# VMS Spot-Check Audit - FILE A: PROBLEMS & PREDICTIONS (150 entries)

**Purpose:** audit corpus for evaluating VMS as a mathematical co-processor for classical
physics. Every problem lies INSIDE the validated range of the classical models; nothing here
claims physics beyond them. VMS supplies the governing equation through its route/closure
chains (F0001-F0031 + pillar dictionaries); the classical derivation reaches the same equation.

**Discipline:** every number is computed ONLY from the locked constants (S0=hbar; SI/CODATA
locks) plus the declared inputs listed per problem. NO experimental result value appears in
this file - comparison lives exclusively in FILE B. Declared exclusions are stated inline.
Entries tagged [family: X] share a physical root with others in that family and are counted
as partially dependent in FILE B's independence accounting.


## Mechanics/Gravity

### M1. Perihelion precession of Mercury
**Problem.** Anomalous periapsis advance of Mercury, arcsec/century.
**VMS chain.** Closed route in the Sun's missing-space profile (F0002); closure point drifts by the curvature correction to the effective potential (Mech SS8): 6piGM/(a(1-e^2)c^2) per orbit.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Stationary solar loop => missing-space profile Phi(r) = -GM/r (F0002); cost map n = 1 + 2|Phi|/c^2.
2. Closed planetary route, Binet closure with the profile's curvature correction (Mech SS8.1):
   u'' + u = GM/l^2 + 3GM u^2/c^2,  u = 1/r.
3. First-order closure drift per orbit: dpsi = 6 pi G M / (a(1-e^2) c^2)
   = 6pi x 6.67430e-11 x 1.98892e+30 / (5.7909e+10 x 0.95772 x 8.98755e+16)
   = 5.0200e-07 rad/orbit = 0.1035 arcsec/orbit.
4. Orbits/century = 36525/87.9691 = 415.20 => 42.99 arcsec/cy.
Classical: same Binet equation from the Schwarzschild geodesic; Newtonian limit (drop u^2 term) gives zero drift.
```
**Classical solution.** GR perihelion shift.
**Declared inputs.** a,e,T of Mercury (ephemeris)
**PREDICTED VALUE:** 42.99 arcsec/century

### M2. Light bending at the solar limb
**Problem.** Deflection of a grazing light route.
**VMS chain.** Route bends toward deeper missing space: theta=4GM/(bc^2), kappa0=2 fixed (Mech SS5).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Grazing light route in n(x) = 1 + 2|Phi|/c^2; ray curvature k = |grad_perp ln n| (Mech SS5, kappa0=2 lock).
2. Integrate along the straight zeroth-order path, impact parameter b: theta = 4GM/(b c^2).
3. theta = 4 x 6.67430e-11 x 1.98892e+30 / (6.9570e+08 x 8.98755e+16) = 8.4922e-06 rad = 1.7516 arcsec.
Classical: GR null geodesic; the Newtonian corpuscle value is half (kappa0=1) - the factor-2 is the route-curvature lock.
```
**Classical solution.** GR null deflection.
**Declared inputs.** b=R_sun
**PREDICTED VALUE:** 1.7516 arcsec

### M3. Shapiro delay, Earth-Mars superior conjunction
**Problem.** Two-way excess radar delay grazing the Sun.
**VMS chain.** Longer route through the profile: one-way (2GM/c^3)ln(4r1r2/b^2) (Mech SS6), doubled.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Route time t = (1/c) Int n ds with n = 1 - 2Phi/c^2 (Mech SS6).
2. Excess over the flat route, one way: dt = (2GM/c^3) ln(4 r1 r2 / b^2).
3. 2GM/c^3 = 9.8535e-06 s; ln(4 x 1.4960e+11 x 2.2784e+11 / 4.8400e+17) = 12.549.
4. One-way dt = 123.6 us; two-way = 247.3 us.
Classical: Shapiro's GR integral, identical logarithm.
```
**Classical solution.** GR Shapiro delay.
**Declared inputs.** r1=1 AU, r2=1.523 AU, b=R_sun
**PREDICTED VALUE:** 247.3 microseconds (two-way)

### M4. Pound-Rebka/Snider frequency shift
**Problem.** Fractional shift over h=22.5 m.
**VMS chain.** Clock depth in the profile (Mech SS7): gh/c^2.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Clock rate vs profile depth: dtau = (1 + Phi/c^2) dt (Mech SS7).
2. Height difference h: dnu/nu = g h / c^2 = 9.80665 x 22.5 / 8.98755e+16 = 2.4551e-15.
Classical: GR gravitational redshift, first order in Phi/c^2.
```
**Classical solution.** GR redshift.
**Declared inputs.** h=22.5 m
**PREDICTED VALUE:** 2.455e-15 (fractional)

### M5. GPS clock rate offset (net)
**Problem.** Net daily offset of a GPS clock vs geoid.
**VMS chain.** Profile-depth speedup minus route-speed slowdown; geoid includes spin term.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Geoid clock depth (spin term included): Phi_s = GM/R_eq + (om R_eq)^2/2 = 6.24955e+07 + 1.0816e+05 = 6.26037e+07 m^2/s^2.
2. Orbit depth: GM/r = 1.50065e+07. Gravitational gain = dPhi/c^2 x 86400 s = +45.8 us/day.
3. Route-speed loss: v^2/2c^2 x 86400 = GM/(2 r c^2) x 86400 = -7.2 us/day.
4. Net = +38.5 us/day.
Classical: identical GR+SR budget used operationally in GPS.
```
**Classical solution.** GR+SR clock budget (GPS ops).
**Declared inputs.** r=26,562 km
**PREDICTED VALUE:** 38.5 microseconds/day

### M6. Gravity Probe B geodetic precession
**Problem.** Gyroscope drift at r=7027.4 km.
**VMS chain.** Frame transport around the profile fails to close: (3/2)(GM)^{3/2}/(c^2 r^{5/2}) (F0002 + PM SS1.5 frame closure).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Frame (tetrad) transport around a closed route in the profile fails to close (PM SS1.5 frame closure + F0002).
2. Circular orbit: Omega_geo = (3/2)(GM)^{3/2}/(c^2 r^{5/2}) = 1.5 x 7.9581e+21 / (8.9876e+16 x 1.3091e+17)
   = 1.0145e-12 rad/s.
3. Per year: x 3.1558e+07 s x 206265 x 1000 = 6604 mas/yr.
Classical: GR de Sitter (geodetic) precession, same expression.
```
**Classical solution.** GR de Sitter precession.
**Declared inputs.** r=7027.4 km
**PREDICTED VALUE:** 6604 milliarcsec/yr

### M7. Geostationary orbit radius
**Problem.** Radius for a sidereal-day period.
**VMS chain.** Closed-route balance: r=(GMT^2/4pi^2)^{1/3} (F0001/F0002).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Closed circular route: GM/r^2 = om^2 r with om = 2pi/T_sid (F0001/F0002 balance).
2. r = (GM T^2/4pi^2)^{1/3} = (3.98603e+14 x 7.42425e+09 / 39.4784)^{1/3} = 42164 km.
Classical: Kepler III.
```
**Classical solution.** Kepler III.
**Declared inputs.** T=86164.09 s
**PREDICTED VALUE:** 42164 km

### M8. Sidereal period of the Moon
**Problem.** Period from a and both masses.
**VMS chain.** Two-loop closure: P=2pi sqrt(a^3/G(M1+M2)).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Two-loop mutual closure: P = 2pi sqrt(a^3/G(M1+M2)).
2. a^3 = 5.67998e+25; G(M1+M2) = 4.03503e+14.
3. P = 2pi x 3.75189e+05 s = 27.285 d (solar perturbation not modeled).
Classical: two-body Kepler III.
```
**Classical solution.** Kepler III (two-body).
**Declared inputs.** a=384,399 km; masses; solar perturbation not modeled
**PREDICTED VALUE:** 27.28 days

### M9. Earth escape speed
**Problem.** From mean surface, non-rotating.
**VMS chain.** Route escapes when kinetic budget = profile depth (F0011): sqrt(2GM/R).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Route escapes when kinetic budget equals profile depth (F0011): (1/2)v^2 = GM/R.
2. v = sqrt(2 x 3.98603e+14 / 6.3710e+06) = 11.19 km/s.
Classical: energy conservation.
```
**Classical solution.** Energy conservation.
**Declared inputs.** R=6371 km
**PREDICTED VALUE:** 11.19 km/s

### M10. Solar gravitational redshift at 1 AU
**Problem.** Line shift of photospheric light received at Earth.
**VMS chain.** Depth difference: GM(1/R_s-1/AU)/c (Mech SS7).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Emission at depth GM/R_s, reception at depth GM/AU; clock-rate ratio = 1 + dPhi/c^2 (Mech SS7).
2. Equivalent velocity: dv = GM(1/R_s - 1/AU)/c = 1.32746e+20 x (1.4374e-09 - 6.6846e-12) / c = 633.5 m/s.
Classical: GR redshift with identical potential difference (theory value ~633.1 m/s).
```
**Classical solution.** GR redshift.
**Declared inputs.** R_sun, 1 AU
**PREDICTED VALUE:** 633.5 m/s (equivalent)

### M11. ISS orbital period
**Problem.** Circular period at 420 km altitude.
**VMS chain.** Closed-route balance v=sqrt(GM/r).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Circular route balance: v = sqrt(GM/r) = sqrt(3.98603e+14/6.7910e+06) = 7661 m/s.
2. T = 2 pi r/v = 92.8 min.
Classical: Newtonian circular orbit.
```
**Classical solution.** Newtonian orbit.
**Declared inputs.** h=420 km
**PREDICTED VALUE:** 92.8 minutes

### M12. Periastron advance of PSR B1913+16
**Problem.** Relativistic periastron rate of the Hulse-Taylor pulsar.
**VMS chain.** Closure drift for a two-loop system (as M1): 3(2pi/Pb)^{5/3}(GM/c^3)^{2/3}/(1-e^2).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Same closure-drift law as M1 for the two-loop system, PN form:
   omega_dot = 3 (2pi/P_b)^{5/3} (G M_tot/c^3)^{2/3} / (1-e^2).
2. (2pi/P_b)^{5/3} = 8.3326e-07; (GM_tot/c^3)^{2/3} = (1.3933e-05 s)^{2/3} = 5.7902e-04; 1/(1-e^2) = 1.6151.
3. omega_dot = 2.3378e-09 rad/s = 4.227 deg/yr.
Classical: GR post-Newtonian periastron advance.
```
**Classical solution.** GR post-Newtonian advance.
**Declared inputs.** P_b, e, masses (timing solution)
**PREDICTED VALUE:** 4.227 deg/yr


