Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks

Aran Nayebi; Dan Yamins

REVIEW 4 major objections 5 minor 19 references

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read For minimal solutions to sufficiently hard tasks, weak affine equivalence between adjacent layers forces unit-level axis equivalence, and terminal weak equivalence alone zippers alignment upstream through every layer.

desk verdict The conditional theorems are real and worth refereeing, but the abstract's 'convergent evolution is inevitable' is not licensed by the paper's own assumptions—terminal weak equivalence is assumed, not proved. read the letter →

arxiv 2607.08561 v2 pith:E5FDQDDQ submitted 2026-07-09 cs.LG q-bio.NC

Dan Yamins , Aran Nayebi This is my paper · ORCID

classification cs.LGq-bio.NC MSC 68T07

keywords contravarianceweakalignmentstrongprivilegedaxeszipperingtheoremtaskhardnessused-axisbudgetrepresentationalgeometry

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

The paper tries to prove a precise version of the contravariance principle: the harder a task is, the fewer distinct internal solutions a neural network can have, so the more similar any two solutions must be. Its central results are two theorem families. The weak–strong equivalence theorems show that if two minimal networks are related by an affine map at two adjacent layers, and the task actually uses most of the nonlinear gates at the lower layer, then the individual coordinate axes of those layers must also match—unit by unit, up to permutation and rescaling. The zippering theorems show that if the two networks are only known to be affine-equivalent at the final output layer, that equivalence propagates backward through all layers, making the earlier layers strongly aligned as well. The paper is explicit that the real-world force of these results depends on two unproven premises: real tasks must force most gates to be used, and training must stay away from the measure-zero exceptional parameter sets where null networks and degeneracies live; both are flagged as open questions in the paper itself.

What carries the argument

The load-bearing mechanism is the task-visible nonlinear signature of a gate. For ReLU, this is a kink trace: the zero set of a unit that the task patch crosses and that the next layer reads out. For softplus, it is the curvature ridge of the smooth activation, identified via pole and monodromy data. The central identity is the commutation lemma: if weak alignment holds at adjacent layers, the affine comparison map commutes with the nonlinear block. Since an injective affine map cannot create or destroy a used nonlinear signature, the singular and curvature sets of the two networks must coincide, forcing coordinate-wise matching. Zippering then upgrades this from adjacent layers to the whole

What would settle it

Retrain many minimal networks on a task whose layer-l gate budget m_l(ε) is known, from bottleneck or pruning experiments, to be a small fraction of d_l. If near-perfect weak alignment at layers l and l+1 is observed but AxisAlign is far above m_l/d_l, or if a task with very low m_l/d_l nevertheless produces near-perfect AxisAlign, the theorem's core inequality fails.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that for ReLU and softplus networks that are minimal in the sense that every counted unit leaves a private task-visible nonlinear signature, weak alignment and strong alignment coincide when the task-hardness ratio m_l(ε)/d_l is near one. Theorem 1 proves that every used A-side axis at layer l is matched to a distinct B-side axis with (E_l z)_{π(j)} = α_j z_j, giving the lower bound AxisAlign ≥ m_l/d_l. An asymptotic version gives the same conclusion when weak-alignment errors shrink. The zippering theorems prove that terminal weak equivalence at some layer s generically forces strong alignment at every upstream layer, using backward induction and one-

Load-bearing premise

The theory's real-world force rests on task hardness—m_l(ε)/d_l near 1, meaning real tasks force most gates to be genuinely used—and on training dynamics not concentrating on the measure-zero exceptional parameter sets; the paper explicitly says both remain unproven (Section 8 and Appendix Remark 5).

Editorial extensions

If this is right

For sufficiently hard tasks, the choice of comparison metric stops mattering: task performance, linear or affine similarity, and unit-level privileged-axis alignment all coincide.
Privileged axes, including Gabor-like tuning structures, arise from end-to-end task optimization rather than architectural accident.
Convergent evolution between independently trained networks—and between artificial networks and brains—is mathematically forced for minimal solutions to hard tasks, conditional on minimality and non-degeneracy.
Raw RSA and CKA comparisons can be dominated by task-irrelevant symmetries; canonicalized versions of these metrics expose a core task-relevant geometry that obeys weak–strong equivalence.
In transformers, privileged axes live in the MLP and attention branches, so unit-level comparisons should look there rather than in the residual stream.

