A lower bound you can move, an upper-bound proof you can inspect
Eighteen Curves Turn the Corner
Turn Gerver's equation-built sofa through a unit-width corner, recompute its area, then operate the injectivity iteration behind Jineon Baek's claimed proof of optimality. The result is presented as a preprint claim because completed peer review could not be verified.
Claimed resolution in arXiv:2411.19826 v1, peer review not verified complete
Layer one: make it fit
Drag first. The sofa stays fixed while the hallway follows Romik's five-phase rotation path. This dual viewpoint is mathematically equivalent to moving the sofa through a fixed hallway. The pale shape is regenerated from the solved path as an intersection of sampled hallway positions, not traced from an image.
Rotation
solving equations
The canvas will report the rotation, phase, active contacts, solved area, and hallway width here.
A C D
corner A C D
corner A C
corner A B C
A B C
area from live quadraturecomputing
active boundary membersA C D
scale checkarea × width²
Gerver's boundary has 18 analytic pieces in the source decomposition: 3 straight segments and 15 curved segments. The live canvas uses the equivalent hallway-intersection definition from Romik's Equation 8. Its finite sampling is a visualization of feasibility, not a continuous collision certificate.
Layer two: fitting is not optimality
A successful turn only supplies a lower bound. It cannot rule out a larger shape with a stranger motion. Baek's preprint closes that gap through an injectivity condition and then a concave quadratic upper-bound functional. The first instrument below runs the actual lower-bound operator from Definitions 6.5.1 and 6.5.2. The second exposes the quadratic identity on a normalized support-mode slice.
Baek iteration
f₀ at t = π/4
A text report of the selected lower-bound function appears here.
The actual recurrence
m₀(x) = x − max(|x−1|, (|x−1|+1)/2) Ff(x) = 1 + ∫₀ˣ m₀(f(π/2−u)) du fₙ₊₁(x) = max(fₙ(x), Ffₙ(x))
The analytic certificate
computing the two closed-form margins
The curves are finite-grid numerical illustrations. The printed 1/12 ladder is Baek's analytic argument. This page evaluates its closed-form margins with ordinary floating point, not directed-rounding interval arithmetic, so the browser display is not itself a rigorous interval proof.
A normalized quadratic support-mode slice
q(0) = area
The selected quadratic value and concavity identity appear here.
This is an exact one-mode demonstration of the quadratic concavity identity, normalized so q(0) equals the computed Gerver area and q′(0) equals zero. It is not a browser evaluation of Baek's full geometric functional Q(K,B,D). That full functional, its enclosing region, and its directional-derivative proof remain in Chapter 8 of the preprint.
Open variant
solving Romik's cubics
X²(X+3)=8, Y(4Y²+3)=1, area = X+atan(Y)
The positive roots and residuals are computed in this browser. This candidate turns both left and right. Neither Romik's paper nor the sources checked for this page prove that it is the largest ambidextrous sofa.
The check
Six independent readouts expose the load-bearing computations. Every value below is filled by the page's own JavaScript after load.
1. Breakpoint system
waiting
2. Path joins
waiting
3. Area cross-check
waiting
4. Motion sampling
waiting
5. Conditional bound
waiting
6. Ambidextrous roots
waiting
Conventions, free choices, approximations, and uncertainties
Units: quoted constants use hallway width 1. Length scales with width and area with width squared.
Viewpoint: the sofa is fixed and the hallway moves. This is Romik's dual convention.
Root finding: damped Newton iteration starts from the freely chosen seed (A,B,φ,θ) = (0.1,1.4,0.04,0.68). Ordinary binary64 arithmetic is used.
Quadrature: the headline area uses 16-point Gauss-Legendre quadrature on 4 panels per subinterval, split at the phase joins. The doubling cross-check repeats it on 2 panels. An adaptive Simpson pass is printed in check 3 only as a comparison, and it disagrees from the eighth decimal.
Equation transcription can be wrong. The offline verifier extracts this page's shipped functions and compares them with separately written formulas and source values.
The rendered shape intersects 121 sampled hallway positions. Finite sampling shows the motion but does not certify every time in a continuum.
The ODE path check covers four phase joins. It does not independently certify positional continuity at every endpoint of the source's 18-piece boundary decomposition.
The f curves use 1,600 grid intervals. The analytic inequalities printed beside them come from Baek's proof, but the browser does not implement directed rounding.
The quadratic slice is a faithful algebraic demonstration, not Baek's complete Q functional.
Publication status: arXiv still exposed only v1 when checked on 29 July 2026. POSTECH notices dated 6 January and 2 February 2026 said peer review was in progress. No completed refereed publication was located, which is not proof that none exists.
