/- 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 2025 problems. The declarations encode the official closed-form answers and numerical answer intervals from the source rows. -/ namespace Ipho20212023Archon.Problems.Y2025 noncomputable section abbrev RangeAnswer := ℝ × ℝ inductive Choice | A | B | C | D deriving DecidableEq, Repr structure ExperimentOutcome where behavior : Choice hstar : Option RangeAnswer fmax : RangeAnswer def sq (x : ℝ) : ℝ := x ^ 2 def cube (x : ℝ) : ℝ := x ^ 3 def fourth (x : ℝ) : ℝ := x ^ 4 /- Problem 1, Hydrogen and galaxies. -/ section Problem1 def bohrCircularVelocity (e eps0 me r : ℝ) : ℝ := Real.sqrt (sq e / (4 * Real.pi * eps0 * me * r)) theorem IPhO_2025_1_A_1 (e eps0 me r : ℝ) : bohrCircularVelocity e eps0 me r = Real.sqrt (sq e / (4 * Real.pi * eps0 * me * r)) := by sorry def bohrOrbitRadius (hbar n alpha me c : ℝ) : ℝ := sq n * hbar / (alpha * me * c) def bohrRadius (hbar alpha me c : ℝ) : ℝ := hbar / (alpha * me * c) def bohrRadiusNumerical : ℝ := (531 : ℝ) / 100 * (10 : ℝ) ^ (-11 : ℤ) def bohrGroundVelocity (alpha c : ℝ) : ℝ := alpha * c theorem IPhO_2025_1_A_2 (hbar n alpha me c : ℝ) : (bohrOrbitRadius hbar n alpha me c, bohrRadius hbar alpha me c, bohrRadiusNumerical, bohrGroundVelocity alpha c) = (sq n * hbar / (alpha * me * c), hbar / (alpha * me * c), (531 : ℝ) / 100 * (10 : ℝ) ^ (-11 : ℤ), alpha * c) := by sorry def bohrOrbitEnergy (e eps0 n r1 : ℝ) : ℝ := -sq e / (8 * Real.pi * eps0 * sq n * r1) def bohrGroundEnergy (alpha me c : ℝ) : ℝ := -((1 : ℝ) / 2) * sq alpha * me * sq c def bohrGroundEnergyEV : ℝ := -((136 : ℝ) / 10) theorem IPhO_2025_1_A_3 (e eps0 n r1 alpha me c : ℝ) : (bohrOrbitEnergy e eps0 n r1, bohrGroundEnergy alpha me c, bohrGroundEnergyEV) = (-sq e / (8 * Real.pi * eps0 * sq n * r1), -((1 : ℝ) / 2) * sq alpha * me * sq c, -((136 : ℝ) / 10)) := by sorry def orbitMagneticField (mu0 e alpha c r1 : ℝ) : ℝ := mu0 * e * alpha * c / (4 * Real.pi * sq r1) theorem IPhO_2025_1_A_4 (mu0 e alpha c r1 : ℝ) : orbitMagneticField mu0 e alpha c r1 = mu0 * e * alpha * c / (4 * Real.pi * sq r1) := by sorry def fineStructureEnergyGap (alpha E1 : ℝ) : ℝ := -4 * sq alpha * E1 def hyperfineWavelength (h c me mp alpha E1 : ℝ) : ℝ := -(h * c) / (((372 : ℝ) / 100) * (me / mp) * 4 * sq alpha * E1) def hyperfineWavelengthCm : ℝ := 21 theorem IPhO_2025_1_A_5 (h c me mp alpha E1 : ℝ) : (fineStructureEnergyGap alpha E1, hyperfineWavelength h c me mp alpha E1, hyperfineWavelengthCm) = (-4 * sq alpha * E1, -(h * c) / (((372 : ℝ) / 100) * (me / mp) * 4 * sq alpha * E1), 21) := by sorry def circularOrbitSpeedFromPotential (r dphi_dr : ℝ) : ℝ := Real.sqrt (r * dphi_dr) theorem IPhO_2025_1_B_1 (r dphi_dr : ℝ) : circularOrbitSpeedFromPotential r dphi_dr = Real.sqrt (r * dphi_dr) := by sorry def ngc6946BulgeMassRange : RangeAnswer := ((675 : ℝ) / 100 * (10 : ℝ) ^ 8, (1125 : ℝ) / 100 * (10 : ℝ) ^ 8) theorem IPhO_2025_1_B_2 : ngc6946BulgeMassRange = ((675 : ℝ) / 100 * (10 : ℝ) ^ 8, (1125 : ℝ) / 100 * (10 : ℝ) ^ 8) := by sorry def haloVelocityK1 (Cm G : ℝ) : ℝ := 4 * Real.pi * Cm * G def haloVelocityK2 (Cm G rm : ℝ) : ℝ := 4 * Real.pi * Cm * G * rm def haloInteriorMass (Cm r rm : ℝ) : ℝ := 4 * Real.pi * Cm * (r - rm * Real.arctan (r / rm)) def haloGravityMagnitude (Cm r rm : ℝ) : ℝ := 4 * Real.pi * Cm * (r - rm * Real.arctan (r / rm)) / sq r def haloVelocityProfile (Cm G r rm : ℝ) : ℝ := Real.sqrt (haloVelocityK1 Cm G - haloVelocityK2 Cm G rm * Real.arctan (r / rm) / r) def haloVelocitySmallRadius (Cm G r rm : ℝ) : ℝ := Real.sqrt (4 * Real.pi * Cm * G * sq r / (3 * sq rm)) def haloVelocityLargeRadius (Cm G : ℝ) : ℝ := Real.sqrt (4 * Real.pi * Cm * G) def haloMassLargeRadius (Cm r : ℝ) : ℝ := 4 * Real.pi * Cm * r def ngc6946VisibleMassRange : RangeAnswer := (Real.rpow 10 ((21 : ℝ) / 2), Real.rpow 10 ((23 : ℝ) / 2)) theorem IPhO_2025_1_B_3 (Cm G r rm : ℝ) : (haloVelocityK1 Cm G, haloVelocityK2 Cm G rm, haloInteriorMass Cm r rm, haloGravityMagnitude Cm r rm, haloVelocityProfile Cm G r rm, haloVelocitySmallRadius Cm G r rm, haloVelocityLargeRadius Cm G, haloMassLargeRadius Cm r, ngc6946VisibleMassRange) = (4 * Real.pi * Cm * G, 4 * Real.pi * Cm * G * rm, 4 * Real.pi * Cm * (r - rm * Real.arctan (r / rm)), 4 * Real.pi * Cm * (r - rm * Real.arctan (r / rm)) / sq r, Real.sqrt (4 * Real.pi * Cm * G - 4 * Real.pi * Cm * G * rm * Real.arctan (r / rm) / r), Real.sqrt (4 * Real.pi * Cm * G * sq r / (3 * sq rm)), Real.sqrt (4 * Real.pi * Cm * G), 4 * Real.pi * Cm * r, (Real.rpow 10 ((21 : ℝ) / 2), Real.rpow 10 ((23 : ℝ) / 2))) := by sorry def verticalGalacticAcceleration (z z0 phi0 r r0 : ℝ) : ℝ := 2 * z / sq z0 * phi0 * Real.log (r / r0) * Real.exp (-(sq (z / z0))) def verticalSmallOscillationFrequency (phi0 z0 r r0 : ℝ) : ℝ := Real.sqrt (2 * phi0 / sq z0 * |Real.log (r / r0)|) theorem IPhO_2025_1_C_1 (z z0 phi0 r r0 : ℝ) : (verticalGalacticAcceleration z z0 phi0 r r0, verticalSmallOscillationFrequency phi0 z0 r r0) = (2 * z / sq z0 * phi0 * Real.log (r / r0) * Real.exp (-(sq (z / z0))), Real.sqrt (2 * phi0 / sq z0 * |Real.log (r / r0)|)) := by sorry inductive RadiusRegime | muchSmaller | muchLarger deriving DecidableEq, Repr def flatRotationRegime : RadiusRegime := RadiusRegime.muchLarger def flatPotentialScale (Cm G : ℝ) : ℝ := 4 * Real.pi * Cm * G def flatCircularSpeed (phi0 : ℝ) : ℝ := Real.sqrt phi0 theorem IPhO_2025_1_C_2 (Cm G phi0 : ℝ) : (flatRotationRegime, flatPotentialScale Cm G, flatCircularSpeed phi0) = (RadiusRegime.muchLarger, 4 * Real.pi * Cm * G, Real.sqrt phi0) := by sorry def radialVelocityEarthSource (vSun RSun R ell : ℝ) : ℝ := vSun * (RSun / R - 1) * Real.sin ell def sourceGalactocentricRadius (RSun vSun ell vr : ℝ) : ℝ := RSun / (1 + vr / (vSun * Real.sin ell)) theorem IPhO_2025_1_C_3 (vSun RSun R ell vr : ℝ) : (radialVelocityEarthSource vSun RSun R ell, sourceGalactocentricRadius RSun vSun ell vr) = (vSun * (RSun / R - 1) * Real.sin ell, RSun / (1 + vr / (vSun * Real.sin ell))) := by sorry def sourceVelocityRange1 : RangeAnswer := ((632 : ℝ) / 100, (634 : ℝ) / 100) def sourceVelocityRange2 : RangeAnswer := ((316 : ℝ) / 10, (318 : ℝ) / 10) def sourceVelocityRange3 : RangeAnswer := ((548 : ℝ) / 10, 55) def sourceRadiusRange1 : RangeAnswer := ((94 : ℝ) / 100, (96 : ℝ) / 100) def sourceRadiusRange2 : RangeAnswer := ((77 : ℝ) / 100, (79 : ℝ) / 100) def sourceRadiusRange3 : RangeAnswer := ((66 : ℝ) / 100, (68 : ℝ) / 100) theorem IPhO_2025_1_C_4 : (sourceVelocityRange1, sourceVelocityRange2, sourceVelocityRange3, sourceRadiusRange1, sourceRadiusRange2, sourceRadiusRange3) = (((632 : ℝ) / 100, (634 : ℝ) / 100), ((316 : ℝ) / 10, (318 : ℝ) / 10), ((548 : ℝ) / 10, 55), ((94 : ℝ) / 100, (96 : ℝ) / 100), ((77 : ℝ) / 100, (79 : ℝ) / 100), ((66 : ℝ) / 100, (68 : ℝ) / 100)) := by sorry def newtonTullyFischerEta (R G : ℝ) : ℝ := R / G def newtonTullyFischerGamma : ℝ := 2 def observedTullyFischerGammaRange : RangeAnswer := ((35 : ℝ) / 10, 4) theorem IPhO_2025_1_D_1 (R G : ℝ) : (newtonTullyFischerEta R G, newtonTullyFischerGamma, observedTullyFischerGammaRange) = (R / G, 2, ((35 : ℝ) / 10, 4)) := by sorry def outerGalaxyAcceleration (vc R : ℝ) : ℝ := sq vc / R def outerGalaxyAccelerationRange : RangeAnswer := (Real.rpow 10 (-((21 : ℝ) / 2)), Real.rpow 10 (-((19 : ℝ) / 2))) theorem IPhO_2025_1_D_2 (vc R : ℝ) : (outerGalaxyAcceleration vc R, outerGalaxyAccelerationRange) = (sq vc / R, (Real.rpow 10 (-((21 : ℝ) / 2)), Real.rpow 10 (-((19 : ℝ) / 