula 1 L235–236

Namespace Notebook.Lorenz.Decided_below_thirteen_hundred

(.atom  .le  (.add  (.var  ⟨0,  by  decide⟩)  (.neg  (.constant  (mkRat  (1300)  1)))))(.\text{atom}\;.\text{le}\;(.\text{add}\;(.\text{var}\;\langle 0,\;\text{by}\;\text{decide}\rangle )\;(.\text{neg}\;(.\text{constant}\;(\text{mkRat}\;(1300)\;1)))))
Lean source · 2 lines
def forsetiPost : Gimle.Forseti.Syntax.Formula 1 :=
  (.atom .le (.add (.var ⟨0, by decide⟩) (.neg (.constant (mkRat (1300) 1)))))

Definition forsetiContract : Prop L238–243

Namespace Notebook.Lorenz.Decided_below_thirteen_hundred

Gimle.Forseti.Trajectory.Contract  Gimle.Forseti.Examples.Lorenz.observed  forsetiTime\text{Gimle}.\text{Forseti}.\text{Trajectory}.\text{Contract}\;\text{Gimle}.\text{Forseti}.\text{Examples}.\text{Lorenz}.\text{observed}\;\text{forsetiTime}  (Gimle.Forseti.Trajectory.Initialized  forsetiTime  (fun  _  =>  True)\;(\text{Gimle}.\text{Forseti}.\text{Trajectory}.\text{Initialized}\;\text{forsetiTime}\;(\text{fun}\;\text{\_}\;=>\;\text{True})  (⋅  =  forsetiInitial))\;(\text{·}\;=\;\text{forsetiInitial}))  (Gimle.Forseti.Trajectory.Always  forsetiTime\;(\text{Gimle}.\text{Forseti}.\text{Trajectory}.\text{Always}\;\text{forsetiTime}  fun  o  =>  forsetiPost.holds  (fun  _  =>  o  forsetiObservation))\;\text{fun}\;\text{o}\;=>\;\text{forsetiPost}.\text{holds}\;(\text{fun}\;\text{\_}\;=>\;\text{o}\;\text{forsetiObservation}))
Lean source · 6 lines
def forsetiContract : Prop :=
  Gimle.Forseti.Trajectory.Contract Gimle.Forseti.Examples.Lorenz.observed forsetiTime
    (Gimle.Forseti.Trajectory.Initialized forsetiTime (fun _ => True)
      (· = forsetiInitial))
    (Gimle.Forseti.Trajectory.Always forsetiTime
      fun o => forsetiPost.holds (fun _ => o forsetiObservation))

Definition forsetiRoot : Prop L245–245

Namespace Notebook.Lorenz.Decided_below_thirteen_hundred

forsetiIdentities  ∧  forsetiContract\text{forsetiIdentities}\;\land \;\text{forsetiContract}
Lean source · 1 line
def forsetiRoot : Prop := forsetiIdentities ∧ forsetiContract

Theorem below_thirteen_hundred Contains sorryAs recorded L247–258

Namespace Notebook.Lorenz.Decided_below_thirteen_hundred

V is output 3 of the compiled circuit, the port obs-v; the input's initial wires (initial-x, initial-y, initial-z) start at (1, 1, 1) at time = 0. The trajectory decider refuted it. Goal (trajectory): trajectory lorenz interpretation continuous-real obligation contract start time = 0 initial initial-x = 1, initial-y = 1, initial-z = 1 observe obs-v as V always : V <= 1300

Statement
have  circuit\text{have}\;\text{circuit}

Unfinished: the statement’s proof contains a placeholder.

Assumptions: propext, sorryAx, Classical.choice, Quot.sound

Lean source & proof · 12 lines
/-- V is output 3 of the compiled circuit, the port obs-v; the input's initial wires (initial-x, initial-y, initial-z) start at (1, 1, 1) at time = 0. The trajectory decider refuted it. Goal (trajectory): trajectory lorenz interpretation continuous-real obligation contract start time = 0 initial initial-x = 1, initial-y = 1, initial-z = 1 observe obs-v as V always : V <= 1300 -/
theorem below_thirteen_hundred :
    have circuit := Gimle.Forseti.Examples.Lorenz.observed
    have time : Gimle.Asgard.Dynamics.TimeDomain := ⟨"time", 0⟩
    ∀ u : Gimle.Asgard.Dynamics.Signal (0 + 3),
      Gimle.Asgard.Dynamics.signalRight u 0 = ![1, 1, 1] →
      (∃ y : Gimle.Asgard.Dynamics.Signal 4, circuit.Rel time u y) ∧
      (∀ y y' : Gimle.Asgard.Dynamics.Signal 4, circuit.Rel time u y → circuit.Rel time u y' →
        ∀ t ≥ 0, y t = y' t) ∧
      (∀ y : Gimle.Asgard.Dynamics.Signal 4, circuit.Rel time u y →
        ∀ t ≥ 0, have V := y t 3; V ≤ 1300) := by
  sorry
end Decided_below_thirteen_hundred
L260–260
Decisionbelow_fourteen_hundred · trajectory · forsetiNever runL263–263
trajectory lorenz from (1, 1, 1) at 0 observe V always : V <= 1400

Not run yet.

What the ball says in coordinates

V≤1600V \le 1600 means ∣x∣≤40|x| \le 40, ∣y∣≤40|y| \le 40 and −2≤z≤78-2 \le z \le 78. The simulation stays well inside: it never sees ∣x∣|x| above 2020, ∣y∣|y| above 2828 or zz above 4848. The gap between the proved ball and the observed attractor is real, and closing it would need a sharper invariant than a single quadratic; that is the next rung, not this one.

#print axioms energy_at_start
L275–275
#print axioms rate_positive_at_origin_side
L276–276
#print axioms storage_inequality
L277–277
end Notebook.Lorenz
L279–279