What is still open
Romik's ambidextrous problem asks for one connected shape that can turn both left and right through unit-width right-angle corners. His exact candidate is computed above, but optimality was not verified. The related T-junction variant may share its optimum, but the checked source does not prove that equality. Baek's theorem concerns the original single right-angle turn. It does not settle general turn angles, curved hallways, several nearby turns, or three-dimensional corridors.
Primary sources and status
Jineon Baek, Optimality of Gerver's Sofa, arXiv:2411.19826v1, submitted 29 November 2024 at 16:37:23 UTC. Preprint, 119 pages. This is the claimed optimality proof and source of the f iteration and Q proof chain.
Joseph L. Gerver, On moving a sofa around a corner, Geometriae Dedicata 42, 267 to 283, June 1992, DOI 10.1007/BF02414066. Peer-reviewed. This is the 18-piece construction.
Dan Romik, Differential Equations and Exact Solutions in the Moving Sofa Problem, Experimental Mathematics 27(3), 316 to 330, online 19 January 2017, DOI 10.1080/10586458.2016.1270858. Peer-reviewed. This is the ODE path, contact equations, area computation, and ambidextrous candidate.
Yoav Kallus and Dan Romik, Improved upper bounds in the moving sofa problem, Advances in Mathematics 340, 960 to 982, 15 December 2018, DOI 10.1016/j.aim.2018.10.022. Peer-reviewed. This is the prior 2.37 upper bound and a published decimal for Gerver's area.
Bogdan Georgiev, Javier Gómez-Serrano, Terence Tao, and Adam Zsolt Wagner, Mathematical exploration and discovery at scale, arXiv:2511.02864v3, revised 22 December 2025. Preprint. Its moving-sofa section still describes Romik's ambidextrous value as a lower bound whose sharpness is open.
Corrected 2026-10-18: The quadrature note under the check said the area came from adaptive Simpson integration stopped at a requested tolerance, cross-checked by an independently coded composite Simpson rule. The page's code takes the headline area from 16-point Gauss-Legendre quadrature on 4 panels and cross-checks it on 2 panels; the composite Simpson routine is never called, and the adaptive Simpson value is shown only as a comparison that disagrees from the eighth decimal (found by the assay line, 2026-10-18).
The claims recordNaN claims re-read against their sources, 18 October 2026
Written 2026-07-29. Claims re-read against their sources on 2026-10-18: NaN checked, 66 confirmed, NaN wrong, 0 unverifiable, 5 first-hand observations checked against their record. By the assay line (research/assay-line/claude-1018b/), Claude fixer (claude-opus-5-5) directed by claude-assaying-boyle-1014; source record by a Claude agent (claude-sonnet-5-5), checked by a Claude agent (claude-opus-5-5).
This pass was the assay line's source-first re-read of this layer (batch claude-1018b, tier B). A Claude agent (claude-sonnet-5-5) wrote a source record for the page's outside claims (research/assay-line/claude-1018b/eighteen-curves-turn-the-corner/N1-sources.json); a second Claude agent (claude-opus-5-5) read each entry against its source (arXiv, Crossref, Springer and ScienceDirect via the Wayback Machine, POSTECH), rendered the page's JavaScript readouts in node, and mutated the layer's verifier in a scratch copy (N2-claude.md); a third, the fixer, re-read the evidence for every finding, decided it (triage.json), and made the one correction: the quadrature note under the check named adaptive Simpson and composite Simpson rules where the code uses Gauss-Legendre. Three MINOR lines were listed, not edited, under the tier B rule.
Claims
CONFIRMED Jineon Baek, Optimality of Gerver's Sofa, arXiv:2411.19826v1, submitted 29 November 2024 at 16:37:23 UTC. https://arxiv.org/abs/2411.19826 : [baek-citation] "Title:Optimality of Gerver's Sofa", "Authors:Jineon Baek", "[v1] Fri, 29 Nov 2024 16:37:23 UTC (3,039 KB)" (fetched 2026-10-19)
CONFIRMED This is the claimed optimality proof and source of the f iteration and Q proof chain. https://arxiv.org/abs/2411.19826 : [baek-claimed-optimality] abstract: "We resolve the moving sofa problem by showing that Gerver's construction with 18 curve sections attains the maximum area 2.2195..."; Definitions 6.5.1, 6.5.2 and Chapter 8 hold the f iteration and Q
CONFIRMED Claimed resolution in arXiv:2411.19826 v1, peer review not verified complete https://arxiv.org/abs/2411.19826 : [status-arxiv-v1] on 2026-10-19 the Submission history lists only "[v1]" and the page has no Journal-ref field
CONFIRMED Baek's preprint closes that gap through an injectivity condition and then a concave quadratic upper-bound functional. https://arxiv.org/html/2411.19826v1 : [baek-injectivity-concave-Q] Section 1.1: "we prove a property called the injectivity condition ... We use Mamikon's theorem to establish the global concavity of Q on L ... Because Q is concave, G also maximizes Q globally"
CONFIRMED The claimed theorem is precise: Gerver's sofa attains the maximum area among connected planar shapes that can negotiate a unit-width right-angle hallway. https://arxiv.org/html/2411.19826v1 : [baek-theorem] "What is the largest area of a connected planar shape that can move around the right-angled corner of a hallway with unit width?"; "Theorem 1.1.1. Gerver's sofa G attains the maximum area of a moving sofa."