Reading between the lines

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

Editorial inference: the paper makes m_l/d_l directly measurable—layerwise bottlenecking, pruning, or width-restricted retraining estimate the gate budget—so the theory's central inequality can be tested empirically without exact weak alignment.
Editorial inference: if task hardness drives axis emergence, then easy and hard versions of the same task should show a discontinuous jump in AxisAlign when m_l/d_l crosses a threshold, and alignment differences between seeds should be most pronounced near task-relevant layers.
Editorial inference: the RSA/CKA analysis implies that raw similarity benchmarks mixing high- and low-performance models may be measuring task-irrelevant symmetry differences; canonicalized comparisons should correlate more strongly with behavior only in the high-performance regime the paper studies.
Editorial inference: the soft zippering bound predicts a downstream-to-upstream error gradient—terminal alignment error should grow with distance upstream, so deep networks should show progressively worse axis alignment in earlier layers.

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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

Colleague,

You should know this before anything else: the paper contains a real conditional result, but its abstract sells you a stronger theorem than it proves. The zippering theorems (exact, asymptotic, soft) assume terminal weak equivalence between the two minimal networks; nothing in the paper shows that two independently trained minimizers of the same hard task actually have affinely related terminal activations. That premise is the load-bearing part of the NeuroAI convergence story, and the paper admits as much in Section 4 when it says the weak-strong mechanism does not explain why re-convergence happens in the first place. So read the formal content as: if terminal weak equivalence holds, and if minimality and the genericity conditions hold, then alignment zippers upstream into strong axis alignment.

The genuinely new pieces are worth your time. The weak–strong equivalence theorem with the used-axis budget m_l(ε) gives a concrete lower bound AxisAlign ≥ m_l/d_l, and the soft/asymptotic versions are competently derived. The ReLU kink-trace and softplus Fourier/ridge arguments in the appendix look internally coherent on inspection. The zippering proof structure—null pairs, MLR/MFR regularity, generic null-net theorem—is a serious piece of work, and they are honest about the exceptional sets and the unproven optimizer-nonconcentration assumption (Remark 5). That is more than most theory papers in this area offer.

The soft spots, in order of importance. First, the abstract's "convergent evolution is probably inevitable" is not licensed. The paper cites Nayebi 2026 for an invertible map between minimal agents, but invertible is not affine, and the zippering machinery specifically needs affine terminal equivalence. Second, the empirical premises—m_l/d_l near 1, minimality, and training not concentrating on exceptional sets—are all deferred. The paper says so plainly, but that means the headline NeuroAI conclusions currently rest on unproven empirical priors. Third, the soft bounds contain uncomputed constants (κ, L_l, c, ε_r), so the quantitative predictions are not yet testable. The RSA/CKA and transformer sections are substantial but deferred to the appendix and I could not fully verify them line-by-line.

Who is this for? Anyone working on representational alignment, model-brain comparison, or the theory of why task-optimized networks converge. It deserves a serious referee, but the referee should push hard on the terminal-equivalence premise and on toning down the abstract. The conditional theorems can stand on their own; the over-claim only weakens the paper.

My recommendation: send it to review, with the expectation of major revision on framing.

Referee Report

4 major / 5 minor

Summary. The paper formalizes a 'contravariance principle' for deep neural networks. For ReLU and softplus networks, it proves two families of theorems. First, exact and asymptotic weak–strong equivalence: if two networks are weakly (affinely) equivalent at two adjacent layers and the layer's axes are task-used, then the used axes are strongly (coordinatewise) aligned, with an alignment lower bound m_ℓ(ε)/d_ℓ (Theorems 1–2 and Appendix Theorems 1–3). Second, zippering: if minimal, regular, generic networks are weakly equivalent at a terminal layer, then alignment propagates upstream to all compared layers, with exact, asymptotic, and soft quantitative versions (Theorems 3–4 and Appendix Theorems 6–8). The appendix develops minimality, MLR/MFR regularity, genericity of the exceptional sets, and extensions to transformers, RSA, and CKA. The paper concludes that for minimal solutions to sufficiently hard tasks, weak and strong alignment coincide and convergent representations are mathematically inevitable.

