Spectral scalar curvature and codimension-two width

Source and verification

This article explains family 336 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: width instead of diameter

Positive scalar curvature does not force a complete manifold to have bounded diameter. The product $S^2(r)\times \mathbb{R}$ has scalar curvature $2/r^2$ and infinite diameter. Projection to the line nevertheless has fibers of diameter $\pi r$.

This motivates Urysohn width. For a metric space $X$, define

$$\begin{array}{rl}& {\mathrm{UW}}_d(X)\\ & =\underset{f:X\to K}{\inf }\underset{y\in K}{\sup }{\mathrm{diam}}_X{f}^{-1}(y),\end{array}$$

where $f$ is continuous and $K$ is a simplicial complex of dimension at most $d$, with its usual polyhedral topology.

The diameter is that of the entire fiber. Bounding each connected component would not bound the distance between two far-apart components mapped to the same point. The family constructs maps for which this full-fiber requirement holds.

The main conclusion loses two dimensions at a controlled scale. The prerequisites are Riemannian curvature, quadratic forms of differential operators, hypersurface stability, and elementary topology of covers and complexes.

🏷️ Three manuscripts and two curvature hypotheses

The pointwise hypothesis is ${\mathrm{Scal}}_g\ge 1$. Its spectral counterpart uses $\Delta =\mathrm{div}\nabla$ and requires

$$\begin{array}{rl}& {\int }_M\left(4|\nabla \varphi {|}^2+{\mathrm{Scal}}_g{\varphi }^2\right)d{V}_g\\ & \ge \lambda {\int }_M{\varphi }^{2}d{V}_g\end{array}$$

for every real compactly supported smooth $\varphi$, where $\lambda >0$. The notation $-4\Delta +\mathrm{Scal}\ge \lambda$ always refers to this quadratic-form inequality. On a noncompact manifold it does not assume that a lowest eigenvalue is attained.

A pointwise lower bound $\mathrm{Scal}\ge \lambda$ implies the spectral inequality, but the spectral condition permits some negative scalar curvature.

Manuscript	Hypothesis and dimensions	Claimed width bound
Positive scalar curvature	Pointwise $\mathrm{Scal}\ge 1$, $n\ge 4$	${\mathrm{UW}}_{n-2}\le C_n$
Spectral curvature, higher dimensions	Spectral bound $\lambda >0$, $n\ge 4$	${\mathrm{UW}}_{n-2}\le C_n/\sqrt{\lambda }$
Spectral curvature in dimension three	Spectral bound $\lambda >0$, $n=3$	${\mathrm{UW}}_1\le 500/\sqrt{\lambda }$

All three statements concern connected complete smooth manifolds without boundary, allowing noncompact and nonorientable examples. They impose no spin or bounded-geometry assumption. The constants are independent of the individual metric and topology. The three-dimensional constant is not optimized. (OpenAI, 2026) (OpenAI, 2026b) (OpenAI, 2026c)

The higher-dimensional spectral paper imports the partition construction of the pointwise companion. Its new work constructs the required weighted and stretched data. The three-dimensional paper gives a direct weighted-surface argument.

🏷️ Scaling and the original metric

Under $g\mapsto \lambda g$, scalar curvature, Laplacians, and squared gradient norms scale by ${\lambda }^{-1}$, while distances scale by $\sqrt{\lambda }$. The spectral lower bound becomes one, explaining the final factor ${\lambda }^{-1/2}$.

Every width bound is measured in the original metric. Producing a conformally related positive-scalar-curvature metric would not suffice without control of its distortion.

The product example extends to $S^2(r)\times {\mathbb{R}}^{n-2}$. Projection has fibers of diameter $\pi r$ and target dimension $n-2$, while the manifold remains unbounded. Width captures precisely the kind of large-scale geometric statement sought here.