CONFIRMED Baek's theorem concerns the original single right-angle turn. https://arxiv.org/html/2411.19826v1 : [baek-theorem-scope] Definition 1.1.1 defines one hallway L = H_L union V_L (one right-angled corner); Definition 1.1.2 a moving sofa in it
CONFIRMED The first instrument below runs the actual lower-bound operator from Definitions 6.5.1 and 6.5.2. https://arxiv.org/html/2411.19826v1 : [baek-defs-F-fn] Definition 6.5.1 "Ff(x)=1+\int_0^x m_0(f(\pi/2-u))du"; Definition 6.5.2 "f_0(x):=0 ... f_{n+1}(x):=max(f_n(x),Ff_n(x))"; buildIterations() in the page applies exactly this with a trapezoid rule on 1,600 intervals
CONFIRMED m₀(x) = x − max(abs(x−1), (abs(x−1)+1)/2) [the page writes the absolute values with vertical bars] https://arxiv.org/html/2411.19826v1 : [formula-m0] Definition 6.3.4: "k_0(x):=max(abs(x-1),(abs(x-1)+1)/2) and m_0(x)=x-k_0(x)"
CONFIRMED The printed 1/12 ladder is Baek's analytic argument. https://arxiv.org/html/2411.19826v1 : [ladder-1-12] Lemma 6.5.3 "Define the constant d_0:=1/12. Then for any c in [0,2/3], we have Fj_c(x) >= j_{c+d_0}(x)"; Lemma 6.5.5 applies it ten times
MINOR For i ≤ 10, fᵢ(x) ≥ max(1−x,(i−1)/12). https://arxiv.org/html/2411.19826v1 : [fi-bound] Lemma 6.5.5 proof: "For any integer 1 <= m <= 10, we will show that f_m >= j_{(m-1)/12}". The page uses the paper's own indexing (its f₀ button is the paper's f_0 := 0), so "i ≤ 10" includes i = 0, where the bound is false: f₀ = 0 < 1−x on [0,1); the page's own readout says "f0(0.2500π) = 0.000000000" while 1−π/4 = 0.2146. Recomputed on the page's grid: f1 to f10 satisfy the bound, f0 misses it by 1 at x = 0. Fix: "For 1 ≤ i ≤ 10"
CONFIRMED Then f₁₀ > 2/3 makes m₀ positive, so f₁₁(x) > 1 for every x > 0. https://arxiv.org/html/2411.19826v1 : [f11-gt-1] Lemma 6.5.5 proof: "So f_10 >= j_{9/12} > 2/3 ... Because m_0(y)>0 for all y>2/3 ... f_11(x) >= Ff_10(x) ... > 1 for all x in (0,pi/2]"; the functions live on [0,pi/2], so "every x > 0" is the proof's (0,pi/2] (the lemma's statement says (0,1])
CONFIRMED The two minimized closed-form margins in Lemma 6.5.3 recompute to 0.127821832 and 0.005643665, both positive. https://arxiv.org/html/2411.19826v1 : [lemma-653-margins] the proof of Lemma 6.5.3 prints both: "the difference is minimized at c=7/6-pi/4 with the value -1/8+3pi/8-3pi^2/32>0" and "minimized at c=5/3-pi/2 with the value -1/2+3pi/4-3pi^2/16>0"; recomputed 0.127821832 and 0.005643665. The source record's note that the paper does not print them is wrong
CONFIRMED That full functional, its enclosing region, and its directional-derivative proof remain in Chapter 8 of the preprint. https://arxiv.org/html/2411.19826v1 : [chapter-8] contents: "8 Optimality of Gerver's Sofa, 8.1 Domain of Q, 8.2 Definition of Q, 8.3 Concavity of Q, 8.4 Gerver's Sofa, 8.5 Directional Derivative of Q"
CONFIRMED It is not a browser evaluation of Baek's full geometric functional Q(K,B,D). https://arxiv.org/html/2411.19826v1 : [Q-KBD] Section 1.8.2: "We now define the upper bound Q(K,B,D) of the area of a monotone sofa S as a quadratic functional on (K,B,D) in L (Definition 8.2.2)"
CONFIRMED This is an exact one-mode demonstration of the quadratic concavity identity, normalized so q(0) equals the computed Gerver area and q′(0) equals zero. public/strata/eighteen-curves-turn-the-corner/index.html : [quadratic-identity] renderQ(): q(x)=area-0.5*x*x*(pi/4), so q(e)+q(-e)-2q(0) = -e^2*pi/4 exactly and q'(0)=0; the page labels it its own slice, and Baek (Section 1.1) supplies only the concavity of Q
CONFIRMED Joseph L. Gerver, On moving a sofa around a corner, Geometriae Dedicata 42, 267 to 283, June 1992, DOI 10.1007/BF02414066. https://api.crossref.org/works/10.1007/BF02414066 : [gerver-citation] Crossref: given "Joseph L.", family "Gerver", Geometriae Dedicata, vol 42, issue 3, pages 267-283, issued 1992-06; Springer page (Wayback 2023-07-31): "Issue Date: June 1992". The sources tool's "DOI MISMATCH (gives Gerver the initial L.)" is a false alarm: the page's "Joseph L. Gerver" matches Crossref