2)))) := by sorry def mondTullyFischerGamma : ℝ := 4 def mondAccelerationScale (vcInf G Mtot : ℝ) : ℝ := fourth vcInf / (G * Mtot) def mondAccelerationScaleRange : RangeAnswer := (Real.rpow 10 (-((21 : ℝ) / 2)), Real.rpow 10 (-((19 : ℝ) / 2))) theorem IPhO_2025_1_D_3 (vcInf G Mtot : ℝ) : (mondTullyFischerGamma, mondAccelerationScale vcInf G Mtot, mondAccelerationScaleRange) = (4, fourth vcInf / (G * Mtot), (Real.rpow 10 (-((21 : ℝ) / 2)), Real.rpow 10 (-((19 : ℝ) / 2)))) := by sorry def mondVelocityExterior (G M a0 r : ℝ) : ℝ := Real.sqrt (G * M / (2 * r) * (1 + Real.sqrt (1 + 4 * a0 * sq r / (G * M)))) def mondVelocityInterior (G M a0 r Rb : ℝ) : ℝ := Real.sqrt (G * M / (2 * r) * cube (r / Rb) * (1 + Real.sqrt (1 + 4 * a0 * sq r / (G * M) * cube (Rb / r)))) theorem IPhO_2025_1_D_4 (G M a0 r Rb : ℝ) : (mondVelocityExterior G M a0 r, mondVelocityInterior G M a0 r Rb) = (Real.sqrt (G * M / (2 * r) * (1 + Real.sqrt (1 + 4 * a0 * sq r / (G * M)))), Real.sqrt (G * M / (2 * r) * cube (r / Rb) * (1 + Real.sqrt (1 + 4 * a0 * sq r / (G * M) * cube (Rb / r))))) := by sorry end Problem1 /- Problem 2, The water column. -/ section Problem2 def waterPressureAtTubeTop (P0 rho g h : ℝ) : ℝ := P0 - rho * g * h def tubeHoldingForceScalar (m rho S h g : ℝ) : ℝ := (m + rho * S * h) * g theorem IPhO_2025_2_A_1 (P0 rho m S h g : ℝ) : (waterPressureAtTubeTop P0 rho g h, tubeHoldingForceScalar m rho S h g) = (P0 - rho * g * h, (m + rho * S * h) * g) := by sorry def tubeExperimentOutcomes : List ExperimentOutcome := [ { behavior := Choice.A, hstar := none, fmax := ((146 : ℝ) / 10, 15) }, { behavior := Choice.A, hstar := none, fmax := (14, (145 : ℝ) / 10) }, { behavior := Choice.B, hstar := some (2, (22 : ℝ) / 10), fmax := (5, (52 : ℝ) / 10) } ] theorem IPhO_2025_2_A_2 : tubeExperimentOutcomes = [ { behavior := Choice.A, hstar := none, fmax := ((146 : ℝ) / 10, 15) }, { behavior := Choice.A, hstar := none, fmax := (14, (145 : ℝ) / 10) }, { behavior := Choice.B, hstar := some (2, (22 : ℝ) / 10), fmax := (5, (52 : ℝ) / 10) } ] := by sorry def vaporPressureRelativeError (Psat P0 m g S : ℝ) : ℝ := Psat / (P0 + m * g / S) def vaporPressureRelativeErrorRange : RangeAnswer := ((10 : ℝ) ^ (-6 : ℤ), 2 * (10 : ℝ) ^ (-6 : ℤ)) theorem IPhO_2025_2_A_3 (Psat P0 m g S : ℝ) : (vaporPressureRelativeError Psat P0 m g S, vaporPressureRelativeErrorRange) = (Psat / (P0 + m * g / S), ((10 : ℝ) ^ (-6 : ℤ), 2 * (10 : ℝ) ^ (-6 : ℤ))) := by sorry def addedMercuryMass (rho St ht Sb