# Nine version4 research/testing dossiers

All physical claims remain unproved. No physical data were generated.

# S4-0039 — Spring-network weighted averaging

A spring-coupled pair of sliders should compute a weighted average position under quasistatic loading.

**Why selected.** A low-cost static test has a closed conditional solution, so friction and manufacturing deviations can be quantified directly rather than hidden inside a complex simulation.

**Design.** Connect a low-friction output slider to two input sliders with independently calibrated linear springs. Measure spring constants over the intended stroke. Set input positions on a two-dimensional grid and read output after settling. Use at least five independent spring/slider assemblies, randomize approach direction, and reserve mixed input pairs for validation. The initial replicate count is a planning choice; a variance pilot must set final precision requirements.

**Endpoints.** Primary: error between measured output and the calibrated stiffness-weighted mean. Secondary: approach-direction hysteresis, settling time, and useful stroke. Proposed criterion: output error below 2% of full-scale stroke for held-out positions, specified before validation.

**Controls.** Single-spring calibration, swapped springs, output load perturbations, reversed approach direction, and a deliberately high-friction slider. Measure output loading explicitly; an attached pointer or sensor is not force-free.

**Conditional analysis.** Assume positive linear spring constants k1,k2, collinear motion, no friction, and no external output force. U(x)=k1*(x-a)^2/2+k2*(x-b)^2/2. Its derivative vanishes only at x=(k1*a+k2*b)/(k1+k2), and U''=k1+k2>0 proves a unique energy minimum. An external load F shifts the result by F/(k1+k2), with sign set by convention. This is an exact conditional statics result, not an original theorem or proof of hardware accuracy.

**Analysis plan.** Fit no free averaging weights after independent spring calibration. Report held-out residuals, hysteresis loops, and sensitivity to output load. Reject the useful device claim if error or hysteresis exceeds the predefined tolerance or if weights drift between assemblies.

**Limitations.** Unproved physical implementation; standard spring equilibrium already establishes the broad principle. No physical data or novelty certified.

**Sources and evidence limits.**

[The quasi-static approximation of the spring-slider motion](https://npg.copernicus.org/articles/3/143/1996/npg-3-143-1996.html). Primary abstract read. The paper studies limits of neglecting inertia in a frictional spring-slider. It is a caution about quasistatic assumptions, not evidence that this proposed averaging fixture performs accurately.


# S4-0123 — Vibrating-aperture gradient readout

A vibrating aperture should estimate a beam's local intensity gradient from its first harmonic.

**Why selected.** The small-modulation expansion supplies a quantitative signal prediction and a strong amplitude-scaling falsification test.

**Design.** Translate a narrow aperture sinusoidally across a stable, safely enclosed low-power beam. Record actual aperture motion and photodetector power synchronously. Use independently calibrated beam profiles with known positive, negative, and near-zero gradients. Sweep modulation amplitude while keeping detector response linear, and repeat with coherent and diffuse illumination.

**Endpoints.** Primary: signed first-harmonic amplitude divided by mechanical amplitude versus independently measured power gradient. Secondary: higher-harmonic contamination and spatial bias. A useful device requires a predeclared gradient error bound on held-out profiles, not merely visible modulation.

**Controls.** Stationary aperture, flat intensity field, reversed displacement phase, source-power modulation without motion, detector linearity check, and several aperture widths. Record stray scattering from the aperture edge.

**Conditional analysis.** Let P(s) be transmitted power through an aperture centered at s. For s=s0+a*cos(wt), smooth P gives P(t)=P(s0)+a*P'(s0)*cos(wt)+O(a^2). For a slit of width d, P'(s)=I(s+d/2)-I(s-d/2), which approaches d*I'(s) only when the intensity is smooth over d. The expansion proves a leading-order relation for the assumed power function, not a universal microscopy theorem. Coherent diffraction and finite aperture can invalidate the simple intensity model.

**Analysis plan.** Check linear scaling with a and sign reversal with gradient. Estimate the amplitude range where the remainder is below uncertainty, then lock it before held-out tests. Reject the claim if first-harmonic response is dominated by phase effects or edge scattering.

