Memory and samples in noiseless Gaussian regression

Source and verification

This article explains family 140 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.

🏷️ Preliminaries: exact equations and finite memory

Let an unknown signal $s$ lie on the unit sphere $S^{d-1}\subset {\mathbb{R}}^d$. At time $t$, a learner receives

$$x_t\sim N(0,I_d),y_t=⟨x_t,s⟩,$$

with independent Gaussian rows. The label is exact: there is no measurement noise or imposed precision limit. If all observations can be retained, $d$ rows determine $s$ almost surely, since the resulting square Gaussian matrix is invertible almost surely.

Family 140 studies a different resource: persistent memory measured in bits. After each observation the learner has at most $2^M$ possible states. It may perform unrestricted computation on the current real-valued pair, but only its next finite state survives. It cannot choose the rows, revisit discarded observations, or retain an uncharged real register.

The learner stops at some index $\tau \le T$, where $T$ is a prescribed deterministic horizon. Its unit-vector output uses only the terminal state, stopping index, and fresh randomness. In particular, even the final observation must pass through the finite state before affecting the output.

Rules may depend on the dimension, accuracy, and time index. A shared random seed may choose them, but that seed and the initial state must be jointly independent of the signal and observations. The source model requires jointly measurable experiments and specifies measurable transition kernels, including its completed-measurability convention. These conditions permit very general randomized algorithms while excluding information hidden in initialization or undefined conditional laws.

Accuracy is angular:

$$\arccos ⟨\hat{s},s⟩\le ϵ,0<ϵ\le 1/10.$$

For unit vectors, angular success implies Euclidean error at most $ϵ$, since the chord length is $2\sin (\theta /2)\le \theta$. The exposition uses graduate probability, linear algebra, conditional distributions, and elementary information theory.

🏷️ The family conclusions and their different scopes

Write ${\sigma }_d$ for uniform probability on the sphere. The basic success assumption is probability at least $2/3$ when $S\sim {\sigma }_d$, including the randomness of the signal, rows, and learner. A guarantee for every fixed signal implies this average guarantee by integration. The converse is not asserted.

Memory-dependent sample lower bound

The projection-moment manuscript claims absolute constants $c>0,d_0$ such that, for every $d\ge d_0$ and every nonnegative integer $M$, success at least $2/3$ in the stated model requires

$$T\ge \frac{cd\log (1/ϵ)}{1+M/d^2}.$$

The constants and dimension threshold are independent of $M$ and $ϵ$. (OpenAI, 2026)

Thus, for every fixed finite $A>0$, memory $M\le A{d}^2$ requires $T\ge c_{A}d\log (1/ϵ)$. The opening manuscript proves this fixed-quadratic-memory conclusion through its own backward argument. (OpenAI, 2026a) For $M(d)=o(d^2)$, the coefficient can be universal; the sufficiently-large-dimension threshold may depend on the memory sequence.

The subsphere manuscript gives the explicit subquadratic-memory coefficient

$$T\ge 2^{-16}d{\log }_2(1/ϵ),$$

at success probability $2/3$, with a dimension threshold uniform in the accuracy. (OpenAI, 2026c) Most formulas below use natural logarithms; this displayed coefficient belongs specifically to base-two logarithms.

The six papers are complementary treatments, not six successive improvements of one exponent.

Manuscript theme	Main mechanism	Distinct scope
Memory and precision	Backward bounds on successful signal mass	Every fixed $M\le A{d}^2$
Posterior replicas	Correlation lost under an independent projection	Conditional-information comparisons and subquadratic memory
Localization	Pay for restoring geometric spread	Cube and sphere priors require different success assumptions
Projection moments	Integrated densities, positive cap domination, and Riesz potentials	Explicit tradeoff for arbitrary $M$
Replacing observations	Compare actual rows with independent rows	Several block comparisons with different conditioning rules
Subspheres	High moments on random residual spheres	Explicit coefficient and dimension accounting

🏷️ A real-register benchmark

The scale $d\log (1/ϵ)$ appears in a simple projection algorithm. Starting at $z_0=0$, update

$$\begin{array}{rl}{r}_t& =y_t-⟨x_t,z_{t-1}⟩,\\ z_t& =z_{t-1}+\frac{r_t}{\Vert x_t{\Vert }^2}{x}_t.\end{array}$$

