proofs
Provedaitherdsposcillatorexact-real

Aither: semi-implicit oscillator equilibrium, modified energy and a physical-energy counterexample

For all h,k,c,x,v∈R\htmlData{mx=1}{h, k, c, x, v \in \htmlData{mx=0}{\mathbb{R}}}:

oscillator⁡(h,k,c,k⋅x,(x,0))=(x,0)∧modifiedEnergy⁡(h,k,oscillator⁡(h,k,0,0,(x,v)))=modifiedEnergy⁡(h,k,(x,v))∧modifiedEnergy⁡(h,k,(x,v))=(v−h⋅k⋅x2)2+k⋅(1−h2⋅k4)⋅x2∧(0≤k  ⟹  h2⋅k≤4  ⟹  0≤modifiedEnergy⁡(h,k,(x,v)))∧physicalEnergy⁡(1,oscillator⁡(1,1,0,0,(0,1)))>physicalEnergy⁡(1,(0,1))\begin{aligned} & \htmlData{mx=10}{\htmlData{mx=1}{\operatorname{oscillator}}(\htmlData{mx=2}{h}, \htmlData{mx=3}{k}, \htmlData{mx=4}{c}, \htmlData{mx=7}{\htmlData{mx=5}{k} \cdot \htmlData{mx=6}{x}}, \htmlData{mx=9}{\left( \htmlData{mx=8}{x}, 0 \right)})} \mathrel{\htmlData{mx=13}{=}} \htmlData{mx=12}{\left( \htmlData{mx=11}{x}, 0 \right)} \\ & \htmlData{mx=0}{\land} \htmlData{mx=25}{\htmlData{mx=15}{\operatorname{modifiedEnergy}}(\htmlData{mx=16}{h}, \htmlData{mx=17}{k}, \htmlData{mx=24}{\htmlData{mx=18}{\operatorname{oscillator}}(\htmlData{mx=19}{h}, \htmlData{mx=20}{k}, 0, 0, \htmlData{mx=23}{\left( \htmlData{mx=21}{x}, \htmlData{mx=22}{v} \right)})})} \\ & \quad \htmlData{mx=14}{=} \htmlData{mx=32}{\htmlData{mx=26}{\operatorname{modifiedEnergy}}(\htmlData{mx=27}{h}, \htmlData{mx=28}{k}, \htmlData{mx=31}{\left( \htmlData{mx=29}{x}, \htmlData{mx=30}{v} \right)})} \\ & \htmlData{mx=0}{\land} \htmlData{mx=40}{\htmlData{mx=34}{\operatorname{modifiedEnergy}}(\htmlData{mx=35}{h}, \htmlData{mx=36}{k}, \htmlData{mx=39}{\left( \htmlData{mx=37}{x}, \htmlData{mx=38}{v} \right)})} \\ & \quad \htmlData{mx=33}{=} \htmlData{mx=61}{\htmlData{mx=49}{\left(\htmlData{mx=48}{\htmlData{mx=41}{v} - \htmlData{mx=47}{\frac{\htmlData{mx=46}{\htmlData{mx=44}{\htmlData{mx=42}{h} \cdot \htmlData{mx=43}{k}} \cdot \htmlData{mx=45}{x}}}{2}}}\right)^{2}} + \htmlData{mx=60}{\htmlData{mx=57}{\htmlData{mx=50}{k} \cdot \left(\htmlData{mx=56}{1 - \htmlData{mx=55}{\frac{\htmlData{mx=54}{\htmlData{mx=52}{\htmlData{mx=51}{h}^{2}} \cdot \htmlData{mx=53}{k}}}{4}}}\right)} \cdot \htmlData{mx=59}{\htmlData{mx=58}{x}^{2}}}} \\ & \htmlData{mx=0}{\land} (0 \mathrel{\htmlData{mx=63}{\leq}} \htmlData{mx=62}{k} \mathrel{\htmlData{mx=78}{\implies}} \htmlData{mx=67}{\htmlData{mx=65}{\htmlData{mx=64}{h}^{2}} \cdot \htmlData{mx=66}{k}} \mathrel{\htmlData{mx=68}{\leq}} 4 \mathrel{\htmlData{mx=77}{\implies}} 0 \mathrel{\htmlData{mx=76}{\leq}} \htmlData{mx=75}{\htmlData{mx=69}{\operatorname{modifiedEnergy}}(\htmlData{mx=70}{h}, \htmlData{mx=71}{k}, \htmlData{mx=74}{\left( \htmlData{mx=72}{x}, \htmlData{mx=73}{v} \right)})}) \\ & \htmlData{mx=0}{\land} \htmlData{mx=84}{\htmlData{mx=80}{\operatorname{physicalEnergy}}(1, \htmlData{mx=83}{\htmlData{mx=81}{\operatorname{oscillator}}(1, 1, 0, 0, \htmlData{mx=82}{\left( 0, 1 \right)})})} \\ & \quad \htmlData{mx=79}{>} \htmlData{mx=87}{\htmlData{mx=85}{\operatorname{physicalEnergy}}(1, \htmlData{mx=86}{\left( 0, 1 \right)})} \end{aligned}
