Random perceptrons: pressure formulas and jamming

Source and verification

This article explains family 222 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

A perceptron weights vectors through their projections onto random patterns. Family 222 groups finite-temperature pressure formulas for Ising and spherical vectors with a separate zero-temperature jamming analysis. The order of limits and the boundedness of the potential determine which result applies.

🏷️ Preliminaries: random constraints and pressure

Let $M=\lfloor \alpha N\rfloor$ and let $g^1,\dots ,g^M$ be independent standard Gaussian vectors in ${\mathbb{R}}^N$. A configuration $x$ lies either in $\{-1,1{\}}^N$ or on the sphere $\Vert x{\Vert }^2=N$. Write ${\pi }_N$ for the uniform probability measure on that space. For a log-potential $f$,

$$\begin{array}{rl}{\mathcal{H}}_N(x)& =\sum _{a=1}^{M}f(g^a\cdot x/\sqrt{N}),\\ p_N& =\frac{1}{N}\log \int e^{{\mathcal{H}}_N(x)}d{\pi }_N(x).\end{array}$$

Writing $f=\beta \varphi$ separates inverse temperature from the reward. Because the prior is a probability measure, $f=0$ gives $p_N=0$ in both models. A counting-measure convention on the cube would add $\log 2$.

For two configurations, the overlap $R=N^{-1}x\cdot y$ is the covariance of their responses to one Gaussian pattern. The Hamiltonian is generally not a Gaussian process: applying a nonlinear $f$ destroys Gaussianity. A mixed-spin Gaussian pressure formula therefore cannot simply be substituted.

🏷️ The four claims

The Ising companion claims a variational formula and convergence in expectation and probability for every bounded Borel real log-potential, at every fixed positive pattern density. Its order parameter is a nondecreasing overlap quantile, with a Gaussian pattern recursion and a dual one-spin entropy term. (OpenAI, 2026)

The Gaussian spherical companion claims the corresponding pressure limit for arbitrary bounded continuous potentials. (OpenAI, 2026b) Another spherical paper replaces iid Gaussian patterns by a matrix invariant under independent left and right orthogonal changes of coordinates. Its singular values may have a compact limiting distribution, with no outliers; the variational formula retains that spectral information. (OpenAI, 2026c)

The jamming paper concerns a different limiting regime: margin $-1$ and a quadratic penalty for constraint violation. It takes system size to infinity, then inverse temperature to infinity, then density down to the feasibility threshold from above. It claims gap and normalized-force laws, including stability under a specified small Gaussian-mixture change in the coordinate law. (OpenAI, 2026a)

A bounded-potential pressure theorem does not automatically justify these unbounded penalties or the successive jamming limits.

🏷️ An explicit spherical variational formula

Let $m:[0,1]\to [0,1]$ be nondecreasing and measurable. Define

$$\begin{array}{c}{D}_m(t)={\int }_t^{1}m(s)ds,\\ S(m)\\ =\frac{1}{2}{\int }_0^1\left(\frac{1}{D_m(t)}-\frac{1}{1-t}\right)dt.\end{array}$$

The entropy cost is a nonnegative extended integral: $D_m(t)\le 1-t$, and infinite values are allowed. With standard Brownian motion $B$, define the controlled pattern value

$$\begin{array}{c}{V}_f(m)\\ =\underset{v}{\sup }\mathbb{E}[f\left(B_1+{\int }_0^{1}m(t)v_{t}dt\right)\\ -\frac{1}{2}{\int }_0^{1}m(t)v_t^{2}dt],\end{array}$$

where controls are progressively measurable and have finite expected displayed cost. The spherical claim is

$$\begin{array}{c}{p}_N\stackrel{\mathbb{P}}{\to }\mathcal{P},\mathbb{E}{p}_N⟶\mathcal{P},\\ \mathcal{P}=\underset{m}{\inf }\{\alpha V_f(m)+S(m)\}.\end{array}$$

The choice $m=1$ has $S(m)=0$ and $V_f(m)=\log \mathbb{E}{e}^{f(G)}$ by the Brownian control representation. It recovers the elementary annealed upper bound. Jensen’s inequality in the other order gives the lower bound

$$\begin{array}{c}\alpha \mathbb{E}f(G)\le \mathcal{P}\le \alpha \log \mathbb{E}{e}^{f(G)},\\ G\sim N(0,1).\end{array}$$

For $f\equiv c$, both sides equal $\alpha c$. These checks fix the normalization; they do not establish the infimum formula for a nonconstant reward.