The zero-row event has probability zero and can be handled by doing nothing. With $e_t=z_t-s$ and $u_t=x_t/\Vert x_t\Vert$, the error satisfies

$$e_t=(I-u_t{u}_t^{\mathsf{T}})e_{t-1}.$$

The fresh direction $u_t$ is uniform on the sphere, so its second-moment matrix is $I_d/d$. Consequently,

$$\begin{array}{rl}\mathbb{E}(\Vert e_t{\Vert }^2\mid e_{t-1})& =(1-1/d)\Vert e_{t-1}{\Vert }^2,\\ \mathbb{E}\Vert e_T{\Vert }^2& =(1-1/d{)}^T.\end{array}$$

Exponential decay on the time scale $d$ explains the logarithmic dependence on accuracy. Markov’s inequality, followed by normalizing a sufficiently accurate estimate, gives a constant-probability angular guarantee at this scale.

This calculation stores real numbers. It is not a finite-bit upper bound matching the family theorem. The research question is a necessity statement for all permitted finite-state learners, including algorithms unlike this projection update.

🏷️ Why exact labels obstruct ordinary posterior arguments

Conditioning a uniform spherical signal on one raw exact equation confines it to a hyperplane section. That conditional law is singular relative to spherical area. A proof that treats it as an ordinary bounded density on the original sphere would fail.

Conditioning on a finite state behaves differently. If a state $w$ has probability $p_w>0$, its posterior density relative to ${\sigma }_d$ is

$$\begin{array}{rl}{f}_w(s)& =\frac{\mathrm{Pr}(W=w\mid S=s)}{p_w}\\ & \le \frac{1}{p_w}.\end{array}$$

Rare states can therefore have very concentrated posteriors. Their geometric concentration must be paid for when a block estimate is averaged over states.

For reference, discrete entropy is $H(W)=-{\sum }_w{p}_w\log p_w$, and $H(W)\le M\log 2$ for a state with at most $2^M$ values. Relative entropy is

$$D(P\Vert Q)=\int \log \left(\frac{dP}{dQ}\right)dP,$$

with value $+\infty$ when absolute continuity fails. Mutual information is relative entropy from the joint law to the product of its marginals. These definitions apply to continuous signals without pretending that an exact real observation has a bounded number of bits.

🏷️ Backward propagation of successful mass

The opening manuscript avoids conditioning on a reached state. Fix a state and an index, and start a fresh future execution there. Let $h(s)\in [0,1]$ be its probability of eventually succeeding at signal $s$. This is defined even when that state would never be reached at that signal.

The invariant bounds the unnormalized measure $h{\sigma }_d$ on every ball. Set $a=(d-1)/2$ and require

$${\int }_{B(z,r)}hd{\sigma }_d\le (Rr{)}^a.$$

Here $z$ lies on the sphere, $r>0$, and $R$ measures concentration of successful mass. It is not a support radius for a posterior. A terminal signal-independent output law satisfies this invariant with $R$ on the accuracy scale, by a spherical cap bound.

Suppose a block of $m\asymp d$ rows can route to one of $N$ destinations, whose continuation functions are $h_1,\dots ,h_N$. Let $g_j(A,As)$ be the probability of routing to $j$. These probabilities may use the entire real-valued block, and ${\sum }_j{g}_j\le 1$.

The one-block lemma gives, for an admissible moment order $q\asymp d$,

$$\begin{array}{rl}{\int }_{B(z,r)}hd{\sigma }_d& \le e^{Cd}{N}^{1/q}(Rr{)}^a,\\ R^{\prime }& =R{e}^{Cd/a}{N}^{1/(aq)}.\end{array}$$

The two appearances of dimension in $aq\asymp d^2$ explain the memory scale. The high moment divides the logarithm of the number of choices by $q$; converting a mass bound into a concentration scale divides by $a$ again.

