Settlement and saturation in the sliding ferroelectric WTe₂

The source’s crystals are bulk WTe₂, a van der Waals semimetal. Its T_d polymorph, stable below 565 K, adopts a trilayer structure that breaks inversion symmetry perpendicular to the layers. Citing earlier studies, the source writes that an uncompensated interlayer vertical charge transfer gives the polymorph a ferroelectric polarisation along the c-axis, normal to the layers, and that ultrafast optical excitation, from the terahertz to the visible, depopulates antibonding orbitals between neighbouring layers and launches an interlayer-sliding mode at f_s = 0.24 THz, which pushes the structure towards a high-symmetry configuration and eventually restores inversion symmetry. Citing a study of few-layer WTe₂, it writes that, owing to antiparallel interlayer sliding between adjacent layers and electronic screening, the effective polarisation of the bulk is dominated by near-surface contributions, for which another cited study observed a robust ferroelectric hysteresis and domains.

The polar T_d phase belongs to the space group Pmn2₁ and the nonpolar, ferroelastic 1T′ phase to P2₁/m. The two, the source writes citing earlier work, are not related by a group–subgroup relationship, so the sliding mode alone provides no symmetry-allowed pathway between them; the source identifies the centrosymmetric Pnma structure as the true high-symmetry reference phase, from which T_d follows by the interlayer-sliding mode (Γ₂⁻) and 1T′ by an intralayer-shear mode (Γ₄⁺). On its calculated potential energy surface the lowest-energy route between T_d and 1T′ bypasses Pnma by exciting both modes at once. Its anharmonic expansion of the free energy about the T_d minimum reads

with 𝖰_A₁ the amplitude of the sliding phonon, which the source writes as proportional to √((𝒫 − 𝒫_z)²), the fluctuation of the polarisation about its minimum, 𝖰_B₁ that of the intralayer-shear phonon, the mode frequencies ω_A₁/2π = f_s = 0.24 THz and ω_B₁/2π = 2.7 THz, and 𝖼_AB the coupling through which shearing reaches the polarisation only by phonon–phonon interaction.

In single-pulse experiments, 130 fs pump pulses of wavelength 1.5 μm and photon energy 0.83 eV, polarised along the b-axis and repeated at 1 kHz, strike the surface at normal incidence over a 280 × 280 μm² spot, at absorbed fluences 𝖥 of 0.4 to 6.5 mJ cm⁻². Probe pulses of 120 fs at 0.8 μm and 1.55 eV, with a fluence of 2.8 mJ cm⁻², arrive at 45° in the ac-plane on an 80 × 40 μm² spot, and their second harmonic, passed by a 400 nm band-pass filter, is read in a cross-polarised configuration. The crystals, commercial single crystals exfoliated to remove oxidised surface layers and kept under nitrogen, are several tens of micrometres thick; every experiment runs at room temperature, the signals degrade by less than 2% over several weeks, each data point averages 5000 pulses, and the measured second-harmonic signal originates predominantly from less than 50 nm below the surface.

After one pulse the second-harmonic intensity drops within 500 fs, with prominent oscillations at f_s. Raising the absorbed fluence from 0.4 to 6.5 mJ cm⁻² progressively shifts the oscillation’s equilibrium position 𝖰₀ towards Pnma and softens the mode, read in a long-lived shift and in fluence-dependent frequency shifts. The sliding-mode amplitude 𝖠_s, the amplitude of the second-harmonic modulation, which the source’s large-amplitude figure normalises to the signal before the pump, “initially increases with fluence, then reaches a maximum around an absorbed fluence of 𝖥_sat ≈ 1.6 mJ cm⁻², and even declines at higher excitation levels”. Together with cited transient reflectivity and X-ray diffraction on WTe₂ and MoTe₂, the source writes that its results suggest such saturation behaviour is a general characteristic of sliding ferroelectrics.