🏷️ Why the matching lower bound is difficult

The Ising proof first treats smooth potentials. Hierarchical Gaussian enrichment and a Poissonized pattern count turn adding one pattern into a pressure derivative. At carefully chosen comparison points, small perturbations enforce identities between spin and hierarchy overlaps, supplying the upper bound.

The lower bound must use the overlap path generated by the actual bulk system. Removing and restoring independent patterns identifies the covariance of the cavity field. Adding finitely many spin coordinates then gives increments expressed through that field. The key compatibility is that the same field supports the entropy dual at the same overlap path. Choosing an unrelated entropy optimizer would not give a valid lower bound. Approximation extends the Ising result to bounded Borel potentials under the source’s stated estimates.

🏷️ What the jamming exponents describe

After removing the contact mass, the claimed small positive-gap cumulative law is $G_J(u)=u^{1-\gamma +o(1)}$. For forces normalized to mean one, the small-force cumulative law is $F_J(s)=s^{1+\theta +o(1)}$, with

$$\begin{array}{c}\gamma =\frac{1}{2+\theta },\\ \gamma \approx 0.412693,\theta \approx \mathrm{0.423108.}\end{array}$$

The paper supplies narrow numerical enclosures through a finite certificate with mathematical error estimates. These notes report the claim and the limiting regime; the certificate was not executed here. Cumulative exponents should not be confused with density exponents, which differ by one when differentiation is justified.

🏷️ Key steps and proof outlines

A cavity field is the random field seen by a small number of new coordinates after the original system has been sampled. The central pressure argument identifies this field from the bulk overlap distribution, then checks that its entropy cost is evaluated at that same distribution. The four manuscripts add different mechanisms to this common idea.

Removing one pattern makes rough potentials manageable

For the Ising model, let $f,g$ be bounded by $K$ in absolute value, and let $\gamma$ be standard Gaussian measure. The useful approximation estimate is

$$\begin{array}{rl}|\mathbb{E}{p}_N(f)-\mathbb{E}{p}_N(g)|& \le \alpha e^{2K}\Vert f-g{\Vert }_{L^1(\gamma )},\\ \underset{q}{\sup }|V_f(q)-V_g(q)|& \le e^{2K}\Vert f-g{\Vert }_{L^1(\gamma )}.\end{array}$$

To prove the first inequality, interpolate linearly between the potentials and differentiate the expected pressure. Each term is a Gibbs average of $(g-f)(g^a\cdot x/\sqrt{N})$. Remove that entire pattern from the Gibbs density. Restoring its weight changes a nonnegative average by at most $e^{2K}$, the ratio between the largest and smallest possible weight. Conditional on the remaining system and a sampled $x$, the omitted pattern is independent and its projection is exactly standard Gaussian. This gives the bound, uniformly in $N$.

For the hierarchical pattern recursion, differentiate the terminal function. The tilted Gaussian transition densities telescope: their full density relative to the unweighted increments is again at most $e^{2K}$, independently of the number of hierarchy levels. The terminal field under the unweighted law has total variance one. This proves the second bound uniformly over overlap paths.

