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arxiv: 2603.03337 · v2 · pith:SS5RO4MLnew · submitted 2026-02-12 · 🧬 q-bio.NC

Does the motor cortex draw on a wire plane?

Patrick Iglesias-Zemmour This is my paper

Pith reviewed 2026-05-16 05:01 UTC · model grok-4.3

classification 🧬 q-bio.NC
keywords motor controltwo-thirds power lawequi-affine metricdiffeologywire diffeologymotor primitivescovariant tensors
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The pith

Equipping the plane with the wire diffeology turns the equi-affine metric into a covariant 3-tensor under the full diffeomorphism group.

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

The paper shows that the two-thirds power law of motor control, which corresponds to constant equi-affine speed, can be realized as a fully covariant geometric invariant. In ordinary differential geometry the equi-affine metric fails to transform as a tensor because it involves acceleration; covariance is restored only by restricting symmetries or adding an affine connection. The proposed alternative equips the Euclidean plane with the wire diffeology, whose smooth structure is generated solely by all smooth curves. This choice matches the biological premise that the motor cortex composes movements from curve primitives rather than two-dimensional patches, so the metric becomes a true 3-tensor without any further structure or symmetry limitation.

Core claim

In the wire diffeology on the Euclidean plane, where smoothness is defined by requiring that every curve pulls back to a smooth function, the equi-affine metric becomes a covariant 3-tensor with respect to every diffeomorphism. No restriction to the affine group and no auxiliary connection are required. The two-thirds power law therefore appears directly as a diffeomorphism-invariant quantity, consistent with the view that motor actions are assembled from a repertoire of elementary curve segments.

What carries the argument

The wire diffeology, the smooth structure on the plane generated by taking all smooth curves as the primitive objects that determine which maps are smooth.

Load-bearing premise

The motor cortex traces curves rather than two-dimensional patches, allowing curves to be taken as the primitive objects that generate the smooth structure.

What would settle it

A diffeomorphism in the wire sense under which the equi-affine metric fails to transform covariantly, or motor data showing that the two-thirds power law breaks for non-affine changes of coordinates in the workspace.

read the original abstract

The two-thirds power law of human motor control ($v \propto \kappa^{-1/3}$) is geometrically equivalent to constant equi-affine speed. In classical differential geometry, however, the equi-affine metric is not a tensor: it depends on acceleration, which does not transform covariantly under arbitrary coordinate changes. To recover tensorial behavior, one must either restrict the symmetry group to the affine group or introduce an affine connection -- sacrificing full diffeomorphism covariance. This article proposes a different geometric setting. We equip the Euclidean plane with the "wire diffeology', the smooth structure generated by all smooth curves. In this diffeological space, the equi-affine metric becomes a true covariant $3$-tensor under the **full** diffeomorphism group -- no restriction of symmetries, no additional structure required. The construction is motivated by a simple fact: the motor cortex traces curves, not two-dimensional patches. Accordingly, curves are taken as primitive, echoing the motor control literature in which movements are built from a repertoire of elementary building blocks -- motor primitives. The wire plane offers a geometric formalization of this idea in which the two-thirds power law emerges as a fully covariant invariant.

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 / 1 minor

Summary. The paper claims that equipping the Euclidean plane with the 'wire diffeology' (the smooth structure generated by taking all smooth curves as plots) renders the equi-affine metric a true covariant 3-tensor under the full diffeomorphism group of the plane. This is motivated by the observation that the motor cortex traces curves rather than two-dimensional patches, allowing the two-thirds power law (v ∝ κ^{-1/3}, equivalent to constant equi-affine speed) to emerge as a fully diffeomorphism-invariant quantity without restricting the symmetry group to affine transformations or introducing an affine connection.

Significance. If the central construction were valid, the result would be significant for geometric motor control: it would supply a setting in which the equi-affine quantity is covariant under arbitrary diffeomorphisms while taking curves as the primitive objects, thereby formalizing the notion of motor primitives without ad-hoc symmetry restrictions and potentially unifying the power-law observation with a broader class of diffeological invariants.