🏷️ From a spectral inequality to a weighted curvature bound

The spectral hypothesis supplies a positive smooth supersolution

$$-4{\Delta }_{g}v+{\mathrm{Scal}}_{g}v\ge v$$

after normalization to $\lambda =1$. Set $t=-2\log v$ and define

$$D(g,t)={\mathrm{Scal}}_g+2{\Delta }_{g}t-|dt{|}_g^2.$$

The logarithmic derivative identity gives

$$\begin{array}{rl}2\Delta t-|dt{|}^2& =-4\Delta v/v,\\ D(g,t)& \ge 1.\end{array}$$

Thus the spectral bound becomes a weighted scalar-curvature inequality. The weight $e^{-t}$ is positive, but $t$ and its derivatives may be unbounded at infinity. The constructions must work without assuming a global bound on them.

🏷️ Higher dimensions: reserve and stretching

The higher-dimensional spectral paper first constructs a graph metric $h^{+}=h+d{u}^2$ and a new signed drift $t^{+}$ satisfying

$$\begin{array}{rl}D(h^{+},t^{+})& \ge D(h,w)+(t^{+}{)}^2\\ & -C(n).\end{array}$$

The added square is a reserve for later deformations. It is obtained without first estimating the arbitrary input drift $w$. Local barriers and an angle estimate control the graph equation on complete manifolds with unbounded geometry.

Next choose a maximal separated set of centers in the original metric. Their inner balls cover the manifold, and their doubled closed balls are locally finite by completeness. A local deformation increases the distance from an inner ball to the exterior of its outer ball.

There is no uniform bound on how many outer balls overlap. Charging a fixed loss for each overlap would destroy a dimension-only conclusion. The proof instead fixes a background metric $h_0$ during one round and charges losses to decreases in

$$Q={\mathrm{tr}}_h{h}_0.$$

As the metric increases, $Q$ decreases, with total available decrease at most $n$. These losses telescope independently of the number of centers.

The drift may be negative. Increasing $t$ need not increase $V(t)=t^2+T^2$. The local estimate therefore retains $(V(t^{\prime })-V(t){)}_{+}$, rather than assuming the difference is positive. For $0\le \Lambda \le 1$,

$$a_{+}\ge \Lambda a$$

is the elementary inequality that closes the round invariant. Small additional errors are chosen summably.

Each label needs only finitely many rounds. Local finiteness makes the final data eventually unchanged near every point, yielding smooth limits. Metric enlargement also preserves completeness.

🏷️ Warped circles and the imported partition theorem

For finitely many logarithmic warp functions $p_{\alpha }$, set $P={\sum }_{\alpha }{p}_{\alpha }$. The scalar curvature of the diagonal circle metric is

$$\begin{array}{rl}R& ={\mathrm{Scal}}_h-2{\Delta }_{h}P-|dP{|}_h^2\\ & -\sum _{\alpha }|d{p}_{\alpha }{|}_h^2.\end{array}$$

Choose the sum of the log warps to be $-t$. Splitting a summand into many identical smaller summands preserves their sum while reducing the sum of squared gradients. This converts a lower bound for $D(h,t)$ into a positive stabilized scalar-curvature bound.

On a noncompact manifold the log warps have locally finite compact supports. Near each point only finitely many circles occur. The argument uses these compatible finite products, not an unexplained Riemannian metric on an infinite-dimensional torus.

The imported partition theorem takes a complete enlarged metric $h\ge g$, sufficiently large prescribed separations, and these stabilized curvature bounds. It returns the codimension-two map with fibers controlled in $g$. Constructing those hypotheses from the spectral inequality is separate from proving the partition theorem.

🏷️ Why the partition has codimension two

The pointwise companion supplies the geometric partition construction. It cuts along compatible walls labeled by the original covering balls. Each cut retains both bulk sides and the wall, with compatible product collars.

On an $m$-dimensional retained piece, positive scalar curvature is required where sufficiently many labels remain undecided. High-dimensional minimizing walls may be singular. Additional pressure profiles force decisions near dangerous singular regions, and regularization restores the smooth curvature control needed for the next step.

An intersection of $n-1$ distinct walls would produce a complete one-dimensional piece. Its positively curved stabilized geometry yields a Riccati inequality incompatible with a complete arclength line. This excludes those intersections.

The countable construction requires more than local finiteness on the original manifold. After an initial radial cut, only finitely many labels and warp factors are nonconstant on each whole analytic piece, including its ends. The remaining potentials are summable spatial constants, reproduced exactly in approximations. This allows the finite arguments to continue through the countable list.

