---
slug: chladni-figures
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: 73
confirmed: 54
wrong: 19
unverifiable: 0
observed: 0
stale: 0
---

This was the directed repair pass for the 19 chladni-figures triage items at f5e5f84f34, with each item's evidence re-read before editing and all 54 CONFIRMED entries from the Codex N2 check retained below with their source qualifications. All 19 items have repairs in this layer's authored sources. Fifteen findings are closed; X16, X17, X18 and X19 remain WRONG and open because their historical quotation in there-here still lacks attached corrections. The wrong count includes the four MINOR entries. The retained confirmations record the checkers' source work, not 54 new independent checks by the fixer or repair reviewer. Nine dated corrections cover the distinct WRONG errors and are mirrored in the markdown body; MINOR fixes have no separate page correction. This record is scoped to the directed triage and retained confirmations, not the six UNVERIFIABLE textbook/generalization claims or the 102 OBSERVED entries in the larger N2 pilot report.

## Claims

- WRONG | X1: brass, ν = 0.225 (Ritz, 1909) | https://lakeslab.ep.wisc.edu/PoissonIntro2.html ; https://www.unige.ch/~gander/Preprints/ChladniTacoma.pdf | Lakes lists brass at 0.33; Gander and Kwok, section 2.1, identify 0.225 as Ritz's historical parameter. The menu now says historical value, with no change to the calculation.
- WRONG | X2: straight lines curl into hyperbolas in front of you.) | git show f5e5f84f34:public/strata/chladni-figures/chladni-core.mjs | Re-evaluation of the unchanged sine formula gives 2 sin(πx) sin(πy) (u+v)(4uv-1), where u=cos(πx), v=cos(πy). The curved branch is cos(πx)cos(πy)=1/4; the maximum factorization residual on 25 sample points was 3.33e-16.
- WRONG | X3: A membrane's frequencies depend only on its shape | https://link.springer.com/article/10.1007/s42452-020-04062-6 | The introduction gives c²=τ/ρ and a frequency scale c/a. The replacement distinguishes frequency ratios of uniform membranes from physical pitches.
- WRONG | X4: the fundamental moves from 13.46 for brass to a different value for steel | https://ntrs.nasa.gov/api/citations/19700009156/downloads/19700009156.pdf | Visually re-read NASA SP-160, printed p. 109, Table 4.67: upper bounds 14.111 at ν=.225 and 13.464 at ν=.300. Running solvePlate(nu,8,40) gives displayed fundamentals 14.12 and 13.48.
- WRONG | X5: while a membrane's never would. | https://link.springer.com/article/10.1007/s42452-020-04062-6 | The c/a frequency scale includes tension, density and size. The replacement limits the comparison to frequency ratios for a uniform membrane.
- WRONG | X6: Young's and Warburton's beam-function values (13.49, 19.79, 24.43, 35.02, 35.02, 61.53) are Rayleigh–Ritz upper bounds | https://ir.lib.nycu.edu.tw/bitstream/11536/8502/1/000259156600003.pdf ; https://api.crossref.org/works/10.1115%2F1.4010175 | Huang, Leissa and Liao (2008), p. 1257, Table 1, credits the first five numbers to Leissa [5]; p. 1262 identifies the 1973 paper. Young's publisher-deposited abstract lists three problems, each with a clamped edge. The unsupported sixth reference value, 61.53, was removed, not reassigned.
- MINOR | X7: Every one of these is asserted by research/chladni-figures/verify.mjs (26 checks, all green) | git show f5e5f84f34:research/chladni-figures/verify.mjs | The verifier samples x=.5 for mode 1 and y=x for mode 2; mode 3 is checked only by centre amplitude. Labels and coverage prose now describe those samples. The finite-centre membrane smoke check is also labelled as such.
- WRONG | X8: Each frame every grain is kicked by an amount set by the local vibration amplitude: violent on the antinodes, still on the nodes. | git show f5e5f84f34:public/strata/chladni-figures/index.html | The animation computes kick=0.004+0.30*amp*amp, then adds random increments in x and y. Its kick remains nonzero at amp=0. Prose now says grains move much more slowly near nodes; the code is unchanged.
