Phase fixation in the stripe phase of a polariton condensate

The source’s microcavity is a Fabry–Pérot resonator between two distributed Bragg reflectors of 15 and 13 SiO₂/Ta₂O₅ layers, centred at 490 nm and deposited on 30 nm ITO electrodes on quartz. Each reflector carries a 35 nm film of the ladder-type polymer MeLPPP, which hosts the exciton resonance, under 20 nm of protective Al₂O₃, and antiparallel orienting layers; glass spacers of 1.5 to 2 μm make the cavity a slight wedge, and the nematic liquid crystal LC2091*, with ordinary and extraordinary indices of about 1.57 and 1.98, fills it. A voltage on the electrodes rotates the liquid-crystal molecules in the yz plane the orienting layers define: modes polarised along y at normal incidence, labelled H, tune with the voltage, modes polarised along x, labelled V, do not, and both tune with the position on the wedge. The source fabricated a series of similar samples because the polymer’s optical quality degrades over a few days to a few weeks.

Angle-resolved photoluminescence of sample 1 shows the modes 15H and 14V entering resonance at 3 V in the typical “Rashba” spectrum, the regime of Rashba–Dresselhaus spin–orbit coupling (RDSOC). The source describes the dispersion and polarisation at small k with a 2 × 2 Hamiltonian written on the circular basis,

with 𝖬_pol the polariton mass, 𝖽_HV the H–V detuning, 𝖻_HV the mass-difference coefficient and 𝒜_RD the RDSOC coupling. As the voltage takes 15H out of that regime toward the exciton resonance at E_X = 2.715 eV, its mass, fitted with a parabola, rises strongly, which the source reads as strong mixing of the photon with the exciton. A Tavis–Cummings model, its excitons spread as a Gaussian of 35 meV standard deviation taken from the polymer’s absorption and its single photonic mode taken from a Berreman calculation, fits the mass against detuning with the Rabi splitting ℰ_R = 93 meV and gives the lower polariton branch an exciton fraction of about 5.5% in the RDSOC regime. Rerun with the polymer’s index held at its background value of 1.7, which removes the exciton, the Berreman calculation changes the photonic mass by about 19% over the tuning range, and the source writes that this simulation “proves that the much bigger mode effective mass change observed in Fig. 2 can be explained solely by exciton-photon coupling”. A separate two–damped-oscillator fit uses a coupling of 90 meV and a damping difference of 70 meV, and at zero detuning a broadening larger than 90 meV leads to the weak-coupling regime, where the real parts of the mode energies are degenerate.

Under pulsed nonresonant excitation, with 181 fs pulses at 425 nm and 500 Hz in the photoluminescence measurements, the emission of the V-polarised mode rises exponentially at a threshold of 95 μJ cm⁻², its linewidth narrows and it shifts weakly to the blue, the source’s evidence of polariton condensation. Condensation, assisted by a vibron resonance, occurs about 200 meV below the excitonic peak in modes of 5% exciton fraction, and its blueshift of 4 meV is almost three times lower than the 11 meV between the lower-branch and bare-photon energies at the exciton–photon detuning of −190 meV, which the source writes “demonstrates that the strong coupling regime holds above the condensation threshold”. In the RDSOC regime the bands show two minima at ±k₀ ≈ 1 μm⁻¹, described by the Hamiltonian with 𝖽_HV = −1.3 meV and 𝒜_RD = 2.7 meV μm. The eigenstates at the two minima are elliptically polarised, with opposite circular and the same linear polarisation degree; the measured linear degree is 16%, against an analytical value of 23%. For an equilibrium condensate in the two minima the source’s model gives polarisation stripes, a spin helix, with a contrast set by the circular degree, about 80%, and density stripes with a contrast set by the linear degree, about 16%, of period π/k₀; stripes placed completely at random from one shot to another would lose contrast as 1/√N over N averaged shots, to 0.7% over 500.