## EM/Optics

### E1. Fine-structure constant [family: constants-lock]
**Problem.** alpha from the locked set; report 1/alpha.
**VMS chain.** Derived, never separately locked (EM Calibration); consistency of the closure constants.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. alpha is never separately locked (EM Calibration): it must fall out of the closure constants.
2. alpha = e^2/(4 pi eps0 hbar c) = (1.602177e-19)^2 / (4pi x 8.854188e-12 x 1.054572e-34 x 2.997925e+08) = 7.2973525737e-03.
3. 1/alpha = 137.0359990.
Classical: QED coupling definition - identical combination.
```
**Classical solution.** QED coupling definition.
**Declared inputs.** locked set
**PREDICTED VALUE:** 137.0359990 (1/alpha)

### E2. Rydberg constant [family: constants-lock]
**Problem.** R_inf = alpha^2 m_e c/2h.
**VMS chain.** Hydrogen ladder scale from the single S0 lock (F0008).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Hydrogen ladder scale from the S0 lock (F0008): R_inf = alpha^2 m_e c/2h
   = 5.325135e-05 x 9.109384e-31 x 2.997925e+08 / (2 x 6.626070e-34) = 1.097373e+07 m^-1.
Classical: Bohr/QM Rydberg constant.
```
**Classical solution.** Bohr/QM.
**Declared inputs.** locked set
**PREDICTED VALUE:** 1.097373158e+07 1/m

### E3. Lyman-alpha vacuum wavelength
**Problem.** H 2->1 line, reduced-mass corrected.
**VMS chain.** Closure ladder (F0008): 1/lam=R_H(1-1/4).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Two-loop reduced mass: mu/m_e = m_p/(m_e+m_p) = 0.999456; R_H = 1.096776e+07 m^-1.
2. 1/lam = R_H (1 - 1/4) => lam = 4/(3 R_H) = 121.5684 nm.
Classical: Bohr/QM with reduced mass.
```
**Classical solution.** Bohr/QM.
**Declared inputs.** m_p (locked)
**PREDICTED VALUE:** 121.5684 nm

### E4. Balmer ratio Ha/Hb
**Problem.** Dimensionless wavelength ratio.
**VMS chain.** Pure integer closure structure: 27/20 exactly (EM SS7.3).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Pure integer closure structure (EM SS7.3): lam_a/lam_b = (1/4-1/16)/(1/4-1/9) = (3/16)/(5/36) = 27/20 = 1.350000 exactly.
Classical: Rydberg-formula ratio - scale-free.
```
**Classical solution.** Rydberg ratio.
**Declared inputs.** none
**PREDICTED VALUE:** 1.350000 (dimensionless)

### E5. Hydrogen 21 cm line
**Problem.** Ground-state hyperfine frequency.
**VMS chain.** Orientation-orientation contact coupling of the two closures: (4/3)g_p(m_e/m_p)alpha^4 m_e c^2. QED/reduced-mass corr. excluded (declared).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Contact coupling of electron-loop and proton-loop orientations (Fermi-contact dictionary):
   dE = (4/3) g_p (m_e/m_p) alpha^4 m_e c^2.
2. = 1.3333 x 5.58569 x 5.446170e-04 x 2.83571e-09 x 8.18711e-14 J = 9.41670e-25 J.
3. nu = dE/h = 1421.2 MHz (QED + reduced-mass corrections excluded, declared).
Classical: hyperfine contact interaction.
```
**Classical solution.** Fermi contact interaction.
**Declared inputs.** g_p (declared anchor)
**PREDICTED VALUE:** 1421.2 MHz

### E6. Thomson cross-section [family: constants-lock]
**Problem.** Low-energy photon-electron scattering.
**VMS chain.** Display-area the driven loop presents to a front: (8pi/3)r_e^2.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Display-area a driven loop presents to a front: sigma_T = (8pi/3) r_e^2 = 8.3776 x (2.81794e-15)^2 = 6.65246e-29 m^2.
Classical: Thomson from Larmor radiation.
```
**Classical solution.** Larmor/Thomson.
**Declared inputs.** locked set
**PREDICTED VALUE:** 6.65246 x10^-29 m^2

### E7. Classical electron radius [family: constants-lock]
**Problem.** r_e from the locked set.
**VMS chain.** Coulomb-budget/rest-budget crossover length (F0003+F0021).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Coulomb budget = rest budget crossover: r_e = e^2/(4pi eps0 m_e c^2) = 2.8179 fm.
Classical: classical electron radius.
```
**Classical solution.** Definition.
**Declared inputs.** locked set
**PREDICTED VALUE:** 2.8179 fm

### E8. Electron cyclotron frequency, 1 T [family: constants-lock]
**Problem.** omega_c non-relativistic.
**VMS chain.** Orientation-gated steering: eB/m_e (F0004).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Orientation-gated steering curves the route: omega_c = eB/m_e = 1.602177e-19 x 1/9.109384e-31 = 1.758820e+11 rad/s (F0004 Eq. 2-8).
Classical: Lorentz-force circular motion.
```
**Classical solution.** Lorentz force.
**Declared inputs.** B=1 T
**PREDICTED VALUE:** 1.758820e+11 rad/s

### E9. Bohr magneton [family: constants-lock]
**Problem.** mu_B = e hbar/2m_e.
**VMS chain.** n=1 loop current x area (PM SS1.3).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. n=1 loop: mu = I A = (e v/2pi r)(pi r^2) = e hbar/2m_e = 9.274010e-24 J/T (PM SS1.3).
Classical: Bohr magneton.
```
**Classical solution.** QM.
**Declared inputs.** locked set
**PREDICTED VALUE:** 9.274010e-24 J/T

### E10. Brewster angle, water
**Problem.** Polarizing angle air->water.
**VMS chain.** Boundary continuity (L2) + polarization projection (L5): tan th=n.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Boundary continuity (L2) + projection drain (L5): reflected p-route vanishes when reflected and refracted
   routes are orthogonal => tan(th_B) = n = 1.3330 => th_B = 53.12 deg.
Classical: Fresnel r_p = 0.
```
**Classical solution.** Fresnel r_p=0.
**Declared inputs.** n=1.3330 (declared)
**PREDICTED VALUE:** 53.12 degrees

### E11. Critical angle, diamond
**Problem.** TIR diamond->air.
**VMS chain.** Tangential closure match fails: sin th_c=1/n (L2).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Tangential route match fails beyond sin(th_c) = 1/n = 0.4137 => th_c = 24.44 deg (L2).
Classical: Snell limit.
```
**Classical solution.** Snell limit.
**Declared inputs.** n=2.417 (declared)
**PREDICTED VALUE:** 24.44 degrees

### E12. HST diffraction limit
**Problem.** Resolution at 550 nm, D=2.4 m.
**VMS chain.** Finite pupil display-area: 1.22 lam/D (L6).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Finite pupil display-area => far-field route-phase transform; first null of the circular pupil:
   theta = 1.22 lam/D = 1.22 x 550e-9/2.4 = 2.796e-07 rad = 0.0577 arcsec (L6).
Classical: Rayleigh/Airy criterion.
```
**Classical solution.** Rayleigh criterion.
**Declared inputs.** D, lambda
**PREDICTED VALUE:** 0.0577 arcsec

### E13. Vacuum speed identity [family: constants-lock]
**Problem.** 1/sqrt(mu0 eps0) vs c.
**VMS chain.** A2 makes c primitive; SI bridge must return it (F0007 acceptance lock).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. c is primitive (A2); the SI bridge must hand it back: 1/sqrt(mu0 eps0) = 1/sqrt(1.256637e-06 x 8.854188e-12) = 299792458.000006 m/s (F0007 acceptance lock).
Classical: Maxwell wave speed.
```
**Classical solution.** Maxwell.
**Declared inputs.** mu0, eps0
**PREDICTED VALUE:** 299792458.000006 m/s


## Particle/Quantum

### P1. de Broglie wavelength, 100 eV electron
**Problem.** Diffraction wavelength.
**VMS chain.** Route-phase period lam=h/p (F0028).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Route phase advances dphi = p dl/hbar (PM SS0.1); spatial period lam = h/p (F0028).
2. p = sqrt(2 m_e E) = sqrt(2 x 9.10938e-31 x 1.60218e-17) = 5.40275e-24 kg m/s.
3. lam = 6.62607e-34/5.40275e-24 = 0.1226 nm.
Classical: de Broglie relation.
```
**Classical solution.** de Broglie.
**Declared inputs.** E=100 eV
**PREDICTED VALUE:** 0.1226 nm

### P2. Compton shift at 90 deg
**Problem.** X-ray wavelength shift.
**VMS chain.** Front-closure exchange (F0003+F0028): h/m_e c at 90 deg.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Front-closure exchange conserving route energy-momentum (F0003+F0028):
   dlam = (h/m_e c)(1 - cos th); at 90 deg dlam = lam_C = h/m_e c.
2. = 6.62607e-34/(9.10938e-31 x 2.99792e+08) = 2.4263 pm.
Classical: Compton kinematics.
```
**Classical solution.** Compton kinematics.
**Declared inputs.** theta=90
**PREDICTED VALUE:** 2.4263 pm

### P3. Photoelectric slope h/e
**Problem.** Universal stopping-potential slope.
**VMS chain.** One quantum per transfer (F0019): slope locked by S0.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. One route quantum pays the escape cost (F0019): eV_stop = h nu - phi.
2. Slope dV/dnu = h/e = 6.626070e-34/1.602177e-19 = 4.1357 x10^-15 V s - universal, locked by S0.
Classical: Einstein photoelectric equation.
```
**Classical solution.** Einstein equation.
**Declared inputs.** locked set
**PREDICTED VALUE:** 4.1357 x10^-15 V s

### P4. Hydrogen 2p fine structure
**Problem.** 2p3/2-2p1/2 interval.
**VMS chain.** Relativistic budget + frame double-cover: Ry alpha^2/16 (F0020). Lamb shift excluded.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Relativistic route budget + frame double-cover coupling combine (F0020):
   E(n,j) = -(Ry/n^2)[1 + (alpha^2/n^2)(n/(j+1/2) - 3/4)].
2. n=2 split (j=3/2 minus j=1/2): dE = Ry alpha^2/16 = 13.6057 x 5.32514e-05/16 = 4.52826e-05 eV.
3. nu = dE e/h = 10.949 GHz (Lamb shift excluded, declared).
Classical: Dirac fine structure.
```
**Classical solution.** Dirac fine structure.
**Declared inputs.** locked set
**PREDICTED VALUE:** 10.949 GHz

### P5. Hydrogen ionization energy
**Problem.** Ground-state binding.
**VMS chain.** Closure ladder n=1, reduced mass (F0008, PM SS7).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Ladder n=1 with two-loop reduced mass: E = Ry x mu/m_e = 13.60569 x 0.999456 = 13.5983 eV.
Classical: Bohr/QM.
```
**Classical solution.** Bohr/QM.
**Declared inputs.** m_p
**PREDICTED VALUE:** 13.5983 eV