CONFIRMED Gerver's boundary has 18 analytic pieces in the source decomposition: 3 straight segments and 15 curved segments. https://arxiv.org/html/1606.08111v3 : [gerver-3-15] Romik Section 1.1: "whose boundary comprises 3 straight line segments and 15 distinct curved segments, each described by a separate analytic expression"
CONFIRMED Dan Romik, Differential Equations and Exact Solutions in the Moving Sofa Problem, Experimental Mathematics 27(3), 316 to 330, online 19 January 2017, DOI 10.1080/10586458.2016.1270858. https://api.crossref.org/works/10.1080/10586458.2016.1270858 : [romik-citation] Crossref: Dan Romik, Experimental Mathematics vol 27 issue 3 pages 316-330, published-online 2017-01-19
CONFIRMED Peer-reviewed. This is the ODE path, contact equations, area computation, and ambidextrous candidate. https://arxiv.org/html/1606.08111v3 : [romik-scope] abstract: "deriving a family of six differential equations"; Section 4 equations (27)-(44); Section 4: "also shows how to compute the area 2.21953166... of Gerver's sofa"; Section 5 the ambidextrous shape
CONFIRMED Romik derives the moving-sofa path from local optimality ODEs. https://arxiv.org/abs/1606.08111 : [romik-local-optimality-ODEs] "deriving a family of six differential equations arising from the area-maximization property"
CONFIRMED Viewpoint: the sofa is fixed and the hallway moves. This is Romik's dual convention. https://arxiv.org/html/1606.08111v3 : [romik-dual] Section 2: "a dual point of view from the frame of reference of the sofa, in which it is the hallway being rotated and translated whereas the sofa stays in a fixed place"
CONFIRMED This dual viewpoint is mathematically equivalent to moving the sofa through a fixed hallway. https://arxiv.org/html/1606.08111v3 : [romik-dual-equivalent] Section 2: "From this point of view we see that a shape S can be moved around the corner if it satisfies the condition (7)"; a change of reference frame
CONFIRMED The live canvas uses the equivalent hallway-intersection definition from Romik's Equation 8. https://arxiv.org/html/1606.08111v3 : [romik-eq8] equation (8): "S_x = L_horiz cap (cap over 0<=t<=pi/2 of x(t)+R_t(L)) cap (x(pi/2)+R_{pi/2}(L_vert))"; pointInSofa() tests exactly these three conditions at 121 sampled t
CONFIRMED Drag first. The sofa stays fixed while the hallway follows Romik's five-phase rotation path. https://arxiv.org/html/1606.08111v3 : [romik-five-phase] equation (24): {A,C,D}, {x,A,C,D}, {x,A,C}, {x,A,B,C}, {A,B,C} with breaks at phi, theta, pi/2-theta, pi/2-phi; the page's phase strip and activeContacts() match
CONFIRMED 28 Romik matching constraints, including four C¹ phase joins: max residual https://arxiv.org/html/1606.08111v3 : [romik-28-equations] "The equations (27)-(44) comprise a total of 28 (scalar) equations in the 22 variables"; (35)-(42) are "the condition that x be continuously differentiable" at the four joins. The page's linearResidual() reproduces (27)-(44) one for one (5 + 3 + 16 + 4), with phi and theta taken from the four-equation system
CONFIRMED It solves the breakpoint equations, regenerates the five-phase path, integrates Gerver's area, and exposes both the numerical and analytic status of the claimed optimality proof. https://arxiv.org/html/1606.08111v3 : [romik-eqs-1-4] equations (1)-(4) for A, B, phi, theta transcribe exactly into gerverResidual(); the verifier's independent transcription agrees (residual 2.151e-16)
CONFIRMED X²(X+3)=8, Y(4Y²+3)=1, area = X+atan(Y) https://arxiv.org/abs/1606.08111 : [romik-ambi-area] "Its area is equal to X+arctan Y, where X and Y are solutions to the cubic equations x^2(x+3)=8 and x(4x^2+3)=1"
CONFIRMED This candidate turns both left and right. https://arxiv.org/html/1606.08111v3 : [ambi-both-ways] Section 1.2: "satisfies the conditions of an ambidextrous moving sofa (i.e., it can move around corners to the left and to the right)"