Significance. If the conditional theorems are correct, this is a substantial contribution to the theory of representational alignment. The paper gives precise, internally coherent mathematical mechanisms showing that affine equivalence across a nonlinear layer forces axis-level equivalence, and that terminal weak equivalence can, under explicit minimality and regularity assumptions, force upstream strong alignment. The theorems are genuinely conditional and the proof structure is detailed, with separate treatments for the nonsmooth ReLU and smooth softplus cases. The explicit use of a hardness budget m_ℓ(ε)/d_ℓ, rather than a fitted constant, is a strength, and the paper candidly identifies the main unverified premises: measuring the used-axis budget for real tasks and proving that trained networks do not concentrate on exceptional sets. However, the central conceptual claim in the abstract goes beyond what the theorems prove, because terminal weak equivalence is assumed rather than derived.

major comments (4)

[Abstract; Theorem 3; Appendix Theorem 6] The abstract states that 'for any two minimal DNN solutions to a sufficiently hard task' weak terminal equivalence zippers to upstream strong alignment. But Theorem 3 and Appendix Theorem 6 assume terminal weak equivalence as a hypothesis; no theorem establishes that two independently obtained minimal solvers of the same hard task are terminally weakly equivalent. The used-axis budget m_ℓ(ε) in Definition 4 constrains the number of used axes, not the solution set up to affine equivalence. Thus the paper proves a conditional implication, not the unconditional convergence claim advertised. This is the load-bearing gap identified in Section 4's open question ('what leads to the repeated re-convergence in the first place?'). Please reframe the main claims as conditional on terminal weak equivalence, or provide a separate theorem deriving terminal weak equivalence from task hardness and minim
[Section 8; Appendix Remark 5] The NeuroAI conclusion that convergent evolution is 'mathematically inevitable' rests on two unproven empirical premises: (1) that m_ℓ(ε)/d_ℓ is near 1 for real tasks, and (2) that training dynamics do not concentrate on the measure-zero exceptional sets. The paper acknowledges both, and the Section 8 discussion correctly says that estimating m_ℓ/d_ℓ 'will require empirical probes.' These are not internal inconsistencies, but they are load-bearing for the paper's central claim. The theorems remain interesting as conditional results; please state the main conclusions with these caveats in the abstract and introduction rather than only in the discussion.
[Appendix Theorem 3 / Eq. (2)] The asymptotic weak–strong theorem involves a constant κ_K(θ) defined as an infimum over a compact family K, and the bound's usefulness depends on κ_K being non-vanishing. The proof shows positivity by contradiction using the exact theorem. This is valid only if the limiting 'zero-defect' member satisfies the exact theorem's usedness assumptions. Since the sequence is allowed to lie in a compact family of tuples, the limit may have trace families that degenerate as n→∞. The paper does not state an explicit uniform usedness margin for Theorem 2. Please clarify what quantitative version of usedness is preserved in the compact limit, or add the needed nondegeneracy condition to the statement of Appendix Theorem 3.
[Section 5; Appendix A.12] The RSA section claims that raw RSA is 'in some sense a stricter metric than linear similarity' and then shows that raw RSA is sensitive to task-irrelevant symmetries. This is not contradictory, but the distinction is central and should be stated more carefully: the strong-alignment results do not control multiplicities, and the paper's own example shows that high AxisAlign and high weak alignment do not imply high raw RSA. The discussion should make clear that the 'bring RSA into the fold' program is conditional on a canonicalization procedure, not a consequence of Theorems 1–4 alone.

minor comments (5)

[Section 3 vs Appendix Section A.2.2] There is a numbering collision: the main text's Theorem 2 is the asymptotic weak–strong theorem, while Appendix A.2.2's Theorem 2 is the exact softplus weak–strong theorem. This will confuse readers; please renumber the appendix results or use a prefix.
[Appendix Remark 5] Typo: 'assupmtion' should be 'assumption'. Also, the remark should be numbered as a limitation in the main text, not only in the appendix, since it directly affects the interpretation of Theorem 3.
[Definition 4 / Eq. (1)] The 'used-axis budget' m_ℓ(ε) is defined via a set F_{ℓ,≤k} of networks with at most k used axes. This is clear, but the phrase 'with at most k used layer-ℓ axes' relies on the informal usedness definition for softplus (Definition 3). Since the formal softplus usedness is only given in Appendix Definition 6, please cross-reference the appendix at the point of Definition 4 in the main text.
[Section 5, 'RSA is in some sense a stricter metric'] This sentence is immediately qualified in the next paragraph, but the initial phrasing is misleading. Consider rewording to avoid the impression that raw RSA is always a stronger similarity notion than linear regression.
[Appendix A.6.4, Lemma 12] The lemma states that character resonances are 'a countable union of proper affine subsets.' Since the parameter space is finite-dimensional, countable union of measure-zero sets is acceptable, but please state explicitly that the union is locally finite or that the parameter space is σ-compact, so that the measure-zero conclusion follows without additional set-theoretic assumptions.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: core theorems are derived in-paper from stated premises; the main gap is an unproven terminal-weak-equivalence antecedent and a non-load-bearing self-citation, not a construction-level reduction.

