---
slug: kapitza-pendulum
passed: 2026-10-14
by: the assay line (research/assay-line/queue-1014c/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex; repair review by Codex, 2026-10-14
checked: 154
confirmed: 126
wrong: 27
unverifiable: 0
observed: 0
stale: 1
---

This was the directed repair pass for the 28 items in queue-1014c/kapitza-pendulum/triage.json against the layer read at c8289bd340. The fixer reread the cited evidence and executed the page's actual physics before editing, retained all 126 checker-confirmed claim IDs with their supported scope, and preserved the original numerical observations. The 27 wrong-count entries comprise 20 WRONG and seven MINOR claims; the modification date is STALE. Shared errors have one dated correction each, mirrored at the end of the markdown; MINOR fixes are recorded only. All triage findings held on rereading. The repair review confirmed three missed-copy problems in other layers. Twelve findings across two shared error classes remain open board notes; their Kapitza copies are fixed, but the portal prose and stored quotations are outside this worker's edit scope. Evidence and note text are in research/assay-line/queue-1014c/kapitza-pendulum/repair-report.md.

## Claims

- WRONG | [X1: interactive-threshold] "exact threshold where the impossible becomes stable." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf ; public/strata/kapitza-pendulum/index.html ; research/kapitza-pendulum/verify-assay.mjs | Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- WRONG | [X2: button-recovery] "the bob wobbles but recovers." | public/strata/kapitza-pendulum/index.html ; research/kapitza-pendulum/verify-assay.mjs | Executing the actual button handlers and RK4 at simulation time zero with 2 cm, 25 cm, 20 Hz puts the first sampled excursion beyond 90 degrees at 0.20 s after an immediate knock. Without the knock, the 10 s control stays within 3.438 degrees. The impulse is unchanged; instructions and verdict no longer promise recovery.
- WRONG | [X3: bob-rises] "The bob rises and stays inverted" | public/strata/kapitza-pendulum/index.html ; research/kapitza-pendulum/verify-assay.mjs | The hanging handler sets theta=0 and angular velocity=0; the acceleration is proportional to sin(theta). A 10 s run leaves both exactly zero. The prose now requires placing the bob near upright and distinguishes stabilization from lifting.
- WRONG | [X4: exact-threshold-heading] "The threshold is exact" | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf ; public/strata/kapitza-pendulum/index.html ; research/kapitza-pendulum/verify-assay.mjs | Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- WRONG | [X5: threshold-iff] "the up state is stable ⟺ (a·ω)² > 2·g·L" | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf ; public/strata/kapitza-pendulum/index.html ; research/kapitza-pendulum/verify-assay.mjs | Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- MINOR | [X6: energy-conservation] "over 40 swings" | research/kapitza-pendulum/verify.py ; research/kapitza-pendulum/numbers.json | energy_drift calls rk4_run with w=sqrt(g/L) and periods=40, so the duration is 40 small-angle periods (40.1213336 s), starting from pi/3. The energy drift remains 3.146076070707211e-8. HTML, notebook and both verifier print labels now name the actual window. MINOR, record only.
- MINOR | [X7: threshold-row-4] "11.85" | research/kapitza-pendulum/numbers.json | numbers.json gives 74.19831016692754 / 6.26418390534633 = 11.844848632812498. The HTML and notebook cell now round to 11.84; all underlying measurements are retained. MINOR, record only.
- WRONG | [X8: threshold-within-percent] "within ~1% of √2 = 1.414" | research/kapitza-pendulum/numbers.json ; research/kapitza-pendulum/verify.py | The five stored percentage differences from sqrt(2) are -3.066985, -0.829310, +0.347015, +0.506873 and +0.766860. The fastest row is farthest below the prediction. At a=0.01 m, K=1.38, the original 60-cycle window ends at 1.7444058093 s; the unchanged RK4 trajectory crosses 90 degrees at 1.9077015754 s when extended to 240 cycles. Kapitza's prose and verifier now describe finite-window observations without claiming convergence; original numeric data are preserved. The portal's convergence claim remains open (fix-checker problem 2).
- WRONG | [X9: excess-next-order] "excess above √2 at the fastest drives" | research/kapitza-pendulum/numbers.json ; https://arxiv.org/html/2308.14441v2 | The fastest K values are 1.37083984375 and 1.4024853515625002, both below sqrt(2). Beneke et al., section I after Figure 3, discusses higher-order theory but does not establish that these observations measure a correction. The sign and unsupported interpretation are corrected in HTML, markdown, notebook and verifier.
- WRONG | [X10: period-agreement] "again converging as the drive speeds up." | research/kapitza-pendulum/numbers.json | The stored period errors are 2.573009% at 180 rad/s, 0.935623% at 140 rad/s and 0.374299% at 120 rad/s. Amplitude and K also vary. The accurate rounded 0.4%-2.6% range remains; the claim that these settings show frequency convergence is removed.
- MINOR | [X11: fusion-confinement] "it confines fusion plasmas" | https://indico.ukaea.uk/event/449/contributions/1444/ | The Novatron abstract describes RF plugging as a proposed improvement and says electrodes are being fitted to compare theory with experiment. The page now says RF ponderomotive plugging is being explored to improve confinement in mirror-fusion designs. MINOR, record only.
- MINOR | [X12: damping-widens] "adding light damping only widens the stable region" | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov, printed pp. 13-14, reports little effect of weak friction on drive-parameter stability boundaries, while the attraction basin can shrink as friction increases. HTML and notebook now make that distinction and remove the blanket widening claim. MINOR, record only.
- WRONG | [X13: verifier-convergence-footer] "true threshold approaching √2 precisely in that limit" | research/kapitza-pendulum/numbers.json ; research/kapitza-pendulum/verify.py | The five stored percentage differences from sqrt(2) are -3.066985, -0.829310, +0.347015, +0.506873 and +0.766860. The fastest row is farthest below the prediction. At a=0.01 m, K=1.38, the original 60-cycle window ends at 1.7444058093 s; the unchanged RK4 trajectory crosses 90 degrees at 1.9077015754 s when extended to 240 cycles. Kapitza's prose and verifier now describe finite-window observations without claiming convergence; original numeric data are preserved. The portal's convergence claim remains open (fix-checker problem 2).
- WRONG | [X14: up-state-stable] "UP STATE: STABLE" | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf ; public/strata/kapitza-pendulum/index.html ; research/kapitza-pendulum/verify-assay.mjs | Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- WRONG | [X15: knock-returns] "knock it and it returns." | public/strata/kapitza-pendulum/index.html ; research/kapitza-pendulum/verify-assay.mjs | Executing the actual button handlers and RK4 at simulation time zero with 2 cm, 25 cm, 20 Hz puts the first sampled excursion beyond 90 degrees at 0.20 s after an immediate knock. Without the knock, the 10 s control stays within 3.438 degrees. The impulse is unchanged; instructions and verdict no longer promise recovery.
- WRONG | [X16: below-above-labels] "above = stands" | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf ; public/strata/kapitza-pendulum/index.html ; research/kapitza-pendulum/verify-assay.mjs | Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- MINOR | [X17: meta-0-3] "Every number is recomputed from the exact equation of motion, no averaging." | public/strata/kapitza-pendulum/index.html | The trajectory uses accel and stepODE on the unaveraged equation; thresholdHz and drawPot use leading-order averaging. Metadata now identifies those separate calculations, including the matching markdown and structured descriptions. MINOR under the directing triage, record only.
- WRONG | [X18: meta-0-4] "The inverted state stabilizes exactly when (aω)² > 2gL" | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf ; public/strata/kapitza-pendulum/index.html ; research/kapitza-pendulum/verify-assay.mjs | Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- MINOR | [X19: meta-1-2] "Every number from the exact equation of motion." | public/strata/kapitza-pendulum/index.html | The trajectory uses accel and stepODE on the unaveraged equation; thresholdHz and drawPot use leading-order averaging. Metadata now identifies those separate calculations, including the matching markdown and structured descriptions. MINOR under the directing triage, record only.