Averaged over 500 shots at 7.6 V, the real-space emission shows vertical stripes parallel to y. Like the emission at 0 V, which shows bright spots on a homogeneous background of 25 μm diameter and no periodicity, it is affected by disorder: the disorder potential acting on the polaritons, read from the inhomogeneous linewidth below threshold, is a few meV, with fluctuations of micrometre size. The Fourier transform of the stripe image peaks at k_x = 2k₀ ≈ 2.1 μm⁻¹. Its ratio to the peak at k_x = 0 estimates the stripe contrast at 2.2%, which the source writes is “three times more than the expected theoretical value, demonstrating the occurrence of a moderate pinning mechanism, either by the pump itself or by the disorder potential”. The polarisation stripes have a similar period and a contrast of 40%.

Single-shot experiments on samples 2 and 3, with pulses of about 50 ps, much longer than all characteristic timescales, show two emission maxima in k-space in the RDSOC regime and one in the parabolic regime, and the spacing of the two, 2k₀, is in full agreement with the source’s theoretical expectation across the voltage. Each peak’s full width at half maximum, 𝗐_k ≈ 0.5 μm⁻¹, sets a minimal coherence length 2π/𝗐_k ≈ 12 μm for each component, and the noncondensed emission is thermal at 313 ± 45 K. Through one linear polariser, where the source’s model gives a contrast of 100%, two shots show their most intense stripes approximately in antiphase, their spatially averaged phases −0.47π and +0.48π. Spatial fluctuations of the two components’ phases, the source writes, would destroy the interference and leave many dislocations, so that the observation of stripes “directly proves” that the relative phase “is constant over the condensate’s spatial and temporal extension”.

Fourier transforms of 500 single-shot images give each shot’s fringe phase. At high condensate density the histograms of samples 2 and 3 show a uniform distribution, near-uniform in the caption’s words, which the source writes “suggests the absence of pinning and fully demonstrates the spontaneous breaking of translational invariance”, and the contrast of the integrated image decays approximately as 1/√N, as expected for completely random phases. At a condensate density about four times smaller, the data corresponding to a change of the pump density by only 10 to 20%, the histogram shows a preferential phase, a partial pinning, and the contrast does not decrease further after averaging of about 10 realisations. Simulations with the hybrid Boltzmann–Gross–Pitaevskii equation, using the dispersion parameters determined from the experiment, reproduce both the pinning at low density and the random positioning of the stripes once the interaction energy becomes comparable with the disorder; under continuous pumping they give gapless, linearly dispersing bogolons, a dispersion the source does not measure.

In one single shot a dislocation of the fringes, and in the phase extracted from its Fourier transform a vortex near the origin, mark a topological defect that, in the source’s words, spontaneously appears in the condensation process; the uncertainty on the vortex position is of the order of the period of the stripes. Vortices stand at random positions, in about 10% of the experiments, with a probability depending on the pumping power, and the source cites the Kibble–Zurek mechanism for their random creation during condensation. It notes that the suppression of the phase’s sensitivity to disorder with increasing density occurs “not for a flow, but in the course of the condensation process”. Its discussion states that the robustness against disorder together with the vortices “suggests the superfluid behavior of the stripe phase, even if this last aspect would require further quantitative analysis”, and that the observations confirm the supersolid nature of the observed phase.

There is one substance, the omnium, and the microcavity, mirrors, films and liquid crystal together, is one of its patterns. 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), and the analysis reads the cavity resonator as the stroke’s own support and its two reflectors as that support’s wall. The Fabry–Pérot photon mode is the one spin-one stroke of the quintuple identity, photon = graviton = gluon = phonon = magnon read in different density and geometry regimes, and the source’s simulated bogolons are the same stroke in the condensate’s regime, simulated as the source simulates them (I-⁠2). The exciton is a matter mode, an eigenmode of the polymer segment’s own support rather than a constituent placed in a separate box; the carrier identity never takes in the braided closures of matter, and the polariton keeps the source’s name for the mixed mode (I-⁠2, I-⁠5).

