Scalar curvature and Ricci-flow singularities: four-dimensional extension and higher-dimensional blowup

Source and verification

This article explains family 351 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.

Ricci flow evolves a metric by its Ricci curvature. A finite-time singularity requires some component of the full curvature tensor to become unbounded. Must the scalar curvature, which is only a trace, also become unbounded?

This family gives two dimension-dependent answers. One manuscript asserts smooth extension under bounded scalar curvature in real dimension four. Another constructs counterexamples in sufficiently high dimension. A third proves the static estimate used in the four-dimensional argument. Read together, they explain both how curvature can escape a trace bound and what additional mechanism prevents that escape in dimension four.

🏷️ Preliminaries: three curvatures and two kinds of continuation

Assume a first course in Riemannian geometry and familiarity with parabolic equations, maximum principles, and integration by parts. A closed manifold is compact without boundary. The Riemann tensor $\mathrm{Rm}$ records sectional curvatures; its contraction is the Ricci tensor $\mathrm{Ric}$, whose trace is scalar curvature $R$.

A Ricci flow is a smooth family satisfying

$${\partial }_{t}g=-2\mathrm{Ric}.$$

The scalar evolution and volume evolution are

$$\begin{array}{rl}({\partial }_t-\Delta )R& =2|\mathrm{Ric}{|}^2,\\ {\partial }_{t}d{\mu }_g& =-Rd{\mu }_g.\end{array}$$

The positive source in the first equation makes the extension question plausible, but bounded values of a solution do not automatically bound its source pointwise. Diffusion and concentration matter. Also, Ricci-flat means $\mathrm{Ric}=0$, not necessarily $\mathrm{Rm}=0$: curvature invisible to Ricci can remain in the full tensor.

Under a length dilation by $s$, the metric becomes $s^{2}g$ and curvature norms acquire the factor $s^{-2}$. The natural time scale is $s^2$. In dimension four,

$$\int |\mathrm{Rm}{|}^{2}d\mu$$

is unchanged by dilation. This scale invariance makes a finite curvature budget compatible with bubbles of very different sizes, while still limiting how many nonflat bubbles can occur.

Here smooth extension means a smooth flow on the same manifold beyond the proposed singular time, agreeing with the original flow before that time. A weak continuation on a singular quotient space would be a different endpoint.

🏷️ The three manuscript claims

Four-dimensional extension

Let $g(t)$ be a smooth Ricci flow on a closed connected smooth four-manifold, defined for $0\le t<T<\infty$ with smooth initial metric. If $|R|$ is uniformly bounded, the manuscript asserts that the flow extends smoothly on the same manifold through $T$. It imposes no orientability, Kähler, Type-I, or curvature-rate hypothesis. (OpenAI, 2026)

The scalar minimum has a uniform lower bound by the maximum principle. Consequently, at a finite maximal existence time the claimed theorem gives

$$\underset{t\uparrow T}{\mathrm{limsup}}\underset{M}{\max }R(\cdot ,t)=+\infty .$$

This is a limsup statement, not an assertion that the maximum tends monotonically to infinity.

Higher-dimensional counterexamples

For sufficiently large fixed integers $q$, the first manuscript constructs a closed flow on $S^2\times S^{q+1}$ with bounded $|R|$ and full curvature tending to infinity at a finite time. For all sufficiently large even integers $k$, its examples satisfy two-sided bounds with exponent $2k/d_q$, where

$$d_q=\frac{q+1-\sqrt{(q+1)(q-7)}}{2}.$$

Thus an unbounded discrete set of blowup exponents is available in one fixed sufficiently high dimension. (OpenAI, 2026a)

More precisely, each chosen flow has positive constants $c,C$ such that, near $T$,

$$\begin{array}{rl}\underset{M}{\max }|\mathrm{Rm}|& \ge c(T-t{)}^{-2k/d_q},\\ \underset{M}{\max }|\mathrm{Rm}|& \le C(T-t{)}^{-2k/d_q}.\end{array}$$

Different choices of $k$ give different flows. The assertion does not cover every $q\ge 10$, every exponent, or dimension four.

