det ≠ 0Lehalle & Livieri, Competition among seaports through Mean Field Games and real-world data, HAL-05380483, preprint submitted 26 Nov 2025. Assessed for what it offers this tree. The answer is one thing above all: it is an instance for a question we already asked and never answered.
Nothing on this page is claimed, certified, enclosed or proved. No kernel, no certificate, no falsifier, no ledger record. It assesses someone else's paper and names what could be built against it. A prospectus that reads like a result is the defect, so this one says so at the top and is not published.
research/spectral-conditioning/ has been a question with no instance since it
was written: “does the smallest singular value of the linearization at the equilibrium
decrease as the crowd-aversion coefficient rises — i.e. does the coupling that flattens the
density push the system a certificate must invert toward singular?”
This paper is an instance, and it publishes its own parameters. Its
equilibrium reduces to a linear system Ω·φ = m (Eq. 17) whose matrix
is built entirely from cost parameters, and its Table 7 prints congestion coefficients
calibrated on ten years of real shipping data. The sweep is the one
research/_frontier/geometry/sigma_min_sweep.js already runs, pointed at a matrix that
comes from ShipFix rather than from a torus.
A discrete-time, finite-state MFG over K seaports and N goods.
A coordinator at each port chooses the transition matrix Qⁿi,j —
the fraction of the fleet waiting for good n at port i that is sent to port
j — trading commercial margin Mi,j = vj − vi
against transportation cost and a congestion term driven by everyone else's arrivals. Under quadratic
costs (ρ = γ = 2) and the mean-field approximation of Proposition 1, Theorem 1
gives an explicit closed form for the control, and Theorem 2 gives existence and uniqueness of
the equilibrium. Then a two-step statistical procedure infers the parameters from the
ShipFix dataset — 416,977 “Dry Coal” records, Jan 2015 to May 2025,
five exporting and five importing countries.
It is close to this tree's existing subject matter and not a stretch: a multi-population congested network equilibrium, solved to a stationary fixed point. That is the Wardrop module's object with goods in place of populations.
Theorem 2 states that the MFE exists and is unique if and only if
det(Ω) ≠ 0 (Eq. 15). The discussion that follows then reasons in the language
of conditioning — the cost matrix being “well-conditioned”, the mapping being
“invertible”, degeneracy arising from symmetric costs or “effective zero-cost
cycles”.
A determinant is not a conditioning measure. det(Ω)
can be 1e-300 and satisfy the theorem exactly while the equilibrium it certifies is
numerically vacuous. The quantity the discussion actually reaches for — how far this cost
structure sits from the degenerate one — is a lower bound on
σmin(Ω). What is proved is a predicate; what is wanted is a
margin. This is not an error in the theorem. The theorem is right. It is a gap between what the
theorem gives and what the paper's own economic reading needs.
For the cost parameters this paper publishes, and over a stated box around them:
σmin(Ω) ≥ β for an explicit
β > 0 — hence the MFE is unique with a certified margin, not merely
with a non-zero determinant.
Ω is 5×5 or 10×10. At that size exact rational arithmetic is free —
eqcert/src/rational.js handles it outright, with no interval widening to fight and no
truncation tail to bound. This is the rare target where the certified answer costs about what the
floating-point answer costs.
And the sweep writes itself, because the paper hands over the axis. Its Table 7 gives the calibrated congestion coefficients:
| Destination | congestion rj | value vj |
|---|---|---|
| China | 1.05 | 1.16 |
| India | 2.10 | 1.28 |
| Japan | 5.24 | 1.40 |
| South Korea | 2.56 | 2.33 |
| Vietnam | 8.73 | 1.51 |
Transportation cost c = 0.92; source values
vi Australia 0.70, Indonesia 1.16, Russia 0.81, South Africa 0.93, US 1.05.
An eight-fold spread in congestion across destinations is exactly the axis
spectral-conditioning wants swept — and here the endpoints are not chosen by us,
they are measured.
Remark 2, in the authors' own words: “We do not address the positivity constraint in the optimization problem, as it can be quite challenging due to its potential to create sparsity in the solutions… The reader may be convinced that negative transitions should not occur in practical situations.”
But Qⁿi,j is a fraction of a
fleet. It must lie in the simplex, and the closed form of Eq. (13) contains nothing that enforces
it — the Lagrange multiplier saturates the sum-to-one constraint and says nothing about signs.
The stated justification for feasibility is an appeal to the reader.
The Wardrop module holds positivity by the Hessian–Riemannian metric and Kirchhoff
by construction (Kϑ̇ ≡ 0, conserved to ~1e-14 along the whole
trajectory), against a projected-gradient rival that repairs feasibility after each step. The house
phrase is “feasibility by geometry vs feasibility by repair”. Here is a published
MFG with neither — feasibility by assertion. The same HRF kernel, pointed at this
instance, would keep the transitions in the simplex by construction.