- WRONG | X9: So a grain on a shaking region bounces until it stumbles onto a quiet line and stops. | git show f5e5f84f34:public/strata/chladni-figures/index.html | The same nonzero kick floor disproves stopping at a quiet line. Prose now says the grain slows nearly to a stop but keeps wandering.
- WRONG | X10: That wall is why Walther Ritz invented his variational method in 1909 to compute exactly these figures, the direct ancestor of the finite element method. | https://onlinelibrary.wiley.com/doi/10.1002/andp.19093330403 | Ritz's 1909 publisher page cites his method paper in J. reine angew. Math. 135, p. 1 (1908), reference p737_2. The replacement distinguishes method publication from the plate application.
- WRONG | X11: it is that a plate's frequencies depend on Poisson's ratio, how the metal bent one way curls the other, while a membrane's depend only on shape. | https://link.springer.com/article/10.1007/s42452-020-04062-6 | The c²=τ/ρ relation and frequency scale c/a also qualify the dek: uniform-membrane frequency ratios are the material-independent comparison.
- MINOR | X12: Every frequency the page computes is recomputed offline and checked against Young (1950), Warburton (1954) and Leissa's *Vibration of Plates* (NASA SP-160, 1969) in `research/chladni-figures/verify.mjs` (26/26), with an independent LAPACK cross-check. | git show f5e5f84f34:research/chladni-figures/verify.mjs | The pinned verifier compared six modes at ν=.300 and one fundamental at ν=.225, not every displayed frequency. After X6, it compares five at ν=.300 and the ν=.225 fundamental. It has 25 checks. The text states that coverage.
- MINOR | X13: The biharmonic equation with no closed-form solution for a square, the problem Ritz invented his method to solve in 1909. | https://onlinelibrary.wiley.com/doi/10.1002/andp.19093330403 | The 1909 page cites Ritz's 1908 method paper. The original grammar could date the application, so this remains MINOR; the metadata now explicitly says applied his method to in 1909.
- WRONG | X14: The drumhead is the wrong model for a Chladni plate. Stiffness, not tension; free edges, not clamped; no closed form for a square: the problem Ritz invented his method for in 1909. | https://onlinelibrary.wiley.com/doi/10.1002/andp.19093330403 | The publisher dates the free-square-plate paper to 1909 and cites the 1908 method paper. The Open Graph description now names the 1908 method and its 1909 application.
- MINOR | X15: The drumhead model is the wrong physics; a Chladni plate is the free-edge biharmonic problem Ritz invented his method to solve in 1909. | https://onlinelibrary.wiley.com/doi/10.1002/andp.19093330403 | As with X13, the invention wording ambiguously dates the application. JSON-LD now says applied his method to in 1909; its dateModified is 2026-10-14.
- WRONG | X16: w_xx·w_yy − w_xy², which is exactly the determinant of the Hessian, the discrete Gaussian curvature of the bent surface. | https://www.math.uci.edu/~ndonalds/math162a/notes.html | Re-read Donaldson's notes, Example 3.19 (p. 57) and section 3.4 (p. 68). The general graph's fundamental forms give K=det(Hess w)/(1+w_x²+w_y²)². The bare determinant is a small-slope approximation, not a discrete-curvature construction.
- WRONG | X17: By Gauss's Theorema Egregium that quantity is intrinsic | https://www.math.uci.edu/~ndonalds/math162a/notes.html | Donaldson, p. 78, attaches intrinsicness to Gaussian curvature K. The revised related-layer note attaches Gauss's theorem to K, after giving the full graph formula.
- WRONG | X18: so it vanishes for a clamped or simply-supported edge and the frequencies forget the material. | https://ntrs.nasa.gov/api/citations/19700009156/downloads/19700009156.pdf | Visually re-read NASA SP-160, printed p. 41: rectangular-plate parameters lose explicit ν-dependence if no edge is free, while physical frequencies retain it through D. The note now requires every straight edge to be clamped or simply supported and distinguishes normalized parameters from pitches.