Static input for dimension four

For metrics converging to one fixed finite tree of Ricci-flat spaces, with specified end decay and dimensionless Ricci error $M\to 0$, the companion proves a renormalized functional estimate $E=o(M)$. All joining lengths tend to infinity, possibly at unrelated rates. The hypotheses include derivative bounds and compatible end identifications. (OpenAI, 2026c)

The rest of the article explains how these claims fit together. The four-dimensional theorem uses the static companion as a substantive input.

🏷️ High dimension: a shrinking cap inside the parabolic scale

The counterexample starts from Stolarski’s doubly warped constructions, in the precise quantitative form recorded in the manuscript. Locally the metric is

$$g=d{s}^2+\varphi (s,t{)}^2{g}_{S^2}+r(s,t{)}^2{g}_{S^q}.$$

At a pole, $r$ vanishes smoothly while $\varphi$ stays positive. The orbit there is an $S^2$. Introduce

$$\begin{array}{c}\delta =T-t,\sigma =k/d_q,\\ \theta ={\delta }^{\sigma },\gamma =r/\sqrt{\delta }.\end{array}$$

The cap scale $\theta$ is much smaller than the parabolic scale $\sqrt{\delta }$. At the latter scale the metric approaches a Ricci-flat cone; the cap resolves its tip before the singular time.

The imported construction supplies profile estimates and signs, including useful control of

$$\begin{array}{c}u=\log \varphi ,z=r_s^2,\\ f=r{u}_r.\end{array}$$

In the inner region, $0\le f\le 1$, $f_r\ge 0$, and $z$ lies between a positive cone constant and one. The scalar upper bound and the cap curvature estimate are proved in this manuscript; they are not assumed from the construction. (OpenAI, 2026a)

Why the cap has the advertised size

The first geometric objective is

$$\begin{array}{c}(r^2+{\theta }^2)|\mathrm{Rm}|\le C,\\ \varphi (0,t)\asymp \theta .\end{array}$$

The lower bound for $\varphi (0,t)$ is particularly revealing. Suppose the pole radius $L_0=\varphi (0,t)$ were much smaller than $\theta$. Set $r_{\ast }=(L_0\theta {)}^{1/2}$. The profile signs and the integral of $(1-f)/r$ force $f$ close to one on the long logarithmic interval $[r_{\ast },\theta ]$.

On that interval, $\varphi /r$ remains strictly above its cone value. In the scalar-curvature formula this creates a negative term; the term involving $f_r$ has the same favorable sign. The already available scalar lower bound then forces $r{z}_r\le -c$. Integrating with respect to $\log r$ makes $z$ decrease by an arbitrarily large amount, contradicting its fixed bounds. Thus the $S^2$ at the pole cannot collapse substantially faster than $\theta$.

To bound full curvature, the proof uses points at which a scale-weighted curvature maximum sets a record over earlier times. Such records provide an actual backward parabolic region with bounded rescaled curvature, so derivative estimates become available before passing to a limit. Bounded warp slopes then contradict a persistent nonzero second derivative of a warp factor. A separate cap argument uses $1-r_s\le Cr/\theta$. This avoids assuming convergence to a particular complete cap model.

At the pole, the sectional curvature tangent to $S^2$ equals $\varphi (0,t{)}^{-2}$. Together with the displayed upper bound, this gives the two-sided full-curvature rate. It remains to show that scalar curvature stays bounded.

The leading reaction coefficient must be retained

For a function expressed in the moving radius $r$, the scalar heat operator becomes

$$\mathcal{D}F=F_t{|}_r-z{F}_{rr}-\frac{q-1+z}{r}{F}_r.$$

The norm evolution gives, wherever $|\mathrm{Ric}|>0$,

$$\mathcal{D}|\mathrm{Ric}|\le c(x,t)|\mathrm{Ric}|.$$

A rough bound $c\le Cq/r^2$ loses too much. The manuscript instead derives

$$\begin{array}{rl}c(x,t)& \le \frac{2(q-1)+C\sqrt{q}}{r^2},\\ c(x,t)& \le C_{\ast }/{\theta }^2.\end{array}$$

The leading coefficient two is essential. To obtain it, the proof writes the curvature action on the three invariant diagonal tensor components, with their multiplicities included in the norm. Its leading eigenvalue is $q-1+O(\sqrt{q})$ before the factor two from the Ricci evolution.

