diff --git a/crates/geopotential/src/coefficients.rs b/crates/geopotential/src/coefficients.rs
index 9d85245..9a522b0 100644
--- a/crates/geopotential/src/coefficients.rs
+++ b/crates/geopotential/src/coefficients.rs
@@ -9,7 +9,11 @@ pub enum TideSystem {
     TideFree,
     /// Includes the permanent deformation (indirect) part, not the direct tidal potential.
     ZeroTide,
-    /// Includes both the direct and the indirect permanent parts.
+    /// Includes both the direct and the indirect permanent parts. The direct part, the tide-generating
+    /// potential itself, then sits in the Earth's own field, falling as `r⁻³` where it grows as `r²`:
+    /// right on the sphere `r = R` only ("there is no mean-tide C̄20", Mäkinen 2021, J. Geod. 95,
+    /// 106). Its gravity is +45.7 µGal at the equator against the true −30.5; take mean-tide gravity
+    /// from `gravimetry::tides::permanent`.
     MeanTide,
     /// Not stated; cannot be converted.
     Unknown,
@@ -17,8 +21,9 @@ pub enum TideSystem {
 
 impl TideSystem {
     /// `C̄20` in this system minus `C̄20` tide-free. Uses the EGM96/EGM2008 convention
-    /// `ΔC̄20 = −3.11080e-8·k/√5` with `k = 0.3` (Rapp 1989), which gives the IERS 2010 value
-    /// −4.1736e-9 for zero-tide minus tide-free.
+    /// `ΔC̄20 = −3.11080e-8·k/√5` with `k = 0.3` (Rapp 1989), which gives −4.1736e-9 for zero-tide
+    /// minus tide-free, the value the IERS Conventions 2010 quote for EGM2008 (their own
+    /// `A₀H₀k₂₀`, Eq. 6.14 with `k₂₀ = 0.30190`, gives −4.2007e-9).
     fn c20_offset(self) -> Option<f64> {
         let direct = -3.11080e-8 / 5f64.sqrt();
         let k = 0.3;
diff --git a/crates/gravimetry/README.md b/crates/gravimetry/README.md
index 55ed73e..3866134 100644
--- a/crates/gravimetry/README.md
+++ b/crates/gravimetry/README.md
@@ -133,6 +133,7 @@ maximum likelihood over many series at once.
 | `atom` | Phase, shot noise, and accelerometer and gradiometer noise of a cold-atom sensor, in orbit (double diffraction) or on the ground (single), its phase sensitivity and the gravitational Aharonov–Bohm rate ([Lévèque et al. 2021](https://arxiv.org/abs/2011.03382); [Stray et al. 2022](https://doi.org/10.1038/s41586-021-04315-3); [CARIOQA](https://arxiv.org/abs/2211.01215); [Trimeche et al. 2019](https://doi.org/10.1088/1361-6382/ab4548)) |
 | `gradiometry` | The error of each coefficient a gradiometer mission like GOCE achieves |
 | `local` | Gravity and gravity gradient of a buried sphere and horizontal cylinder, the shapes a ground survey fits, and the averaging time an instrument needs to see them |
+| `tides` | The lunar and solar tide at a gravimeter, the gravimetric and diminishing factors of the Love numbers, and the permanent tide that separates mean-tide, zero-tide and tide-free gravity and geoids ([Longman 1959](https://doi.org/10.1029/JZ064i012p02351), as [LongmanTide](https://github.com/jrleeman/LongmanTide) codes it; [Mäkinen 2021](https://doi.org/10.1007/s00190-021-01541-5)) |
 | `sim` | A closed-loop simulation of a GRACE-type pair: simulate, estimate the field back, and compare its errors with the predicted ones |
 | `noise` | Noise with any spectrum, its whitening, and the FFT |
 | `technotes` | NASA's TN-14 (C̄20 and C̄30 from laser ranging, [Loomis et al. 2020](https://doi.org/10.1029/2019GL085488)) and TN-13 (degree 1, [Sun et al. 2016](https://doi.org/10.1002/2016JB013073)), from [PO.DAAC](https://archive.podaac.earthdata.nasa.gov/podaac-ops-cumulus-docs/gracefo/open/docs/) |
@@ -161,6 +162,8 @@ errors of a real month.*
   files; `prompt_lead` gives how far ahead of the P-wave a light-speed gravity signal arrives.
 - **Site surveys and monitoring.** `local` gives the gravity and gradient of a buried void or
   dense body, and the averaging time a quantum gravimeter needs to resolve it.
+- **Absolute gravimetry.** `tides` corrects a reading for the lunar and solar tide and moves it
+  between the zero-tide convention of gravity and the mean-tide convention of the IHRS.
 - **Mission design.** `missions`, `atom` and `sim` size the sensors and simulate what field a
   satellite pair recovers, including cold-atom instruments.
 
@@ -179,6 +182,9 @@ errors of a real month.*
   equivalent mass sums to zero); on the 2025 Kamchatka fault model, its uplift matches the
   USGS's own surface displacement with r = 0.89.
 - **Point masses:** their potential matches Newton's law.
+- **Tides:** LongmanTide's test vector to 3×10⁻¹¹ mGal; averaged over the 18.61-year nodal cycle,
+  Longman's tide is δ₂ times the permanent tide to 0.2 µGal; the permanent tide matches Mäkinen's
+  (2021) Eqs. 19–21 to their printed digits.
 - **Kalman smoother:** it matches the exact Gaussian-process solution to 1e-10, its fit
   recovers the settings of simulated data, and on the real record withheld months are predicted
   with calibrated errors.
diff --git a/crates/gravimetry/examples/buried_tunnel.rs b/crates/gravimetry/examples/buried_tunnel.rs
index 1c4bf6b..da5571e 100644
--- a/crates/gravimetry/examples/buried_tunnel.rs
+++ b/crates/gravimetry/examples/buried_tunnel.rs
@@ -1,30 +1,51 @@
 //! Finding a void from the surface: the gravity and gradient of a buried tunnel, and how long a
-//! quantum gravimeter must average to see it. The geometry is the Birmingham survey's
-//! ([Stray et al. 2022](https://doi.org/10.1038/s41586-021-04315-3)), which found a tunnel at
-//! about 1.9 m depth.