major comments (2)
  1. [Abstract] Abstract and the central proposal: the claim that the equi-affine metric (built from det(ẋ, ẍ) or its 1/3 power) becomes a covariant 3-tensor under the full diffeomorphism group does not follow from the wire-diffeology construction. The wire diffeology is defined as the smallest diffeology containing all C^∞ curves as plots; every C^∞ diffeomorphism of R² therefore remains smooth and invertible in this structure. The standard chain-rule transformation ẍ_new = Dφ · ẍ + D²φ(ẋ, ẋ) therefore still applies, introducing Hessian terms that prevent det(ẋ, ẍ) from transforming as a tensor density.
  2. [The construction] The construction section: no explicit computation is supplied showing how the diffeological plots cancel the second-derivative contributions under a general (non-affine) coordinate change. The invariance is asserted rather than derived; the allowed diffeomorphism group remains identical to the standard Diff(R²), so the non-tensorial terms survive.
minor comments (1)
  1. The manuscript would benefit from an explicit transformation formula (under a sample non-affine diffeomorphism) that demonstrates the claimed cancellation of Hessian terms.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading and for identifying the absence of an explicit derivation in our manuscript. We respond point by point below and will revise the paper to incorporate the requested calculations while adjusting the scope of the central claim.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the central proposal: the claim that the equi-affine metric (built from det(ẋ, ẍ) or its 1/3 power) becomes a covariant 3-tensor under the full diffeomorphism group does not follow from the wire-diffeology construction. The wire diffeology is defined as the smallest diffeology containing all C^∞ curves as plots; every C^∞ diffeomorphism of R² therefore remains smooth and invertible in this structure. The standard chain-rule transformation ẍ_new = Dφ · ẍ + D²φ(ẋ, ẋ) therefore still applies, introducing Hessian terms that prevent det(ẋ, ẍ) from transforming as a tensor density.

    Authors: We agree that the manuscript asserts covariance without supplying the explicit transformation law. The wire diffeology does not alter the local coordinate expression of diffeomorphisms of R², so the standard chain rule and Hessian contributions remain. In the revised version we will add a dedicated subsection deriving the pullback of det(ẋ, ẍ) along a general diffeomorphism φ. This derivation will show that the extra term det(Dφ · ẋ, D²φ(ẋ, ẋ)) does not vanish for non-affine φ. Accordingly we will revise the abstract and introduction to state that the wire diffeology renders the equi-affine speed reparametrization-invariant along curve plots and covariant under the affine subgroup, while acknowledging that full diffeomorphism covariance still requires an affine connection. revision: yes

  2. Referee: [The construction] The construction section: no explicit computation is supplied showing how the diffeological plots cancel the second-derivative contributions under a general (non-affine) coordinate change. The invariance is asserted rather than derived; the allowed diffeomorphism group remains identical to the standard Diff(R²), so the non-tensorial terms survive.

    Authors: The referee correctly notes the lack of explicit computation. We will insert the missing derivation in the construction section, computing the transformed equi-affine quantity both in coordinates and intrinsically along the generating curve plots. The calculation confirms that the non-tensorial terms persist under arbitrary diffeomorphisms. The revised text will therefore limit the invariance claim to reparametrizations of individual plots and to affine transformations of the plane, while retaining the motivation that the wire diffeology formalizes curve-based motor primitives without introducing extraneous structure beyond the plots themselves. revision: yes

standing simulated objections not resolved
  • A derivation in which the Hessian terms cancel for arbitrary (non-affine) diffeomorphisms, since no such cancellation occurs.

Circularity Check

1 steps flagged

Wire diffeology defined to make equi-affine quantity covariant by construction

specific steps
  1. self definitional [Abstract]
    "We equip the Euclidean plane with the 'wire diffeology', the smooth structure generated by all smooth curves. In this diffeological space, the equi-affine metric becomes a true covariant 3-tensor under the full diffeomorphism group -- no restriction of symmetries, no additional structure required."

    The equi-affine quantity is constructed from det(ẋ, ẍ) (or its 1/3 power). The wire diffeology is the smallest diffeology containing all smooth curves as plots; every C^∞ diffeomorphism of R² therefore remains a diffeomorphism. The chain-rule second-derivative terms therefore survive, yet the paper asserts covariance without exhibiting a cancellation mechanism. The tensoriality is thus asserted as a consequence of the definition rather than independently verified.

full rationale

The paper introduces the wire diffeology precisely so that the equi-affine metric (already known to encode the two-thirds power law) transforms as a tensor under the full diffeomorphism group. The construction takes curves as primitive plots, which by definition includes all C^∞ maps of the plane; no independent calculation is shown that cancels the Hessian terms in the transformation law of det(ẋ, ẍ). The covariance therefore reduces to the choice of diffeology rather than being derived from it.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the definition of the wire diffeology and the domain assumption that curves are the appropriate primitives for motor cortex modeling.

axioms (2)
  • standard math The equi-affine metric is geometrically equivalent to constant equi-affine speed and to the two-thirds power law.
    Stated as a classical fact in the abstract.
  • domain assumption The motor cortex traces curves rather than two-dimensional patches, so curves are the primitive objects.
    Explicitly invoked to motivate the wire diffeology.
invented entities (1)
  • wire diffeology no independent evidence
    purpose: Smooth structure on the plane generated by all smooth curves that renders the equi-affine metric a covariant 3-tensor under full diffeomorphisms.
    New structure introduced in the paper.

pith-pipeline@v0.9.0 · 5503 in / 1287 out tokens · 84542 ms · 2026-05-16T05:01:33.783770+00:00 · methodology

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