zl : ℝ) : ℝ := rho * (St * ht + Sb * (zl - ht)) theorem IPhO_2025_2_B_1 (rho St ht Sb zl : ℝ) : addedMercuryMass rho St ht Sb zl = rho * (St * ht + Sb * (zl - ht)) := by sorry def addedMassSegmentSlopes (rho Sb St : ℝ) : ℝ × ℝ × ℝ × ℝ := (rho * Sb, rho * St, -rho * (Sb - St), 0) def addedMassAngularPoints (zStar Hb : ℝ) : ℝ × ℝ × ℝ := (0, zStar - Hb, zStar) theorem IPhO_2025_2_B_2 (rho Sb St zStar Hb : ℝ) : (addedMassSegmentSlopes rho Sb St, addedMassAngularPoints zStar Hb) = ((rho * Sb, rho * St, -rho * (Sb - St), 0), (0, zStar - Hb, zStar)) := by sorry def addedMassAmplitude (Sb A g : ℝ) : ℝ := Sb * A / g def addedMassAmplitudeRange : RangeAnswer := (1, (11 : ℝ) / 10) theorem IPhO_2025_2_B_3 (Sb A g : ℝ) : (addedMassAmplitude Sb A g, addedMassAmplitudeRange) = (Sb * A / g, (1, (11 : ℝ) / 10)) := by sorry def restThresholdXi : ℝ := 2 theorem IPhO_2025_2_C_1 : restThresholdXi = 2 := by sorry def cableTensionScalar (Sc Sb St rho g X : ℝ) : ℝ := 4 * Sc * (Sb - St) / (Sb + Sc - St) * rho * g * X def cableTensionApproxScalar (Sc Sb rho g X : ℝ) : ℝ := 4 * Sb * Sc / (Sb + Sc) * rho * g * X theorem IPhO_2025_2_C_2 (Sc Sb St rho g X : ℝ) : (cableTensionScalar Sc Sb St rho g X, cableTensionApproxScalar Sc Sb rho g X) = (4 * Sc * (Sb - St) / (Sb + Sc - St) * rho * g * X, 4 * Sb * Sc / (Sb + Sc) * rho * g * X) := by sorry def firstRegimeCondition (xi lambda : ℝ) : Prop := xi + 2 * lambda > 2 def secondRegimeCondition (xi lambda : ℝ) : Prop := xi + 2 * lambda < 2 theorem IPhO_2025_2_C_3 (xi lambda : ℝ) : (firstRegimeCondition xi lambda ↔ xi + 2 * lambda > 2) ∧ (secondRegimeCondition xi lambda ↔ xi + 2 * lambda < 2) := by sorry def springForceLimit (A Sc : ℝ) : ℝ := A * Sc / 2 def displacementLimit (A rho g : ℝ) : ℝ := A / (4 * rho * g) def maximumRecoverableWork (A Sc rho g : ℝ) : ℝ := sq A * Sc / (2 * rho * g) def maximumRecoverableWorkRange : RangeAnswer := (19, 21) theorem IPhO_2025_2_C_4 (A Sc rho g : ℝ) : (springForceLimit A Sc, displacementLimit A rho g, maximumRecoverableWork A Sc rho g, maximumRecoverableWorkRange) = (A * Sc / 2, A / (4 * rho * g), sq A * Sc / (2 * rho * g), (19, 21)) := by sorry def practicalMaximumWork (Sc A rho g : ℝ) : ℝ := Sc * sq A / (rho * g) def recoverableToPracticalWorkRatio : ℝ := (1 : ℝ) / 2 theorem IPhO_2025_2_C_5 (Sc A rho g : ℝ) : (practicalMaximumWork Sc A rho g, recoverableToPracticalWorkRatio) = (Sc * sq A / (rho * g), (1 : ℝ) / 2) := by sorry end Problem2 /- Problem 3, Champagne. -/ section