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 analysis’s lift the imaginary unit is the unit bivector of that phase plane, and the projected oscillator turns as a single-angle rotor,

Two such phases make the stripes. The source’s density for components at ±k₀,

places the fringes by the relative phase η alone, while the global phase 𝖿_g drops out of the density. The source’s statement that the relative phase is constant over the condensate’s spatial and temporal extension is, in the analysis’s reading, phase fixation across one macroscopic pattern, and the near-uniform histograms are the image of ignorance of each shot’s initial helical phases: experimental evidence of I-⁠4’s quantum face, at the precision of the source’s integrated images.

Every step is a system-wide settlement: the global optimisation is solved at every step from the state the system is in, events do not commute, and responses are system-wide and many-to-many, never one-to-one, particle by particle (0.1a). One relative phase places the stripes across the whole condensate in each shot, and the analysis reads that shot’s pattern, with the vortex that, in the source’s words, spontaneously appears in the condensation process, as one such settlement. The core names no objective for the optimisation: the source’s Gross–Pitaevskii, gain and relaxation terms remain its method, and its cited account of vortex creation remains the Kibble–Zurek mechanism.

Observed structure is nodal architecture: condensation, crystallisation and percolation are global eigenmode snaps of the standing-wave network, and the mode geometry is set by the boundary conditions, the clamps, not by the medium (P-⁠2). The stripes are the interference of the two k-components, their period π/k₀ fixed by the two minima, and the mirrors and the liquid crystal’s optical axis, which the voltage turns in the yz plane, set those boundary conditions. In the analysis’s reading the stripe phase in the source’s microcavity, a cavity the source describes as allowing “the achievement of room-temperature supersolidity for a quantum fluid of light”, is experimental evidence of condensation and crystallisation as one snap. P-⁠2’s medium is the omnium, and the liquid crystal is a material pattern of it.

A magnetic field is a macroscopic helical density gradient of the omnium: imposing a physical twist 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 (P-⁠5, I-⁠2). The source traces the RDSOC to “the chirality of the system as a whole”: drawn as three lines, the two mirrors and the tilted optical axis look like a letter Z to a beam with a nonzero in-plane wave vector, a figure different from its mirror image. The analysis reads that as the cavity’s geometry acting on the stroke’s spin, its momentum E/c carrying spin ħ at the helix radius λ̄ = ħc/E = c/ω (I-⁠2). A liquid crystal that rotates in one yz plane between antiparallel orienting layers carries no physical twist, so the first clause of P-⁠5’s twist sentence, silent on an untwisted material, establishes by twist no magnetic field in the source’s cavity, and its two minima stay degenerate; the twisted cavities and cells of T-⁠1 and T-⁠2 supply the twist.

P-⁠5 also reads vortex lines as bivector topological defects. The analysis reads the fringe vortex of the source’s Fig. 5 as one, at the source’s vortex-position uncertainty, of the order of the stripe period: its winding lies on the helical phase axis at fixed field identity, and it is neither a braid count nor a measured magnetic moment.

Every density cone carries a flow direction, the cone’s own circulation sense: the same flow direction repels, the opposite attracts, and chirality is one carrier of a flow direction, not the only one (P-⁠4). The analysis reads each circular component’s rotation sense as its flow direction. The source cites same-spin and opposite-spin interaction constants 𝔤₁ and 𝔤₂, with 𝔤₂ < 0 and |𝔤₂| ≪ 𝔤₁ at negative detunings, and sets 𝔤₂ = 0 in its model; in this reading the same-sense constant repels and the opposite-sense one attracts. That sign reading rests on a cited input, never an experimental datum, the pairs are those the condensate’s one settlement joins, and the magnitude, an I-⁠1 derivation with a declared kernel, enters no number here (O-⁠5, O-⁠6).