For the published parameters, the closed form of Eq. (13) does — or does not —
yield a Q lying in the simplex. Infeasible is a finding; feasible is a certificate
the paper does not have. Either way the question is decided rather than assumed, and it is decided by
exact arithmetic on a matrix small enough to hold in the hand.
There is already sign trouble visible in their own tables, which
is why this is worth checking rather than assuming: five negative Ai,j in
Table 5 (Australia→China −21,331; Australia→South Korea −1,539;
Australia→Vietnam −38,075; Indonesia→Vietnam −20,895; South
Africa→Vietnam −54,015), and negative non-diagonal Bi,j in Table 6
which the authors themselves flag as “raises red flags” for Japan. Those are
regression coefficients rather than transitions directly — they do not prove infeasibility and
must not be quoted as if they did — but they are the reason the question is live.
Charles-Albert Lehalle (Polytechnique, CMAP, IP Paris; Fellow of Institut Louis Bachelier) and Giulia Livieri (LSE). Serious authors, current work, and a paper that states its own two gaps out loud — a uniqueness criterion weaker than the reading it is given, and a positivity constraint deferred by name.
That is precisely the gaps-plus-fixes, honour-and-validate posture that worked for the Wardrop approach: their theorem, their instance, their published parameters, plus a certified margin and a feasibility guarantee they left open. Coal shipping is also congestion-priced network flow calibrated on real commodity data — the same mathematical product as the energy track, though it is not the commercial fleet product and should never be presented as if it were.
The empirical half is not a certification target, and proposing it would be
the overclaim this tree's standard exists to prevent. Sections 3–4 rest on: 958 data points
per regression across K(K−1) separate regressions; non-stationarity acknowledged and
handled by selecting a stable window (June 2018 – January 2021) from a plot; relative margins
assumed constant across the whole dataset; a congestion proxy with s = 60 and
m = 20 chosen by inspecting a time series; and ports aggregated to countries with
Ti,j averaged over 100 port-pair routes — which the authors state breaks
down for Russia and the US, whose ports sit on different oceans.
The authors are honest about every one of these. They label them limitations and name them as future work. Certifying statistics that the people who produced them have already qualified would add rigour where none was claimed and none is owed. Certify the algebra; leave the econometrics alone.
| Requisite | State |
|---|---|
| Exact rational linear algebra at 10×10 | eqcert/src/rational.js — exists |
| A σmin sweep harness | research/_frontier/geometry/sigma_min_sweep.js — exists, written for a different matrix |
| HRF / simplex-preserving flow | exists as MWD's kernel, validated by research/mfg-lab/tests/test-wardrop.js |
The distance matrix Ti,j | NOT PUBLISHED AS NUMBERS. Figure 4 shows dispersion as boxplots only. Table 4 does publish every port's latitude and longitude, and Searoutes is open (Eurostat, github.com/eurostat/searoute), so it is recomputable — but that is a build with a dependency, not a read. Ω cannot be assembled without it. |
| A certified lower bound on σmin over a parameter BOX | DOES NOT EXIST. Sweeping at points is not a bound between them. This is the actual mathematical work and it should not be under-described. |
SCORING.json, owner's correction 2026-07-30: score on academic novelty
and artifact value, never novelty alone.
ZERO, and it should be stated that way in any approach to the authors. Certified bounds on the smallest singular value of a small rational matrix are classical. Nothing proposed here is new mathematics.
Provisionally HIGH, and unverified. The claim would be that no re-runnable, zero-dependency certificate exists for this equilibrium's uniqueness margin or its feasibility. That claim has not been through a literature gate and must not be stated until it has.
Ω may be comfortably well-conditioned, making the certificate true and
dull. If σmin is 0.5 across the whole realistic parameter box, the
finding is “their theorem was fine and so was their arithmetic”, which is worth one line
and no paper. The result only becomes interesting if the margin degrades along the congestion
axis — which is the spectral-conditioning hypothesis, and which may simply be
false here.
That is the correct reason to run it, not a reason to avoid it: the hypothesis is currently untested against any instance at all, and a measured refutation on real calibrated parameters is a result this programme is happy to publish. It has published a refutation before.
This is a PREPRINT, and this tree has a scar exactly there. HAL-05380483, submitted 26 Nov 2025, not stated as published. The costliest documented error in this repository came from treating a preprint sentence as “the paper” when the published version said the opposite. Any citation pins the HAL identifier and version explicitly, and Citations are fetched at source before a word of this goes outward.
Read from the PDF text as supplied to the session on 2026-07-31; not independently retrieved from HAL, and no version comparison performed. Every figure quoted above — Table 5, Table 6, Table 7, the sample counts, the window — is transcribed from that text and has not been re-derived.