- STALE | [X20: modified-date] ""dateModified":"2026-06-29"" | git log c8289bd340 -- public/strata/kapitza-pendulum/index.html | The read-only path history at c8289bd340 includes revisions dated 2026-09-27 and 2026-10-01 after the original 2026-06-29 date. dateModified is set to the requested pass date, 2026-10-14. The Corrected lines date this content revision; the explicit dateModified exception requires no Updated line.
- WRONG | [X21: md-threshold-dek] "The threshold is exact" | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf ; public/strata/kapitza-pendulum/index.html ; research/kapitza-pendulum/verify-assay.mjs | Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- WRONG | [X22: md-verifier-dek] "within ~1% of √2, converging to it as the drive speeds up." | research/kapitza-pendulum/numbers.json ; research/kapitza-pendulum/verify.py | The five stored percentage differences from sqrt(2) are -3.066985, -0.829310, +0.347015, +0.506873 and +0.766860. The fastest row is farthest below the prediction. At a=0.01 m, K=1.38, the original 60-cycle window ends at 1.7444058093 s; the unchanged RK4 trajectory crosses 90 degrees at 1.9077015754 s when extended to 240 cycles. Kapitza's prose and verifier now describe finite-window observations without claiming convergence; original numeric data are preserved. The portal's convergence claim remains open (fix-checker problem 2).
- WRONG | [X23: md-accelerators-dek] "focuses every particle accelerator's beam." | https://www.bnl.gov/newsroom/news.php?a=217140&lang=en | Brookhaven's 2020 account says strong focusing is used in the vast majority of present-day accelerators and all high-energy colliders. The dek now says most present-day accelerators; the body and HTML use high-energy colliders. X23 has a correction line; X28 is the triage's MINOR scope fix, with no separate line.
- WRONG | [X24: md-threshold-relation] "stable exactly when (aω)² beats 2gL." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf ; public/strata/kapitza-pendulum/index.html ; research/kapitza-pendulum/verify-assay.mjs | Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- WRONG | [X25: md-bob-return] "The bob rises, stays inverted, balanced on nothing" | public/strata/kapitza-pendulum/index.html ; research/kapitza-pendulum/verify-assay.mjs | The hanging handler sets theta=0 and angular velocity=0; the acceleration is proportional to sin(theta). A 10 s run leaves both exactly zero. The prose now requires placing the bob near upright and distinguishes stabilization from lifting.
- WRONG | [X26: md-threshold-results] "within ~1% of √2 and tightens toward it as the drive gets faster" | research/kapitza-pendulum/numbers.json ; research/kapitza-pendulum/verify.py | The five stored percentage differences from sqrt(2) are -3.066985, -0.829310, +0.347015, +0.506873 and +0.766860. The fastest row is farthest below the prediction. At a=0.01 m, K=1.38, the original 60-cycle window ends at 1.7444058093 s; the unchanged RK4 trajectory crosses 90 degrees at 1.9077015754 s when extended to 240 cycles. Kapitza's prose and verifier now describe finite-window observations without claiming convergence; original numeric data are preserved. The portal's convergence claim remains open (fix-checker problem 2).
- WRONG | [X27: md-paul-static] "Earnshaw's theorem forbids any static field from holding" | https://simion.com/info/earnshaws_theorem.html ; https://www.nobelprize.org/uploads/2018/06/paul-lecture.pdf | SIMION restricts Earnshaw to stable stationary equilibrium using electrostatic forces alone in charge-free space. Paul, printed pp. 613-614, describes a Penning trap with a static electric field and axial magnetic field, with no radiofrequency drive. The markdown and the HTML escape-direction wording now preserve the electrostatic and stationary qualifications.
- MINOR | [X28: md-every-large-accelerator] "every large particle accelerator runs on." | https://www.bnl.gov/newsroom/news.php?a=217140&lang=en | Brookhaven's 2020 account says strong focusing is used in the vast majority of present-day accelerators and all high-energy colliders. The dek now says most present-day accelerators; the body and HTML use high-energy colliders. X23 has a correction line; X28 is the triage's MINOR scope fix, with no separate line.
- CONFIRMED | [rest-down] "A pendulum has exactly one place it will rest: hanging straight down." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov equation (2), with drive and damping zero, has down and up equilibria. Linearizing restores down and destabilizes up; rest means stable rest in this context.
- CONFIRMED | [up-unstable] "The upside-down position is the textbook example of unstable" ... "the faintest breath topples it." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov equation (2), with drive and damping zero, has down and up equilibria. Linearizing restores down and destabilizes up; rest means stable rest in this context.
- CONFIRMED | [shake-stabilizes] "Yet shake the pivot up and down fast enough and a second stable resting place appears, pointing straight up." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 1 supports local upright stability in suitable amplitude and frequency intervals. This does not promise spontaneous lifting or recovery from the button impulse; those are corrected separately.
- CONFIRMED | [push-return] "Push it and it springs back." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 1 supports local upright stability in suitable amplitude and frequency intervals. This does not promise spontaneous lifting or recovery from the button impulse; those are corrected separately.
- CONFIRMED | [slow-landscape] "The slow, time-averaged landscape the bob really feels." | https://arxiv.org/html/2308.14441v2 | Beneke et al. equation (13), with angle shifted by pi, gives the displayed leading-order potential. Its curvature is g/L+a²ω²/(2L²) at down and -g/L+a²ω²/(2L²) at up. These are properties of the averaged potential.
- CONFIRMED | [down-always-valley] "Down (θ=0) is always a valley." | https://arxiv.org/html/2308.14441v2 | Beneke et al. equation (13), with angle shifted by pi, gives the displayed leading-order potential. Its curvature is g/L+a²ω²/(2L²) at down and -g/L+a²ω²/(2L²) at up. These are properties of the averaged potential.
- CONFIRMED | [second-valley] "The fast shake adds the green bump that, past threshold, digs a second valley at the top (θ=180°)." | https://arxiv.org/html/2308.14441v2 | Beneke et al. equation (13), with angle shifted by pi, gives the displayed leading-order potential. Its curvature is g/L+a²ω²/(2L²) at down and -g/L+a²ω²/(2L²) at up. These are properties of the averaged potential.
- CONFIRMED | [green-bump] "That is the green bump in the panel above." | public/strata/kapitza-pendulum/index.html | drawPot evaluates the total averaged potential with a positive sin-squared shaking term. Its color follows the averaged threshold; it is not a separate force plot.
- CONFIRMED | [bob-dot] "The dot is the bob now." | public/strata/kapitza-pendulum/index.html | drawPot reduces the integrated angle modulo 2pi and draws the dot at that angle on the potential curve.
- CONFIRMED | [reduced-motion] "Motion is paused because your system requests reduced motion. Press Stand it up or ▶ Play to run the simulation when you choose to." | public/strata/kapitza-pendulum/index.html | The boot handler respects prefers-reduced-motion and leaves the animation paused. verify-assay.mjs also checks that state; the named buttons call start.
- CONFIRMED | [below-line-falls] "Now drag the drive frequency below the line" ... "it falls." | public/strata/kapitza-pendulum/index.html | The default 20 Hz control remains within 3.438 degrees for 10 s, while the 8 Hz control crosses 90 degrees at the 0.725 s sample. This supports the qualitative demo in those settings, not a universal exact boundary.
- CONFIRMED | [line-computed] "The line is not a setting; it is computed from the equation of motion, live." | public/strata/kapitza-pendulum/index.html | thresholdHz recomputes sqrt(2gL)/(2pi*a) when amplitude or length changes. This is the leading-order averaged formula, not a boundary measured from the trajectory.