The required radial derivative bounds come from one-dimensional parabolic estimates on cylinders of width $r_c/\sqrt{q}$. Moving the cylinder center by

$$R_c(t{)}^2=r_c^2+2(q-1)(t_c-t)$$

cancels the large radial drift. The transformed coefficients have bounds independent of $q$, giving $|r{z}_r|\le C{q}^{-1/2}$ and $r^2|u_{rr}|\le C\sqrt{q}$. This is why a high-dimensional parameter can improve the final comparison instead of simply enlarging every constant.

From a Ricci barrier to a scalar barrier

Choose $a=5/2$, then $q$ sufficiently large that $2<d_q<a$ and the reaction comparison has a positive margin. Next choose a sufficiently large even $k$. Put

$$\begin{array}{c}L=k-1,e=a-L/\sigma ,\\ 0<e<1,\\ H={\delta }^L(r^2+b_0{\theta }^2{)}^{-a/2}.\end{array}$$

Only after the flow and its constant $C_{\ast }$ are fixed is $b_0>0$ chosen small. Using both reaction bounds, $H$ is a supersolution on a small parabolic neighborhood of the cap. The profile controls its lateral boundary values. Comparison and a split into $r\le \theta$ and $r\ge \theta$ give

$$|\mathrm{Ric}|\le CH\le C{r}^{-e}.$$

For example, when $r\ge \theta ={\delta }^{\sigma }$, the inequality ${\delta }^L\le r^{L/\sigma }$ gives the last bound directly. On $r\le \theta$, use $H\le C{\delta }^{-\sigma e}\le C{r}^{-e}$.

The exponent $e<1$ is the decisive threshold. For any power $\alpha$,

$$\mathcal{D}{r}^{\alpha }=-\alpha (q-1+\alpha z)r^{\alpha -2}.$$

Set $\ell =2-2e>0$. The function

$$S_{\epsilon }=C_1-C_2{r}^{\ell }+\epsilon /r$$

has a positive heat-operator contribution of order $r^{-2e}$, sufficient to dominate $2|\mathrm{Ric}{|}^2$. The term $\epsilon /r$ protects the comparison at the pole. One first compares on a punctured region over a fixed smooth time interval, removes the inner boundary, and finally lets $\epsilon \downarrow 0$. There is no assumption that $r^{\ell }$ is smooth at the pole.

A separate exterior Ricci barrier and the profile-controlled intervening annulus complete the scalar upper bound globally. The maximum principle already supplies the lower bound. Thus rapid full-curvature concentration coexists with bounded scalar curvature. (OpenAI, 2026a)

🏷️ Dimension four: follow the outermost bubble

Assume now that a four-dimensional closed flow has bounded scalar curvature but develops curvature concentration. The curvature radius $l(x,t)$ is the supremum of radii $0<r\le l_0$ for which

$$\underset{B_{g(t)}(x,r)}{\sup }|\mathrm{Rm}|\le r^{-2}.$$

It tests a whole ball, not just curvature at its center, and is spatially Lipschitz. The geometric estimates in the manuscript give controlled volume ratios, a total $L^2$ curvature budget, and scaled derivative bounds. Small-scale limits are Ricci-flat spaces, smooth apart from isolated genuine quotient singularities.

An ALE end is an end approaching Euclidean space modulo a finite group acting freely on the unit sphere. A bubble tree organizes the nonflat limits: a smaller bubble attaches near a singular point of a larger one. Each nonflat vertex consumes a definite amount of the curvature budget, so only finitely many vertices occur. Long necks encode their separating scales. These compactness, neck, and removability arguments are substantial geometric parts of the manuscript. (OpenAI, 2026b)

The relevant scale is the outermost nonflat bubble, not necessarily the smallest curvature radius. With a fixed nonnegative detection cutoff $\psi$, the paper uses a localized maximum of the form

$$a(t)=\underset{y\in U}{\max }[l\psi (l^2|\mathrm{Rm}|)].$$

The quantities inside the maximum are evaluated at $(y,t)$. This scale is positive, continuous, and tends to zero at the concentrating cluster. A maximizing point is regular at scale comparable to $a(t)$. Much smaller bubbles may lie inside it.