To go beyond the linear model of displacive excitation of coherent phonons (DECP), which ties the amplitude linearly to the excited-carrier density, the source computes the drive state by state. Density functional theory with spin–orbit coupling, electron–phonon couplings interpolated on Wannier functions, and time-dependent Boltzmann equations for the electrons, started as a hot Fermi–Dirac distribution whose temperature matches the absorbed energy, give each mode ν its equation of motion and its drive,

with 𝗋_ν the damping rate, 𝗀 the electron–phonon coupling of band 𝖻 at wave vector 𝐤 and Δ𝗈 the change of that state’s occupation from the Fermi–Dirac distribution before excitation. At low electronic temperature, the source writes, changes in occupation of electron- and hole-like states near the Fermi energy generate net positive sliding forces; with increasing fluence, higher-energy states become populated and counteract the directed sliding motion. Its state-resolved figure shows the drive at 2000 K and 4000 K. The calculation predicts saturation at T_sat = 3100 K, which corresponds to an absorbed fluence of 𝖥_sat = 1.6 mJ cm⁻², and beyond it experiment and theory both decline; two higher-frequency modes, at 3.5 and 6.4 THz, grow monotonically. The static limit of Eq. 2, 𝖰_max = 𝒟_ν/ω_ν², is in the source’s words “an upper theoretical limit to the maximum lattice displacement that the crystal can undergo for a given electronic excitation”.

A Mach–Zehnder interferometer then splits each pump into two cross-polarised pulses, which suppresses coherent artefacts near zero delay, and the second harmonic is recorded against the pump–pump delay Δt and the pump–probe delay at a combined absorbed fluence of 2.1 mJ cm⁻². Strong modulations of the amplitude with the delay, shown in traces at 4.28, 5.78 and 6.53 ps, demonstrate in the source’s words coherent control over the sliding-mode amplitude in the excited state, corroborated by ab initio calculations: a simulation that adds the two drives, 𝒟_ν(t) + 𝒟_ν(t − Δt), and assumes an amplitude-dependent sliding frequency. Comparing Δt = 0, where both pulses act as one, with Δt = 8.3 ps, two periods, the source finds that “for equal total optical energy, sequential optical excitation yields up to 50% higher sliding amplitudes”, extending the linear relation between optical energy and amplitude. It attributes this, most likely, to the pair avoiding electronic temperatures above T_sat and so minimising the population of the higher-lying states that drive the mode the opposite way, and writes that for in-phase excitation the two vibrational wave packets interfere constructively. At short delays the amplitude after the second pulse is strongly suppressed against the source’s expectation without saturation, for delays up to about 2 ps, a timescale that agrees, the source writes, with carrier lifetimes from time- and angle-resolved photoemission; it judges saturable absorption too weak to account for the saturation and its decline.

Double-pulse excitation, the source writes, controls the amplitude while keeping other key degrees of freedom, such as the electronic excitation and the temperature, constant. The damping 𝗋_s = 1/𝗍_s rises linearly with 𝖠_s across the full range, and the slope of the line itself increases with the combined fluence, compared at 2.2 and 4.1 mJ cm⁻²; the source writes that this behaviour “indicates that damping is mediated by the coherent vibrational amplitude”, which is not observable under equilibrium conditions, where nuclear wave packets remain centred at their equilibrium positions irrespective of temperature. It considers a Klemens process, the sliding mode decaying into two acoustic phonons of opposite momentum, the most probable decay mechanism, citing the prediction that stimulated phonon decay strongly enhances optical-phonon damping under non-equilibrium conditions. The frequency softens as the amplitude grows, on a frequency axis from 0.21 to 0.23 THz; the source excludes a significant contribution from electronic softening, both the lattice temperature and the level of electronic excitation being, it writes, fixed in double-pulse experiments, and attributes the shift to intrinsic lattice anharmonicity, which its simulation carries as a dressed frequency ω̃_ν = ω_ν − 𝗄_dr|𝖰_ν|, with 𝗄_dr a positive fitting parameter.

