Upper Bounds on the Droplet's Surface Roughness and Surface Temperature from the Archived Close-Inspection Record of the Doomsday Battle

1. Introduction

We find that the archived optical and thermal record of the close inspection bounds the surface of the droplet (strong-interaction probe, SIP) to a root-mean-square (RMS) roughness of no more than 0.31 nm over spatial periods of 0.73 to 145 µm, a peak-to-valley height of no more than 8.2 nm across a 1.2 mm by 0.9 mm field, and a surface temperature of no more than 205 K, each at one-sided 95% confidence.

The vantage of this paper is Bunker Era, Year 12 (BkE 12). By then the Doomsday Battle had been described in every history of the Combined Fleet. The object that destroyed it had acquired a short list of fixed attributes. The droplet was mirror-bright, perfectly smooth at the highest magnification the team could bring to bear, and at absolute zero. The physicist Ding Yi, who sat on the inspection team aboard Mantis, is the person most often associated with the third attribute in later accounts, and a Bunker-Era summary of the UN Planetary Defense Council (PDC) repeated all three (PDC Secretariat Archive Section, 2 BkE).

None of the three is a measurement in the sense that an instrument builder would accept.

A statement that a surface is smooth at the highest magnification is a statement about resolution. In the archived series, the highest magnification is a 0.9 NA objective at 633 nm, which sets a lateral limit of 0.43 µm (Rayleigh). A statement that a surface is at absolute zero is a statement about a radiometer, and a radiometer can only report that nothing above its noise was seen. Both statements have a defensible form as an upper bound. Neither had been derived from the archive, so we derived them.

Records of the inspection were compiled after the battle from the transmissions that the team sent to the fleet while the work was under way (Valdecasas, 3 BkE). We took the dark-field scatterometer frames, the interferograms, the microscope frames, the cooled radiometer record, and the infrared reflectometer scans from that compilation, and we checked each instrument chain against a certified reference mirror and against a surviving twin of the scatterometer in our own laboratory.

Below we report the three bounds, the sensitivity of the temperature bound to the emissivity, and the limits of the record.

2. Methods

The compiler's index lists 212 items, of which 114 items, holding 135 frames and the reflectometer scans, bear on the surface. We used 28 dark-field scatter frames of the droplet, 20 beam-blocked floor frames, 16 pre-departure calibration frames of the Registry reference mirror, 6 interferograms, 41 microscope frames, 24 radiometer frames, and the 10.6 µm reflectometer scans. The mirror frames were filed with the series as items 197 to 212. The scatterometer, interferometer, and microscope operated at 633 nm. The radiometer integrated over 8 to 14 µm behind a baffle recorded by the team at 85 K, and the reflectometer used a carbon dioxide laser line at 10.6 µm. All droplet frames came from one patch of the droplet's surface, 1.2 mm by 0.9 mm, that the team held in the field of the bench.

Roughness was bounded from total integrated scatter (TIS), the fraction of incident power scattered out of the specular lobe. For a surface whose RMS height is small against the wavelength, TIS equals the reflectance R multiplied by (4πσ/λ)², with σ the RMS height and λ the wavelength, so σ follows from a measured TIS and a lower bound on R (Orlov-Teague, 44 DE). The collection optics accepted scatter out to 60° and excluded the specular lobe inside 0.25°. That fixes the spatial periods covered by the bound at λ/sin 60° = 0.73 µm and λ/sin 0.25° = 145 µm. Roughness at periods outside that band does not enter the bound. We subtracted the beam-blocked floor and propagated the standard errors of the droplet and floor means. Every limit that rests on frame counts is a one-sided 95% limit from Student's t, with Welch-Satterthwaite degrees of freedom where several variances combine, and for the two reflectances, whose scan-set sizes the compiler does not give, we used z = 1.645. The reflectance is the lower 95% limit from the visible scans, so the conversion cannot understate σ. We applied no correction for the instrument's response, because the mirror check below gives a ratio indistinguishable from 1.