full rationale

The central derivations are not circular. Theorem 1 assumes exact weak equivalence at adjacent layers plus usedness, and proves, by commuting the diagram through the nonlinearity and matching kink traces / curvature ridges, that every used A-side axis is matched to a B-side axis; the lower bound AxisAlign >= m_l(eps)/d_l is then read off from the independently defined used-axis budget m_l(eps), not from fitted alignment data. Theorem 3 / Appendix Theorem 6 similarly proves a genuine conditional: if terminal weak equivalence holds, then minimality plus MLR/MFR plus genericity lets a backward induction identify scalar arguments layer by layer. That implication is contentful and self-contained. The abstract's stronger, unconditional phrasing — 'for any two minimal DNN solutions to a sufficiently hard task ... alignment zippers up' — is not licensed by Theorem 3, which begins by supposing terminal weak equivalence; the paper never proves that two independently obtained minimal solvers of the same hard task are terminally weakly equivalent. That is a scope gap or unproven premise, not an equation-level circularity. The self-citation to Nayebi [2026, Corollary 5] asserts only an arbitrary invertible isomorphism, not affine weak equivalence, and is not used inside the zippering proof, so it is not load-bearing for the theorem chain; the empirical zippering figure is also cited only as motivation. The paper itself flags the open problem of what causes repeated re-convergence and the unproven nonconcentration of training on exceptional sets (Section 4 and Appendix Remark 5). These are honest limitations rather than circular steps. Hence the derivation is largely self-contained, and the modest score reflects the unsupported advertisement and self-citation presence, not a reduction of the conclusions to the assumptions.

Assumptions & free parameters 5 free parameters · 7 assumptions · 3 invented entities

The paper's contribution is conditional: the theorems rest on new definitional constructs (usedness, minimality, MLR/MFR) plus standard analytic/algebraic background, and the real-world conclusions additionally rest on three unproven empirical premises (m_ℓ/d_ℓ≈1, minimality of real systems, learning dynamics avoiding exceptional sets) that the authors explicitly defer to future work. The soft bounds contain uncomputed constants (κ_K, L_ℓ, c, ε_r), so the quantitative content is currently schematic.

free parameters (5)

κ_K(θ) (asymptotic weak–strong constant)
Appears in eq. (2) and Appendix Theorem 3; an existence constant over the compact family K and threshold θ. The paper states constants of this type are not explicitly calculated and computing them is future work (Section 8).
L_ℓ (soft-zippering Lipschitz constants)
Appears in eq. (3) and Appendix Theorem 8 via Lemma 11; per-layer amplification constants governing upstream error propagation. Not computed; 'we do not explicitly calculate them here' (Section 8).
c (quantitative minimality margin)
Quantitative minimality margins in Appendix Definitions 18–19; enter the soft-zippering bound (1−θ)²c²d_r; values unspecified.
ε_r (Lemma 11 local error thresholds)
Per-layer validity radii in the quantitative one-step error inversion; needed to state the regime where soft zippering applies.
m_ℓ(ε)/d_ℓ (used-axis budget ratio)
Task-hardness measure (Definition 4/5). Not fitted, but its presumed magnitude is load-bearing: Theorem 1's conclusion is vacuous unless the ratio is near 1. Section 8 asserts only an 'empirical prior' that mℓ/dℓ 'is often not tiny' and defers measurement.

assumptions (7)

