A congestion-averse crowd in a single-well cost landscape can settle into TWO peaks — or THREE. The mechanism is a band-pass response with an exact, discount-free crossover at σ* = 1/(8π²); the phenomenon is proved — two computer-assisted theorems and a bracket table of certified instances whose enclosure balls fix the exact peak count of the exact solution, bracketing every predicted threshold. The crowd re-weights a harmonic the potential already contains. Floats draw the map below; the diamonds are the territory.
Published, not peer-reviewed, not independently rerun. Two priority claims, each fenced: to our knowledge these are the first certified equilibria of a mean-field game whose exact peak count strictly EXCEEDS the potential's well count — the nearest published phenomena are spontaneous instability (arXiv:2605.20213) and peaks that mirror the potential (arXiv:1705.10741), and we found no validated-numerics equilibrium enclosures for MFG anywhere in the literature; and the base instance of the program (the companion page, mfg-congest.html, released together with this atlas) is to our knowledge the first validated-numerics enclosure of an MFG equilibrium at all. Refutations of either claim are invited. Every claim on this page re-runs from this repository: the enclosures come from two independent implementations of the radii-polynomial argument that agree on the certified radius to the last digit, the peak counts derive only from certified region signs, and every instrument carries red controls that fire. Grid dots are float candidates and are labeled as such; nothing on this page is a forecast.
Dots: 224 float equilibrium solves (N = 32, fresh seed each, pinned by sha256) · dashed line: the closed-form linear-response boundary r_c(σ) · diamonds: the certified theorems — filled where EVERY density in the enclosure ball has exactly two maxima, open where exactly one. The T7/T8 pair straddles the boundary.
The potential (dim, dashed, right scale — a positional comparison, never a magnitude one) has a single well at x = ½. The certified equilibrium density peaks at x ≈ 0.363 and 0.637 with a saddle between — the exact solution provably carries two strict maxima, with the certification margin six orders above the enclosure pads.
The mechanism does not stop at one extra peak. With the third harmonic inside its predicted window — the Chebyshev U₂ law gives (1/27, 1/3) at k = 3 — the crowd splits three ways at the bottom of the same single well.
Honest counting: T1 and T6 are the theorems; the rest are rows of a bracket table under them — negatives below the amplitude threshold and past the crossover, replications, and the threshold pin. Each row is an enclosure with full-ball local uniqueness (even AND odd blocks — evenness is a corollary, not an assumption), and every bound is displayed: the radius, the contraction bound Z₁, the closure margin, and the certified density floor. Records in certs/.
| instance | σ | A₂/A₁ (or A₃/A₁) | N | ν | ball radius | Z₁ | margin | min m | peaks / wells | record |
|---|---|---|---|---|---|---|---|---|---|---|
| T5 | 0.001 | 0.15 | 176 | 1.015 | 5.31e-13 | 0.9260 | 2.11e-6 | 0.9649 | 2 / 1 | terra-recert-t5.json |
| T6 (3-peak) | 0.002 | A₃: 0.25 | 96 | 1.02 | 2.73e-13 | 0.9046 | 3.95e-6 | 0.8823 | 3 / 1 | terra-recert-t6.json |
| T2 | 0.002 | 0.12 | 96 | 1.02 | 2.25e-13 | 0.8889 | 5.54e-6 | 0.9059 | 1 / 1 | terra-recert-t2.json |
| T7 | 0.002 | 0.13 | 96 | 1.02 | 2.30e-13 | 0.8899 | 5.43e-6 | 0.9044 | 1 / 1 | terra-recert-t7.json |
| T8 | 0.002 | 0.14 | 96 | 1.02 | 2.32e-13 | 0.8909 | 5.32e-6 | 0.9030 | 2 / 1 | terra-recert-t8.json |
| T4 | 0.002 | 0.15 | 96 | 1.02 | 2.37e-13 | 0.8919 | 5.21e-6 | 0.9016 | 2 / 1 | terra-recert-t4.json |
| T1 (main) | 0.002 | 0.2 | 96 | 1.02 | 2.52e-13 | 0.8970 | 4.69e-6 | 0.8944 | 2 / 1 | terra-recert-t1.json |
| T3 | 0.02 | 0.2 | 64 | 1.05 | 2.97e-14 | 0.5235 | 4.99e-4 | 0.9621 | 1 / 1 | terra-recert-t3.json |
Ergodic quadratic MFG, V ≡ 0, σ = ½. Past the pitchfork c* = −σ²(2π)² the constant state, the symmetry-broken branch and its half-shift mirror are enclosed in PAIRWISE DISJOINT ℓ¹_ν uniqueness balls with certified positive density, at each of six couplings c = −11 … −24: AT LEAST THREE distinct exact solutions at every listed coupling, exactly where Lasry–Lions monotonicity makes no claim. The radii grow and the density digs toward vacuum as the coupling deepens — min m down to 4.45e-4 at c = −24 — and the half-shift symmetry is thereby proved to produce a genuinely different solution, not a relabeling. At c = −9.5, inside the monotone regime, the branch collapses onto the constant and no claim is made — the recorded boundary. (certs/mfg-cap-multiplicity.json)
Krawczyk exhaustion over an explicit printed box B: every subbox of an adaptive partition is eliminated by an interval-residual or Krawczyk exclusion, and each surviving box is isolated by Moore–Krawczyk K(X) ⊂ int(X) — existence AND uniqueness per box. Box-bounded, truncation-level; the PDE-level count is the stated open problem.
