| /- |
| 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 2023 theory problems. The definitions |
| below isolate the official closed-form expressions; the named theorems expose |
| one answer statement for each requested subquestion. |
| -/ |
|
|
| open scoped BigOperators |
|
|
| namespace Ipho20212023Archon.Problems.Y2023 |
|
|
| noncomputable section |
|
|
| section Problem1 |
|
|
| def brownianInitialVelocity (I0 M : β) : β := I0 / M |
|
|
| def brownianRelaxationTime (M gamma : β) : β := M / gamma |
|
|
| theorem IPhO_2023_1_A_1 (I0 M gamma : β) : |
| (brownianInitialVelocity I0 M, brownianRelaxationTime M gamma) = |
| (I0 / M, M / gamma) := by |
| sorry |
|
|
| def collisionVelocity {ΞΉ : Type*} (s : Finset ΞΉ) (impulse time : ΞΉ β β) |
| (M tau t : β) : β := |
| β i β s, impulse i / M * Real.exp (-(t - time i) / tau) |
|
|
| def collisionTimeInRange (t ti : β) : Prop := 0 < ti β§ ti < t |
|
|
| theorem IPhO_2023_1_A_2 {ΞΉ : Type*} (s : Finset ΞΉ) (impulse time : ΞΉ β β) |
| (M tau t : β) : |
| collisionVelocity s impulse time M tau t = |
| β i β s, impulse i / M * Real.exp (-(t - time i) / tau) β§ |
| (β i, collisionTimeInRange t (time i) β 0 < time i β§ time i < t) := by |
| sorry |
|
|
| def brownianMeanDisplacement : β := 0 |
|
|
| def discreteMeanSquareDisplacement (N : β) (C delta : β) : β := |
| (N : β) * C * delta ^ 2 |
|
|
| def brownianMeanSquareDisplacement (C delta t : β) : β := C * delta * t |
|
|
| theorem IPhO_2023_1_B_1 (N : β) (C delta t : β) |
| (ht : t = (N : β) * delta) : |
| brownianMeanDisplacement = 0 β§ |
| discreteMeanSquareDisplacement N C delta = brownianMeanSquareDisplacement C delta t := by |
| sorry |
|
|
| def velocityVarianceDeltaExponent : β€ := -1 |
|
|
| def meanSquareTimeExponent : β€ := 1 |
|
|
| theorem IPhO_2023_1_B_2 : |
| (velocityVarianceDeltaExponent, meanSquareTimeExponent) = (-1, 1) := by |
| sorry |
|
|
| def positiveCrossingRate (n0 dn v delta : β) : β := |
| ((1 : β) / 2) * n0 * v - ((1 : β) / 4) * dn * v ^ 2 * delta |
|
|
| theorem IPhO_2023_1_C_1 (n0 dn v delta : β) : |
| positiveCrossingRate n0 dn v delta = |
| ((1 : β) / 2) * n0 * v - ((1 : β) / 4) * dn * v ^ 2 * delta := by |
| sorry |
|
|
| def negativeCrossingRate (n0 dn v delta : β) : β := |
| ((1 : β) / 2) * n0 * v + ((1 : β) / 4) * dn * v ^ 2 * delta |
|
|
| def diffusionFlux (C delta dn : β) : β := -((1 : β) / 2) * dn * C * delta |
|
|
| def diffusionCoefficient (C delta : β) : β := ((1 : β) / 2) * C * delta |
|
|
| theorem IPhO_2023_1_C_2 (C delta dn t : β) : |
| diffusionFlux C delta dn = -((1 : β) / 2) * dn * C * delta β§ |
| diffusionCoefficient C delta = ((1 : β) / 2) * C * delta β§ |
| brownianMeanSquareDisplacement C delta t = 2 * diffusionCoefficient C delta * t := by |
| sorry |
|
|
| def electricConcentrationGradient (n kT Q E : β) : β := n / kT * Q * E |
|
|
| theorem IPhO_2023_1_C_3 (n kT Q E : β) : |
| electricConcentrationGradient n kT Q E = n / kT * Q * E := by |
| sorry |
|
|
| def meanElectrophoreticVelocity (Q E gamma tau t : β) : β := |
| Q * E / gamma * (1 - Real.exp (-t / tau)) |
|
|