domain assumption The compared networks are affine-ReLU or affine-softplus feedforward networks of transparent layer blocks
Main theorems are stated for these architectures; extension to CNNs, RNNs, transformers, GELU/ELU is argued in the appendix (§A.8, §A.11) rather than proven for the main results.
domain assumption Task hardness in the sense of a high used-axis budget: m_ℓ(ε)/d_ℓ close to 1
Theorem 1 gives AxisAlign ≥ m_ℓ/d_ℓ; the real-world 'hard task' interpretation requires most axes to be used. Section 8 calls this an empirical prior and says estimation requires future layerwise bottlenecking/pruning/ablation probes.
ad hoc to paper Networks are minimal (Softplus pole-separated minimality, Def 10; ReLU EFF-minimality, Def 12)
Minimality is the paper's own strengthened usedness condition. Its real-world status in trained DNNs and brains is stated as an open question (Section 8), supported only by a speculative 'energetic efficiency' argument for brains.
ad hoc to paper Regularity: MLR (Def 16) for softplus cross-matches and coherent MFR (Def 17) for ReLU
MLR/MFR are the paper's own nondegeneracy conditions. Genericity proofs (§A.6) show they fail only on measure-zero sets, but only conditional on rank conditions holding somewhere on each stratum.
domain assumption Learning dynamics do not concentrate on the measure-zero exceptional sets
Appendix Remark 5: the paper explicitly states it 'implicitly assumes' trained-net parameter distributions avoid the exceptional sets, and that proving this is an important topic for future work. This is the key bridge from generic-parameter claims to real optimized networks.
domain assumption For real brains, minimality holds
Section 8: 'We speculate that minimality in real brains is an energetic efficiency that is evolutionarily advantageous' — explicit speculation.
standard math Standard analytic/algebraic background: ridge-function identifiability via line-restriction and Fourier decay; UFD property of Laurent polynomial rings; algebraic-geometry dimension counting
Invoked in §A.2.2 (Fourier ridge isolation) and §A.4.3 (Laurent binomials, primitivity, divisor-faithfulness) with citations to Pinkus 2015; Griffiths–Harris 1978; Cox–Little–Schenck 2011.

invented entities (3)

used-axis budget m_ℓ(ε)
purpose: Formal measure of task hardness; proposed 'gate budget' scale for task constraint difficulty
No empirical measurement for real tasks yet; the paper itself calls for layerwise bottlenecking/pruning/ablation or width-restricted retraining to estimate it (Section 8).
exceptional sets of parameters (null networks; MLR/MFR failures; Jacobian-singular loci)
purpose: Loci where zippering/identifiability may fail; declared measure-zero to make genericity arguments work
Genericity is proven with respect to uniform parameter measure; whether training distributions avoid these sets is unproven and flagged in Remark 5.
formalized contravariance principle
purpose: Explains solution-set dispersion decreasing with task hardness, hence convergent evolution between networks and brains
Original informal concept from Cao-Yamins [2024] (co-authored). The formalization is this paper's contribution, but the principle has no independent falsifiable handle beyond the theorems that embed it.

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Pith. "Pith review of Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks." pith.science (2026). https://pith.science/paper/E5FDQDDQ

@misc{pith2026260708561,
  author       = {Pith},
  title        = {Pith review of: Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E5FDQDDQ}},
  note         = {Machine review of arXiv:2607.08561}
}

read the original abstract

A series of results from the NeuroAI over the past fifteen years have raised core questions both about how to compare Deep Neural Network (DNN) models to the brain, and about how much convergent evolution to expect between artificial networks and real brain networks. Here, we show that for any two minimal DNN solutions to a sufficiently hard task: (i) "weak" alignment of network representations based on affine mappings guarantees "strong" alignment of privileged axes, and (ii) alignment "zippers" up the network hierarchy, causing the emergence of privileged axes from end-to-end task optimization. These results formalize the notion of contravariance from Cao and Yamins [2024], and illustrate important consequences for the theory of NeuroAI: with sufficiently strong tasks, choice of metric for inter-network comparison is not all that sensitive, and that convergent evolution is probably inevitable.

Figures

Figures reproduced from arXiv: 2607.08561 by the authors.

**Figure 1.** Why do these core NeuroAI results arise? What can we infer from them? (A) In a variety of domains, from visual cortex responses to static images and movies, auditory cortex responses to sounds and spoken words, to language-area responses to text, there turns out to be a correlation across a wide range of DNN models, between a DNN model’s performance on AI tasks and fit of that model to neural data, using linear mapp… view at source ↗

**Figure 2.** The contravariance principle. The work of Cao and Yamins [2024] notes that, given a fixed architecture size, the dispersion of a set of solutions to an easy task (upper left of figure) is larger than the dispersion of a set of solutions to an hard task (lower right). In other words, the solution set dispersion is contravariant in (inversely related to) the constraint strength. This idea provides an informal understa… view at source ↗

**Figure 3.** Two notions of similarity. A: Weak alignment. B: Strong axis alignment. originally described by Cao and Yamins [2024]. The contravariance concept described in that work notes that the harder the task that a network is constrained to do, the fewer the solutions there will be to it, and therefore the more likely it will be that any two solutions to it will be similar to each other in important respects ( [PITH_FULL_I… view at source ↗

Figures from the paper (27 more)