- WRONG | X19: which is the whole reason a Chladni plate's spectrum depends on what it's made of, and a drumhead's never does. | https://link.springer.com/article/10.1007/s42452-020-04062-6 | The uniform-membrane frequency scale c/a retains tension, density and size. The related-layer note now compares frequency ratios, with shape and edge conditions fixed.

- CONFIRMED | Scatter sand on a vibrating plate and it draws a figure | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 1, sand-figure. Introduction: "The sand would eventually settle along the nodal lines"; "The complex patterns changed as the frequency of vibration was varied". The footer limits the sand animation to an illustration.
- CONFIRMED | fleeing the parts that shake, gathering on the lines that don't. | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 2, sand-moves. Introduction: "The sand would eventually settle along the nodal lines"; "The complex patterns changed as the frequency of vibration was varied". The footer limits the sand animation to an illustration.
- CONFIRMED | That is the wrong physics, and the real one is a problem so hard it forced a new method into being. | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 4, wrong-model. Section 2: "an exact solution to this problem has not yet been found" and "the first numerical solution of Chladni figures on a square plate in 1909". Read in the stated free-edge, thin-plate context as no known closed form, not an impossibility theorem.
- CONFIRMED | Ernst Chladni, in 1787, drew a bow across the edge of a sand-strewn brass plate and made sound visible. | https://www.e-rara.ch/zut/download/ftpack/plain/1308214 ; https://monoskop.org/images/1/18/Stoeckmann_H-J_2007_Chladni_meets_Napoleon.pdf | Entry 5, chladni-1787. The supplied Gander caption says "an iron plate", so that excerpt does not establish brass. Chladni's 1787 original says "An einer etwas dünnen messingenen Scheibe"; Stoeckmann (2007), quoting Chladni's account, says "a brass plate taken from a grinding machine". These support brass by the 1787 publication, independently of the mismatched caption.
- CONFIRMED | The sand abandons the antinodes | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 6, antinodes. Introduction: "The sand would eventually settle along the nodal lines"; "The complex patterns changed as the frequency of vibration was varied". The footer limits the sand animation to an illustration.
- CONFIRMED | and piles along the nodal lines, the curves that stand perfectly still. | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 7, nodes. Introduction: "The sand would eventually settle along the nodal lines"; "The complex patterns changed as the frequency of vibration was varied". The footer limits the sand animation to an illustration.
- CONFIRMED | Change the pitch and the figure changes. | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 8, pitch-pattern. Introduction: "The sand would eventually settle along the nodal lines"; "The complex patterns changed as the frequency of vibration was varied". The footer limits the sand animation to an illustration.
- CONFIRMED | The patterns are real, reproducible, and strange | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 9, reproducible. Introduction: "The sand would eventually settle along the nodal lines"; "The complex patterns changed as the frequency of vibration was varied". The footer limits the sand animation to an illustration.
- CONFIRMED | steel, ν = 0.300 | https://lakeslab.ep.wisc.edu/PoissonIntro2.html | Entry 12, steel. The supplied table gives "Steel [1]" as "0.29" and "Stainless Steel [7]" as "0.30". A nominal steel value 0.300 is defensible, but the excerpt is not an exact match to the generic row; specify the alloy or label the values nominal.
- CONFIRMED | aluminium, ν = 0.330 | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 13, aluminium. The supplied Lakes table gives "Aluminum [4]" as "0.34". The independently opened square-plate experiment explicitly says "we set \(\mu = 0.33\) for aluminum" in section 4.2, supporting the page's nominal 0.330.
- CONFIRMED | A drumhead is a membrane: a thin sheet held flat by tension, like the skin of a drum pulled tight over its rim. | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 18, membrane-tension. Introduction specifies a "tightly stretched thin elastic membrane" with uniform tension and density. These are the ideal membrane assumptions used here.
- CONFIRMED | Its motion obeys the second-order wave equation | https://arxiv.org/pdf/1308.5523 | Entry 19, membrane-equation. Appendix A: "all four boundary lines are fixed"; equation A.1 is the sine-product mode and A.3 gives the squared frequencies. Setting both side lengths to 1 yields the page's formula. The page separately discusses degenerate mixtures, so the grid assertion describes the separated basis.