**Limitations.** Unproved geometry-specific sensor; substantial vibration-microscopy prior art. No new optical discovery or measured performance asserted.

**Sources and evidence limits.**

[Coherent and incoherent operation in optical vibration microscopy](https://opg.optica.org/ao/abstract.cfm?uri=ao-36-17-3920). Primary publisher abstract read. Korpel and Snyder analyze aperture-dependent vibration microscopy and report different harmonic information in coherent and incoherent limits. This directly warns that a harmonic interpretation depends on the optical regime; it is not generic proof of the proposed first-harmonic intensity-gradient channel.


# S4-0210 — Void-array pulse splitting

A patterned void array should make an impact pulse split into two delayed arrivals.

**Why selected.** Two distinct arrivals and their energy content can be measured directly; the strongest model prediction is separable from claims about a fabricated lattice.

**Design.** Fabricate elastic strips containing a designed void region that creates two mechanically coupled propagation routes. Launch a reproducible low-energy impact and measure full-field displacement or multiple accelerometers before and after the region. Compare at least five independent specimens with a solid strip and a mass-matched nonbranching void control.

**Endpoints.** Primary: separation between two identifiable arrival peaks divided by input pulse width. Secondary: energy in each arrival, transmitted energy, and reproducibility. Predefine a split as two peaks each exceeding a noise-based energy threshold, separated by more than the instrument impulse width.

**Controls.** Solid strip; equal-mass geometry; reversed launch direction; multiple pulse amplitudes; support-reflection timing; independent sensor impulse calibration. Verify that a second peak is not an end reflection.

**Conditional analysis.** In an assumed linear two-path model y(t)=A1*f(t-t1)+A2*f(t-t2). Distinct arrivals can occur when |t2-t1| exceeds pulse duration and both amplitudes exceed noise. This expression follows from linear superposition, but it assumes paths and coupling coefficients that the void structure may not realize. Nonlinear solitons, dispersion, damping, and mode conversion require different models; the cited nonlinear lattice result cannot be imported as proof.

**Analysis plan.** Estimate arrival windows from independently measured path speeds before inspecting the final output. Report raw waveforms and energy accounting, including lost or reflected energy. Reject the device claim if the apparent split disappears after reflection control or if one branch carries negligible energy.

**Limitations.** Unproved physical structure. The broad pulse-separation phenomenon is already published; no global novelty or original theorem claimed.

**Sources and evidence limits.**

[Focusing and Mode Separation of Elastic Vector Solitons in a 2D Soft Mechanical Metamaterial](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.123.024101). Primary publisher abstract read. Deng and colleagues report experimental, numerical, and analytical pulse-mode separation in a soft mechanical metamaterial. This establishes closely related prior art and does not verify the proposed void geometry or two-arrival timing.


# S4-0307 — Contact-line sorting by rolling mobility

A moving contact line should sort inert particles by rolling versus sliding mobility at equal size.

**Why selected.** The proposal links a directly measurable single-particle property to an observable deposition endpoint and has several clear competing explanations.

**Design.** Use equal-diameter inert particles with independently measured rolling resistance and similar density/wettability, distinguishable by nonreactive optical marks. Recede a water meniscus at controlled speed on a characterized substrate and track particle rotation as well as translation. Use multiple independent substrate preparations and mixed populations; analyze particle trajectories blind to mobility class.

**Endpoints.** Primary: difference in final deposition position between mobility classes at matched size. Secondary: observed rolling fraction, escape probability, and sorting purity versus recovery. Proposed useful separation must persist after controlling contact angle and density.

**Controls.** Pinned contact line; no evaporation with externally moved meniscus; rotation-constrained particles; pure populations; substrate rotation; matched wetting measurements. Surface modification used to change friction may also change capillary forces and must be measured.

**Conditional analysis.** A simple force comparison can only suggest regimes: capillary force competes with translational friction and rolling resistance torque divided by radius. No universal ordering theorem follows because immersion, contact-line pinning, roughness, and hydrodynamic drag change simultaneously. The physical sorting claim remains unproved. A fitted trajectory model is not evidence of the mechanism unless independent mobility measurements predict held-out trajectories.