+//! quantum gravimeter or gradiometer must average to see it. The geometry and the gradiometer are
+//! the Birmingham survey's ([Stray et al. 2022](https://doi.org/10.1038/s41586-021-04315-3)),
+//! which found a tunnel at about 1.9 m depth.
 //!
 //! ```text
 //! cargo run -p gravimetry --example buried_tunnel
 //! ```
 
+use gravimetry::atom::AtomInterferometer;
 use gravimetry::local::{EOTVOS, MICROGAL, averaging_time, cylinder};
 
 fn main() {
     let (radius, depth, contrast) = (1.0, 1.9, -2000.0); // an air-filled tunnel in soil
-    println!("{:>8} {:>12} {:>12}", "offset/m", "Δg/µGal", "Γuu/E");
+
+    // Stray et al.'s gradiometer: two single-diffraction interferometers 1 m apart vertically,
+    // T = 85 ms, 10⁵ atoms at 0.7 Hz. It reads the difference in gravity between its sensors.
+    let stray = AtomInterferometer { t: 0.085, contrast: 1.0, atoms: 1e5, cycle: 1.0 / 0.7, diffraction: 1 };
+    let baseline = 1.0;
+    let difference = |offset: f64| (cylinder(radius, depth, contrast, offset).0 - cylinder(radius, depth + baseline, contrast, offset).0) / baseline;
+
+    println!("{:>8} {:>10} {:>10} {:>12} {:>12}", "offset/m", "Δg/µGal", "Γuu/E", "Δg/1 m in E", "phase/mrad");
     for offset in [0.0, 1.0, 2.0, 3.0, 5.0] {
         let (gz, gradient) = cylinder(radius, depth, contrast, offset);
-        println!("{offset:>8.1} {:>12.1} {:>12.1}", gz / MICROGAL, gradient / EOTVOS);
+        let d = difference(offset);
+        println!("{offset:>8.1} {:>10.1} {:>10.1} {:>12.1} {:>12.2}", gz / MICROGAL, gradient / EOTVOS, d / EOTVOS, stray.phase(d * baseline) * 1e3);
     }
     let (peak, _) = cylinder(radius, depth, contrast, 0.0);
     println!("peak anomaly {:.0} µGal", peak / MICROGAL);
 
     // An Exail-class gravimeter at 500 nm/s²/√Hz needs this long per station for a 10:1 signal.
     let target = peak.abs() / 10.0;
-    println!("to resolve it 10:1 ({:.0} nm/s²): {:.0} s per station", target * 1e9, averaging_time(500e-9, target));
+    println!("gravimeter, to resolve it 10:1 ({:.0} nm/s²): {:.0} s per station", target * 1e9, averaging_time(500e-9, target));
+
+    // The gradiometer at its projection limit, and at the 466 E/√Hz it reached outdoors.
+    let target = difference(0.0).abs() / 10.0;
+    for (what, asd) in [("projection limit", stray.gradient_asd(baseline)), ("outdoors", 466.0 * EOTVOS)] {
+        println!("gradiometer, {what} {:.0} E/√Hz: {:.0} s per station", asd / EOTVOS, averaging_time(asd, target));
+    }
+
     assert!((peak / MICROGAL + 44.1).abs() < 0.1);
     // The gradient changes sign away from the tunnel, which is what localises it.
     assert!(cylinder(radius, depth, contrast, 0.0).1 < 0.0 && cylinder(radius, depth, contrast, 3.0).1 > 0.0);
+    // Over a line mass the difference across the baseline is the point gradient at the geometric
+    // mean of the two depths, 2Gλ/(z₁z₂): about Stray et al.'s 150 E and 17.5 mrad.
+    let mid = cylinder(radius, (depth * (depth + baseline)).sqrt(), contrast, 0.0).1;
+    assert!((difference(0.0) / mid - 1.0).abs() < 1e-12);
+    assert!((difference(0.0) / EOTVOS + 152.2).abs() < 0.1 && (stray.phase(difference(0.0)) * 1e3 + 17.7).abs() < 0.1);
 }
 
 #[test]
diff --git a/crates/gravimetry/src/lib.rs b/crates/gravimetry/src/lib.rs
index 321a55f..967ec18 100644
--- a/crates/gravimetry/src/lib.rs
+++ b/crates/gravimetry/src/lib.rs
@@ -12,6 +12,7 @@ pub mod missions;
 pub mod noise;
 pub mod sim;
 pub mod technotes;
+pub mod tides;
 
 /// Area of the world ocean, m²: 3.618×10⁸ km², so 361.8 Gt of water raise sea level 1 mm.
 pub const OCEAN_AREA: f64 = 3.618e14;
diff --git a/crates/gravimetry/src/tides.rs b/crates/gravimetry/src/tides.rs
new file mode 100644
index 0000000..08e7110
--- /dev/null
+++ b/crates/gravimetry/src/tides.rs
@@ -0,0 +1,224 @@
+//! The Earth tide at a gravimeter: Longman's lunar and solar tide, the Love-number factors that
+//! scale a rigid Earth's tide to the observed one, and the permanent tide that separates the
+//! mean-tide, zero-tide and tide-free conventions.
+//!
+//! ```
+//! use geopotential::julian_epoch;
+//! use gravimetry::tides::{delta, longman, permanent};
+//!
+//! // State College, Pennsylvania, at 2015-04-23 00:00 UT: 3.5 µGal of lunar and solar tide
+//! // (LongmanTide's test vector), with Longman's own Love numbers.
+//! let (lat, lon) = (40.7914f64.to_radians(), 282.1414f64.to_radians());
+//! let (moon, sun, total) = longman(lat, lon, 370.0, julian_epoch(2015, 4, 23, 0, 0));
+//! assert!((total / 1e-8 - 3.5347).abs() < 1e-4 && (moon + sun - total).abs() < 1e-18);
+//! assert!((delta(2, 0.612, 0.303) - 1.1575).abs() < 1e-15);
+//! // The permanent tide lowers gravity 30.49 µGal at the equator (Mäkinen 2021, Eq. 20).
