Hand-considered hypotheses, 2401–2500
Precision engineering and optical manufacturing. These entries cover removal processes, surface metrology, alignment, tolerancing, machine-tool error, polishing, and mount-induced distortion. Each was written on its own and given the logical form that its argument actually takes; the form is printed above the entry so that structural repetition would be visible. None has been tested. Parenthetical numbers are estimates, not measurements. This is the twenty-fifth hundred of the series.
Removal processes and tooling
Misattribution
H2401. Astigmatic figure error that appears only after a mirror is clamped is routinely charged to the polishing run, but on ring-mounted blanks it is usually the mount's own out-of-plane flatness printed through the part. Measuring the same optic on three supports of different stiffness should settle it: the astigmatism's axis follows the mount, not the polish tool path, and the amplitude tracks support flatness rather than the dwell schedule.
Hidden term
H2402. Preston-type removal models leave out the lap's thermal expansion, yet a pitch lap warmed by friction along the work track develops a bump that removes extra material at the track centre. Running an identical dwell schedule at two spindle speeds, matching the Preston product while changing the frictional heating, isolates the term; its magnitude should follow the measured lap temperature rise and not the nominal pressure.
Identity
H2403. The tool-mark band and the residual-waviness band that shops report separately after sub-aperture polishing are one continuous spectrum seen through two different instrument transfer functions. A single measurement with the transfer function stated should reproduce both bands from one power spectral density, and the claim fails where the two bands do not join at their common spatial frequency.
Bound
H2404. No interferometer can certify structure finer than twice its pupil sampling, and stitching does not evade this because stitching enlarges the field, not the per-frame band limit. Comparing a stitched map against a single-frame map over their overlap should show the stitched map's extra content vanishing above the frame's diffraction cutoff.
Threshold
H2405. Brittle materials grind ductilely below a depth-of-cut threshold governed more by the wheel's effective edge radius than by the material's fracture toughness. Two wheels of identical grit size but different bond hardness should therefore show thresholds differing by a factor of two or more, an estimate a plunge-grinding series would settle.
Reversal
H2406. Using the last element as the alignment compensator reverses the sign of the system's sensitivity to the first element's decenter once the beam is faster than roughly f/1.5, so the conventional instruction to adjust the last lens doubles coma instead of removing it. Nodding the first element and recording the last element's required motion exposes the sign change without any wavefront model. The crossover aperture is a property of the design rather than of the alignment state, so it can be computed before the hardware exists.
Count
H2407. A three-axis diamond turning machine carries at most twenty-one independent geometric error terms, and a calibration table with more rows is a re-parameterization of those twenty-one. A rank test on the measured error matrix shows the redundancy, and the null space names which rows were invented rather than measured.
Artefact mimics effect
H2408. Stylus roughness on fine-ground glass can imitate a fracture-layer signature, because a two-micrometre tip cannot enter valleys narrower than its radius and the recorded profile is a dilation of the true surface. Measuring with two tip radii and extrapolating to zero radius separates tip geometry from real skewness.
Scale argument
H2409. Gravity sag of a mounted optic goes as the fourth power of clear aperture over the cube of thickness, so a mount validated at 100 mm is not merely worse at 200 mm but worse by roughly sixteen times. This is why scaling up a proven mount, rather than re-analysing it, is the most common mounting failure in the record.
Inversion of standard practice
H2410. Polishing after mounting is normally forbidden, yet for thin meniscus mirrors the better sequence may be to mount first and figure last, because the mount's distortion is then removed by the figuring run instead of being added to it. Two identical blanks figured before and after mounting would show which sequence leaves the smaller mounted figure error.
Selection effect
H2411. Reported polishing yields are biased upward because parts that fail inspection are reworked before the formal measurement, truncating the error distribution at the tolerance limit. Recording the first measurement of every part, unreworked, should move the distribution's shoulder outward and widen it.