- CONFIRMED | [rod-bob-pivot] "The setup is the simplest in mechanics: a rigid rod, a weight on the end, a pivot at the top." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 2 specifies a light rigid rod with mass concentrated in its small heavy bob.
- CONFIRMED | [unforced-swings] "Leave it alone and it hangs. Lift it to the side and it swings." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov equation (2), with drive and damping zero, has down and up equilibria. Linearizing restores down and destabilizes up; rest means stable rest in this context.
- CONFIRMED | [two-equilibria] "There are only two places it can balance without moving" ... "hanging down, which is stable, and pointing up, which is not." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov equation (2), with drive and damping zero, has down and up equilibria. Linearizing restores down and destabilizes up; rest means stable rest in this context.
- CONFIRMED | [unstable-disturbance] "The upward balance is the standard picture of instability: it exists in principle, but any real disturbance grows, and over it goes." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov equation (2), with drive and damping zero, has down and up equilibria. Linearizing restores down and destabilizes up; rest means stable rest in this context.
- CONFIRMED | [top-stable] "The top has become a stable resting place." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 1 supports local upright stability in suitable amplitude and frequency intervals. This does not promise spontaneous lifting or recovery from the button impulse; those are corrected separately.
- CONFIRMED | [history-description] "This is Kapitza's pendulum" ... "described by A. Stephenson in 1908 and explained by the physicist Pyotr Kapitza in 1951." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf ; https://api.crossref.org/works/10.1080/14786440809463763 ; https://ufn.ru/ru/articles/1951/5/b/ | Butikov printed p. 1 credits Stephenson and Kapitza. Crossref gives February 1908 for Stephenson; the UFN publisher gives volume 44, pages 7-20 (1951) for Kapitza.
- CONFIRMED | [stephenson-reference] "A. Stephenson (1908)" | https://api.crossref.org/works/10.1080/14786440809463763 | The publisher-deposited Crossref record dates the original paper to February 1908; the 2009 date is later online publication.
- CONFIRMED | [kapitza-reference] "P. L. Kapitza (1951)" | https://ufn.ru/ru/articles/1951/5/b/ | The UFN publisher record identifies Kapitza and the 1951 paper, volume 44, pages 7-20.
- CONFIRMED | [not-fingertip-balance] "Nothing is balancing on the fast jiggle in the way a broom balances on a fingertip." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 3 separates small rapid vibrations from slow motion under a prescribed periodic pivot drive.
- CONFIRMED | [fast-slow-average] "When one motion is much faster than another, you can average over the fast one and ask what slow landscape is left behind." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 3 separates small rapid vibrations from slow motion under a prescribed periodic pivot drive.
- CONFIRMED | [restoring-effective-potential] "Do that here and the rapid shaking contributes a real, time-averaged restoring force" ... "an effective potential." | https://arxiv.org/html/2308.14441v2 | Beneke et al. equation (13), with angle shifted by pi, gives the displayed leading-order potential. Its curvature is g/L+a²ω²/(2L²) at down and -g/L+a²ω²/(2L²) at up. These are properties of the averaged potential.
- CONFIRMED | [pivot-drive] "For a pivot driven as s(t) = a·cos(ωt), the slow motion of the bob behaves as if it lived in" | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov equation (2), with damping set to zero and z=a*cos(ωt), matches the displayed equation including both length denominators.
- CONFIRMED | [effective-potential-equation] "U(θ) ∝ −(g/L)·cos θ + (a²ω² / 4L²)·sin²θ" | https://arxiv.org/html/2308.14441v2 | Beneke et al. equation (13), with angle shifted by pi, gives the displayed leading-order potential. Its curvature is g/L+a²ω²/(2L²) at down and -g/L+a²ω²/(2L²) at up. These are properties of the averaged potential.
- CONFIRMED | [angle-origin] "where θ is measured from straight down." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 2 measures the angle from the lower equilibrium position; the page uses the same origin.
- CONFIRMED | [gravity-valley] "The first term is ordinary gravity: a single valley at the bottom (θ=0)." | https://arxiv.org/html/2308.14441v2 | Beneke et al. equation (13), with angle shifted by pi, gives the displayed leading-order potential. Its curvature is g/L+a²ω²/(2L²) at down and -g/L+a²ω²/(2L²) at up. These are properties of the averaged potential.
- CONFIRMED | [shaking-term-shape] "The second term" ... "the gift of the shaking" ... "is largest where sin²θ is largest, at the horizontal, and it pushes the bob away from sideways and toward the vertical, either vertical." | https://arxiv.org/html/2308.14441v2 | Beneke et al. equation (13), with angle shifted by pi, gives the displayed leading-order potential. Its curvature is g/L+a²ω²/(2L²) at down and -g/L+a²ω²/(2L²) at up. These are properties of the averaged potential.
- CONFIRMED | [hilltop-to-valley] "Crank the shaking up and the bump grows until the top, once a hilltop, caves into a genuine valley. The bob can sit in it." | https://arxiv.org/html/2308.14441v2 | Beneke et al. equation (13), with angle shifted by pi, gives the displayed leading-order potential. Its curvature is g/L+a²ω²/(2L²) at down and -g/L+a²ω²/(2L²) at up. These are properties of the averaged potential.
- CONFIRMED | [potential-panel] "This is exactly the morphing you can watch in the U(θ) panel as you raise the drive." | public/strata/kapitza-pendulum/index.html | drawPot evaluates the total averaged potential with a positive sin-squared shaking term. Its color follows the averaged threshold; it is not a separate force plot.
- CONFIRMED | [minimum-criterion] "The top (θ=180°) flips from hilltop to valley at a precise condition. Differentiate the effective potential, demand that θ=180° be a minimum, and out drops a clean inequality:" | https://arxiv.org/html/2308.14441v2 | Differentiating the displayed potential at pi gives -g/L+a²ω²/(2L²). Positivity gives the stated inequality for that approximate potential.
- CONFIRMED | [peak-pivot-speed] "In words: the peak speed of the pivot, a·ω, must beat √(2gL)." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Differentiating a*cos(ωt) gives peak pivot speed aω. K=aω/sqrt(gL); the stated parameter directions and sqrt(2) threshold follow within the averaged model.
- CONFIRMED | [parameter-directions] "Make the shake bigger or faster, or the pendulum shorter, and you cross the line." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Differentiating a*cos(ωt) gives peak pivot speed aω. K=aω/sqrt(gL); the stated parameter directions and sqrt(2) threshold follow within the averaged model.
- CONFIRMED | [dimensionless-threshold] "Written without units, with ω₀ = √(g/L) the natural swing rate, the same threshold is just K = (a/L)(ω/ω₀) > √2 ≈ 1.414" ... "the number in the readout." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Differentiating a*cos(ωt) gives peak pivot speed aω. K=aω/sqrt(gL); the stated parameter directions and sqrt(2) threshold follow within the averaged model.
- CONFIRMED | [slider-threshold-formula] "The drive-frequency slider's "threshold" mark is this formula solved for ω, recomputed whenever you change the amplitude or length." | public/strata/kapitza-pendulum/index.html | thresholdHz recomputes sqrt(2gL)/(2pi*a) when amplitude or length changes. This is the leading-order averaged formula, not a boundary measured from the trajectory.
- CONFIRMED | [averaging-assumption] "But that derivation assumed the averaging is valid." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 4 restricts the leading-order approximation to small a/L and high ω/ω0. The asymptotic statement is not evidence of convergence in the five measured rows.
- CONFIRMED | [simulation-not-formula] The bob's trajectory is computed without the averaged stability formula. | public/strata/kapitza-pendulum/index.html | accel and stepODE integrate the unaveraged ODE. The threshold and potential panels do use averaging; the revised prose makes this scope explicit.
- CONFIRMED | [true-step-integration] The simulation integrates the unaveraged equation of motion for an undamped point bob on a massless rod. | public/strata/kapitza-pendulum/index.html | accel implements the idealized equation; stepODE performs substepped RK4. The equation is exact for this model; the numerical trajectory is approximate and does not establish an exact stability boundary.
