Dirac spectral flow and Floer theory of hyperbolic three-manifolds

Francesco Lin; Michael Lipnowski

REVIEW 4 major objections 5 minor 20 references

Dirac spectral flow and Floer theory of hyperbolic three-manifolds

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Dirac crossing signs fix Floer homology of hyperbolic 3-manifolds

desk verdict Genuinely new geometric computation of piercing sequences for hyperbolic 3-manifolds, with the Floer dictionary deferred to an unpublished companion; conditional but worth refereeing carefully. read the letter →

arxiv 2506.07238 v1 pith:PBXRFEH3 submitted 2025-06-08 math.GT math.DGmath.NTmath.SP

Francesco Lin , Michael Lipnowski This is my paper

classification math.GTmath.DGmath.NTmath.SP MSC 57R5853C2758J2858J5011F72

keywords monopoleFloerhomologyhyperbolic3-manifoldsDiracoperatorspectralflowSelbergtraceformulapiercingsequencespin-cstructuresSeiberg-Wittenequations

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Monopole Floer homology is a powerful algebraic invariant of three-manifolds that is usually hard to compute. This paper shows that for certain hyperbolic three-manifolds with first Betti number one, the invariant is determined by a simple geometric datum: the sequence of signs with which small Dirac eigenvalues cross zero as one travels around the circle of flat spin-c connections. Under the hypotheses that the manifold is spectrally large and the crossings are transverse, this 'piercing sequence' completely describes the Floer chain complex. The authors verify these hypotheses from hyperbolic geometry alone — geodesic lengths, spin-c holonomies, and a norm bound on a closed one-form — and thereby produce the first Floer chain complexes with non-trivial homology on hyperbolic three-manifolds. For the manifolds worked out, the twisted Floer homology is 0, R, or R⊕R according as the piercing sequence is empty, (+,-), or (+,-,+,-).

What carries the argument

The main technical tool is the odd Selberg trace formula for the family of Dirac operators D_{Bτ}, together with its derivative in τ. The derivative formula (14) expresses a weighted sum of eigenvalue derivatives as a geometric sum over closed geodesics involving spin-c holonomies; Lemma 2.1 bounds each individual derivative by 2π C_Y, where C_Y is the minimal $C^{0}$ norm of a closed one-form representing the generator of $H^{1}$(Y;Z). Combining this with Fourier-analytic test functions (the sixth or eighth convolution power of the interval indicator) gives certified intervals containing a unique small eigenvalue and proves its derivative never vanishes, establishing transversality. The piercing sequence — the ordered list of ± signs of these transverse crossings around the circle — is the object that carries the Floer homology.

What would settle it

Compute the Floer homology of a listed manifold by an independent method, such as a surgery exact triangle or a Heegaard Floer package, and compare with the predicted 0 or R (or R⊕R for #10867); a mismatch would indicate that either the piercing sequence computation or the dictionary from [Lin24b] is wrong. Alternatively, numerically evaluate the derivative of the small eigenvalue at the certified crossing point of, say, the self-conjugate spin-c structure on #357 and search for a sign change of s'_0(τ) or a zero, which would contradict transversality.

Watch

Extended reading notes

Core claim

The central claim is that the Floer homology of a spectrally large b1=1 hyperbolic three-manifold with a torsion spin-c structure is governed by the one-parameter family of Dirac operators associated to flat spin-c connections, not by curvature or irreducible solutions. Generically the kernel locus K consists of finitely many points with signs; the sequence of signs around the circle is the piercing sequence. The authors prove, for a dozen census manifolds, that K is transverse and exactly two or four points, compute the piercing sequence using a derivative of the odd Selberg trace formula, and then apply the dictionary that turns the piercing sequence into explicit chain complexes. The resulting Floer homology groups are the first examples with non-trivial homology for hyperbolic manifolds, including a rank-two example on #10867.

Load-bearing premise

The load-bearing premise is the unpublished result [Lin24b], which asserts that under Conditions 1 and 2 one can perturb the Seiberg-Witten equations so that the Floer chain complex is explicitly the one described from the piercing sequence; the present paper cites this dictionary without proving it.

