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Transformation Laws in Neural Representations: Structure, Realisability, and Construction

Yuan Sun

arXiv:2609.18190v1cs.LGcs.NE

TL;DR

It remains unclear whether and where neural features preserve input transformations’ composition as feature-space actions, a question tied to what observers can enact from those features. The paper characterizes when these actions exist and how they fail to be realized, showing that a carrier has a fixed linear action exactly when its retained blocks are closed.

  • Problem

    It remains unclear whether and where neural features preserve input transformations’ composition as feature-space actions, a question tied to what observers can enact from those features.

    Source: A. The operational question · p. 1, A. Actions, encoding, and operable… · p. 5

  • Method

    The paper gives criteria for transformations to descend through encoders and analyzes linear, rectifier, and measured-carrier realisations, including a rank-consistency account of rectifier sharing cost.

    Source: C. Two sources of realisation error · p. 8, §IX · p. 26

  • Results

    A fixed linear action on a carrier exists exactly when its retained blocks are closed; on rectifiers, realization error separates source-regime loss from changed-gate sharing cost.

    Source: C. Two sources of realisation error · p. 8, §IX · p. 26

  • Takeaways & Limitations

    Transformation laws offer a concrete object linking understanding of neural representations to their design.

    Source: §IX · p. 27

    The defect law is not a calibrated predictor for real backbones, and the paper builds no carrier for the dissipative family.

    Source: §IX · p. 27

Abstract

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How neural representations preserve the structure of input changes connects representation analysis with internal intervention. We study operable representational content through compatible actions of reference transformations on neural features. We characterise when a transformation descends through an encoder, and give a linear setting in which the defect is governed by the transformation's demand for discarded information, measured in the metric the representation induces. On a rectifier the failure to realise a transformation has two distinguishable sources --- what the source region has already made unrecoverable, and what it costs to satisfy every region the transformation visits with one operator --- and for a \textit{measured} harmonic carrier the same question has a closed answer: a linear realisation exists exactly when the retained harmonic blocks are invariant under the action. Using colour as the in-depth instance, we find that hue orbits in frozen visual features concentrate 84--88\% of their energy in the first two harmonics with rotation planes shared across shapes, that this organisation is substantially inherited from input and architecture and is reshaped by training and depth, and that the measured structure supports prediction, transport from new starting states, and composition --- with global and local realisations differing sharply in which they achieve. Guided by the measurements, we construct a compact interface whose rotation action is fixed by the structure and never fitted: it reads hue zero-shot at 3.4$^\circ$ median error on unseen shapes. Theory, structural measurement, and construction together establish transformation laws as a concrete object connecting the understanding of neural representations to their design.

I. INTRODUCTION … B. Symmetry-based and equivariant representations

The paper asks when input transformations can be represented as compatible actions on neural features, then connects realisability theory to measurements of colour structure and a structure-fixed interface. It distinguishes measuring an existing network’s transformation laws from building symmetry into an architecture.

A. The operational question

  • The operational question is whether a physical transformation algebra survives encoding as operators on features, preserving its composition law rather than merely supplying isolated feature edits.The paper frames this as whether a feature-space action makes the encoder–transformation diagram commute.Source: A. The operational question · p. 1

B. Three results

  • An induced action exists exactly when a transformation preserves encoder fibres; linear failure comes from mixing discarded information back, while rectifier failure separates source-region loss from cross-region consistency.For measured harmonic carriers, a fixed linear realisation exists exactly when retained blocks are invariant under the action; chain error follows a three-regime law set by the truth action’s Lipschitz constant.Source: B. Three results · p. 1, F. Contributions · p. 3

C. Reading is not transforming

  • Task-sufficient readout does not imply transformation-sufficient features: an encoder may preserve an attribute while discarding information needed after a state change.The paper also warns that geometric proxies can rank an unfaithful operator above a correctly fitted one, so capability claims are graded against ground-truth targets and downstream behaviour.Source: C. Reading is not transforming · p. 2