+//! assert!((permanent(0.0, 0.0).1 / 1e-8 + 30.49).abs() < 0.01);
+//! ```
+
+use core::f64::consts::TAU;
+
+use geopotential::geocentric;
+use metrology::WGS84_A;
+
+/// The time average of the tide-generating potential in Mäkinen's (2021, Eq. 18) form
+/// `W_T = A(r/a)²(sin²ψ − 1/3)`, m²/s²: −2.9166, from the IERS Conventions 2010, which Mäkinen
+/// proposes for the International Height Reference Frame. Spectral expansions from Doodson's to
+/// Kudryavtsev's agree to 0.0002 (his Table 1).
+pub const PERMANENT_A: f64 = -2.9166;
+
+/// The gravimetric factor `δₙ = 1 + (2/n)hₙ − ((n+1)/n)kₙ` of degree `n` for Love numbers `h`
+/// and `k`: the observed tide in gravity over a rigid Earth's. It sums the direct pull `nW/r` of
+/// a tide potential `W ∝ rⁿ`, the free-air change `2hW/r` of the surface lifted `hW/g`, and the
+/// pull `−(n + 1)kW/r` of the Earth's own redistributed mass. With the IERS Conventions' nominal
+/// `h₂ = 0.6078` and `k₂₀ = 0.30190` it is 1.155, the conventional 1.16.
+pub fn delta(n: u32, h: f64, k: f64) -> f64 {
+    let n = f64::from(n);
+    1.0 + 2.0 * h / n - (n + 1.0) * k / n
+}
+
+/// The diminishing factor `γ = 1 + k − h` for Love numbers `h` and `k` of one degree: the
+/// observed tide in tilt, or in a potential difference measured between points that rise with
+/// the ground, over a rigid Earth's. About 0.69 at degree 2 (Mäkinen 2021, Eq. 52).
+pub fn gamma(h: f64, k: f64) -> f64 {
+    1.0 + k - h
+}
+
+/// The lunar and solar tide in gravity at geodetic latitude `lat` and east longitude `lon` (rad),
+/// height `h` (m) and Julian epoch `epoch` (UT, as from [`geopotential::julian_epoch`]), m/s²:
+/// `(moon, sun, total)`, positive upward. A tide that pulls up lowers gravity, so it is also the
+/// correction to add to a reading.
+///
+/// [Longman's (1959)](https://doi.org/10.1029/JZ064i012p02351) closed form from the mean
+/// orbital elements: the lunar tide to degree 3, the solar to degree 2, both times
+/// [`delta`]`(2, 0.612, 0.303)` = 1.1575, with Longman's own cgs constants and Love numbers, as
+/// [LongmanTide](https://github.com/jrleeman/LongmanTide) (MIT) codes it. Its masses make the
+/// lunar tide 0.09% and the solar 0.2% larger than today's `GM`s, and its degree-2 factor also
+/// scales the degree-3 term, whose own `δ₃` is 1.07. Averaged over the 18.61-year nodal cycle it
+/// is `δ₂` times the permanent tide of [`permanent`], to 0.2 µGal.
+pub fn longman(lat: f64, lon: f64, h: f64, epoch: f64) -> (f64, f64, f64) {
+    // Longman's constants, cgs: G, the lunar and solar masses, the lunar eccentricity, the ratio
+    // of the mean motions of the sun and moon, the mean distances of the moon and sun, the
+    // Earth's equatorial radius, the lunar orbit's inclination to the ecliptic and the obliquity.
+    let (mu, moon, sun) = (6.673e-8, 7.3537e25, 1.993e33);
+    let (e, m, c, c1, a) = (0.05490, 0.074804, 3.84402e10, 1.495e13, 6.378270e8);
+    let (i, omega): (f64, f64) = (0.08979719, 23.452f64.to_radians());
+    // Julian centuries from 1899 December 31, 12h, J2000.0 less one century, and the hour angle
+    // of the mean sun west of the station, from the days since J2000.0, 12h.
+    let tc = (epoch - 2000.0) / 100.0 + 1.0;
+    let day = (epoch - 2000.0) * 365.25;
+    let t = TAU * (day - day.floor()) + lon;
+    let (tc2, tc3) = (tc * tc, tc * tc * tc);
+
+    // The moon: mean longitude, perigee, the sun's mean longitude and the ascending node.
+    let s = 4.72000889397 + 8399.70927456 * tc + 3.45575191895e-05 * tc2 + 3.49065850399e-08 * tc3;
+    let p = 5.83515162814 + 71.0180412089 * tc + 0.000180108282532 * tc2 + 1.74532925199e-07 * tc3;
+    let hs = 4.88162798259 + 628.331950894 * tc + 5.23598775598e-06 * tc2;
+    let node = 4.52360161181 - 33.757146295 * tc + 3.6264063347e-05 * tc2 + 3.39369576777e-08 * tc3;
+    let incl = (omega.cos() * i.cos() - omega.sin() * i.sin() * node.cos()).acos();
+    let nu = (i.sin() * node.sin() / incl.sin()).asin();
+    let chi = t + hs - nu;
+    let (cos_alpha, sin_alpha) = (node.cos() * nu.cos() + node.sin() * nu.sin() * omega.cos(), omega.sin() * node.sin() / incl.sin());
+    let sigma = s - (node - 2.0 * (sin_alpha / (1.0 + cos_alpha)).atan());
+    let l = sigma + 2.0 * e * (s - p).sin() + 1.25 * e * e * (2.0 * (s - p)).sin()
+        + 3.75 * m * e * (s - 2.0 * hs + p).sin() + 11.0 / 8.0 * m * m * (2.0 * (s - hs)).sin();
+
+    // The sun: perigee, the Earth's eccentricity and the sun's longitude.
+    let p1 = 4.90822941839 + 0.0300025492114 * tc + 7.85398163397e-06 * tc2 + 5.3329504922e-08 * tc3;
+    let e1 = 0.01675104 - 0.00004180 * tc - 0.000000126 * tc2;
+    let chi1 = t + hs;
+    let l1 = hs + 2.0 * e1 * (hs - p1).sin();
+
+    // Zenith angles, the station's radius and the two distances.