CONFIRMED Neither Romik's paper nor the sources checked for this page prove that it is the largest ambidextrous sofa. https://arxiv.org/html/1606.08111v3 : [ambi-not-proved] Romik: "a plausible candidate to be the solution"; Georgiev et al. Problem 6.63: "It remains open whether this bound is sharp"
CONFIRMED Romik's ambidextrous problem asks for one connected shape that can turn both left and right through unit-width right-angle corners. https://arxiv.org/html/1606.08111v3 : [ambi-statement] Romik Section 1.2: "the planar shape of maximal area that can negotiate right-angled turns both to the right and to the left in a hallway of width 1"; "connected" is not Romik's word (0 hits for "connected" in his text) but is Georgiev et al.'s ("a connected planar shape", Problem 6.63); same problem
CONFIRMED The related T-junction variant may share its optimum, but the checked source does not prove that equality. https://arxiv.org/html/1606.08111v3 : [t-junction] Figure 3 caption: "another version of it asking for the largest shape that can turn completely around in a T-junction (the answer to both versions may be the same)"
CONFIRMED Yoav Kallus and Dan Romik, Improved upper bounds in the moving sofa problem, Advances in Mathematics 340, 960 to 982, 15 December 2018, DOI 10.1016/j.aim.2018.10.022. https://web.archive.org/web/20230731145311/https://www.sciencedirect.com/science/article/pii/S000187081830416X : [kr-citation] "Advances in Mathematics Volume 340, 15 December 2018, Pages 960-982 ... Yoav Kallus a, Dan Romik b"; Crossref agrees (vol 340, pages 960-982, 2018-12)
CONFIRMED Peer-reviewed. This is the prior 2.37 upper bound and a published decimal for Gerver's area. https://arxiv.org/html/1706.06630 : [kr-237-decimal] "We prove a new and improved upper bound of 2.37"; "Its area is Gerver's constant mu_G=2.21953166..."
CONFIRMED Jineon Baek, A Conditional Upper Bound for the Moving Sofa Problem, arXiv:2406.10725, submitted 15 June 2024. https://arxiv.org/abs/2406.10725 : [cond-citation] "[Submitted on 15 Jun 2024 (v1), last revised 2 Dec 2024 (this version, v2)] Title:A Conditional Upper Bound for the Moving Sofa Problem Authors:Jineon Baek"
MINOR Preprint. This is the injectivity-conditional bound recomputed below. https://arxiv.org/abs/2406.10725 : [cond-status] the arXiv page says "This paper has been withdrawn by Jineon Baek", "[v2] Mon, 2 Dec 2024 03:02:31 UTC (1 KB) (withdrawn)", Comments: "This work has been superseded by arXiv:2411.19826 that resolves the moving sofa problem completely. Do not cite this work". The page's "Primary sources and status" list gives its status only as "Preprint." although the withdrawal predates the page (29 July 2026) by 20 months; the bound itself is in the v1 abstract and is not retracted as false
CONFIRMED (conditional, not the final claimed optimum) [1 + π²/8 = 2.233700550136170] https://arxiv.org/abs/2406.10725 : [cond-bound] "we prove that any moving sofa satisfying a property named the injectivity condition has an area of at most 1 + pi^2/8 = 2.2337..."; 1 + pi^2/8 = 2.2337005501361697 recomputed
CONFIRMED Bogdan Georgiev, Javier Gómez-Serrano, Terence Tao, and Adam Zsolt Wagner, Mathematical exploration and discovery at scale, arXiv:2511.02864v3, revised 22 December 2025. https://arxiv.org/abs/2511.02864 : [georgiev-citation] "Authors:Bogdan Georgiev, Javier Gómez-Serrano, Terence Tao, Adam Zsolt Wagner"; "[v3] Mon, 22 Dec 2025 12:49:01 UTC"; v3 is still the latest on 2026-10-19
CONFIRMED Its moving-sofa section still describes Romik's ambidextrous value as a lower bound whose sharpness is open. https://arxiv.org/html/2511.02864v3 : [georgiev-ambi] Problem 6.63: "Romik [243] introduced the 'Romik sofa' that produced a lower bound C >= 1.6449... It remains open whether this bound is sharp."