Problem3 def bubblePressure (P0 sigma a : ℝ) : ℝ := P0 + 2 * sigma / a theorem IPhO_2025_3_A_1 (P0 sigma a : ℝ) : bubblePressure P0 sigma a = P0 + 2 * sigma / a := by sorry def criticalBubbleRadius (sigma P0 cl c0 : ℝ) : ℝ := 2 * sigma / (P0 * (cl / c0 - 1)) def criticalBubbleRadiusMicrometer : ℝ := (3 : ℝ) / 10 theorem IPhO_2025_3_A_2 (sigma P0 cl c0 : ℝ) : (criticalBubbleRadius sigma P0 cl c0, criticalBubbleRadiusMicrometer) = (2 * sigma / (P0 * (cl / c0 - 1)), (3 : ℝ) / 10) := by sorry def bubbleMoles (a P0 R T0 : ℝ) : ℝ := (4 : ℝ) / 3 * Real.pi * cube a * P0 / (R * T0) def diffusionModelRadiusSquared (a0 D R T0 P0 cl c0 t : ℝ) : ℝ := sq a0 + (2 * D * R * T0 / P0) * (cl - c0) * t def transferModelRadius (a0 K R T0 P0 cl c0 t : ℝ) : ℝ := a0 + (K * R * T0 / P0) * (cl - c0) * t def selectedBubbleGrowthModel : Choice := Choice.B def bubbleGrowthKRange : RangeAnswer := ((9 : ℝ) / 10 * (10 : ℝ) ^ (-4 : ℤ), (11 : ℝ) / 10 * (10 : ℝ) ^ (-4 : ℤ)) theorem IPhO_2025_3_A_3 (a P0 R T0 a0 D K cl c0 t : ℝ) : (bubbleMoles a P0 R T0, diffusionModelRadiusSquared a0 D R T0 P0 cl c0 t, transferModelRadius a0 K R T0 P0 cl c0 t, selectedBubbleGrowthModel, bubbleGrowthKRange) = ((4 : ℝ) / 3 * Real.pi * cube a * P0 / (R * T0), sq a0 + (2 * D * R * T0 / P0) * (cl - c0) * t, a0 + (K * R * T0 / P0) * (cl - c0) * t, Choice.B, ((9 : ℝ) / 10 * (10 : ℝ) ^ (-4 : ℤ), (11 : ℝ) / 10 * (10 : ℝ) ^ (-4 : ℤ))) := by sorry def bubbleBuoyancyForce (a rhoL g0 : ℝ) : ℝ := (4 : ℝ) / 3 * Real.pi * cube a * rhoL * g0 def bubbleStokesDrag (eta a v : ℝ) : ℝ := 6 * Real.pi * eta * a * v def bubbleTerminalVelocity (eta a rhoL g0 : ℝ) : ℝ := 2 / (9 * eta) * sq a * rhoL * g0 def champagneViscosityRange : RangeAnswer := ((10 : ℝ) ^ (-3 : ℤ), 4 * (10 : ℝ) ^ (-3 : ℤ)) theorem IPhO_2025_3_A_4 (a rhoL g0 eta v : ℝ) : (bubbleBuoyancyForce a rhoL g0, bubbleStokesDrag eta a v, bubbleTerminalVelocity eta a rhoL g0, champagneViscosityRange) = ((4 : ℝ) / 3 * Real.pi * cube a * rhoL * g0, 6 * Real.pi * eta * a * v, 2 / (9 * eta) * sq a * rhoL * g0, ((10 : ℝ) ^ (-3 : ℤ), 4 * (10 : ℝ) ^ (-3 : ℤ))) := by sorry def freeSurfaceBubbleRadius (qa eta H rhoL g0 : ℝ) : ℝ := Real.rpow (27 * qa * eta * H / (2 * rhoL * g0)) ((1 : ℝ) / 3) def freeSurfaceBubbleRadiusMmRange : RangeAnswer := ((36 : ℝ) / 100, (49 : ℝ) / 100) theorem IPhO_2025_3_A_5 (qa eta H rhoL g0 : ℝ) : (freeSurfaceBubbleRadius qa eta H rhoL g0, freeSurfaceBubbleRadiusMmRange) = (Real.rpow (27 * qa * eta * H / (2 * rhoL * g0)) ((1 : ℝ) / 3), ((36 : ℝ) / 100, (49 : ℝ) / 100)) := by sorry def concentrationDecayCoefficient (Nb fb eta K H rhoL g0 Vl : ℝ) : ℝ := 18 * Real.pi * Nb * fb * eta * K * H / (rhoL * g0 * Vl) def concentrationDecayResidual (dc_dt cL c0 Nb fb eta K H rhoL g0 Vl : ℝ) : ℝ := dc_dt + concentrationDecayCoefficient Nb fb eta K H rhoL g0 Vl * (cL - c0) def concentrationDecayEquation (dc_dt cL c0 Nb fb eta K H rhoL g0 Vl : ℝ) : Prop := concentrationDecayResidual dc_dt cL c0 Nb fb eta K H rhoL g0 Vl = 0 def concentrationDecayTime (rhoL g0 Vl Nb fb eta K H : ℝ) : ℝ := rhoL * g0 * Vl / (18 * Real.pi * Nb * fb * eta * K * H) theorem IPhO_2025_3_A_6 (dc_dt cL c0 Nb fb eta K H rhoL g0 Vl : ℝ) : (concentrationDecayEquation dc_dt cL c0 Nb fb eta K H rhoL g0 Vl ↔ dc_dt + 18 * Real.pi * Nb * fb * eta * K * H / (rhoL * g0 * Vl) * (cL - c0) = 0) ∧ concentrationDecayTime rhoL g0 Vl Nb fb eta K H = rhoL * g0 * Vl / (18 * Real.pi * Nb * fb * eta * K * H) := by sorry def filmRetractionSpeed (sigma rhoL h : ℝ) : ℝ := Real.sqrt (2 * sigma / (rhoL * h)) theorem IPhO_2025_3_B_1 (sigma rhoL h : ℝ) : filmRetractionSpeed sigma rhoL h = Real.sqrt (2 * sigma / (rhoL * h)) := by sorry def pistonOscillationFrequency (S P0 gamma mp V0 : ℝ) : ℝ := 1 / (2 * Real.pi) * Real.sqrt (sq S * P0 * gamma / (mp * V0)) theorem IPhO_2025_3_B_2 (S P0 gamma mp V0 : ℝ) : pistonOscillationFrequency S P0 gamma mp V0 = 1 / (2 * Real.pi) * Real.sqrt (sq S * P0 * gamma / (mp * V0)) := by sorry def champagneFilmRadiusMmRange : RangeAnswer := ((5 : ℝ) / 10, (6 : ℝ) / 10) def champagneFilmThicknessMicrometer : ℝ := (37 : ℝ) / 10 theorem IPhO_2025_3_B_3 : (champagneFilmRadiusMmRange, champagneFilmThicknessMicrometer) = (((5 : ℝ) / 10, (6 : ℝ) / 10), (37 : ℝ) / 10) := by sorry def initialBottlePressure6CBar : ℝ := (481 : ℝ) / 100 def initialBottlePressure20CBar : ℝ := (776 : ℝ) / 100 theorem IPhO_2025_3_C_1 : (initialBottlePressure6CBar, initialBottlePressure20CBar) = ((481 : ℝ) / 100, (776 : ℝ) / 100) := by sorry def finalGasTemperature6C : ℝ := (1953 : ℝ) / 10 def finalGasTemperature20C : ℝ := (1843 : ℝ) / 10 def trueFogStatements : List Choice := [Choice.A, Choice.D] theorem IPhO_2025_3_C_2 : (finalGasTemperature6C, finalGasTemperature20C, trueFogStatements) = ((1953 : ℝ) / 10, (1843 : ℝ) / 10, [Choice.A, Choice.D]) := by sorry def corkRiseHeight6C : ℝ := (77 : ℝ) / 10 theorem IPhO_2025_3_C_3 : corkRiseHeight6C = (77 : ℝ) / 10 := by sorry end Problem3 end end Ipho20212023Archon.Problems.Y2025