- CONFIRMED | [unaveraged-equation] "θ″ = −( g − a·ω²·cos ωt ) / L · sin θ" | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov equation (2), with damping set to zero and z=a*cos(ωt), matches the displayed equation including both length denominators.
- CONFIRMED | [threshold-sweep-method] The verifier bisects on drive frequency using finite-window trajectories. | research/kapitza-pendulum/verify.py | find_threshold uses brackets 2 and 60 times ω0, relative tolerance 1e-3, and stays_up. The latter has a finite observation window; it does not prove lasting stability.
- CONFIRMED | [verifier-numpy-rk4] "The verifier (research/kapitza-pendulum/verify.py, pure NumPy, fixed-step RK4) integrates the true equation of motion" ... "no averaging anywhere in the simulation." | research/kapitza-pendulum/verify.py | The source imports NumPy and implements four RK stages. The layer verifier runs with installed NumPy; the source and published copy are synchronized in this repair.
- CONFIRMED | [bisection-release] "Then, for each amplitude, it bisects on drive frequency to find where the inverted bob (released with a 0.10 rad tilt) stops toppling:" | research/kapitza-pendulum/verify.py | stays_up starts at pi+0.10 rad with zero angular velocity; find_threshold bisects the frequency using that finite-window classification.
- CONFIRMED | [table-column-meanings] The table columns are a/L, finite-window ω/ω0, K and K/sqrt(2). | research/kapitza-pendulum/verify.py | numbers.json stores amplitude ratio, drive angular frequency, K and its ratio to sqrt(2). Dividing the frequency by the stored ω0 gives the displayed second column; X7 fixes its fourth-row rounding.
- CONFIRMED | [fast-limit-exact] "which is exactly when the "fast-slow" averaging is supposed to become exact." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 4 restricts the leading-order approximation to small a/L and high ω/ω0. The asymptotic statement is not evidence of convergence in the five measured rows.
- CONFIRMED | [slow-period-formula] The averaged potential predicts the small-oscillation period 2pi/sqrt(a²ω²/(2L²)-g/L). | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | The curvature at pi gives the displayed small-oscillation frequency. Butikov printed p. 5 restricts this parabolic-well approximation to small slow oscillations.
- CONFIRMED | [rerun-table] The page table is reproduced from the verifier after converting drive frequency to ω/ω0. | research/kapitza-pendulum/verify.py | The verifier emits rad/s, not the dimensionless second column. Reproduction and preservation of numbers.json are reported below; the original claim is retained in that conversion scope.
- CONFIRMED | [ponderomotive-name] "It even has a name when it acts on charges: the ponderomotive force." | https://s3.cern.ch/inspire-prod-files-9/941adf195058a18f57eb0a3b9c3a4e2f | The CERN-hosted plasma-wakefield paper, p. 3, describes the ponderomotive expulsion of plasma electrons by an intense laser pulse.
- CONFIRMED | [earnshaw-forbids] Electrostatic fields alone cannot give a charged particle a stable stationary equilibrium in charge-free space. | https://simion.com/info/earnshaws_theorem.html | SIMION states the electrostatic and stationary restriction through Laplace's equation. Paul describes dynamic confinement with static electric and magnetic fields in a Penning trap.
- CONFIRMED | [paul-oscillating-saddle] "Wolfgang Paul's solution (Nobel Prize, 1989) was to make the saddle oscillate." | https://www.nobelprize.org/uploads/2018/06/paul-lecture.pdf ; https://www.nobelprize.org/prizes/physics/1989/paul/facts/ | Paul's lecture describes high-frequency quadrupole confinement and the effective well (printed pp. 602-611); the Nobel Foundation records the 1989 prize.
- CONFIRMED | [paul-nobel-year] "Nobel Prize, 1989" | https://www.nobelprize.org/prizes/physics/1989/paul/facts/ | The Nobel Foundation dates Paul's ion-trap prize to 1989.
- CONFIRMED | [paul-effective-well] "Exactly as with the pendulum, the fast switching turns the unstable saddle into a stable effective well," | https://www.nobelprize.org/uploads/2018/06/paul-lecture.pdf ; https://www.nobelprize.org/prizes/physics/1989/paul/facts/ | Paul's lecture describes high-frequency quadrupole confinement and the effective well (printed pp. 602-611); the Nobel Foundation records the 1989 prize.
- CONFIRMED | [ion-hours] "a lone ion sits in vacuum for hours." | https://www.nature.com/articles/ncomms1944 | The Nature paper describes an ionized ytterbium atom held in a Paul trap for several hours. Atom here includes an ionized atom; this is not neutral-atom confinement.
- CONFIRMED | [rotating-saddle] "The mechanical cartoon of this is literally a ball on a saddle that spins" ... "same averaging, same math." | https://www.nobelprize.org/uploads/2018/06/paul-lecture.pdf | Paul's lecture, printed pp. 610-611, describes a rotating saddle disc with a ball remaining near its center when the rotation frequency is in the suitable range.
- CONFIRMED | [ions-clocks-computers] "Trapped ions are now a leading platform for atomic clocks and quantum computers." | https://www.nist.gov/physics/nists-quantum-research-highlights | NIST describes trapped ions as qubits and atomic clocks and discusses research and commercial quantum-computing use.
- CONFIRMED | [force-expels-charges] "The same time-averaged force pushes charges out of regions of strong oscillating field" | https://s3.cern.ch/inspire-prod-files-9/941adf195058a18f57eb0a3b9c3a4e2f | The CERN-hosted plasma-wakefield paper, p. 3, describes the ponderomotive expulsion of plasma electrons by an intense laser pulse.
- CONFIRMED | [laser-wake] "structures the wake behind an intense laser pulse." | https://s3.cern.ch/inspire-prod-files-9/941adf195058a18f57eb0a3b9c3a4e2f | The CERN-hosted plasma-wakefield paper, p. 3, describes the ponderomotive expulsion of plasma electrons by an intense laser pulse.
- CONFIRMED | [strong-focusing-principle] "And strong focusing, the alternating-gradient principle that lets a particle accelerator's magnets focus a beam that each magnet alone would defocus, is the Kapitza idea applied in space rather than time: alternate the "wrong" force fast enough and the average becomes a restoring one." | https://www.nobelprize.org/uploads/2018/06/paul-lecture.pdf | Paul printed p. 602 connects alternating focusing and defocusing forces to accelerator strong focusing. A single magnet defocuses in one transverse plane; the sequence focuses in both.
- CONFIRMED | [all-large-colliders] Strong focusing underlies high-energy particle colliders. | https://www.bnl.gov/newsroom/news.php?a=217140&lang=en | Brookhaven's 2020 account explicitly gives the all-high-energy-colliders scope, distinct from its majority-of-accelerators scope.
- CONFIRMED | [model-assumptions] The simulation integrates the unaveraged equation of motion for an undamped point bob on a massless rod. | public/strata/kapitza-pendulum/index.html | accel implements the idealized equation; stepODE performs substepped RK4. The equation is exact for this model; the numerical trajectory is approximate and does not establish an exact stability boundary.
- CONFIRMED | [leading-order-scope] "The (aω)²>2gL threshold and the effective potential are leading-order results of the method of averaging, valid when the drive is fast (ω≫ω₀) and the shake small (a≪L);" | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 4 restricts the leading-order approximation to small a/L and high ω/ω0. The asymptotic statement is not evidence of convergence in the five measured rows.
- CONFIRMED | [playback-real-time] "Playback can be slowed to reveal the fast buzz; "real time" steps the ODE at the true rate." | public/strata/kapitza-pendulum/index.html | The frame handler multiplies elapsed frame time by the playback factor. Factor 1 gives real time for frame intervals below the 50 ms clamp; longer gaps are clamped.