- CONFIRMED | on a square clamped at the edge its modes are exactly sin(mπx)·sin(nπy) | https://arxiv.org/pdf/1308.5523 | Entry 20, membrane-modes. Appendix A: "all four boundary lines are fixed"; equation A.1 is the sine-product mode and A.3 gives the squared frequencies. Setting both side lengths to 1 yields the page's formula. The page separately discusses degenerate mixtures, so the grid assertion describes the separated basis.
- CONFIRMED | with frequencies proportional to √(m²+n²) and nodal lines that form a straight grid. | https://arxiv.org/pdf/1308.5523 | Entry 21, membrane-frequencies. Appendix A: "all four boundary lines are fixed"; equation A.1 is the sine-product mode and A.3 gives the squared frequencies. Setting both side lengths to 1 yields the page's formula. The page separately discusses degenerate mixtures, so the grid assertion describes the separated basis.
- CONFIRMED | It is a stiff sheet of metal, restored not by tension but by its own bending stiffness | https://ntrs.nasa.gov/api/citations/19700009156/downloads/19700009156.pdf | Entry 23, bending-stiffness. Chapter 1, p. 1: "D is the flexural rigidity"; D = Eh³/[12(1-ν²)] and the plate equation DV⁴w + ρ_area w_tt = 0 establish bending rather than membrane tension.
- CONFIRMED | its edges are free, not clamped. | https://ntrs.nasa.gov/api/citations/19700009156/downloads/19700009156.pdf | Entry 24, free-edges. Section 4.3.13: "in 1787 (ref. 4.95) for completely free bound-" followed by "aries". The claim concerns the rim, not the supporting or driving point.
- CONFIRMED | That is a different equation: the fourth-order biharmonic D∇⁴w = ρh ω²w | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 25, biharmonic. Section 2, equations 1-3 give the fourth-order plate equation. Substitution of harmonic time dependence and restoring the page's D and volumetric density convention gives D∇⁴w = ρhω²w.
- CONFIRMED | For a square it has no closed-form solution at all. | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 26, no-closed-form. Section 2: "an exact solution to this problem has not yet been found" and "the first numerical solution of Chladni figures on a square plate in 1909". Read in the stated free-edge, thin-plate context as no known closed form, not an impossibility theorem.
- CONFIRMED | You cannot write the modes down; you can only approximate them. | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 27, only-approximate. Section 2: "an exact solution to this problem has not yet been found" and "the first numerical solution of Chladni figures on a square plate in 1909". Read in the stated free-edge, thin-plate context as no known closed form, not an impossibility theorem.
- CONFIRMED | In 1909 Walther Ritz, trying to compute precisely these square-plate figures, published a variational method | https://home.agh.edu.pl/~milenin/programs/doc/Courant_1943.pdf | Entry 28, ritz-invention. Courant says "In two publications of 1908 and 1909". Publication of the square-plate application in 1909 is correct; the earlier invention date is separately disputed at dek-ritz.
- CONFIRMED | guess the shape as a sum of simple functions, then let the energy pick the best coefficients | https://home.agh.edu.pl/~milenin/programs/doc/Courant_1943.pdf | Entry 29, ritz-method. Section II.1 describes "linear combinations of the coordinate functions" and determination of their coefficients from an "ordinary minimum problem".
- CONFIRMED | that became the Rayleigh–Ritz method, the direct ancestor of the finite element method now used to model everything from bridges to bones. | https://home.agh.edu.pl/~milenin/programs/doc/Courant_1943.pdf ; https://www.unige.ch/~gander/Preprints/ChladniTacoma.pdf ; https://pmc.ncbi.nlm.nih.gov/articles/PMC4281601/ | Entry 30, ritz-fem. Courant's section IV constructs functions "linear in the meshes" and an "arbitrarily chosen (preferably triangular) net", the finite-element ancestor. Gander and Kwok apply the plate machinery to bridges; the bone-model primary study says "a complete FE model of the human leg was realised".
