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arxiv: 2606.06403 · v1 · pith:SS73A3Q7new · submitted 2026-06-04 · 🧮 math.DG · math-ph· math.MP· math.SP

Second-Jet Equivariant η Separations on Lens Spaces

Sanchita Sharma This is my paper

Pith reviewed 2026-06-27 23:31 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.MPmath.SP
keywords lens spacesequivariant eta invariantDirac operatorspectral geometryspin-Fourier residuesresidual circle actionsecond derivative
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The pith

The residual-circle equivariant η germ separates lens spaces L(25,4) and L(25,9) with normalized second derivative -6080 while their ordinary η invariants agree.

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

Lens spaces serve as test cases in spectral geometry because their spin Dirac eigenspaces have explicit congruence descriptions. The paper computes a residual-circle equivariant η germ by retaining the spin-Fourier character of each eigenspace under the standard torus action rather than collapsing to the scalar η value. For the square family of lens spaces L(ℓ², ℓ-1) and L(ℓ², 2ℓ-1) with odd ℓ at least 5, the ordinary η values match between each pair and the first derivative of the germ vanishes by symmetry. For the concrete pair L(25,4) versus L(25,9) the normalized second derivative equals -6080. This shows that the germ detects a distinction invisible to the ordinary η invariant.

Core claim

For three-dimensional lens spaces with the round metric and standard coordinate-torus action, the residual-circle equivariant η germ is formed by keeping the spin-Fourier character of the Dirac eigenspaces. In the square family L(ℓ²,ℓ-1) and L(ℓ²,2ℓ-1) for odd ℓ≥5 the ordinary η invariants of each pair agree, the first derivative vanishes by symmetry, and the second derivative is nonzero. The normalized value of this second derivative for L(25,4) versus L(25,9) is -6080. The computation uses the spin-Fourier residues directly.

What carries the argument

The residual-circle equivariant η germ, formed by retaining the spin-Fourier character of each Dirac eigenspace and differentiating with respect to the residual circle parameter.

If this is right

  • The ordinary η invariant is insufficient to distinguish the paired lens spaces in the square family.
  • The second derivative of the residual-circle equivariant η germ provides a nonzero separation for every such pair with odd ℓ≥5.
  • The distinction between the spaces is carried entirely by the spin-Fourier residues of the eigenspaces.
  • Equivariant extensions of the η invariant can capture geometric information beyond the scalar case.

Where Pith is reading between the lines

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

  • The same second-jet construction could be applied to other families of lens spaces or different torus actions to test for further separations.
  • If the method extends, higher-order derivatives of equivariant spectral invariants may refine the classification of three-manifolds that share the same ordinary η value.
  • Analogous germs might be defined for other spectral invariants such as the rho invariant on the same spaces.

Load-bearing premise

The spin-Fourier residues of the Dirac eigenspaces can be used directly to compute the second derivative of the residual-circle equivariant η germ without additional correction terms from the coordinate-torus action.

What would settle it

An independent computation of the second derivative for L(25,4) versus L(25,9) that incorporates possible torus-action corrections and produces a value other than -6080.

read the original abstract

Lens spaces are useful test examples in spectral geometry because their spin Dirac eigenspaces admit explicit congruence descriptions. We use these descriptions to study equivariant $\eta$ invariants for three-dimensional lens spaces with the round metric and the standard coordinate-torus action, retaining the spin-Fourier character of each eigenspace rather than only the ordinary scalar $\eta$ value. For the square family $L(\ell^2,\ell-1)$ and $L(\ell^2,2\ell-1)$, with $\ell\geq 5$ odd, we obtain a residual-circle equivariant $\eta$ separation: the ordinary $\eta$ values agree, and the first derivative of the residual $\eta$ germ vanishes by symmetry, but the second derivative is nonzero. For $L(25,4)$ versus $L(25,9)$, the normalized second derivative is $-6080$. Thus, the residual-circle equivariant $\eta$ germ detects a distinction invisible to the ordinary $\eta$ invariant. The calculation uses spin-Fourier residues directly; perturbative Hessian signs serve only as motivation and are not part of the 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