| def electrophoreticDrift (Q E gamma : β) : β := Q * E / gamma |
|
|
| theorem IPhO_2023_1_C_4 (Q E gamma tau t : β) : |
| (meanElectrophoreticVelocity Q E gamma tau t, electrophoreticDrift Q E gamma) = |
| (Q * E / gamma * (1 - Real.exp (-t / tau)), Q * E / gamma) := by |
| sorry |
|
|
| def einsteinDiffusion (kT gamma : β) : β := kT / gamma |
|
|
| theorem IPhO_2023_1_C_5 (kT gamma : β) : |
| einsteinDiffusion kT gamma = kT / gamma := by |
| sorry |
|
|
| def meanSquareAvogadroRelation (R T deltaT a eta NA : β) : β := |
| R * T * deltaT / (3 * Real.pi * a * eta * NA) |
|
|
| def avogadroFromMSD (R T deltaT a eta msd : β) : β := |
| R * T * deltaT / (3 * Real.pi * a * eta * msd) |
|
|
| def avogadroEstimate : β := ((56 : β) / 10) * (10 : β) ^ 23 |
|
|
| theorem IPhO_2023_1_D_1 (R T deltaT a eta NA msd : β) : |
| meanSquareAvogadroRelation R T deltaT a eta NA = |
| R * T * deltaT / (3 * Real.pi * a * eta * NA) β§ |
| avogadroFromMSD R T deltaT a eta msd = |
| R * T * deltaT / (3 * Real.pi * a * eta * msd) β§ |
| avogadroEstimate = ((56 : β) / 10) * (10 : β) ^ 23 := by |
| sorry |
|
|
| def driftDiffusionMSD (u D t : β) : β := (u * t) ^ 2 + 2 * D * t |
|
|
| def driftDiffusionCrossoverTime (D u : β) : β := 2 * D / u ^ 2 |
|
|
| theorem IPhO_2023_1_D_2 (u D t : β) : |
| (driftDiffusionMSD u D t, driftDiffusionCrossoverTime D u) = |
| ((u * t) ^ 2 + 2 * D * t, 2 * D / u ^ 2) := by |
| sorry |
|
|
| def runTumbleSmallMSD (D t : β) : β := 2 * D * t |
|
|
| def runTumbleIntermediateMSD (u0 t : β) : β := u0 ^ 2 * t ^ 2 |
|
|
| def runTumbleLargeMSD (u0 delta0 t : β) : β := u0 ^ 2 * delta0 * t |
|
|
| theorem IPhO_2023_1_D_3 (D u0 delta0 t : β) : |
| (runTumbleSmallMSD D t, runTumbleIntermediateMSD u0 t, |
| runTumbleLargeMSD u0 delta0 t) = |
| (2 * D * t, u0 ^ 2 * t ^ 2, u0 ^ 2 * delta0 * t) := by |
| sorry |
|
|
| def criticalCoagulationConcentration (B eps kT NA A q : β) : β := |
| 8 * B ^ 2 * eps ^ 3 * kT ^ 5 / (Real.exp 4 * NA * A ^ 2 * q ^ 6) |
|
|
| theorem IPhO_2023_1_E_1 (B eps kT NA A q : β) : |
| criticalCoagulationConcentration B eps kT NA A q = |
| 8 * B ^ 2 * eps ^ 3 * kT ^ 5 / (Real.exp 4 * NA * A ^ 2 * q ^ 6) := by |
| sorry |
|
|
| end Problem1 |
|
|
| section Problem2 |
|
|
| def symmetricNucleusOptimalA (aS aC : β) : β := 2 * aS / aC |
|
|
| def symmetricNucleusOptimalARounded : β := 50 |
|
|
| theorem IPhO_2023_2_A_1 (aS aC : β) : |
| (symmetricNucleusOptimalA aS aC, symmetricNucleusOptimalARounded) = |
| (2 * aS / aC, 50) := by |
| sorry |
|
|
| def massNumberTwoThirds (A : β) : β := Real.rpow A ((2 : β) / 3) |
|
|
| def mostStableAtomicNumber (A aC asym : β) : β := |
| (A / 2) / (1 + aC / (4 * asym) * massNumberTwoThirds A) |
|
|
| def goldStableAtomicNumberEstimate : β := 79 |
|
|
| theorem IPhO_2023_2_A_2 (A aC asym : β) : |
| (mostStableAtomicNumber A aC asym, goldStableAtomicNumberEstimate) = |
| ((A / 2) / (1 + aC / (4 * asym) * Real.rpow A ((2 : β) / 3)), 79) := by |
| sorry |
|
|
| def symmetricFissionCoefficient : β := |
| (Real.rpow (2 : β) ((1 : β) / 3) - 1) / |
| (1 - Real.rpow (2 : β) (-(2 : β) / 3)) |
|
|