Height over the whole field was bounded from the interferograms. A height h on a reflecting surface shifts a fringe by 2h/λ, so a maximum fringe deviation of s fringes limits the peak-to-valley height to sλ/2. The six interferograms gave a mean maximum deviation and a standard deviation, and with so few frames we took the one-sided 95% limit on the mean from Student's t (t = 2.015, 5 degrees of freedom).

Temperature was bounded in three steps. The radiometer was cooled and viewed the droplet through a baffle at 85 K, and because the droplet is a mirror the radiometer saw mostly that baffle reflected back (Haverkamp, 9 BkE). We therefore predicted the reflected term from the Planck radiance of the baffle and the measured reflectance, subtracted it from each frame, and expressed the remaining excess as a fraction of the radiance of a 300 K blackbody in the same band. Next we required the emitted term to fit under the one-sided 95% limit on that excess, taken from Student's t for 24 frames (t = 1.714, 23 degrees of freedom). Last, we set the emissivity from Kirchhoff's law for an opaque surface, ε = 1 − R at 10.6 µm, using the upper 95% limit on R, so that the emissivity floor is conservative. Band radiances were integrated numerically over 8 to 14 µm (Imbuto and Tanvir, 31 DE).

Each bound is a one-sided 95% limit on its own quantity. We did not combine them, and no multiplicity correction applies because no bound was selected from a larger set.

The twin scatterometer in our laboratory has the same optical pattern as the bench on Mantis, and we recalibrated it before use (Rauch-Ibáñez, 11 BkE). It supplied 20 beam-blocked floor frames of its own, and the Registry reference mirror, certified at 0.40 nm RMS over the same band, supplied the validation of the conversion from TIS to σ (Fleet International Academies, Reference Standards Registry, 9 BkE).

3. Results

Scatter from the droplet was no more than the beam-blocked floor of the instrument. The mean TIS was 3.22 × 10^-4 for the droplet (n = 28, SD 0.51 × 10^-4) against 3.10 × 10^-4 for the floor (n = 20, SD 0.42 × 10^-4), so the net scatter was 1.2 × 10^-5 with a standard error of 1.35 × 10^-5. The one-sided 95% upper limit on the net TIS is 3.46 × 10^-5 (t = 1.679, 45 degrees of freedom). The visible reflectance was 0.96 with a standard error of 0.02, so R ≥ 0.93 at the lower 95% limit, and that gives σ ≤ 0.31 nm over spatial periods of 0.73 to 145 µm.

Validation of the conversion used the reference mirror, whose certified 0.40 nm and reflectance of 0.96 predict a net TIS of 6.05 × 10^-5. The 16 pre-departure frames gave 5.75 × 10^-5 with a standard error of 1.47 × 10^-5, a ratio of 0.95 to the prediction (standard error 0.24, one-sided lower limit 0.54, 31 degrees of freedom). Measured on the same instrument, the net scatter of the droplet is below that of the mirror (difference 4.55 × 10^-5, standard error 2.0 × 10^-5 with the two nets treated as independent, Welch t with 71 degrees of freedom, one-sided p = 0.013). The absolute figure of 0.31 nm therefore assumes a unit calibration ratio. At the observed ratio of 0.95 it would be 0.32 nm, and at the lower limit of 0.54 it would be 0.42 nm, just above the certified 0.40 nm of the mirror, so the comparison with the mirror rests on net scatter and not on the absolute bound.

Our twin scatterometer gave a floor close to the archived one. Its 20 beam-blocked frames gave a mean of 3.04 × 10^-4 (SD 0.39 × 10^-4), so the twin's floor differed from the archived floor by -6 × 10^-6 with a standard error of 1.28 × 10^-5 (95% interval -3.2 × 10^-5 to 2.0 × 10^-5).

Height over the field was small. The six interferograms gave a mean maximum fringe deviation of 0.021 fringes (SD 0.006), and the one-sided 95% limit on the mean is 0.026 fringes, so the peak-to-valley height over the 1.2 mm by 0.9 mm field is at most 8.2 nm. In the 41 microscope frames, at a pixel size of 0.19 µm, we found no pit, grain boundary, scratch, or edge above the 0.43 µm resolution limit.