For topological reconstruction, long cylinders are used only through finite truncations and compressed into collars on a fixed copy of $M$. The collapse map $q:M\to M$ satisfies

$$d_g(q(x),x)<1.$$

Continuous sign coordinates $a_i\in [0,1]$ equal one on the collapsed inner region and zero beyond the corresponding outer ball. Wall neighborhoods are thickened simultaneously so that at most $n-2$ coordinates are fractional at any point. Some coordinate is always one.

These coordinate vectors lie in a cubical complex of dimension at most $n-2$. Compatible triangulations produce the simplicial target. Local finiteness places the map locally in a finite subcomplex, proving continuity in the ordinary polyhedral topology.

If two points have the same image, a common coordinate equals one. Their collapse images lie in the same original outer ball of radius $2D$. Therefore

$$\begin{array}{rl}{d}_g(x,x^{\prime })& <1+4D+1\\ & =4D+2.\end{array}$$

This is a bound on the entire fiber, regardless of its connectedness. It also explains why the original positions must be tracked throughout the analytic metric changes.

🏷️ Dimension three: weighted stable surfaces

The three-dimensional proof uses the same weighted inequality $D(g,t)\ge 1$ but constructs a graph through surfaces. A two-sided surface stable for weighted area minus weighted volume with pressure $\mu$ satisfies

$$H-\nu t=\mu .$$

Here $\nu$ is a chosen unit normal and $H$ is mean curvature. A positive function for the stability operator transfers the ambient estimate to a two-dimensional weighted inequality. If

$$D(g,t)+{\mu }^2-2|d\mu {|}_g\ge c>0$$

along a closed connected stable surface, the manuscript derives the intrinsic diameter bound $2\pi /\sqrt{c}$ in its original induced metric.

Integrating the transferred inequality and applying Gauss–Bonnet gives positive Euler characteristic. Thus the surface is a sphere or a projective plane. A local version of the estimate also prevents minimizing sheets from escaping a sufficiently large fixed neighborhood.

Cut along a locally finite maximal family of closed stable weighted minimal surfaces. The resulting pieces have a separation property for appropriate compact two-sided surfaces. The nonorientable case is treated on the orientation cover with data that retain the required coorientations downstairs.

To construct separating bubbles near cut boundaries, double each piece and smooth the metric and density compatibly with reflection. The useful smoothing variable is the logarithm of the weighted area density of normal slices. Weighted minimality makes its first normal derivative vanish at the seam, allowing the curvature inequality to survive.

The bounded-diameter interfaces then form a tree of complementary regions: each interface separates. A common interface meeting every path to a distant slab controls that slab’s diameter. Overlapping slabs give a cover of multiplicity at most two, and a partition of unity maps the piece to a graph.

🏷️ Gluing graphs without enlarging whole fibers

The cut surfaces are spheres or projective planes, so their maps to finite graphs are nullhomotopic. This allows the graph maps of adjacent pieces to be joined.

An arbitrary nullhomotopy could identify remote collar pieces and enlarge a fiber. The construction keeps each homotopy inside a finite subgraph whose vertices correspond to cover sets meeting the relevant collar. Collar pieces mapped to one old graph point therefore remain near one common cover set. New edges join only the two collars of a single cut.

This additional locality is what upgrades connected-piece diameter estimates to a bound on full fibers. The bookkeeping yields the stated constant $500$ at spectral normalization one.

🏷️ Consequences and scope

For closed manifolds, the pointwise paper also applies the comparison between filling radius and Urysohn width:

$$\begin{array}{rl}\mathrm{FillRad}(M,g)& \le \frac{1}{2}{\mathrm{UW}}_{n-1}(M,g)\\ & \le \frac{1}{2}{\mathrm{UW}}_{n-2}(M,g).\end{array}$$

It gives a filling-radius bound at the curvature scale. Its separate universal-cover consequences track the topology of the target and the uniform simplex metric. The general width theorem itself asserts a continuous map; it should not silently be upgraded to a Lipschitz statement.

The weighted, singular-descent, and gluing mechanisms are substantive parts of the manuscripts’ claims. The elementary scaling and fiber calculation explain the conclusion once those constructions are supplied.