The outermost property is essential. If two tiny tips remained far apart relative to $a(t)$, their separation-scale limit would contain two genuine tips. A flat limit has at most one; a nonflat limit would be a larger bubble, contradicting the definition of the root scale.

Normalize a fixed-factor change

The proof selects intervals on which $a(t)$ changes by a fixed factor, with enough time on both sides for cutoff estimates. No monotonicity of $a(t)$ is assumed. Short-time motion estimates show that an interval of duration $d_i$ realizing such a change has $a^2/d_i\to 0$.

Rescale lengths by $d_i^{-1/2}$ and time by $d_i^{-1}$. The selected interval becomes $[0,1]$, with a buffer on either side; its root scale $a_i$ tends to zero and its scalar bound also tends to zero. Write $a=a_i$ below. The contradiction will show that the two endpoint root geometries are isometric at scale $a$, so their root scales could not have changed by the prescribed factor.

🏷️ The static functional and why it is small

For a metric with one ALE end, define a renormalized Einstein–Hilbert functional. In exterior coordinates let

$$\begin{array}{c}{b}_i={\partial }_j{g}_{ij}-{\partial }_i{g}_{jj},\\ {\mathcal{B}}_D(g)={\int }_{r=D}{b}_i{n}_{i}dS,\\ E_D(g)={\int }_{r<D}Rd\mu -{\mathcal{B}}_D(g),\\ E(g)=\underset{D\to \infty }{\lim }{E}_D(g).\end{array}$$

All integrals on a quotient end are quotient integrals. The two terms need not converge separately. If $g=\delta +p$ with $p=O(r^{-q})$, $q>1$, the linear part of the scalar density is exactly the divergence subtracted by ${\mathcal{B}}_D$. The remainder has size $O(r^{-2q-2})$; in dimension four its radial integral is controlled by $\int r^{1-2q}dr$, which converges.

The same cancellation gives the first variation

$$D{E}_g(h)=\int ⟨\frac{1}{2}Rg-\mathrm{Ric},h⟩d\mu .$$

The functional has length degree two: a global length dilation by $s$, with exterior coordinates dilated accordingly, multiplies $E$ by $s^2$.

In a tree normalized to root scale one, an edge of length $L_e$ joins parent and child scales by $a_c=a_p{e}^{-2L_e}$. The error $M$ controls ten scaled derivatives of ${\rho }^2\mathrm{Ric}$, with positive exponential decay on joining half-necks and decay $r^{-q}$ at the root. Metric convergence and twenty-derivative end bounds are also assumed. Direct first variation gives $|E|\le CM$; the companion improves this to

$$E=o(M).$$

This is a sequential assertion for one fixed limiting tree. It supplies neither a universal rate nor an exponent uniform over all trees. (OpenAI, 2026c)

Reduction to a finite analytic family

After fixing coordinates to make the equations elliptic, the companion retains the finite-dimensional kernel and cokernel of each vertex operator as parameters and auxiliary data. They are not assumed absent. At a joining neck, matching values and normal derivatives is supplemented by the critical spherical mode. Explicit radial mode estimates then produce an inverse uniform in long neck lengths.

This constructs a family $h_{\ast }(L,z)$, where $L$ records lengths and $z$ records bounded finite-dimensional data. For an actual metric with error $M$,

$$\begin{array}{rl}\Vert g-h_{\ast }\Vert & \le CM,\\ \Vert \mathrm{Ric}(h_{\ast })\Vert & \le CM,\\ |E(g)-E(h_{\ast })|& \le C{M}^2.\end{array}$$

The last estimate pairs an $O(M)$ Ricci tensor with an $O(M)$ metric difference in the first variation. Thus the finite-dimensional family captures the functional to better than its conjectured linear error scale. No assumption that every infinitesimal Ricci-flat deformation integrates to a family of Ricci-flat metrics is needed.