I-⁠1 holds at every scale, F = F_maxα₁α₂ with F_max = c⁴/4G, and a driven observable vanishes at zero drive and never recovers the static force. The blueshift and the pumped interaction energy are driven observables, the 95 μJ cm⁻² threshold is a measured drive level, and none is an I-⁠1 force; at condensed-matter compactness the bare law returns about 10⁻⁴¹ N, a correct small number, and the measured binding scale is reached only through a declared response kernel. Every energy of the cavity enters its density ledger once (I-⁠7), rest mass M, stored non-rest energy E_st, pressure work PV and thermal radiation aT⁴V each one channel:

The absorbed pump enters E_st/c² once, as reservoir energy, condensate energy and the film’s heat after absorption, and the liquid crystal’s stored electric and elastic energies enter it once under the AC drive; a splitting, blueshift, detuning or linewidth reads these channels and adds none. The 313 K of the noncondensed emission is the source’s fitted tail temperature, neither a lattice thermometer nor an emission clock’s frequency: a medium’s clock factor rescales the metric in which its flow is free, never the time kept by a clock the medium carries (0.1c).

Mass is a confinement eigenvalue,

with Ω the support domain, B its boundary data, n the strand count, χ the orientation and 𝗍_chart the position on the inside–outside chart, and no form of D, its domains, boundary data or spectrum is written here (O-⁠9). The source’s dispersive masses, of the lower branch, of the H and V modes and of the photon mode, are optical responses of the cavity’s modes, never rest masses, and electron patterns are n = 3 braids. From the two-oscillator eigenvalues the analysis reconstructs the lower-band photon gap and the exciton share,

the source’s 11 meV and 5% at its displayed precision: identity maps on the source’s inputs, corroborating no invariant, beside its separate 5.5% fit and its model interval of 5 to 10%.

Each probe reads a weighted mean over its own region, never a point density (I-⁠3): photoluminescence weights the pump spot and the collection aperture, the Fourier contrast the imaged area and the ensemble of shots, the phase histogram each realisation equally, absorption the illuminated film, reflectivity a spot about 10 μm across, and the interferogram the condensate’s area in both arms of the Michelson interferometer. The brightness of an emission image is never the omnium’s local density, and no map from the one to the other is written here (O-⁠4).

X-⁠1, exploratory. The stripe pattern read on the source’s own inputs: the period 2π/2k₀ is 3.0 μm from the source’s 2.1 μm⁻¹, the random-phase contrast 16%/√500 is 0.7% at one figure, the source’s own identity and no independent test of phase ignorance, and the width-derived coherence scale 2π/𝗐_k carries the single figure of the 0.5 μm⁻¹ width, about 10 μm, where the source prints the bound at two, ≈ 12 μm. The coordinate that places the fringes is a phase, never a location on the density chart. An in-sample re-description, never counted with the tests.

Five tests stand where a core clause supplies what the consensus calculation lacks: the field a physical twist establishes (P-⁠5), the density input an isotope substitution adds at fixed geometry (I-⁠7), or the magnetic projection of a circulation (P-⁠5, I-⁠3a). 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: P-⁠5’s twist-to-moment forms run through I-⁠1 and I-⁠2 with a declared kernel that enters no number here (O-⁠6), and no isotope sensitivity is computed here (I-⁠7), so each test predicts an existence or, under a declared positive sensitivity, a sign. The apparatus, settings and error models are the analysis’s design, not measurements in the source, and every numerical test point is a design assumption, never an amplitude of the core.