The source concludes that the linear relationship between excited-carrier density and coherent vibrational amplitude assumed in DECP is an oversimplification, that its results point towards a dynamical extension of DECP in which the photo-induced driving force evolves with the time- and energy-dependent carrier distribution, rather than arising from an instantaneous displacement of the equilibrium nuclear positions, and it expects the sequential scheme to be broadly applicable, particularly to other sliding ferroelectrics. It names ferroelectric bi- and trilayers, in which sliding is predicted to follow modified interlayer interactions, as a natural next step.

There is one substance, the omnium: matter, energy and spacetime (0.1). The chain E/m = c² = 1/(ε₀μ₀) = (d/t)² is one state read four ways, and c is the strand rate of the omnium, derived from its two response coefficients (0.1a, 0.1d). One formula unit of the source’s crystals, 183.84 + 2 × 127.60 = 439.04 u, read on the exterior branch at the analysis’s declared field radius, the interlayer spacing of 0.70 nm, has the gravitational window α_ext = 𝓜c²/(2F_maxr_f) = 1.5 × 10⁻⁴², and two neighbouring units at contact carry the pair force F = F_maxα₁α₂ = 7.2 × 10⁻⁴¹ N (I-⁠1). Both are correct small numbers, the window the curvature of general relativity for this crystal, and neither is the interlayer restoring force of the sliding mode; nonspherical, rotating, overlapping or dynamical configurations, the sliding crystal among them, take derived evaluations through multipoles and the overlap complex, and none is written here (O-⁠5).

Every step is a system-wide settlement: the global optimisation is solved at every step from the state the system is in, the expected behaviour is not necessarily the evolving system’s optimum, events do not commute, and responses are system-wide and many-to-many, never one-to-one, particle by particle (0.1a). The source’s drive is one sum over every electronic state’s change of occupation, with contributions of both signs: coupling to low-energy states drives the lattice towards Pnma, coupling to higher-lying states favours motion in the opposite direction. In the analysis’s own reading that sum is one system-wide, many-to-many settlement, and the source’s outlook, a driving force that evolves with the time- and energy-dependent carrier distribution rather than an instantaneous displacement of the equilibrium positions, is each step settled from the state the system is in. The two-pulse gain at fixed energy is read as the expected behaviour not being the system’s optimum: the expectation that one pulse carrying the whole energy moves the mode at least as far as two, which the system, settled each step from the state it is in, does not meet. The source’s Eq. 4 adds two drives, each computed as if the other pulse were absent, while its own account needs the second pulse to meet the electrons the first one left; both stay the source’s. The core names no objective for the optimisation, and the source’s density functional, Wannier and Boltzmann calculations, with the lowest-energy route they find, remain its method.

The second line is the analysis’s arithmetic for Eq. 2: a drive switched on suddenly peaks at twice its static limit when undamped. The source’s 𝖰_max stays its own quantity.

There are no independent force carriers: photon = graviton = gluon = phonon = magnon is the same spin-one excitation of the density field at different density and geometry regimes, interactions are curvature plus geodesic motion, and the braided closures of matter are never inside the identity (I-⁠2). The pump at 0.83 eV, the probe at 1.55 eV, its second harmonic at 3.10 eV, the sliding stroke at 0.24 THz, whose quantum is hf_s = 0.99 meV, and the acoustic branches are that one excitation in different regimes. The energy passes from the optical to the lattice regime through the occupation of the electrons, which are n = 3 closures, never through an exchanged carrier, and “mediated” stays the source’s word, in its anharmonic coupling mediated by coherent nuclear motion and its damping mediated by the coherent amplitude.

Energy injection excites standing density waves; the loci where the transverse density gradient vanishes are the nodal lines and surfaces, the mode geometry is set by the boundary conditions, the clamps, not by the medium, and a load is taken by the interstitial spacetime between the anchors, whose packing changes by −ΔV/V (P-⁠2, P-⁠3). The coherent sliding is read as a standing density wave the electromagnetic injection excites, its amplitude set by the injection’s timing and its geometry by the stacking of the W and Te pinning sites; P-⁠2’s medium is the omnium, never the crystal. The orbitals the pump depopulates between neighbouring layers and the long-lived shift of 𝖰₀ towards Pnma are read as load taken by the interstitial between the anchors. The source prints no ΔV/V, so no number is attached.