D. What the measurements find

  • Hue orbits place 84–88% of their energy in the first two harmonics, share rotation planes across shapes (cosine ≥0.96), and predict re-rendered features at 0.17–0.40% relative error.This prediction is more than an order of magnitude tighter than the copy baseline; the organisation is substantially inherited from input and architecture and erodes with depth along the readout–transformation pairing.Source: D. What the measurements find · p. 2, F. Contributions · p. 3

E. From structure to implementation

  • Operability is uneven: local phase-conditioned transport succeeds where global families fail, while depth erosion follows the readout–transformation pairing and measured structure fixes an action that is never fitted.The interface works on unseen shapes and never-fitted parameters; its action is fixed by the measured structure rather than learned as an architectural prior.Source: E. From structure to implementation · p. 2, B. Three results · p. 2, F. Contributions · p. 3

F. Contributions

  • Intervention evaluation grades feature-target error, chain targets, composition defect, and consumer error only after checking reference validity, consumer-class survival, and matched displacement.Baselines are specified by construction: copy for transport, a trivial depth rule for attribution, and random and shuffled operators for capability.Source: F. Contributions · p. 3

II. RELATED WORK; A. Transformations in existing representations

  • Related work separately measures equivariance, builds symmetry into networks, derives reference transformations, or studies causal correspondence; this paper joins existence conditions, realisation form, composition cost, and construction.Existing measurement work scores transformation response but lacks an information condition and a chain score against the truth-level composite.Source: §II · p. 3, A. Transformations in existing… · p. 3

B. Symmetry-based and equivariant representations

  • Equivariant architectures assume a transformation group in advance, whereas this work measures which actions an opaque trained network supports and uses the measured structure as an interface blueprint.An architectural equivariance claim concerns the whole network; a measurement can instead specify an action at a particular site, for a consumer, and with a measured defect.Source: B. Symmetry-based and equivariant… · pp. 3–4

C. Colour and perceptual transformations … B. Compression–transformation compatibility

The paper asks when input transformations remain operable after neural encoding, distinguishing whether an action exists from how it can be realised. It connects algebraic properties of transformations, information merged by the encoder, and measured structure used to construct operators.

C. Colour and perceptual transformations

  • Hue rotation, diffusion, and clipped scaling exemplify a group, semigroup, and monoids with degenerate points, whose composition behaviour—not perceptual uniformity—frames the analysis.The pixel-level hue rotation and diffusion semigroup are treated as exact for the rendered domain; the paper takes no position on perceptual uniformity or spectral rendering.Source: C. Colour and perceptual transformations · p. 4

D. Causal abstraction and latent linearisation

  • Transformation actions extend intervention-level conditions to the whole algebra, quantify composition costs, identify surviving structure through kernels and stabilisers, and address realisation and construction.The paper also proves the natural per-dimension class cannot realise a closed orbit and measures when linear realisation exists.Source: D. Causal abstraction and latent… · p. 5

E. What none of the four lines contains

  • The paper joins an existence condition, a decomposition that prices realisation, measured structure that fixes operator form, and a construction whose action is not fitted.It frames this as the missing connection across equivariance measurement, prescribed-symmetry architectures, colour and scale-space theory, and intervention-level accounts.Source: E. What none of the four lines contains · p. 5

III. TRANSFORMATION LAWS THROUGH NEURAL ENCODING

  • The theory makes operable content depend on explicit structural conditions: how an encoder merges states determines whether an input action can be defined on features at all.Determining which transformations remain distinguishable and the form in which an action can be realised requires further structure.Source: §III · p. 5

A. Actions, encoding, and operable content

  • Feature-level actions and consumer-level actions have independent conditions, while kernels and stabilisers specify which transformations the representation merges or leaves indistinguishable.For example, a second-harmonic-only orbit cannot distinguish hues 180° apart, and every hue rotation stabilises the grey axis.Source: A. Actions, encoding, and operable… · pp. 5–6