Unit trap
H2412. A drawing specified in waves at 633 nm and a shop that works in nanometres differ by a factor of 1.24 in the same sentence, enough to consume a typical figure tolerance. Auditing a shop's records usually turns up both units on one drawing, with the tighter of the two applied to the looser number. An audit at the point of use, rather than at the point of issue, finds these before the part does.
Derived quantity
H2413. The slope tolerance quoted for a mirror is not independent of its height tolerance; it is the height error's derivative at a stated spatial period, so specifying both is over-specification. One of the two is always slack, and a specification written as a power spectral density shows which one is doing no work.
Misattribution
H2414. Progressive waviness in single-point diamond turning is often booked as tool wear, but at constant chip load it is more often the spindle's asynchronous error growing with speed. Cutting the same path at several spindle speeds with the tool unchanged holds wear constant while varying synchronous error.
Hidden term
H2415. Long-path interferometry carries a term from the horizontal air-temperature gradient that midpoint sensing does not remove, because the gradient is not linear along the beam. The residual appears as a slow figure drift whose sign flips when the measurement is repeated from the opposite direction.
Identity
H2416. Edge roll-off after sub-aperture polishing and the turned edge of a machined part are the same convolution of a tool footprint with a step in the workpiece boundary. The claim is testable by checking whether the roll-off width equals the tool footprint at half maximum in both processes.
Bound
H2417. Deterministic figuring cannot converge below the reproducibility of its own removal function, so the achievable figure is bounded by run-to-run variation of the tool influence function rather than by metrology. Iterating a known-good part while requesting zero correction, and recording the residual that appears, measures that floor.
Threshold
H2418. An adhesive bond line prints cure shrinkage through as figure error only below a thickness set by the ratio of the bond's shear compliance to the optic's bending stiffness. For hundred-millimetre class optics the threshold is a few tens of micrometres (estimate), which is why a thicker glue joint is sometimes the more stable one. The threshold rises with the adhesive's modulus, so a stiffer glue tolerates a thinner line.
Aliasing
H2419. Running a smaller polishing tool faster to remove mid-spatial frequency error can increase that error, because the tool's footprint period aliases against the dwell grid. The reversal should occur where the dwell grid period approaches twice the tool footprint, a condition a dwell-pitch series would locate.
Count
H2420. A rotationally symmetric system with N surfaces has a sensitivity matrix of rank at most 2N−1, so a tolerance analysis that adjusts more compensators than that is fitting metrology noise and calling it alignment. A singular-value count of the matrix settles how many adjustments are real.
Scale argument
H2421. Subaperture stitching error accumulates as the square root of the subaperture count only when overlap errors are independent; when the same reference surface is reused, accumulation is linear and the stitched map inherits the reference's low-order error at full amplitude. Stitching with two different references separates the two behaviours.
Inversion of standard practice
H2422. Choosing the flattest reference flat is wrong when the part's error shares low orders with the reference, because the test then reports their difference and hides their sum. A reference whose error is deliberately orthogonal to the expected figure, even with larger RMS, gives the smaller measurement uncertainty.
Selection effect
H2423. Diamond-turned roughness is usually quoted from a small central patch, where cutting speed and synchronous error are most favourable. Full-aperture maps typically show roughness rising toward the edge by a factor near two (estimate), so the quoted number is a best-case sample rather than a process average.
Unit trap
H2424. Converting peak-to-valley to RMS with the factor 3.5 assumes a particular spatial distribution; for a surface dominated by one Zernike term the true ratio runs from about two to four. A tolerance converted this way can be wrong by forty per cent, always in the optimistic direction for surfaces dominated by low orders. The multiplier is not a constant, and treating it as one is the whole error.
Misattribution
H2425. Slow monotonic drift in a precision stage is usually logged as piezoelectric creep, but on a leadscrew stage it is more often the screw's thermal expansion. A step-and-hold at two thermal states separates them, since creep scales with step size and expansion does not.
Hidden term
H2426. Flatness measured on a vacuum chuck contains the chuck's own figure as an additive term, which is why lapping a part while it sits on the chuck improves the measurement without improving the part. Measuring the same part on and off the chuck gives the term directly.