| def symmetricFissionCoefficientTwoSig : β := (7 : β) / 10 |
|
|
| theorem IPhO_2023_2_A_3 : |
| (symmetricFissionCoefficient, symmetricFissionCoefficientTwoSig) = |
| ((Real.rpow (2 : β) ((1 : β) / 3) - 1) / |
| (1 - Real.rpow (2 : β) (-(2 : β) / 3)), (7 : β) / 10) := by |
| sorry |
|
|
| def neutronStarGravityCoefficient (hbarc mN R0 MP : β) : β := |
| ((3 : β) / 5) * hbarc * mN ^ 2 / (R0 * MP ^ 2) |
|
|
| def neutronStarCriticalMassNumber (aV asym agrav : β) : β := |
| Real.rpow ((asym - aV) / agrav) ((3 : β) / 2) |
|
|
| def neutronStarGravityCoefficientEstimate : β := (6 : β) / (10 : β) ^ 37 |
|
|
| def neutronStarCriticalMassNumberEstimate : β := (4 : β) * (10 : β) ^ 55 |
|
|
| theorem IPhO_2023_2_B_1 (hbarc mN R0 MP aV asym agrav : β) : |
| neutronStarGravityCoefficient hbarc mN R0 MP = |
| ((3 : β) / 5) * hbarc * mN ^ 2 / (R0 * MP ^ 2) β§ |
| neutronStarCriticalMassNumber aV asym agrav = |
| Real.rpow ((asym - aV) / agrav) ((3 : β) / 2) β§ |
| (neutronStarGravityCoefficientEstimate, neutronStarCriticalMassNumberEstimate) = |
| ((6 : β) / (10 : β) ^ 37, (4 : β) * (10 : β) ^ 55) := by |
| sorry |
|
|
| def freeFallClockFactor (dphi c : β) : β := 1 - dphi / c ^ 2 |
|
|
| def freeFallClockDelay (dphi c tauI : β) : β := freeFallClockFactor dphi c * tauI |
|
|
| theorem IPhO_2023_2_C_1 (dphi c tauI : β) : |
| freeFallClockDelay dphi c tauI = (1 - dphi / c ^ 2) * tauI := by |
| sorry |
|
|
| def shapiroDelay (G MWD c xN xE d : β) : β := |
| 2 * G * MWD / c ^ 3 * Real.log (4 * |xN| * xE / d ^ 2) |
|
|
| theorem IPhO_2023_2_C_2 (G MWD c xN xE d : β) : |
| shapiroDelay G MWD c xN xE d = |
| 2 * G * MWD / c ^ 3 * Real.log (4 * |xN| * xE / d ^ 2) := by |
| sorry |
|
|
| def shapiroDelaySwing (G MWD c eps : β) : β := |
| 2 * G * MWD / c ^ 3 * Real.log (4 / eps ^ 2) |
|
|
| theorem IPhO_2023_2_C_3 (G MWD c eps : β) : |
| shapiroDelaySwing G MWD c eps = |
| 2 * G * MWD / c ^ 3 * Real.log (4 / eps ^ 2) := by |
| sorry |
|
|
| def whiteDwarfMassRatioEstimate : β := (1 : β) / 2 |
|
|
| theorem IPhO_2023_2_C_4 : |
| whiteDwarfMassRatioEstimate = (1 : β) / 2 := by |
| sorry |
|
|
| def keplerAngularFrequencyExponent : β := -(3 : β) / 2 |
|
|
| theorem IPhO_2023_2_C_5 : |
| keplerAngularFrequencyExponent = -(3 : β) / 2 := by |
| sorry |
|
|
| inductive WaveProfile |
| | a |
| | b |
| | c |
| deriving DecidableEq, Repr |
|
|
| def chosenGravitationalWaveProfile : WaveProfile := WaveProfile.b |
|
|
| theorem IPhO_2023_2_C_6 : |
| chosenGravitationalWaveProfile = WaveProfile.b := by |
| sorry |
|
|
| end Problem2 |
|
|
| section Problem3 |
|
|
| def dropSurfaceEnergy (a gamma : β) : β := 4 * Real.pi * a ^ 2 * gamma |
|
|
| def mergedDropEnergyDifference (a gamma : β) : β := |
| 4 * Real.pi * (2 - Real.rpow (2 : β) ((2 : β) / 3)) * a ^ 2 * gamma |
|
|
| def mergedDropMass (a rho : β) : β := 8 * Real.pi * a ^ 3 * rho / 3 |
|
|
| def mergedDropJumpSpeedSquared (k gamma rho a : β) : β := |
| 3 * (2 - Real.rpow (2 : β) ((2 : β) / 3)) * k * gamma / (rho * a) |
|
|
| def mergedDropJumpSpeed (k gamma rho a : β) : β := |
| Real.sqrt (mergedDropJumpSpeedSquared k gamma rho a) |
|
|