B. Compression–transformation compatibility

  • An action on encoded features exists exactly when inputs merged by the encoder remain merged after each transformation, and its composition law then follows automatically.The same descent condition applies to a consumer of features; failure there can prevent an action on visible outputs even when a feature-space operation succeeds.Source: B. Compression–transformation… · p. 6
  • A network-level composition experiment tests the defect of a fitted operator family, not the group axiom, because the induced action inherits composition whenever it exists.Accordingly, the paper grades transformation chains against both the truth-level composite and a direct refit.Source: B. Compression–transformation… · p. 6
  • At q = 0.01, transformed base-equal pairs remain equal at rates of 0.25–0.46 for hue and 0.58–0.82 for heat across four frozen backbones.These rates exceed the chance rate fixed by the tolerance quantile; approximate existence is stronger for heat, which merges states rather than moving them.Source: B. Compression–transformation… · p. 6

C. Two sources of realisation error · D. A quantitative linear model

The sections distinguish irrecoverable information loss from restrictions imposed by a chosen operator family, then derive exact existence and quantitative defect results for linear representations. They localize rectifier error into within-cell and cross-cell costs and state when the linear defect law does—and does not—generalize.

C. Two sources of realisation error

  • A nonlinear transformation can be realized exactly even when every linear map fails, leaving zero irrecoverable error but nonzero family error.A fitted residual measures the sum of these distinct sources, so attributing it between them requires an experiment.Source: C. Two sources of realisation error · p. 6
  • TF = 0.602 and TRMS = 0.603 at the first ResNet-50 site, but Theorem 2 concerns TRMS because it follows from squared-error normal equations.Both normalized scores use 1 for the transformed route and 0 for no better than a no-op; they differ in whether averaging occurs before or after the ratio.Source: C. Two sources of realisation error · p. 6
  • On a rectifier, pooled realization error splits orthogonally into within-cell error and the sharing cost of using one operator across cells.The split is a within-cell versus between-cell decomposition of pooled regression.Source: C. Two sources of realisation error · pp. 7–8
  • Within-cell error can reflect either information already discarded by the source rule or a family too restricted to express the cell relation.It vanishes when the source rule determines the target rule on that cell and the operator family can express it.Source: C. Two sources of realisation error · p. 8
  • A fixed linear map realizes a rectifier transformation on visited cells exactly when the joint linear system is consistent, with its least-squares residual measuring sharing cost.A zero finite-sample residual implies a matrix identity only when cells cover a region with non-empty interior or span it; the criterion concerns visited cells.Source: C. Two sources of realisation error · p. 8

D. A quantitative linear model

  • A linear operator realizes a transformation on retained features exactly when the transformation preserves the encoder’s kernel.The quantitative defect is driven by mixing discarded features back into retained ones: under an isometric retained representation, κ = ||C||_F / ||A - I||_F; the metric W sets its magnitude.Source: D. A quantitative linear model · p. 8
  • +0.40 to +0.65 is the high bias of the law’s prediction across 42 reachable depth-grid sites, where it orders no better than naive baselines.The closed form is exact under isotropy, but real sites are strongly anisotropic; the transferable result is the alignment mechanism, not a calibrated magnitude.Source: D. A quantitative linear model · p. 8

E. The piecewise-linear layer: what a real encoder does … G. Operational measurements

A fixed action on a piecewise-linear encoder must satisfy every realized region-transition rule; operational measurements therefore evaluate consumer-visible changes against explicit baselines and task constraints.

E. The piecewise-linear layer: what a real encoder does

  • A fixed linear action exists on a rectifier exactly when one operator satisfies the joint system of rules across every realized region transition.When the gate pattern is unchanged, only diagonal transitions are realized, and the conditions reduce to ρA_r = A_rT and ρb_r = b_r for each visited region.Source: E. The piecewise-linear layer: what a… · pp. 8–10
  • The measured defect-law deviation peaks at 0.0091 near κ = 1.2, while transfer rises from 0.069 to 0.429 as input anisotropy changes.The defect measurement is compared with the isometric closed form κ/(1 + κ^2); across anisotropy, the lowest-variance half carries 0.43–0.49 of displacement, indicating that transfer tracks split geometry rather than discarded energy.Source: Figure 3 · p. 9

Consequence.