1 major / 1 minor

Summary. The paper claims that for the family of lens spaces L(ℓ², ℓ-1) and L(ℓ², 2ℓ-1) with ℓ odd and ≥5, equipped with the round metric and standard coordinate-torus action, the ordinary η invariants agree and the first derivative of the residual-circle equivariant η germ vanishes by symmetry, while the normalized second derivative is nonzero; explicitly, this second derivative equals -6080 for the pair L(25,4) versus L(25,9). The computation is performed directly from the spin-Fourier residues of the Dirac eigenspaces under the residual-circle action, with perturbative Hessian signs used only for motivation.

Significance. If the residue calculation is shown to capture the full second jet without omitted torus corrections, the result supplies a concrete, parameter-free example of a refined equivariant η invariant that distinguishes lens spaces invisible to the ordinary η invariant. The explicit numerical distinction on these congruence-described spinor spaces is a strength, as is the absence of free parameters or fitting in the derivation.

major comments (1)
  1. [abstract, final paragraph] Abstract, final paragraph: the assertion that the second derivative of the residual η germ is obtained directly from spin-Fourier residues of the Dirac eigenspaces, without additional correction terms from the retained coordinate-torus action, is load-bearing for the claimed numerical distinction of -6080; an explicit argument is required showing that any torus-action contributions to the second jet either vanish or cancel identically between L(25,4) and L(25,9).
minor comments (1)
  1. The phrase 'residual-circle equivariant η germ' is used repeatedly without an explicit local definition or coordinate chart in the opening paragraphs; a brief expansion would improve accessibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to make explicit that torus-action contributions cancel in the second-jet comparison. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [abstract, final paragraph] Abstract, final paragraph: the assertion that the second derivative of the residual η germ is obtained directly from spin-Fourier residues of the Dirac eigenspaces, without additional correction terms from the retained coordinate-torus action, is load-bearing for the claimed numerical distinction of -6080; an explicit argument is required showing that any torus-action contributions to the second jet either vanish or cancel identically between L(25,4) and L(25,9).

    Authors: We agree that an explicit statement is required. Both lens spaces carry the identical round metric and the same standard coordinate-torus action. The residual-circle equivariant η germ is obtained by fixing the torus parameters and differentiating only along the residual S¹. Any second-jet contribution arising from the retained torus action is therefore identical for the two spaces and cancels exactly in the difference. The nonzero normalized value −6080 is produced solely by the differing spin-Fourier residues of the Dirac eigenspaces under this residual action. In the revised manuscript we will insert a short clarifying paragraph immediately after the final sentence of the abstract, stating the cancellation explicitly and referencing the common torus action. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct residue computation is self-contained

full rationale

The paper derives the second-jet distinction for L(25,4) versus L(25,9) by direct evaluation of spin-Fourier residues of the Dirac eigenspaces under the residual-circle action. The abstract states explicitly that 'The calculation uses spin-Fourier residues directly' and that perturbative Hessian signs are 'only as motivation and are not part of the invariant.' No parameter fitting, self-definitional reduction, or load-bearing self-citation chain is present; the reported normalized second derivative of -6080 is obtained from the residue data rather than forced by the inputs or prior author results. The derivation therefore remains independent of its target claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the explicit congruence descriptions of spin Dirac eigenspaces on lens spaces and on the ability to extract second derivatives of the equivariant eta germ from spin-Fourier residues. No free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption Lens spaces with the round metric admit explicit congruence descriptions of their spin Dirac eigenspaces.
    Invoked in the first sentence of the abstract as the starting point for the calculation.
  • domain assumption The residual-circle equivariant η germ is well-defined and its second derivative can be extracted directly from spin-Fourier residues.
    Stated in the final sentence of the abstract as the method used.

pith-pipeline@v0.9.1-grok · 5728 in / 1502 out tokens · 17797 ms · 2026-06-27T23:31:56.417225+00:00 · methodology

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