+    let (sl, cl) = lat.sin_cos();
+    let cos_theta = sl * incl.sin() * l.sin() + cl * ((0.5 * incl).cos().powi(2) * (l - chi).cos() + (0.5 * incl).sin().powi(2) * (l + chi).cos());
+    let cos_phi = sl * omega.sin() * l1.sin() + cl * ((0.5 * omega).cos().powi(2) * (l1 - chi1).cos() + (0.5 * omega).sin().powi(2) * (l1 + chi1).cos());
+    let r = (1.0 / (1.0 + 0.006738 * sl * sl)).sqrt() * a + h * 100.0;
+    let (ap, ap1) = (1.0 / (c * (1.0 - e * e)), 1.0 / (c1 * (1.0 - e1 * e1)));
+    let d = 1.0 / (1.0 / c + ap * e * (s - p).cos() + ap * e * e * (2.0 * (s - p)).cos()
+        + 15.0 / 8.0 * ap * m * e * (s - 2.0 * hs + p).cos() + ap * m * m * (2.0 * (s - hs)).cos());
+    let big_d = 1.0 / (1.0 / c1 + ap1 * e1 * (hs - p1).cos());
+
+    let gm = mu * moon * r / (d * d * d) * (3.0 * cos_theta * cos_theta - 1.0)
+        + 1.5 * mu * moon * r * r / (d * d * d * d) * (5.0 * cos_theta.powi(3) - 3.0 * cos_theta);
+    let gs = mu * sun * r / (big_d * big_d * big_d) * (3.0 * cos_phi * cos_phi - 1.0);
+    // Gal to m/s², times the gravimetric factor.
+    let k = 1e-2 * delta(2, 0.612, 0.303);
+    (gm * k, gs * k, (gm + gs) * k)
+}
+
+/// The permanent tide at geodetic latitude `lat` (rad) and ellipsoidal height `h` (m):
+/// `(W_T, g_T)`, the time-averaged tide-generating potential (m²/s²) and its part of gravity along
+/// the ellipsoidal normal (m/s²), from [`PERMANENT_A`] (Mäkinen 2021, Eq. 18).
+///
+/// [Mäkinen (2021)](https://doi.org/10.1007/s00190-021-01541-5): `g_T` is what to add to
+/// zero-tide gravity (the IAG convention) for mean-tide gravity, −30.49 µGal at the equator and
+/// +60.77 at the poles (his Eq. 20); `W_T/γ`, with `γ` normal gravity, is the height of the
+/// mean-tide geoid above the zero-tide one, +99.4 mm at the equator and −196.4 mm at the poles
+/// (Eq. 21). In Cartesian coordinates `W_T = (A/3a²)(2z² − p²)`, so this is exact for any height,
+/// and his polynomials in `sin²φ` (Eqs. 19–21) agree with it to their printed digits. A tide-free
+/// value differs from the zero-tide one by the Earth's response: `k` times `W_T`, and `δ − 1`
+/// times `g_T` for the [`delta`] its tide correction used.
+///
+/// Mean-tide gravity cannot be taken from a field whose `C̄20` was moved to
+/// [`TideSystem::MeanTide`](geopotential::TideSystem::MeanTide): there `W_T` is a degree-2
+/// term of the Earth's own field, falling as `r⁻³` instead of growing as `r²`, which matches it on
+/// the sphere `r = R` but gives +45.7 µGal at the equator instead of −30.5.
+pub fn permanent(lat: f64, h: f64) -> (f64, f64) {
+    let (r, psi) = geocentric(lat, h);
+    let (p, z) = (r * psi.cos(), r * psi.sin());
+    let k = PERMANENT_A / (3.0 * WGS84_A * WGS84_A);
+    // g_T = −∂W_T/∂h, along the normal (cos φ, sin φ) in the meridian plane.
+    (k * (2.0 * z * z - p * p), 2.0 * k * (p * lat.cos() - 2.0 * z * lat.sin()))
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use geopotential::{Coefficients, TideSystem, julian_epoch};
+    use metrology::{GM_EARTH, WGS84_F};
+
+    const MICROGAL: f64 = crate::local::MICROGAL;
+
+    #[test]
+    fn longmans_tide() {
+        // LongmanTide's test vector (tests/test_basics.py), mGal. The Julian epoch holds the time
+        // to about 4 µs, which is 3×10⁻¹¹ mGal.
+        let (lat, lon) = (40.7914f64.to_radians(), 282.1414f64.to_radians());
+        let (moon, sun, total) = longman(lat, lon, 370.0, julian_epoch(2015, 4, 23, 0, 0));
+        for (got, want) in [(moon, 0.0324029651226), (sun, -0.0288682178454), (total, 0.00353474727722)] {
+            assert!((got / 1e-5 - want).abs() < 1e-10, "{} against {want}", got / 1e-5);
+        }
+    }
+
+    #[test]
+    fn averaged_over_the_nodal_cycle_it_is_the_permanent_tide() {
+        // The mean of the routine tide is not zero (Honkasalo 1964): over 18.61 years Longman's
+        // tide is δ₂ times −g_T, Mäkinen's average from the spectral expansions, to 0.2 µGal (0.19
+        // at 45°).
+        let step = 1.37 / 24.0 / 365.25; // years, incommensurate with the day and month
+        let n = (18.61 / step) as usize;
+        for deg in [0.0, 45.0, 90.0] {
+            let lat = f64::to_radians(deg);
+            let mean = (0..n).map(|k| longman(lat, 0.0, 0.0, 2000.0 + k as f64 * step).2).sum::<f64>() / n as f64;
+            let want = -delta(2, 0.612, 0.303) * permanent(lat, 0.0).1;
+            assert!((mean - want).abs() < 0.25 * MICROGAL, "{deg}°: {} against {} µGal", mean / MICROGAL, want / MICROGAL);
+        }
+    }
+
+    #[test]
+    fn love_number_factors() {
+        // Longman's own Love numbers, and the IERS Conventions 2010 nominal values: h₂ = 0.6078
+        // (ch. 7), k₂₀ = 0.30190 (Table 6.3), h₃ = 0.292 (ch. 7), k₃ = 0.093 (Table 6.3).