A driven observable vanishes at zero drive and never recovers the static force; the susceptibility maps force to observable and is not the force law (I-⁠1). Eq. 2’s oscillator maps the drive of Eqs. 3 and 4 to 𝖰_ν through its Green function, a susceptibility, and the drive stands in no force slot of I-⁠1. The amplitude 𝖠_s vanishes at zero drive, and its saturation and decline are properties of the drive, of the oscillator’s fluence-dependent response and of the second-harmonic transduction, never of a force law. The source prints no map from 𝖠_s to a displacement, and in the analysis’s arithmetic, for a second-harmonic intensity quadratic in the polar displacement measured from Pnma, the modulation per unit oscillation, the transduction, is proportional to the equilibrium’s distance from Pnma: it falls as the long-lived shift approaches Pnma and carries part of the fall of 𝖠_s. No amplitude, damping, frequency or shift is run through the bounded map of 0.4, which belongs to a named application and never to a response (I-⁠6).

Quantum mechanics is helical geodesics on truncated density cones: the wavefunction ψ = Ae^iφ is the projection of a helix onto the plane normal to the cone axis, superposition is ignorance of the helical phase φ₀ at observation, the universe is deterministic, and there are no probability amplitudes (I-⁠4). In the cylindrical lift the oscillator’s phase is a single-angle rotor, φ_h(t) = φ₀ − ωt, with the imaginary unit the unit bivector of the phase plane, and each pulse fixes the φ₀ of its own oscillation. The two pulses’ oscillations are two rotors of one mode, adding at the phase difference ωΔt: in phase when ωΔt is a whole number of turns at the frequency the mode runs between the pulses. The source’s 8.3 ps is 2/f_s; its delays of 4.28, 5.78 and 6.53 ps are 1.0, 1.4 and 1.6 periods at 0.24 THz; across the 0.21 to 0.23 THz axis of its large-amplitude fits, 8.3 ps is 1.74 to 1.91 turns, and the phases differ. The source’s vibrational and nuclear wave packets stay its words.

Every consensus datum about an interior is a profile-weighted mean (I-⁠3), and curvature, a tide or a density gradient is read between neighbours (I-⁠3a, Use 7). Each second-harmonic datum is a mean weighted by the depth profile of the signal within the 50 nm below the surface, so no point value is read off it. The polarisation of van der Waals sliding ferroelectrics, the source writes citing earlier work, arises from the relative displacement of adjacent layers; the second harmonic reads that displacement, which the analysis reads as a comparison of neighbouring layers, and none of curvature, tide or density gradient is read off the polarisation. The analysis reads the cited near-surface dominance of the bulk polarisation as a statement about that probe weight.

Mass, energy, temperature and pressure enter one R_s, each channel once (I-⁠7, 0.5):

The absorbed pump enters once, as E/c² = 5.7 × 10⁻²⁵ kg per pulse in the column under the probe spot at 1.6 mJ cm⁻², an energy of 5.1 × 10⁻⁸ J, whatever it becomes: the electrons’ heat, the coherent mode or the lattice’s heat. The radiation term of the probed skin at the source’s room temperature, taken as 295 K, is at most 2 × 10⁻³⁸ kg and never the electrons’ energy. The electronic temperatures enter 𝓜 as channels. In 0.1c a medium’s clock factor, on a stress tensor of perfect form, is the reading of its flow, never the time kept by a clock the medium carries, and a solid’s shear stress lies outside that form: no frequency is read off a temperature. The fixed combined fluence fixes the deposited energy and not its sharing in time between electrons and lattice, which the source’s own mechanism makes depend on the delay.

Mass is a confinement eigenvalue, m_qc² = ħω_q, and no form of the confinement operator, its domains, boundary data or spectrum is written here (O-⁠9). The electrons are n = 3 closures, and in the analysis’s reading the pump’s excitation and the relaxation after it are changes of mode inside that class with no rebraiding; no relaxation rate or decay kernel is written here (O-⁠12). The source’s higher-lying states are higher modes of the same closures on the crystal’s modes, with no address on the confinement axis.

