Horizon formation in a driven matter-wave cavity

The source builds a cavity of light and fills it with matter. A Bose–Einstein condensate of 2 × 10⁵ ⁷Li atoms in the state |F, m_F⟩ = |1, 1⟩, cooled to negligible thermal fraction in a 1064 nm crossed optical dipole trap, sits in a uniform vertical field of 543.6 G that sets the s-wave scattering length to zero. Over 200 ms the atoms move adiabatically into a horizontal 1064 nm lattice of spacing 532 nm and depth 𝒱_L = 12.32 E_R, with E_R = ħ²k_L²/2M the recoil energy of a lattice photon, populating its lowest band, S. A field gradient along the lattice, ramped up over 25 ms, drives Bloch oscillations of period 𝔗_B = 10.70 ms, and a 4 ms pulse of 270 kHz amplitude modulation of the lattice opens an avoided crossing that carries the atoms into the second excited band, D; after 8.6 ms more of Bloch oscillation the gradient ramps back to zero over another 25 ms. The main text gives the modulation depth as 1.58 E_R and the residual quasimomentum as 𝔮₀ = 0.74 ħk_L, near the most linear part of the D band’s dispersion. The band’s calculated curvature changes sign at 0.8104 ħk_L, above which, the source notes, a larger quasimomentum gives a smaller group velocity.

Two 532 nm light sheets, Gaussian barriers of height 𝒱_M = 6.36 E_R, root-mean-square width 𝖻_M = 10 μm and a diameter of four such widths, 40 μm, confine the moving wavepacket, and the source writes that the probability of tunnelling through them is negligible. The first sheet turns on 1.2 mm from the atoms, which reflect from it about 1 ms after the gradient ramp ends; the second turns on 4 ms later, 0.6 mm from the atoms’ original position. The average cavity length between the sheets is 𝓛̄ = 680.7 μm, and an acousto-optic deflector translates the second sheet, x₂(t) = x₂,₀ + 𝖠𝓛̄ cos(ω_dt − Φ), with 𝖠 the fractional amplitude and Φ the phase. The source’s Eq. 3 sets the resonance by the group velocity at 𝔮₀, ω₀ = πv_g/𝓛̄, and a triangle wave fitted to the trajectory of a localised wavepacket in the undriven cavity gives ω₀/2π = 95.1(5) Hz.

A uniformly filled static cavity shows no special dynamics. Driven near resonance, at 95.5 Hz with 𝖠 = 1.45 × 10⁻², a mirror speed of “more than a percent of the effective speed of light” in the source’s words, the diffuse cloud concentrates onto a single trajectory in a few drive cycles. That trajectory meets the moving boundary when it is at the average length and moving inward, the predicted stable fixed point of the one-cycle Floquet map, and an additional fainter trajectory, not predicted by the photonic theory, also emerges in this series of images and in some later data. Starting a more localised wavepacket at phases Φ stepped by π/6 across the cycle, three repeats each over 80 ms, the trajectories leave their different initial conditions and coalesce around one trajectory. Those that start close to the stable fixed point experience, counter-intuitively in the source’s words, stronger dissipation than those that start further away, and a faint extra trajectory appears for a few starts near Φ = π/6 and 5π/3. Averaged over every start, the images show depletion around the unstable fixed point and accumulation around the stable one.

Imaged stroboscopically at the drive frequency, atoms from a range of starting points drift away from the unstable fixed point and toward the stable one. Between the two, the stroboscopic velocity takes only one sign, in agreement, the source writes, with the curved-spacetime interpretation and the identification of the fixed points with black-hole and white-hole event horizons. A numerical integration of the time-dependent Schrödinger equation reproduces the approach to the stable fixed point for most initial conditions, quantitatively in the source’s word. Near drive phase 0 the wavepacket smears and the sharp trajectories disappear, which the source attributes to repeated encounters with an approaching mirror that blue-shift the wavepacket to higher quasimomenta until it meets the Brillouin-zone edge and leaves the D band by Bragg scattering; its simulated D-band fraction supports that account.