**Figure 3.** Figure 3: Two notions of similarity. A: Weak alignment. B: Strong axis alignment. Definition 1 (Weak alignment). A layer-ℓ comparison map from A to B is an injective affine map Eℓ(z) = Tℓz + aℓ , Tℓ : R d A ℓ → R d B ℓ . The networks are weakly equivalent at layer ℓ on the task …

**Figure 4.** Figure 4: Used gates and kink preservation. A: Only a crossed, downstream-used zero set contributes a task-visible nonlinear signature. B: A used ReLU boundary creates a slope change, and an injective affine map cannot erase that change. ℓ + 1, they will be different enough that…

**Figure 5.** Figure 5: Asymptotic weak–strong alignment along optimization. In a common use case, two [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: (A) Same-layer predictivity between different initialization seeds of an AlexNet-like model seeds. Post the nonlinearity at each layer (dark green bars), the gap to weak equivalence is quite large; but at each pre-nonlinearity layer (light green bars), the gap to weak …

**Figure 7.** Figure 7: Softplus usedness as a curvature-based analogue of ReLU usedness. Panel A shows the [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]

**Figure 7.** Figure 7: Transformer weak–strong structure in the updated residual-peeling formulation. Residual [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: Softplus and ReLU use different nonlinear signatures to identify hidden axes. For [PITH_FULL_IMAGE:figures/full_fig_p036_8.png]

**Figure 8.** Figure 8: Softplus usedness as a curvature-based analogue of ReLU usedness. Panel A shows the [PITH_FULL_IMAGE:figures/full_fig_p030_8.png]

**Figure 9.** Figure 9: Softplus pole signatures used in the zippering proof. Panel A shows the complex pole [PITH_FULL_IMAGE:figures/full_fig_p037_9.png]

**Figure 9.** Figure 9: Softplus and ReLU use different nonlinear signatures to identify hidden axes. For [PITH_FULL_IMAGE:figures/full_fig_p039_9.png]

**Figure 10.** Figure 10: EFF-minimality for ReLU one-step expansions. Panel A illustrates task-visible nonlinear [PITH_FULL_IMAGE:figures/full_fig_p039_10.png]

**Figure 10.** Figure 10: Softplus pole signatures used in the zippering proof. Panel A shows the complex pole [PITH_FULL_IMAGE:figures/full_fig_p040_10.png]

**Figure 11.** Figure 11: Depth-warped singleton alignment. A layer [PITH_FULL_IMAGE:figures/full_fig_p070_11.png]

**Figure 11.** Figure 11: EFF-minimality for ReLU one-step expansions. Panel A illustrates task-visible nonlinear [PITH_FULL_IMAGE:figures/full_fig_p042_11.png]

**Figure 12.** Figure 12: Transformer weak–strong structure. Residual states may zipper weakly across depth [PITH_FULL_IMAGE:figures/full_fig_p075_12.png]

**Figure 12.** Figure 12: Depth-warped singleton alignment. A layer [PITH_FULL_IMAGE:figures/full_fig_p074_12.png]

**Figure 13.** Figure 13: A shifted identity block. The correct place to test the construction is the first backward [PITH_FULL_IMAGE:figures/full_fig_p086_13.png]

**Figure 13.** Figure 13: A shifted identity block. The correct place to test the construction is the first backward [PITH_FULL_IMAGE:figures/full_fig_p100_13.png]

**Figure 14.** Figure 14: A postactivation-shift symmetry. The hidden activated coordinates move in a downstream [PITH_FULL_IMAGE:figures/full_fig_p087_14.png]

**Figure 14.** Figure 14: A postactivation-shift symmetry. The hidden activated coordinates move in a downstream [PITH_FULL_IMAGE:figures/full_fig_p102_14.png]

**Figure 15.** Figure 15: The regular discrete Softplus resonance. The identity is exact, but it requires a special [PITH_FULL_IMAGE:figures/full_fig_p089_15.png]

**Figure 15.** Figure 15: The regular discrete Softplus resonance. The identity is exact, but it requires a special [PITH_FULL_IMAGE:figures/full_fig_p103_15.png]

**Figure 16.** Figure 16: Why one exposed facet is not enough for ReLU. MFR records enough labelled facets [PITH_FULL_IMAGE:figures/full_fig_p090_16.png]

**Figure 16.** Figure 16: Why one exposed facet is not enough for ReLU. MFR records enough labelled facets [PITH_FULL_IMAGE:figures/full_fig_p104_16.png]

