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Mark Walsh

Publications and source records attributed to Mark Walsh.

12 recordsLinked to original sources
arXiv2026

Support sufficiency as action-sufficient compression: a single-cycle rate-regret formulation

Robust decision-making requires compression. A system that forms a rich support state cannot usually preserve its full structure at the point of action. It must retain only those distinctions needed to act, verify, abstain, or defer under the current consequence geometry. This paper formalizes support sufficiency as action-sufficient compression. Let $H$ denote a full support state, $\mathcal{A}$ a finite action set, and $Z$ a consequence geometry specifying payoff structure. For fixed $Z$, the coarsest exactly action-sufficient compression is the quotient of support space by policy equivalence. Two support states may be merged exactly when they require the same optimal action. This clarifies why content-only and scalar-confidence-only arbitration fail whenever their induced partitions cross action boundaries. Approximate sufficiency is then defined by bounded expected policy regret. In the finite single-cycle setting, this yields a rate-regret problem with source $H$, reproduction alphabet $\mathcal{A}$, and distortion given by consequence-sensitive regret. The optimal stochastic action channel inherits the standard rate-distortion Gibbs form, applied here to support states with regret distortion. The contribution is interpretive: action adequacy is distinguished from reconstruction fidelity, information-bottleneck prediction, and rational inattention. Robust single-cycle arbitration does not require preserving all support, but it does require preserving the distinctions that consequence geometry makes action-relevant.

cs.IT↗
arXiv2026

Support Sufficiency as Consequence-Sensitive Compression in Belief Arbitration

When a system commits to a hypothesis, much of the evidential structure behind that commitment is lost to compression. Standard accounts assume that selected content and scalar confidence suffice for downstream control. This paper argues that they do not, and that determining what must survive compression is itself a consequence-sensitive problem. We develop a recurrent arbitration architecture in which active constraint fields jointly determine a hypothesis geometry over candidates. Rather than carrying that geometry forward in full, the system compresses it into a support-aware control state whose resolution is regulated by current consequence geometry, arbitration memory, and resource constraints. A bounded objective formalizes the tradeoff. Too little retained support collapses policy-relevant distinctions, producing controllers that select content adequately while misrouting verification, abstention, and recovery. Too much retained support fragments learning across overly fine contexts, degrading adaptation even as discrimination improves. These failure modes yield ordered controller predictions confirmed by a minimal repeated-interaction simulation. Adaptive controllers that regulate support resolution outperform all fixed-resolution controllers in cumulative utility. Agile adaptive control outperforms sluggish adaptive control. Fixed high-resolution control achieves the best commitment accuracy but still trails adaptive controllers because resource cost and learning fragmentation offset the gains from richer retention. Support sufficiency should be understood not as a static representational threshold, but as a dynamic compression criterion. Robust arbitration depends on preserving the smallest support structure adequate for policy under the current consequence landscape, and on regulating that structure as conditions change across repeated cycles of inference and action.

cs.AI↗
arXiv2026

Audited calibration under regime shift as a computational test of support-structured broadcast

A central prediction of the accompanying theoretical framework is that metacognitive calibration can vary even when content-level performance is held approximately fixed, depending on whether support structure is preserved in a globally reusable broadcast state. We provide a minimal computational test of this claim using a two-channel probabilistic cue-integration task with regime shifts that induce systematic miscalibration in one channel. We compare content-dominated architectures, in which confidence is calibrated by a single global mapping from evidence strength to probability, to an auditor architecture that learns a regime-conditioned calibration mapping from an audit trail of outcomes. We then couple confidence to control by implementing a policy that either acts immediately or requests one additional sample when confidence falls below a threshold. Across matched evidence streams, the auditor substantially improves calibration, particularly in the degraded regime, and produces qualitatively different control behavior by selectively requesting additional evidence under low-support conditions. These results demonstrate a concrete, testable dissociation between content performance and system-level confidence and policy that arises from globally reusable support summaries.