Both twist tests use the source’s LC2091* at equal chiral-dopant concentration in each hand, eight samples of each hand and four untwisted controls with antiparallel orienting layers. For T-⁠1, sixteen microcavities of the source design carry one 90° twist of the liquid crystal between orienting layers rubbed at that angle. Without a pump, the source’s white-light reflectivity in wave-vector space reads each minimum’s dip, fitted independently of the comparator and labelled by its circular polarisation degree, with each cavity mounted at 0° and at 180° about its normal and a coin fixing the sign of the spectrometer’s tilt drift between the mountings. Each valley takes opposite laboratory sides in the two mountings, so the laboratory-fixed tilt cancels and the sample’s splitting stays in

For T-⁠2, sixteen 2 μm cells rubbed at 90° on both plates, each between one ITO-coated and one uncoated quartz plate on its own gold back reflector, are read by a zero-area-loop Sagnac interferometer at 1550 nm, which reads the double-pass nonreciprocal rotation and is blind by construction to reciprocal birefringence and natural optical activity. Each cell is read normally, ITO toward the beam, and turned over about the ITO plate’s rubbing direction, so that its ITO-to-uncoated direction reverses against the beam, with a coin fixing the sign of the offset drift; with ϖ_K the measured Kerr rotation, the fixed instrumental offset and the Earth’s field along the beam cancel in 𝗊_K = ½(ϖ_K,normal − ϖ_K,turned), and 𝔡_h,K is its right-minus-left contrast.

1. T-⁠1
   In a twisted cavity the hand splits the two minima. In the analysis’s reading of P-⁠5 the field lies along the twist axis, its sense set by the hand against the cavity’s own polar direction, the order of its 15- and 13-layer reflectors, and the consensus magneto-optical response, independently calibrated nonzero, couples that field to the circular components. FDT: 𝔡_h,E ≠ 0. Comparator: zero, the consensus premise of reciprocity for a nonmagnetic linear cavity, which gives one eigenvalue at opposite wave vectors with natural optical activity included. Detection is the exact two-sided permutation test over all 12870 balanced relabelings of the hands at p ≤ 0.0027, of attainable size 34/12870 ≈ 0.0026. A splitting of 25 μeV or more per hand is excluded when the contrasts shifted against the hand by that size put 𝔡_h,E in the lower tail and the opposite shifts put it in the upper tail, each one-sided at p ≤ 0.00135, of exact size 17/12870; otherwise the result is inconclusive.
2. T-⁠2
   A twisted cell turns the polarisation nonreciprocally. The analysis reads the cell’s axis from ITO to the uncoated plate, its field sense set by the hand against that polar direction. FDT: 𝔡_h,K ≠ 0. Comparator: zero, the consensus premise that a nonmagnetic layer gives no nonreciprocal rotation; the measured Sagnac rotation is independent of that input. The same permutation tests decide, and a rotation of 100 nrad or more per hand is excluded by the same shifted rule.

A detection in either twist test is read as the twist’s field, with the four untwisted controls’ contrasts reported beside it, since the first clause of P-⁠5’s twist sentence establishes by twist no magnetic field in them. The assumed errors are 10 μeV per minimum per mounting, a 5 μeV tilt reset per mounting and a coin-signed 3 μeV tilt drift for T-⁠1, so that 𝗄_qE² = 10² + 5²/2 + 3²/4 gives a per-cavity scatter of 10.7 μeV and a hand-difference scatter of 5.4 μeV; and 20 nrad per reading, a 50 nrad offset reset per reading and a coin-signed 10 nrad drift for T-⁠2, 38 nrad per cell and 19 nrad for the hand difference, a 100 nrad laboratory term cancelling. The resolved scales, the null medians of the permutation critical contrasts, are about 10 μeV and 50 nrad at one figure.