X-⁠1, exploratory. The order of an unequal pump pair: at the source’s delay of 8.3 ps and combined fluence of 2.1 mJ cm⁻², the amplitude with the strong pulse first less that with the weak pulse first. The comparator is the source’s Eqs. 2 to 4 with each pulse’s own drive, under which the order changes the first pulse’s damping during the delay and, through the fitted dressed frequency, the frequency the mode runs at between the pulses; the reading is 0.1a’s “events do not commute”. An in-sample re-description, never counted with the tests, and never run through the bounded map.

Five tests stand where a core clause supplies what the consensus calculation lacks: a constitutive input, the density channel of I-⁠7 for T-⁠1 and T-⁠2 and the skin of 0.6 for T-⁠5, or an additional source, the magnetic projection of a circulation and the field of a twist, both of P-⁠5, for T-⁠3 and T-⁠4. Each compares readings arranged so that a shared conventional response cancels or stands at its specified conventional limit, and each fixes its observable, its comparator, its decision rules and its rates before any data. The core assigns none of these observables a magnitude: no isotope sensitivity is computed here (I-⁠7), P-⁠5’s quantitative twist-to-moment forms are derived through I-⁠1 and I-⁠2 with a declared kernel, and none is derived here (O-⁠6), and no form of the local profile of 0.6 is written here (O-⁠4). T-⁠1, T-⁠2 and T-⁠5 therefore predict a sign, under a declared positive sensitivity or the analysis’s map from a denser skin to a narrower gap, and T-⁠3 and T-⁠4 an existence. The apparatus, settings and error models are the analysis’s design, never measurements in the source, and every numerical test point is a design assumption, never an amplitude of the core. Every Student test is one-sided at the three-sigma rate 0.00135, with the quantile 4.530 at seven degrees of freedom; every permutation test runs over all 12870 balanced relabelings of sixteen readings.

I-⁠7: for purely density-coupled equilibrium observables a heavier isotope gives a larger R_s, a higher α and a larger response, opposite in sign to conventional reduced-mass scaling, and a sign competing with other channels is written as a sensitivity-weighted sum with every sensitivity defined. The isotope tests grow eight pairs of crystals, one of each pair from tellurium enriched to at least 99% in ¹³⁰Te and the other in ¹²⁶Te, with natural tungsten, one growth route and one stoichiometry target, the two of a pair grown back to back and each crystal’s tellurium content measured; they are exfoliated and mounted as the source’s, and each pair is measured back to back at one stage temperature in an order a fair coin draws. At fixed pressure, radius and packing the density input is the move of the A-⁠1 window with the formula mass, 443.65 u against 435.65 u,

and the analysis reads I-⁠7’s larger response as a higher frequency and a higher transition temperature, a declared positive sensitivity, so the density contribution is positive and its magnitude unfixed.