CONFIRMED POSTECH Department of Mathematics, institutional notice dated 6 January 2026. https://math.postech.ac.kr/en/bbs/board.php?bo_table=m08_04&wr_id=15 : [postech-jan] "[Alumnus] Jineon Baek Solves a 60-Year-Old Mathematical Problem, Recognized as a Global Mathematical Breakthrough ... 2026-01-06"
CONFIRMED Announcement, not peer review. It said the paper was undergoing peer review. https://math.postech.ac.kr/en/bbs/board.php?bo_table=m08_04&wr_id=15 : [postech-jan-review] "The corresponding paper has been submitted to one of the most prestigious journals in mathematics and is currently undergoing peer review."
CONFIRMED POSTECH notices dated 6 January and 2 February 2026 said peer review was in progress. https://math.postech.ac.kr/en/bbs/board.php?bo_table=m08_04&wr_id=16 : [postech-both] "2026-02-02 ... The results are presented in a comprehensive research paper that is currently undergoing international peer review."; the arXiv half of the sentence is the next line
OBSERVED arXiv still exposed only v1 when checked on 29 July 2026. research/eighteen-curves-turn-the-corner/README.md : [obs-arxiv-v1-0729] README at 9ddc1e9045, "Checked 29 July 2026", item 1: "The arXiv record exposed one version and no journal reference when checked."; consistent with today's page, which still lists only v1 (a version cannot be removed, so none existed in July)
CONFIRMED No completed refereed publication was located, which is not proof that none exists. https://api.crossref.org/works?query.bibliographic=Optimality+of+Gerver%27s+Sofa : [no-refereed-pub] Crossref title query: no record of Baek's paper (positive control: the same query returns Gerver 1992 and Leng et al., Symmetry 2024); Crossref author query "Jineon Baek" returns his other refereed papers (positive control) but not this one; arXiv has no Journal-ref; https://annals.math.princeton.edu/toappear lists forthcoming papers (positive control: "Anschütz", "Hopkins") with no "Baek" or "sofa" (2026-10-19). OpenAlex was rate-limited and not used
CONFIRMED The f curves use 1,600 grid intervals. public/strata/eighteen-curves-turn-the-corner/index.html : [data-1600] buildIterations(): "var count=1600,h=(Math.PI/2)/count"
CONFIRMED Grid size 1,600 intervals. public/strata/eighteen-curves-turn-the-corner/index.html : [data-1600-built] renderIteration() alt text, same grid
CONFIRMED Root finding: damped Newton iteration starts from the freely chosen seed (A,B,φ,θ) = (0.1,1.4,0.04,0.68). public/strata/eighteen-curves-turn-the-corner/index.html : [data-seed] solveGerver(): "var z = [0.1, 1.4, 0.04, 0.68];" with halving damping; rendered readout "3 Newton steps", phi and theta match Romik Table 1
CONFIRMED active contact-to-wall equations across 41 sampled times, max signed-distance residual public/strata/eighteen-curves-turn-the-corner/index.html : [data-41] runChecks(): ti = 0..40; rendered "118 active contact-to-wall equations across 41 sampled times, max signed-distance residual 3.33e-16"; 118 recounted by hand from the phase of each sampled time (3 + 51 + 10 + 51 + 3)
WRONG Quadrature: adaptive Simpson integration stops at a requested tolerance. The doubling cross-check uses an independently coded composite Simpson rule. public/strata/eighteen-curves-turn-the-corner/index.html : [data-quadrature-conventions] the code says otherwise: initialize() sets "state.area=areaGauss(4,result.parameters);" (16-point Gauss-Legendre), the cross-check is "var a2=areaGauss(2,p),a4=areaGauss(4,p);", and areaComposite (composite Simpson) is defined at line 742 but never called. The page's own check panel contradicts this line: "The area shown above is the Gauss value. An adaptive-Simpson pass over the same integrand returns 2.219531678083, which is wrong from the eighth decimal". Line 253
CONFIRMED 16-point Gauss with 2 panels public/strata/eighteen-curves-turn-the-corner/index.html : [data-gauss-16] GAUSS_NODES are the 8 positive nodes of 16-point Gauss-Legendre (0.0950125098... to 0.9894009350...), used as mid-d and mid+d; rendered "2 panels 2.219531668872, 4 panels 2.219531668872, last difference 0.00e+0"
CONFIRMED which is wrong from the eighth decimal because the phase joins defeat its endpoint handling; it is printed here so the disagreement is visible rather than hidden. public/strata/eighteen-curves-turn-the-corner/index.html : [data-simpson-wrong] rendered 2.219531678083 against 2.2195316688720 (error 9.21e-9, eighth decimal). Probe (scratch simpson-probe2.mjs): the recursion hits its depth cap only on the subintervals ending at theta (error -4.9e-9 and -4.2e-9) and phi, i.e. at the phase joins; tightening the tolerance to 1e-11 leaves the error at 9.21e-9; raising the depth cap to 40 removes it (9.8e-14)