Set $F=E(h_{\ast })$, and let $J\ge 0$ be the squared Ricci integral with an additional integrable weight at the root end. The central differential estimate is

$$\begin{array}{c}|dF|\le C\sqrt{J}\omega ,\\ \omega =\sum _{\alpha }|d{z}_{\alpha }|+\sum _e{e}^{-c{L}_e}|d{L}_e|.\end{array}$$

The exponentially small cost of varying a neck length reflects decay on the neck and the shrinking physical volumes of its descendants.

A path to the degenerating boundary

Ordinary finite-dimensional analyticity alone is insufficient because $L_e\to \infty$. Replacing $L_e$ by $e^{-L_e}$ does not automatically create an analytic extension to zero.

The companion instead proves that $F$ and $J$ belong to a precise asymptotic class: they extend holomorphically to quadratic complex length regions, have exponential-polynomial expansions to every finite order, and have recursively analytic coefficients on one common ordinary-parameter neighborhood. Remainders are strictly better than the requested exponential order. These expansions are not asserted to converge as one infinite series.

The construction opens one neck at a time, expands the resulting end solutions, and cancels successive matching errors by linear solves. Larger prescribed growing coefficients enter those linear solves; they do not require a new nonlinear neighborhood at every order. Finite-part integration transfers the expansions to $F$ and $J$.

The resulting path-selection theorem realizes the relevant signs and limits on a real analytic path whose actual length coordinates and bounded ordinary coordinates are eventually monotone. It therefore has finite $\omega$-length. This selection theorem, including the treatment of different rates of divergence, is a major analytic input, not a consequence of a formal expansion alone.

The final contradiction is short enough to reproduce. Failure of $E=o(M)$, combined with the comparison estimates, produces a sequence with

$$F\ne 0,F\to 0,F^2\ge c_{0}J.$$

Path selection preserves these conditions. Along that path,

$$|d\log |F||\le C{c}_0^{-1/2}\omega .$$

Finite weighted variation makes the total change in $\log |F|$ finite, whereas $F\to 0$ requires it to tend to $-\infty$. This contradiction proves the static improvement. (OpenAI, 2026c)

🏷️ Turn static smallness into vanishing Ricci energy

The actual flow cluster has only a finite exterior neck, whereas $E$ requires a complete end with decay exponent greater than one. The manuscript constructs an analytic extension from a fixed material collar at radius $b=Na$, where $N$ is large and fixed before taking the sequence limit.

The extension depends only on the collar metric; the longer exterior is used to estimate its error. This distinction lets the extension vary in time without differentiating a changing choice of entire bubble tree. A finite outer-boundary curvature term is retained even when the exterior Ricci tensor vanishes. Directly cutting the metric off to flat would not provide the required error estimate. (OpenAI, 2026b)

An extra power of decay on the neck

Let $\eta$ be a time cutoff supported in the buffered interval and positive on $[0,1]$, and put $G=\eta \mathrm{Ric}$. Define

$$K^2=a^{-4}\int {\int }_{B(x_t,N_{1}a)}|G{|}^{2}d\mu dt.$$

The parabolic estimates give an exterior decay contribution of order $(a/r{)}^{2+\alpha }K$, for some $\alpha >0$. Why is there an improvement over the borderline power two?

The contracted Bianchi identity says

$$\mathrm{div}(G-\frac{1}{2}g{\mathrm{tr}}_{g}G)=0.$$

On the flat four-dimensional model, a componentwise harmonic critical field would have the form $r^{-2}A$ for a constant symmetric tensor $A$. Substitution into this identity forces $A=0$. Once the spherical mean is separated, the first nonconstant spherical eigenvalue is three, with radial harmonic powers $r$ and $r^{-3}$. This exclusion of the critical decaying mode supplies the extra decay.

Spatially constant fields do remain. The proof controls their values at an outer boundary and sums only positive-order derivatives across shells. Summing a fixed small value error over arbitrarily many shells would not be valid.