🏷️ The two Hölder inequalities and radius cancellation

Restrict each continuation measure to the test ball:

$${\mu }_j=1_{B(z,r)}{h}_j{\sigma }_d.$$

A projection theorem supplies exact joint matrix-label densities $f_j$ with

$$\Vert f_j{\Vert }_{L^q}\le e^{Cd}{R}^a{r}^{a-m(1-1/q)}.$$

The density estimate loses a power of $r$. But all projected measures occupy the same matrix-dependent ellipsoid, whose joint reference measure is at most $e^{Cd}{r}^m$.

First apply Hölder on matrix-label space to the product $f_j{g}_j$. Then apply Hölder over the $N$ destinations. Since $0\le g_j\le 1$ and their sum is at most one, the latter step costs $N^{1/q}$. The support-volume term appears to the power $1-1/q$, giving the exact cancellation

$$\begin{array}{rl}& r^{a-m(1-1/q)}(r^m{)}^{1-1/q}\\ & =r^a.\end{array}$$

This preserves the invariant at every radius, allowing iteration. A bound only at a single radius would not supply the same closure.

Stopping must also be counted. A destination is either a continuing state or a terminal state tagged by its stopping index within the block. Hence

$$N\le (m+2)2^M,$$

and therefore

$$\log (R^{\prime }/R)\le C(1+M/d^2).$$

The proof pays for a local stopping index at each block, while still preserving the output rule’s dependence on the global index. It does not silently discard the information carried by the stopping time.

🏷️ From block growth to the sample tradeoff

The integrated-frame argument in the projection manuscript gives an especially transparent success estimate. Set

$$\begin{array}{c}k=q=\lfloor d/4\rfloor ,b=\lceil T/k\rceil ,\\ N_{\ast }=(k+2)2^M.\end{array}$$

This version uses ball-growth exponent $\beta =d-1-q$. For an absolute $C\ge 1$, its bound is

$$\mathrm{Pr}(\mathrm{success})\le C^d{ϵ}^q[C^d{N}_{\ast }^{1/q}{]}^b.$$

The terminal cap gives the small factor ${ϵ}^q$; every preceding block can enlarge it by only the displayed factor. Taking logarithms at success probability $2/3$, using $q\asymp d$ and $b\le 8T/d+1$, yields

$$\begin{array}{rl}d\log (1/ϵ)\le & C_1(T+d)\\ & \times (1+M/d^2).\end{array}$$

The additive $d$ comes from rounding to a full block. It cannot simply be dropped when accuracy is constant.

A separate full-data argument resolves this endpoint. Give the estimator all rows and labels and let $U$ be their row span. Write $S=u+v$ with $u\in U$ and $v\perp U$. Reflection sends $u+v$ to $u-v$ and preserves both the prior and the observed data. If $\Vert v\Vert >ϵ$, the same estimate cannot lie within $ϵ$ of both signals.

Since $\mathbb{E}\Vert u{\Vert }^2\le T/d$, reflection and Markov’s inequality give

$$\begin{array}{c}\mathrm{Pr}(\Vert \hat{S}-S\Vert \le ϵ)\\ \le \frac{1}{2}+\frac{T}{2d(1-{ϵ}^2)}.\end{array}$$

For $ϵ\le 1/10$, this forces a linear number of samples at success $2/3$, even with unrestricted memory. Substituting this endpoint bound into the preceding inequality absorbs the extra $d$ and gives the stated tradeoff.

🏷️ Projection geometry: three analytic interfaces

The projection-moment manuscript develops three distinct ways to control the same finite-state process. (OpenAI, 2026b)

An integrated density moment starts from a finite measure with ball growth $\nu (B(x,r))\le K{r}^{\beta }$. Under the specified dimension margins and projected-density hypotheses, it bounds the $q$th moment averaged over an orthonormal frame and its entire label space. Expanding the moment introduces $q$ signal points. Gram–Schmidt turns their difference determinant into successive distances to affine spans. Covering thin tubes by balls and integrating the local mass bound controls the inverse distances.

The Gaussian matrix also has a radial factor, which the learner is permitted to inspect. Separating that factor from an independent orthonormal frame and averaging the routing probabilities retains this permitted dependence. The projected support volume cancels the density’s dimensional and radius losses in the backward argument.

