lean_problems/Y2024.lean

/-
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 2024 theory problems. The definitions
record the official closed-form or numerical answers from the HF JSONL source;
the named theorems expose one answer statement for each source row.
-/

namespace Ipho20212023Archon.Problems.Y2024

noncomputable section

def fourthRoot (x : ℝ) : ℝ := Real.sqrt (Real.sqrt x)

def closedInterval (lo hi x : ℝ) : Prop := lo ≤ x ∧ x ≤ hi

section Problem1

def solarConstant (sigma TS RS d : ℝ) : ℝ := sigma * TS ^ 4 * (RS / d) ^ 2

def solarConstantNumerical : ℝ := 1350

theorem IPhO_2024_1_A_1 (sigma TS RS d : ℝ) :
    (solarConstant sigma TS RS d, solarConstantNumerical) =
      (sigma * TS ^ 4 * (RS / d) ^ 2, 1350) := by
  sorry

def radiativeEquilibriumTemperature (S0 sigma : ℝ) : ℝ :=
  fourthRoot (S0 / (4 * sigma))

def radiativeEquilibriumTemperatureFromSun (RS d TS : ℝ) : ℝ :=
  Real.sqrt (RS / (2 * d)) * TS

def radiativeEquilibriumTemperatureNumerical : ℝ := 278

theorem IPhO_2024_1_A_2 (S0 sigma RS d TS : ℝ) :
    (radiativeEquilibriumTemperature S0 sigma,
        radiativeEquilibriumTemperatureFromSun RS d TS,
        radiativeEquilibriumTemperatureNumerical) =
      (fourthRoot (S0 / (4 * sigma)), Real.sqrt (RS / (2 * d)) * TS, 278) := by
  sorry

def planckPeakAuxiliary (x : ℝ) : ℝ := 5 * (1 - Real.exp (-x)) - x

theorem IPhO_2024_1_A_3 (x : ℝ) :
    planckPeakAuxiliary x = 5 * (1 - Real.exp (-x)) - x := by
  sorry

def wienPeakX : ℝ := 496 / 100

def wienConstantNmK : ℝ := 2890000

theorem IPhO_2024_1_A_4 :
    closedInterval (496 / 100) (497 / 100) wienPeakX ∧
      closedInterval (2890000 : ℝ) 2900000 wienConstantNmK := by
  sorry

def sunPeakWavelengthNm : ℝ := 501

def earthPeakWavelengthNm : ℝ := 10400

theorem IPhO_2024_1_A_5 :
    closedInterval (501 : ℝ) 502 sunPeakWavelengthNm ∧
      earthPeakWavelengthNm = 10400 := by
  sorry

def spectralFluxRatioByTemperature (d RS TE TS : ℝ) : ℝ :=
  (d / RS) ^ 2 * (TE / TS) ^ 5

def spectralFluxRatioByWavelength (d RS lambdaS lambdaE : ℝ) : ℝ :=
  (d / RS) ^ 2 * (lambdaS / lambdaE) ^ 5

def spectralFluxRatioNumerical : ℝ := 12 / 1000

theorem IPhO_2024_1_A_6 (d RS TE TS lambdaS lambdaE : ℝ) :
    spectralFluxRatioByTemperature d RS TE TS =
        (d / RS) ^ 2 * (TE / TS) ^ 5 ∧
      spectralFluxRatioByWavelength d RS lambdaS lambdaE =
        (d / RS) ^ 2 * (lambdaS / lambdaE) ^ 5 ∧
        closedInterval (120 / 10000) (121 / 10000) spectralFluxRatioNumerical := by
  sorry

def oneLayerSurfaceTemperature (rA S0 sigma : ℝ) : ℝ :=
  fourthRoot (((1 - rA) * (S0 / 2)) / sigma)

def oneLayerAtmosphereTemperature (rA S0 sigma : ℝ) : ℝ :=
  fourthRoot (((1 - rA) * (S0 / 4)) / sigma)