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. The isotope tests use eight pairs of source-design cavities, one member made with MeLPPP synthesised from carbon-13-enriched monomers, at least 99% on the backbone, and one from natural-abundance monomers, by one route and to one molar-mass target, each pair fabricated back to back in one deposition run and measured back to back within a day of filling, a coin fixing each pair’s order against run drift and the polymer’s ageing. The density input is the whole cavity’s independently determined A-⁠1 window at fixed pressure, field radius and packing, never constructed from the observable it predicts; the heavier member’s odds enter through I-⁠6’s bounded map, α_ext(𝔶) = α_ext,nat(1 + 𝔶)/(1 + α_ext,nat𝔶) for a relative odds enhancement 𝔶 ≥ 0, which keeps the window on its open interval, and the density contribution to the log splitting is the integral along the natural-to-heavy path,

with S_dens = ∂ ln ℰ_R/∂ ln α_ext at fixed pressure, field radius, packing and the other channels, positive under the declared positive sensitivity, which is a premise to test and not a measured derivative. Each test reads the eight pair differences, heavy less natural, and detects when their mean less the comparator’s upper end exceeds 4.530 times its standard error, the one-sided Student criterion at seven degrees of freedom and a rate of 0.00135.

1. T-⁠3
   Under a positive splitting sensitivity, the carbon-13 cavity has the larger Rabi splitting. Each splitting is extracted independently through the source’s parabola fits to the lower-branch masses and its Tavis–Cummings fit of mass against detuning, with each film’s own exciton width from absorption, and the observable is the pair difference of ln ℰ_R. Comparator: the zero-point volume, the oscillator strength, the reduced mass and the residual film density and thickness join to 𝔠_R ∈ [−0.02, +0.01] in log splitting, the reduced-mass term −0.020 times the Huang–Rhys factor from carbon-13 lowering the C=C stretch by ½ ln(13/12) ≈ 0.040 in its logarithm at fixed reorganisation energy. At the error model’s pair scatter, cavity and fit errors of 0.01 each and run and ageing drifts of 0.005 each with the pair coin’s sign, the precision criterion is 4.530 × 0.022/√8 ≈ 0.04, and an added contribution of 0.08 above the interval’s lower end is excluded when the statistic below its bound of 0.06 rejects at the same rate.
2. T-⁠4
   Under a positive index sensitivity, the carbon-13 witness film has the higher optical index. Eight witness-film pairs, each film spun on quartz in its cavity’s deposition run, read back to back by ellipsometry at 700 nm, below the exciton absorption, with the source’s background index of 1.7 as the comparator’s evaluation point. With the Lorentz–Lorenz function 𝔏 = (𝔫² − 1)/(𝔫² + 2), d𝔫/d ln 𝔏 = (𝔫² − 1)(𝔫² + 2)/6𝔫 is 0.91 at 𝔫 = 1.7. Comparator: zero-point number density [0, +0.0003], polarisability [−0.001, 0] and packing ±0.001 in log 𝔏 give [−0.0018, +0.0012] in index, rounded outward to [−0.002, +0.002]. At film scatter 0.001, ellipsometric error 0.0005 and a coin-signed drift of 0.0005 the precision criterion is about 0.003, and an added 0.009 above the interval’s lower end is excluded when the statistic below its bound of 0.007 rejects.

A detection in either isotope test is read as the density channel when it survives correcting each pair difference by the pair’s measured chromophore-density difference, from the exciton absorbance per unit thickness by the source’s absorption method. At the illustrative lower comparator end the added contributions give predicted pair differences of 0.06 and 0.007 against ablated values of −0.02 and −0.002.

The fifth test reads the circulation. P-⁠5: mechanical vorticity and magnetic moments are different grade projections of one density-gradient topography, with no carrier exchange; I-⁠3a: the antisymmetric part of the deviation map is the vorticity, the bivector grade, the mechanical circulation whose magnetic projection is that of P-⁠5. In the source-design cavity under its single-shot RDSOC excitation, a weak 700 nm transmission probe, outside the stop band and below the exciton absorption, covers the condensate at a delay inside its pulse, and a balanced polarimetric camera at 1 μm resolution reads the Faraday rotation in a 1 μm diameter disc at each vortex core against an annulus centred 6 μm from it, about two stripe periods at 2π/2.1 ≈ 3.0 μm and beyond the source’s uncertainty on the vortex position. Each core is located from the interferogram’s fork, and the circulation sign 𝔴 = ±1 is read from the global-phase winding in a retroreflected Michelson interferogram, the source’s zero-voltage method carried to the RDSOC regime: 2000 accepted cores per sign, about 2 × 2000/0.1 = 40000 shots at the source’s frequency of vortices.