+        assert!((delta(2, 0.612, 0.303) - 1.1575).abs() < 1e-15);
+        assert!((delta(2, 0.6078, 0.30190) - 1.15495).abs() < 1e-14);
+        assert!((delta(2, 0.6078, 0.30190) - 1.16).abs() < 0.01); // the conventional factor
+        assert!((delta(3, 0.292, 0.093) - 1.070_666_666_666_667).abs() < 1e-15);
+        // Mäkinen's Eq. 52, 1 + k − h ≈ 0.69, with k ≈ 0.30 and the nominal h.
+        assert!((gamma(0.6078, 0.30190) - 0.6941).abs() < 1e-12 && (gamma(0.6078, 0.30190) - 0.69).abs() < 0.005);
+        // A rigid Earth (h = k = 0) shows the tide as it is; a fluid one (h = 1 + k) shows no tilt.
+        assert_eq!((delta(2, 0.0, 0.0), gamma(0.0, 0.0), gamma(1.25, 0.25)), (1.0, 1.0, 0.0));
+    }
+
+    #[test]
+    fn makinens_permanent_tide() {
+        // Eq. 18's A follows from the IERS H₀ = −0.31460 m (Eq. 16): A = (3/2)H₀√(5/4π)g_e(a/R_e)².
+        let a_iers = 1.5 * -0.31460 * (5.0 / (4.0 * core::f64::consts::PI)).sqrt() * 9.79828685 * (WGS84_A / 6378136.55f64).powi(2);
+        assert!((a_iers - PERMANENT_A).abs() < 5e-5, "{a_iers}");
+        // Eqs. 19–21 in sin²φ, to their printed digits, with GRS80 normal gravity (Moritz 2000).
+        let e2 = WGS84_F * (2.0 - WGS84_F);
+        let gamma0 = |s2: f64| 9.780_326_771_5 * (1.0 + 0.001_931_851_353 * s2) / (1.0 - e2 * s2).sqrt();
+        for deg in (0..=90).step_by(5) {
+            let lat = f64::to_radians(f64::from(deg));
+            let s2 = lat.sin().powi(2);
+            let (w, g) = permanent(lat, 0.0);
+            assert!((w - (0.9722 - 2.8841 * s2 - 0.0195 * s2 * s2)).abs() < 5e-5, "{deg}°: W_T {w}");
+            assert!((g / MICROGAL - (-30.49 + 90.95 * s2 + 0.31 * s2 * s2)).abs() < 0.005, "{deg}°: g_T {}", g / MICROGAL);
+            assert!((w / gamma0(s2) * 1e3 - (99.40 - 295.41 * s2 - 0.42 * s2 * s2)).abs() < 0.005, "{deg}°: H_T");
+        }
+        // The spherical form −2A/a·(sin²φ − 1/3) at the equator: −30.485 µGal; the mean-tide geoid
+        // 9.9 cm above the zero-tide one there.
+        assert!((permanent(0.0, 0.0).1 / MICROGAL + 30.485).abs() < 1e-3);
+        assert!((permanent(0.0, 0.0).0 / 9.780_326_771_5 - 0.0994).abs() < 1e-4);
+        // Eq. 19's height factor 1 + 2h/a, to 4×10⁻⁶ at 1 km.
+        let lat = 45f64.to_radians();
+        assert!((permanent(lat, 1e3).0 / permanent(lat, 0.0).0 - (1.0 + 2e3 / WGS84_A)).abs() < 5e-6);
+    }
+
+    #[test]
+    fn the_mean_tide_c20_is_right_only_on_its_sphere() {
+        // Moving C̄20 to the mean-tide system adds W_T as part of the Earth's own field. At r = R on
+        // the equator its potential is Mäkinen's to 2×10⁻⁴ (EGM's −3.11080×10⁻⁸/√5 against
+        // IERS's A₀H₀ = −1.39141×10⁻⁸); its gravity is 3W/r where the tide's is −2W/r.
+        let mut zero = Coefficients::zeros(GM_EARTH, WGS84_A, 2);
+        zero.tide_system = TideSystem::ZeroTide;
+        let mean = zero.to_tide_system(TideSystem::MeanTide).unwrap();
+        let v = |c: &Coefficients, r: f64| c.partials(r, 0.0, 0.0).v;
+        let added = v(&mean, WGS84_A) - v(&zero, WGS84_A);
+        let (w, g) = permanent(0.0, 0.0);
+        assert!((added / w - 1.0).abs() < 2e-4, "{}", added / w);
+        let dr = 1.0;
+        let pull = -((v(&mean, WGS84_A + dr) - v(&zero, WGS84_A + dr)) - added) / dr;
+        assert!((pull / g + 1.5).abs() < 1e-3 && (pull / MICROGAL - 45.7).abs() < 0.1, "{}", pull / MICROGAL);
+    }
+}
diff --git a/crates/gravitics/README.md b/crates/gravitics/README.md
index f949f09..ecc9b79 100644
--- a/crates/gravitics/README.md
+++ b/crates/gravitics/README.md
@@ -99,6 +99,7 @@ assert!((near * 1e3 - 13.28).abs() < 0.01);
 | `gem` | Gravitoelectromagnetism in three conventions with the spin-2 factor, the gravitomagnetic dipole as a `Field`, the extremal charge-to-mass ratio, and Blackett's magnetism-from-spin |
 | `anomalies` | CODATA 2022's sixteen G measurements with their correlations, their generalised-least-squares mean and Birge ratio; Anderson's flyby formula; column densities and shielding against Majorana and lunar laser ranging; Pioneer's thermal decay |
 | `modified` | The radial acceleration relation ([McGaugh et al. 2016](https://arxiv.org/abs/1609.05917)), Tully–Fisher velocity and MOND radius; Yukawa forces with the current limits ([Lee et al. 2020](https://arxiv.org/abs/2002.11761)), the dark-energy length and boson mass; Einstein–Cartan density; Weber-like perihelion and mass shift |
+| `equivalence` | Tests of the equivalence principle: the Eötvös parameter η, the MICROSCOPE, Eöt-Wash and atom-interferometer results, the coupling to B, L or B − L that a bound on η allows for a vector or scalar boson, and the Yukawa form factor of a layered sphere ([Touboul et al. 2022](https://arxiv.org/abs/2209.15487); [Fayet 2025](https://arxiv.org/abs/2507.02723)) |
 | `quantum` | Bose–Marletto–Vedral entanglement phases ([Bose et al. 2017](https://arxiv.org/abs/1707.06050)); the Diósi–Penrose X-ray rate ([Donadi et al. 2021](https://arxiv.org/abs/2111.13490)) |
 | `hfgw` | High-frequency gravitational waves: the Li–Baker detector (flux conventions, graviton rate, the interference signal, the shot-noise time) and the Pais patents (radiated flux, surface field, magnetic conversion as he uses it and in SI) |
 | `superconductors` | The London moment; frame dragging of a ring in general relativity against Tajmar's report; the Cooper-pair density and gravitational London depth; a perfect shield's shadow; Podkletnov's pendulum and Marx bank; the measured bounds |
@@ -152,6 +153,8 @@ Every test reproduces a published number from the paper's own inputs, mostly to
 - **Alternatives:** the RAR's published shape and deep-MOND limit; Eöt-Wash's 5.1 meV boson;
   Ruggiero & Tartaglia's Einstein–Cartan densities; the Weber perihelion ratios against a direct
   orbit integration.