| def mergedDropJumpSpeedEstimate : β := (23 : β) / 100 |
|
|
| theorem IPhO_2023_3_A_1 (a gamma k rho : β) : |
| mergedDropEnergyDifference a gamma = |
| 4 * Real.pi * (2 - Real.rpow (2 : β) ((2 : β) / 3)) * a ^ 2 * gamma β§ |
| mergedDropMass a rho = 8 * Real.pi * a ^ 3 * rho / 3 β§ |
| (mergedDropJumpSpeed k gamma rho a, mergedDropJumpSpeedEstimate) = |
| (Real.sqrt (3 * (2 - Real.rpow (2 : β) ((2 : β) / 3)) * |
| k * gamma / (rho * a)), (23 : β) / 100) := by |
| sorry |
|
|
| def hydrostaticPressure (P0 rho g z : β) : β := P0 - rho * g * z |
|
|
| theorem IPhO_2023_3_B_1 (P0 rho g z : β) : |
| hydrostaticPressure P0 rho g z = P0 - rho * g * z := by |
| sorry |
|
|
| def horizontalPressureForce (rho g z1 z2 : β) : β := |
| ((1 : β) / 2) * rho * g * (z2 ^ 2 - z1 ^ 2) |
|
|
| theorem IPhO_2023_3_B_2 (rho g z1 z2 : β) : |
| horizontalPressureForce rho g z1 z2 = |
| ((1 : β) / 2) * rho * g * (z2 ^ 2 - z1 ^ 2) := by |
| sorry |
|
|
| def surfaceTensionHorizontalForce (gamma theta1 theta2 : β) : β := |
| gamma * Real.cos theta1 - gamma * Real.cos theta2 |
|
|
| theorem IPhO_2023_3_B_3 (gamma theta1 theta2 : β) : |
| surfaceTensionHorizontalForce gamma theta1 theta2 = |
| gamma * Real.cos theta1 - gamma * Real.cos theta2 := by |
| sorry |
|
|
| def capillaryConservedQuantity (rho g gamma z theta : β) : β := |
| ((1 : β) / 2) * rho * g * z ^ 2 + gamma * Real.cos theta |
|
|
| def capillaryDimensionlessExponent : β := 2 |
|
|
| def capillaryLength (gamma rho g : β) : β := Real.sqrt (gamma / (rho * g)) |
|
|
| theorem IPhO_2023_3_B_4 (rho g gamma z theta : β) : |
| (capillaryConservedQuantity rho g gamma z theta, capillaryDimensionlessExponent, |
| capillaryLength gamma rho g) = |
| (((1 : β) / 2) * rho * g * z ^ 2 + gamma * Real.cos theta, 2, |
| Real.sqrt (gamma / (rho * g))) := by |
| sorry |
|
|
| def capillarySurfaceSmallSlope (ell theta0 x : β) : β := |
| -ell * Real.tan theta0 * Real.exp (-x / ell) |
|
|
| theorem IPhO_2023_3_B_5 (ell theta0 x : β) : |
| capillarySurfaceSmallSlope ell theta0 x = |
| -ell * Real.tan theta0 * Real.exp (-x / ell) := by |
| sorry |
|
|
| def rodHorizontalForce (rho g gamma za zb thetaA thetaB : β) : β := |
| ((1 : β) / 2) * rho * g * (zb ^ 2 - za ^ 2) + |
| gamma * (Real.cos thetaB - Real.cos thetaA) |
|
|
| theorem IPhO_2023_3_C_1 (rho g gamma za zb thetaA thetaB : β) : |
| rodHorizontalForce rho g gamma za zb thetaA thetaB = |
| ((1 : β) / 2) * rho * g * (zb ^ 2 - za ^ 2) + |
| gamma * (Real.cos thetaB - Real.cos thetaA) := by |
| sorry |
|
|
| def rodAttractionForce (rho g z0 : β) : β := -((1 : β) / 2) * rho * g * z0 ^ 2 |
|
|
| theorem IPhO_2023_3_C_2 (rho g z0 : β) : |
| rodAttractionForce rho g z0 = -((1 : β) / 2) * rho * g * z0 ^ 2 := by |
| sorry |
|
|
| def midpointHeightBetweenRods (za xa ell : β) : β := |
| 2 * za / (Real.exp (xa / ell) + Real.exp (-xa / ell)) |
|
|
| theorem IPhO_2023_3_C_3 (za xa ell : β) : |
| midpointHeightBetweenRods za xa ell = |
| 2 * za / (Real.exp (xa / ell) + Real.exp (-xa / ell)) := by |
| sorry |
|
|
| end Problem3 |
|
|
| end |
|
|
| end Ipho20212023Archon.Problems.Y2023 |
|
|