At twice the fundamental resonance, 191 Hz, with 𝖠 = 7.27 × 10⁻³, given as half the previous amplitude so as to preserve the wall’s maximum velocity, an initially homogeneous cavity develops two stable fixed-point trajectories, and a fit to two doubly reflected Gaussians recovers the expected π phase difference between them. With 𝖠 = 2.9 × 10⁻² and a sharp π step in the modulation phase after four complete drive cycles, a wavepacket started at the unstable fixed point first moves toward the stable one and, once the two fixed points switch roles, returns to its original trajectory. The source calls this a demonstration of stroboscopic time reversal.

The source traces every departure from the simplest driven-cavity model to imperfectly linear dispersion. Near the zone edge it smears the wavepackets close to the stable fixed point, and because any curvature in the dispersion makes reflection from a moving barrier change the group velocity, the evolution depends on the wall’s velocity as well as its position, which the source writes explains the extra trajectories at the nodes of wall velocity, phases π/2 and 3π/2. Its Supplementary Information maps the cavity with a moving boundary conformally onto the static cavity and writes the resonant map after 𝔫 cycles in closed form. It adds that the stroboscopic velocity dx/d𝔫 can be interpreted as originating from a nontrivial spacetime curvature, and that its perturbative solution to a sinusoidal drive does not match any canonical general relativistic model, “even disregarding the lack of a singularity”. The simulation starts from a D-band superposition of Bloch waves with a quasimomentum spread of 0.1 ħk_L full width at half maximum, expanded freely to the measured 106 μm full width at half maximum, and fixes 𝔮₀ = 0.736 ħk_L by matching its static cavity to what it calls the experimental value of 95.5 Hz.

There is one substance, the omnium, and quantum mechanics, electromagnetism and general relativity read it as one identity, E/m = c² = 1/(ε₀μ₀) = (d/t)² (0.1). c is the strand rate, the rate at which the one stroke runs, measured on the spot (0.1d). A room, box, cell, domain or grain is that pattern’s own support region, with the pattern’s own boundary data for a wall, never a vessel holding a separate occupant (0.1b): the analysis reads the source’s cavity, 680.7 μm between the sheet centres, as the cloud’s own support region and the two sheets as its boundary data, the B of I-⁠5. The lattice beams and the sheets are open strokes, and the ⁷Li cloud is a closed pattern. Mass is a confinement eigenvalue, m_qc² = ħω_q, and no form of the confinement operator D, its domains, boundary data or spectrum is written here (O-⁠9). The transfer from the S band to the D band and the loss from the D band to the higher F band are the same closure taking another mode of the lattice: no strand count changes, and each is the system-wide settlement of the state it starts from, never a reaction particle by particle (I-⁠5, 0.1a).

The identity photon = graviton = gluon = phonon = magnon is one spin-1 excitation of the density field, read at different density and geometry regimes, and it is carrier-only: the braided closures of matter are never inside it (I-⁠2). A closed pattern’s phase rate exceeds its run, their product c², and only the open stroke has the two equal at every frequency. The source’s map rests on characteristics defined only by initial position and direction of propagation, which is the open stroke’s non-dispersive run. The cloud in the D band runs at v_g = 0.1221 m s⁻¹ at 𝔮₀ = 0.74 ħk_L, nearly uniformly over a window of quasimomentum, and its run turns over at the inflection, 0.8104 ħk_L, where it reaches 0.1243 m s⁻¹. The analysis reads the photonic map as holding where that run is uniform and departing where reflection from the moving sheet changes it: the faint trajectories and the smearing that the source attributes to residual curvature in the dispersion are the closed pattern’s.

On the source’s inputs its Eq. 3 gives the hard-wall resonance 89.7 Hz at 𝔮₀ = 0.74 ħk_L, and no quasimomentum gives more than 91.3 Hz. The Gaussian walls shorten the round trip: in a semiclassical count the cloud turns 14.9 μm from a sheet centre, where the sheet’s potential reaches the 2.10 E_R the cloud carries above the bottom of the D band, and the round trip comes to 92.4 Hz, against the fitted 95.1(5) Hz. Eq. 3 is the hard-wall idealisation and never calibrates 𝔮₀, and the remaining gap is the source’s own.

The source’s resonant map, its Eq. 1, written in the analysis’s names, carries a starting position x through 𝔫 cycles:

At 𝖠 = 1.45 × 10⁻² the rate 2π𝖠 = 0.091 per cycle doubles the stable point’s density in 7.6 cycles, and the source’s 80 ms at 95.5 Hz are 7.64 cycles. Replacing Φ by Φ + π exchanges the two fixed points and returns the map with e^2π𝖠𝔫 replaced by e^{−2π𝖠𝔫}: at leading order the shifted map is the inverse of the first.