Editorial extensions

If this is right

For every manifold in Table 1, the twisted Floer homology HM_*(Y,s; Γ_η) is now known to be 0 or R, and for #10867 it is R⊕R; these are the first non-trivial computations on hyperbolic three-manifolds.
The method converts the Main Question into finitely many numerical checks on the length spectrum for any spectrally large b1=1 manifold, with the practical caveat that the required cutoff R grows quickly.
When the piercing sequence has four crossings, the reduced Floer homology HM_*(Y,s) itself is non-trivial (Z in degree -1 in the example), so the phenomenon is not an artifact of local coefficients.
The dictionary in Section 1 shows that the piercing sequence determines not only homology groups but the whole chain complex and the action of γ ∈ H1(Y;Z)/tors.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the dictionary from [Lin24b] is correct, the same recipe would determine Floer homology for every spectrally large fibered census manifold whose length spectrum is computed to the needed cutoff, making the Main Question effectively decidable within that class.
The lower bound C_Y ≥ π·Th(Y)/vol(Y) (Remark A.1) suggests that manifolds with large Thurston norm relative to volume cannot be spectrally large, which would bound the reach of the method.
One could test the transversality assumption numerically for the non-self-conjugate spin-c structures on #10867, predicting trivial homology; a failure there would pinpoint the limit of the approach.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know about arXiv:2506.07238. The geometric work is real: the authors compute, from lengths and holonomies of geodesics, the full piercing sequence of the Dirac family for a dozen hyperbolic 3-manifolds, and those computations are described with explicit numerical certificates. The Floer conclusion drawn from those piercing sequences is conditional: the dictionary from piercing sequence to Floer chain complex is cited to an unpublished preprint [Lin24b] and is not proved in this paper.

What is actually new is the derivative of the odd Selberg trace formula and the certification that the small Dirac eigenvalue crosses zero exactly once, transversely, in each interval. The authors make real efforts to keep the bounds explicit: Weyl-law estimates, explicit C_Y bounds on closed 1-forms, interval arithmetic for eigenvalue locations, and tail bounds. The piercing-sequence computations for the census manifolds are described with enough care that a determined reader could reimplement them. That part is solid and honestly reported.

The soft spots are proportional, and the paper mostly names them itself. The load-bearing one is the dependency on [Lin24b]. Examples 1.1 to 1.3 simply state the outputs of the dictionary; if the perturbation result fails for non-self-conjugate spin-c structures or for the four-crossing case in #10867, Theorems 1 and 2 would not follow, even though the piercing-sequence computations stand. This is not a hidden flaw—the citation is explicit—but it is a substantial condition. Second, no code or data artifacts are included, so the numerical certificates cannot be independently checked as shipped; for computations that took days of CPU time, that matters. The tail estimate in Remark 4.2 is sketched rather than fully written out, which is a minor but real gap.

Overall, this is careful, substantial work with a clearly stated external dependency. The right readers are people working at the spectral geometry / Floer theory interface. It deserves a serious referee, and I would send it out, asking the referee to treat [Lin24b] as part of the proof: either the authors should post it or include its core statements in an appendix. I would also press them to release the computational pipeline.

Referee Report

4 major / 5 minor

Summary. The paper studies the correspondence between hyperbolic geometry and monopole Floer homology for closed oriented hyperbolic three-manifolds Y with b1=1 and torsion spin-c structures s. It proposes to determine HM_*(Y,s;Γ_η) and the corresponding chain complexes from the 'piercing sequence' of transverse eigenvalue crossings of the family of Dirac operators D_{Bτ} around the circle of flat connections. The main theorems assert explicit Floer chain complexes for a list of census manifolds (Theorem 1) and for the manifold #10867 (Theorem 2), with Floer homology equal to 0, R, or R^2 depending on whether the piercing sequence is empty, (+,−), or (+,−,+,−). The proof combines the spectral-largeness condition λ_1^*>2, interval certificates for small eigenvalues, an odd Selberg trace formula and its derivative to prove transversality, and an a priori derivative bound via an explicit constant C_Y. The Floer-theoretic dictionary that converts a piercing sequence into a chain complex is cited to the unpublished preprint [Lin24b]; the geometric computations of the piercing sequences themselves are the main new contribution presented here.