1. T-⁠1
   The ¹³⁰Te crystal slides faster than the consensus isotope shift allows. Each crystal’s f_s comes from single-pulse traces taken as the source’s, at its lowest absorbed fluence, 0.4 mJ cm⁻², matched within each pair, and the pair contrast is 𝔡 = ln f_s(¹³⁰Te) − ln f_s(¹²⁶Te). Comparator: the same sum without the density channel, the consensus calculation, of a rigid-layer inertial term −½Δln M_f = −0.009105, a zero-point volume term from 0 to +0.0012 for a Grüneisen parameter up to 8, ±0.00005 from the enrichment and an anharmonic term at the 10⁻⁶ level: −0.0092 to −0.0078 in ln f_s, rounded outward. With a fit error and a crystal scatter of 0.0010 and a fluence-matching error of 0.0004 per crystal and a coin-signed drift of 0.0002 per pair, the pair scatter is 0.00209 and detection needs the mean contrast above −0.0045, the upper end plus 4.530 × 0.00209/√8 = 0.0033. A contribution of 0.007 above the lower end is excluded when the one-sided test of the mean below −0.0022 rejects at the same rate.
2. T-⁠2
   The ¹³⁰Te crystal leaves the T_d phase at a higher temperature. The T_d ⇌ 1T′ transition temperature is the midpoint of its onsets on heating and on cooling at 2 K min⁻¹, read from calorimetry and from the loss and return of the static second harmonic in the source’s probe geometry, each crystal corrected to its pair’s mean tellurium content, at the source’s 565 K; the pair contrast is 𝔡′ = ln T_tr(¹³⁰Te) − ln T_tr(¹²⁶Te). The phases are not related by a group–subgroup relationship, so the transition is first order, and the midpoint cancels the symmetric part of each onset’s kinetic isotope term. Comparator: a phase equilibrium is independent of the isotopic masses in the classical limit, the consensus premise, so its isotope channel at 565 K is the quantum Wigner–Kirkwood correction, within ±3 × 10⁻⁴, beside a tellurium-content residual of ±3 × 10⁻⁴: −0.0006 to +0.0006 in ln T_tr, the antisymmetric kinetic part, under 10⁻⁵, lying inside the interval’s last figure. A calorimetric reproducibility of 0.3 K, a crystal scatter of 0.5 K and a coin-signed drift of 0.1 K give a pair scatter of 0.0015, and detection needs the mean above 0.0030; a contribution of 0.005 above the lower end is excluded when the test of the mean below 0.0044 rejects.

A T-⁠1 detection is read as the density channel when it survives with each pair difference corrected by the pair’s measured tellurium-content difference times the measured dependence of ln f_s on that content, and a T-⁠2 detection when the calorimetric and the static second-harmonic determinations each reject. The two tests read one mechanism of I-⁠7 through one substitution, an oscillator frequency and a phase equilibrium alike, so their outcomes are one test of the density channel’s input sign in two observables.

P-⁠5: mechanical vorticity and magnetic moments are different grade projections of one density-gradient topography, with no carrier exchange, and imposing a physical twist on a material is establishing a magnetic field. I-⁠3a splits the deviation map of a family of patterns into a trace, density; an antisymmetric part, the vorticity, whose magnetic projection is that of P-⁠5; and a symmetric trace-free part, the shear, the tidal stretch and compression. The family of layers slides antiparallel, neighbour against neighbour, so its deviation map has an antisymmetric part, a circulation of alternating sense from gap to gap, whose probe-weighted mean over the skin is that of the uncompensated near-surface gaps: the analysis’s reading of the projections of I-⁠3a, read between neighbours.