- CONFIRMED | [historical-attribution-footer] "Historical attribution: A. Stephenson (1908) noticed the stability; P. L. Kapitza (1951) gave the averaging theory the effect is named for." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf ; https://api.crossref.org/works/10.1080/14786440809463763 ; https://ufn.ru/ru/articles/1951/5/b/ | Butikov printed p. 1 credits Stephenson and Kapitza. Crossref gives February 1908 for Stephenson; the UFN publisher gives volume 44, pages 7-20 (1951) for Kapitza.
- CONFIRMED | [independent-check] "This layer's check works the answer out again from scratch, independently of this page." | research/kapitza-pendulum/verify.py | verify.py computes from its own parameters and never reads the page. Its only file operation writes numbers.json.
- CONFIRMED | [cannot-notice-page-drift] The standalone numerical verifier cannot detect a changed figure on the page. | research/verifier-independence/census-pairs.tsv ; research/kapitza-pendulum/verify.py | The census records that it does not read the layer's shipped files; inspecting verify.py confirms this limitation. The new control probe is separate.
- CONFIRMED | [published-check] The numerical verifier is published under /checks/research/kapitza-pendulum/verify.py. | public/checks/research/kapitza-pendulum/verify.py ; public/checks/manifest.json | The public file exists and the existing manifest names that URL. This repair synchronizes the file; shared manifest metadata is left for the site publisher.
- CONFIRMED | [verifier-citation] "research/kapitza-pendulum/verify.py" | research/kapitza-pendulum/verify.py | The source imports NumPy and implements four RK stages. The layer verifier runs with installed NumPy; the source and published copy are synchronized in this repair.
- CONFIRMED | [nearby-dynamic-stability] "Two stabilities that look like they shouldn't exist, both born from motion rather than from a static balance." | https://arendschwab.com/research/stablebicycle/ | The bicycle researchers describe a self-steering moving bicycle. Together with the driven-pendulum source, this supports the analogy within the respective stable ranges.
- CONFIRMED | [nearby-page-recomputations] "Both take an ordinary piece of hardware and recover the exact physics hiding in it, ending on a number the page recomputes in front of you." | public/strata/as-hangs-the-chain/index.html ; public/strata/kapitza-pendulum/index.html | The catenary page implements solveCatenary with Newton iteration; the pendulum computes its potential and averaged threshold live. This does not make the pendulum threshold exact.
- CONFIRMED | [nearby-gait-analogy] "The same inverted pendulum, held up two different ways." | src/content/strata/lightest-in-the-middle-of-a-step.md | The related gait layer uses the stance-leg inverted-pendulum model and distinguishes it from driven-pivot stabilization.
- CONFIRMED | [portal-first-rung] "Rung 1 of the portal" ... "the term motion adds is a time-averaged EFFECTIVE POTENTIAL." | src/content/strata/steady-only-in-motion.md | The portal lists the Kapitza effective potential first, followed by its other mechanisms.
- CONFIRMED | [up-state-unstable] Below K=sqrt(2), the leading-order averaged potential has an unstable upright maximum. | public/strata/kapitza-pendulum/index.html | Its curvature at pi is negative below the threshold. At equality it vanishes, and the new label uses ≤ with the precise statement that up is not a minimum.
- CONFIRMED | [top-valley-knock] The averaged potential has an upright valley when K>sqrt(2). | public/strata/kapitza-pendulum/index.html | Its second derivative at pi is positive under this inequality. That does not guarantee recovery after the actual Knock it impulse.
- CONFIRMED | [hilltop-fall] Below K=sqrt(2), the leading-order averaged potential has an unstable upright maximum. | public/strata/kapitza-pendulum/index.html | Its curvature at pi is negative below the threshold. At equality it vanishes, and the new label uses ≤ with the precise statement that up is not a minimum.
- CONFIRMED | [readout-threshold] "threshold K = <b>√2 ≈ 1.414</b>" | public/strata/kapitza-pendulum/index.html | sqrt(2)=1.4142135623730951. With a=0.02 m, L=0.25 m and g=9.81, the averaged drive threshold is 17.62420930435013 Hz, displayed as 17.6 Hz.
- CONFIRMED | [default-frequency-threshold] "stand-up threshold ≈ <span id="thr-f">17.6</span> Hz" | research/kapitza-pendulum/numbers.json | sqrt(2)=1.4142135623730951. With a=0.02 m, L=0.25 m and g=9.81, the averaged drive threshold is 17.62420930435013 Hz, displayed as 17.6 Hz.
- CONFIRMED | [amplitude-definition] "how far the pivot travels (a)" | public/strata/kapitza-pendulum/index.html | s=a*cos(ωt) makes a the maximum distance from the midpoint, as the amplitude label states. Total peak-to-peak travel is 2a.
- CONFIRMED | [meta-0-0] "A pendulum has only one stable resting place, hanging down." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov equation (2), with drive and damping zero, has down and up equilibria. Linearizing restores down and destabilizes up; rest means stable rest in this context.
- CONFIRMED | [meta-0-1] "Shake its pivot up and down fast enough and a second one appears: it stands on its head and springs back when you push it." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 1 supports local upright stability in suitable amplitude and frequency intervals. This does not promise spontaneous lifting or recovery from the button impulse; those are corrected separately.
- CONFIRMED | [meta-0-2] "Drag the drive frequency across the threshold and watch the upside-down state switch from impossible to stable." | public/strata/kapitza-pendulum/index.html | The default 20 Hz control remains within 3.438 degrees for 10 s, while the 8 Hz control crosses 90 degrees at the 0.725 s sample. This supports the qualitative demo in those settings, not a universal exact boundary.
- CONFIRMED | [meta-1-0] "Shake a pendulum's pivot fast enough and a second stable resting place appears: standing straight up." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 1 supports local upright stability in suitable amplitude and frequency intervals. This does not promise spontaneous lifting or recovery from the button impulse; those are corrected separately.
- CONFIRMED | [meta-1-1] "Drag past the threshold and watch the impossible become stable." | public/strata/kapitza-pendulum/index.html | The default 20 Hz control remains within 3.438 degrees for 10 s, while the 8 Hz control crosses 90 degrees at the 0.725 s sample. This supports the qualitative demo in those settings, not a universal exact boundary.
- CONFIRMED | [meta-2-0] "Shake the pivot fast enough and a pendulum stands on its head." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 1 supports local upright stability in suitable amplitude and frequency intervals. This does not promise spontaneous lifting or recovery from the button impulse; those are corrected separately.
- CONFIRMED | [meta-2-1] "Drag past the threshold and watch the upside-down state turn stable." | public/strata/kapitza-pendulum/index.html | The default 20 Hz control remains within 3.438 degrees for 10 s, while the 8 Hz control crosses 90 degrees at the 0.725 s sample. This supports the qualitative demo in those settings, not a universal exact boundary.
- CONFIRMED | [plain-stabilization] "Kapitza's pendulum: shake a pendulum's pivot up and down fast enough and the upside-down position, normally the picture of instability, becomes a genuinely stable resting place that springs back when pushed." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 1 supports local upright stability in suitable amplitude and frequency intervals. This does not promise spontaneous lifting or recovery from the button impulse; those are corrected separately.
- CONFIRMED | [plain-simulation] The simulation integrates the unaveraged equation of motion for an undamped point bob on a massless rod. | public/strata/kapitza-pendulum/index.html | accel implements the idealized equation; stepODE performs substepped RK4. The equation is exact for this model; the numerical trajectory is approximate and does not establish an exact stability boundary.
- CONFIRMED | [structured-stabilization] "Shake a pendulum's pivot up and down fast enough and a second stable resting place appears: standing straight up." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 1 supports local upright stability in suitable amplitude and frequency intervals. This does not promise spontaneous lifting or recovery from the button impulse; those are corrected separately.