Significance. If the dictionary from [Lin24b] is correct, the paper achieves a notable first: explicit monopole Floer chain complexes with nontrivial homology for hyperbolic three-manifolds, computed from geometric data such as geodesic lengths, spin-c holonomies, and the norm bound C_Y. The technique of differentiating the family of Selberg trace formulas in the spin-c parameter in order to prove transversality of Dirac eigenvalue crossings is new and likely to be useful beyond these examples. The explicit interval certificates, the use of spectral density bounds, and the careful treatment of C_Y are strengths, and the results give concrete, falsifiable predictions linking spectral geometry to Floer homology. However, the central Floer conclusions are not self-contained in this manuscript, and the proof of Theorem 1 as written covers only a subset of the listed manifolds in detail. The significance is therefore conditional, although the geometric part stands on its own.

major comments (4)

[Section 1 (Examples 1.1–1.3; Remark 1.2)] Theorems 1 and 2 rest on the assertion, cited to the unpublished preprint [Lin24b], that under Conditions 1 and 2 the piercing sequence completely determines the Floer chain complex. This is the load-bearing step of the paper, but no statement of this dictionary is proved here, and Examples 1.1–1.3 simply quote the output. In particular, the claimed vanishing of the irreducible contribution B_u^s in Remark 1.2, the tower grading shifts, and the treatment of non-self-conjugate spin-c structures are all asserted rather than demonstrated. If [Lin24b] is not available or contains a subtlety (for example irreducible solutions created by the perturbation, or different grading shifts for non-self-conjugate structures), the Floer conclusions of Theorems 1 and 2 do not follow even though the piercing-sequence computations are correct. The manuscript should either prove the relevant cases of the dictionary in an appendix or state Theorems 1 and 2 as conditional on a precise theorem in a publicly available companion paper.
[Theorem 1 / Section 5] Theorem 1 asserts results for all twelve manifolds in Table 1, but the body of the paper gives detailed piercing-sequence computations only for the self-conjugate and a conjugate pair on #357, for #3250, and for #10867 (the latter being Theorem 2). No piercing-sequence data, interval certificates, or even summary tables are provided for #356, #381, #734, #735, #790, #882, #1155, #1280, #1284, or #3673. The sentence in the introduction that the strategy 'can be applied' is not a proof of a numerical theorem. The authors should include the computed piercing sequences for all spin-c structures appearing in Theorem 1, or at least the certification data for each manifold, or restrict the theorem to the cases actually worked out.
[Section 5.2] The proof of transversality for #3250 is incomplete as written. After stating that the strategy proves s_0'(τ)≠0 on [0.4467,0.4480], the text says 'focusing for simplicity on the midpoint τ0' and verifies the inequality (23) only at τ0=0.44735. A bound at a single point does not rule out a second crossing or a tangency elsewhere in the interval. To establish Condition 2 for this spin-c structure, the authors need uniform bounds over the whole interval (as they do for #357 and #10867) or an explicit argument that the midpoint check suffices.
[Section 4.2 / Remark 4.2] The tail estimates used in proving the inequalities behind (23) are not fully specified. Remark 4.2 states that the infinite sum can be bounded explicitly by evaluating the first forty terms and bounding the remainder using S_Y, but the text does not give the actual tail bound, the values of S_Y, or the required numerical data. Since these bounds are the certificates for transversality in Sections 4 and 5, the reader cannot verify the numerical proofs. Please provide the precise inequalities and the certified numerical values, or a reproducible script, used to obtain numbers such as 0.0470 in Section 4.3 and 0.7099 in Section 5.2.

minor comments (5)