### P6. Positronium binding energy
**Problem.** e+e- ground state.
**VMS chain.** Equal-mass two-loop closure: Ry/2 (PM SS7).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Equal-mass two-loop closure, mu = m_e/2: E = Ry/2 = 6.8028 eV.
Classical: hydrogen-like QM.
```
**Classical solution.** QM mu=m_e/2.
**Declared inputs.** locked set
**PREDICTED VALUE:** 6.8028 eV

### P7. Muonic hydrogen K-alpha
**Problem.** 2p->1s X-ray energy.
**VMS chain.** Same ladder, muon loop: (3/4)Ry mu(mu-p)/m_e - no retune (PM SS7).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Muon loop in the proton profile, mu(mu-p)/m_e = 185.841 - same ladder, no retune.
2. E1 = Ry x 185.841 = 2528.5 eV; K-alpha = (3/4)|E1| = 1.896 keV.
Classical: hydrogenic QM with muon reduced mass.
```
**Classical solution.** Hydrogen-like QM.
**Declared inputs.** m_mu
**PREDICTED VALUE:** 1.896 keV

### P8. H-D isotope shift of H-alpha
**Problem.** Deuterium discovery shift.
**VMS chain.** Reduced-mass dependence of the ladder (PM SS7).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Ladder depends on mu: lam_Ha(H) = 656.4696 nm; lam_Ha(D) = 656.2910 nm.
2. dlam = 0.1786 nm - the shift that revealed deuterium.
Classical: reduced-mass isotope shift.
```
**Classical solution.** Bohr isotope shift.
**Declared inputs.** m_d
**PREDICTED VALUE:** 0.1786 nm

### P9. Electron spin resonance, 1 T
**Problem.** Free-electron ESR frequency.
**VMS chain.** Orientation flip cost: g_e mu_B B/h (PM SS1.5).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Loop orientation flip in B (PM SS1.5): nu = g_e mu_B B/h = 2.002319 x 9.27401e-24/6.62607e-34 = 28.025 GHz.
Classical: ESR resonance.
```
**Classical solution.** Zeeman/ESR.
**Declared inputs.** g_e (declared)
**PREDICTED VALUE:** 28.025 GHz

### P10. Muon lifetime at gamma=29.3
**Problem.** Storage-ring lab lifetime.
**VMS chain.** Moving closure's internal circulation slows by gamma (F0003): gamma tau_0.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. The moving closure's internal circulation slows by gamma (F0003 dispersion applied to the loop clock).
2. tau_lab = gamma tau_0 = 29.30 x 2.1969811 = 64.37 us.
Classical: SR time dilation.
```
**Classical solution.** SR dilation.
**Declared inputs.** gamma=29.30, tau_0=2.1969811 us (PDG)
**PREDICTED VALUE:** 64.37 microseconds

### P11. Thermal neutron wavelength
**Problem.** lambda at 0.0253 eV.
**VMS chain.** Route-phase period, composite closure (F0028).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. lam = h/sqrt(2 m_n E) = 6.62607e-34/sqrt(2 x 1.67493e-27 x 4.0535e-21) = 0.1798 nm (F0028).
Classical: de Broglie.
```
**Classical solution.** de Broglie.
**Declared inputs.** E=0.0253 eV
**PREDICTED VALUE:** 0.1798 nm

### P12. Fe-56 binding energy per nucleon [family: SEMF]
**Problem.** B/A of Fe-56.
**VMS chain.** Budget build (PM SS6 L6): volume/surface/asymmetry/Coulomb/pairing, single K + ratios.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first (budget build, PM SS6 Lemma L6):
1. B = aV A - aS A^(2/3) - aC Z^2/A^(1/3) - aA (N-Z)^2/A + aP/sqrt(A)  [declared set 15.75/17.8/0.711/23.7/11.18 MeV]
2. Fe-56 (Z=26, N=30): volume 882.0 - surface 260.5 - Coulomb 125.6 - asym 6.8 + pair 1.49
   = 490.6 MeV => B/A = 8.760 MeV.
Classical: semi-empirical mass formula.
```
**Classical solution.** SEMF liquid drop.
**Declared inputs.** declared SEMF set
**PREDICTED VALUE:** 8.760 MeV

### P13. Pb-208 binding energy per nucleon [family: SEMF]
**Problem.** B/A, doubly magic.
**VMS chain.** Same budgets, no retune; shell micro-term = 0 declared.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Same budgets, no retune. Pb-208 (Z=82, N=126): volume 3276 - surface 624.9 - Coulomb 806.9 - asym 220.6 + pair 0.78
   = 1624.4 MeV => B/A = 7.810 MeV (shell term = 0 declared; Pb-208 is doubly magic).
Classical: SEMF.
```
**Classical solution.** SEMF.
**Declared inputs.** same set
**PREDICTED VALUE:** 7.810 MeV


## Thermodynamics

### T1. Speed of sound in air, 20 C
**Problem.** Adiabatic sound speed.
**VMS chain.** Adiabatic compression of the route ensemble: sqrt(gamma R T/M) (5Y.4).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Route-count EoS (F0009/F0010) + adiabatic compression of the ensemble: c_s = sqrt(gamma R T/M).
2. = sqrt(1.400 x 8.31446 x 293.15/0.0289645) = 343.2 m/s.
Classical: Laplace sound speed.
```
**Classical solution.** Laplace formula.
**Declared inputs.** gamma=1.400, M_air, T=293.15 K
**PREDICTED VALUE:** 343.2 m/s

### T2. Molar volume at STP
**Problem.** V_m at 273.15 K, 101.325 kPa.
**VMS chain.** Dilute route-count closure PV=nRT (F0009/F0010).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Dilute closure of the route count: V = RT/P = 8.31446 x 273.15/101325 = 22.4140 L/mol.
Classical: ideal gas law.
```
**Classical solution.** Ideal gas.
**Declared inputs.** T,P
**PREDICTED VALUE:** 22.4140 L/mol

### T3. Stefan-Boltzmann constant [family: constants-lock]
**Problem.** sigma from locked set.
**VMS chain.** Mode counting + ensemble weights (F0031 chain): pi^2 kB^4/(60 hbar^3 c^2).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Mode counting (F0013/F0028) + ensemble weights (F0029) + zeta(4) = pi^4/90:
   sigma = pi^2 kB^4/(60 hbar^3 c^2) = 5.6703744e-08 W m^-2 K^-4.
Classical: Planck-spectrum integral.
```
**Classical solution.** Planck integral.
**Declared inputs.** locked set
**PREDICTED VALUE:** 5.6703744 x10^-8 W m^-2 K^-4

### T4. Total solar irradiance [family: solar-T]
**Problem.** Solar constant at 1 AU.
**VMS chain.** sigma T^4 diluted by display-area ratio (F0031+L7).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Surface flux sigma T^4 = 6.2939e+07 W/m^2; display-area dilution (R_s/AU)^2 = 2.16268e-05.
2. S = 1361.2 W/m^2.
Classical: Stefan-Boltzmann + inverse square.
```
**Classical solution.** SB + inverse square.
**Declared inputs.** T_eff=5772 K
**PREDICTED VALUE:** 1361.2 W/m^2

### T5. Wien peak, solar spectrum [family: solar-T]
**Problem.** lambda_max at 5772 K.
**VMS chain.** Peak of route-mode occupancy: b/T.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Occupancy-spectrum maximum: lam_max = b/T = 2.897772e-03/5772 = 502.0 nm.
Classical: Wien displacement.
```
**Classical solution.** Wien law.
**Declared inputs.** T=5772 K
**PREDICTED VALUE:** 502.0 nm

### T6. CMB spectral peak (wavelength) [family: CMB]
**Problem.** lambda_max at 2.7255 K.
**VMS chain.** Same law at the background temperature.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Same law at T = 2.7255 K: lam_max = 1.0632 mm.
Classical: Wien displacement.
```
**Classical solution.** Wien law.
**Declared inputs.** T=2.7255 K (FIRAS input)
**PREDICTED VALUE:** 1.0632 mm

### T7. RMS speed of N2, 300 K
**Problem.** sqrt(3RT/M).
**VMS chain.** Equipartition over route modes (F0009).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Equipartition over 3 route modes: (1/2)m<v^2> = (3/2)kT => v_rms = sqrt(3RT/M) = 516.8 m/s.
Classical: Maxwell-Boltzmann.
```
**Classical solution.** Maxwell-Boltzmann.
**Declared inputs.** M_N2
**PREDICTED VALUE:** 516.8 m/s

### T8. Mean free path of N2
**Problem.** At 300 K, 101.325 kPa.
**VMS chain.** Route-interruption length kT/(sqrt2 pi d^2 P).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Route interrupted when another closure's display-area pi d^2 is swept: lam = kT/(sqrt2 pi d^2 P)
   = 4.1419e-21/(1.4142 x pi x 1.369e-19 x 101325) = 67.2 nm.
Classical: kinetic mean free path.
```
**Classical solution.** Kinetic theory.
**Declared inputs.** d=0.370 nm (declared)
**PREDICTED VALUE:** 67.2 nm

### T9. Johnson-Nyquist noise
**Problem.** 1 kOhm, 300 K, 10 kHz.
**VMS chain.** Fluctuation identity (SS5): sqrt(4kTRdf).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Fluctuation identity (F0029 SS5, F0016): <V^2> = 4 kB T R df.
2. V_rms = sqrt(4 x 1.38065e-23 x 300 x 1000 x 10^4) = 0.407 uV.
Classical: Nyquist theorem.
```
**Classical solution.** Nyquist theorem.
**Declared inputs.** R,T,df
**PREDICTED VALUE:** 0.407 microvolts

### T10. Stokes-Einstein diffusion
**Problem.** 1 um sphere in water, 20 C.
**VMS chain.** D=mu_d kB T with Stokes mobility (F0016, T3 strategy).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Einstein relation D = mu_d kB T (F0016, T3 strategy) with Stokes mobility 1/(6 pi eta r):
   D = 4.0474e-21/(6pi x 0.001002 x 0.5e-6) = 4.2858e-13 m^2/s.
Classical: Stokes-Einstein.
```
**Classical solution.** Stokes-Einstein.
**Declared inputs.** eta, r (declared)
**PREDICTED VALUE:** 4.286 x10^-13 m^2/s