- CONFIRMED | [structured-exact-ode] The simulation integrates the unaveraged equation of motion for an undamped point bob on a massless rod. | public/strata/kapitza-pendulum/index.html | accel implements the idealized equation; stepODE performs substepped RK4. The equation is exact for this model; the numerical trajectory is approximate and does not establish an exact stability boundary.
- CONFIRMED | [md-rest] "A pendulum has exactly one stable resting place: hanging down." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov equation (2), with drive and damping zero, has down and up equilibria. Linearizing restores down and destabilizes up; rest means stable rest in this context.
- CONFIRMED | [md-up] "The upside-down position is the textbook picture of instability: the faintest nudge topples it." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov equation (2), with drive and damping zero, has down and up equilibria. Linearizing restores down and destabilizes up; rest means stable rest in this context.
- CONFIRMED | [md-shake] "Yet shake the pivot up and down fast enough and a second stable resting place appears, pointing straight up; push it and it springs back." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 1 supports local upright stability in suitable amplitude and frequency intervals. This does not promise spontaneous lifting or recovery from the button impulse; those are corrected separately.
- CONFIRMED | [md-immersive] "This is Kapitza's pendulum, and the immersive page lets you build it: drag the drive frequency across the line where the impossible becomes stable and watch the bob refuse to fall." | public/strata/kapitza-pendulum/index.html | The default 20 Hz control remains within 3.438 degrees for 10 s, while the 8 Hz control crosses 90 degrees at the 0.725 s sample. This supports the qualitative demo in those settings, not a universal exact boundary.
- CONFIRMED | [md-force] "The fast shaking adds a real, time-averaged restoring force (an effective potential) that, past a threshold, digs a valley at the top where there was a hill." | https://arxiv.org/html/2308.14441v2 | Beneke et al. equation (13), with angle shifted by pi, gives the displayed leading-order potential. Its curvature is g/L+a²ω²/(2L²) at down and -g/L+a²ω²/(2L²) at up. These are properties of the averaged potential.
- CONFIRMED | [md-formula-never] The bob's trajectory is computed without the averaged stability formula. | public/strata/kapitza-pendulum/index.html | accel and stepODE integrate the unaveraged ODE. The threshold and potential panels do use averaging; the revised prose makes this scope explicit.
- CONFIRMED | [md-paul-dek] "It is the same trick that traps a single ion in a Paul trap" | https://www.nobelprize.org/uploads/2018/06/paul-lecture.pdf ; https://www.nobelprize.org/prizes/physics/1989/paul/facts/ | Paul's lecture describes high-frequency quadrupole confinement and the effective well (printed pp. 602-611); the Nobel Foundation records the 1989 prize.
- CONFIRMED | [md-plasma-dek] RF ponderomotive plugging has been observed in a streaming plasma. | https://indico.ukaea.uk/event/449/contributions/1444/ ; https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.46.999 | Fader et al. (1981) report that experimental effect. This supports the general plasma application, while the Novatron fusion claim is limited to a proposed confinement improvement.
- CONFIRMED | [md-riderless] "A moving bicycle rights itself with no rider;" | https://arendschwab.com/research/stablebicycle/ | The bicycle researchers describe a self-steering moving bicycle. Together with the driven-pendulum source, this supports the analogy within the respective stable ranges.
- CONFIRMED | [md-pivot-relation] "a vibrating pivot lets a pendulum balance upside-down." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 1 supports local upright stability in suitable amplitude and frequency intervals. This does not promise spontaneous lifting or recovery from the button impulse; those are corrected separately.
- CONFIRMED | [md-dynamics-restoring] "in both, the dynamics supply a restoring force that a still picture can't see" | https://arendschwab.com/research/stablebicycle/ | The bicycle researchers describe a self-steering moving bicycle. Together with the driven-pendulum source, this supports the analogy within the respective stable ranges.
- CONFIRMED | [md-both-interactive] "both pages let you cross the line where it switches on." | public/strata/the-bike-that-rights-itself/index.html ; public/strata/kapitza-pendulum/index.html | The bicycle page's speed input calls setSpeed; the pendulum frequency input changes its ODE drive. Both let readers change a stability-controlling parameter.
- CONFIRMED | [md-chain-shape] "There a hanging chain proves it is a catenary, not a parabola;" | https://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-handout3.pdf | The Cambridge worked example derives y=lambda+c*cosh((x-x0)/c), a catenary, with higher powers absent from a parabola.
- CONFIRMED | [md-mass-damper] "A tuned mass damper adds a deliberate oscillation to drain a building's sway;" | https://www.taipei-101.com.tw/en/observatory/feature/damper | The Taipei 101 operator describes its tuned mass damper as a passive system designed to reduce building sway in wind.
- CONFIRMED | [md-pivot-control] "Kapitza's pivot adds a deliberate oscillation to manufacture a stability that gravity alone denies." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 1 supports local upright stability in suitable amplitude and frequency intervals. This does not promise spontaneous lifting or recovery from the button impulse; those are corrected separately.
- CONFIRMED | [md-two-equilibria] "A pendulum is the simplest balance in mechanics, and it has exactly two equilibria: hanging straight **down**, which is stable, and pointing straight **up**, which is the standard textbook example of *unstable*" ... "it exists in principle, but any real disturbance grows and over it goes." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov equation (2), with drive and damping zero, has down and up equilibria. Linearizing restores down and destabilizes up; rest means stable rest in this context.
- CONFIRMED | [md-history] "is **Kapitza's pendulum**, noticed by A. Stephenson in 1908 and explained by Pyotr Kapitza in 1951." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf ; https://api.crossref.org/works/10.1080/14786440809463763 ; https://ufn.ru/ru/articles/1951/5/b/ | Butikov printed p. 1 credits Stephenson and Kapitza. Crossref gives February 1908 for Stephenson; the UFN publisher gives volume 44, pages 7-20 (1951) for Kapitza.
- CONFIRMED | [md-averaging] "When one motion is far faster than another, you can average over the fast one and ask what slow landscape it leaves behind" ... "and the rapid shaking contributes a genuine, time-averaged restoring force, an **effective potential** `U(θ) ∝ −(g/L)cos θ + (a²ω²/4L²)sin²θ`." | https://arxiv.org/html/2308.14441v2 | Beneke et al. equation (13), with angle shifted by pi, gives the displayed leading-order potential. Its curvature is g/L+a²ω²/(2L²) at down and -g/L+a²ω²/(2L²) at up. These are properties of the averaged potential.
- CONFIRMED | [md-potential-terms] "The gravity term gives the single valley at the bottom; the shaking term grows where the rod is horizontal and pushes the bob toward *either* vertical. Crank the drive and the hilltop at the top caves into a real valley. The bob can sit in it." | https://arxiv.org/html/2308.14441v2 | Beneke et al. equation (13), with angle shifted by pi, gives the displayed leading-order potential. Its curvature is g/L+a²ω²/(2L²) at down and -g/L+a²ω²/(2L²) at up. These are properties of the averaged potential.
- CONFIRMED | [md-peak-speed] "Demanding that the top be a minimum of that potential gives **(aω)² > 2gL**" ... "the *peak pivot speed* aω must beat √(2gL); in dimensionless form, `K = (a/L)(ω/ω₀) > √2 ≈ 1.414`." | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Differentiating a*cos(ωt) gives peak pivot speed aω. K=aω/sqrt(gL); the stated parameter directions and sqrt(2) threshold follow within the averaged model.
- CONFIRMED | [md-equation-integration] The simulation integrates the unaveraged equation of motion for an undamped point bob on a massless rod. | public/strata/kapitza-pendulum/index.html | accel implements the idealized equation; stepODE performs substepped RK4. The equation is exact for this model; the numerical trajectory is approximate and does not establish an exact stability boundary.