1. T-⁠5
   The winding of a vortex core shows in a probe’s Faraday rotation. The observable 𝔡_w is the mean disc-minus-annulus rotation at positive winding less that at negative winding, which cancels winding-even rotation. FDT: 𝔡_w − 𝖺_mis𝖢_w ≠ 0. Comparator 𝖢_w: the conventional winding-odd contrast for the same spatial weights, measured independently and never fitted, which holds the orbital moment driven in the ITO and the polymer by absorbed angular-momentum flux, 𝔴ħ per absorbed photon, and the winding-odd part of the condensate’s own spin density at the core, through the TE–TM coupling, called in the source’s introduction the most prominent SOC, which its Eq. 1 omits. A misread share of 0.02 ± 0.005 dilutes signal and comparator alike by 𝖺_mis = 0.96. Detection: |𝗓_w| > 3, with 𝗓_w the contrast less the diluted comparator over its full standard error; a contrast of 20 μrad or more is excluded when the absolute difference plus three standard errors, the misread-calibration term evaluated at 𝖢_w + 20 μrad, stays below 0.96 × 20 μrad. No clause fixes its sign relative to the winding, or its magnitude without a response kernel (O-⁠6).

A detection is read as the projection of circulation when the same statistic, with the core replaced by a vortex-free point in the same shots and the comparator recomputed there, returns none. At per-shot noise of 100 μrad, winding-even scatter of 30 μrad, a location contribution of 20 μrad and a comparator error of 1 μrad, the three-standard-error scale rounds to 10 μrad, and at the test point of 20 μrad the diluted prediction is 20 μrad at one figure, against an ablated zero. Executed under the declared error models, the rules reject a null at 0.00265 ± 0.00016 for T-⁠1 and T-⁠2 in 100000 trials each, at 0.00138 and 0.00137 ± 0.00001 for T-⁠3 and T-⁠4 at the comparators’ upper ends in 10⁷ trials, and at 0.00262 ± 0.00004 for T-⁠5 in 2 × 10⁶ trials; the supplied design benchmarks give a power of 0.531 at 10 μeV per hand for T-⁠1, 0.665 at 40 nrad per hand for T-⁠2 and 0.87 at 15 μrad for T-⁠5. In every test, removing the added input returns the conventional value, zero for the twist and vortex tests and the comparator’s end for the isotope tests.

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 its effective Hamiltonian, its photonic-mass simulation, its Tavis–Cummings fit, its fitted absorption and tail temperature and its phase-to-stripe estimator, each in its own class: reported, fitted, estimated and simulated values stay what the source made them. The reconstructed gap of about 11 meV and exciton share of about 5%, the 3.0 μm period, the 0.7% random-phase contrast and the coherence scale are identity maps on the source’s inputs, each corroborating no invariant. The twist input tends to zero, the added isotope response tends to zero, and the vortex’s extra magnetic projection tends to zero with the conventional orbital and spin comparator kept: limits of new-source responses, never closed endpoints of a curvature ratio. In the gravitational window the cavity 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), with I-⁠4’s weak-field, slow-motion reduction to the Newtonian pair force at leading order. No changed observable follows from a branch tag alone.

The source states that the suppression of the phase’s sensitivity to disorder occurs “not for a flow, but in the course of the condensation process”, and after its explicit superfluid qualifier its conclusion states that the observations confirm the supersolid nature of the observed phase. The analysis keeps both statements as the source’s, adds no measured frictionless flow, vortex circulation or magnetic signal to that record, and names in the five tests where the prospective observables differ.