### T11. Dulong-Petit heat capacity
**Problem.** Classical lattice C.
**VMS chain.** Equipartition, 6 modes per closure: 3R. Quantum freeze-out excluded (declared).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. 6 quadratic route modes per lattice closure x kB/2: C = 3R = 24.94 J/(mol K) (freeze-out excluded, declared).
Classical: Dulong-Petit.
```
**Classical solution.** Dulong-Petit.
**Declared inputs.** classical limit
**PREDICTED VALUE:** 24.94 J/(mol K)

### T12. Clausius-Clapeyron slope, water 100 C
**Problem.** dP/dT on coexistence.
**VMS chain.** Phase-slope strategy T4: L/(T dv).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Phase route-count balance (T4 strategy): dP/dT = L/(T dv) = 2.256e+06/(373.124 x 1.6708) = 3.619 kPa/K.
Classical: Clausius-Clapeyron.
```
**Classical solution.** Clausius-Clapeyron.
**Declared inputs.** L, dv (steam properties)
**PREDICTED VALUE:** 3.619 kPa/K


---
# SET 2 - EXTENSION (100 additional, maximally independent)


## Mechanics/Gravity

### M13. Foucault pendulum, Paris
**Problem.** Precession rate at 48.85 N.
**VMS chain.** Route plane stays fixed while Earth's closure rotates under it: 15.041 deg/hr x sin(lat).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. The swing route's plane is inertial; Earth's closure rotates beneath at 360.9856 deg/day (sidereal+solar).
2. Projected rate at latitude L: rate = 15.041 deg/hr x sin(L) = 15.041 x 0.7529 = 11.33 deg/hr.
Classical: rotating-frame kinematics (Coriolis).
```
**Classical solution.** Rotating-frame kinematics.
**Declared inputs.** lat=48.846
**PREDICTED VALUE:** 11.33 deg/hr

### M14. Meter pendulum period
**Problem.** L=1.000 m, small amplitude.
**VMS chain.** Linearized closure oscillation (F0027): 2pi sqrt(L/g).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Small departure from the stable hang: linearized closure oscillation (F0027), omega^2 = g/L.
2. T = 2pi sqrt(L/g) = 2pi sqrt(1/9.80665) = 2.0064 s.
Classical: SHM small-angle pendulum.
```
**Classical solution.** SHM.
**Declared inputs.** g=9.80665
**PREDICTED VALUE:** 2.0064 s

### M15. Orbital period of Io [family: Kepler-GMJ]
**Problem.** From a and GM_Jupiter.
**VMS chain.** Closed route in Jupiter's profile: Kepler III.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Closed route in Jupiter's profile: P = 2pi sqrt(a^3/GM_J) = 2pi sqrt(7.4991e+25/1.2669e+17) = 1.769 d.
Classical: Kepler III.
```
**Classical solution.** Kepler III.
**Declared inputs.** a=421,700 km, GM_J
**PREDICTED VALUE:** 1.769 days

### M16. Period of Halley's comet
**Problem.** From a=17.834 AU.
**VMS chain.** Eccentric closed route: P=a^{3/2} in solar units.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Eccentric closed route, same closure law; in solar units P[yr] = a[AU]^{3/2} = 17.834^1.5 = 75.3 yr.
Classical: Kepler III.
```
**Classical solution.** Kepler III.
**Declared inputs.** a (orbit solution)
**PREDICTED VALUE:** 75.3 years

### M17. Earth mean orbital speed
**Problem.** 2 pi AU / year.
**VMS chain.** Route length per closure period.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Route length per closure period: v = 2 pi AU / yr = 2pi x 1.49598e+11/3.15582e+07 = 29.78 km/s.
Classical: orbital kinematics.
```
**Classical solution.** Kepler kinematics.
**Declared inputs.** AU, year
**PREDICTED VALUE:** 29.78 km/s

### M18. Sun-Earth L1 distance
**Problem.** Balance point for SOHO.
**VMS chain.** Three-route balance: d=AU (M_E/3M_sun)^{1/3}.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Three-route balance on the Sun-Earth line (Hill): d = AU (M_E/3M_sun)^{1/3}.
2. (M_E/3M_sun)^{1/3} = (1.0009e-06)^{1/3} = 0.01000 => d = 1.4964 x10^9 m from Earth.
Classical: restricted three-body L1.
```
**Classical solution.** Restricted 3-body (Hill).
**Declared inputs.** masses
**PREDICTED VALUE:** 1.4964 x10^6 km /1e3

### M19. Eastward deflection of a dropped mass
**Problem.** Reich 1831: h=158.5 m at lat 51.33.
**VMS chain.** Dropped route keeps the tower top's larger closure speed: d=(omega g cos(lat) t^3)/3.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Released at the tower top, the route keeps the top's larger closure speed; relative drift integrates to
   d = (1/3) om g cos(L) t^3 with t = sqrt(2h/g) = 5.685 s.
2. d = (1/3) x 7.2921e-05 x 9.81 x 0.6248 x 183.7 = 27.4 mm eastward.
Classical: Coriolis deflection.
```
**Classical solution.** Coriolis kinematics.
**Declared inputs.** h, lat (declared)
**PREDICTED VALUE:** 27.4 mm

### M20. Schwarzschild precession of star S2
**Problem.** Per-orbit precession around Sgr A*.
**VMS chain.** Same closure-drift law as M1, 4 million solar masses: 6piGM/(a(1-e^2)c^2).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Same closure-drift law as M1 at Sgr A*: a = 0.12540 arcsec x 8277 pc = 1.553e+14 m; e = 0.884; M = 8.473e+36 kg.
2. dpsi = 6 pi G M/(a(1-e^2)c^2) = 3.4951e-03 rad/orbit = 12.0 arcmin/orbit.
Classical: GR Schwarzschild precession (paper's GR value 12.1 arcmin).
```
**Classical solution.** GR precession.
**Declared inputs.** a from 0.1254 arcsec at 8.277 kpc; e=0.884; M=4.26e6 Msun (GRAVITY solution)
**PREDICTED VALUE:** 12.0 arcmin/orbit

### M21. Speed of gravitational waves
**Problem.** Fractional difference (v_GW-c)/c.
**VMS chain.** A2: ALL deformations of space propagate at invariant c - zero difference, no dispersion.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. A2: ALL deformations of space propagate at the single invariant speed c - gravitational and
   electromagnetic fronts share one route speed by axiom. Predicted (v_GW - c)/c = 0 exactly, no dispersion.
Classical: GR also predicts v_GW = c; alternatives (massive graviton, extra dispersion) predict nonzero.
```
**Classical solution.** GR: GWs travel at c.
**Declared inputs.** none
**PREDICTED VALUE:** 0.0 (fractional; predicted exactly 0)

### M22. Moon escape speed
**Problem.** From the lunar surface.
**VMS chain.** sqrt(2GM_M/R_M) (F0011).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. v = sqrt(2GM_M/R_M) = sqrt(2 x 4.9003e+12/1.7374e+06) = 2.38 km/s (F0011).
Classical: energy conservation.
```
**Classical solution.** Energy conservation.
**Declared inputs.** M_Moon, R_Moon
**PREDICTED VALUE:** 2.38 km/s

### M23. Orbital period of Charon
**Problem.** Pluto-Charon mutual orbit.
**VMS chain.** Two-loop closure, comparable masses: Kepler III with M1+M2.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Comparable-mass two-loop closure: P = 2pi sqrt(a^3/G(M_P+M_C)) = 2pi sqrt(7.5249e+21/9.7552e+11) = 6.387 d.
Classical: two-body Kepler III.
```
**Classical solution.** Two-body Kepler.
**Declared inputs.** a=19,596 km; masses (New Horizons)
**PREDICTED VALUE:** 6.387 days

### M24. Solar mass from the year [family: consistency]
**Problem.** M=4pi^2 AU^3/(G yr^2).
**VMS chain.** Inverting the Earth-route closure.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Invert Earth's route closure: M = 4pi^2 AU^3/(G yr^2) = 1.98842e+30 kg = 1.9884 x10^30 kg.
Classical: Kepler III inverted.
```
**Classical solution.** Kepler III.
**Declared inputs.** AU, yr
**PREDICTED VALUE:** 1.9884 x10^30 kg

### M25. Orbital period of Titan
**Problem.** From a and GM_Saturn.
**VMS chain.** Closed route in Saturn's profile.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. P = 2pi sqrt(a^3/GM_Sat) = 2pi sqrt(1.8242e+27/3.7931e+16) = 15.95 d.
Classical: Kepler III.
```
**Classical solution.** Kepler III.
**Declared inputs.** a=1,221,870 km, GM_Sat
**PREDICTED VALUE:** 15.95 days

### M26. Light bending at Jupiter's limb
**Problem.** Deflection grazing Jupiter.
**VMS chain.** Same route-bending law as M2, planetary scale: 4GM_J/(R_J c^2).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Route bending at Jupiter's limb, same law as M2: theta = 4GM_J/(R_J c^2)
   = 4 x 1.2669e+17/(7.1492e+07 x 8.9876e+16) = 7.887e-08 rad = 16.27 mas.
Classical: GR deflection, planetary scale.
```
**Classical solution.** GR deflection.
**Declared inputs.** GM_J, R_J
**PREDICTED VALUE:** 16.27 milliarcsec

### M27. Cassini test: route-bending coefficient
**Problem.** PPN gamma from Doppler tracking at conjunction.
**VMS chain.** kappa0=2 is fixed by the profile normalization (Mech SS5) => gamma=1 exactly, no freedom.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. The bending coefficient kappa0 = 2 is fixed by the profile normalization (Mech SS5) - not adjustable.
   In PPN language: gamma = 1 exactly. Prediction: gamma = 1.000000.
Classical: GR gamma = 1; Cassini measures (1+gamma)/2 via the Shapiro phase.
```
**Classical solution.** GR gamma=1.
**Declared inputs.** none
**PREDICTED VALUE:** 1.000000 (gamma)

### M28. Free-air gravity gradient
**Problem.** dg/dh at Earth's surface.
**VMS chain.** Profile steepness: -2GM/R^3.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Profile steepness just above the surface: dg/dh = -2GM/R^3 = -2 x 3.9860e+14/2.5860e+20 = -3.0828e-06 s^-2
   = 0.3083 mGal/m (magnitude).
Classical: Newtonian inverse-square gradient.
```
**Classical solution.** Newtonian gradient.
**Declared inputs.** GM_E, R_E
**PREDICTED VALUE:** 0.3083 mGal/m (magnitude x10^5 SI)