- CONFIRMED | [md-verifier-method] The verifier bisects on drive frequency using finite-window trajectories. | research/kapitza-pendulum/verify.py | find_threshold uses brackets 2 and 60 times ω0, relative tolerance 1e-3, and stays_up. The latter has a finite observation window; it does not prove lasting stability.
- CONFIRMED | [md-exact-limit] "exactly the limit where fast-slow averaging becomes exact" | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Butikov printed p. 4 restricts the leading-order approximation to small a/L and high ω/ω0. The asymptotic statement is not evidence of convergence in the five measured rows.
- CONFIRMED | [md-ponderomotive] "Acting on charges it is the **ponderomotive force**:" | https://s3.cern.ch/inspire-prod-files-9/941adf195058a18f57eb0a3b9c3a4e2f | The CERN-hosted plasma-wakefield paper, p. 3, describes the ponderomotive expulsion of plasma electrons by an intense laser pulse.
- CONFIRMED | [md-plasma] RF ponderomotive plugging has been observed in a streaming plasma. | https://indico.ukaea.uk/event/449/contributions/1444/ ; https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.46.999 | Fader et al. (1981) report that experimental effect. This supports the general plasma application, while the Novatron fusion claim is limited to a proposed confinement improvement.
- CONFIRMED | [md-line-live] "the line is not a setting, it is computed from the equation of motion, live." | public/strata/kapitza-pendulum/index.html | thresholdHz recomputes sqrt(2gL)/(2pi*a) when amplitude or length changes. This is the leading-order averaged formula, not a boundary measured from the trajectory.
- CONFIRMED | [single-atom-heading] "Trapping a single atom" | https://www.nature.com/articles/ncomms1944 | The Nature paper describes an ionized ytterbium atom held in a Paul trap for several hours. Atom here includes an ionized atom; this is not neutral-atom confinement.
- CONFIRMED | [four-applications-heading] "The same trick, four places you'd never expect" | public/strata/kapitza-pendulum/index.html | The two boxes name four applications: ion traps, plasma confinement, accelerator focusing and laser wakes. The fusion example is now qualified as exploratory.
- CONFIRMED | [length-hint] "L · longer = harder to stand up" | https://butikov.faculty.ifmo.ru/ImprovedCriterion.pdf | Differentiating a*cos(ωt) gives peak pivot speed aω. K=aω/sqrt(gL); the stated parameter directions and sqrt(2) threshold follow within the averaged model.
- CONFIRMED | [frequency-default] "Drive frequency <span class="val" id="freq-val">20.0 Hz</span>" | public/strata/kapitza-pendulum/index.html | The input defaults and state are f=20 Hz, a=0.02 m and L=0.25 m. syncFromControls displays 20.0 Hz, 2.0 cm and 25 cm.
- CONFIRMED | [amplitude-default] "Shake amplitude <span class="val" id="amp-val">2.0 cm</span>" | public/strata/kapitza-pendulum/index.html | The input defaults and state are f=20 Hz, a=0.02 m and L=0.25 m. syncFromControls displays 20.0 Hz, 2.0 cm and 25 cm.
- CONFIRMED | [length-default] "Pendulum length <span class="val" id="len-val">25 cm</span>" | public/strata/kapitza-pendulum/index.html | The input defaults and state are f=20 Hz, a=0.02 m and L=0.25 m. syncFromControls displays 20.0 Hz, 2.0 cm and 25 cm.
- CONFIRMED | [canvas-description] The drawn pivot vibrates vertically and the bob uses the integrated angle. | public/strata/kapitza-pendulum/index.html | drawStage uses a*cos(ωt) for pivot displacement and st.theta for rod orientation. Boot or Stand it up places the bob near upright.
- CONFIRMED | [frequency-range] (added) "min="0" max="32" step="0.1"" | public/strata/kapitza-pendulum/index.html | The HTML controls declare 0-32 Hz by 0.1 Hz, 0.4-7.0 cm amplitude by 0.1 cm, and 12-45 cm length by 1 cm. Unit conversion is explicit in syncFromControls.
- CONFIRMED | [amplitude-range] (added) "min="0.4" max="7" step="0.1"" | public/strata/kapitza-pendulum/index.html | The HTML controls declare 0-32 Hz by 0.1 Hz, 0.4-7.0 cm amplitude by 0.1 cm, and 12-45 cm length by 1 cm. Unit conversion is explicit in syncFromControls.
- CONFIRMED | [length-range] (added) "min="12" max="45" step="1"" | public/strata/kapitza-pendulum/index.html | The HTML controls declare 0-32 Hz by 0.1 Hz, 0.4-7.0 cm amplitude by 0.1 cm, and 12-45 cm length by 1 cm. Unit conversion is explicit in syncFromControls.
- CONFIRMED | [playback-factors] (added) "'×0.3', '×0.1', '×0.03'" | public/strata/kapitza-pendulum/index.html | The labels match the factors [1, 0.3, 0.1, 0.03], which scale elapsed physical time in the frame handler.
- CONFIRMED | [image-dimensions] The social card has dimensions 1200 by 630 pixels. | public/strata/kapitza-pendulum/index.html | Read the existing PNG IHDR directly: width 1200, height 630, matching the HTML metadata. The image was not rendered or re-encoded.
- CONFIRMED | [tilt-readout] (added) "tilt from up" | public/strata/kapitza-pendulum/index.html | tiltFromUp subtracts 180 degrees and folds modulo 360 to [0,180], yielding unsigned distance from upright.

## Findings

- [open note:7b1799] X1 (interactive-threshold, WRONG): Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- [fixed] X2 (button-recovery, WRONG): Executing the actual button handlers and RK4 at simulation time zero with 2 cm, 25 cm, 20 Hz puts the first sampled excursion beyond 90 degrees at 0.20 s after an immediate knock. Without the knock, the 10 s control stays within 3.438 degrees. The impulse is unchanged; instructions and verdict no longer promise recovery.
- [fixed] X3 (bob-rises, WRONG): The hanging handler sets theta=0 and angular velocity=0; the acceleration is proportional to sin(theta). A 10 s run leaves both exactly zero. The prose now requires placing the bob near upright and distinguishes stabilization from lifting.
- [open note:7b1799] X4 (exact-threshold-heading, WRONG): Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- [open note:7b1799] X5 (threshold-iff, WRONG): Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- [fixed] X6 (energy-conservation, MINOR): energy_drift calls rk4_run with w=sqrt(g/L) and periods=40, so the duration is 40 small-angle periods (40.1213336 s), starting from pi/3. The energy drift remains 3.146076070707211e-8. HTML, notebook and both verifier print labels now name the actual window. MINOR, record only.
- [fixed] X7 (threshold-row-4, MINOR): numbers.json gives 74.19831016692754 / 6.26418390534633 = 11.844848632812498. The HTML and notebook cell now round to 11.84; all underlying measurements are retained. MINOR, record only.
- [open note:6c2430] X8 (threshold-within-percent, WRONG): The five stored percentage differences from sqrt(2) are -3.066985, -0.829310, +0.347015, +0.506873 and +0.766860. The fastest row is farthest below the prediction. At a=0.01 m, K=1.38, the original 60-cycle window ends at 1.7444058093 s; the unchanged RK4 trajectory crosses 90 degrees at 1.9077015754 s when extended to 240 cycles. Kapitza's prose and verifier now describe finite-window observations without claiming convergence; original numeric data are preserved. The portal's convergence claim remains open (fix-checker problem 2).
- [fixed] X9 (excess-next-order, WRONG): The fastest K values are 1.37083984375 and 1.4024853515625002, both below sqrt(2). Beneke et al., section I after Figure 3, discusses higher-order theory but does not establish that these observations measure a correction. The sign and unsupported interpretation are corrected in HTML, markdown, notebook and verifier.