🏷️ Key steps and proof outlines

The trace invariant explains why arbitrary overlap is affordable

During one stretching round, keep the background metric $h_0$ fixed, even as the current metric $H$ increases. Define $Q={\mathrm{tr}}_H{h}_0$ and let $\Sigma$ be the sum of the small errors already spent. The invariant is

$$D(H,z)\ge c+\Lambda V(z),\Lambda =1-\delta (n-Q)-\Sigma ,V(z)=z^2+T^2.$$

Initially $Q=n$. Since enlargement makes $Q$ decrease and $0<Q\le n$, the total trace loss is at most $n$, regardless of how many covering sets overlap. Choose the separate errors to sum to less than $\delta$, so $\Lambda \ge 1-(n+1)\delta >0$.

At the next step, the local theorem provides $(V(z^{\prime })-V(z){)}_{+}$ and charges $[\delta (Q-Q^{\prime })+d]V(z^{\prime })$. Since $a_{+}\ge \Lambda a$ for $0\le \Lambda \le 1$, the new estimate is exactly $D(H^{\prime },z^{\prime })\ge c+{\Lambda }^{\prime }V(z^{\prime })$. This proof handles negative drift, where increasing $z$ can decrease $z^2$. Local finiteness ensures eventual stabilization near each point; monotone metric enlargement preserves completeness. (OpenAI, 2026b)

A one-dimensional test recovers length in the original surface metric

The three-dimensional companion transfers stability to a weighted surface inequality $D(k,U)\ge c$. Let a minimizing geodesic for $\tilde{k}=e^{-2U}k$ have original $k$-arclength parameter $s\in [0,L]$. Its index form and the conformal curvature formula give

$$0\le {\int }_0^L{e}^U\left[({\varphi }^{\prime }{)}^2-\frac{1}{2}(c+|dU{|}_k^2){\varphi }^2\right]ds.$$

Set $\varphi =e^{-U/2}\psi$. Completing squares and using $|U^{\prime }|\le |dU{|}_k$ bounds the integrand above by $2({\psi }^{\prime }{)}^2-c{\psi }^2/2$. Approximation of $\sin (\pi s/L)$ by compactly supported tests yields $2{\pi }^2/L^2\ge c/2$, hence $L\le 2\pi /\sqrt{c}$.

The parameter $s$ is the original arclength throughout. Thus this is a bound on original-metric distance, even if the conformal factor is not uniformly controlled. Integrating the same weighted curvature inequality gives $4\pi \chi (\Sigma )>0$, providing the sphere/projective-plane alternatives used later in graph gluing. (OpenAI, 2026c)

Analytic cuts become a low-dimensional target only after compatibility

The pointwise companion’s analytic induction must retain collars on both sides of every wall, remove dangerous singular regions, and restore the curvature inequality before the next cut. Its dimension drop rules out an intersection of $n-1$ walls through the final one-dimensional curvature contradiction. To turn that fact into a map, all wall neighborhoods are thickened together so that at most $n-2$ sign coordinates are fractional at any point.

These coordinates define a cubical target of dimension at most $n-2$, and a common coordinate equal to one anchors each whole fiber in one controlled original ball. In dimension three, collar homotopies must similarly remain in graph subcomplexes associated with nearby cover sets. Nullhomotopy by itself has no diameter information. These compatibility requirements explain why the final full-fiber bound is a separate theorem from the stable-surface diameter estimate. (OpenAI, 2026a)

See Also

Integral scalar curvature and simplicial volume studies a different global consequence of curvature. Both families use weighted geometry and degree or partition constructions, but their invariants and imported inputs remain distinct.

🏷️ Manuscripts and verification scope

• Spectral scalar curvature and uniform Urysohn width — pinned manuscript and companion source.
• Spectral scalar curvature and Urysohn width in dimension three — pinned manuscript and companion source.
• Positive scalar curvature and uniform codimension-two width — pinned manuscript and companion source.

No family scope page was present at lean/docs/336.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. Positive scalar curvature and uniform codimension-two width.

🐻  OpenAI 2026b. Spectral scalar curvature and uniform Urysohn width.

🐻  OpenAI 2026c. Spectral scalar curvature and Urysohn width in dimension three.