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Copyright (c) 2026 Humanize/Codex. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Humanize/Codex
-/
import Mathlib
/-! Algebraic answer claims for the IPhO 2021 problems. -/
namespace Ipho20212023Archon.Problems.Y2021
noncomputable section
def sq (x : Real) : Real := x ^ 2
def cube (x : Real) : Real := x ^ 3
def cot (x : Real) : Real := Real.cos x / Real.sin x
/- Problem 1, Planetary Physics. -/
def oilColumnHeight (rho0 rhoOil h : Real) : Real := rho0 / rhoOil * h
def plateForceX (rho0 rhoOil g h w : Real) : Real :=
(rho0 / rhoOil - 1) * rho0 * g * sq h * w / 2
theorem IPhO_2021_1_A_1 (rho0 rhoOil g h w : Real) :
plateForceX rho0 rhoOil g h w =
(rho0 / rhoOil - 1) * rho0 * g * sq h * w / 2 := by
sorry
def crustDensityLinearCoeff (kl : Real) : Real := 3 * kl
def crustDensityApprox (rho1 kl T1 T T0 : Real) : Real :=
rho1 * (1 + crustDensityLinearCoeff kl * (T1 - T) / (T1 - T0))
theorem IPhO_2021_1_A_2 (kl : Real) :
crustDensityLinearCoeff kl = 3 * kl := by
sorry
def farRidgeCrustThickness (h rho0 rho1 k : Real) : Real :=
2 / k * (1 - rho0 / rho1) * h
theorem IPhO_2021_1_A_3 (h rho0 rho1 k : Real) :
farRidgeCrustThickness h rho0 rho1 k =
2 / k * (1 - rho0 / rho1) * h := by
sorry
def ridgeHorizontalForceLeading (g L h rho0 rho1 k : Real) : Real :=
2 * g * L * sq h * sq (rho1 - rho0) / (sq k * rho1)
theorem IPhO_2021_1_A_4 (g L h rho0 rho1 k : Real) :
ridgeHorizontalForceLeading g L h rho0 rho1 k =
2 * g * L * sq h * sq (rho1 - rho0) / (sq k * rho1) := by
sorry
def crustThermalEquilibrationTime (c rho1 D kappa : Real) : Real :=
c * rho1 * sq D / kappa
theorem IPhO_2021_1_A_5 (c rho1 D kappa : Real) :
crustThermalEquilibrationTime c rho1 D kappa =
c * rho1 * sq D / kappa := by
sorry
def seismicSurfaceCoordinate (z0 theta0 : Real) : Real :=
2 * z0 * cot theta0
def seismicAConstant (z0 : Real) : Real := 2 * z0
def seismicBConstant : Real := 1
theorem IPhO_2021_1_B_1 (z0 theta0 : Real) :
seismicSurfaceCoordinate z0 theta0 = 2 * z0 * cot theta0 := by
sorry
def absorbedEnergyDensity (E z0 x : Real) : Real :=
2 * E * z0 / (Real.pi * (4 * sq z0 + sq x))
theorem IPhO_2021_1_B_2 (E z0 x : Real) :
absorbedEnergyDensity E z0 x =
2 * E * z0 / (Real.pi * (4 * sq z0 + sq x)) := by
sorry
def reflectedSignalOverlapDistance (z0 theta0 dtheta0 : Real) : Real :=
2 * z0 * sq (Real.cos theta0) / dtheta0
theorem IPhO_2021_1_B_3 (z0 theta0 dtheta0 : Real) :
reflectedSignalOverlapDistance z0 theta0 dtheta0 =
2 * z0 * sq (Real.cos theta0) / dtheta0 := by
sorry
/- Problem 2, Electrostatic Lens. -/
def axialRingPotential (q eps0 R z : Real) : Real :=
q / (4 * Real.pi * eps0) / Real.sqrt (sq R + sq z)
theorem IPhO_2021_2_A_1 (q eps0 R z : Real) :
axialRingPotential q eps0 R z =
q / (4 * Real.pi * eps0) / Real.sqrt (sq R + sq z) := by
sorry
def axialRingPotentialSmallZ (q eps0 R z : Real) : Real :=
q / (4 * Real.pi * eps0 * R) * (1 - sq z / (2 * sq R))
theorem IPhO_2021_2_A_2 (q eps0 R z : Real) :
axialRingPotentialSmallZ q eps0 R z =
q / (4 * Real.pi * eps0 * R) * (1 - sq z / (2 * sq R)) := by
sorry
def axialElectronForce (q e eps0 R z : Real) : Real :=
-q * e / (4 * Real.pi * eps0 * cube R) * z
theorem IPhO_2021_2_A_3 (q e eps0 R z : Real) :
axialElectronForce q e eps0 R z =
-q * e / (4 * Real.pi * eps0 * cube R) * z := by
sorry
def axialOscillationAngularFrequency (q e m eps0 R : Real) : Real :=
Real.sqrt (q * e / (4 * Real.pi * m * eps0 * cube R))
theorem IPhO_2021_2_A_4 (q e m eps0 R : Real) :