Energy injection, acoustic, electromagnetic, thermal or parametric, excites standing density waves, and mode geometry is set by the boundary conditions, not by the medium (P-⁠2). The source drives one sheet’s position. Its diffuse cloud collects on the trajectory that meets the moving boundary at the average length moving inward and thins around the one that meets it at the same position moving outward; a drive at twice the resonance gives two stable trajectories; and, in the source’s words, the waveform “can vary the number of fixed points and exchange their stability character”. The analysis reads this as the structure a parametric energy injection excites, its geometry set by the boundary conditions. It places no trajectory on P-⁠2’s nodal loci and no sheet on P-⁠3’s clamps, and evaluates no α, gradient of α or chart for the structure, which the source’s map returns.

A pattern follows its geodesic and carries no reading of the curvature it crosses; two neighbouring patterns do, and so do a pattern and a reference path (I-⁠3a, Use 7). The source’s offset trajectories rephase after several cycles of boundary modulation and coalesce around a single trajectory, and its stability criterion turns on whether small deviations compound or decay; the map’s slopes, e^{−2π𝖠𝔫} at the stable point and e^+2π𝖠𝔫 at the unstable one, are separation rates of neighbouring trajectories. The analysis reads these as the between-neighbours structure the core gives curvature. One trajectory’s stroboscopic drift alone reads none, and the source’s “flow” stays its word for the map’s drift: the cavity’s map carries no clock factor.

The flat special-relativistic cone is the uniform-density idealisation, and refraction, lensing and horizon formation are one phenomenon (P-⁠4); there is no singularity anywhere (0.8). Between the source’s fixed points the stroboscopic velocity takes one sign only, and the source writes that its perturbative solution does not match any canonical general relativistic model, “even disregarding the lack of a singularity”. The analysis reads the one-way region as horizon formation at the cloud’s own run, with no singularity. It computes no aperture index and no α for the cavity: the band’s run is a closed pattern’s run, and the dispersion curve in the (𝔮, ℰ) plane is no density cone.

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 a process run backward is never the mirror of the forward event (0.1a). After the sudden π phase step the wavepacket returns to its original trajectory. The analysis reads the return as a new forward settlement under the flipped drive, whose leading-order map is the inverse of the first; the exact return that the map expects is not necessarily the evolving system’s optimum, and the source prints no return fidelity.

The analysis reads the source’s “effective speed of light” as the cloud’s band run, a closed pattern’s run in the frame of the lattice at rest in the laboratory: 0.1221 m s⁻¹, never c and never an effective c. The wall’s peak speed 𝖠𝓛̄ω_d = 5.92 mm s⁻¹ at 95.5 Hz, with 𝖠𝓛̄ = 9.87 μm, is 0.0485 of that run, and π𝖠 = 0.0456 is its ratio to the round-trip run 2𝓛̄ω_d/2π = 0.130 m s⁻¹. The static cavity’s 2𝓛̄ω₀/2π = 0.1295 m s⁻¹ exceeds v_g: the soft walls shorten the round trip, and the rest of the difference is the source’s own. Its “Lorentz boost” is the boost of its model wave equation, whose wave speed it sets to one; closed patterns share every symmetry of the stroke equation (0.1d).

I-⁠1 holds always, at all scales, for two bodies: F = (c⁴/4G)α₁α₂, with 0 < α_i < 1 so that F < F_max. Two ⁷Li patterns one lattice site apart, d = 532 nm, have Schwarzschild radii R_s = 1.730 × 10⁻⁵³ m and exterior inputs α_ext(d) = R_s/d = 3.25 × 10⁻⁴⁷ each, and attract with F = G𝓜₇²/d² = 3.20 × 10⁻⁵⁰ N, with 𝓜₇ the I-⁠7 total of one pattern: small, and a correct prediction. The tuning of contact interactions to zero sets a contact channel, a response reached through a declared kernel. The sheets’ push, the band loss and the drive’s term of the stroboscopic drift are driven observables, zero without their drive, and none recovers the static force.