- [fixed] X10 (period-agreement, WRONG): The stored period errors are 2.573009% at 180 rad/s, 0.935623% at 140 rad/s and 0.374299% at 120 rad/s. Amplitude and K also vary. The accurate rounded 0.4%-2.6% range remains; the claim that these settings show frequency convergence is removed.
- [fixed] X11 (fusion-confinement, MINOR): The Novatron abstract describes RF plugging as a proposed improvement and says electrodes are being fitted to compare theory with experiment. The page now says RF ponderomotive plugging is being explored to improve confinement in mirror-fusion designs. MINOR, record only.
- [fixed] X12 (damping-widens, MINOR): Butikov, printed pp. 13-14, reports little effect of weak friction on drive-parameter stability boundaries, while the attraction basin can shrink as friction increases. HTML and notebook now make that distinction and remove the blanket widening claim. MINOR, record only.
- [open note:6c2430] X13 (verifier-convergence-footer, WRONG): The five stored percentage differences from sqrt(2) are -3.066985, -0.829310, +0.347015, +0.506873 and +0.766860. The fastest row is farthest below the prediction. At a=0.01 m, K=1.38, the original 60-cycle window ends at 1.7444058093 s; the unchanged RK4 trajectory crosses 90 degrees at 1.9077015754 s when extended to 240 cycles. Kapitza's prose and verifier now describe finite-window observations without claiming convergence; original numeric data are preserved. The portal's convergence claim remains open (fix-checker problem 2).
- [open note:7b1799] X14 (up-state-stable, WRONG): Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- [fixed] X15 (knock-returns, WRONG): Executing the actual button handlers and RK4 at simulation time zero with 2 cm, 25 cm, 20 Hz puts the first sampled excursion beyond 90 degrees at 0.20 s after an immediate knock. Without the knock, the 10 s control stays within 3.438 degrees. The impulse is unchanged; instructions and verdict no longer promise recovery.
- [open note:7b1799] X16 (below-above-labels, WRONG): Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- [fixed] X17 (meta-0-3, MINOR): The trajectory uses accel and stepODE on the unaveraged equation; thresholdHz and drawPot use leading-order averaging. Metadata now identifies those separate calculations, including the matching markdown and structured descriptions. MINOR under the directing triage, record only.
- [open note:7b1799] X18 (meta-0-4, WRONG): Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- [fixed] X19 (meta-1-2, MINOR): The trajectory uses accel and stepODE on the unaveraged equation; thresholdHz and drawPot use leading-order averaging. Metadata now identifies those separate calculations, including the matching markdown and structured descriptions. MINOR under the directing triage, record only.
- [dated] X20 (modified-date, STALE): The read-only path history at c8289bd340 includes revisions dated 2026-09-27 and 2026-10-01 after the original 2026-06-29 date. dateModified is set to the requested pass date, 2026-10-14. The Corrected lines date this content revision; the explicit dateModified exception requires no Updated line.
- [open note:7b1799] X21 (md-threshold-dek, WRONG): Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- [open note:6c2430] X22 (md-verifier-dek, WRONG): The five stored percentage differences from sqrt(2) are -3.066985, -0.829310, +0.347015, +0.506873 and +0.766860. The fastest row is farthest below the prediction. At a=0.01 m, K=1.38, the original 60-cycle window ends at 1.7444058093 s; the unchanged RK4 trajectory crosses 90 degrees at 1.9077015754 s when extended to 240 cycles. Kapitza's prose and verifier now describe finite-window observations without claiming convergence; original numeric data are preserved. The portal's convergence claim remains open (fix-checker problem 2).
- [fixed] X23 (md-accelerators-dek, WRONG): Brookhaven's 2020 account says strong focusing is used in the vast majority of present-day accelerators and all high-energy colliders. The dek now says most present-day accelerators; the body and HTML use high-energy colliders. X23 has a correction line; X28 is the triage's MINOR scope fix, with no separate line.
- [open note:7b1799] X24 (md-threshold-relation, WRONG): Butikov, printed p. 4, equation (5), identifies an approximate criterion requiring a/L much smaller than 1 and drive frequency much greater than the natural frequency. The potential algebra is retained. The unchanged page ODE at 7 cm, 12 cm, 32 Hz crosses 90 degrees at the 0.09167 s sample despite K > sqrt(2). The Kapitza layer's authored threshold copies and badge now identify the leading-order averaged prediction. Shared copies remain in steady-only-in-motion and the unannotated there-here quotations (fix-checker problems 1 and 3).
- [fixed] X25 (md-bob-return, WRONG): The hanging handler sets theta=0 and angular velocity=0; the acceleration is proportional to sin(theta). A 10 s run leaves both exactly zero. The prose now requires placing the bob near upright and distinguishes stabilization from lifting.
- [open note:7b1799] X26 (md-threshold-results, WRONG): The five stored percentage differences from sqrt(2) are -3.066985, -0.829310, +0.347015, +0.506873 and +0.766860. The fastest row is farthest below the prediction. At a=0.01 m, K=1.38, the original 60-cycle window ends at 1.7444058093 s; the unchanged RK4 trajectory crosses 90 degrees at 1.9077015754 s when extended to 240 cycles. Kapitza's prose and verifier now describe finite-window observations without claiming convergence; original numeric data are preserved. The portal's convergence claim remains open (fix-checker problem 2).
- [fixed] X27 (md-paul-static, WRONG): SIMION restricts Earnshaw to stable stationary equilibrium using electrostatic forces alone in charge-free space. Paul, printed pp. 613-614, describes a Penning trap with a static electric field and axial magnetic field, with no radiofrequency drive. The markdown and the HTML escape-direction wording now preserve the electrostatic and stationary qualifications.
- [fixed] X28 (md-every-large-accelerator, MINOR): Brookhaven's 2020 account says strong focusing is used in the vast majority of present-day accelerators and all high-energy colliders. The dek now says most present-day accelerators; the body and HTML use high-energy colliders. X23 has a correction line; X28 is the triage's MINOR scope fix, with no separate line.

## Validation

- `node scripts/check-claims-pass.mjs`: exit 1. This record has twelve required `open board notes` placeholders that the gate cannot accept until the directing instance obtains board note IDs. Other failures were doppler-siren, the-price-of-forgetting and the-snows-of-yesteryear; those layers were left alone.
- The gate's exported `checkRecord` reports only the twelve pending-note syntax errors and no other record errors. Counts remain 154 checked, 126 confirmed, 27 wrong-count (20 WRONG plus seven MINOR), one STALE. The 28 findings comprise 15 fixed, one dated and twelve pending across two shared error classes.
- `OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 PYTHONDONTWRITEBYTECODE=1 python3 -u research/kapitza-pendulum/verify.py`: exit 0. The original numbers.json is byte-identical (2,566 bytes); the 240-cycle comparison finds the missed 90-degree excursion at 1.9077 s after the 60-cycle window ends at 1.7444 s.
- `node research/kapitza-pendulum/verify-assay.mjs`: exit 0. The actual inline ODE and handlers reproduce the default control, immediate-knock fall at the 0.20 s sample, large-drive fall at the 0.09167 s sample, exactly hanging control and reduced-motion pause. This is a numerical control check, not a browser rendering test.
- `GIT_NO_LAZY_FETCH=1 GIT_OPTIONAL_LOCKS=0 node scripts/check-em-dashes.mjs --base 9306f663bb5bb7f447014082bd7f342ea22e982d`: exit 0. The explicit local merge base avoids the checker's default git fetch.
- Source and public verifier copies agree. All eight correction bodies match HTML and markdown, appear above the check placard in HTML, and identify the assay line and requested date. All eight were checked against the original claims and evidence and left unchanged. No Updated line is present; X20 retains the triage's dateModified exception.
- The repair changed this record and wrote research/assay-line/queue-1014c/kapitza-pendulum/repair-report.md. The full fixer report remains research/kapitza-pendulum/fix-report-2026-10-14.md; its zero-open conclusion is superseded by this repair review.