def oneLayerSurfaceTemperatureNumerical : ℝ := 307

def oneLayerAtmosphereTemperatureNumerical : ℝ := 258

theorem IPhO_2024_1_B_1 (rA S0 sigma : ℝ) :
    (oneLayerSurfaceTemperature rA S0 sigma,
        oneLayerAtmosphereTemperature rA S0 sigma,
        oneLayerSurfaceTemperatureNumerical,
        oneLayerAtmosphereTemperatureNumerical) =
      (fourthRoot (((1 - rA) * (S0 / 2)) / sigma),
        fourthRoot (((1 - rA) * (S0 / 4)) / sigma), 307, 258) := by
  sorry

def effectiveAlbedo (rA rE : ℝ) : ℝ :=
  rA + ((1 - rA) ^ 2 * rE) / (1 - rA * rE)

def effectiveAlbedoNumerical : ℝ := 313 / 1000

theorem IPhO_2024_1_B_2 (rA rE : ℝ) :
    effectiveAlbedo rA rE =
        rA + ((1 - rA) ^ 2 * rE) / (1 - rA * rE) ∧
      effectiveAlbedoNumerical = 313 / 1000 := by
  sorry

def greySurfaceTemperature (alpha epsilon S0 sigma : ℝ) : ℝ :=
  fourthRoot (((1 - alpha) * S0) / (2 * sigma * (2 - epsilon)))

def greyEmissivityNumerical : ℝ := 81 / 100

theorem IPhO_2024_1_B_3 (alpha epsilon S0 sigma : ℝ) :
    greySurfaceTemperature alpha epsilon S0 sigma =
        fourthRoot (((1 - alpha) * S0) / (2 * sigma * (2 - epsilon))) ∧
      closedInterval (807 / 1000) (811 / 1000) greyEmissivityNumerical := by
  sorry

def greySurfaceTemperatureDerivative (alpha epsilon S0 sigma : ℝ) : ℝ :=
  (1 / 4) * fourthRoot (((1 - alpha) * S0) / (2 * sigma * (2 - epsilon))) *
    (1 / (2 - epsilon))

def onePercentGreyTemperatureIncrease : ℝ := 49 / 100

theorem IPhO_2024_1_B_4 (alpha epsilon S0 sigma : ℝ) :
    greySurfaceTemperatureDerivative alpha epsilon S0 sigma =
        (1 / 4) * fourthRoot (((1 - alpha) * S0) / (2 * sigma * (2 - epsilon))) *
          (1 / (2 - epsilon)) ∧
      closedInterval (487 / 1000) (492 / 1000) onePercentGreyTemperatureIncrease := by
  sorry

def coupledEmissivity (TE TA sigma alpha S0 : ℝ) : ℝ :=
  (sigma * TE ^ 4 - (1 - alpha) * (S0 / 4)) /
    (sigma * (TE ^ 4 - TA ^ 4))

def conductiveCoefficient (TE TA sigma alpha S0 : ℝ) : ℝ :=
  ((2 * TA ^ 4 - TE ^ 4) * (sigma * TE ^ 4 - (1 - alpha) * (S0 / 4))) /
    ((TE ^ 4 - TA ^ 4) * (TE - TA))

def coupledEmissivityNumerical : ℝ := 85 / 100

def conductiveCoefficientNumerical : ℝ := 36 / 100

theorem IPhO_2024_1_B_5 (TE TA sigma alpha S0 : ℝ) :
    coupledEmissivity TE TA sigma alpha S0 =
        (sigma * TE ^ 4 - (1 - alpha) * (S0 / 4)) /
          (sigma * (TE ^ 4 - TA ^ 4)) ∧
      conductiveCoefficient TE TA sigma alpha S0 =
        ((2 * TA ^ 4 - TE ^ 4) * (sigma * TE ^ 4 - (1 - alpha) * (S0 / 4))) /
          ((TE ^ 4 - TA ^ 4) * (TE - TA)) ∧
        closedInterval (847 / 1000) (852 / 1000) coupledEmissivityNumerical ∧
          closedInterval (357 / 1000) (366 / 1000) conductiveCoefficientNumerical := by
  sorry