Every energy of the cloud enters its total gravitating energy once (I-⁠7). The wall’s work, the blue shift of repeated encounters with an approaching mirror, enters as non-rest energy; the Zeeman energy of |1, 1⟩ at 543.6 G enters once, lowering each atom’s rest energy by 3.94 × 10⁻¹⁶ of Mc²; the lattice and sheet light shifts enter once each; and the contact channel enters once, at its zero.

Moving at constant speed, on a pattern’s own clocks and rulers, through a density gradient is gravity (0.1c), and two readings follow for this record. The source measures no gravity: its lattice is horizontal, and its “curved spacetime” and stroboscopic drift are read between neighbours. And the Bloch-oscillation force on the |1, 1⟩ cloud in the field gradient along the lattice is the slow limit of the helical window, in which a pattern whose rest energy depends on where it sits in a twist takes that energy as its clock factor, 𝔪 = E/E₀:

With 𝔗_B = 10.70 ms the push is 1.164 × 10⁻²⁵ N, a gradient of 1.33 G cm⁻¹ on the |1, 1⟩ moment of 0.943 Bohr magnetons at 543.6 G, and the weight, across the horizontal axis, has no component along it. A uniform twist turns a held moment and pushes nothing, and the universality of free fall is the gravitational window’s, where it stands.

Quantum mechanics is helical geodesics on truncated density cones: 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, and the dynamics is deterministic (I-⁠4). The source’s Schrödinger simulation is accepted with its numbers. Each absorption image is a column density integrated along the imaging axis, a profile-weighted mean, so no point value is read off it (I-⁠3), and the source’s normalised density names no normalisation.

X-⁠1, exploratory. The order of two drive segments: a segment at phase Φ followed by one at Φ + π, against the reverse order, from one initial state, compared by their final column densities at equal total cycle count. On the source’s leading-order map the two orders give one final state, the initial state when the two segments are equal; beyond that order reflection from the moving sheet changes the closed pattern’s run, and the two orders are two settlements, read through “events do not commute” (0.1a). The source prints no reverse order and no return fidelity. An in-sample re-description, never counted with the tests.

Five tests stand where a core clause supplies what general relativity and quantum mechanics lack: I-⁠7’s density channel, a constitutive input, for an isotope pair, and P-⁠5’s twist, an additional source. Each fixes its referent, observable, comparator, design, precision criterion, decision rules and an ablation before any data, and a fair coin drawn before each pair sets the order of its two members, which gives any slow drift between them a random sign. No clause fixes the size of an isotope sensitivity, no twist-to-moment form is written here, and no map from a circulating charge or a twist to an I-⁠1 input is written here (O-⁠6), so T-⁠1 and T-⁠2 predict a sign under a declared positive sensitivity and T-⁠3 to T-⁠5 an existence. The apparatus, settings and error models are the analysis’s design, not measurements in the source.

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. Isotope substitution at fixed geometry moves compactness only: for one atom at fixed field radius, the ⁷Li value less the ⁶Li value, Δ ln α_ext = Δ ln 𝓜 = Δ ln M = 0.153917 from the AME2020 masses 7.016003437 u and 6.015122887 u. The partner is ⁶Li, a different-n closure from ⁷Li (I-⁠5), spin-polarised in its lowest hyperfine state at the same field, identical fermions with no s-wave collisions: 10² atoms at T/T_F ≤ 0.24 in the same beams and sheets, the intensities scaled by M₇/M₆ so that 𝒱_L/E_R = 12.32 and 𝒱_M/E_R = 6.36 on the beams’ axis for each isotope, set from measured beam powers and computed polarisabilities, never from a cavity or band frequency. Each test reads eight pairs, back to back, and with O its observable and S_dens the response of ln O to the density channel at the held dimensionless settings,