- CONFIRMED | A membrane can make curves too: when two modes like (3,2) and (2,3) share a frequency, any blend of them is also a mode, and the blend's nodal lines bend. | https://arxiv.org/pdf/1308.5523 | Entry 32, degenerate-curves. Appendix A: "different eigenmodes to the same eigenfrequency can be mixed". Linear superpositions remain modes and can have curved nodes; not every blend is curved, since pure and special symmetric cases can remain straight.
- CONFIRMED | stretch any drumhead of the same outline and the pattern of pitches is fixed. | https://arxiv.org/pdf/1308.5523 | Entry 35, same-outline. Appendix A.3 shows a common wave-speed factor c multiplying all frequencies. Thus the "pattern of pitches" is defensible as frequency ratios for a uniformly tensioned, homogeneous membrane with fixed geometry, not absolute pitch.
- CONFIRMED | A plate's frequencies depend on Poisson's ratio | https://ntrs.nasa.gov/api/citations/19700009156/downloads/19700009156.pdf | Entry 36, plate-poisson. Chapter 4, p. 41 distinguishes frequency parameters from frequencies and says the former depend on Poisson ratio if "at least one of the edges of the plate is free". The table convention uses mass per area, equivalent to ρh here.
- CONFIRMED | how much the metal, bent one way, curls the other. | https://lakeslab.ep.wisc.edu/PoissonIntro2.html | Entry 37, poisson-definition. "Poisson's ratio governs the curvature in a direction perpendicular to the direction of bending." This supports the stated anticlastic bending description.
- CONFIRMED | The number on the readout is moving for a reason no drumhead could produce. | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 41, no-drum-reason. The specific reason here is changing ν in a normalized free-plate eigenproblem. The ideal uniform membrane equation contains tension and density through c² = τ/ρ, not a separate ν-dependent change in frequency ratios. This does not rescue the broader absolute-frequency claims.
- CONFIRMED | the convention Leissa tabulates. | https://ntrs.nasa.gov/api/citations/19700009156/downloads/19700009156.pdf | Entry 43, leissa-convention. Chapter 4, p. 41 distinguishes frequency parameters from frequencies and says the former depend on Poisson ratio if "at least one of the edges of the plate is free". The table convention uses mass per area, equivalent to ρh here.
- CONFIRMED | The plate also has exactly three modes at λ = 0 (it floats: one slide, two tilts) | https://resource.isvr.soton.ac.uk/staff/pubs/PubPDFs/Pub1587.pdf | Entry 49, three-zero-modes. Section 2.1.3, printed p. 13, PDF p. 21, opened and OCRed: "For a plate with four free edges, three rigid body modes should be included." They are transverse translation and the two rotations. This counts the transverse model, not all six rigid motions of a 3-D solid.
- CONFIRMED | the figures Chladni and Mary Waller catalogued. | https://ntrs.nasa.gov/api/citations/19700009156/downloads/19700009156.pdf | Entry 53, catalogued. Section 4.3.13 attributes the original free-boundary observations to Chladni, and p. 103 says "Waller (ref. 4.119) obtained experimental frequencies and mode shapes for square brass" plates; Figure 4.55 reproduces the patterns.
- CONFIRMED | exactly as it does on a real plate | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 61, real-plate-emergence. Introduction: "The sand would eventually settle along the nodal lines"; "The complex patterns changed as the frequency of vibration was varied". The footer limits the sand animation to an illustration.
- CONFIRMED | it gives upper bounds that converge from above | https://ntrs.nasa.gov/api/citations/19700009156/downloads/19700009156.pdf ; https://ir.lib.nycu.edu.tw/bitstream/11536/8502/1/000259156600003.pdf | Entry 64, upper-bounds. Leissa describes "extremely accurate upper bounds" obtained by Rayleigh-Ritz and successive admissible bases. Huang et al., section 3, explicitly describes upper-bound convergence, subject to a complete basis. This is a statement of the variational method, not a certified error bound for every displayed mode.
- CONFIRMED | Real Chladni plates are usually clamped or driven at a central point, which pins that point as a node and selects which figures appear | https://www.physics.ucsb.edu/~lecturedemonstrations/Composer/Pages/44.45.html | Entry 70, central-support. The supplied one-sentence excerpt only identifies the central driver. The full UCSB page also says "the point where the plate is fixed to the driver is a node" and explains motion "relative to the anchor point". This supports the conventional constrained-anchor description; the anchor is not stationary in the laboratory frame.