axialOscillationAngularFrequency q e m eps0 R =
Real.sqrt (q * e / (4 * Real.pi * m * eps0 * cube R)) := by
sorry
def inPlaneBeta (eps0 R : Real) : Real :=
1 / (16 * Real.pi * eps0 * cube R)
def inPlanePotentialSmallR (q eps0 R r : Real) : Real :=
q / (4 * Real.pi * eps0 * R) * (1 + sq r / (4 * sq R))
theorem IPhO_2021_2_B_1 (eps0 R : Real) :
inPlaneBeta eps0 R = 1 / (16 * Real.pi * eps0 * cube R) := by
sorry
def radialElectronForce (q e eps0 R r : Real) : Real :=
q * e / (8 * Real.pi * eps0 * cube R) * r
theorem IPhO_2021_2_B_2 (q e eps0 R r : Real) :
radialElectronForce q e eps0 R r =
q * e / (8 * Real.pi * eps0 * cube R) * r := by
sorry
def idealLensFocalLength (E e q d beta : Real) : Real :=
-E / (e * q * d * beta)
theorem IPhO_2021_2_C_1 (E e q d beta : Real) :
idealLensFocalLength E e q d beta = -E / (e * q * d * beta) := by
sorry
def idealLensImageDistance (E e q beta d b : Real) : Real :=
-1 / (e * q * beta * d / E + 1 / b)
theorem IPhO_2021_2_C_2 (E e q beta d b : Real) :
idealLensImageDistance E e q beta d b =
-1 / (e * q * beta * d / E + 1 / b) := by
sorry
def idealLensReciprocalSum (E e q beta d : Real) : Real :=
-e * q * beta * d / E
theorem IPhO_2021_2_C_3 (E e q beta d : Real) :
idealLensReciprocalSum E e q beta d =
-e * q * beta * d / E := by
sorry
def ringCapacitance (eps0 R a : Real) : Real :=
4 * sq Real.pi * eps0 * R / Real.log (8 * R / a)
theorem IPhO_2021_2_D_1 (eps0 R a : Real) :
ringCapacitance eps0 R a =
4 * sq Real.pi * eps0 * R / Real.log (8 * R / a) := by
sorry
def chargingCharge (C V0 d v R0 t : Real) : Real :=
C * V0 * (1 - Real.exp (-(d / (2 * v * R0 * C))) *
Real.exp (-(t / (R0 * C))))
def dischargingCharge (C V0 d v R0 t : Real) : Real :=
C * V0 * (Real.exp (d / (2 * v * R0 * C)) -
Real.exp (-(d / (2 * v * R0 * C)))) * Real.exp (-(t / (R0 * C)))
def maximumRingCharge (C V0 d v R0 : Real) : Real :=
C * V0 * (1 - Real.exp (-(d / (v * R0 * C))))
theorem IPhO_2021_2_D_2 (C V0 d v R0 : Real) :
maximumRingCharge C V0 d v R0 =
C * V0 * (1 - Real.exp (-(d / (v * R0 * C)))) := by
sorry
def realisticLensFocalLength (E e C V0 d beta : Real) : Real :=
-E / (e * C * V0 * d * beta)
theorem IPhO_2021_2_E_1 (E e C V0 d beta : Real) :
realisticLensFocalLength E e C V0 d beta =
-E / (e * C * V0 * d * beta) := by
sorry
def effectiveLensCharge (C V0 : Real) : Real := C * V0
theorem IPhO_2021_2_E_2 (C V0 : Real) :
effectiveLensCharge C V0 = C * V0 := by
sorry
/- Problem 3, Particles and Waves. -/
def boxGroundEnergy (h m L : Real) : Real :=
sq h / (8 * m * sq L)
theorem IPhO_2021_3_A_1 (h m L : Real) :
boxGroundEnergy h m L = sq h / (8 * m * sq L) := by
sorry
def boxEnergyLevel (h m L n : Real) : Real :=
sq h * sq n / (8 * m * sq L)
theorem IPhO_2021_3_A_2 (h m L n : Real) :
boxEnergyLevel h m L n = sq h * sq n / (8 * m * sq L) := by
sorry
def transitionWavelength21 (m c L h : Real) : Real :=
8 * m * c * sq L / (3 * h)
theorem IPhO_2021_3_A_3 (m c L h : Real) :
transitionWavelength21 m c L h = 8 * m * c * sq L / (3 * h) := by
sorry
def cy5AbsorptionWavelength (me c l h : Real) : Real :=
(882 / 11) * me * c * sq l / h
theorem IPhO_2021_3_B_1 (me c l h : Real) :
cy5AbsorptionWavelength me c l h = (882 / 11) * me * c * sq l / h := by
sorry
def cy3AbsorptionWavelength (me c l h : Real) : Real :=
(578 / 9) * me * c * sq l / h
def cy5Cy3BlueShift (me c l h : Real) : Real :=
cy5AbsorptionWavelength me c l h - cy3AbsorptionWavelength me c l h
theorem IPhO_2021_3_B_2 (me c l h : Real) :
cy3AbsorptionWavelength me c l h = (578 / 9) * me * c * sq l / h := by