1. T-⁠1
   Under a positive density sensitivity, the lithium-7 cavity frequency lies above the recoil scaling. The observable is ω₀ by the source’s triangle-wave fit, each isotope at 𝔮₀ = 0.74 ħk_L. Comparator: at matched dimensionless settings the band and the soft-wall round trip scale as 1/M, so Δ ln ω₀ = −Δ ln M = −0.153917, and the transverse occupation of the ⁶Li atoms and the polarisability term make it [−0.1570, −0.1553]. Detection: 𝗍₁ = √8 (𝔡̄ + 0.1553)/𝒮_𝔡 > 4.530, the one-sided 0.00135 quantile of Student’s t at seven degrees of freedom, with 𝔡̄ the mean contrast and 𝒮_𝔡 their standard deviation; at a pair scatter of 0.00842 and a coin-signed drift of 0.0010 the rule resolves a contribution of 𝒵₁ = 0.0136 above the interval. A contribution of 0.025 above the interval’s lower end is excluded when the contrasts reject below −0.1320; otherwise the result is inconclusive. Executed exactly, the size is 0.0014 at the interval’s upper end, and the power 0.26, 0.66 and 0.93 at 0.010, 0.015 and 0.020 above it.
2. T-⁠2
   Under the same sensitivity, the lithium-7 S–D band separation lies above the recoil scaling. The observable is the separation 𝔤_SD at 𝔮 = 0 in the static lattice, read as the first moment of the lattice’s amplitude-modulation line, resolved in quasimomentum by band mapping: 9.059 E_R = 227.6 kHz for ⁷Li at 12.32 E_R, which the source’s 270 kHz meets at 𝔮 = 0.655 ħk_L. Comparator: −Δ ln M with the S band’s transverse term, [−0.1522, −0.1506]. Detection: 𝗍₂ = √8 (𝔡̄ + 0.1506)/𝒮_𝔡 > 4.530, resolving 𝒵₂ = 0.0093 at a pair scatter of 0.00569; a contribution of 0.015 above the lower end is excluded when the contrasts reject below −0.1372. The size is 0.0014, and the power 0.18, 0.52 and 0.83 at 0.006, 0.009 and 0.012 above the upper end.

A detection in either isotope test is read as the density channel when it survives correcting each contrast by its pair’s on-axis depth ratio, measured by Kapitza–Dirac diffraction of each isotope’s cloud released from a 10 kHz transverse trap and analysed beyond the Raman–Nath regime with each isotope’s own recoil, through the depth sensitivity S_V = −0.474 of ω₀ and +0.762 of 𝔤_SD.

P-⁠5: a magnetic field is a macroscopic helical density gradient of the omnium, and vortex lines are bivector topological defects; imposing a physical twist, torsion or helical geometry, on a material is establishing a magnetic field; and mechanical vorticity and magnetic moments are different grade projections of one density-gradient topography, with no carrier exchange. Each test reads the twist between neighbours, the two senses of a winding, a hand or a turn, or against a reference path (0.1c, I-⁠3a), so a conventional response that acts on both alike cancels in the contrast.