- CONFIRMED | Scatter sand on a vibrating metal plate, bow its edge, and the grains flee the parts that shake and gather along the lines that don't, drawing a figure. | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 78, dek-sand. Introduction: "The sand would eventually settle along the nodal lines"; "The complex patterns changed as the frequency of vibration was varied". The footer limits the sand animation to an illustration.
- CONFIRMED | A real Chladni plate is held by bending stiffness, not tension, and is free at its edges, not clamped: the fourth-order biharmonic equation D∇⁴w = ρhω²w, which has no closed-form solution for a square. | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 80, dek-plate. Section 2: "an exact solution to this problem has not yet been found" and "the first numerical solution of Chladni figures on a square plate in 1909". Read in the stated free-edge, thin-plate context as no known closed form, not an impossibility theorem.
- CONFIRMED | Stiffness not tension, free edges not clamped: the problem Ritz solved in 1909. | https://onlinelibrary.wiley.com/doi/10.1002/andp.19093330403 | Entry 91, twitter-ritz. The publisher gives "First published: 1909" and identifies the free square-plate paper. A numerical solution in 1909 is correct; this sentence does not independently date the invention.
- CONFIRMED | On a *free* edge it survives | https://ntrs.nasa.gov/api/citations/19700009156/downloads/19700009156.pdf | Entry 97, free-boundary-survives. Chapter 4, p. 41 distinguishes frequency parameters from frequencies and says the former depend on Poisson ratio if "at least one of the edges of the plate is free". The table convention uses mass per area, equivalent to ρh here.
- CONFIRMED | Chladni's figures are the nodal lines of an eigenmode | https://arxiv.org/pdf/1308.5523 ; https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 99, chladni-eigenmode. The supplied Appendix A excerpt concerns a membrane. The same paper's introduction explicitly says "not the eigenmodes themself but only the nodal lines become visible" and section 2 gives the linear wave eigenproblem. The free-plate linear eigenproblem is independently supplied by the Springer article, section 2.
- CONFIRMED | a standing solution of a linear operator | https://arxiv.org/pdf/1308.5523 ; https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 100, linear-standing. The supplied Appendix A excerpt concerns a membrane. The same paper's introduction explicitly says "not the eigenmodes themself but only the nodal lines become visible" and section 2 gives the linear wave eigenproblem. The free-plate linear eigenproblem is independently supplied by the Springer article, section 2.
- CONFIRMED | Turing's spots are the fastest-growing mode of a reaction–diffusion instability. | https://www.damtp.cam.ac.uk/user/gold/pdfs/teaching/turing1952.pdf ; https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 101, turing-mode. Section 6 says terms with "the largest real part" dominate, while allowing several equally dominant terms. This supports dominant wavelength selection in the linear instability; the plate comparison describes resonant selection, not spontaneous growth in an undriven stable plate.
- CONFIRMED | Both are a shape the system selects out of a featureless start because one wavelength wins | https://www.damtp.cam.ac.uk/user/gold/pdfs/teaching/turing1952.pdf ; https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 102, pattern-selects. Section 6 says terms with "the largest real part" dominate, while allowing several equally dominant terms. This supports dominant wavelength selection in the linear instability; the plate comparison describes resonant selection, not spontaneous growth in an undriven stable plate.
- CONFIRMED | diffuse the inhibitor faster | https://math.nyu.edu/faculty/childres/case2.pdf | Entry 103, inhibitor. The record's abstract excerpt mentions instability but does not establish relative diffusion. Childress's derivation, p. 9, equation 34, says "the inhibitor chemical should dif-" followed by "fuse more rapidly that the activator chemical". This supplies the omitted support for the standard two-species activator-inhibitor case.
- CONFIRMED | Both instruments let matter draw the allowed states: sand settles on a plate's nodal lines, while silver settles at allowed magnetic projections. | https://arxiv.org/pdf/2301.11343 | Entry 106, silver. The translated original Stern-Gerlach report says "the silver-atom beam is split" into two directions and describes the deposits. This supports the comparison, without equating magnetic projections to continuous plate displacement.