Now approximate a bounded Borel $f$ by smooth functions in Gaussian $L^1$, keeping the same bound $K$. Both the finite-dimensional pressure and its proposed variational value converge uniformly under this approximation. Since the entropy term does not depend on $f$, the infimum passes to the limit as well. Finally, replacing one whole pattern changes $\log Z_N$ by at most $2K$, so independence of the patterns gives

$$\mathrm{Var}(p_N)\le \frac{4M{K}^2}{N^2}=O(N^{-1}).$$

The expected-pressure formula therefore yields convergence in probability. This argument explains why bounded Borel data are allowed here: uniform approximation on the real line is unnecessary. (OpenAI, 2026b)

The spherical constraint identifies the cavity covariance

At the smooth approximation stage, take $f\in C_c^{\infty }(\mathbb{R})$, so its derivatives are bounded. Denote the limiting overlap law by $\mu$, and let $a(r)$ be the off-diagonal covariance profile obtained by removing and restoring two patterns. The missing step is to identify it with the stationary field in the spherical entropy formula.

At finite $N$, integrate tangential divergence on the sphere using

$$u=x^2-R_{12}{x}^1,$$

which is tangent at $x^1$. Its divergence is $-(N-1)R_{12}$. For a smooth overlap test $G$, integration by parts gives

$$\begin{array}{r}(N-1)\mathbb{E}⟨R_{12}G⟩=\mathbb{E}⟨G^{\prime }(R_{12})(1-R_{12}^2)+Gu\cdot \nabla {\mathcal{H}}_N⟩,\end{array}$$

with the small perturbation included in the full log density when applying the identity. Gaussian integration by parts on each pattern then expresses the last term in overlaps of three replicas. The overlap identities and ultrametricity evaluate the conditional three-replica expression.

Here is the resulting one-dimensional calculation. In this paragraph use the overlap threshold $r$, rather than the quantile parameter of the earlier variational formula, and define

$$\begin{array}{rl}D(r)& =1-r-\int (s-r{)}_{+}\mu (ds),\\ U(r)& =r\left(1+\int (1-s)a(s)\mu (ds)\right)\\ & -\int (r-s{)}_{+}a(s)\mu (ds).\end{array}$$

The tangential identity becomes $a(r)D(r)=U(r)$ for $\mu$-almost every $r$. Since $U(1)=1$, $D(1)=0$, and $a$ is bounded, the overlap support stays a positive distance below one. In the distributional sense,

$$D^{\prime \prime }=-\mu ,U^{\prime \prime }=-a\mu .$$

Hence the derivative of the Wronskian is

$$d(U^{\prime }D-U{D}^{\prime })=(U-aD)\mu =0.$$

This includes atoms: their jump contributions vanish by the same almost-everywhere identity. Above the overlap support, $D(r)=1-r$ and $U$ is affine with $U(1)=1$, so the constant Wronskian equals one. Integrating the derivative of $U/D$ gives

$$a(r)=\frac{U(r)}{D(r)}={\int }_0^r\frac{dt}{D(t{)}^2}\text{for }\mu \text{-almost every }r.$$

This is the covariance required by the stationary spherical field. It closes the compatibility between the physical cavity increment and the entropy variational term; it is more than an optimization over an independently chosen field. (OpenAI, 2026c)

Keeping the singular-value distribution in the answer

For bi-orthogonally invariant disorder, condition on the singular values and split them into finitely many blocks. Let $v_i$ be the fraction of input dimensions in block $i$, and ${\rho }_i$ the fraction of input squared norm assigned there. Uniform spherical measure makes these fractions Dirichlet distributed, with parameters equal to half the block dimensions. Stirling’s formula gives the logarithmic density rate per input coordinate

$$I_v(\rho )=\frac{1}{2}\sum _i{v}_i\log \frac{{\rho }_i}{v_i}.$$

If the block singular value is $s_i$, its output squared norm per output coordinate is $s_i^2{\rho }_i/\alpha$. Conditional on these radii, the angular theorem evaluates the pressure in the orthogonal Haar subspaces. A Laplace principle over $\rho$ therefore adds $I_v(\rho )$ to that angular pressure and takes the supremum. A structural input null space still contributes to $I_v$, even though it produces no output field.