  • Gate changes are neither necessary nor sufficient for failure: a coordinate-swap action realizes ReLU(x), ReLU(−x) under sign reversal, but differing regional rules can defeat one shared operator without gate changes.The obstruction is whether one operator satisfies every rule the transformation visits; enlarging the operator family improves scores by at most 0.087 at ResNet-50 sites and 0.11 at ConvNeXt sites, with a median gap of 0.005.Source: Consequence · p. 10

Consequence (which quantity governs)

  • Consumer-visible gate-change share orders transfer most tightly (ρ = −0.886), followed by gate-change share (ρ = −0.83), while flip count does not order scores (ρ = +0.49, not significant).These measured associations are proxies for the sharing cost, not the cost itself; trained-site deviations are changes of local rule, and depth growth means visiting more distinct rules.Source: Consequence (which quantity governs) · p. 10

F. From existence to form

  • A continuous finite-dimensional circle action decomposes, in an appropriate basis, into invariant and harmonic rotation blocks, but theory does not predict which harmonics dominate measured features.Coordinates may be elliptical, frequencies need not be orthogonal, and multiple copies of one frequency may mix.Source: F. From existence to form · p. 10

G. Operational measurements

  • Operator evaluation uses feature-target, chain-target, composition-defect, and consumer-error measures, with copy, trivial-depth, random, and shuffled baselines appropriate to each task.Sites where the transformation does not move what the consumer reads are excluded rather than scored.Source: G. Operational measurements · pp. 10–11
  • Evaluations first check reference identifiability, consumer-class integrity, and cross-transformation comparability, because each depends on the encoding and task rather than the candidate operator.TF is normalized by the no-op’s own error, so comparisons across transformations must report or match displacement.Source: G. Operational measurements · p. 11

IV. THE SHARED STRUCTURE OF COLOUR ORBITS … F. When the measured carrier supports a linear action: a design condition

The measured colour orbits have a shared low-order harmonic structure, while their usable domain is bounded by degeneracies and by which carrier coordinates are retained. A linear action exists exactly when the retained carrier is invariant under the transformation; measurements also show that closure alone does not ensure one global map works across content.

IV. THE SHARED STRUCTURE OF COLOUR ORBITS

  • Colour variation in frozen features has a shared, low-order organisation that provides a candidate foundation for acting across objects.The study frames this as its central empirical finding.Source: §IV · p. 11

A. Controlled orbit measurements

  • Controlled hue orbits are transformations in rendering parameter space, and the synthetic-to-real gap is measured rather than assumed away.The reported median error is 3.4° on synthetic data versus 8.0° on high-concentration real regions.Source: A. Controlled orbit measurements · p. 11

C. A shared scaffold with content-dependent envelopes

D. Nontriviality and degenerate states

  • A second-harmonic-only orbit cannot distinguish hues 180° apart, and grey and black are stabilised by hue rotation.Discriminative range contracts as saturation approaches the grey or black axis, bounding the operable domain.Source: D. Nontriviality and degenerate states · p. 11

E. Inheritance and reorganisation

  • The orbit structure is present before training: untrained networks preserve its algebraic shape more purely, and affine remixing cannot synthesise it.The paper attributes the structure to input statistics and architecture, with training adding selection and reorganisation rather than creation.Source: E. Inheritance and reorganisation · pp. 11–12

B. Low-order harmonic organisation

  • The first two harmonics carry 84–88% of orbit energy in every shape, with shared low-harmonic planes aligned at 0.97–1.00.The planes are shared across shapes, while alignment falls to 0.54–0.92 for harmonics k ≥7; Figure 4 also shows orbit energy weakening near low saturation while the plane survives.Source: B. Low-order harmonic organisation · p. 11, Figure 4 · p. 12

