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arxiv: 2605.22118 · v1 · pith:DIPBDSH6new · submitted 2026-05-21 · 🧮 math.AG

On defective spans of singular vector tuples beyond the boundary format

Ettore Teixeira Turatti , Emanuele Ventura This is my paper

Pith reviewed 2026-05-22 03:00 UTC · model grok-4.3

classification 🧮 math.AG
keywords tensorssingular vector tuplescritical spacesdefective spansboundary formatcohomology mapKoszul cohomologyalgebraic geometry
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The pith

For tensor spaces beyond the boundary format by one, the codimension of the general span of singular vector tuples in the critical space equals the kernel dimension of a cohomology map.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines tensor spaces that exceed the boundary format by one in an arbitrary number of factors. It relates the codimension of the general span of singular vector tuples inside the critical space directly to the dimension of the kernel of a map in cohomology. The authors construct an infinite family of order-three tensors where this span behaves defectively, reaching the maximum possible codimension rather than the expected minimum. They conjecture a classification of behaviors in this regime and link the setup to Koszul cohomology.

Core claim

In tensor spaces exceeding the boundary format by one factor in any number of modes, the codimension of the general span of singular vector tuples within the general critical space equals the dimension of the kernel of a certain cohomology map. Moreover, there is an infinite family of order-three tensors for which the general span is defective and reaches the maximum codimension instead of the minimum.

What carries the argument

The cohomology map whose kernel dimension equals the codimension gap between the general span of singular vector tuples and the critical space.

If this is right

  • The codimension difference becomes computable from cohomology data for all formats one beyond the boundary.
  • Defective maximum-codimension behavior occurs for an infinite family of order-three tensors.
  • The behavior of critical spaces in this regime admits a conjectured classification.
  • Koszul cohomology supplies an algebraic tool for studying these spans and defects.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Explicit computation of the cohomology kernels would yield exact dimension formulas for the spans across many formats.
  • The persistence of defects one step beyond the boundary suggests similar phenomena may appear in formats further outside it.
  • The Koszul cohomology connection could let homological algebra classify the defective cases completely.

Load-bearing premise

The general critical space is well-defined and the span of singular vector tuples can be compared to it via a cohomology map whose kernel dimension directly gives the codimension difference.

What would settle it

A concrete example of a tensor space one beyond the boundary format in which the codimension of the general span of singular vector tuples fails to equal the kernel dimension of the stated cohomology map.

read the original abstract

In this paper, we study tensor spaces beyond the boundary format and analyze whether the general critical space coincides with the general span of singular vector tuples. For all tensor spaces exceeding the boundary format by one in an arbitrary number of factors, we relate the codimension of this span within the critical space to the dimension of the kernel of a map in cohomology. Furthermore, we exhibit an infinite family of order-three tensors with a defective behavior: the general span of singular vector tuples achieves the maximum possible codimension rather than the expected minimum. Finally, we conjecture a classification of the behavior of critical spaces in this regime and draw a connection to Koszul cohomology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies tensor spaces beyond the boundary format and examines whether the general critical space coincides with the general span of singular vector tuples. For formats exceeding the boundary by one with arbitrary numbers of factors, it relates the codimension of the span inside the critical space to the dimension of the kernel of a cohomology map. It constructs an explicit infinite family of order-three tensors where the general span achieves maximal codimension (defective behavior), conjectures a classification of critical-space behavior in this regime, and connects the results to Koszul cohomology.

Significance. If the central codimension relation holds and the order-three family is rigorously verified, the work would clarify defective phenomena for singular vector tuples outside the boundary format and strengthen links between tensor geometry and cohomology. The explicit infinite family for order three provides concrete evidence of maximal codimension, which is a positive feature; however, the general claim for arbitrary factors rests on an unverified vanishing assumption in the cohomology sequence.

major comments (2)
  1. [§3] §3 (derivation of the main relation): the equality between codim(span of singular vector tuples inside critical space) and dim(ker of the cohomology map) is obtained from a long exact sequence whose higher terms (H^i for i>1) are asserted to vanish. No independent vanishing theorem or spectral-sequence argument is supplied that covers arbitrary numbers of factors; this vanishing is load-bearing for the claimed relation when the number of factors exceeds three, as noted in the abstract.
  2. [§5] §5 (infinite family of order-three tensors): while explicit tensors are exhibited and the maximal-codimension claim is stated, the proof that the defect holds for the general member of the family (rather than special members) requires additional detail on the generality of the construction and the computation of the actual codimension; this is central to the defective-behavior assertion.
minor comments (2)
  1. Notation for the critical space and the span of singular vector tuples should be introduced with a clear definition before the main theorems; the current usage in the abstract and early sections assumes familiarity that may not be universal.
  2. The conjecture on classification of critical spaces (final section) would benefit from a precise statement of the conjectured dichotomy or trichotomy, including any expected codimension formulas.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address the two major points below and indicate the revisions that will be made to clarify the arguments.

read point-by-point responses

  1. Referee: [§3] §3 (derivation of the main relation): the equality between codim(span of singular vector tuples inside critical space) and dim(ker of the cohomology map) is obtained from a long exact sequence whose higher terms (H^i for i>1) are asserted to vanish. No independent vanishing theorem or spectral-sequence argument is supplied that covers arbitrary numbers of factors; this vanishing is load-bearing for the claimed relation when the number of factors exceeds three, as noted in the abstract.

    Authors: We agree that the vanishing of H^i for i>1 in the long exact sequence is essential to the codimension relation for an arbitrary number of factors. The manuscript derives the relation from the sequence but does not supply a separate vanishing argument that applies uniformly when the number of factors exceeds three. In the revision we will add a short subsection to §3 that justifies the vanishing via the structure of the Koszul complex on the product of projective spaces and a standard spectral-sequence comparison with the known vanishing range for the relevant line bundles; this will make the argument self-contained for the general case. revision: yes

  2. Referee: [§5] §5 (infinite family of order-three tensors): while explicit tensors are exhibited and the maximal-codimension claim is stated, the proof that the defect holds for the general member of the family (rather than special members) requires additional detail on the generality of the construction and the computation of the actual codimension; this is central to the defective-behavior assertion.

    Authors: We accept that the current write-up of the family in §5 states the maximal-codimension result but leaves the generality argument somewhat implicit. The construction is parametrized so that the general member lies in a Zariski-open subset of the parameter space; we will expand the proof to identify this open set explicitly, show that the codimension computation (via direct evaluation of the kernel dimension) is constant on that open set, and verify that the special loci where the codimension could drop have positive codimension. These details will be inserted into the revised §5. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central relation derived from exact sequence rather than by definition or self-fit.

full rationale

The paper derives the codimension relation from a long exact sequence in cohomology for formats exceeding the boundary by one, relating it to the kernel dimension of a map. This is not a self-definitional reduction or a fitted input renamed as prediction, as the equality follows from the sequence structure and any required vanishing statements are mathematical claims about the Koszul complex rather than tautological inputs. The exhibited infinite family of order-three examples provides explicit defective cases independent of the general argument. No load-bearing self-citation chain or ansatz smuggling is present; the derivation remains self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard facts from algebraic geometry of tensors and Koszul cohomology; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption The general critical space for tensors beyond boundary format is a well-defined variety whose dimension can be compared to the span of singular vector tuples via a cohomology map.
    Invoked when relating codimension to kernel dimension.

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