Threshold
H2427. Scratching begins at a slurry particle size that depends on the pad's asperity spacing rather than on particle size alone, so a finer slurry on a worn pad scratches more than a coarser one on a fresh pad. A pad-conditioning series should show the crossing point.
Reversal
H2428. Hardening a pitch lap improves figure convergence and degrades roughness at the same time, and the trade is not monotonic: past a hardness set by the pitch's penetration number both worsen, because the lap stops conforming. The optimum is a saddle, so a one-dimensional hardness sweep will mislead. A hardness sweep that brackets the saddle looks like scatter rather than like an optimum.
Identity
H2429. Print-through in thin mirrors and quilting in lightweighted mirrors are the same elastic response of a plate to periodic support or periodic thickness. The spatial period identifies the cause, and the two differ only in whether the period matches the rib pitch or the support pitch.
Scale argument
H2430. Subsurface damage depth scales with the abrasive's penetration depth, which goes as the square root of load per particle, so doubling the load deepens damage by only about 1.4 times while widening its lateral extent by about two. Depth-only measurements therefore under-report the damaged volume.
Surface metrology and how a surface is specified
Bound
H2431. The decenter tolerance of a surface is bounded by the coma it generates per unit decenter, and that sensitivity rises with the cube of numerical aperture. A system aligned at f/4 and re-housed at f/2 therefore has one eighth of the tolerance, not one half.
Misattribution
H2432. Faint scatter spots seen after final cleaning are often logged as polishing damage but are dried slurry residue. A solvent rinse series with dark-field inspection at each step separates them, since residue count falls monotonically while true damage does not.
Threshold
H2433. Regenerative chatter in diamond turning has a threshold set by the ratio of tool stiffness to workpiece stiffness, and below a critical ratio the process is stable at any spindle speed. A stiffness series at fixed depth of cut locates that ratio. The ratio can be measured on a static rig before any cutting, which makes the threshold cheap to check.
Artefact mimics effect
H2434. A linear phase-shift calibration error in an interferometer produces a fringe-order-dependent error indistinguishable from spherical aberration in a single measurement. Measuring the same part at two cavity lengths separates them, because true spherical aberration is invariant.
Derived quantity
H2435. The polishing rate quoted in nanometres per minute is a derived quantity that hides the pressure and velocity used to obtain it, so two shops quoting the same rate can differ by a factor in removed thickness for the same dwell. Quoting the Preston coefficient instead makes the numbers comparable.
Reversal
H2436. Increasing dwell to remove a local bump can deepen it when the tool is smaller than the bump's width, because the tool's edge removes more at the shoulders than at the crest. The failure is the bull-nose divergence of deterministic figuring and appears as a ring around the original bump.
Overconstraint
H2437. A kinematic mount has exactly six constraints, so any mount with more than six adjustable pads is overconstrained by construction and its figure will depend on the order of tightening. A tightening-order experiment exposes the dependence in a way that a single measurement cannot.
Scale argument
H2438. Bond-induced distortion from CTE mismatch scales with bond length and temperature excursion, but its effect on figure scales with the cube of the diameter-to-thickness ratio. A mount that is benign on a thick blank is destructive on a thin one at the same temperature change.
Hidden term
H2439. An interferometer used with a vertical test beam measures the part in a different gravity state than one used horizontally, and for large optics this adds a term of the order of the part's self-weight deflection. Two laboratories can therefore disagree by an amount neither can see in its own data.
Inversion of standard practice
H2440. Centring a lens by minimizing coma is standard, but in a cemented doublet the minimum-coma position is not the minimum-decentre position, because the cement wedge contributes coma of the opposite sign. Centring on the wedge's own signature, not on total coma, is the correct procedure.
Selection effect
H2441. Published diamond tool-life figures are gathered on machines with hydrostatic spindles, so they overstate life on aerostatic machines where synchronous error is larger. The selection is in the machine population, not in the tool.