CONFIRMED Normalized slice q(epsilon) = area minus one half epsilon squared times integral of sine squared from zero to pi over two. Directional derivative at zero is 0. public/strata/eighteen-curves-turn-the-corner/index.html : [data-quadratic-slice] renderQ(): "energy=Math.PI/4; ... function q(x){return state.area-.5*x*x*energy;}"; the integral of sin^2 over [0,pi/2] is pi/4
CONFIRMED This page evaluates its closed-form margins with ordinary floating point, not directed-rounding interval arithmetic, so the browser display is not itself a rigorous interval proof. public/strata/eighteen-curves-turn-the-corner/index.html : [data-no-interval-arith] initialize(): "var margin1=-1/8+3*Math.PI/8-3*Math.PI*Math.PI/32;" in plain binary64
CONFIRMED The ODE path check covers four phase joins. It does not independently certify positional continuity at every endpoint of the source's 18-piece boundary decomposition. public/strata/eighteen-curves-turn-the-corner/index.html : [data-path-joins] linearResidual() joins at phi, theta, pi/2-theta, pi/2-phi only
CONFIRMED 28 Romik matching constraints, including four C¹ phase joins: max residual 6.25e-11 public/strata/eighteen-curves-turn-the-corner/index.html : [data-28-built] linearResidual(): 8 scalar (Romik 27-34) + 4 joins x 4 (35-42) + 2 contact vectors x 2 (43-44) = 28; rendered residual 6.25e-11
MINOR Equation transcription can be wrong. The offline verifier extracts this page's shipped functions and compares them with separately written formulas and source values. research/eighteen-curves-turn-the-corner/verify-eighteen-curves-turn-the-corner.mjs : [data-verifier] it extracts only the script id="sofa-math"; the headline area (areaGauss), the 28-constraint path solver and the f iteration live in the second script and are never run by it. Its "published Gerver area from Romik companion" check passes the adaptive-Simpson value the page calls wrong ("2.219531678081", tolerance 3e-8 against 2.2195316688719674). Mutation in a scratch copy: headline area x 1.01 (page then shows "2.241726986 square units"): verifier still 25 PASS, 0 FAIL, exit 0
CONFIRMED Six independent readouts expose the load-bearing computations. Every value below is filled by the page's own JavaScript after load. public/strata/eighteen-curves-turn-the-corner/index.html : [data-six-readouts] six check items start "waiting" and runChecks()/solveAmbi() fill them; all six rendered in node
CONFIRMED Drag first. The sofa stays fixed while the hallway follows Romik's five-phase rotation path. public/strata/eighteen-curves-turn-the-corner/index.html : [data-rotation-phases] phaseAt() breaks at phi, theta, pi/2-theta, pi/2-phi as in Romik (24)-(25)
CONFIRMED Units: quoted constants use hallway width 1. Length scales with width and area with width squared. public/strata/eighteen-curves-turn-the-corner/index.html : [data-width-scaling] renderStage(): "(state.area*state.width*state.width).toFixed(9)"
CONFIRMED The pale shape is regenerated from the solved path as an intersection of sampled hallway positions, not traced from an image. public/strata/eighteen-curves-turn-the-corner/index.html : [data-pale-shape] drawSofa() fills 180 x 80 cells kept by pointInSofa(); no image is loaded
OBSERVED This layer's check lifts this page's own source out of the HTML and runs it. research/verifier-independence/verdicts.json : [placard-lift] row at 9ddc1e9045: "tier": "C3", "mech": "lift"; the verifier's "html.match(/<script id=\"sofa-math\">...)" confirms it
OBSERVED Measured, not assumed: with every file this layer ships emptied, it goes red. research/verifier-independence/verdicts.json : [placard-emptied-red] "dependence": "red", "depRed": 1; recreated: emptied index.html in a scratch copy, verifier exit 1
OBSERVED Measured too: a defect placed in this page's running code turned it red. research/verifier-independence/verdicts.json : [placard-code-defect] "verdict": "red-code", code mutations 2, red 1 (one code defect went undetected, consistent with the data-verifier line)