def coupledSensitivityEquationOne (epsilon TE TA dTE dTA : ℝ) : Prop :=
  epsilon * (1 / (TE - TA) + 4 * TE ^ 3 / (2 * TA ^ 4 - TE ^ 4)) * dTE =
    1 + epsilon * (8 * TA ^ 3 / (2 * TA ^ 4 - TE ^ 4) + 1 / (TE - TA)) * dTA

def coupledSensitivityEquationTwo
    (epsilon TE TA sigma alpha S0 dTE dTA : ℝ) : Prop :=
  1 + epsilon *
      (4 * TE ^ 3 / (TE ^ 4 - TA ^ 4) -
        4 * sigma * TE ^ 3 / (sigma * TE ^ 4 - (1 - alpha) * (S0 / 4))) * dTE =
    (4 * TA ^ 3 / (TE ^ 4 - TA ^ 4)) * epsilon * dTA

def coupledOnePercentTemperatureIncrease : ℝ := 525 / 1000

theorem IPhO_2024_1_B_6 (epsilon TE TA sigma alpha S0 dTE dTA : ℝ) :
    (coupledSensitivityEquationOne epsilon TE TA dTE dTA ↔
        epsilon * (1 / (TE - TA) + 4 * TE ^ 3 / (2 * TA ^ 4 - TE ^ 4)) * dTE =
          1 + epsilon *
            (8 * TA ^ 3 / (2 * TA ^ 4 - TE ^ 4) + 1 / (TE - TA)) * dTA) ∧
      (coupledSensitivityEquationTwo epsilon TE TA sigma alpha S0 dTE dTA ↔
        1 + epsilon *
          (4 * TE ^ 3 / (TE ^ 4 - TA ^ 4) -
            4 * sigma * TE ^ 3 / (sigma * TE ^ 4 - (1 - alpha) * (S0 / 4))) *
            dTE =
          (4 * TA ^ 3 / (TE ^ 4 - TA ^ 4)) * epsilon * dTA) ∧
        closedInterval (521 / 1000) (528 / 1000) coupledOnePercentTemperatureIncrease := by
  sorry

end Problem1

section Problem2

def ringElectricField (lambda eps0 R x y z : ℝ) : ℝ × ℝ × ℝ :=
  (-(lambda * x) / (4 * eps0 * R ^ 2),
    -(lambda * y) / (4 * eps0 * R ^ 2),
    lambda * z / (2 * eps0 * R ^ 2))

def ringOscillationFrequency (Q lambda eps0 R m : ℝ) : ℝ :=
  Real.sqrt (Q * lambda / (4 * eps0 * R ^ 2 * m))

theorem IPhO_2024_2_A_1 (lambda eps0 R x y z Q m : ℝ) :
    (ringElectricField lambda eps0 R x y z,
        ringOscillationFrequency Q lambda eps0 R m,
        ringOscillationFrequency Q lambda eps0 R m) =
      ((-(lambda * x) / (4 * eps0 * R ^ 2),
          -(lambda * y) / (4 * eps0 * R ^ 2),
          lambda * z / (2 * eps0 * R ^ 2)),
        Real.sqrt (Q * lambda / (4 * eps0 * R ^ 2 * m)),
        Real.sqrt (Q * lambda / (4 * eps0 * R ^ 2 * m))) := by
  sorry

def paulTrapK (Q lambda0 eps0 R m : ℝ) : ℝ :=
  Real.sqrt (Q * lambda0 / (2 * eps0 * R ^ 2 * m))

def paulTrapA (Q u eps0 R m Omega : ℝ) : ℝ :=
  Q * u / (2 * eps0 * R ^ 2 * m * Omega ^ 2)

theorem IPhO_2024_2_A_2 (Q lambda0 eps0 R m u Omega : ℝ) :
    (paulTrapK Q lambda0 eps0 R m, paulTrapA Q u eps0 R m Omega) =
      (Real.sqrt (Q * lambda0 / (2 * eps0 * R ^ 2 * m)),
        Q * u / (2 * eps0 * R ^ 2 * m * Omega ^ 2)) := by
  sorry