**Analysis plan.** Use device/substrate-level replication rather than counting every particle as independent. Fit deposition position with mobility, size, wetting, and local flow covariates; publish null and failed tracks. Reject rolling-specific sorting if its effect disappears after these controls or if rotation-constrained controls separate equally.

**Limitations.** No physical data. Related contact-line deposition and friction research already exists. Useful purity, scalability, and novelty remain unknown.

**Sources and evidence limits.**

[Characterizing sliding and rolling contacts between single particles](https://pubmed.ncbi.nlm.nih.gov/40048270/). Primary article abstract and metadata read. The study measures rolling and sliding at particle contacts and finds traction-dependent rolling behavior. It supports measuring mobility separately, not sorting by a receding meniscus.

[An experimental and theoretical study of the inward particle drift in contact line deposits](https://pubs.rsc.org/en/content/articlelanding/2022/sm/d2sm00142j). Primary publisher abstract read. The study links evaporation-driven interface descent, particle/substrate friction, and deposit motion. This is strong related work and a confounding mechanism for any proposed rolling-specific separation.


# S4-0414 — Impulse-order memory in a bistable oscillator

A bistable oscillator should retain the order of two impulse magnitudes in its final state.

**Why selected.** Reversing an otherwise matched pulse pair is a clean intervention, while equal-impulse and equal-energy controls expose trivial explanations.

**Design.** Construct an elastic bistable beam with two well-characterized stable states. Starting from the same state, apply a large and a small calibrated force impulse separated by a controlled delay, then reverse their order. Measure force history and full motion, not nominal actuator commands. Repeat across independent beams and randomized sequences after a verified reset.

**Endpoints.** Primary: difference in final-state probability between AB and BA sequences. Secondary: dependence on delay, pulse width, total impulse, and total work. A useful memory requires a preregistered separation above reset variability over a bounded operating region.

**Controls.** Single pulses, equal pulses, long delays allowing full relaxation, time-reversed force traces, symmetric monostable beam, and equal-work pulse pairs. Initial velocity must be measured or bounded.

**Conditional analysis.** For m*xdd+c*xd+U'(x)=F(t), two impulses apply velocity jumps while free evolution changes position and velocity between them. The resulting state maps need not commute; AB and BA can therefore differ. Noncommutation alone does not prove different final wells: both trajectories may relax to the same basin. No geometry-independent order-memory theorem is claimed. A model prediction must use independently calibrated U, damping, and measured pulses.

**Analysis plan.** Map final-state probabilities versus delay on a training region, then lock an operating point and test new beams. Report state-classification uncertainty and reset failures. Reject memory if order effects are explained entirely by unequal delivered work or uncontrolled initial motion.

**Limitations.** Unproved mechanical claim; bistable memory and nonlinear phase selection are established topics. No successful memory or novel theorem asserted.

**Sources and evidence limits.**

[Bistable phase control via rocking in a nonlinear electronic oscillator](https://pubmed.ncbi.nlm.nih.gov/17199404/). Primary abstract read. Buldu and colleagues demonstrate phase bistability under a phase-alternating electronic drive. It supports nonlinear state selection as background but does not establish mechanical impulse-order memory.


# S4-0616 — Mechanical gap modulation for conductivity/lift-off separation

A rotating slotted conductor should produce distinguishable drag signatures for surface and through-thickness cracks.

**Why selected.** A practical nuisance parameter can be deliberately modulated, allowing a quantitative identifiability test before claiming better measurements.

**Design.** Place an inductive probe on a calibrated small-amplitude translation stage above nonmagnetic conductive standards of independently measured conductivity and thickness. Record complex impedance and its synchronous change during gap modulation. Vary mean lift-off, conductivity, tilt, and temperature independently. Reserve new plates and gaps for held-out validation.

**Endpoints.** Primary: conductivity error under unknown lift-off using modulated versus unmodulated measurements. Secondary: estimated gap error, sensitivity singular values, and sensitivity to tilt. Predefine the acceptable uncertainty after instrument-noise calibration.

**Controls.** Fixed-gap measurement, direct optical gap reference, nonconductive target, known thickness changes, tilt perturbations, and cable-motion-only control. Mechanical modulation can itself change wiring impedance.