1. T-⁠3
   A sensor filling a quantized vortex reads its winding. In the source’s condensate at 543.6 G, a two-photon Raman pulse with a Laguerre–Gauss beam of winding 𝔴 = ±1 moves about half the atoms to |2, 1⟩ with circulation 𝔴 about the vertical bias, and the rest stay in |1, 1⟩ without circulation and fill the vortex core as the sensor. Observable: the sensor’s |1, 1⟩ → |2, 2⟩ line, 2187.0 MHz at 543.6 G with a slope of 2.719 MHz G⁻¹, read by Rabi spectroscopy; 4000 pairs, one shot at each winding, 𝔡_i = ν_i(+1) − ν_i(−1). FDT: 𝔡 ≠ 0, the circulation’s magnetic projection adding to the bias with a sense the winding sets. Comparator: zero for the winding-odd part; the internal line is blind to the sense of a centre-of-mass circulation, and the interspecies shift and the vortex component’s own dipole field are winding-even. Detection: |𝗍₃| > 3.0019, resolving 𝒵₃ = 0.0923 kHz at 1.374 kHz per shot, of size 0.0027 and power 0.35, 0.82 and 0.97 at 0.040, 0.060 and 0.075 kHz per sense; an effect of 0.100 kHz per sense or more is excluded when both shifted one-sided tests reject.
2. T-⁠4
   Quartz enantiomers beside the cavity shift the cloud’s line oppositely. Sixteen quartz cylinders, eight left-handed and eight right-handed, 10 mm across and 40 mm long with the optic axis along the cylinder, each in turn held 15 mm from the cavity’s centre, its axis vertical along the bias, outside the glass cell. Observable: the |1, 1⟩ → |2, 2⟩ line of the cloud held in the static cavity, 500 shots with the crystal in its holder and 500 with the holder empty, so each crystal’s reading 𝗊 is its line in the holder less its line with the holder empty, the reference path, and 𝔡 = 𝗊̄_left − 𝗊̄_right. FDT: 𝔡 ≠ 0, the helical geometry establishing a field whose sense the hand sets. Comparator: zero for the hand-odd part; the crystals are nonmagnetic, time-reversal-invariant mirror images, and their diamagnetic field at the cloud, about 0.4 kHz of line shift, is the same for both hands. Detection: the exact two-sided permutation test over all 12870 hand relabelings at p ≤ 0.0027, of size 34/12870 ≈ 0.0026; at a hand scatter of 44.1 Hz it resolves 132 Hz, with power 0.4, 0.8 and 0.95 at 75, 100 and 125 Hz per hand, and 150 Hz per hand or more is excluded when both shifted tails give p ≤ 0.00135.
3. T-⁠5
   A twisted rod beside the cavity moves the Bloch period. A rod of polyether ether ketone, 3 mm across and 60 mm long, stands vertical 15 mm from the cavity’s centre on the horizontal line at 45° to the lattice axis, clear of every beam, its lower end clamped and its upper end turned by ±0.20 rad: a surface shear of 0.0050, a surface stress of 6.5 MPa and a torque of 0.034 N m. Observable: the S-band Bloch period 𝔗_B of the source’s cloud in its field gradient, fitted over five periods; 400 pairs, one run at each sense, 𝔡_i = 𝔗_B(+) − 𝔗_B(−). FDT: 𝔡 ≠ 0, the twist establishing a field whose gradient along the lattice axis moves the push −∂_zE. Comparator: zero for the twist-odd part within ±0.02 μs; torsion changes the rod’s diamagnetism azimuthally about its axis, which leaves no field outside the rod, and the clamp region’s strain is bounded at about 0.01 μs, doubled. Detection: |𝔡̄| > 3.0189 𝒮_𝔡/√400 + 0.02 μs, 𝒵₅ = 0.2489 μs at 1.07 μs per run, of size 0.0011 and power 0.26, 0.75 and 0.98 at 0.10, 0.15 and 0.20 μs per sense; 0.30 μs per sense or more is excluded when both shifted tests reject.

A detection is read as the circulation’s when it survives correcting each pair for the difference in the circulating component’s imaged atom number, as the hand’s when the hand-even part matches the computed diamagnetic shift within three standard errors, and as the twist’s when the sense-even part matches the untwisted rod and the sense-odd part vanishes as the unclamped rod turns rigidly. At each test’s power point, removing the added input returns the comparator: 𝔡 = −0.1403 against −0.1553 for T-⁠1, −0.1386 against −0.1506 for T-⁠2, 0.120 kHz against 0 for T-⁠3, 200 Hz against 0 for T-⁠4 and 0.30 μs against 0 for T-⁠5, each difference above its test’s resolved scale.

FDT agrees with the measured record of general relativity and quantum mechanics without reservation. On the source’s own inputs and dictionary the limiting map returns its values: the resonant map, its Eq. 1 and Eqs. S16–S23, with the fixed points of its Eq. S20; its Eq. 3; the time-dependent Schrödinger equation of Eqs. S24–S26; the fitted 95.1(5) Hz and π phase difference; the sheets’ 40 μm diameter as four root-mean-square widths of 10 μm; and the fixed points 𝓛̄/6 and 5𝓛̄/6 of the Supplementary Information’s off-resonant example, its Fig. S2. Each is the identity map, corroborating no invariant, and for the same stress tensor and data I-⁠3 containment returns the same curvature as general relativity. A print that differs from what its own inputs give stays the source’s: the fitted 95.1(5) Hz against the 89.7 Hz its Eq. 3 gives on those inputs, 7.27 × 10⁻³ against half of 1.45 × 10⁻², and the simulation’s “experimental value of 95.5 Hz”.

The tests converge on their comparators, T-⁠1 on [−0.1570, −0.1553], T-⁠2 on [−0.1522, −0.1506], T-⁠3 and T-⁠4 on zero and T-⁠5 on zero within ±0.02 μs, and they diverge only through I-⁠7’s constitutive input or P-⁠5’s additional source, never through α, a chart or a branch tag alone. Where FDT, general relativity and quantum mechanics converge, FDT reproduces their results; where it declines their assumptions, here an empty vacuum, a curvature read off one trajectory’s drift and a return read as time run backward, it is under no obligation to reproduce them.