F. When the measured carrier supports a linear action: a design condition

  • A fixed linear action exists on carrier z = Bc exactly when A∆ker B ⊆ ker B; retained coordinates must form an invariant subspace.For a continuous rotation with one two-dimensional block per frequency, this means retaining or dropping each conjugate pair together; repeated frequencies or a single fixed angle require the invariant-subspace criterion.Source: F. When the measured carrier supports… · p. 13
  • A three-dimensional code missing sin 2h leaves residual 0.96, while adding that coordinate makes the action numerically exact.On measured features, closure is numerically zero for injective synthesis, yet a global map fitted on half the hues has large error on the rest because the carrier-to-feature map depends on content; the criterion specifies which coordinates a reduced carrier may keep, not whether one global map serves all content.Source: F. When the measured carrier supports… · p. 14

G. Beyond hue: the protocol’s reach … D. Composable transport and its price

The section extends the transformation protocol beyond hue and compares global and local ways of realising measured actions. It separates prediction, transport, and composition, and characterises how realised-action error accumulates under repeated transformations.

G. Beyond hue: the protocol’s reach

  • The protocol’s reach includes heat as a second full family and clipped scalings as a measured composition failure; spatial rotation has zero power for an invariant head.Rotation’s zero power is an evaluation-premise failure, not a negative result about representations; measuring it requires consumers that are rotation-sensitive as well as low-power controls.Source: G. Beyond hue: the protocol’s reach · p. 14

A. Prediction, transport, and composition

  • Prediction tests orbit approximation, transport tests increments from starting states never used in fitting, and composition tests whether operator products match the true composite transformation.Each task has separate holdouts and grades a different aspect of the transformation law.Source: A. Prediction, transport, and… · p. 14

B. Global realisations

  • A coordinate-wise monotone family cannot exactly realise a closed non-degenerate orbit; on the reference orbit, its best fit has L2 error 998 versus 924 for the best fixed linear map.The theorem gives an error lower bound of A^2ℓ/4 for each coordinate decreasing by A over intervals of length ℓ; the carrier family retaining the conjugate pair is exact.Source: B. Global realisations · p. 15

D. Composable transport and its price

  • Theorem 4 bounds n-step error by regime: L < 1 gives bounded geometric accumulation, L = 1 linear accumulation, and L > 1 exponential growth.The tested heat degradation reflects growing single-step error on later states, not truth-action amplification; hue’s flat chains indicate small single-step error. The expansive regime and independent estimates of ε and L were not tested.Source: D. Composable transport and its price · pp. 15–17

B. Global realisations

  • 0.17–0.40% relative MSE: the band-limited global family predicts re-rendered inputs, up to 16.6× tighter than the copy baseline.Content conditioning did not rescue global forms across 48 matched-capacity contrasts, including an oracle partition by displacement direction.Source: B. Global realisations · p. 15, Table 7 · p. 16

C. Local state-conditioned realisations

  • 0.4%: the local phase-conditioned transporter closes loops on unseen shapes and starting states, fitting an order of magnitude tighter than the best global family.It divides the hue circle into 40° segments, reads phase from the representation, and applies a segment generator estimated from paired endpoint features.Source: C. Local state-conditioned realisations · p. 15, Table 8 · p. 16

D. Composable transport and its price

  • For fixed-total paths, chains come within a few points of direct fits for hue and heat, while single-step substitutes do not; heat degrades with chain length.Path consistency and fidelity differ: the global linear family agrees most tightly between paths but transports worst. Heat’s high single-step score is qualified by a total near saturation.Source: D. Composable transport and its price · p. 15, Figure 6 · p. 16, Table 9 · p. 17
  • In ResNet-50 consumer-response metrics, typical stretch is slightly above one for both hue and heat at every depth, despite feature-distance ratios below one for heat and near one for hue.Thus input-space contraction does not establish contraction in a consumer’s seminorm; the accumulation regime must be measured there.Source: D. Composable transport and its price · p. 17

E. Functional consequences of the realised action … D. Learning in a controlled linear setting

Operability declines with depth because transformations must be served by one operator across local rules, and this decline depends on the readout–transformation pairing rather than algebra alone. Geometry and controlled learning further show how that reach changes, while capability evaluation must combine fidelity, composition, and downstream effect.