Positive cap domination gives a stronger kind of object than an integrated estimate: an actual countable positive sum of normalized cap measures dominating a routed signal measure. Weighted costs penalize small caps. Finite dictionaries at each scale and a vanishing fine-scale $L^1$ tail provide compactness; separation by bounded nonnegative tests produces the dominating sum. These positive measures then propagate joint signal-state laws forward, with their masses left unnormalized.

The Riesz-potential argument instead controls density evaluation at specified labels. It begins with

$$V(h)=\underset{z\in {\mathbb{R}}^d}{\sup }\int \frac{h(s)}{\Vert s-z{\Vert }^{d/2}}d{\sigma }_d(s).$$

An integrated density estimate alone cannot determine values on a matrix-dependent graph, because changing a density on a null set leaves its integrated norm unchanged. This argument chooses one lower-limit density version and proves its estimate at every fixed affine offset before using it in the recursion. Its shell-count and stopping-time costs are separately controlled.

🏷️ Posterior replicas and correlation loss

The replica manuscript studies a block with prior $p=f{\sigma }_d$, where $0\le f\le L$, and a finite message $W$ generated from $(A,AS)$. Draw several signals $S_1,\dots ,S_t$ independently from the posterior given the same block data, independently of the message given those data. (OpenAI, 2026b)

Each pair $(S_i,W)$ has the original signal-message law. After the block data are forgotten, the replicas are dependent: they have matching labels for the common hidden matrix. Matching their separate signal marginals would not be enough for the information argument; the pair laws with $W$ must agree.

Let $K$ be the relative entropy of the joint replica law from $p^{\otimes t}$. Let $K^{\prime }$ be the corresponding total correlation after an independent matrix $B$ projects every replica. Writing $J_B=I(S;W\mid B,BS)$, the information lemma is

$$t{J}_B\le H(W)+K-K^{\prime }.$$

To see the mechanism, decompose tuple-message relative entropy in two orders, both before and after projection. Subtract the identities. The remaining tuple information is at most $H(W)$, while conditional data processing makes the discarded conditional-correlation difference nonnegative.

The task is therefore to bound correlation lost, $K-K^{\prime }$, rather than all of $K$. The geometry of equal labels produces an inverse Gram determinant of the differences $S_i-S_1$. A normalized comparison on projected replicas retains a corresponding inverse-volume term, canceling much of the original geometric dependence.

For the paper’s specified row counts, with $t\asymp d$, this gives

$$\begin{array}{rl}{J}_B\le & \frac{H(W)}{t}+Cd\\ & +C\log (2+\log L).\end{array}$$

The weak dependence on $L$ makes it possible to average over rare entering states. The independent projection is supplied to the analyst, not to the learner. It leaves a residual sphere of dimension proportional to $d$, on which accurate recovery still demands order $d\log (1/ϵ)$ conditional information. Different companion comparisons condition either on the current state or on the full state history; these conditioning variables cannot be interchanged for free.

🏷️ Localization and the cost of revealing a cell

The localization manuscript gives a direct way to restore geometric spread after messages concentrate a posterior. Its first complete route uses a cube prior mapped onto a spherical patch. With $n=d-1$, let $Z$ be uniform on

$$[-1/(2\sqrt{n}),1/(2\sqrt{n}){)}^n,$$

and set $S=(Z,\sqrt{1-\Vert Z{\Vert }^2})$. This is the image of uniform cube volume, not uniform spherical area on the patch. (OpenAI, 2026a)

Inside a dyadic parent cell, sufficient spread means that a descendant at relative depth $j$ has posterior probability at most a constant times $2^{-nj/2}$. After a message, reveal the deepest cell along the sampled point’s chain whose mass is at least $2^{-nj/2}$.

At depth $j$ there are at most $2^{nj/2}$ such heavy cells, so naming one costs roughly $(nj/2)\log 2$, plus the entropy of its depth. Yet confinement to a depth-$j$ cell certifies $nj\log 2$ information relative to the initial uniform cube. The difference pays for the auxiliary revelation.