Compare two estimates for functional variation

On the true interior, substituting the flow velocity into the first variation gives exactly

$$\begin{array}{rl}& ⟨\frac{1}{2}Rg-\mathrm{Ric},-2\mathrm{Ric}⟩\\ & =2|\mathrm{Ric}{|}^2-R^2.\end{array}$$

The scalar term is negligible in the normalized intervals. The artificial tail costs at most a constant times

$$b^4(a/b{)}^{4+2\alpha }{K}^2=a^4{N}^{-2\alpha }{K}^2,$$

up to terms tending to zero. Its radial integrability uses the extension decay exponent $q>1$.

A collection of time patches supplies fixed collars and extended functionals $E_j(t)$. A subordinate partition satisfies ${\sum }_j{\chi }_j=1$ and ${\sum }_j|{\chi }_j^{\prime }|\le C(1+B_1)$, where $B_1$ controls collar Ricci derivatives. Different patches may give different functional values; the proof integrates each patch by parts and estimates the partition derivatives rather than identifying those values.

Applying the static theorem after extracting a fixed tree gives a competing upper bound of order $o(a^4)(K+1{)}^2$. Uniformity over active times follows by a subsequence contradiction; it does not require one static exponent for every possible tree. Combining the two bounds yields, after choosing $N$ large once,

$$K^2\le {\gamma }_i(K+1{)}^2,{\gamma }_i\to 0.$$

This already implies $K\to 0$: for $\sqrt{{\gamma }_i}<1$,

$$K\le \frac{\sqrt{{\gamma }_i}}{1-\sqrt{{\gamma }_i}}.$$

No a priori bound for $K$ is needed. This is the main dynamic consequence of the static companion. (OpenAI, 2026b)

🏷️ Small energy prevents a fixed-factor change of scale

One cannot immediately turn $K\to 0$ into a uniform pointwise Ricci bound throughout the cluster: there may be much smaller child bubbles. The final argument instead constructs a common set of good material points.

The parabolic estimates yield an envelope $Q_i(t)$ with $\Vert Q_i{\Vert }_{L^2(0,1)}\to 0$ and

$$|\mathrm{Ric}|(y,t)\le Q_i(t)\left[1+\frac{a^2}{l(y,t{)}^2}\right].$$

A geometric estimate along minimizing geodesics converts this into a distance-variation bound involving $a/l$ at the endpoints. Meanwhile, the bad-set estimate

$$\mathrm{Vol}\{l<s\}\le C{s}^4$$

controls the spatial integral of $a/l$ on the region $l<a$. Bounded scalar curvature makes volume forms uniformly comparable on fixed material sets. Integrating in time and discarding points with large accumulated endpoint error removes only $o(a^4)$ volume.

Volume lower bounds then make the retained set $o(a)$-dense at every time. Distances between its points change by $o(a)$, so the endpoint root limits are isometric, with the same material points supplying dense corresponding nets.

It remains to recover the particular scale definition, not just an unspecified isometry. The proof shows that the whole-ball curvature radius converges to its intrinsic limiting counterpart. At a quotient tip, its genuine singularity prevents treating the tip as an ordinary regular point. At regular points, smooth convergence controls admissible balls and curvature violations. Endpoint maximizers can therefore compete in each other’s root-scale maximum, forcing equal limiting values.

That contradicts the selected fixed-factor change. With concentration excluded, full curvature stays bounded; the metric converges smoothly to a positive metric on the original manifold, and short-time existence extends the flow. (OpenAI, 2026b)

🏷️ What to retain from the family

The higher-dimensional construction separates the cap scale from the parabolic scale and uses a sharp reaction estimate to make the scalar source controllable. The four-dimensional argument uses the outermost bubble, the absence of a critical Ricci mode, and a renormalized functional whose static size is too small to pay for a fixed amount of dynamic Ricci energy.

The static companion explains why infinitely long necks require more than an ordinary analytic-family argument. Its finite-variation contradiction is elementary once the analytic family, uniform inverses, and path-selection class have been established; establishing those inputs is the substantial work.

For nearby geometric ideas, the Einstein pinching family discusses rigidity under curvature conditions, while the Clifford-volume family uses scale limits and geometric detection in a different variational setting.