What the notation means · or point at any symbol
RReal numbers
The real numbers: every point on the number line. (ℝ)
x ∈ TVariables
h, k, c, x, v range over real numbers: the statement must hold for every choice. ((h k c x v : ℝ))
oscillator
oscillator: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (oscillator)
f(x)Function application
oscillator applied to h, k, c, k · x, (x, 0). Lean writes f x; papers write f(x). (oscillator h k c (k * x) (x, 0))
=Equality
oscillator(h, k, c, k · x, (x, 0)) and (x, 0) are the same. (oscillator h k c (k * x) (x, 0) = (x, 0))
·Multiplication
k times x. (k * x)
(a, b)Tuple
The ordered pair (or tuple) of x, 0. ((x, 0))
modifiedEnergy
modifiedEnergy: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (modifiedEnergy)
aᵇPower
v − h · k · x/2 raised to the power 2. ((v - h * k * x / 2) ^ 2)
+Addition
(v − h · k · x/2)^2 plus k · (1 − h^2 · k/4) · x^2. ((v - h * k * x / 2) ^ 2 + k * (1 - h ^ …)
-Subtraction
v minus h · k · x/2. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (v - h * k * x / 2)
a/bDivision
h · k · x divided by 2. In Lean, dividing by 0 gives 0. (h * k * x / 2)
∧And
Both hold: 0 ≤ k → (h^2 · k ≤ 4 → 0 ≤ modifiedEnergy(h, k, (x, v))), and physicalEnergy(1, oscillator(1, 1, 0, 0, (0, 1))) > physic…. ((0 ≤ k → h ^ 2 * k ≤ 4 → 0 ≤ modifiedEn…)
≤At most
0 is less than or equal to k. (0 ≤ k)
⟹Implies
If h^2 · k ≤ 4, then 0 ≤ modifiedEnergy(h, k, (x, v)). (Between types, A → B is the type of functions from A to B.) (h ^ 2 * k ≤ 4 → 0 ≤ modifiedEnergy h k …)
physicalEnergy
physicalEnergy: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (physicalEnergy)
>Greater than
physicalEnergy(1, oscillator(1, 1, 0, 0, (0, 1))) is strictly larger than physicalEnergy(1, (0, 1)). (physicalEnergy 1 (oscillator 1 1 0 0 (0…)

Exact-real model of nDho with velocity updated before position. Proves constant-force equilibrium, modified-energy invariance for unforced undamped motion, completed-square identity, nonnegativity under k>=0 and h^2*k<=4, and an explicit increase in ordinary physical energy. Does not establish Float64 behavior, damped/forced global stability, or positive definiteness at the boundary.

How it was proved

Proved
1 October 2026
By
Ajax
Checked by
Lean's kernel (Lean 4.33.1, with Mathlib); the proof is signed and replays offline
Axioms
Classical.choice, Quot.sound, propext
Proof
30 lines of Lean
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