[Equation (14)] Please check the displayed sign in Equation (14): with the stated convention φτ(γ)=e^{2πi τ [γ]}φ0(γ), differentiating cos(φτ(γ)) introduces a minus sign; the later absolute-value estimates in the paper are insensitive to this, but the displayed identity should be correct as written.
[Example 1.3] The notation T_+^{⊕2}⟨−1⟩ in Example 1.3 would benefit from a short explanation of how the grading shift acts on a direct sum of towers and how the listed chain complex is obtained from the two maxima/two minima Morse picture.
[Section 3.2] The conclusion that the interval contains exactly one small eigenvalue is justified by 'simply looking at plots' and continuity; please replace this with the interval-based Step 1 and Step 2 certificates at several points or with a uniform argument over the interval.
[References] Since the main theorems depend on [Lin24b], that reference should include a public identifier or version once available; as it stands, the reader cannot access the companion result.
[Throughout] Minor typographical issues: 'spin c' is sometimes written with a visible space, and there are a few missing superscripts in phrases such as 'spin^c structure' in the abstract and introduction.

Circularity Check

1 steps flagged · score 4.0 of 10

The Floer chain-complex dictionary is imported from the unpublished same-author preprint [Lin24b], making Theorems 1-2 conditional, while the geometric piercing-sequence computations are independent.

self citation load bearing [Section 1, after Definition 1.1; quoted output used in Examples 1.1-1.3 and Theorems 1-2.]
"In [Lin24b] it is shown how to suitably perturb the Seiberg-Witten equations to achieve transversality while not introducing irreducible solutions when Conditions 1 and 2 hold. ... A consequence of this is that one can compute the Floer chain complex in a completely explicit fashion; the key phenomenon that gives rise to interesting Floer homology is that the presence of points of K causes the various towers associated to reducible critical points to shift in grading because of spectral flow."

Theorems 1 and 2 assert explicit Floer chain complexes (0, R, or R⊕R) determined by the piercing sequence, but the implication 'piercing sequence ⇒ Floer chain complex' is not proved here. Section 1 cites the unpublished, same-author preprint [Lin24b] for it, and Examples 1.1-1.3 simply state the outputs. Sections 3-5 only certify Conditions 1 and 2 and compute piercing sequences from hyperbolic data; they do not derive the Floer dictionary. Thus the central Floer conclusions are forced by a self-citation chain rather than by an argument in this paper. If [Lin24b]'s perturbation or grading-shift claims fail, Theorems 1-2 would not follow even though the geometric computations are correct.

full rationale

The genuinely new and self-contained part of the paper is the geometric side: using the published LL21/LL22b trace formulas, SnapPy data, and the Appendix A bound on C_Y to certify spectral largeness, locate intervals with small Dirac eigenvalues, and prove single transverse crossings, thereby computing piercing sequences for the listed census manifolds. That part is not circular, and it does not depend on the Floer conclusions. The step connecting piercing sequences to Floer chain complexes, however, is delegated to [Lin24b], an unpublished preprint by the first author; Examples 1.1-1.3 are stated consequences, not proved here. This is a load-bearing self-citation, so the Floer outputs of Theorems 1 and 2 are conditional on [Lin24b]. I score this 4 rather than higher because the cited dictionary has explicit hypotheses (Conditions 1 and 2) independent of the target Floer groups and the paper verifies those hypotheses on the geometric side; there is no fitted parameter renamed as a prediction and no definitional identity forcing the result. The earlier same-author trace formulas are published mathematical tools used as inputs, not circular support.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a chain of previously established results, mostly by the same authors. The main new computational inputs are numerical cutoffs and test functions. No new physical entities are introduced; the piercing sequence is a defined combinatorial object rather than an invented entity.

free parameters (4)

Length spectrum cutoff R = 7, 7.5, 8.5 depending on example
The certification uses only closed geodesics up to this length; larger R would give sharper bounds but costs more computation. The choice is a proof parameter, not fitted to the Floer output.
Test functions H6, K7, K8 with scaling = H6 = (1/2 1_{[-1,1]})^{*6}; K7 = x(1/2 1_{[-1,1]})^{*7}; K8 scaled by 1.0625
Chosen by hand for positivity and decay properties of their Fourier transforms; the proofs of eigenvalue counting and crossing existence depend on these choices.
Spectral density interval centers nu = 2.5, 2.7, 1.9541, 2.2941, 2.4741, 2.7041, varying by example
Used in the explicit Weyl law bound (21) to count eigenvalues in intervals; values are selected based on the location of the next eigenvalue around the crossing.
C_Y optimization seeds = not specified
The upper bounds on C_Y in Table 2 are obtained by a random descent over vertex values; different seeds give different bounds. The reported bounds are optimization outputs, not theoretical values.

