/- 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