Unit trap
H2442. Mixing arcseconds and microradians in an alignment budget costs about five per cent per conversion, an error invisible when the two units live in different documents. Budgets that close on paper and not in the laboratory very often contain exactly this mixture.
Artefact mimics effect
H2443. Air turbulence along a long horizontal path produces low-order aberration dominated by astigmatism and coma, closely mimicking a misaligned system. The discriminator is temporal: turbulence decorrelates in seconds while alignment error is stationary over hours.
Threshold
H2444. A coating changes a mirror's figure only above a stress-thickness product, below which the substrate's stiffness absorbs the stress without measurable bending. The threshold follows from plate stiffness and explains why small mirrors survive coatings that visibly warp large ones. The product, not either factor alone, is the design variable.
Reversal
H2445. Averaging more interferometric frames does not reduce phase noise below the cavity's vibration floor, and beyond a certain number of frames the average is dominated by drift, so noise falls and then rises with averaging length. The minimum locates the vibration floor without a seismometer.
Definition artefact
H2446. The boundary between roughness and waviness is a convention set by the instrument's cutoff, not a property of the surface. Changing the cutoff moves the same topography between the two categories, so a drawing that fixes both amplitudes without fixing the cutoff is not a specification.
Aliasing
H2447. Every interferometer has a maximum measurable slope set by fringe density at the sampling limit of the fringe pattern, and structure steeper than that appears as spurious flattened plateaus rather than as noise. Tilting the part until fringes alias reveals the plateau, whose height is a diagnostic of the limit.
Scale argument
H2448. Pad asperity contact is a micron-scale phenomenon while the pressure distribution over a hundred-millimetre part is a millimetre-scale one, and the two couple only through the pad's bulk compliance. A model that resolves one scale and not the other will fit the centre well and mispredict edge roll-off.
Count
H2449. RMS figure error is one number and manufacturability is not: two surfaces carrying the same RMS can need three times the polishing time, set by how many Zernike terms exceed a threshold amplitude. A term-count statistic, not RMS, should be the acceptance number.
Misattribution
H2450. Volumetric error measured on a coordinate machine is often assigned to the machine's geometry when it belongs to the fixture's stiffness. Measuring the same artefact on two fixtures of different stiffness leaves geometric error invariant while fixture error moves.
Hidden term
H2451. Every touch-probe measurement carries a term from the tip radius and the calibration sphere's form error, and on curved surfaces the term enters as a scale error rather than an offset. It is therefore invisible on flats and conspicuous on spheres of small radius. Calibrating the probe on a flat therefore cannot reveal it.
Overconstraint
H2452. A thin mirror becomes overconstrained at a support count that depends on thickness-to-diameter ratio rather than on weight. Below the threshold, adding supports improves figure; above it, adding supports worsens it, and the crossing point is measurable by a support-count series.
Reversal
H2453. Increasing the removal rate in ion beam figuring shortens dwell per pass and therefore raises sensitivity to beam-current drift, so past a rate set by the beam's stability the figure worsens as throughput improves. A throughput-versus-figure curve should show the turnover.
Artefact mimics effect
H2454. Subaperture stitching can leave a seam that appears as low-order astigmatism aligned with the stitch grid. Rotating the part beneath a fixed stitch pattern separates them, since true astigmatism rotates with the part and the seam does not.
Scale argument
H2455. The thermal time constant of a precision machine scales with the square of its characteristic length, so a machine scaled up by two takes four times as long to reach equilibrium. Its warm-up specification cannot be scaled linearly, and a scaled prototype will always look thermally better than the full-size machine. A half-scale prototype will understate the full machine's warm-up by a factor of four.
Alignment, tolerancing, and assembly
Inversion of standard practice
H2456. Metrology is conventionally performed after thermal soak, but for machines whose error is dominated by a steady gradient the more reproducible state is the warm steady state, not the cold one. Repeated measurement at a fixed time after start, against measurement after overnight soak, decides which state to specify.