More precisely, if ${\Pi }_t$ is the complete augmented transcript and $J_t$ its final depth, the proof gives

$$\begin{array}{rl}I(Z;{\Pi }_t)& \le Cdt\\ & +(n/2+2)\log 2\mathbb{E}{J}_t,\\ I(Z;{\Pi }_t)& \ge n\log 2\mathbb{E}{J}_t.\end{array}$$

For sufficiently large $n$, subtraction bounds $\mathbb{E}{J}_t$ by $Ct$, then gives $I(Z;{\Pi }_t)\le Cdt$. The permitted block alphabet satisfies the paper’s explicit entropy condition; every fixed alphabet bound $\exp (A{d}^2)$ fits with constants depending on $A$.

A delicate point is normalization by the probability of selecting a cell, not merely the probability of lying in it. Selection of the deepest heavy ancestor is a stricter event. Its entropy also pays for the spread constant needed by the next block.

The paper’s other routes include finite-entropy regularization, grid and span refinements, and corrected information functionals involving inverse-distance kernels. They use different conditioning histories and analytic inputs. The cube-based learner conclusions use an every-signal guarantee to obtain both the cube experiment and a separate spherical constant-accuracy endpoint. The spherical routes can begin directly from average success under ${\sigma }_d$.

🏷️ Replacing selected rows by independent rows

The replacement manuscript isolates a common conditioning error. Although the original Gaussian rows are independent of the signal, conditioning on a selected state generally biases those rows. Replacing them by fresh rows preserves certain marginals but changes the full joint law. (OpenAI, 2026e)

One comparison takes an actual matrix $A$ with $\lfloor d/10\rfloor$ rows and a message of entropy at most $d^2$ nats. Supplement $A$ by independent rows $C$ to obtain $\lfloor d/2\rfloor$ rows in total. Let $G$ be a wholly independent matrix of that latter size. The theorem gives

$$\begin{array}{c}I(S;W\mid G,GS)\\ \le I(S;W\mid A,C,AS,CS)+Kd.\end{array}$$

Here the prior is uniform spherical, and the message may have arbitrary joint dependence on $(S,A)$ subject to its entropy bound. The actual and independent experiments share the relevant signal-message and signal-matrix marginals; they need not share the joint triple law.

Subtracting the information identities produces a negative conditional row-information term. Data processing uses that term to pay for the discrepancy between the actual conditional label law and the projection of the state posterior. Dropping its sign or pretending the actual rows remain independent would miss the comparison mechanism.

A mixed projection moment provides the analytic bound: actual rows are shared among moment factors, while auxiliary rows are averaged separately. Local mass controls inverse lengths and inverse Gram determinants. The proof changes the row law at a positive smoothing scale before taking the exact-label limit under the actual law.

Other comparisons replace rows one at a time or use a finite two-signal equal-label measure, normalized before probability data processing. Their assumptions differ. The paper’s critical-radius streaming consequence allows success $3/5$ under subquadratic memory, but that threshold is not automatically part of every other manuscript or formalization selection.

🏷️ Subspheres and the remaining dimension budget

The subsphere manuscript tests a continuation on every suitable affine sphere

$$\begin{array}{c}z+rS(H),\\ z\perp H,\Vert z{\Vert }^2+r^2=1.\end{array}$$

Here the test dimension is $D=\dim H$. A block of fewer than $D$ Gaussian rows determines a projection $p$ of the residual signal and leaves a uniform conditional law on

$$\begin{array}{c}z+p+\rho S(K),\\ {\rho }^2=r^2-\Vert p{\Vert }^2.\end{array}$$

For continuation $v$, let $F_v(K)$ be its average on this residual sphere. If the block may choose among $W$ states, a high moment gives

$$\begin{array}{c}\mathbb{E}\underset{v}{\max }{F}_v(K)\\ \le {\left(\sum _v\mathbb{E}{F}_v(K{)}^h\right)}^{1/h}.\end{array}$$

State selection costs $W^{1/h}$, with $h\asymp d$. Expanding the moment samples several directions in the same random subspace. Their unconditional dependence is the geometric obstacle.

A two-component comparison dominates each next-direction law by a factor times a uniform sphere law and a mixture of latitude laws. The same continuation estimates apply in dimensions $D-1$ and $D-2$. Crucially, the proof does not spend a new test dimension for every sampled direction: the whole block costs only two dimensions. Beta-distribution inverse-radius moments supply the quantitative constants.