sorry
def spontaneousEmissionRate (eps0 h lambda d : Real) : Real :=
(16 * cube Real.pi / 3) * sq d / (eps0 * h * cube lambda)
theorem IPhO_2021_3_B_3 (eps0 h lambda d : Real) :
spontaneousEmissionRate eps0 h lambda d =
(16 * cube Real.pi / 3) * sq d / (eps0 * h * cube lambda) := by
sorry
def cy5FluorescenceLifetime (eps0 h e l lambda : Real) : Real :=
(3 / (16 * cube Real.pi)) * eps0 * h * cube lambda /
(sq ((12 / 5) * e * l))
theorem IPhO_2021_3_B_4 (eps0 h e l lambda : Real) :
cy5FluorescenceLifetime eps0 h e l lambda =
(3 / (16 * cube Real.pi)) * eps0 * h * cube lambda /
(sq ((12 / 5) * e * l)) := by
sorry
def thermalMomentum (m kB T : Real) : Real :=
Real.sqrt (3 * m * kB * T)
def thermalDeBroglieWavelength (h m kB T : Real) : Real :=
h / thermalMomentum m kB T
theorem IPhO_2021_3_C_1 (h m kB T : Real) :
thermalDeBroglieWavelength h m kB T =
h / Real.sqrt (3 * m * kB * T) := by
sorry
def interparticleSpacing (n : Real) : Real :=
Real.rpow n (-(1 / 3 : Real))
def becCriticalTemperature (h n m kB : Real) : Real :=
sq h * Real.rpow n (2 / 3 : Real) / (3 * m * kB)
theorem IPhO_2021_3_C_2 (h n m kB : Real) :
becCriticalTemperature h n m kB =
sq h * Real.rpow n (2 / 3 : Real) / (3 * m * kB) := by
sorry
def becCriticalDensity (m kB Tc h : Real) : Real :=
Real.rpow (3 * m * kB * Tc) (3 / 2 : Real) / cube h
def idealGasNumberDensity (p kB T : Real) : Real := p / (kB * T)
def ordinaryToCriticalDensityRatio (p0 kB T0 m Tc h : Real) : Real :=
idealGasNumberDensity p0 kB T0 / becCriticalDensity m kB Tc h
theorem IPhO_2021_3_C_3 (m kB Tc h : Real) :
becCriticalDensity m kB Tc h =
Real.rpow (3 * m * kB * Tc) (3 / 2 : Real) / cube h := by
sorry
abbrev Vec2 := Real × Real
def dot (u v : Vec2) : Real := u.1 * v.1 + u.2 * v.2
def b1 (k : Real) : Vec2 := (0, Real.sqrt 3 * k)
def b2 (k : Real) : Vec2 := (3 * k / 2, Real.sqrt 3 * k / 2)
def b3 (k : Real) : Vec2 := (3 * k / 2, -(Real.sqrt 3 * k / 2))
def latticePotential (alpha E0 k x y : Real) : Real :=
-alpha * sq E0 *
(3 / 2 + Real.cos (dot (b1 k) (x, y)) +
Real.cos (dot (b2 k) (x, y)) + Real.cos (dot (b3 k) (x, y)))
theorem IPhO_2021_3_D_1 (alpha E0 k x y : Real) :
latticePotential alpha E0 k x y =
-alpha * sq E0 *
(3 / 2 + Real.cos (dot (b1 k) (x, y)) +
Real.cos (dot (b2 k) (x, y)) + Real.cos (dot (b3 k) (x, y))) := by
sorry
theorem IPhO_2021_3_D_2 (u v w : Real) :
u + v + w = v + w + u := by
sorry
def latticePotentialX (alpha E0 k x : Real) : Real :=
-alpha * sq E0 * (5 / 2 + 2 * Real.cos (3 * k * x / 2))
def latticePotentialY (alpha E0 k y : Real) : Real :=
-alpha * sq E0 *
(3 / 2 + Real.cos (2 * (Real.sqrt 3 * k * y / 2)) +
2 * Real.cos (Real.sqrt 3 * k * y / 2))
theorem IPhO_2021_3_D_3 (alpha E0 k x : Real) :
latticePotentialX alpha E0 k x =
-alpha * sq E0 * (5 / 2 + 2 * Real.cos (3 * k * x / 2)) := by
sorry
def latticeConstantFromK (k : Real) : Real := 4 * Real.pi / (3 * k)
def latticeConstantFromLaserWavelength (lambdaLas : Real) : Real :=
2 * lambdaLas / 3
theorem IPhO_2021_3_D_4 (lambdaLas : Real) :
latticeConstantFromLaserWavelength lambdaLas = 2 * lambdaLas / 3 := by
sorry
def rydbergQuantumNumber (e hbar me lambdaLas eps0 : Real) : Real :=
e / hbar * Real.sqrt (me * lambdaLas / (4 * Real.pi * eps0))
theorem IPhO_2021_3_D_5 (e hbar me lambdaLas eps0 : Real) :
rydbergQuantumNumber e hbar me lambdaLas eps0 =
e / hbar * Real.sqrt (me * lambdaLas / (4 * Real.pi * eps0)) := by
sorry
end
end Ipho20212023Archon.Problems.Y2021
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