assumptions (6)

domain assumption Selberg trace formulas for Dirac operators on hyperbolic 3-manifolds with flat spin-c connections, including the differentiated odd formula (14).
Proved in [LL21] and extended in Section 2; the paper relies on these identities to convert spectral information into geometric side sums. Published separately by the same authors.
domain assumption Spectral largeness lambda_1^* > 2 rules out irreducible Seiberg-Witten solutions.
From [LL22b]; used to restrict Floer theory to reducible solutions and the torus of flat connections.
ad hoc to paper The dictionary from piercing sequence to monopole Floer chain complexes, under Conditions 1 and 2, holds.
Cited to the unpublished preprint [Lin24b] by one of the authors; this is the load-bearing bridge from spectral data to Floer homology and is not reproven here.
domain assumption Census manifolds under consideration are branched double covers of links in S^3, and the covering involution acts as -1 on H^1(Y;Z) = Z.
Used in Section 5.1 to assert symmetry of spectral intervals for non-self-conjugate spin-c structures, reducing the number of intervals to check. Asserted without proof.
standard math Rellich's theorem and Kato perturbation theory apply to the family D_{B_tau} = D_{B_0} + 2 pi i tau rho(alpha).
Used to justify eigenvalue differentiability and the bound |s_j'(tau)| <= 2 pi C_Y in Lemma 2.1.
ad hoc to paper The tail of the infinite spectral sums in the derivative trace formula can be bounded explicitly as stated in Remark 4.2.
The bound is asserted but the full calculation is not displayed, leaving a small gap in the rigor of the transversality certification.

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Cite this review

Pith. "Pith review of Dirac spectral flow and Floer theory of hyperbolic three-manifolds." pith.science (2026). https://pith.science/paper/PBXRFEH3

@misc{pith2026250607238,
  author       = {Pith},
  title        = {Pith review of: Dirac spectral flow and Floer theory of hyperbolic three-manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PBXRFEH3}},
  note         = {Machine review of arXiv:2506.07238}
}

abstract

We study the interplay between hyperbolic geometry and monopole Floer homology for a closed oriented three-manifold $Y$ with $b_1=1$ equipped with a torsion spin$^c$ structure $\mathfrak{s}$. We show that, under favorable circumstances, one can completely describe the Floer theory of $(Y,\mathfrak{s})$ purely in terms of geometric data such as the lengths and holonomies of closed geodesics. In particular, we perform the first computations of monopole Floer chain complexes with non-trivial homology for hyperbolic three-manifolds. The examples we consider admit no irreducible solutions to the Seiberg-Witten equations, and the non-triviality of the Floer homology groups is a consequence of the geometry of the $1$-parameter family of Dirac operators associated to flat spin$^c$ connections. The main technical challenge is to understand explicitly how the Dirac eigenvalues with small absolute value cross the value zero in this family; we tackle this using Fourier analytic tools via the corresponding $1$-parameter family of odd Selberg trace formulas and its derivative.

Figures

Figures reproduced from arXiv: 2506.07238 by the authors.

**Figure 1.** On the left, the small eigenvalue (i.e. the eigenvalue with smallest absolute value) crosses zero with sign `, while on the right it crosses it three times with signs `, ´, `. From hyperbolic geometry to spectral theory. Most of the work in the present paper is devoted to developing techniques to determine the geometry of the locus K and in particular the piercing sequence of pY, sq. The key tool we will use is a ve… view at source ↗

**Figure 2.** The output of the techniques developed in [LL22b] for Census 357 at cutoff R “ 6.5: if λ is an eigenvalue of the Hodge Laplacian on coexact 1-forms, then the blue plot is ě 1 at ? λ. In particular, λ ˚ 1 ą 2. The determination of the intervals in T for which the Dirac operator DB admits a small eigenvalue involves two steps: first, determining intervals potentially containing small eigenvalues, and then certifying … view at source ↗