### M29. Orbital decay of PSR B1913+16
**Problem.** Period derivative from gravitational radiation.
**VMS chain.** Mutually orbiting loops radiate deformation at c; quadrupole route-loss (GR dictionary import, declared): peters formula.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first (declared GR-dictionary import):
1. Mutually orbiting loops lose route budget to outgoing deformation at c; quadrupole loss (Peters):
   Pb_dot = -(192pi/5)(2pi/P_b)^{5/3} (T_sun)^{5/3} m1 m2 (m1+m2)^{-1/3} f(e), T_sun = GM_sun/c^3 = 4.92675e-06 s.
2. f(e) = (1 + 73/24 e^2 + 37/96 e^4)/(1-e^2)^{7/2} = 11.857.
3. Pb_dot = -2.4031e-12 = -2.403 x10^-12 s/s.
Classical: GR quadrupole formula (identical).
```
**Classical solution.** GR quadrupole formula.
**Declared inputs.** P_b, e, masses (timing)
**PREDICTED VALUE:** -2.403 x10^-12 s/s

### M30. Eotvos effect
**Problem.** Weight change, 10 m/s eastward at equator.
**VMS chain.** Moving with the closure rotation raises route speed: dg=2 omega v.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Moving with the closure spin raises the route's rotation rate: dg = 2 om v = 2 x 7.2921e-05 x 10 = 1.458e-03 m/s^2 = 145.8 mGal.
Classical: Eotvos correction in marine gravimetry.
```
**Classical solution.** Rotating frame.
**Declared inputs.** v=10 m/s
**PREDICTED VALUE:** 145.8 mGal

### M31. Sidereal year [family: consistency]
**Problem.** From AU and GM_sun.
**VMS chain.** Earth-route closure period: 2pi sqrt(AU^3/GM).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. P = 2pi sqrt(AU^3/GM_sun) = 2pi sqrt(3.34793e+33/1.32712e+20) = 365.257 d.
Classical: Kepler III.
```
**Classical solution.** Kepler III.
**Declared inputs.** AU, GM_sun (ephemeris)
**PREDICTED VALUE:** 365.26 days

### M32. De Sitter precession of the lunar orbit
**Problem.** Geodetic precession of the Earth-Moon system in the Sun's profile.
**VMS chain.** Same frame-closure failure as M6, heliocentric: (3/2)(GM/c^2 AU) n.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Earth-Moon frame transported around the Sun's profile (same failure as M6):
   Omega = (3/2)(GM/c^2 AU) n, n = sqrt(GM/AU^3) = 1.9910e-07 rad/s.
2. Omega = 1.5 x 9.8706e-09 x n = 2.9478e-15 rad/s = 19.19 mas/yr.
Classical: de Sitter precession, detected in lunar laser ranging.
```
**Classical solution.** GR geodetic.
**Declared inputs.** GM_sun, AU
**PREDICTED VALUE:** 19.19 milliarcsec/yr

### M33. g at ISS altitude [family: consistency]
**Problem.** Field strength at r=6791 km.
**VMS chain.** Profile gradient GM/r^2.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Profile gradient at r = 6.7910e+06 m: g = GM/r^2 = 8.643 m/s^2.
Classical: Newtonian field.
```
**Classical solution.** Newtonian.
**Declared inputs.** r
**PREDICTED VALUE:** 8.643 m/s^2

### M34. Orbital period of Europa [family: Kepler-GMJ]
**Problem.** From a and GM_J.
**VMS chain.** Same profile as M15, different closure.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Same Jupiter profile as M15: P = 2pi sqrt(3.0198e+26/1.2669e+17) = 3.550 d.
Classical: Kepler III.
```
**Classical solution.** Kepler III.
**Declared inputs.** a=670,900 km
**PREDICTED VALUE:** 3.550 days

### M35. Period of Neptune
**Problem.** From a=30.07 AU.
**VMS chain.** P=a^{3/2} in solar units.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. P = a^{3/2} = 30.07^1.5 = 164.9 yr.
Classical: Kepler III.
```
**Classical solution.** Kepler III.
**Declared inputs.** a
**PREDICTED VALUE:** 164.9 years

### M36. Earth equatorial rotation speed
**Problem.** omega R_eq.
**VMS chain.** Closure surface speed.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Closure surface speed: v = om R_eq = 7.2921e-05 x 6.3781e+06 = 465.1 m/s.
Classical: kinematics.
```
**Classical solution.** Kinematics.
**Declared inputs.** omega, R_eq
**PREDICTED VALUE:** 465.1 m/s

### M37. Solar escape speed at 1 AU
**Problem.** For outbound probes.
**VMS chain.** sqrt(2GM_sun/AU) (F0011).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. v = sqrt(2GM_sun/AU) = sqrt(2 x 1.32712e+20/1.49598e+11) = 42.12 km/s (F0011).
Classical: energy conservation.
```
**Classical solution.** Energy conservation.
**Declared inputs.** GM_sun, AU
**PREDICTED VALUE:** 42.12 km/s


## EM/Optics

### E14. Rayleigh scattering ratio 400/700 nm
**Problem.** Blue-to-red scattered power.
**VMS chain.** Small-closure re-radiation of the front scales as 1/lam^4 (dipole route response).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Small-closure re-radiation amplitude ~ omega^2 (driven-dipole route response) => power ~ 1/lam^4.
2. Ratio = (700/400)^4 = 9.38.
Classical: Rayleigh scattering.
```
**Classical solution.** Rayleigh 1/lam^4.
**Declared inputs.** two wavelengths
**PREDICTED VALUE:** 9.38 (ratio)

### E15. Malus law at 30 deg
**Problem.** Transmitted fraction.
**VMS chain.** Closure drain to the polarizer axis (L5): cos^2.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Closure drain to the pass axis (L5): I = I0 cos^2(30 deg) = 0.7500.
Classical: Malus law.
```
**Classical solution.** Malus.
**Declared inputs.** theta=30
**PREDICTED VALUE:** 0.7500 (fraction)

### E16. Young double slit
**Problem.** Fringe spacing: 632.8 nm, d=0.25 mm, L=1 m.
**VMS chain.** Route-phase superposition (L3): lam L/d.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Route-phase superposition (L3): bright at dL = m lam; spacing dy = lam L/d = 632.8e-9 x 1.0/2.5e-4 = 2.531 mm.
Classical: Young interference.
```
**Classical solution.** Two-slit interference.
**Declared inputs.** geometry
**PREDICTED VALUE:** 2.531 mm

### E17. Grating angle, sodium D
**Problem.** 589.3 nm, 600 lines/mm, first order.
**VMS chain.** Multi-route phase closure: sin th = lam/d.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Multi-route closure (L3): sin th = lam/d = 589.3e-9 x 600e3 = 0.35358 => th = 20.71 deg.
Classical: grating equation.
```
**Classical solution.** Grating equation.
**Declared inputs.** d=1/600 mm
**PREDICTED VALUE:** 20.71 degrees

### E18. Bragg angle, NaCl (Cu K-alpha)
**Problem.** First-order, d=282.0 pm, 154.06 pm.
**VMS chain.** Layered route-phase closure: sin th=lam/2d.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Layered route-phase closure: 2d sin th = lam => th = asin(0.27315) = 15.85 deg.
Classical: Bragg law.
```
**Classical solution.** Bragg law.
**Declared inputs.** d, lam (declared)
**PREDICTED VALUE:** 15.85 degrees

### E19. Moseley: Mo K-alpha [family: Moseley]
**Problem.** Z=42 K-alpha wavelength.
**VMS chain.** Inner closure sees (Z-1) screened charge: 1/lam=R(Z-1)^2(3/4). Screening approx declared.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Inner closure orbits (Z-1) screened charge; hydrogenic ladder x (Z-1)^2 (declared screening approx):
   1/lam = R (Z-1)^2 (3/4), Z=42 => lam = 1/(1.0974e+07 x 1681 x 0.75) = 72.28 pm.
Classical: Moseley's law.
```
**Classical solution.** Moseley's law.
**Declared inputs.** Z=42
**PREDICTED VALUE:** 72.28 pm

### E20. Moseley: Cu K-alpha [family: Moseley]
**Problem.** Z=29 K-alpha wavelength.
**VMS chain.** Same screened-closure law.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Same law, Z=29: lam = 1/(1.0974e+07 x 784 x 0.75) = 154.98 pm.
Classical: Moseley's law.
```
**Classical solution.** Moseley.
**Declared inputs.** Z=29
**PREDICTED VALUE:** 154.98 pm

### E21. Balmer series limit
**Problem.** n->inf edge of Balmer lines.
**VMS chain.** Ladder edge: 4/R_H.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Ladder edge n->inf: lam = 4/R_H = 364.71 nm.
Classical: Balmer limit.
```
**Classical solution.** Rydberg formula.
**Declared inputs.** m_p
**PREDICTED VALUE:** 364.71 nm

### E22. Paschen-alpha wavelength
**Problem.** H 4->3 line, vacuum.
**VMS chain.** Ladder: 1/lam=R_H(1/9-1/16).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. 1/lam = R_H(1/9 - 1/16) = 5.33155e+05 => lam = 1875.6 nm.
Classical: Rydberg formula.
```
**Classical solution.** Rydberg.
**Declared inputs.** m_p
**PREDICTED VALUE:** 1875.6 nm

### E23. He+ Lyman-alpha
**Problem.** Hydrogenic Z=2, 2->1, vacuum.
**VMS chain.** Same closure rule, Z=2 profile: lam(H)/4 with He reduced mass - no retune.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Z=2 profile, He reduced mass mu/m_e = 0.999863: 1/lam = 4 R_He (3/4) => lam = 30.3797 nm - no retune.
Classical: hydrogenic QM.
```
**Classical solution.** Hydrogenic QM.
**Declared inputs.** m_alpha
**PREDICTED VALUE:** 30.3797 nm

### E24. Positronium 1S-2S interval
**Problem.** Two-photon transition frequency.
**VMS chain.** Equal-mass ladder: (3/4)(Ry/2)/h. QED recoil excluded (declared).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Equal-mass pair, mu = m_e/2: E(1S-2S) = (3/4)(Ry/2) = 5.1021 eV.
2. nu = E/h = 1233.691 THz (QED recoil excluded, declared).
Classical: QM with mu = m_e/2.
```
**Classical solution.** QM mu=m_e/2.
**Declared inputs.** locked set
**PREDICTED VALUE:** 1233.691 THz

### E25. Speed of light in water
**Problem.** c/n.
**VMS chain.** Route slowness in the medium's cost map: c/n (L4).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Route slowness in the medium's cost map (L4): v = c/n = 2.2490e+08 = 2.2490 x10^8 m/s.
Classical: refractive index.
```
**Classical solution.** Refraction.
**Declared inputs.** n=1.333
**PREDICTED VALUE:** 2.2490 x10^8 m/s

### E26. Half-wave dipole at 100 MHz
**Problem.** Resonant length.
**VMS chain.** Standing route-phase on the conductor: lam/2.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Standing route-phase on the conductor: L = lam/2 = c/(2f) = 1.499 m.
Classical: dipole resonance.
```
**Classical solution.** Antenna theory.
**Declared inputs.** f=100 MHz
**PREDICTED VALUE:** 1.499 m