def rapidCoordinateAcceleration (p a Omega t : ℝ) : ℝ :=
  p * a * Omega ^ 2 * Real.cos (Omega * t)

def rapidCoordinateSolution (p a Omega t : ℝ) : ℝ :=
  -p * a * Real.cos (Omega * t)

theorem IPhO_2024_2_A_3 (p a Omega t : ℝ) :
    (rapidCoordinateAcceleration p a Omega t, rapidCoordinateSolution p a Omega t) =
      (p * a * Omega ^ 2 * Real.cos (Omega * t), -p * a * Real.cos (Omega * t)) := by
  sorry

def slowCoordinateAcceleration (k a Omega p : ℝ) : ℝ :=
  (k ^ 2 - a ^ 2 * Omega ^ 2 / 2) * p

def paulTrapStable (Omega k a : ℝ) : Prop := Omega > Real.sqrt 2 * k / a

theorem IPhO_2024_2_A_4 (k a Omega p : ℝ) :
    slowCoordinateAcceleration k a Omega p =
        (k ^ 2 - a ^ 2 * Omega ^ 2 / 2) * p ∧
      (paulTrapStable Omega k a ↔ Omega > Real.sqrt 2 * k / a) := by
  sorry

def paulTrapKNumerical : ℝ := 200000

def paulTrapOmegaMinNumerical : ℝ := 7000000

theorem IPhO_2024_2_A_5 :
    (paulTrapKNumerical, paulTrapOmegaMinNumerical) = (200000, 7000000) := by
  sorry

def naturalLinewidth (tau : ℝ) : ℝ := 1 / tau

theorem IPhO_2024_2_B_1 (tau : ℝ) :
    naturalLinewidth tau = 1 / tau := by
  sorry

def photonCollisionRatePlus (sL alpha omegaL v c : ℝ) : ℝ :=
  sL + alpha * omegaL * (v / c)

def photonCollisionRateMinus (sL alpha omegaL v c : ℝ) : ℝ :=
  sL - alpha * omegaL * (v / c)

def momentumAbsorptionPlus (sPlus hbar kPlus : ℝ) : ℝ := sPlus * (-hbar * kPlus)

def momentumAbsorptionMinus (sMinus hbar kMinus : ℝ) : ℝ := sMinus * (hbar * kMinus)

def opticalMolassesForce (alpha hbar kL v : ℝ) : ℝ := -(2 * alpha * hbar * kL ^ 2) * v

theorem IPhO_2024_2_B_2 (sL alpha omegaL v c sPlus sMinus hbar kPlus kMinus kL : ℝ) :
    (photonCollisionRatePlus sL alpha omegaL v c,
        photonCollisionRateMinus sL alpha omegaL v c,
        momentumAbsorptionPlus sPlus hbar kPlus,
        momentumAbsorptionMinus sMinus hbar kMinus,
        opticalMolassesForce alpha hbar kL v) =
      (sL + alpha * omegaL * (v / c),
        sL - alpha * omegaL * (v / c),
        sPlus * (-hbar * kPlus),
        sMinus * (hbar * kMinus),
        -(2 * alpha * hbar * kL ^ 2) * v) := by
  sorry

def recoilHeatingPower (hbar kL m tau : ℝ) : ℝ := hbar ^ 2 * kL ^ 2 / (m * tau)

theorem IPhO_2024_2_B_3 (hbar kL m tau : ℝ) :
    recoilHeatingPower hbar kL m tau = hbar ^ 2 * kL ^ 2 / (m * tau) := by
  sorry

def opticalCoolingPower (alpha hbar kL v : ℝ) : ℝ := -2 * alpha * hbar * kL ^ 2 * v ^ 2

def equilibriumMeanSquareVelocity (hbar Gamma alpha m : ℝ) : ℝ :=
  hbar * Gamma / (2 * alpha * m)

def dopplerTemperature (hbar Gamma alpha kB : ℝ) : ℝ :=
  hbar * Gamma / (2 * alpha * kB)