+- **Equivalence principle:** MICROSCOPE's final η; Fayet's coupling limits on B, L and B − L for
+  both spins, his form factors (homogeneous and five-shell Earth) and their weakening at 710 km.
 - **Quantum gravity:** Bose et al.'s phases; Donadi et al.'s germanium rate.
 
 The formulas were independently re-derived from their sources and every test value recomputed;
diff --git a/crates/gravitics/src/equivalence.rs b/crates/gravitics/src/equivalence.rs
new file mode 100644
index 0000000..e1f7710
--- /dev/null
+++ b/crates/gravitics/src/equivalence.rs
@@ -0,0 +1,214 @@
+//! Tests of the weak equivalence principle: the Eötvös parameter η, what the best experiments
+//! measured, and how strongly a new long-range force may couple to baryon or lepton number
+//! before MICROSCOPE would have seen it.
+//!
+//! A new force from a boson coupled to a charge `Q` (baryon number `B`, lepton number `L`, or
+//! `B − L`) pulls on each body in proportion to its charge per unit mass, which differs between
+//! elements, so two bodies of different composition fall differently. MICROSCOPE compared
+//! titanium and platinum in orbit for two years and found no difference at 10⁻¹⁵
+//! ([Touboul et al. 2022](https://arxiv.org/abs/2209.15487)); in
+//! [Fayet's (2025)](https://arxiv.org/abs/2507.02723) reading this bounds a `B − L` coupling at
+//! 1.1×10⁻²⁵, 10²⁵ times weaker than electromagnetism.
+//!
+//! The identifiers say `eta`, since `gravimetry::local::EOTVOS` is the unit of gravity gradient.
+//!
+//! ```
+//! use gravitics::equivalence::{MICROSCOPE_LIMITS, Mediator, TESTS, coupling_bound};
+//!
+//! // MICROSCOPE's η(Ti, Pt) = (−1.5 ± 2.3 ± 1.5)×10⁻¹⁵: 2.7×10⁻¹⁵ in all.
+//! assert!((TESTS[0].combine() - 2.746e-15).abs() < 1e-18);
+//! // The Earth's (B − L) per atomic mass unit, and Ti's less Pt's (Fayet 2025, Eqs. 6–7).
+//! let g = coupling_bound(Mediator::Vector, 0.5138, -0.05625, MICROSCOPE_LIMITS);
+//! assert!((g / 1.067e-25 - 1.0).abs() < 1e-3); // "|g_B−L| < 1.1×10⁻²⁵"
+//! ```
+
+use metrology::{ALPHA, AMU, C, G, HBAR};
+
+/// The Eötvös parameter `η = 2(a_A − a_B)/(a_A + a_B)` of two bodies falling with accelerations
+/// `a_a` and `a_b` (Touboul et al. 2022, Eq. 1).
+pub fn eta(a_a: f64, a_b: f64) -> f64 {
+    2.0 * (a_a - a_b) / (a_a + a_b)
+}
+
+/// One measurement of η with its statistical and systematic errors (1σ). Where a paper quotes
+/// one combined error it is in `stat` and `syst` is zero.
+#[derive(Clone, Copy, Debug, PartialEq)]
+pub struct Test {
+    /// The experiment and the two materials.
+    pub name: &'static str,
+    /// η of the first material against the second.
+    pub eta: f64,
+    /// The statistical error.
+    pub stat: f64,
+    /// The systematic error.
+    pub syst: f64,
+    /// Where it comes from.
+    pub source: &'static str,
+}
+
+impl Test {
+    /// The two errors in quadrature.
+    pub fn combine(&self) -> f64 {
+        self.stat.hypot(self.syst)
+    }
+}
+
+/// Published tests of the equivalence principle, the tightest first: in orbit (MICROSCOPE's
+/// Ti–Pt result and its Pt–Pt control), a torsion balance, and atoms in free fall.
+pub const TESTS: [Test; 4] = [
+    Test { name: "MICROSCOPE Ti–Pt", eta: -1.5e-15, stat: 2.3e-15, syst: 1.5e-15, source: "https://arxiv.org/abs/2209.15487" },
+    Test { name: "MICROSCOPE Pt–Pt (control)", eta: 0.0, stat: 1.1e-15, syst: 2.3e-15, source: "https://arxiv.org/abs/2209.15487" },
+    Test { name: "Eöt-Wash Be–Ti, towards the Earth", eta: 0.3e-13, stat: 1.8e-13, syst: 0.0, source: "https://arxiv.org/abs/0712.0607" },
+    Test { name: "Stanford ⁸⁵Rb–⁸⁷Rb", eta: 1.6e-12, stat: 1.8e-12, syst: 3.4e-12, source: "https://arxiv.org/abs/2005.11624" },
+];
+
+/// MICROSCOPE's one-sided 95% limits on η(Ti, Pt): below 4.5×10⁻¹⁵ if positive, below 6.5×10⁻¹⁵
+/// in size if negative. The result leans negative, so a positive η is the more constrained
+/// (Fayet 2025, Eq. 16).