1. T-⁠3
   The sliding circulation turns a probe’s polarisation with the domain’s sign. Sixteen single-domain regions of the source’s crystals, eight of each polar sign, read by second-harmonic interference with a reference, are excited as the source’s single pulse at 1.6 mJ cm⁻². A zero-area Sagnac loop retraces the probe’s path at the source’s 45° incidence, blind to birefringence and natural optical activity, and reads the non-reciprocal Kerr rotation at f_s in quadrature with the displacement oscillation; the contrast 𝔡_P is the mean over the eight positive regions less that over the eight negative, so every domain-independent term cancels. FDT: 𝔡_P ≠ 0, an existence; no clause fixes its sign. Comparator: the rotation (Barnett) term of the same circulation. For an interlayer displacement amplitude up to 2 pm and a probe depth of 25 nm the circulation’s curl is 6.0 × 10⁷ s⁻¹, its rotation rate, half the curl, a spin–rotation field of 1.7 × 10⁻⁴ T at the electron’s gyromagnetic ratio and, with the orbital coupling twice the spin’s, 3.4 × 10⁻⁴ T; at up to 3 μrad T⁻¹ for the same geometry it is at most 1.1 nrad per domain, of either sign. The electrons’ magnetisation linear in the sliding velocity and the residual-delay term of the loop widen that interval before the tests run; the executed figures below are the Barnett interval’s. Detection: both permutation tests on the contrasts shifted by each end of the interval give p ≤ 0.0027, with the two shifted contrasts of one sign, a rule of size 17/12870 = 0.0013 with the truth at either end. At a reading error of 3 nrad and a region scatter of 2 nrad, the contrast scatters by 1.8 nrad, and detection needs |𝔡_P| above 2 × 1.1 + 3.0 × 1.8 = 7.6 nrad. A projection of 10 nrad per domain beyond the interval is excluded when both one-sided tails reject at 0.00135.
2. T-⁠4
   A twisted bilayer rotates the polarisation non-reciprocally with its hand. Sixteen twisted WTe₂ bilayers at 5°, eight of each hand, encapsulated in hexagonal boron nitride on one substrate type, and four aligned controls, at room temperature and zero applied field, are read by polar Kerr rotation at 1550 nm in a zero-area Sagnac interferometer, each device read normally and turned over together with its substrate, the order drawn by a coin. The bilayer is never turned alone: its two identical layers keep an in-plane two-fold axis, while the stack’s asymmetry, substrate below and encapsulation above, admits a field along the normal, which turning the chip over reverses against the beam. Per device 𝗊_K = ½(ϖ_n − ϖ_t), the normal reading less the turned, in which the instrument’s offset and the Earth’s field along the beam cancel. FDT: the twist establishes a field, its sense set in the analysis’s reading by the hand against the stack’s asymmetry, an existence. Comparator: zero, the consensus premise that a nonmagnetic, time-reversal-invariant structure rotates no polarisation non-reciprocally, whatever its twist. A reading error of 20 nrad, an offset reset of 50 nrad per reading and a coin-signed drift of 10 nrad give a per-device scatter of 38.4 nrad and a hand-contrast scatter of 19.2 nrad, and the two-sided permutation test detects at p ≤ 0.0027, of size 34/12870 = 0.0026, which the design scatter places at |𝔡_K| above 3.0 × 19.2 = 58 nrad; 100 nrad per hand is excluded when both shifted one-sided tails reject at 0.00135.

A T-⁠3 detection is read as the projection of the sliding circulation when the same statistic at a negative pump–probe delay, with no coherent motion, returns none. A T-⁠4 detection is read as the twist’s field when the twisted devices’ hand-even mean agrees with the four aligned controls’ within three standard errors of their difference, one standard error being 21 nrad under the stated noise, the controls’ values reported beside it. The two tests read the two faces of P-⁠5, a circulation’s magnetic projection and a helical geometry’s field, both as comparisons whose consensus deviation cancels or vanishes.

0.6: a bound configuration’s local density is highest at its skin on both branches, the exterior and the interior, and its centre is the dilute region. The analysis reads the crystal as a bound configuration with its skin at its surface, and at fixed areal mass a denser skin is a narrower outer gap, the analysis’s map.

1. T-⁠5
   The outermost van der Waals gap is narrower than the bulk gap. On eight freshly exfoliated cleavage surfaces, crystal-truncation-rod fits of surface X-ray diffraction at room temperature give 𝗌_gap = (d₁₂ − d_bulk)/d_bulk, the outermost gap against the bulk gaps of the same crystal, its skin against its own interior. FDT: 𝗌_gap < 0, a sign with no magnitude. Comparator: the van der Waals-corrected relaxation of the same slab across functionals at room temperature, with the outer gap’s thermal expansion in excess of the bulk gap’s from the slab’s quasi-harmonic free energy, −0.008 to +0.008. A fit error of 0.003 and a surface scatter of 0.004 give 0.0050 per surface, and detection needs the mean below −0.016, the lower end less 4.530 × 0.0050/√8. A contraction of 0.030 below the upper end is excluded when the one-sided test of the mean above −0.022 rejects; a detection is read as the skin’s packing when it survives the removal of surfaces with an oxide coverage above 1%.