**Conditional analysis.** Let measured complex impedance be Z(sigma,h). Small gap modulation h=h0+a*cos(wt) gives a first-harmonic approximately a*dZ/dh. The observable pair (Z,dZ/dh) can locally identify (sigma,h) only if its real-valued sensitivity matrix has rank two at the noise level. Full rank gives local linear inversion; it does not guarantee global uniqueness, numerical stability, or correctness of the electromagnetic model. No unconditional separation theorem follows.

**Analysis plan.** Estimate sensitivities from independent standards, quantify conditioning, then evaluate held-out plates without fitting their true conductivity. Compare against a frequency-based baseline and publish all high-error cases. Reject the proposal if modulation adds wiring artifacts or fails to improve uncertainty.

**Limitations.** Unproved protocol; conductivity/lift-off inversion has substantial prior art. No measured accuracy or novelty certified.

**Sources and evidence limits.**

[A dual-stage dimensionless method for simultaneous estimation of electrical conductivity, lift-off, and thickness in Eddy Current Testing](https://doi.org/10.1016/j.measurement.2026.120640). Primary publisher search-accessible abstract read; direct page denied access. The authors report simultaneous conductivity/lift-off estimation using frequency-dependent measurements. This is close prior art and shows that the broad inverse problem is already studied; it does not validate mechanical gap modulation.

[Study of Lift-Off Invariance for Pulsed Eddy-Current Signals](https://doi.org/10.1109/TMAG.2008.2006246). Author-institution abstract read. The work studies lift-off invariance and its dependence on probe geometry and conductivity. It cautions that compensation features are not universal.


# S4-0731 — Moisture-dependent pressure-relaxation tails

A porous chamber wall should reveal absorbed moisture through a changed pressure-relaxation tail.

**Why selected.** Moisture can be independently weighed, and a pressure-decay protocol makes competing leak and permeability explanations testable.

**Design.** Condition porous wall coupons to known gravimetric moisture contents, mount them around a small reference chamber, and apply a modest repeatable pressure step. Measure full pressure decay and specimen mass. Use independent coupons, randomized moisture states where reversible, and blank chamber tests before each run. Track temperature and any liquid motion.

**Endpoints.** Primary: held-out moisture-estimation error from a preregistered decay-tail feature. Secondary: fitted fast and slow timescales, leak-rate sensitivity, and recovery after drying. A useful indicator must distinguish adjacent moisture states beyond chamber drift.

**Controls.** Nonporous wall, calibrated pinhole leaks, dry porous coupons with changed permeability, equal-mass surface water, temperature steps, and sealed/vented pore boundaries. Moisture changes both storage and transport and can alter chamber volume.

**Conditional analysis.** In a lumped linear model Ceff*dp/dt=-G*p, the decay time is tau=Ceff/G. Added sorption storage could increase Ceff, but moisture may decrease or increase conductance G. Thus a longer tail is not uniquely attributable to moisture, and no monotonicity theorem is defensible without independently constraining G. A multiexponential fit can describe data while remaining physically nonidentifiable.

**Analysis plan.** Compare moisture-only predictions against leak-only and permeability controls using held-out coupons. Publish parameter covariance and residual structure, not just fit quality. Reject the diagnostic claim if different dry/leak conditions reproduce the same tail or if calibration fails after wet-dry cycling.

**Limitations.** Unproved sensor concept; primary related work establishes coupled transport only. No experimental moisture estimates or novelty claim.

**Sources and evidence limits.**

[Effect of air pressure on moisture transfer inside porous building materials](https://onlinelibrary.wiley.com/doi/full/10.1002/2475-8876.12047). Primary publisher abstract read. Fukui and colleagues compare water uptake when air escape is restricted and model coupled air/moisture transfer. This supports coupling between moisture and pore-gas pressure, not uniqueness of a relaxation-tail moisture indicator.


# S4-0849 — Equalized clamping and post-release distortion

A self-equalizing clamp should reduce dimensional error after unclamping a thin workpiece.

**Why selected.** The intuitive claim may be false; a simple matched-fixture study can determine whether equal forces actually reduce springback for a declared geometry.