The induction must still have enough dimensions left. There are at most $(T+1)2^M$ terminal index-state pairs. Integrating each signal-independent randomized output law against the uniform signal bounds its contribution by one spherical cap mass, giving

$$\mathrm{Pr}(\mathrm{success})\le (T+1)2^M(2ϵ{)}^{d-1}.$$

This does not require outputs themselves to take only finitely many vector values; their conditional laws are indexed by finitely many terminal pairs.

Under a hypothetical successful run with $T=O(d\log (1/ϵ))$ and $M=o(d^2)$, the sharper cap count forces $\log (1/ϵ)=o(d)$. Hence the number of blocks is $o(d)$, leaving room for the two-dimension loss per block. This conclusion holds under the hypothetical short-run assumption; it is not an extra restriction on the theorem’s accuracy range. Extremely fine accuracy can already force a much larger horizon through terminal counting alone.

🏷️ Interpretation and related notes

The common obstruction is limited transfer of information across observations. Exact equations may contain unlimited real precision, yet a finite state cannot necessarily preserve the geometry needed to exploit that precision efficiently. The arguments charge state selection, concentration, auxiliary revelations, and stopping information explicitly.

The sample bounds concern isotropic Gaussian rows and angular recovery of a signal fixed before those rows are drawn. They do not establish the separate condition-number conjecture for ill-conditioned designs, nor do noisy lower bounds alone imply these exact-observation results. Conversely, an exact-data learner can simulate a prescribed independent noise distribution internally, which explains the direction of some noisy-model consequences.

One-tape simulation also measures persistent memory in bits while allowing expensive recomputation. Log-concave sampling queries uses exact real oracles with unrestricted computation; its query model highlights why a real-valued interface and a finite-bit memory constraint are separate assumptions.

🏷️ Key steps and proof outlines

The replica inequality follows from two exact chain rules

Let $S_1,\dots ,S_t$ have the common signal marginal $p$, with total correlation $K=D(Q\Vert p^{\otimes t})$. Let $K_W$ be the average total correlation after conditioning on the finite message $W$, measured against the product of the conditional marginals. Assuming the matching signal-message pair laws and finite $K$, expanding one relative entropy in two orders gives

$$tI(S;W)=I((S_i{)}_{i=1}^t;W)+K-K_W.$$

For an independent matrix $B$, put $U_i=B{S}_i$, and define $K^{\prime },K_W^{\prime }$ similarly after projection. A second expansion gives the corresponding identity for $tI(BS;W\mid B)$. Subtracting yields

$$\begin{array}{rl}tI(S;W\mid B,BS)=& I((S_i);W\mid B,(U_i))\\ & +(K-K^{\prime })-(K_W-K_W^{\prime }).\end{array}$$

Coordinatewise data processing gives $K_W^{\prime }\le K_W$, while the remaining tuple-message information is at most $H(W)$. This proves the replica bound. The negative conditional-correlation term is useful; estimating it in absolute value would lose the argument. Posterior resampling supplies equality of the pair laws $(S_i,W)$, since both the replicas and the message are conditionally generated from the same data. Equal separate signal marginals would not suffice. (OpenAI, 2026c)

Equal labels produce inverse-volume weights

For a fixed tuple $u_1,\dots ,u_t$, form the difference matrix $D=[u_2-u_1,\dots ,u_t-u_1]$. Each Gaussian row of $AD$ has covariance $D^{\mathsf{T}}D$. Its density at zero is proportional to $\det (D^{\mathsf{T}}D{)}^{-1/2}$, and $k$ independent rows give the power $-k/2$. Thus conditioning several signals to have the same labels weights nearly affinely dependent tuples strongly.

Gram–Schmidt writes the square-root determinant as the product of distances from each new point to the preceding affine span. A tube estimate of the form $\mu \{\mathrm{dist}(S,L)\le r\}\le C{r}^{\gamma }$ controls the inverse moments when the inverse-power exponent is below $\gamma$: integrate the tail, or sum dyadic shells. The source’s dimension margins ensure this for every successive span. This supplies the analytic bridge from geometric spread to finite replica correlation; it also explains why the number of replicas and observed rows cannot consume the whole ambient dimension.