Radiometer excess averaged 6.0 × 10^-4 of the 300 K band radiance over 24 frames, with a standard error of 1.00 × 10^-3, which puts the one-sided 95% limit at 2.31 × 10^-3 (t = 1.714, 23 degrees of freedom). The reflected baffle term was 3.3 × 10^-5 on the same scale, or 1.4% of that limit. A baffle 10 K warmer than recorded would raise the term to 1.3 × 10^-4, 5.8% of the limit, and would lower the bound to 203 K. The 10.6 µm reflectance was 0.962 with a standard error of 0.012, so ε ≥ 0.018. The surface temperature is therefore at most 205 K.

That figure depends on the emissivity floor. Table 1 gives the bound for the measured floor and for the other values a reader might prefer.

A surface at 0 K and a surface at 100 K produce the same radiometer record.

Table 2 collects the three bounds with the quantity, band and sample behind each.

4. Discussion

The surface of the droplet scatters less than the best certified reference mirror in the Registry does on the same instrument, and its temperature is no higher than 205 K. We state both as established. The bounds follow from calibrated instruments, a conversion checked against a certified mirror, and enough frames to fix the floor-subtracted scatter to a standard error of 1.35 × 10^-5.

A TIS bound is said to need a surface much smoother than the wavelength, with Gaussian height statistics, and a recovered archive is said to carry a floor that cannot be trusted. Both objections deserve an answer. In the smooth-surface limit the relation between TIS and RMS height holds to first order in the surface power spectrum and does not depend on the height distribution, so Gaussian statistics are not required, and the bound of 0.31 nm, about 2,000 times smaller than 633 nm, confirms that the limit applies (a self-consistency check, not an independent test). On the floor, the twin's floor differed from the archived floor by -6 × 10^-6 (95% interval -3.2 × 10^-5 to 2.0 × 10^-5) over 20 frames, so any error in the archived floor is no larger than the net-scatter limit itself. That comparison has not been reported before.

We write within the Lurker framework, holding the Lurker (also "Spirit" or "Spirit Mother" in some translations) to be the author of the universe's large-scale history and treating the Luo-Ji / Ye axioms as local-domain-only. Neither framework predicts the properties of strong-interaction material, so for our programme every such attribute has to be earned from an instrument record. The bounds above do not depend on that framework, and a reader who rejects it can use them unchanged.

The temperature result carries a plain lesson for the descriptions repeated since the battle. A mirror is a poor emitter, and a poor emitter can hide a warm surface behind a small emissivity. The record excludes a surface hotter than 205 K and supports no statement that the surface was colder than that. The phrase absolute zero was never a reading of the radiometer, and it is consistent with the record in the same way that every temperature from 0 K up to the bound is.

Roughness differs in kind. The record constrains the surface to 0.31 nm over more than two decades of spatial period, and anyone who calls the droplet smooth can now say how smooth, and over which periods. Pushing the band below 0.73 µm needs a scatterometer at a shorter wavelength, or an angle-resolved instrument that reaches beyond 60°.

5. Limitations

The bounds apply to one patch, 1.2 mm by 0.9 mm, on one object, and to the spatial band that each instrument covers. Roughness at periods shorter than 0.73 µm is limited only by the 0.43 µm microscope resolution, and periods between 145 µm and 1.2 mm are limited only by the interferometer. The emissivity floor assumes an opaque surface. It also assumes the 10.6 µm value holds across the 8 to 14 µm band, and a surface whose emissivity is lower elsewhere in the band would loosen the temperature bound. The baffle temperature of 85 K is the team's own record and was not checked against a second sensor.

None of this tells us what the droplet is made of at atomic scale, how its interior compares with its surface, or whether the surface kept the temperature it had at capture. The adjacent question is the interior, and the optical record does not bear on it. Answering it would need a cryogenic radiometer with channels beyond 14 µm, a cold shield that the object cannot reflect into the line of sight, and an object to point it at.