**Design.** Hold thin nonhazardous test plates in a three-jaw fixture with either a passive force-equalizing linkage or fixed independent jaws. Match total force and nominal contact positions. Measure geometry before clamping, under load, and after release using a calibrated surface scan. Use multiple independent plates and fixtures; vary one controlled thickness or curvature imperfection.

**Endpoints.** Primary: root-mean-square post-release shape error relative to the initial plate. Secondary: in-load deformation, force equality, residual strain, and contact marks. Proposed benefit must hold at equal total force and equal restraint against a standardized disturbance.

**Controls.** Equal-force active reference, deliberately unequal forces, matched contact-area pads, reversed plate orientation, and a purely elastic low-load condition. Separate elastic recovery from plastic indentation and residual-stress release.

**Conditional analysis.** Equal forces do not generally minimize deformation. Even a simple beam has bending moments depending on force locations, so equal magnitudes at unequal lever arms can increase curvature. Linear elasticity gives u=K^-1*f under specified supports, but minimizing a shape-error norm requires geometry-dependent optimization of f. No general equal-force theorem supports the physical hypothesis. The experiment is needed to identify a useful restricted regime or reject it.

**Analysis plan.** Compare paired geometry changes with force maps and uncertainty. Report null results as useful constraints. Reject the broad claim if equalization reduces force variation but fails to reduce post-release shape error; any narrower supported claim must state the plate/support geometry explicitly.

**Limitations.** Unproved and potentially false engineering hypothesis. Existing clamp-distortion literature is related background; no performance or novelty certified.

**Sources and evidence limits.**

[The application of tool path compensation for the reduction of clampinginduced geometric errors](https://doi.org/10.1080/002075497194165). Author-institution abstract read. De Meter and Hockenberger discuss clamp-related displacement, measurement, modeling, and compensation. Their work establishes the practical problem, but does not prove that equal clamp forces minimize post-release distortion.


# S4-0012 — Pulley shaping of a sinusoidal input

A shaped pulley should transform a sinusoidal input into a nearly triangular displacement waveform.

**Why selected.** Kinematic waveform shaping permits a direct mathematical target, while finite acceleration and cable mechanics supply decisive practical limits.

**Design.** Drive a shaped pulley with a measured sinusoidal input angle and read cable displacement with an encoder. Choose a smoothed triangular target to avoid an ideal corner's infinite acceleration. Compare with a circular pulley under equal amplitude and speed. Build independent profiles and measure cable tension, slip, and output over several frequencies.

**Endpoints.** Primary: waveform root-mean-square error relative to the declared smoothed triangle. Secondary: harmonic distortion, peak tension, phase lag, and wear drift. Proposed benefit is lower waveform error than a circular pulley without loss of cable contact.

**Controls.** Quasistatic angle-position scan; circular pulley; direct displacement-stage target; reversed travel; tension variation; cable stretch calibration. Include motor tracking error in measured input rather than attributing it to the pulley.

**Conditional analysis.** For normalized sinusoidal input x=sin(theta), the function y=(2/pi)*arcsin(x) yields a triangular periodic waveform when evaluated over full theta. However dy/dx=2/(pi*sqrt(1-x^2)) diverges at x=+/-1, so a finite, smooth physical transmission cannot implement the ideal mapping exactly at turning points. A bounded-slope approximation is required. This is a conditional kinematic identity and a realizability limitation, not proof of a manufacturable pulley.

**Analysis plan.** Publish the designed bounded-slope mapping before tests, compare measured quasistatic and dynamic errors, and quantify how smoothing trades corner sharpness for tension. Reject the device claim if slip or cable elasticity dominates or if no useful waveform improvement survives at finite speed.

**Limitations.** Unproved implementation; standard waveform transformations and pulley mechanics have extensive prior art. No new theorem or experimental success claimed.

**Sources and evidence limits.**

[Investigation of a laboratory mechanical system with fibre and pulley](https://doi.org/10.1007/s40435-014-0064-y). Research-registry description of primary work read. The described study measures and models a driven fibre/pulley system under several waveforms. It supports attention to cable dynamics, not the proposed triangular-output pulley profile.
