verification: snap-through of a shallow arch by arc length

arc-shallow-arch: a sinusoidal arch of 40 × 2 × 1 hex20 under a sinusoidal dead load scaled to
the one-mode closed form's upper limit load: the upper limit point at 0.992 (−0.8 %), the lower
at 0.219 (0.8 % of the peak load) and exactly one negative eigenvalue between them. ROADMAP and
parity updated, docs/oracle.md regenerated.

Co-Authored-By: Claude Opus 5.5 <[email protected]>
Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo

docs/oracle.md

@@ -2,12 +2,15 @@
 
 Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the policy is [verification.md](verification.md). Every case is our own model, built with FieldCraft commands from the problem's published description, and compared with the reference value its `expected.json` cites, within the stated tolerance and for the stated reason. Accepted results are frozen in each case's `fixture.json`: a later change that moves one beyond its band (10⁻⁹ of its size) fails CI even while the reference still passes. No reference value was produced with a commercial product.
 
-**196 of 202 quantities pass** (6 known failure(s)) in 60 active cases: 51 theory, 9 published benchmark, 0 experiment; 10 pending cases wait for capabilities still being built.
+**199 of 205 quantities pass** (6 known failure(s)) in 61 active cases: 52 theory, 9 published benchmark, 0 experiment; 10 pending cases wait for capabilities still being built.
 
 ## Theory: closed-form solutions, patch tests and mesh invariants
 
 | Case | Quantity | Reference | FieldCraft | Error | Tolerance | Result | Fixture |
 |---|---|---|---|---|---|---|---|
+| [arc-shallow-arch](../tests/oracle/arc-shallow-arch/) | upper limit load factor (snap-through) | 1 | 0.99183 | -0.817% | 2% | pass | matches |
+| [arc-shallow-arch](../tests/oracle/arc-shallow-arch/) | lower limit load factor | 0.21104 | 0.21895 | 7.9e-3 | ± 0.02 | pass | matches |
+| [arc-shallow-arch](../tests/oracle/arc-shallow-arch/) | negative eigenvalues of the tangent past the upper limit point | 1 | 1 | ≈ 0 | exact | pass | matches |
 | [bar-pressure-wave](../tests/oracle/bar-pressure-wave/) | displacement of the loaded end at t = 0.06 | -6.0000e-4 | -5.9980e-4 | 0.034% | 1% | pass | matches |
 | [bar-pressure-wave](../tests/oracle/bar-pressure-wave/) | particle velocity at x = 5, 20 elements behind the front, at t = 0.06 | -0.01 | -0.0098552 | 1.45% | 2% | pass | matches |
 | [bar-pressure-wave](../tests/oracle/bar-pressure-wave/) | energy balance error | 0 | 0.001527 | 1.5e-3 | ± 0.01 | pass | matches |
@@ -218,6 +221,7 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
 
 ## Sources and tolerances
 
+- **[arc-shallow-arch](../tests/oracle/arc-shallow-arch/)** (theory): Snap-through of a shallow arch by arc length. Source: One-mode closed form of a shallow sinusoidal arch y₀ = a sin(πx/L) on pins at a fixed span under q₀ sin(πx/L): q₀ = EI k⁴ [c + (a − c)(2ac − c²)/(4r²)], k = π/L, limit points at c = a ∓ r √((a²/r² − 4)/3) (S. P. Timoshenko, J. M. Gere, Theory of Elastic Stability, 2nd ed., McGraw-Hill 1961, §7.6; Y. C. Fung, A. Kaplan, NACA TN 2840, 1952): L = 100, h = 1, unit width, E = 10⁴, ν = 0, a/r = 3.5; the load is scaled to the upper limit load, so the peak load factor is 1 and the valley 0.211035. Tolerances — peak: 40 × 2 × 1 hex20 against shallow Euler–Bernoulli arch theory, the load piecewise constant per element, and the limit load picked from converged states (second order in the arc); valley: as the peak; the valley is a fifth of the peak load and a small difference of the bending and axial terms, so its tolerance is stated as 2 % of the peak load; negative: the inertia of the LDLᵀ factor: exactly one unstable direction between the limit points
 - **[bar-pressure-wave](../tests/oracle/bar-pressure-wave/)** (theory): Elastic wave in a bar from a pressure step (explicit). Source: d'Alembert's solution of the one-dimensional wave equation: c = √(E/ρ), particle velocity v = p/(ρc); Graff, Wave Motion in Elastic Solids, Dover 1991, §2.1. Tolerances — u_end: the end moves at the particle velocity p/(ρc) from the start; the discrete wave's ringing behind the front changes the end's displacement by well under 1%; v_mid: central differences ring behind a step front (dispersion); bulk viscosity damps it within a few elements, leaving about 1% twenty elements behind; balance: external work = kinetic + internal + dissipated energy; the solver warns above 1%
 - **[bar-rigid-wall-impact](../tests/oracle/bar-rigid-wall-impact/)** (theory): Elastic bar striking a rigid wall (explicit). Source: One-dimensional elastic impact on a rigid wall: contact time 2L/c, rebound at the impact speed, contact stress ρcv₀; Graff, Wave Motion in Elastic Solids, Dover 1991, §2.3; Johnson, Impact Strength of Materials, Arnold 1972, §1.6. Tolerances — u_front: the contact ends at 2L/c in the elastic theory; the discrete release front is spread over a few elements, a few tenths of a percent of the time, and the rebound speed is v₀ to about 1%; u_back: as for the struck end; ke: an elastic impact keeps the kinetic energy; quadratic bulk viscosity in the compression front and the vibration left in the bar take about 1%
 - **[buckle-euler-cantilever](../tests/oracle/buckle-euler-cantilever/)** (theory): Euler buckling of a clamped–free column of solids. Source: Euler's column with Engesser's shear correction P = P_e/(1 + P_e/(κ G A)), κ = 5/6 (S. P. Timoshenko, J. M. Gere, Theory of Elastic Stability, 2nd ed., McGraw-Hill 1961, §2.1, §2.17): square section 1 × 1, E = 1000, ν = 0, length 10 clamped–free, effective length 20, P_e = π² E I/20², so σ_cr = P/A = 2.046071. Tolerances — lambda: hex20, 2 × 2 across and 10 along: the 3D solid's bending and shear against beam theory, and the preload concentrated at the held face