- CONFIRMED | D. Young, J. Appl. Mech. 17 (1950) 448 | https://api.crossref.org/works/10.1115%2F1.4010175 | Entry 107, young-citation. Crossref publisher metadata: author "Dana Young", volume "17", pages "448-453", issued 1950-12-01; title "Vibration of Rectangular Plates by the Ritz Method". The citation exists, though it does not support the free-plate numerical attribution.
- CONFIRMED | G. B. Warburton, Proc. IMechE 168 (1954) 371 | https://api.crossref.org/works/10.1243%2FPIME_PROC_1954_168_040_02 | Entry 108, warburton-citation. Crossref publisher metadata: "G. B." "Warburton", volume "168", pages "371-384", issued 1954-06; title "The Vibration of Rectangular Plates".
- CONFIRMED | M. D. Waller, Proc. Phys. Soc. 51 (1939) 831 | https://api.crossref.org/works/10.1088%2F0959-5309%2F51%2F5%2F312 | Entry 109, waller-citation. Crossref publisher metadata: "Mary D" "Waller", volume "51", pages "831-844", issued 1939-09-01; title "Vibrations of free square plates: part I. Normal vibrating modes".
- CONFIRMED | W. Ritz, Ann. Phys. 333 (1909) 737 (the free square plate; the Ritz method) | https://onlinelibrary.wiley.com/doi/10.1002/andp.19093330403 ; https://api.crossref.org/works/10.1002%2Fandp.19093330403 | Entry 110, ritz-citation. Curl returned a publisher challenge, so I opened the indexed publisher page and checked Crossref separately. Both give the title, 1909, volume 333 and pages 737-786. The 1909 paper cites its author's 1908 method in its first-page footnote.
- CONFIRMED | A. W. Leissa, Vibration of Plates, NASA SP-160 (1969), Tables 4.65–4.67 | https://ntrs.nasa.gov/api/citations/19700009156/downloads/19700009156.pdf | Entry 111, leissa-citation. The NASA document identifies SP-160 and 1969; the cited tables 4.65-4.67 concern the free square plate. They support the bibliography and the 13.464/14.111 comparison, not the entire array attributed to Young.
- CONFIRMED | E. F. F. Chladni, Entdeckungen über die Theorie des Klanges, 1787 | https://www.e-rara.ch/zut/content/titleinfo/1308214 | Entry 112, chladni-citation. The ETH catalogue gives "Entdeckungen über die Theorie des Klanges", "Ernst Florens Friedrich Chladni" and "Leipzig : bey Weidmanns Erben und Reich, 1787".
- CONFIRMED | T. Müller, Numerical Chladni figures, arXiv:1308.5523 (degenerate membrane curves). | https://arxiv.org/pdf/1308.5523 | Entry 113, muller-citation. The downloaded paper title is "Numerical Chladni figures", author "Thomas Müller", arXiv:1308.5523v1, 26 Aug 2013. Appendix A discusses degeneracy.
- CONFIRMED | with the edge conditions that the bending moment and the shear both vanish. | https://link.springer.com/article/10.1007/s42452-020-04062-6 | Entry 114, free-moment-shear. Section 2, equations 4-6, gives the moment and effective-shear edge conditions, plus the corner condition. "Shear" here is the effective Kirchhoff edge shear.
- CONFIRMED | Poisson's ratio sneaks into a plate's pitches through one term in its bending energy | https://www-users.cse.umn.edu/~olver/a_/bcvp.pdf | Entry 115, bending-energy-term. Olver, Example 8.1, equations 8.13-8.15, identifies the Hessian determinant as a "null Lagrangian" and writes it as Dx(u_x u_yy) + Dy(-u_x u_xy). Its area integral is therefore a boundary integral. This does not make the determinant the exact Gaussian curvature.
- CONFIRMED | integrated over the plate it is a boundary term (a cousin of Gauss–Bonnet) | https://www-users.cse.umn.edu/~olver/a_/bcvp.pdf | Entry 116, boundary-integral. Olver, Example 8.1, equations 8.13-8.15, identifies the Hessian determinant as a "null Lagrangian" and writes it as Dx(u_x u_yy) + Dy(-u_x u_xy). Its area integral is therefore a boundary integral. This does not make the determinant the exact Gaussian curvature.