**Figure 17.** Figure 17: Split-facet matching. Coherent MFR requires all marked facets of one source argument [PITH_FULL_IMAGE:figures/full_fig_p090_17.png]

**Figure 17.** Figure 17: Split-facet matching. Coherent MFR requires all marked facets of one source argument [PITH_FULL_IMAGE:figures/full_fig_p105_17.png]

**Figure 18.** Figure 18: Schematic model-by-model scatter plots. (A) The canonical ridge–RSA relation is one-sided: sufficiently high ridge similarity forces high canonical RSA, so the comparison models lie below an increasing frontier. (B) Raw symmetries or unequal spatial sampling can rewei…

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Works this paper leans on

19 extracted references · 1 linked inside Pith

[1]

Ifv j = 0, thej-th term may be deleted
[2]

Ifg j≡γonV, thej-th term may be deleted and the affine part replaced by a(z)↦−→a(z) +vjσ(γ)
[3]

usedness

doi: 10.18653/v1/2020.blackboxnlp-1.4. 22 Jason Phang, Haokun Liu, and Samuel R. Bowman. Fine-tuned transformers show clusters of similar representations across layers. InProceedings of the Fourth BlackboxNLP Workshop on An- alyzing and Interpreting Neural Networks for NLP, pages 529–538. Association for Computational Linguistics, 2021. doi: 10.18653/v1/2...

arXiv 2020
[4]

Sincegj is affine, the residual−vigj can be absorbed into the affine parta

If gi =−gj on V, then the two terms may be replaced by one Softplus term plus an affine residual: viσ(gi) +vjσ(gj) = (vi +vj)σ(gj)−v igj. Sincegj is affine, the residual−vigj can be absorbed into the affine parta. Ifvi +vj = 0, no nonlinear Softplus term remains from this pair
[5]

generically

More generally, if for a nonempty setS ∑ j∈S vjgj is constant onC, then simultaneously replacinggj by−gj for allj∈S changes the expansion only by a bias term. Consequently, modulo affine residuals, every finite Softplus expansion can be reduced to one in which all remaining nonlinear terms have nonzero coefficient, nonconstant argument, and no two remaini...

2015
[6]

Ifv i +vj = 0, both nonlinear terms disappear

Ifg i =g j onV, then the two terms may be replaced by one term: viσ(gi) +vjσ(gj) = (vi +vj)σ(gj). Ifv i +vj = 0, both nonlinear terms disappear
[9]

Minimality is an intrinsic condition on each expansion: every counted unit leaves a “private” task-visible nonlinear signature

First strengthen the notion of usedness from single axes/gates tominimalityof entire layer-wise expansions (§A.4.2). Minimality is an intrinsic condition on each expansion: every counted unit leaves a “private” task-visible nonlinear signature. From equality of two one-step expansions, those private signatures one side of the quality must be reproduced on...
[10]

We then seek to understand what happens when two minimal layers collide. We show that the property ofidentifiability– that is, the recovery of equal internal constituent operations – can be ensured by a set of very simple conditions onregularity(§A.4.3 and §A.4.4)

Show all 19 references

[11]

obvious self-collisions

When then show how identifiability, based on minimality and regularity, leads to exact zippering (§A.4) – assuming perfect terminal equivalence of last-layer representations. Looking ahead, in the next sections that follow, we show that (when one more additional condition on t...

1978
[12]

Intrinsic minimality fails only on analytic or algebraic degeneracy sets, conditional on task- usedness and task exposure
[13]

The regularity mechanisms fail only on resonance, incidence, or rank-defect sets, provided the required condition holds at one point on each relevant stratum
[14]

verification tool

Jacobian regularity, after fixing the standard hidden-unit order and ReLU normalization, fails on the zero set of an analytic Gram determinant, provided the full-rank condition holds at least at one point. 62 The first two give generic identifiability. The third controls the g...

2012
[15]

First, intrinsic minimality removes representations that are already redundant inside one expansion
[16]

Second, MLR or MFR rules out a nontrivial equal-function pair at a point where the relevant nonlinear signatures are identifiable
[17]

all ends well

Third, by not actually ruling them out but showing that they are rare. The generic null-net theorem doesnotclaim that every exceptional identity is impossible: it allows resonant nontrivial collisions on lower-dimensional subsets, but prevents them from filling a full- dimensi...