### E27. Skin depth, copper at 60 Hz
**Problem.** Field penetration depth.
**VMS chain.** Front decay into a lossy closure sea: sqrt(2 rho/(omega mu0)).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Front decays into the conductor's closure sea: delta = sqrt(2 rho/(omega mu0))
   = sqrt(2 x 1.678e-08/(377.0 x 1.2566e-06)) = 8.42 mm.
Classical: skin effect.
```
**Classical solution.** EM skin effect.
**Declared inputs.** rho_Cu=1.678e-8 (declared)
**PREDICTED VALUE:** 8.42 mm

### E28. Calculable capacitor
**Problem.** Parallel plates 1 m^2, 1 mm gap.
**VMS chain.** Display-area flux per volt: eps0 A/d.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Display-area flux per volt across the gap: C = eps0 A/d = 8.854188e-12 x 1/0.001 = 8.8542 nF.
Classical: parallel-plate capacitance (Thompson-Lampard metrology).
```
**Classical solution.** Electrostatics.
**Declared inputs.** A, d
**PREDICTED VALUE:** 8.8542 nF

### E29. Solenoid field
**Problem.** 1000 turns/m, 1 A.
**VMS chain.** Circulating closure transport: mu0 n I.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Circulating transport (Ampere closure): B = mu0 n I = 1.256637e-06 x 1000 x 1 = 1.2566 mT.
Classical: solenoid field.
```
**Classical solution.** Ampere's law.
**Declared inputs.** n, I
**PREDICTED VALUE:** 1.2566 mT

### E30. Vacuum impedance [family: constants-lock]
**Problem.** Z0=sqrt(mu0/eps0).
**VMS chain.** Front's E/B transport ratio in free space.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Front's E/B transport ratio: Z0 = sqrt(mu0/eps0) = sqrt(1.256637e-06/8.854188e-12) = 376.730 ohm.
Classical: vacuum wave impedance.
```
**Classical solution.** Wave impedance.
**Declared inputs.** locked set
**PREDICTED VALUE:** 376.730 ohm

### E31. Proton cyclotron frequency, 1 T [family: constants-lock]
**Problem.** nu=eB/2pi m_p.
**VMS chain.** Steering of the composite closure.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Composite-closure steering: nu = eB/2pi m_p = 15.245 MHz at 1 T.
Classical: cyclotron frequency.
```
**Classical solution.** Lorentz force.
**Declared inputs.** B=1 T
**PREDICTED VALUE:** 15.245 MHz

### E32. Proton NMR frequency, 1 T
**Problem.** nu=g_p mu_N B/h.
**VMS chain.** Orientation flip of the three-loop closure.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Three-loop orientation flip: nu = g_p mu_N B/h = 5.58569 x 5.05078e-27/6.62607e-34 = 42.577 MHz at 1 T.
Classical: NMR resonance.
```
**Classical solution.** NMR resonance.
**Declared inputs.** g_p (declared)
**PREDICTED VALUE:** 42.577 MHz

### E33. Faraday constant [family: constants-lock]
**Problem.** Charge per mole of closures.
**VMS chain.** N_A x e.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Charge per mole of unit closures: F = N_A e = 6.022141e+23 x 1.602177e-19 = 96485.332 C/mol.
Classical: Faraday constant.
```
**Classical solution.** Electrochemistry.
**Declared inputs.** locked set
**PREDICTED VALUE:** 96485.332 C/mol

### E34. Silver deposited per ampere-hour
**Problem.** Electrolysis mass.
**VMS chain.** One closure charge per ion: M_Ag x 3600/F.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. One closure charge per Ag+ ion: m = M_Ag x Q/F = 107.8682 x 3600/96485.33 = 4.0247 g per A h.
Classical: Faraday electrolysis.
```
**Classical solution.** Faraday's law.
**Declared inputs.** M_Ag=107.868 g/mol
**PREDICTED VALUE:** 4.0247 g

### E35. Ives-Stilwell quadratic Doppler
**Problem.** Coefficient of beta^2 in the transverse shift.
**VMS chain.** Moving closure's internal circulation slows (F0003): dlam/lam=beta^2/2 => coefficient 1/2.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Moving closure's internal circulation slows by gamma (F0003): received lam = lam0 gamma at 90 deg
   => dlam/lam = beta^2/2 + O(beta^4). Predicted quadratic coefficient = 1/2 exactly.
Classical: SR transverse Doppler.
```
**Classical solution.** SR time dilation.
**Declared inputs.** dimensionless
**PREDICTED VALUE:** 0.500 (coefficient)

### E36. Michelson-Morley fringe shift
**Problem.** Expected orientation-dependent shift.
**VMS chain.** A2: c invariant for every route - predicted shift exactly zero (classical ether: ~0.4 fringe).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. A2: c is route-invariant - the two interferometer arms accumulate identical phase for any orientation.
   Predicted fringe shift = 0 exactly. (Classical ether kinematics: ~0.4 fringe for v_E = 30 km/s.)
Classical: SR null.
```
**Classical solution.** SR: null.
**Declared inputs.** none
**PREDICTED VALUE:** 0.00 (fringes; predicted exactly 0)

### E37. Stellar aberration constant
**Problem.** Annual aberration angle.
**VMS chain.** Receiver route tilt: v_E/c.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Receiver route tilt: kappa = v_E/c = 29784.7/2.9979e+08 = 9.9351e-05 rad = 20.493 arcsec.
Classical: Bradley aberration.
```
**Classical solution.** Bradley aberration.
**Declared inputs.** v_E from M17
**PREDICTED VALUE:** 20.493 arcsec

### E38. Wettzell G ring laser Sagnac frequency
**Problem.** Earth-rotation beat of a 4x4 m HeNe ring at lat 49.144.
**VMS chain.** Counter-routes in a rotating closure pick up path difference (L3 route-phase): f=4A omega sin(lat)/(lam P).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Counter-routes around a rotating closure differ in path (L3 route-phase): beat f = 4 A om sin(L)/(lam P).
2. = 4 x 16 x 7.2921e-05 x 0.7564 / (633e-9 x 16) = 348.5 Hz.
Classical: Sagnac effect.
```
**Classical solution.** Sagnac effect.
**Declared inputs.** A=16 m^2, P=16 m, lam=633 nm
**PREDICTED VALUE:** 348.5 Hz


## Particle/Quantum

### P14. He+ ionization energy
**Problem.** Z=2 hydrogenic ground state.
**VMS chain.** Z^2 ladder with He reduced mass - no retune.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Z=2 ladder with He reduced mass: E = 4 Ry x 0.999863 = 54.415 eV - no retune.
Classical: hydrogenic QM.
```
**Classical solution.** Hydrogenic QM.
**Declared inputs.** m_alpha
**PREDICTED VALUE:** 54.415 eV

### P15. Electron wavelength at 300 keV
**Problem.** TEM relativistic wavelength.
**VMS chain.** Route-phase period with relativistic p (F0003+F0028): h c/pc.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Relativistic route momentum (F0003): pc = sqrt(T^2 + 2 T m c^2) = sqrt(9.000e+10 + 3.0660e+11) = 629761.4 eV.
2. lam = hc/pc = 1239.8420 eV nm / 629761.4 eV = 1.969 pm (F0028).
Classical: relativistic de Broglie.
```
**Classical solution.** Relativistic de Broglie.
**Declared inputs.** T=300 keV
**PREDICTED VALUE:** 1.969 pm

### P16. Alpha-decay ratio Po-210/Po-212
**Problem.** log10 of the half-life ratio from barrier actions.
**VMS chain.** Action-gap law (PM SS5, PM2 strategy): ln ratio = DeltaG, G=2pi Z1 Z2 alpha c/v - the loader's log-linear diagnostic across 13 orders of magnitude.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first (action-gap law, PM SS5 / PM2 strategy):
1. Escape rate ~ exp(-2 dS/S0); pure-Coulomb barrier action G = 2pi Z1 Z2 alpha c/v, v = c sqrt(2E/m_alpha c^2).
2. Po-210 (E=5.304 MeV): v/c = 0.05335, G = 140.95. Po-212 (E=8.785): v/c = 0.06866, G = 109.52.
3. ln(t210/t212) = G210 - G212 = 31.43 => log10 ratio = 13.65.
   The loader's ln-linear diagnostic (SS5.5) across ~13.6 orders of magnitude; prefactors cancel in the ratio.
Classical: Gamow/WKB tunneling.
```
**Classical solution.** Gamow tunneling.
**Declared inputs.** E_alpha 5.304/8.785 MeV (declared)
**PREDICTED VALUE:** 13.65 (log10 of ratio)

### P17. Franck-Hertz spacing (Hg)
**Problem.** Accelerating-voltage period.
**VMS chain.** First closure excitation of Hg at 253.65 nm: hc/lam.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. First Hg closure excitation at 253.65 nm: dE = hc/lam = 4.888 eV - the accelerating-voltage period.
Classical: quantized levels (Franck-Hertz).
```
**Classical solution.** Quantized levels.
**Declared inputs.** lam (declared)
**PREDICTED VALUE:** 4.888 eV

### P18. STM current decay per angstrom
**Problem.** Factor for phi=4.5 eV.
**VMS chain.** Escape-gap decay through the barrier (PM SS5/WKB dictionary): exp(2 kappa d), kappa=sqrt(2m phi)/hbar.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Escape-gap decay (PM SS5 = WKB dictionary): kappa = sqrt(2 m phi)/hbar = 1.0868e+10 /m for phi = 4.5 eV.
2. Current ~ exp(-2 kappa d): factor per 0.1 nm = exp(2.174) = 8.79.
Classical: STM tunneling.
```
**Classical solution.** WKB tunneling.
**Declared inputs.** phi=4.5 eV (declared)
**PREDICTED VALUE:** 8.79 (factor per 0.1 nm)

### P19. Annihilation photon energy
**Problem.** e+e- at rest -> 2 gamma.
**VMS chain.** Loop budget fully released: m_e c^2 each (F0003).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Loop budget fully released (F0003): E = m_e c^2 = 510.999 keV per photon.
Classical: annihilation energetics.
```
**Classical solution.** Mass-energy.
**Declared inputs.** locked set
**PREDICTED VALUE:** 510.999 keV

### P20. Deuteron binding energy
**Problem.** From locked masses.
**VMS chain.** Composite closure budget deficit: (m_p+m_n-m_d)c^2 (F0003 bookkeeping).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Composite closure holds less budget than its parts: B = (m_p + m_n - m_d) c^2
   = (1.672622e-27 + 1.674927e-27 - 3.343584e-27) x c^2 = 2.2246 MeV.
Classical: mass defect.
```
**Classical solution.** Mass defect.
**Declared inputs.** locked masses
**PREDICTED VALUE:** 2.2246 MeV

### P21. LHC proton speed shortfall
**Problem.** c - v at 6.5 TeV.
**VMS chain.** Dispersion (F0003): c-v ~ c/2gamma^2.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Dispersion (F0003): gamma = E/m c^2 = 6.5e12/9.3827e+08 = 6928; c - v = c/2gamma^2 = 3.123 m/s.
Classical: SR kinematics.
```
**Classical solution.** SR kinematics.
**Declared inputs.** E=6.5 TeV
**PREDICTED VALUE:** 3.123 m/s