q-bio.NC↗
arXiv2021

Positive $(p, n)$-intermediate scalar curvature and cobordism

In this paper we consider a well-known construction due to Gromov and Lawson, Schoen and Yau, Gajer, and Walsh which allows for the extension of a metric of positive scalar curvature over the trace of a surgery in codimension at least $3$ to a metric of positive scalar curvature which is a product near the boundary. We generalize this construction to work for $(p,n)$-intermediate scalar curvature for $0\leq p\leq n-2$ for surgeries in codimension at least $p+3$. We then use it to generalize a well known theorem of Carr. Letting ${\cal R}^{s_{p,n}>0}(M)$ denote the space of positive $(p, n)$-intermediate scalar curvature metrics on an $n$-manifold $M$, we show for $0\leq p\leq 2n-3$ and $n\geq 2$, that for a closed, spin, $(4n-1)$-manifold $M$ admitting a metric of positive $(p,4n-1)$-intermediate scalar curvature, ${\cal R}^{s_{p,4n-1}>0}(M)$ has infinitely many path components.

math.DG↗
arXiv2020

H-Space and Loop Space Structures for Intermediate Curvatures

For dimensions $n\geq 3$ and $k\in\{2, \cdots, n\}$, we show that the space of metrics of $k$-positive Ricci curvature on the sphere $S^{n}$ has the structure of an $H$-space with a homotopy commutative, homotopy associative product operation. We further show, using the theory of operads and results of Boardman, Vogt and May that the path component of this space containing the round metric is weakly homotopy equivalent to an $n$-fold loop space.

math.DG↗
arXiv2014

The Space of Positive Scalar Curvature Metrics on a Manifold with Boundary

We study the space of Riemannian metrics with positive scalar curvature on a compact manifold with boundary. These metrics extend a fixed boundary metric and take a product structure on a collar neighbourhood of the boundary. We show that the weak homotopy type of this space is preserved by certain surgeries on the boundary in co-dimension at least three. Thus, there is a weak homotopy equivalence between the space of such metrics on a simply connected spin manifold $W$, of dimension $n\geq 6$ and with simply connected boundary, and the corresponding space of metrics of positive scalar curvature on the standard disk $D^{n}$. Indeed, for certain boundary metrics, this space is weakly homotopy equivalent to the space of all metrics of positive scalar curvature on the standard sphere $S^{n}$. Finally, we prove analogous results for the more general space where the boundary metric is left unfixed.

math.DG↗
arXiv2013

H-Spaces, Loop Spaces and the Space of Positive Scalar Curvature Metrics on the Sphere

For dimensions n greater than or equal to 3, we show that the space of metrics of positive scalar curvature on the n-sphere is homotopy equivalent to a subspace which takes the form of a H-space with a homotopy commutative, homotopy associative product operation. This product operation is based on the connected sum construction. We then exhibit an action of the little n-disks operad on this subspace which, using results of Boardman, Vogt and May implies that when n=3 or n is at least 5, the space of metrics of positive scalar curvature on the n-sphere is weakly homotopy equivalent to an n-fold loop space.

math.DG↗
arXiv2011

Metrics of positive scalar curvature and generalised Morse functions, part II

The surgery technique of Gromov and Lawson may be used to construct families of positive scalar curvature metrics which are parameterised by Morse functions. This has played an important role in the study of the space of metrics of positive scalar curvature on a smooth manifold and its corresponding moduli spaces. In this paper, we extend this technique to work for families of generalised Morse functions, i.e. smooth functions with both Morse and birth-death singularities.

math.DG↗
arXiv2009

Homotopy groups of the moduli space of metrics of positive scalar curvature

We prove that for many degrees in a stable range the homotopy groups of the moduli space of metrics of positive scalar curvature on S^n and on other manifolds are non-trivial. This is achieved by further developing and then applying a family version of the surgery construction of Gromov-Lawson to an exotic smooth families of spheres due to Hatcher. As described, this works for all manifolds of suitable dimension and for the quotient of the space of metrics of positive scalar curvature by the (free) action of the subgroup of diffeomorphisms which fix a point and its tangent space. We also construct special manifolds where the quotient of the space of metrcis of positive scalar curvature by the full diffeomorphism group has non-trivial higher homotopy groups.

math.GT↗
arXiv2008

Metrics of positive scalar curvature and generalised Morse functions, part 1

It is well known that isotopic metrics of positive scalar curvature are concordant. Whether or not the converse holds is an open question, at least in dimensions greater than four. We show that for a particular type of concordance, constructed using the surgery techniques of Gromov and Lawson, this converse holds in the case of closed simply connected manifolds of dimension at least five.

math.DG↗