+pub const MICROSCOPE_LIMITS: (f64, f64) = (4.5e-15, 6.5e-15);
+
+/// `α/(G u²) = e²/(4πε₀ G u²)`, 1.2536×10³⁶: the electric repulsion of two unit charges over the
+/// gravitational attraction of two atomic mass units, the scale of Fayet's Eq. 7.
+pub const ALPHA_OVER_G_U2: f64 = ALPHA * HBAR * C / (G * AMU * AMU);
+
+/// The spin of the boson that carries a new force.
+#[derive(Clone, Copy, Debug, PartialEq, Eq)]
+pub enum Mediator {
+    /// Spin 1, like the photon: like charges repel.
+    Vector,
+    /// Spin 0: like charges attract.
+    Scalar,
+}
+
+/// The η a new long-range force puts between two bodies falling towards the Earth (Fayet 2025,
+/// Eq. 7): `η = ∓(α/(G u²))ε²(Q/A_r)_⊕Δ(Q/A_r)`, the upper sign for a vector mediator. `epsilon`
+/// is the coupling in units of `e`, `earth` the Earth's charge per atomic mass unit and
+/// `difference` the first body's less the second's. For a range much longer than the Earth's
+/// radius; a finite range `λ` multiplies η by `(1 + r/λ)e^(−r/λ)` at distance `r`
+/// ([`yukawa_factor`](crate::modified::yukawa_factor) less one) and by the Earth's form factor
+/// (see [`sphere_form_factor`]) (Fayet's Eq. 94).
+pub fn eta_from_coupling(mediator: Mediator, epsilon: f64, earth: f64, difference: f64) -> f64 {
+    let sign = match mediator {
+        Mediator::Vector => -1.0,
+        Mediator::Scalar => 1.0,
+    };
+    sign * ALPHA_OVER_G_U2 * epsilon * epsilon * earth * difference
+}
+
+/// The largest coupling `|g| = |ε|√(4πα)` that one-sided limits `(positive, negative)` on η
+/// allow, for the same inputs as [`eta_from_coupling`]: the mediator and the sign of the charge
+/// difference fix the sign of η, and so which limit applies. With [`MICROSCOPE_LIMITS`] and a
+/// `B − L` charge it is 1.07×10⁻²⁵ for a vector and 1.28×10⁻²⁵ for a scalar, which Fayet rounds
+/// to 1.1 and 1.3 (his Eqs. 24–25). A finite range raises it by `F^(−1/2)`, where `F` is the
+/// factor that range puts on η.
+pub fn coupling_bound(mediator: Mediator, earth: f64, difference: f64, (positive, negative): (f64, f64)) -> f64 {
+    let per_epsilon2 = eta_from_coupling(mediator, 1.0, earth, difference);
+    let limit = if per_epsilon2 > 0.0 { positive } else { negative };
+    (limit / per_epsilon2.abs()).sqrt() * (4.0 * core::f64::consts::PI * ALPHA).sqrt()
+}
+
+/// How much more strongly a homogeneous sphere of radius `R` sources a Yukawa force of range `λ`
+/// than a point of the same charge, at `x = R/λ`: `φ(x) = 3(x cosh x − sinh x)/x³` (Fayet 2025,
+/// Eq. 37). It is 1 for a long range and grows as `(3/2)eˣ/x²`, since the charge near the surface
+/// is nearer. A layered body is the mass-weighted sum of its shells' terms (his Eq. 80).
+pub fn sphere_form_factor(x: f64) -> f64 {
+    if x.abs() < 1.0 {
+        // Σ 3x²ᵏ/((2k + 1)!(2k + 3)): the closed form loses every digit to cancellation as x → 0.
+        let (mut term, mut sum) = (1.0, 0.0);
+        for k in 0..12 {
+            sum += 3.0 * term / f64::from(2 * k + 3);
+            term *= x * x / f64::from((2 * k + 2) * (2 * k + 3));
+        }
+        sum
+    } else {
+        3.0 * (x * x.cosh() - x.sinh()) / x.powi(3)
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use crate::modified::{boson_mass_ev, yukawa_factor};
+
+    #[test]
+    fn eotvos_parameters() {
+        // η is antisymmetric, and Δa/g to first order in Δa.
+        let g = 7.95;
+        assert!(eta(3.0, 1.0) == 1.0 && eta(1.0, 2.0) == -eta(2.0, 1.0) && eta(g, g) == 0.0);
+        assert!((eta(g + 1e-6, g) / (1e-6 / g) - 1.0).abs() < 1e-7);
+        // Touboul et al. (2022), Eq. 5 and the control: η(Ti, Pt) = [−1.5 ± 2.3 ± 1.5]×10⁻¹⁵,
+        // 2.746×10⁻¹⁵ in quadrature, a differential pull of 2.2×10⁻¹⁴ m/s² at 710 km.
+        let [ti_pt, pt_pt, eot_wash, stanford] = TESTS;
+        assert!((ti_pt.combine() / 2.745_906e-15 - 1.0).abs() < 1e-6);
+        assert!((ti_pt.combine() * metrology::GM_EARTH / (6371e3 + 710e3f64).powi(2) / 2.18e-14 - 1.0).abs() < 0.01);
+        assert!(ti_pt.eta.abs() < ti_pt.combine() && pt_pt.eta.abs() < pt_pt.combine());
+        // Schlamminger et al. (2008), Eq. 2, and Asenbaum et al. (2020): 3.85×10⁻¹² in all.
+        assert!(eot_wash.eta.abs() < eot_wash.combine() && (stanford.combine() / 3.847e-12 - 1.0).abs() < 1e-3);
+    }
+
+    #[test]
+    fn fayets_coupling_bounds() {
+        assert!((ALPHA_OVER_G_U2 / 1.2536e36 - 1.0).abs() < 5e-5);
+        // Eq. 6: Δ(Q/A_r) of Ti less Pt. Eq. 7 prints the combined coefficients ∓1.00×10³³ (B),
+        // ±3.623×10³⁴ (B − L) and ∓3.482×10³⁴ (L); they give the Earth's charges per mass. Its L
+        // and B − L add up to 1.0008 nucleons per u, between ¹⁶O's 1.0003 and ⁵⁶Fe's 1.0012; B's
+        // own coefficient and Δ(B/A_r) carry two or three digits, so its 1.010 is good to 1%.