The spectral approximation is controlled quantitatively. For Lipschitz $f$,

$$|F_N(A,f)-F_N(\tilde{A},f)|\le \mathrm{Lip}(f)\sqrt{M/N}\Vert A-\tilde{A}{\Vert }_{\mathrm{op}}.$$

Indeed, bound the sum of coordinate errors by $\sqrt{M}$ times their Euclidean norm and use $\Vert x\Vert =\sqrt{N}$. The no-outlier assumption permits approximating every singular slot within a vanishing operator-norm error. Weak convergence of the empirical spectral law alone would not do this. For bounded continuous $f$, bounded operator norm makes the fraction of large output coordinates uniformly small; compact-interval approximation then removes the Lipschitz assumption. (OpenAI, 2026d)

How one similarity exponent gives both jamming laws

The jamming manuscript first establishes the ordered thermodynamic, zero-temperature, and threshold limits. Its endpoint analysis introduces a susceptibility $\chi (1-\tau )$ and an auxiliary scale

$$w(1-\tau )={\int }_{1-\tau }^1\chi (t{)}^{-2}dt.$$

The difficult selection step identifies a unique coefficient $a_{\ast }$ in logarithmic time. Compactness produces entire limits of the rescaled profiles. Analytic barriers and the finite certificate place their coefficient $A$ near $a_{\ast }$. The final rigidity estimate has the form

$$d=\underset{v}{\sup }|A(v)-a_{\ast }|,|A(v)-a_{\ast }|\le c_{\mathrm{loc}}d,c_{\mathrm{loc}}<\mathrm{0.95.}$$

Taking the supremum forces $d=0$. To make this a contraction, the tail errors must be proportional to $d$; an additive numerical error would only give a neighborhood of the root. The source obtains that proportionality by differentiating the difference between input and reference tail integrals. The finite certificate supplies the remaining middle-interval bounds and a positive denominator. That certificate has not been rerun for this post.

Rigidity yields $\chi (1-\tau )={\tau }^{a_{\ast }+o(1)}$ and $w(1-\tau )={\tau }^{1-2a_{\ast }+o(1)}$. The endpoint mass comparisons give

$$\begin{array}{rl}\mathbb{P}(0<Y\le \sqrt{\tau })& ={\tau }^{a_{\ast }+o(1)},\\ \mathbb{P}(0<R\le \sqrt{w(1-\tau )})& ={\tau }^{a_{\ast }+o(1)}.\end{array}$$

Substitute $u=\sqrt{\tau }$ for gaps and invert the continuous force scale for forces. Conditioning on contact and multiplying forces by a fixed normalization do not change logarithmic exponents. Therefore

$$1-\gamma =2a_{\ast },1+\theta =\frac{2a_{\ast }}{1-2a_{\ast }},\gamma =\frac{1}{2+\theta }.$$

The relation between the two exponents is thus a consequence of the shared endpoint scale, while their numerical values depend on the certified selection of $a_{\ast }$. (OpenAI, 2026a)

See Also

• Family 221 develops finite hierarchical cavity messages in a sparse interaction model.
• Family 217 concerns fluctuations of a different model after its mean pressure has already been identified.

🏷️ Manuscripts and verification scope

• The free energy of the Ising random perceptron — pinned manuscript and companion source.
• Microscopic jamming in the negative spherical perceptron — pinned manuscript and companion source.
• The spherical perceptron with bi-orthogonally invariant disorder — pinned manuscript and companion source.
• The free energy of the spherical random perceptron — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The Ising selection is only finiteness of the variational value for bounded continuous potentials, not convergence of the pressure or its bounded-Borel extension. The spherical selection describes the Gaussian pressure formula and a supporting field duality statement. It does not cover the bi-orthogonally invariant extension or microscopic jamming. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. Microscopic jamming in the negative spherical perceptron.

🐻  OpenAI 2026b. The free energy of the Ising random perceptron.

🐻  OpenAI 2026c. The free energy of the spherical random perceptron.

🐻  OpenAI 2026d. The spherical perceptron with bi-orthogonally invariant disorder.