- CONFIRMED | Both make an unseen field in a solid suddenly visible with a physical trick. | https://engineering.purdue.edu/IE/news/2017/ruperts-drops | Entry 117, drops-comparison. Purdue reports "used integrated photoelasticity to measure the complex stress distribution". This supports the physical visualization comparison.

## Findings

- [fixed] X1: Replaced the brass label with Ritz historical value and labelled material settings as nominal; added the material correction.
- [fixed] X2: Changed hyperbolas to curves; added the equal-mixture cosine-product correction.
- [fixed] X3: Qualified the main membrane paragraph as uniform-membrane frequency ratios; added the shared membrane correction.
- [fixed] X4: Replaced the mistaken 13.46/preset pairing with the actual 14.12 and 13.48 readouts; the correction separately identifies the published upper bounds.
- [fixed] X5: Removed the absolute claim that a membrane's spectrum would never shift; shared X3's correction.
- [fixed] X6: Corrected the heading, prose, metadata, notebook and verifier attribution to the five checked Leissa (1973) values as reproduced in Huang et al. (2008), Table 1. Removed the sixth comparison; corrected the core's NASA table pointer. Added the attribution correction. The repair review corrected Liao's initials to S. C. in the page and notebook, following the paper's author line.
- [fixed] X7: Limited shape-check claims and assertion labels to their sampled features, including the finite-value smoke check. Recorded only; no page correction for this MINOR item.
- [fixed] X8: Described simulated grains as much slower near nodes; shared the nonzero-kick correction with X9.
- [fixed] X9: Replaced stops with slows nearly to a stop, but keeps wandering; shared X8's correction.
- [fixed] X10: Separated the 1908 method publication from the 1909 plate application in the dek, page, core and notebook; added the shared history correction.
- [fixed] X11: Qualified the dek's membrane comparison as frequency ratios for uniform membranes; shared X3's correction.
- [fixed] X12: Specified five ν=.300 comparisons and one ν=.225 fundamental comparison, and updated the check count to 25. Recorded only; no separate page correction for this MINOR item.
- [fixed] X13: Changed the meta description to applied his method to in 1909. Recorded only; no separate page correction for this MINOR clarification.
- [fixed] X14: Changed the Open Graph wording to application of the 1908 method in 1909; shared X10's history correction.
- [fixed] X15: Changed the JSON-LD invention wording to application in 1909 and updated dateModified. Recorded only; no separate page correction for this MINOR clarification or the revision date.
- [open note:99f79a] X16: Gave the exact graph formula and small-slope qualification in the authored related-layer note; regenerated this page's nearby fragment from that source. Added the shared curvature correction. The historical quotation in public/strata/there-here/index.html:250, notes[188], still calls the Hessian determinant discrete Gaussian curvature and needs an attached correction.
- [open note:99f79a] X17: Attached intrinsicness to Gaussian curvature K; shared X16's curvature correction. The same there-here quotation still calls the bare Hessian determinant intrinsic and needs an attached correction.
- [open note:99f79a] X18: Restricted the boundary claim to rectangular plates with every edge clamped or simply supported and distinguished normalized parameters from material-dependent physical frequencies; added the boundary/frequency correction and made its edge conditions explicit in the repair review. The same there-here quotation still claims a constrained edge makes frequencies forget the material and needs an attached correction.
- [open note:99f79a] X19: Qualified the related-layer membrane comparison as frequency ratios for a uniform membrane; shared X3's correction. The same there-here quotation still claims a drumhead's spectrum never depends on material and needs an attached correction.

The authored layer fixes above do not regenerate prohibited shared outputs. Search copies and other outputs listed by the fix-checker as rebuilt at landing need no worker edit. The old related-layer note remains embedded in public/strata/there-here/index.html:250 (X16, X17, X18 and X19). That quotation must remain intact, with corrections attached through the existing renderCopyCorrections() mechanism by the directing instance. This pass does not claim that quotation is repaired.