Executed under the declared error models, the Student rules reject a null at exactly 0.00135 with the truth at the comparator’s end on the detection side, at 0.00010 and 4 × 10⁻⁶ for T-⁠1 at the interval’s centre and far end, and at 5 × 10⁻⁵ and 8 × 10⁻⁷ for T-⁠2; T-⁠5’s rule holds 0.00135 at the lower end and stays below 10⁻¹⁰ at the centre and the upper end. Their powers are 0.75, 0.94 and 0.99 for T-⁠1 at 0.004, 0.005 and 0.006 above the upper end; 0.82, 0.93 and 0.98 for T-⁠2 at 0.0030, 0.0035 and 0.0040; and 0.80, 0.94 and 0.99 for T-⁠5 at contractions of 0.010, 0.012 and 0.014 beyond the lower end. The permutation rules hold their exact sizes, and their powers are 0.74, 0.94 and 0.99 for T-⁠3 at 4, 5 and 6 nrad per domain, with the comparator at its upper end, and 0.34, 0.7 and 0.9 for T-⁠4 at 30, 40 and 50 nrad per hand, in 3 × 10⁵ trials each. Removing the added input changes each decision: at the test points the expected T-⁠1 contrast moves from −0.0028 to the comparator’s −0.0078, a shift of 0.0050 against the detection margin of 0.0033; T-⁠2’s from 0.0041 to 0.0006 against 0.0024; T-⁠3’s contrast from 12.2 nrad to the Barnett term’s 2.2 nrad against three standard deviations, 5.4 nrad; T-⁠4’s hand contrast at 50 nrad per hand from 100 nrad to zero against 58 nrad; and T-⁠5’s gap change from −0.020 to −0.008 against 0.0080. The source expects its sequential scheme to apply broadly, particularly to other sliding ferroelectrics, and the five designs carry to any sliding ferroelectric with enrichable constituents, polar domains, a cleavable surface or a twistable bilayer, their comparators recomputed for that material.

FDT agrees with the measured record of general relativity and quantum mechanics without reservation. On the source’s inputs and dictionary the limiting map returns the values those inputs give: its fits return their own amplitudes, dampings, frequencies, shifts, slopes and dressed-frequency coefficient, and its calculations their own potential energy surface, state-resolved drive, saturation temperature, static limit and two-pulse map, each in its own class. The derived values recompute: 2/f_s = 8.3 ps, hc/λ_p = 0.83 eV at 1.5 μm, the probe’s 1.55 eV against hc/λ at the one figure of 0.8 μm, and 5000 pulses at 1 kHz in 5 s, each returning its own input and corroborating no invariant. The source’s close agreement between experiment and theory is an agreement in shape between the calculated 𝖰_max and the fitted 𝖠_s, plotted on two ordinates with no printed scale between them, and its corroboration of the coherent control by ab initio calculations rests on a simulation carrying a parameter fitted to the source’s own data, an in-sample reading.

A print that differs from what its own inputs give stays the source’s, kept apart: the pump’s photon at 0.82 eV in the main text and 0.83 eV in the Methods; the period written 2π/f_s, 26 ps, against 1/f_s = 4.2 ps; 2000 K and 4000 K printed as above and below T_sat = 3100 K, which in that order they are below and above; the probe’s 2.8 mJ cm⁻² printed under the pump’s symbols; the damping slopes at 2.2 and 4.1 mJ cm⁻², printed as extracted from fits to the two-pulse data, which are shown at 2.1 mJ cm⁻² only; the Methods’ “Figs. 2c”, a panel the state-resolved figure does not have; the magnitude |𝒫 − 𝒫_z| where the sliding coordinate is signed; and Eq. 1, which prints no ½ on its quadratic terms and no energy scale, so that for an adimensional amplitude it has the dimension of a squared frequency. In the gravitational window the crystal reads laboratory-small compactness on A-⁠1, and for the same stress tensor and data FDT and general relativity solve the same field equation, G_μν = κT_μν (I-⁠3). Divergences stand only where a core clause supplies what the consensus calculation lacks, a constitutive input for T-⁠1, T-⁠2 and T-⁠5 or an additional source for T-⁠3 and T-⁠4, never the coordinate, a chart or a branch tag alone.