theorem IPhO_2024_2_B_4 (alpha hbar kL v Gamma m kB : ℝ) :
    (opticalCoolingPower alpha hbar kL v,
        equilibriumMeanSquareVelocity hbar Gamma alpha m,
        dopplerTemperature hbar Gamma alpha kB) =
      (-2 * alpha * hbar * kL ^ 2 * v ^ 2,
        hbar * Gamma / (2 * alpha * m),
        hbar * Gamma / (2 * alpha * kB)) := by
  sorry

def dopplerTemperatureNumerical : ℝ := 2 / 10000

theorem IPhO_2024_2_B_5 :
    dopplerTemperatureNumerical = 2 / 10000 := by
  sorry

end Problem2

section Problem3

def binaryEffectivePotential (M1 M2 G a x y : ℝ) : ℝ :=
  -G * M1 / Real.sqrt ((x + M2 / (M1 + M2) * a) ^ 2 + y ^ 2) -
    G * M2 / Real.sqrt ((x - M1 / (M1 + M2) * a) ^ 2 + y ^ 2) -
      (1 / 2) * (G * (M1 + M2) / a ^ 3) * (x ^ 2 + y ^ 2)

theorem IPhO_2024_3_A_1 (M1 M2 G a x y : ℝ) :
    binaryEffectivePotential M1 M2 G a x y =
      -G * M1 / Real.sqrt ((x + M2 / (M1 + M2) * a) ^ 2 + y ^ 2) -
        G * M2 / Real.sqrt ((x - M1 / (M1 + M2) * a) ^ 2 + y ^ 2) -
          (1 / 2) * (G * (M1 + M2) / a ^ 3) * (x ^ 2 + y ^ 2) := by
  sorry

def rocheOverflowRatio : ℝ := 36 / 100

theorem IPhO_2024_3_A_3 :
    rocheOverflowRatio = 36 / 100 := by
  sorry

def binarySeparationRate (beta a M1 M2 : ℝ) : ℝ :=
  -2 * beta * a * (1 / M1 - 1 / M2)

def binaryPeriodRate (beta a G M M1 M2 : ℝ) : ℝ :=
  -6 * Real.pi * Real.sqrt (a ^ 3 / (G * M)) * beta * (1 / M1 - 1 / M2)

theorem IPhO_2024_3_A_4 (beta a G M M1 M2 : ℝ) :
    (binarySeparationRate beta a M1 M2, binaryPeriodRate beta a G M M1 M2) =
      (-2 * beta * a * (1 / M1 - 1 / M2),
        -6 * Real.pi * Real.sqrt (a ^ 3 / (G * M)) * beta * (1 / M1 - 1 / M2)) := by
  sorry

def accretionDiscTemperature (G M1 beta sigma r : ℝ) : ℝ :=
  fourthRoot (G * M1 * beta / (8 * Real.pi * sigma * r ^ 3))

theorem IPhO_2024_3_A_5 (G M1 beta sigma r : ℝ) :
    accretionDiscTemperature G M1 beta sigma r =
      fourthRoot (G * M1 * beta / (8 * Real.pi * sigma * r ^ 3)) := by
  sorry

def accretionDiscTemperatureNumerical : ℝ := 9000

theorem IPhO_2024_3_A_6 :
    accretionDiscTemperatureNumerical = 9000 := by
  sorry

def maximumPostExplosionVelocity (G M1p M2 a : ℝ) : ℝ :=
  Real.sqrt (2 * G * (M1p + M2) / a)

def minimumPostExplosionMass (M1 M2 : ℝ) : ℝ := (M1 - M2) / 2

theorem IPhO_2024_3_A_7 (G M1p M2 a M1 : ℝ) :
    (maximumPostExplosionVelocity G M1p M2 a, minimumPostExplosionMass M1 M2) =
      (Real.sqrt (2 * G * (M1p + M2) / a), (M1 - M2) / 2) := by
  sorry

def centralGravity (G rhoC r : ℝ) : ℝ := -(4 * Real.pi * G * rhoC * r) / 3

theorem IPhO_2024_3_B_1 (G rhoC r : ℝ) :
    centralGravity G rhoC r = -(4 * Real.pi * G * rhoC * r) / 3 := by
  sorry

