/- 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