### P22. Pair-production threshold
**Problem.** Photon -> e+e- (nuclear field).
**VMS chain.** Two loop budgets must be paid: 2 m_e c^2.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Two loop budgets must be created: E_th = 2 m_e c^2 = 1.0220 MeV (nuclear recoil negligible, declared).
Classical: QED pair threshold.
```
**Classical solution.** QED threshold.
**Declared inputs.** locked set
**PREDICTED VALUE:** 1.0220 MeV

### P23. Radiation force per watt
**Problem.** Absorbed beam.
**VMS chain.** Front momentum transport: p=E/c (F0015/F0025).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Front momentum transport (F0015/F0025): p = E/c => F/P = 1/c = 3.3356 nN/W (full absorption).
Classical: Maxwell stress / radiation pressure.
```
**Classical solution.** Maxwell stress.
**Declared inputs.** locked set
**PREDICTED VALUE:** 3.3356 nN/W

### P24. Neutron-proton mass difference
**Problem.** (m_n-m_p)c^2.
**VMS chain.** Closure budget difference from locked masses.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Closure budget difference: (m_n - m_p) c^2 = (1.674927e-27 - 1.672622e-27) c^2 = 1.2933 MeV.
Classical: mass bookkeeping.
```
**Classical solution.** Mass bookkeeping.
**Declared inputs.** locked masses
**PREDICTED VALUE:** 1.2933 MeV

### P25. Hydrogen 1S-2S frequency
**Problem.** Two-photon interval.
**VMS chain.** Ladder: (3/4)Ry(mu/m_e)/h. QED/Lamb excluded (declared).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Ladder interval: E = (3/4) Ry (mu/m_e) = 0.75 x 13.60569 x 0.999456 = 10.1987 eV.
2. nu = 2466.038 THz (QED/Lamb excluded, declared).
Classical: QM two-photon 1S-2S.
```
**Classical solution.** QM.
**Declared inputs.** m_p
**PREDICTED VALUE:** 2466.038 THz

### P26. Deuterium hyperfine frequency
**Problem.** D ground-state splitting.
**VMS chain.** Same contact coupling as E5, spin-1: nu_H x (3/2)(g_d/g_p).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Same contact coupling as E5; spin-1 deuteron gives (3/2)(g_d/g_p) x nu_H:
   nu_D = 1421.2 x 1.5 x 0.15351 = 327.2 MHz.
Classical: Fermi contact, I=1.
```
**Classical solution.** Fermi contact, I=1.
**Declared inputs.** g_d (declared)
**PREDICTED VALUE:** 327.2 MHz

### P27. Muonium hyperfine frequency
**Problem.** mu+e- ground splitting.
**VMS chain.** Contact coupling with muon loop: nu_H x (mu_mu/mu_p) x reduced-mass^3 ratio.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Muon loop replaces proton loop: nu = nu_H x (mu_mu/mu_p) x [(1+m_e/m_p)/(1+m_e/m_mu)]^3
   = 1421.2 x 3.183345 x 0.98724 = 4466.3 MHz.
Classical: Fermi contact for muonium.
```
**Classical solution.** Fermi contact.
**Declared inputs.** mu_mu/mu_p=3.183345 (declared)
**PREDICTED VALUE:** 4466.3 MHz

### P28. Sn-120 binding energy per nucleon [family: SEMF]
**Problem.** Mid-mass check.
**VMS chain.** Same budgets, no retune (PM SS6).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Same budgets, Sn-120 (Z=50, N=70): B = 1018.6 MeV => B/A = 8.488 MeV. No retune.
Classical: SEMF.
```
**Classical solution.** SEMF.
**Declared inputs.** same set
**PREDICTED VALUE:** 8.488 MeV

### P29. U-238 binding energy per nucleon [family: SEMF]
**Problem.** Heavy-mass check.
**VMS chain.** Same budgets, no retune.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Same budgets, U-238 (Z=92, N=146): B = 1804.2 MeV => B/A = 7.581 MeV. No retune.
Classical: SEMF.
```
**Classical solution.** SEMF.
**Declared inputs.** same set
**PREDICTED VALUE:** 7.581 MeV

### P30. Most stable Z at A=100 [family: SEMF]
**Problem.** Valley of stability.
**VMS chain.** Budget minimum in Z: Z*=(A/2)/(1+(aC/4aA)A^{2/3}).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Budget minimum in Z at fixed A (dB/dZ = 0): Z* = (A/2)/(1 + (aC/4aA) A^(2/3))
   = 50/(1 + 0.00750 x 21.54) = 43.0.
Classical: SEMF valley of stability.
```
**Classical solution.** SEMF stationarity.
**Declared inputs.** same set
**PREDICTED VALUE:** 43.0 (Z)

### P31. D-T fusion energy release
**Problem.** Q value from masses.
**VMS chain.** Budget difference of closures: (m_D+m_T-m_He4-m_n)c^2.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Budget difference of closures: Q = (m_D + m_T - m_He4 - m_n) c^2
   = (2.0141017781+3.0160492779-4.0026032545) u c^2 - m_n c^2 + ... = 17.589 MeV.
Classical: mass defect (D+T -> He-4 + n).
```
**Classical solution.** Mass defect.
**Declared inputs.** AME masses (declared)
**PREDICTED VALUE:** 17.589 MeV

### P32. Compton edge of Cs-137
**Problem.** 662 keV backscatter edge.
**VMS chain.** Maximum front-closure energy transfer: 2E^2/(mc^2+2E).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Maximum front-closure transfer at backscatter: T_max = 2E^2/(m_e c^2 + 2E)
   = 2 x 661.66^2/(510.999 + 1323.32) = 477.3 keV.
Classical: Compton edge.
```
**Classical solution.** Compton kinematics.
**Declared inputs.** E=661.66 keV
**PREDICTED VALUE:** 477.3 keV

### P33. Duane-Hunt limit at 30 kV
**Problem.** Shortest X-ray wavelength.
**VMS chain.** Full route budget into one quantum: hc/eV.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Entire route budget into one quantum: lam_min = hc/eV = 1239.8420 eV nm/30000 eV = 41.33 pm.
Classical: Duane-Hunt limit.
```
**Classical solution.** Duane-Hunt law.
**Declared inputs.** V=30 kV
**PREDICTED VALUE:** 41.33 pm

### P34. Speed of a 1 MeV electron
**Problem.** beta at T=1 MeV.
**VMS chain.** Dispersion (F0003): gamma=1+T/mc^2.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. gamma = 1 + T/m c^2 = 1 + 1000/511.0 = 2.9569; beta = sqrt(1-1/gamma^2) = 0.9411 (F0003).
Classical: SR kinematics.
```
**Classical solution.** SR kinematics.
**Declared inputs.** T=1 MeV
**PREDICTED VALUE:** 0.9411 (beta)

### P35. C60 de Broglie wavelength
**Problem.** 720 u at 210 m/s.
**VMS chain.** Route-phase period of a 60-atom composite closure: h/mv - same rule, x10^6 the mass.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Same route-phase rule at 10^6 x the electron mass: lam = h/mv = 6.6261e-34/(1.1967e-24 x 210) = 2.64 pm.
Classical: de Broglie for C60 (matter-wave interference).
```
**Classical solution.** de Broglie.
**Declared inputs.** m=720.66 u, v=210 m/s (declared)
**PREDICTED VALUE:** 2.64 pm

### P36. He+ 2p fine structure
**Problem.** Z^4-scaled splitting.
**VMS chain.** F0020 x Z^4: 16 x (Ry alpha^2/16).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. F0020 scales as Z^4: split = 16 x Ry alpha^2/16 = Ry alpha^2 = 175.19 GHz.
Classical: Dirac fine-structure scaling.
```
**Classical solution.** Dirac FS scaling.
**Declared inputs.** Z=2
**PREDICTED VALUE:** 175.19 GHz

### P37. Nuclear radius ratio Pb-208/Fe-56 [family: SEMF]
**Problem.** (208/56)^{1/3}.
**VMS chain.** Volume budget ~ A (PM SS6): R ~ A^{1/3}; diffuse-surface caveat declared.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Volume budget ~ A (PM SS6): R ~ A^(1/3) => R(Pb)/R(Fe) = (208/56)^(1/3) = 1.549.
   (Diffuse-surface caveat declared: rms radii feel the surface term.)
Classical: liquid-drop radius law.
```
**Classical solution.** Liquid-drop radius law.
**Declared inputs.** A values
**PREDICTED VALUE:** 1.549 (ratio)

### P38. Ra-226 alpha-decay Q value
**Problem.** Energy release.
**VMS chain.** Budget difference: (m_Ra-m_Rn-m_He)c^2.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Q = (m_Ra - m_Rn - m_He) c^2 = (226.0254103 - 222.0175763 - 4.0026032545) x 931.494 MeV/u = 4.872 MeV.
Classical: alpha-decay energetics.
```
**Classical solution.** Mass defect.
**Declared inputs.** AME masses (declared)
**PREDICTED VALUE:** 4.872 MeV


## Thermodynamics

### T13. Maxwell peak/rms speed ratio
**Problem.** v_p/v_rms.
**VMS chain.** Mode-occupancy maximum vs second moment: sqrt(2/3).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Mode-occupancy maximum vs second moment: v_p = sqrt(2RT/M), v_rms = sqrt(3RT/M) => ratio sqrt(2/3) = 0.8165.
Classical: Maxwell-Boltzmann.
```
**Classical solution.** Maxwell-Boltzmann.
**Declared inputs.** none
**PREDICTED VALUE:** 0.8165 (ratio)

### T14. Atmospheric scale height
**Problem.** RT/Mg at 288 K.
**VMS chain.** Route-count vs profile depth balance (F0009 + M-profile).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Route count vs profile depth (F0009 + M-profile): H = RT/Mg = 8.3145 x 288.15/(0.0289645 x 9.80665) = 8.43 km.
Classical: barometric formula.
```
**Classical solution.** Barometric formula.
**Declared inputs.** T=288.15 K
**PREDICTED VALUE:** 8.43 km

### T15. Speed of sound in helium
**Problem.** At 293.15 K.
**VMS chain.** Monatomic route ensemble: gamma=5/3.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Monatomic ensemble gamma = 5/3: c_s = sqrt((5/3) R T/M_He) = 1007 m/s.
Classical: Laplace sound speed.
```
**Classical solution.** Laplace formula.
**Declared inputs.** M_He
**PREDICTED VALUE:** 1007 m/s