def stellarH1 (rho r gamma : ℝ) : ℝ := r ^ 2 * Real.rpow rho (gamma - 2)

def stellarH2 (G K gamma r : ℝ) : ℝ := 4 * Real.pi * G * r ^ 2 / (K * gamma)

theorem IPhO_2024_3_B_2 (rho r gamma G K : ℝ) :
    (stellarH1 rho r gamma, stellarH2 G K gamma r) =
      (r ^ 2 * Real.rpow rho (gamma - 2), 4 * Real.pi * G * r ^ 2 / (K * gamma)) := by
  sorry

def stellarLengthScale (G pc rhoC : ℝ) : ℝ := Real.sqrt pc / (Real.sqrt G * rhoC)

theorem IPhO_2024_3_B_3 (G pc rhoC : ℝ) :
    stellarLengthScale G pc rhoC = Real.sqrt pc / (Real.sqrt G * rhoC) := by
  sorry

def dimensionlessA1 (u x gamma : ℝ) : ℝ := x ^ 2 * Real.rpow u (gamma - 2)

def dimensionlessA2 (x gamma : ℝ) : ℝ := 4 * Real.pi * x ^ 2 / gamma

theorem IPhO_2024_3_B_4 (u x gamma : ℝ) :
    (dimensionlessA1 u x gamma, dimensionlessA2 x gamma) =
      (x ^ 2 * Real.rpow u (gamma - 2), 4 * Real.pi * x ^ 2 / gamma) := by
  sorry

def gammaTwoProfileF (x : ℝ) : ℝ :=
  Real.sin (Real.sqrt (2 * Real.pi) * x) / Real.sqrt (2 * Real.pi)

theorem IPhO_2024_3_B_5 (x : ℝ) :
    gammaTwoProfileF x =
      Real.sin (Real.sqrt (2 * Real.pi) * x) / Real.sqrt (2 * Real.pi) := by
  sorry

def stellarGammaNumerical : ℝ := 167 / 100

theorem IPhO_2024_3_B_6 :
    closedInterval (164 / 100) (170 / 100) stellarGammaNumerical := by
  sorry

def perturbedDensity (rho epsilon : ℝ) : ℝ := rho * (1 - 3 * epsilon)

def perturbedGravity (g epsilon : ℝ) : ℝ := g * (1 - 2 * epsilon)

theorem IPhO_2024_3_B_7 (rho g epsilon : ℝ) :
    (perturbedDensity rho epsilon, perturbedGravity g epsilon) =
      (rho * (1 - 3 * epsilon), g * (1 - 2 * epsilon)) := by
  sorry

def shellAcceleration (gTilde K gamma rhoTilde densityGradient : ℝ) : ℝ :=
  gTilde - K * gamma * Real.rpow rhoTilde (gamma - 2) * densityGradient

theorem IPhO_2024_3_B_8 (gTilde K gamma rhoTilde densityGradient : ℝ) :
    shellAcceleration gTilde K gamma rhoTilde densityGradient =
      gTilde - K * gamma * Real.rpow rhoTilde (gamma - 2) * densityGradient := by
  sorry

def radialPerturbationAcceleration (G rhoC gamma epsilon : ℝ) : ℝ :=
  -(4 * Real.pi * G * rhoC / 3) * (3 * gamma - 4) * epsilon

def stableGammaMinimum : ℝ := 4 / 3

def stellarOscillationFrequency (G rhoC gamma : ℝ) : ℝ :=
  Real.sqrt ((4 * Real.pi * G * rhoC / 3) * (3 * gamma - 4))

theorem IPhO_2024_3_B_9 (G rhoC gamma epsilon : ℝ) :
    (radialPerturbationAcceleration G rhoC gamma epsilon,
        stableGammaMinimum,
        stellarOscillationFrequency G rhoC gamma) =
      (-(4 * Real.pi * G * rhoC / 3) * (3 * gamma - 4) * epsilon,
        4 / 3,
        Real.sqrt ((4 * Real.pi * G * rhoC / 3) * (3 * gamma - 4))) := by
  sorry

end Problem3

end

end Ipho20212023Archon.Problems.Y2024