The effective Hamiltonian band

For H = p²/2 + V(x) on the circle, the corrector u solves (P + u′)²/2 + V = H̄, so P + u′ = ±√(2(H̄ − V)). If H̄ = max V the root can change sign where V peaks, u′ can be arranged with any mean between −P₀ and P₀, and H̄ stays at max V: the flat part. Beyond P₀ the sign is fixed and the mean of P + u′ is the integral Q(H̄) = ∫√(2(H̄ − V)), so H̄ is the number with Q(H̄) = |P|. The paper quotes this (§6.1, from Cacace–Camilli) for V = sin 2πx with P₀ = 4/π.

Everything on this page is that formula evaluated as intervals. Q is strictly increasing (its derivative is the period ∫1/√(2(c − V)) > 0), so bisection brackets c whenever Q can be bounded above and below; Q is bounded by the midpoint rule with its remainder w³/24·sup|f″|, the second derivative carried through every operation as an interval. The trouble is where the formula is interesting: as c ↓ max V the integrand √(2(c − V)) develops a cusp at the peak of V and f″ grows like (c − V)^{−3/2}. Cells where the base is not positive by a margin fall back to a plain Riemann bound and the enclosure of Q widens toward P₀. Bisection stops when it cannot tell which side of |P| the integral is on; the mean value theorem, with the period ∫1/√(2(c − V)) as the derivative of Q, then turns the last straddling enclosure into a bracket on c. P₀ itself has the cusp on the nose; its enclosure is 1.5e-6 wide, it contains 4/π, and the one sampled P inside it is left undecided.

On the rotating part the projected Mather measure is the time the orbit spends: m(x) = 1/(T(c)√(2(c − V(x)))) with T(c) the period ∫1/√(2(c − V)). Both are enclosed at P = 1.5 and P = 2; the density integrates to 1 to 1e−5 on the enclosure midpoints. On the flat part it is the Dirac mass at the maximum of V, x = ¼ for the sine — Mather's theorem, and the paper's own statement at P = P₀ (δ at ¾ for their −sin). The support of the Mather measure is therefore decided at every sampled P except the undecided one.

Table 1. The two-dimensional Hamiltonian |p|²/2 + cos 2πx₁ + cos 2πx₂ is separable, so H̄(P₁, P₂) = H̄(P₁) + H̄(P₂) with the one-dimensional cosine formula — the paper says so and quotes H̄(1.5, 2.5) = 4.4099660 from Gomes–Oberman (2004), then reports its own flow and Newton values at k = 10 to 10⁴. Both one-dimensional values are bracketed here to below 1e−6 with 8192 cells: c(1.5) ∈ [1.244637626, 1.244637655], c(2.5) ∈ [3.165327608, 3.165327639], sum [4.409965234, 4.409965294]. The quoted 4.4099660 lies above the enclosure by 7.1e-7: its seventh significant digit is not the value's. The paper's own numbers all lie below the enclosure, which is what they must do — the entropy-penalized H̄^k is a soft maximum and never exceeds H̄ — and its k = 10⁴ value 4.40996 agrees with the enclosure to every digit it prints.

Table 2. At P = 0.5 on −sin 2πx the paper says H̄ = 1 and reports H̄◇, the flow's large-time value at k = 100, for four meshes: 0.964609 to 0.96476. The exact value is decided here — 0.5 is below the lower end of the P₀ enclosure, so the point is FLAT and H̄ = 1 exactly — and the printed values sit 0.0354 to 0.0352 below it. The gap does not close from N = 60 to N = 120; it is the entropy penalization at k = 100, not the mesh, and the paper's Figure 1 shows the same gap shrinking with k.

• Decided. The enclosures of P₀ (containing 4/π), of every rotating H̄(P), of the separable Table 1 value and of the periods and Mather densities at two P; the flat values; the placement of every printed number against its enclosure. All interval statements, re-derived at every build.
• Assumed. The one-dimensional formula itself (classical; Cacace–Camilli as the paper cites it; the sign argument in §1 is the sketch, not a proof). The Mather measure on the flat part is Mather's theorem for a single nondegenerate maximum, not re-proved. Q strictly increasing in c, from its derivative being a positive integral.
• Not claimed. Anything at the undecided P; anything in two dimensions beyond the separable case — the non-separable Hamiltonian of §6.2 has no formula and is not touched; a new method: the formula is textbook, and the page's contribution is the enclosure and the two placements. The width of the band is the quadrature's, as the red control shows by halving the cells.
• On the seventh digit. The 4.4099660 is quoted by the paper from another paper; it may be a rounding, a typo, or a six-digit computation written with eight. Nothing here says which. What is said is that the value with a rigorous enclosure is 4.409965 and that 4.4099660 is outside it.

node instruments/hbar/battery.js       # 18 checks, 5 red controls: P₀ vs 4/π, brackets vs the trapezoid rule, the Mather mass, the tables
node instruments/hbar/run.js --check   # the record, re-derived and compared byte for byte

The instrument is instruments/hbar/exact.js with derive.js; the interval jet is instruments/interval/taylor2.js; the record is certs/hbar-band.json.

• D. A. Gomes, X. Yang, The Hessian Riemannian flow and Newton's method for effective Hamiltonians and Mather measures, arXiv:1810.03483v2; ESAIM M2AN 54 (2020) 1883–1915 — §6.1 (the formula and P₀ = 4/π), §6.2 (Table 1 and the quoted 4.4099660), §6.4 (Table 2).
• D. A. Gomes, A. M. Oberman, Computing the effective Hamiltonian using a variational approach, SIAM J. Control Optim. 43 (2004) — the source of the quoted value, not consulted here; the number is taken as the paper prints it.
• S. Cacace, F. Camilli, A generalized Newton method for homogenization of Hamilton–Jacobi equations, SIAM J. Sci. Comput. 38 (2016) — the one-dimensional formula as the paper cites it.
• The regularization, measured, the first-order game by its current — the machine's other first-order pages; the ergodic constants there are enclosed by radii polynomials, this one by the classical integral.