Sampling artefact
H2457. Peak-to-valley is not well defined on a sampled surface: it is the range of the samples, so it grows with sampling density on a rough surface. Two laboratories reporting peak-to-valley for the same part at different pixel counts can differ by a factor of two and both be correct.
Misattribution
H2458. Removing subsurface damage by polishing is often treated as depth-controlled, but at the crack tips the removal is pad-pressure-controlled, so a light polish can leave damage that a heavier polish removes at the same nominal depth. Wedge-polishing followed by etch-rate mapping distinguishes the two.
Count
H2459. A ring mount excites at most three low-order figure modes under clamping, and any observed fourth mode indicates a load other than the mount. Modal decomposition of the mounted figure identifies which mode is unaccounted for, and therefore which load is missing from the model.
Hidden term
H2460. Tool radius compensation in diamond turning carries a term from the tool's edge radius rather than its nose radius, and the two differ by the projection of the rake angle. The resulting error is a form error that appears only on convex surfaces, which is why it survives flat-part qualification.
Threshold
H2461. Every sub-aperture process has an edge exclusion zone wider than the tool radius, set by the tool's inability to hold pressure at the boundary. Dwelling on the edge and observing the sign of the change measures that width, and polishing inside the zone produces roll-off rather than correction. The width belongs to the tool and the edge geometry together, not to the tool alone.
Reversal
H2462. Adding a null lens to test an asphere can make the test less sensitive to the asphere's error when the null's own residual is of the same order. The correct null has a residual orthogonal to the asphere's error, not the smallest residual.
Redundant specification
H2463. Figure and form are the same measurement at different spatial bandwidths, so a drawing that lists both with independent tolerances specifies a band-pass filter twice and usually inconsistently. One of the two tolerances is then redundant and the other is doing all the work.
Scale argument
H2464. Mounting distortion and gravity sag have the same functional form for a plate on discrete supports, so a mount that compensates gravity at one orientation doubles it at another. Rotating the part within its mount separates the two without changing either.
Artefact mimics effect
H2465. Phase-unwrapping errors produce step discontinuities that mimic real tool marks, but their heights quantize to the wavelength and their edges follow the unwrapping path rather than the tool path. Measuring at two wavelengths exposes the quantization.
Selection effect
H2466. Figure measurements are often taken at whichever orientation gives the best number, biasing reported astigmatism low by the projection of the true value. Averaging four rotations removes the bias; reporting the best rotation preserves it.
Bound
H2467. Deterministic polishing cannot correct structure at a spatial period shorter than twice the tool footprint, and requesting correction there adds tool marks of the tool's own period. The correctable band is therefore bounded by tool size, not by the metrology that reveals the error.
Hidden term
H2468. Roughness on a curved surface includes a term from the instrument's form removal, because subtracting a sphere from a short scan leaves a quadratic residual that biases amplitude parameters low on convex parts. Scanning the same patch at two evaluation lengths shows the bias.
Threshold
H2469. Coating adhesion fails above a deposited stress-thickness product that depends on the substrate's surface energy, so two identical substrates with different cleaning histories have thresholds differing by up to a factor of two (estimate). Paired coupons and a scratch series would test it.
Inversion of standard practice
H2470. Cleaning a part between grinding and polishing is standard, but some glasses hydrate on the cleaned surface and polish more slowly, so leaving the slurry film in place shortens the run. A paired run with and without intermediate cleaning decides which behaviour the glass has.
Overparameterization
H2471. A five-axis machine needs at most seven geometric error parameters per axis, and a model with more is absorbing thermal or load effects into geometry. A thermal-state experiment, run at two soak times, shows which parameters are actually geometry. The seventh parameter is usually the thermal one, and it is identifiable only by its time constant.
Artefact mimics effect
H2472. A ghost reflection in a Fizeau cavity produces a periodic ripple whose period equals the ghost's optical path difference, closely mimicking a polishing ripple. Changing cavity length moves the ghost ripple and leaves the real one fixed.