The projection-moment route uses the same geometry without the replica information identity. Expanding a smoothed projected-density moment introduces several signals and the same difference determinant. All powers of the smoothing width cancel between the common-label volume and Gaussian small-ball probability. A uniform moment bound then permits passage to an exact projected measure. Pointwise density evaluation, used in the Riesz route, requires its separately specified density version and cannot be deduced merely by changing an almost-everywhere representative. (OpenAI, 2026d)

A selected cell is conditioned on its selection event

In the localization route, let $U$ be the deepest heavy dyadic ancestor chosen along the sampled point’s chain. For a selected cell $R$, write $p_R=\mathrm{Pr}(U=R)$, rather than $\mathrm{Pr}(Z\in R)$. The conditional signal law is restricted not just to $R$ but to points for which no deeper heavy descendant was selected. Its normalization is therefore $p_R$.

At relative depth $j$, at most $2^{\alpha nj}$ cells can be heavy when the threshold is $2^{-\alpha nj}$. Hence

$$H(U)\le H(j(U))+\alpha n\log 2\mathbb{E}j(U).$$

Comparing the depth distribution to the geometric probability $2^{-(j+1)}$ gives $H(j(U))\le (\mathbb{E}j(U)+1)\log 2$. The source also bounds the expected logarithm of the next spread constant by the depth entropy plus a constant. These costs account for both naming the cell and restoring a usable concentration bound. Confinement to depth $j$ certifies $nj\log 2$ information relative to the uniform cube; choosing $\alpha =1/2$ leaves enough difference to pay for the auxiliary revelation. (OpenAI, 2026a)

The complementary methods preserve different invariants

The opening backward method preserves ball-mass growth for unnormalized continuation-success measures. Its Hölder step charges destination selection by $N^{1/q}$, and the common projected support cancels the radius loss in the density norm. The subsphere method instead preserves continuation bounds on a class of residual affine spheres; its latitude comparison spends two test dimensions per block, rather than one per replica. These are different closures of the same resource problem and require their respective terminal cap bounds. (OpenAI, 2026b; OpenAI, 2026f)

The row-replacement companion handles a further obstruction: conditioning on a message biases the actual observation matrix. Its information comparison retains a negative conditional row-information term to pay for replacing that biased experiment by an independent one. Smoothing is introduced before changing the row law, and the exact-label limit is taken under the actual experiment. Independence of the original rows before observing the message cannot justify a replacement after conditioning. Together these interfaces keep memory, stopping information, posterior concentration, and exact real labels in the same quantified model. (OpenAI, 2026e)

🏷️ Manuscripts and verification scope

• Memory and precision in noiseless Gaussian regression — pinned manuscript and companion source.
• Posterior replicas and conditional information in Gaussian regression — pinned manuscript and companion source.
• Localization costs and information growth for exact Gaussian observations — pinned manuscript and companion source.
• Projection moments, positive cap domination, and Riesz estimates on the sphere — pinned manuscript and companion source.
• Replacing Gaussian observations in memory-constrained inference — pinned manuscript and companion source.
• Subsphere methods for memory–sample lower bounds in noiseless Gaussian regression — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. It describes fixed-quadratic and subquadratic memory bounds, selected geometric and information interfaces, and the explicit subsphere coefficient. Its selected streaming threshold is 2/3; the stronger 3/5 variant is outside that selection. The replacement comparison selection excludes its separate streaming consequences and several intermediate constructions. The auxiliary dimension-two randomized-model reduction is also outside the projection-moment selection. These documented scopes do not independently verify every quantitative strengthening in the manuscripts. 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. Localization costs and information growth for exact Gaussian observations.

🐻  OpenAI 2026b. Memory and precision in noiseless Gaussian regression.

🐻  OpenAI 2026c. Posterior replicas and conditional information in Gaussian regression.

🐻  OpenAI 2026d. Projection moments, positive cap domination, and Riesz estimates on the sphere.

🐻  OpenAI 2026e. Replacing Gaussian observations in memory-constrained inference.

🐻  OpenAI 2026f. Subsphere methods for memory–sample lower bounds in noiseless Gaussian regression.