2008
[18]

Combining with the previous bound yields ∥P[u]−P[v]∥ F≤ 2 √ 2δ√ 1−δ 2 = 2 √ 2(1−θ)√ 1−(1−θ) 2

A direct calculation gives ∥P[u]−P[v]∥ 2 F = 2(1−c 2)≤4(1−c). Combining with the previous bound yields ∥P[u]−P[v]∥ F≤ 2 √ 2δ√ 1−δ 2 = 2 √ 2(1−θ)√ 1−(1−θ) 2. The sign ofαis irrelevant becauseP[αv] =P[v]. Choose the injective matching used in Appendix Theorem 3. There arem−B θ g...
[19]

Orthogonal projection is nonexpansive, so the same bound holds after applyingΠ H

Averaging the projector differences and using the triangle inequality gives ∥Rax(A)−R ax(B)∥F≤ 1 m [ (m−B θ) 2 √ 2(1−θ)√ 1−(1−θ) 2 +Bθ √ 2 ] ≤ 2 √ 2(1−θ)√ 1−(1−θ) 2 + √ 2Bθ m ≤ 2 √ 2(1−θ)√ 1−(1−θ) 2 + √ 2 m e2 weak κ(θ)2, where the final line uses(82). Orthogonal projection is...

2024
[2020]

Amil Merchant, Elahe Rahimtoroghi, Ellie Pavlick, and Ian Tenney

doi: 10.1162/tacl_a_00349. Amil Merchant, Elahe Rahimtoroghi, Ellie Pavlick, and Ian Tenney. What happens to BERT embeddings during fine-tuning? InProceedings of the Third BlackboxNLP Workshop on Analyzing and Interpreting Neural Networks for NLP, pages 33–44. Association for ...

doi
[2024]

21 Ansh Soni, Sudhanshu Srivastava, Marvin Maechler, Konrad Kording, and Meenakshi Khosla

Version 2 posted September 14, 2025. 21 Ansh Soni, Sudhanshu Srivastava, Marvin Maechler, Konrad Kording, and Meenakshi Khosla. Conclusions drawn from neural network to brain alignment depend strongly on the chosen similarity measure.bioRxiv, 2024. doi: 10.1101/2024.08.07.6070...

2025

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Ifv j = 0, thej-th term may be deleted

[2] [2]

Ifg j≡γonV, thej-th term may be deleted and the affine part replaced by a(z)↦−→a(z) +vjσ(γ)

[3] [3]

usedness

doi: 10.18653/v1/2020.blackboxnlp-1.4. 22 Jason Phang, Haokun Liu, and Samuel R. Bowman. Fine-tuned transformers show clusters of similar representations across layers. InProceedings of the Fourth BlackboxNLP Workshop on An- alyzing and Interpreting Neural Networks for NLP, pages 529–538. Association for Computational Linguistics, 2021. doi: 10.18653/v1/2...

arXiv 2020

[4] [4]

Sincegj is affine, the residual−vigj can be absorbed into the affine parta

If gi =−gj on V, then the two terms may be replaced by one Softplus term plus an affine residual: viσ(gi) +vjσ(gj) = (vi +vj)σ(gj)−v igj. Sincegj is affine, the residual−vigj can be absorbed into the affine parta. Ifvi +vj = 0, no nonlinear Softplus term remains from this pair

[5] [5]

generically

More generally, if for a nonempty setS ∑ j∈S vjgj is constant onC, then simultaneously replacinggj by−gj for allj∈S changes the expansion only by a bias term. Consequently, modulo affine residuals, every finite Softplus expansion can be reduced to one in which all remaining nonlinear terms have nonzero coefficient, nonconstant argument, and no two remaini...

2015

[6] [6]

Ifv i +vj = 0, both nonlinear terms disappear

Ifg i =g j onV, then the two terms may be replaced by one term: viσ(gi) +vjσ(gj) = (vi +vj)σ(gj). Ifv i +vj = 0, both nonlinear terms disappear

[7] [9]

Minimality is an intrinsic condition on each expansion: every counted unit leaves a “private” task-visible nonlinear signature

First strengthen the notion of usedness from single axes/gates tominimalityof entire layer-wise expansions (§A.4.2). Minimality is an intrinsic condition on each expansion: every counted unit leaves a “private” task-visible nonlinear signature. From equality of two one-step expansions, those private signatures one side of the quality must be reproduced on...

[8] [10]

We then seek to understand what happens when two minimal layers collide. We show that the property ofidentifiability– that is, the recovery of equal internal constituent operations – can be ensured by a set of very simple conditions onregularity(§A.4.3 and §A.4.4)