+        let (db, dl): (f64, f64) = (0.00079, 0.05704);
+        let dbl = db - dl;
+        assert!((dbl + 0.05625).abs() < 1e-15);
+        let earth = |coefficient: f64, d: f64| coefficient / (ALPHA_OVER_G_U2 * d.abs());
+        let (eb, ebl, el) = (earth(1.00e33, db), earth(3.623e34, dbl), earth(3.482e34, dl));
+        assert!((ebl + el - 1.0008).abs() < 1e-4 && (eb - (ebl + el)).abs() < 0.01);
+        assert!((eta_from_coupling(Mediator::Vector, 1.0, ebl, dbl) / 3.623e34 - 1.0).abs() < 1e-12);
+        // Eqs. 24–25 (95% CL), and note 40's unrounded 1.07 and 1.28 for B − L.
+        for (mediator, q, d, want) in [
+            (Mediator::Vector, ebl, dbl, 1.1e-25), (Mediator::Vector, eb, db, 7.7e-25), (Mediator::Vector, el, dl, 1.3e-25),
+            (Mediator::Scalar, ebl, dbl, 1.3e-25), (Mediator::Scalar, eb, db, 6.4e-25), (Mediator::Scalar, el, dl, 1.1e-25),
+        ] {
+            let got = coupling_bound(mediator, q, d, MICROSCOPE_LIMITS);
+            assert!((got / want - 1.0).abs() < 0.05, "{mediator:?}: {got:e} against {want:e}");
+        }
+        let vector = coupling_bound(Mediator::Vector, ebl, dbl, MICROSCOPE_LIMITS);
+        let scalar = coupling_bound(Mediator::Scalar, ebl, dbl, MICROSCOPE_LIMITS);
+        assert!((vector / 1.067e-25 - 1.0).abs() < 1e-3 && (scalar / 1.283e-25 - 1.0).abs() < 1e-3);
+        // The two differ by √(6.5/4.5), the dissymmetry of the result.
+        assert!((scalar / vector - (6.5f64 / 4.5).sqrt()).abs() < 1e-12);
+    }
+
+    #[test]
+    fn the_form_factor_of_a_sphere() {
+        // Fayet's Table IV, homogeneous sphere, to its printed digits.
+        for (x, want) in [(0.5, 1.0252), (1.0, 1.1036), (2.0, 1.4616), (3.0, 2.2428), (5.0, 7.1243), (10.0, 297.36), (20.0, 1.728e6), (30.0, 1.722e10), (50.0, 3.049e18), (100.0, 3.992e39), (200.0, 2.696e82)] {
+            assert!((sphere_form_factor(x) / want - 1.0).abs() < 2.5e-4, "x = {x}: {}", sphere_form_factor(x));
+        }
+        // 1 + x²/10 + x⁴/280 + x⁶/15120 near zero, where the closed form has no digits left; the
+        // series and the closed form meet at x = 1.
+        for x in [0.0, 1e-8, 1e-3, 0.1] {
+            let x2: f64 = x * x;
+            let want = 1.0 + x2 / 10.0 + x2 * x2 / 280.0 + x2 * x2 * x2 / 15120.0;
+            assert!((sphere_form_factor(x) - want).abs() < 1e-8 * x2 * x2 * x2 + 2e-16, "{x}");
+        }
+        let (below, above) = (sphere_form_factor(1.0 - 1e-15), sphere_form_factor(1.0));
+        assert!((below - above).abs() < 4e-15);
+        assert!((sphere_form_factor(-2.0) - sphere_form_factor(2.0)).abs() < 1e-15);
+    }
+
+    #[test]
+    fn a_layered_earth_at_satellite_height() {
+        // Fayet's five-shell Earth (Eq. 80): Φ(x) = Σ yᵢ φ(Rᵢx/R), Table IV's Φ(3) = 2.0021.
+        let shells = [(0.5, 1.0), (0.1333, 0.9953), (0.1715, 0.8948), (0.1929, 0.5462), (0.0023, 0.1916)];
+        let earth = |x: f64| shells.iter().map(|&(y, s)| y * sphere_form_factor(s * x)).sum::<f64>();
+        assert!((earth(3.0) / 2.0021 - 1.0).abs() < 1e-5 && (earth(0.0) - 1.0).abs() < 1e-15);
+        // Table V: at MICROSCOPE's 710 km, a boson of mass m weakens η by F = (1 + r/λ)e^(−r/λ)Φ,
+        // so the coupling limits grow by G = F^(−1/2).
+        let (radius, r) = (6371e3, 6371e3 + 710e3);
+        let hbar_c = HBAR * C / metrology::EV; // eV m
+        for (m, f, g) in [(1e-14, 0.957, 1.022), (2e-14, 0.867, 1.074), (5e-14, 0.574, 1.320), (1e-13, 0.281, 1.886), (2e-13, 0.0927, 3.28)] {
+            let lambda = hbar_c / m;
+            assert!((boson_mass_ev(lambda) / m - 1.0).abs() < 1e-12);
+            let big_f = (yukawa_factor(r, 1.0, lambda) - 1.0) * earth(radius / lambda);
+            // To the table's three digits.
+            assert!((big_f / f - 1.0).abs() < 2e-3 && (big_f.powf(-0.5) / g - 1.0).abs() < 2e-3, "m = {m:e}: F = {big_f}");
+        }
+    }
+}
diff --git a/crates/gravitics/src/lib.rs b/crates/gravitics/src/lib.rs
index 070b296..fef4ce7 100644
--- a/crates/gravitics/src/lib.rs
+++ b/crates/gravitics/src/lib.rs
@@ -3,6 +3,7 @@
 pub mod anomalies;
 pub mod electrodes;
 pub mod electrogravity;
+pub mod equivalence;
 pub mod gem;
 pub mod hfgw;
 pub mod modified;