**Figure 3.** The plot of J0pτ q for the unique self-conjugate spinc structure on Census 357, computed at cutoff R “ 7. where vs is an n-column vector of functions of the spectral variable s explicitly determined in terms of F; in our specific implementation, these are elementary functions involving sines and cosines. With the goal of letting τ vary in r0, 1s, one can then compute for any even compactly supported test function H … view at source ↗

Figures from the paper (13 more)

**Figure 4.** Figure 4: The plot of Jspτ0q at τ0 “ 0.15465 for the unique self-conjugate spinc structure on Census 357, computed at cutoff R “ 7. 3.2. Certifying a unique small eigenvalue. We now show that for our examples, in each of the intervals in T determined above the Dirac operator adm…

**Figure 5.** Figure 5: The plot of the first derivative of sinc7 . This function is useful to detect the sign of ‘medium’ sized eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

**Figure 6.** Figure 6: In green, the plot of ΓK7 pτ q for unique self-conjugate spinc structure on Census #357. The vertical red lines are really bars denoting the interval r0.1537, 0.1556s and its mirror. ΓK7 pτ q goes from very positive to very negative is quite suggestive of the fact tha…

**Figure 7.** Figure 7: The plots of Jsp0.1q and Jsp0.2q for the unique self-conjugate spinc structure on Census 357, computed at cutoff R “ 7. fit in a family that for τ in the interval r0.1537, 0.1556s is the peak near 0 as in [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

**Figure 8.** Figure 8: The plot of the second derivative of sinc7 for t ě 0. This function is very small for t Á 2.5. At τ “ 0.2, we have that the next eigenvalue has absolute value at least 2.4 and obtain correspondingly that ˇ ˇ ˇ ˇ ˇ ÿ i‰0 psinc7 q 1 psip0.2qq ˇ ˇ ˇ ˇ ˇ ď 0.0104, and conc…

**Figure 9.** Figure 9: The plots of J0pτ q and ΓK7 pτ q for the pair of self conjugate spinc structures under consideration on #357, computed at cutoff R “ 7 [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]

**Figure 10.** Figure 10: The plot of Jspτ1q for the pair of self conjugate spinc structures under consideration on #357, computed at cutoff R “ 7. The plot of Jspτ1q, where τ1 is the midpoint of the first interval, is shown in [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]

**Figure 11.** Figure 11: For one of the (self-conjugate) spinc structures on #3250, the plot of J0pτ q computed at cutoff R “ 7.5, and the plot of ΓK7 pτ q. there is exactly one small eigenvalue as in Section 3; the plot of the family of odd trace formula (again computed with K7) is given in …

**Figure 12.** Figure 12: The plots of Jsp0.4q and Jsp0.5q for the spinc structure on #3250 under consideration, computed at cutoff R “ 7.5 [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]

**Figure 13.** Figure 13: The plot of Jsp0.44735q for the spinc structure on #3250 under consideration, computed at cutoff R “ 7.5. methods of Subsection 4.1 (using the fact that sinc6 pxq ě 0.9977 for |x| ď 0.04715, the latter being half the length of the interval) prove that there is at most…

**Figure 14.** Figure 14: The plots of J0pτ q and ΓKpτ q for one of the self-conjugate spinc structures on #10867, computed using R “ 8.5. where the first summand, the maximum of psinc7 q 2 in r1.5856, 1.6799s, comes from the second smallest eigenvalue, while the second one comes from the tail…

**Figure 15.** Figure 15: The plots of Jspτ q for the spinc structure under consideration on # 10867 at the midpoints of the intervals I1 and I2, computed at cutoff R “ 8.5. significantly more challenging because of the smaller spectral gap and the higher eigenvalue density, and we will work i…

**Figure 16.** Figure 16: A schematic picture of the map P ´1 R . so that the pullback of the hyperbolic metric to the horizontal disk is given by P ˚ Rpghypq “ a 2 ` 1 p1 ` a 2 ´ t 2q 2 dt2 ` t 2 1 ` a 2 ´ t 2 gSn´1 . From this we see that the differential of the inverse P ´1 R multiplies len…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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