| Galerkin N | dimensions | boxes processed | seconds | solutions in B | matching |
|---|---|---|---|---|---|
| N = 2 | 5 | 2,077 | 0.1 | EXACTLY 3 | constant · branch · mirror, one-to-one |
| N = 3 | 7 | 21,642 | 0.8 | EXACTLY 3 | constant · branch · mirror, one-to-one |
| N = 4 | 9 | 335,862 | 17.7 | EXACTLY 3 | constant · branch · mirror, one-to-one |
| N = 5 | 11 | 6,954,073 | 586.7 | EXACTLY 3 | constant · branch · mirror, one-to-one |
Coupling plane (s, d): s = symmetric cross-interaction, d = attack–defense asymmetry. Every rectangle of coupling matrices is decided UNIFORMLY over its whole cell — 11,628 cells certified MULTIPLE (two exact solutions for EVERY parameter in the cell, disjoint balls, positive densities), 24 UNIQUE, and the undecided region drawn as itself — hatched, not rounded away. The exact area identity (decided in rationals) confirms the three regions tile the rectangle. (certs/mfg2p-regime-map.json)
When the cost is class-independent, the two-population equilibrium is a FACE — a positive-dimensional set the equilibrium conditions cannot pin to a point — and its exact tangent dimension obeys a purely combinatorial law: k = |shared| − cons + z, where z counts exit-free components of the shared subgraph. The natural shortcut (drop z) is correct exactly when every shared component touches an exit, and undercounts by exactly z otherwise.
The law is decided against the exact ℚ null space on two seeded 4,000-network ensembles — the shortcut fails on 572 and 571 instances respectively, precisely the z > 0 cases, and every failing instance is ENUMERATED in the record so any reader can re-run any one. A constructed exit-free-cycle family realizes every deficit. Exact over ℚ; combinatorics on the constraint matrices, not an enclosure. (certs/facelaw-theorem.json)
Rational-kernel token dynamics on the sphere — (1 + β⟨x_i,x_j⟩)^p, chosen so equilibrium and stability are decidable in exact ℚ; never a Transformer/softmax claim. Three theorems: the consensus spectrum {0, −1} is β- AND p-free (the kernel slope cancels identically — decided by exact dual-number expansion); the ⟨u,v⟩ = −1/β two-cluster family’s cross-weights vanish identically, to first order for every p ≥ 2 (and NOT at p = 1 — the model’s own honest candidate for a true bifurcation); and the reduced flow below. Plus a four-artifact catalogue of how finite float budgets manufacture bifurcations — one artifact re-demonstrated live at every battery run, with its exact refutation. (certs/attnflow-theorems.json)
Linearize the discounted congestion MFG about its flat equilibrium: the density response to a cost mode of frequency κ is
— a band-pass, peaked at κ = 1/σ. The second harmonic overtakes the fundamental exactly when σκ₁ = ½, and the γ-terms cancel IDENTICALLY (an exact polynomial fact, certs/terra-sigmastar.json), giving σ* = 1/(8π²), discount-free. The equilibrium counts wells by gain-weighted harmonic amplitudes; the potential counts them flat; the splitting window is exactly the gap between the two counts.
Mechanism is linear response, derived in advance and confirmed — Turing-adjacent mode selection in a driven, monotone-regime system, distinguished from spontaneous instability (Karuturi, arXiv:2605.20213) and from peaks that mirror the potential (Cesaroni–Cirant, arXiv:1705.10741). The theorems certify existence, full-ball uniqueness (both parities), density positivity and the branch selection; the peak counts never trust a float sign.
Every certificate re-runs from this repository:
python3 instruments/mfgcap/run_recert.py t1 # the T1 enclosure, nine falsifiers node instruments/critcount/run.js t1 # EXACTLY 2 maxima over the whole ball python3 instruments/mfgcap/sigmastar.py # sigma* = 1/(8pi^2), exact rationals python3 instruments/mfgcap/bracket_table.py # the bracket table + threshold pin node labs/mfg/census.js --N 5 --c -12 # EXACTLY 3 (6.95M boxes) node labs/mfg/multiplicity.js # >= 3 solutions per coupling make test # every battery, every red control
The write-up: paper/terra-peaks.md · PDF: paper/terra-peaks.pdf — generated from the same certificates as this page. Refutations and independent re-runs are invited: carlos@carlostoledo.co.