🏷️ Key steps and proof outlines

The missing radial mode can be excluded directly

On a flat four-dimensional neck, the borderline componentwise harmonic field would be $r^{-2}A$, with $A$ a constant symmetric matrix. Its trace-reversed divergence is

$$-2r^{-3}\left(A-\frac{1}{2}I\mathrm{tr}A\right)n,n=x/r.$$

Vanishing for every direction $n$ forces $A-\frac{1}{2}I\mathrm{tr}A=0$. Taking the trace in dimension four then gives $\mathrm{tr}A=0$, hence $A=0$. This uses the contracted Bianchi constraint, not a curvature decay assumption.

After subtracting spherical averages, the first nonconstant eigenvalue on $S^3$ is three. For a field $H(r,\omega ,t)$ with zero spherical mean and compact support in time, choose $1<p\le 2$ and put $z(r)=(\int \Vert H(r,\cdot ,t){\Vert }_{L^2(S^3)}^{p}dt{)}^{1/p}$. Pair the heat equation with the normalized field, use convexity of its Hilbert norm, and integrate its $p$th power in time. Compact time support removes the integrated time derivative. The resulting radial inequality is

$$-z^{\prime \prime }-\frac{3}{r}{z}^{\prime }+\frac{3}{r^2}z\le \text{source norm}.$$

Its homogeneous powers solve $3-\beta (\beta +2)=0$, so they are $r$ and $r^{-3}$. This supplies a decay gain beyond the excluded $r^{-2}$ mode. Spherical mean derivatives are recovered from the divergence equation; constant tensors remain controlled by the outer boundary value. (OpenAI, 2026b)

The functional estimate must improve a linear error to a little-o error

The static first variation alone gives $|E|\le CM$, which is not small enough to force the normalized Ricci energy to vanish. The analytic reduction approximates the metric by a finite-dimensional family to $O(M)$ and its functional to $O(M^2)$. The remaining functions satisfy $|dF|\le C\sqrt{J}\omega$, where changing long neck lengths costs exponentially little.

If $F^2\ge cJ$ along a selected path toward the degenerating boundary, division gives bounded variation of $\log |F|$ in the finite weighted path length. But $F\to 0$ would require infinite logarithmic variation. The contradiction yields $E=o(M)$. The special path-selection theorem supplies a path retaining the inequalities and controlled length; ordinary analyticity on a bounded parameter set would not address diverging neck lengths. (OpenAI, 2026c)

Static and dynamic estimates close only after choosing the collar scale

The dynamic first variation has positive leading term $2\int |\mathrm{Ric}{|}^2$, while bounded scalar curvature makes the $R^2$ contribution negligible at the bubble scale. The improved neck decay bounds the artificial tail by a factor $N^{-2\alpha }$ times the normalized interior energy. Choose the collar multiplier $N$ large once, then take the degenerating sequence. The static little-o estimate now produces $K^2\le {\gamma }_i(K+1{)}^2$ with ${\gamma }_i\to 0$, which forces $K\to 0$ without a prior bound on $K$.

The higher-dimensional companion follows a different mechanism. Its dimension-explicit reaction estimate produces a Ricci barrier $|\mathrm{Ric}|\le C{r}^{-e}$ with $e<1$. The scalar source is then bounded by $C{r}^{-2e}$, which a positive heat contribution from $-r^{2-2e}$ can absorb. A punctured comparison and an auxiliary $\epsilon /r$ term handle the pole before limits are taken. Thus the contrast between dimensions lies in the specific decay and concentration mechanisms, not in the scalar evolution identity alone. (OpenAI, 2026a)

🏷️ Manuscripts and verification scope

• A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup — pinned manuscript and companion source.
• Bounded scalar curvature and smooth extension of four-dimensional Ricci flow — pinned manuscript and companion source.
• Path selection and an elliptic inequality on degenerating Ricci-flat trees — pinned manuscript and companion source.

No family scope page was present at lean/docs/351.md in the pinned snapshot. This does not establish that all related formalization code is absent. 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. A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup.

🐻  OpenAI 2026b. Bounded scalar curvature and smooth extension of four-dimensional Ricci flow.

🐻  OpenAI 2026c. Path selection and an elliptic inequality on degenerating Ricci-flat trees.