Scale argument
H2473. Floor vibration affects a measurement as the square root of averaging time only for white noise; machine tools produce narrowband vibration, for which averaging does not help and can make things worse by aliasing. A noise spectrum taken during the measurement distinguishes the two regimes. White-noise intuition is the wrong intuition for a machine tool.
Misattribution
H2474. Periodic waviness at the screw's pitch frequency is usually assigned to servo error but is more often the screw's own periodic error. Commanding the same path at a different feed changes servo error and leaves screw error unchanged.
Hidden term
H2475. The measured sag of a large mirror includes a term from the support's compliance, which enters as added flexibility and makes the part appear thinner than it is. Loading the part with a known mass, and watching how sag changes, gives the term directly.
Machine error, mounting, and the cost of a tolerance
Observability
H2476. A scratch becomes visible in dark-field inspection above a width set by the illumination's numerical aperture rather than by its depth, so inspection thresholds are instrument thresholds. Inspecting the same part under two apertures changes the count without changing the surface.
Reversal
H2477. Adding thickness to a protective coating to improve durability can reduce figure stability, because the coating's stress moment grows with thickness squared. The optimum is where durability gain equals figure loss, and it is usually thinner than the durability specification alone would demand. Durability testing alone will not find the optimum, because it measures only one side of the trade.
Identity
H2478. The pad print seen after polishing and the mount print seen after clamping have the same spatial signature because both are contact-pressure maps. Whether the pattern survives demounting separates them.
Selection effect
H2479. Tool wear rates in the literature are measured on tools that survived the test, so the reported rate is biased low. A survival analysis over a tool set that includes failures raises the estimate, and the size of that raise is the size of the bias.
Bound
H2480. The stability of a metrology frame is bounded by the temperature coefficient of its longest dimension, and averaging cannot remove it because it is a bias rather than noise. For Invar the bound is a few nanometres per kelvin per metre (estimate). No averaging scheme changes a bias, which is why the frame is designed rather than filtered.
Hidden term
H2481. Every linear interferometer measurement carries a cosine error from beam misalignment that is second order in the angle, so a milliradian of misalignment gives about half a part per million of scale error. It is small but systematic, and on long-range calibration it is often the largest single term.
Threshold
H2482. Etch-rate measurements reveal subsurface damage only above a depth set by the etch's selectivity, and below that depth damage is present but invisible. That threshold is why a polish certified damage-free can still fail in service.
Reversal
H2483. Slowing a polishing run to improve control can increase roughness, because the pad stays in contact longer and the slurry chemistry has time to etch. The reversal occurs at a dwell beyond which chemical removal dominates mechanical.
Count
H2484. A three-point mount under gravity excites exactly two low-order figure modes, astigmatism and trefoil. Observing coma or spherical aberration after mounting therefore means something besides the supports is loading the part, and the mode observed names that load.
Artefact mimics effect
H2485. Periodic error in a rotary encoder at the scale's graduation period mimics a real azimuthal figure error on a turned part, but its period is fixed in encoder counts rather than in part angle. Re-zeroing the encoder moves the pattern. The pattern is fixed in the encoder's frame, which is exactly what makes it identifiable.
Scale argument
H2486. Polishing time to remove a given figure error scales roughly with the fourth power of part diameter at fixed tool size, because both the area to be covered and the removal per pass change with size. This is why large optics are not polished by scaling up small ones.
Inversion of standard practice
H2487. The stiffest mount is not the most stable for a thin optic: a compliant mount absorbs differential expansion and reduces figure change, so a deliberately soft mount can outperform a rigid one across a thermal cycle. Two mounts of known stiffness, cycled together, would show the crossing.
Misattribution
H2488. Potato-chip distortion after coating is usually attributed to coating stress but is often the substrate's residual stress being released by the coating's heat. A low-temperature coating process has the stress but not the release, so the figure change should be absent.
