The concentration frontier, as a measurement

The instance is the discounted Gomes–Mitake congestion system with exponent a = ½, potential A cos 2πx + γm with γ = ½, the system whose one equilibrium the congestion page encloses and re-proves at every build. The certifier is a radii-polynomial contraction in a weighted sequence space; it returns an enclosure with an explicit radius, or a refusal with a name: Z1 ≥ 1 when the approximate inverse stops being one, a non-positive mean of √m when the candidate itself has gone bad, a non-finite candidate when Newton diverges. Only the first of those is the certificate's boundary; the other two are the solver's, and the data keeps them apart.

A ladder is the certifier run at sixteen amplitudes, from 0.05 to 32, at one viscosity. Its pattern must be monotone, certified then refused, and on every ladder it is. The boundary A⋆ is bracketed by the last certified and the first refused amplitude, then bisected to a tenth of a percent, every trial warm-started from the last certificate so a cold-start failure can never be mistaken for a refusal.

Refine the truncation order N from 14 to 20, 28 and 40 at three viscosities, holding the candidate grid fixed at 1024 points so the two cannot be confounded, and the boundary rises every time. At σ = 0.1 it rises ten percent; at σ = 0.5 and 1.2, about six. It is still rising at N = 40. A boundary measured at one N is therefore a lower bound on the frontier of this method, and a published A⋆(σ) at N = 14 would be a resolution artefact at the five-to-ten-percent level. The source note says this first, and this page repeats it before anything else.

The Z1 bound is a maximum over explicit columns and an analytic tail. In the tail, every term but one carries a denominator that grows with N. The one exception is the reciprocal block's tail inverse, the scalar 1/(2s̄₀), because that block's linear part has no derivative; its term, cMrecip = ‖w∗w‖_ν/(2s̄₀), is measured to be N-free. Since Z1 ≥ cMrecip, the implication cMrecip ≥ 1 ⇒ refusal holds at EVERY N. Define A_rec(σ) as the amplitude where cMrecip reaches 1: it is computable from the candidate alone, before any interval arithmetic, and it came out bit-identical across N = 14, 20, 28 and 40 at every σ. A⋆(N) approaches it from below.

The ceiling depends on the weight ν of the sequence space by as much as the boundary depends on N — six percent across ν ∈ [1.00, 1.10] at σ = 0.5. It is a property of this approximate-inverse construction with this ν, not of the equation. Nothing here bounds where solutions exist, and the refused region was not probed for existence.

• Claimed. The ladders and their patterns; the brackets; the named modes; the rise of A⋆ with N; the N-invariance of A_rec; the ratios. Each is re-derived from the pinned files at every build and gated.
• Not claimed. A⋆(σ) as “the boundary” (it is the boundary of the N = 14 certificate); the limit of A⋆(N) (N > 40 is unevaluated); an order for the decay of the gap (the local slope drifts from about −0.8 to −0.7 and is drawn dashed); anything on the a-axis (the kernel hard-codes a = ½); existence of solutions past the boundary.
• Not re-run. The frontier itself. The three data files come from the source lab through the lift, are sha-pinned, and are re-hashed at every build; the kernel that produced them is named by its hash in the source note. The one published point of the same certifier is re-run at every build of its own page. This is the house convention: published, not independently rerun, and said so.
• Occupied, and cited. Certified continuation over a parameter set is the validated-numerics literature's (Day–Lessard–Mischaikow; van den Berg–Lessard–Mischaikow; Gameiro–Lessard–Pugliese). What is drawn here is narrower: a refusal named and mechanised, and a ceiling read off the code path.

node instruments/frontier/battery.js     # pins, ladders, brackets, ceilings; 19 checks, 5 red controls
make drift                               # re-hash the lifted data against the source lab
node instruments/frontier/run.js --check  # the record, re-derived and compared

The data is corpus/refusal-frontier/ with the source note beside it; the record is certs/frontier-measurement.json.

• D. A. Gomes, H. Mitake, Existence for stationary mean-field games with congestion and quadratic Hamiltonians, NoDEA 22 (2015) 1897–1910 — the existence theory the instance sits in.
• S. Day, J.-P. Lessard, K. Mischaikow, Validated continuation for equilibria of PDEs, SIAM J. Numer. Anal. 45 (2007); M. Gameiro, J.-P. Lessard, A. Pugliese, Computation of smooth manifolds via rigorous multi-parameter continuation in infinite dimensions, Found. Comput. Math. 16 (2016) — certified continuation over parameters, cited as the occupied method.
• The source note: corpus/refusal-frontier/REFUSAL_FRONTIER.md (sin-mfg, 2026-07-28, addendum 2026-08-21), lifted verbatim.
• A congestion mean-field game, enclosed — the one point, re-proved at every build.