### T16. Speed of sound in water
**Problem.** From bulk modulus.
**VMS chain.** Compression-front speed: sqrt(K/rho).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Compression front in the liquid: c = sqrt(K/rho) = sqrt(2.210e+09/998.0) = 1488 m/s.
Classical: Newton-Laplace.
```
**Classical solution.** Newton-Laplace.
**Declared inputs.** K=2.21 GPa, rho=998 (declared)
**PREDICTED VALUE:** 1488 m/s

### T17. Monatomic gas kappa/(eta c_v)
**Problem.** Kinetic-theory ratio for Ar.
**VMS chain.** Same route carriers transport momentum and energy: ratio 5/2 exactly.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. The same route carriers transport momentum and energy; Chapman-Enskog closure gives
   kappa = (5/2) eta c_v exactly for a monatomic gas => ratio = 2.50.
Classical: kinetic theory.
```
**Classical solution.** Chapman-Enskog.
**Declared inputs.** none
**PREDICTED VALUE:** 2.50 (ratio)

### T18. Graham effusion factor U-235/238 F6
**Problem.** Single-stage enrichment.
**VMS chain.** Route flux ~ 1/sqrt(M): sqrt(352.04/349.03).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Route flux ~ 1/sqrt(M): factor = sqrt(352.04/349.03) = 1.00430 per stage.
Classical: Graham's law.
```
**Classical solution.** Graham's law.
**Declared inputs.** molar masses
**PREDICTED VALUE:** 1.00430 (factor)

### T19. Van't Hoff osmotic pressure
**Problem.** 1 mM ideal solute, 298 K.
**VMS chain.** Solute route-count pressure: cRT.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Solute route-count pressure: Pi = cRT = 1 x 8.3145 x 298.15 = 2479.0 Pa = 2.479 kPa.
Classical: van't Hoff.
```
**Classical solution.** Van't Hoff.
**Declared inputs.** c=1 mol/m^3
**PREDICTED VALUE:** 2.479 kPa

### T20. Cryoscopic constant of water
**Problem.** K_f.
**VMS chain.** Phase route-count balance: RT0^2 M/DH_fus.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Phase balance shift per solute closure: K_f = R T0^2 M/DH_fus = 8.3145 x 74610.9 x 0.0180153/6009.5 = 1.860 K kg/mol.
Classical: cryoscopy.
```
**Classical solution.** Raoult/Clausius.
**Declared inputs.** DH_fus (declared)
**PREDICTED VALUE:** 1.860 K kg/mol

### T21. Ebullioscopic constant of water
**Problem.** K_b.
**VMS chain.** Same balance at the boiling point: RT_b^2 M/DH_vap.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Same at the boiling point: K_b = R T_b^2 M/DH_vap = 0.5129 K kg/mol.
Classical: ebullioscopy.
```
**Classical solution.** Raoult.
**Declared inputs.** DH_vap (declared)
**PREDICTED VALUE:** 0.5129 K kg/mol

### T22. Human radiative output
**Problem.** Skin 306 K vs room 293 K, A=1.8, eps=0.98.
**VMS chain.** sigma T^4 exchange (F0031).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. sigma T^4 exchange (F0031): P = eps A sigma (T_s^4 - T_a^4) = 0.98 x 1.8 x 5.6704e-08 x (8.7849e+09 - 7.3852e+09) = 140 W.
Classical: Stefan-Boltzmann exchange.
```
**Classical solution.** Stefan-Boltzmann exchange.
**Declared inputs.** declared geometry
**PREDICTED VALUE:** 140 W

### T23. CMB energy density [family: CMB]
**Problem.** aT^4 at 2.7255 K.
**VMS chain.** Mode-count energy of the background: 4 sigma T^4/c.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Mode-count energy density: u = (4 sigma/c) T^4 = 7.56573e-16 x 55.180 = 4.1748e-14 J/m^3.
Classical: radiation constant.
```
**Classical solution.** Radiation constant.
**Declared inputs.** T (FIRAS)
**PREDICTED VALUE:** 4.175 x10^-14 J/m^3

### T24. CMB photon number density [family: CMB]
**Problem.** Photons per cm^3.
**VMS chain.** Occupancy integral with zeta(3): (2 zeta3/pi^2)(kT/hbar c)^3.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Occupancy integral with zeta(3): n = (2 zeta3/pi^2)(kT/hbar c)^3 = 0.24359 x (1190.23)^3 = 410.7 /cm^3.
Classical: Bose photon counting.
```
**Classical solution.** Bose statistics.
**Declared inputs.** T (FIRAS)
**PREDICTED VALUE:** 410.7 per cm^3

### T25. Diesel compression temperature
**Problem.** 20:1 from 300 K, ideal air.
**VMS chain.** Adiabatic route-count compression: T V^{gamma-1} const.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Adiabatic route-count compression: T2 = T1 r^(gamma-1) = 300 x 20^0.4 = 994 K.
Classical: adiabatic law.
```
**Classical solution.** Adiabatic law.
**Declared inputs.** gamma=1.40
**PREDICTED VALUE:** 994 K

### T26. Adiabatic index of argon
**Problem.** Monatomic gamma.
**VMS chain.** 3 translational route modes: 5/3.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. 3 translational route modes: gamma = (3/2+1)/(3/2) = 5/3 = 1.6667.
Classical: equipartition.
```
**Classical solution.** Equipartition.
**Declared inputs.** none
**PREDICTED VALUE:** 1.6667 (gamma)

### T27. Adiabatic index of N2
**Problem.** Diatomic gamma at 300 K.
**VMS chain.** 5 active modes (vibration frozen, declared): 7/5.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. 5 active modes (vibration frozen, declared): gamma = 7/5 = 1.4000.
Classical: equipartition.
```
**Classical solution.** Equipartition.
**Declared inputs.** none
**PREDICTED VALUE:** 1.4000 (gamma)

### T28. Mercury barometer height
**Problem.** 1 atm column.
**VMS chain.** Pressure = profile weight: P/(rho g).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Pressure = column weight: h = P/(rho g) = 101325/(13595.1 x 9.80665) = 760.0 mm.
Classical: hydrostatics.
```
**Classical solution.** Hydrostatics.
**Declared inputs.** rho_Hg (declared)
**PREDICTED VALUE:** 760.0 mm

### T29. Entropy of vaporization, benzene
**Problem.** At its boiling point.
**VMS chain.** Route-count jump per closure: DH/T_b.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Route-count jump per closure across the transition: dS = DH/T_b = 30720.0/353.25 = 87.0 J/(mol K).
Classical: entropy of vaporization (Trouton-scale).
```
**Classical solution.** Trouton-scale entropy.
**Declared inputs.** DH, T_b (declared)
**PREDICTED VALUE:** 87.0 J/(mol K)

### T30. Gas expansivity at 0 C
**Problem.** (1/V)(dV/dT) at constant P.
**VMS chain.** Route-count linear in T: 1/273.15.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Route count linear in T: (1/V)(dV/dT)_P = 1/T = 1/273.15 = 3.6610 x10^-3 /K.
Classical: Gay-Lussac.
```
**Classical solution.** Gay-Lussac.
**Declared inputs.** T=273.15 K
**PREDICTED VALUE:** 3.6610 x10^-3 /K

### T31. CMB spectral peak (frequency) [family: CMB]
**Problem.** nu_max at 2.7255 K.
**VMS chain.** Frequency-form Wien: 2.8214 kT/h.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Frequency-form occupancy maximum (x = 2.8214): nu_max = 2.8214 kT/h = 160.2 GHz.
Classical: Wien (frequency form).
```
**Classical solution.** Wien (frequency).
**Declared inputs.** T (FIRAS)
**PREDICTED VALUE:** 160.2 GHz

### T32. Solar luminosity [family: solar-T]
**Problem.** 4 pi R^2 sigma T^4.
**VMS chain.** Total display-area flux of the solar surface.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Total display-area flux: L = 4pi R_s^2 sigma T^4 = 6.0821e+18 x 6.2939e+07 = 3.8280e+26 W.
Classical: Stefan-Boltzmann luminosity.
```
**Classical solution.** Stefan-Boltzmann.
**Declared inputs.** R_sun, T_eff
**PREDICTED VALUE:** 3.828 x10^26 W

### T33. Loschmidt number
**Problem.** n at 0 C, 1 atm.
**VMS chain.** Route density P/kT.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Route density: n = P/kT = 101325/(1.38065e-23 x 273.15) = 2.6868e+25 /m^3.
Classical: ideal gas.
```
**Classical solution.** Ideal gas.
**Declared inputs.** T, P
**PREDICTED VALUE:** 2.6868 x10^25 /m^3

### T34. AFM cantilever thermal amplitude
**Problem.** k=0.1 N/m at 300 K.
**VMS chain.** Equipartition per mode: sqrt(kT/k) (F0016/F0029 SS5).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Equipartition per mode (F0029 SS5): (1/2)k<x^2> = (1/2)kB T => x_rms = sqrt(kB T/k) = sqrt(4.142e-21/0.1) = 0.204 nm.
Classical: thermal noise of a harmonic mode.
```
**Classical solution.** Equipartition noise.
**Declared inputs.** k (declared)
**PREDICTED VALUE:** 0.204 nm

### T35. Speed of sound in argon
**Problem.** At 293.15 K.
**VMS chain.** Monatomic ensemble, heavier closure: sqrt(5RT/3M).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Monatomic, heavier closure: c_s = sqrt((5/3) R T/M_Ar) = 319 m/s.
Classical: Laplace sound speed.
```
**Classical solution.** Laplace.
**Declared inputs.** M_Ar
**PREDICTED VALUE:** 319 m/s

### T36. Molar C_v of argon
**Problem.** Monatomic constant-volume capacity.
**VMS chain.** 3 modes x R/2: 12.47.

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. 3 modes x R/2: C_v = (3/2)R = 12.47 J/(mol K).
Classical: equipartition.
```
**Classical solution.** Equipartition.
**Declared inputs.** none
**PREDICTED VALUE:** 12.47 J/(mol K)

### T37. Water vapor pressure at 25 C
**Problem.** Integrated Clausius-Clapeyron from the 100 C anchor.
**VMS chain.** Phase-slope law integrated with mean L (declared constant-L approximation; expect %-level band).

**Worked derivation (VMS-first, numbers substituted):**
```
VMS-first:
1. Integrate the phase-slope law with mean L (declared constant-L approx):
   ln(P/P0) = -(L M/R)(1/T - 1/T0) = -5156.8 x (3.354016e-03 - 2.680074e-03) = -3.4754.
2. P = 101.325 x exp(...) = 3.136 kPa (%-level band declared).
Classical: integrated Clausius-Clapeyron.
```
**Classical solution.** Clausius-Clapeyron integral.
**Declared inputs.** L_mean=2.38e6 J/kg (declared)
**PREDICTED VALUE:** 3.136 kPa