Disordered systems are hard because the randomness is baked in. You cannot average it away, you cannot solve around it, and almost nothing about them is exactly computable. So it is genuinely surprising that there exists a single curve through the random-bond magnet's phase diagram along which the energy is known in closed form, at every lattice size, with nothing fitted. It is called the Nishimori line, and M14 walks along it.
Here is the setup. Take a square lattice of magnets and make a fraction p
of the bonds hostile — those pairs want to disagree, the rest want to agree. Higher
p means more frustration. Normally you would be free to set the temperature
independently, but the Nishimori line ties them together:
tanh(1/T) = 1 − 2p. Pick a p and the temperature is
forced. That constraint is not a convenience; it is the condition under which a
hidden gauge symmetry of the model kicks in, and the symmetry is what makes the energy
computable.
What it says is that the disorder-averaged energy per magnet must be exactly
−2·tanh(1/T), equivalently −2(1 − 2p). Every point on the line
is therefore a prediction the simulation was never told, and every measurement is a
checkable falsification. Because it is a symmetry identity rather than a critical
temperature, it holds at any lattice size — no finite-size shift to correct, no
extrapolation to argue about. That is what makes it verify cheaply and robustly. It is
also, honestly, what makes it a weaker claim than pinning the interesting point on that
line would be.
The interesting point is where the line crosses out of the ordered phase — the
multicritical Nishimori point, at roughly p ≈ 0.1094,
T ≈ 0.9528 in the literature. This run brackets it but does not pin it: the
ferromagnetic order visibly collapses as p rises along the line, passing
half strength near p ≈ 0.126. Pinning it properly needs a much larger run.
The panel below gives you one slider for p, which drags the temperature
with it, and shows the measured energy landing on a line nobody fitted.
left · the fabric — ember bonds are hostile, moss ones friendly; raising p reddens it and the order dissolves. right · top: the p–T plane with the Nishimori line and where the slider has put you; bottom: measured energy against the exact −2·tanh(1/T). The satisfying moment is the measured dot riding the theory line while everything else in the picture falls apart.
temperature — (set by p — you cannot move it) · measured E/N — · predicted — · deviation — · ⟨|m|⟩ — · realizations averaged 1
Press off the line and the constraint is released: the temperature is pushed
away from the value p demands, and the measured energy walks straight off
the prediction curve while nothing else about the picture changes. Nothing demonstrates
as economically that the line is special — it is not that this model is easy, it is that
this one path through it is exactly solvable.
The graded claim is the first number and only the first number. Across
p ∈ [0.04, 0.16] at L = 24 over 64 disorder realizations, the measured
disorder-averaged energy never departs from −2·tanh(1/T) by more than
0.015 — a prediction with nothing fitted, reproduced at every point on the walk.
That is what earns the green leaf.
The second number is reported, not graded. The ferromagnetic order does collapse
along the line, from 0.85 down to 0.22, and it passes half strength near
p ≈ 0.126 — which brackets the literature's multicritical point at
pc ≈ 0.1094, Tc ≈ 0.9528, but does not
pin it. Bracketing and pinning are different claims and this page keeps them apart on
purpose.
The bond machinery here is M11's, reused verbatim — same checkerboard, same J-weighted neighbour sum, same Metropolis half-sweep — so the random-bond engine can never drift away from the spin-glass one. The three-dimensional version of that spin glass, still looking for its transition, is M12.