Correlation error
H2489. A budget in nanometres RMS per surface does not add in RMS across surfaces unless the errors are uncorrelated. For correlated errors from a common polishing process the sum is linear, making the budget optimistic by the square root of the surface count.
Threshold
H2490. White-light profilometry aliases when surface slope exceeds the objective's acceptance angle, and aliased data look like a valid surface with a characteristic staircase. The threshold is a slope, not a height, so it moves with the objective rather than with the part.
Threshold migration
H2491. Improving metrology resolution can appear to reduce yield without any process change, because parts previously binned as good are now resolved as marginal. The apparent loss is a property of the instrument, and re-binning the old data at the new resolution predicts it exactly. The parts were not worse; the ruler was finer.
Hidden term
H2492. A part polished while clamped carries a preload that is released on demounting, so the polished figure is correct only in the clamped state. Measuring before and after demount separates the preload's contribution from the polishing error.
Identity
H2493. Mid-spatial frequency error and ripple are one band with two names in two industries, and specifications written in both are usually inconsistent in amplitude because the stated bandwidths differ. Converting one to the other through the power spectral density should make them agree, and where they do not, the bandwidth rather than the surface is wrong. Where the two disagree, the bandwidth definition is the suspect, not the surface.
Calibration circularity
H2494. Alignment cannot be better than the wavefront sensor's own calibration, so a system aligned to 10 nm RMS with a sensor calibrated to 20 nm RMS is aligned to the sensor rather than to the system. A round-robin between two sensors would place the true floor.
Threshold
H2495. The transition from smoothing to scratching in polishing occurs at a particle concentration threshold rather than at a load threshold, because contact changes from many particles to few. A concentration series at fixed load would show the transition without changing pressure.
Artefact mimics effect
H2496. Residual magnetism in a chuck can deflect a thin part during measurement, mimicking a figure error that vanishes on a non-magnetic support. Measuring the same part on both supports, in the same orientation, gives the deflection directly.
Scale argument
H2497. The cost of a tolerance scales roughly as the inverse cube of the tolerance for figuring processes, since convergence time grows as the inverse square and the risk of a fatal scratch grows linearly. The last factor of two in a specification therefore dominates the price. Loose tolerances are cheap because they remove time, not because they remove skill.
Inversion of standard practice
H2498. Polishing to final figure before coating is conventional, but coating stress will move the figure, so figuring to a deliberately biased figure that the coating corrects is the cheaper route for large mirrors. A paired run, one biased and one nominal, would show which ends closer to specification.
Redundancy
H2499. A two-mirror system has four independent alignment degrees of freedom that affect image quality; adjusting a fifth, usually the secondary's axial position, trades defocus against spherical aberration without improving the image. The trade appears as a straight line in aberration space, and its slope identifies the redundant adjustment.
No-go
H2500. Any material-removal process must remove material, so its statistical nature sets a figure floor of the order of the atomic step times the square root of the removed volume in monolayers (an estimate). Below that floor polishing is stochastic, and no deterministic control can help, which bounds every figuring claim above. That is a bound on the art, not on the ambition.
On the forms used
Twenty-five distinct logical forms appear here, some named in the brief and some not: misattribution, hidden term, identity, bound, threshold, reversal, count, artefact that mimics an effect, scale argument, inversion of standard practice, selection effect, unit trap, derived quantity, aliasing, overconstraint, redundancy, overparameterization, sampling artefact, definition artefact, calibration circularity, correlation error, threshold migration, observability, redundant specification, and no-go.
100 of 2,500. The precision-engineering block is deliberately separate from the source-and-detector entries, which belong to another part of the series and are not repeated here. Each hypothesis is stated so that a negative result would be informative; that is the only test of whether a hypothesis was written or merely generated.
If one of these is worth an afternoon, it is H2403 — that the tool-mark band and the waviness band are one spectrum seen through two instruments. Two measurements on one part, one with each instrument, and one stated transfer function would settle it.
Incoprea Research · hand-considered hypotheses 2401–2500 · 3 November 2026 · ~/Desktop/Nov3/