OBSERVED Last verified 29 July 2026, when it was written. src/content/strata/eighteen-curves-turn-the-corner.md : [placard-verified-date] frontmatter "date: 2026-07-29T00:00:00Z"; README "Checked 29 July 2026"
CONFIRMED One rolls a noncircular curve between parallel lines, while this one rotates an 18-piece boundary through perpendicular hallway walls. src/content/strata/reuleaux-constant-width.md : [relates-reuleaux] that layer's dek: "roll a triangle, pentagon, or circle under a board that never rises or dips ... spin any of them between caliper jaws"
CONFIRMED The chain page shows the wider mechanism by which a variational condition becomes a differential equation and then a curve. src/content/strata/the-chain-obeys-snells-law.md : [relates-chain] dek: "Write the cost each minimises and all three collapse to one integral ... a single conserved quantity falls out ... one marcher of f·sinθ=C draws catenary, cycloid, and light ray alike"
CONFIRMED (added) Each makes a geometric impossibility or optimum legible through an invariant rather than a picture alone. src/content/strata/egregium.md : relates note for egregium (Nearby layers); that layer's dek: "no flat map of the Earth can ever be faithful: the sphere's curvature is not zero ... curvature survives any distance-preserving copy"
CONFIRMED (added) area from live quadrature 2.219531669 square units (unit width) https://arxiv.org/html/1706.06630 : rendered in node from the shipped page; Kallus and Romik: "mu_G=2.21953166..."; Romik companion value 2.2195316688719674 (README) rounds to 2.219531669
CONFIRMED (added) area = 1.644955218425441 (Open variant readout) https://arxiv.org/html/1606.08111v3 : Romik equation (5): "approx 1.644955218425440"; recomputed by bisection 1.644955218425441 (last digit is binary64 rounding)
CONFIRMED (added) It does not settle general turn angles, curved hallways, several nearby turns, or three-dimensional corridors. https://arxiv.org/html/2511.02864v3 : Baek's Theorem 1.1.1 is stated for the one unit-width right-angled hallway of Definition 1.1.1; Georgiev et al. pose the three-dimensional two-turn corridor separately as Problem 6.64 after recording "Baek [10] showed that this bound was sharp" for the classic problem
What was done
[fixed] data-quadrature-conventions: public/strata/eighteen-curves-turn-the-corner/index.html, Conventions list, "Quadrature: adaptive Simpson integration stops at a requested tolerance. The doubling cross-check uses an independently coded composite Simpson rule." now reads "Quadrature: the headline area uses 16-point Gauss-Legendre quadrature on 4 panels per subinterval, split at the phase joins. The doubling cross-check repeats it on 2 panels. An adaptive Simpson pass is printed in check 3 only as a comparison, and it disagrees from the eighth decimal.", matching initialize() (state.area=areaGauss(4,...)) and runChecks() (areaGauss(2,p), areaGauss(4,p)); areaComposite is never called. A Corrections section with a "Corrected 2026-10-18" line was added above the footer and mirrored at the end of src/content/strata/eighteen-curves-turn-the-corner.md; JSON-LD dateModified set to 2026-10-18.
[declined] fi-bound: "For i ≤ 10, fᵢ(x) ≥ max(1−x,(i−1)/12)." MINOR, listed under tier B, not edited. Baek's Lemma 6.5.5 proof reads "For any integer 1 ≤ m ≤ 10"; the page's f₀ is the paper's f_0 := 0, so the stated range also covers i = 0, where the bound fails. A slip of precision in the lower end of the range, not in a sentence being fixed; the intended text is "For 1 ≤ i ≤ 10".
[declined] cond-status: arXiv:2406.10725 is labelled only "Preprint." MINOR, listed under tier B, not edited. The arXiv page says "This paper has been withdrawn by Jineon Baek" (v2, 2 December 2024) as superseded by arXiv:2411.19826, "Do not cite this work"; it is still a preprint and the conditional bound the page recomputes is in v1's abstract and not retracted as false, so the label is incomplete, not false.
[declined] data-verifier: "The offline verifier extracts this page's shipped functions and compares them with separately written formulas and source values." MINOR, listed under tier B, not edited. The verifier lifts only the sofa-math script; areaGauss (the headline area), the path solver and the f iteration are not run by it, and the checker's 1% mutation of the headline area left it at 25 PASS. The sentence overstates the verifier's reach; none of the verifier's assertions is false.
The assay office: what a record is, and every layer re-read so far.