E. Functional consequences of the realised action

  • A random orthogonal operator achieves the best projection score at 15 of 42 sites but has negative transfer at every site and detector agreement F1 = 0.000.Within a family, projection correlates with transfer (ρ = +0.93), and feature-residual ratio correlates with the correct sign; evaluation therefore jointly tests feature-target fidelity, composition, and downstream effect against their own baselines.Source: E. Functional consequences of the… · p. 17

B. The geometry of changing realisability

  • At the deepest measured layer, orbit bending averages 729° and reaches 760° in one cell, while closed-loop error worsens to 0.24 normalised.Per-shape readouts become switch-like as the continuous rotation field compresses toward discrete sign-like clusters; interpreting this as a symbolisation transition remains a hypothesis.Source: B. The geometry of changing… · p. 17

A. Depth profiles of structure and performance

C. The readout–transformation pairing

  • Consumer-visible gate-change share orders transfer more closely (ρ = −0.886) than raw gate-change share (ρ = −0.83), while flip count does not order it.The mechanism decomposes transformation effects into within-regime and gate-changing-unit terms; the consumer-visible share is a proxy for sharing cost, not the cost itself. An exactly orthogonal target keeps TF = 0.9999993 across depths 1, 4, and 8, unlike an ill-conditioned random target.Source: C. The readout–transformation pairing · p. 18
  • The hue readout declines under hue on three of four backbones (median ρ = −0.80) and is flat under heat (two of four, median ρ = +0.00); the high-frequency readout declines under heat on all four (median ρ = −0.60) and is flat under hue.The crossed design uses chance-corrected consumer-visible share; ViT-B/16 is a floor case whose flatness is uninformative.Source: C. The readout–transformation pairing · p. 18, Table 11 · p. 19

D. Learning in a controlled linear setting

  • With κ fixed, controlled linear training leaves optimal transfer unchanged despite near-total retained-subspace reorganisation; transfer changes track Δκ (ρ = −0.833), not row-space angle (ρ = −0.146).Weight decay raises transfer only where κ moves, showing that induced mixing, rather than rank or task loss, describes operability.Source: D. Learning in a controlled linear… · p. 19

E. A unified interpretation … E. What the structure buys

These sections connect measured transformation structure to operable interfaces, while distinguishing what the linear theory establishes from what measurements diagnose and construction requires.

E. A unified interpretation

  • The unified account places compatibility at encoder, orbit, and readout, while nonlinear-site deviations diagnose failed premises rather than predict them.The exact law remains the linear theory’s result; its deviation ordering is no better than naive baselines, and measured structure–operability links are not extrapolated.Source: E. A unified interpretation · p. 19

F. Where the realisation error goes: a unit-level diagnostic

  • Held out, sharing cost is negligible, pooled channel relations degrade with depth, and cross-channel coupling helps shallow layers but harms deep layers.The full operator cuts error substantially at the first layer but is worse than the diagonal restriction at the last; the diagnostic concerns per-cell affine and channel-diagonal families, not the global operator across layers.Source: F. Where the realisation error goes: a… · p. 19, Table 13 · p. 20

A. Design from the measured organisation

  • Measured low-order spectrum sets K = 2, shared phase sets a shared action, content-dependent amplitude calls for a dependent envelope, and degeneracy marks the operating domain.K = 2 is the measured design choice, not a theoretical minimum; raw-feature complexity motivates an optional context branch.Source: A. Design from the measured organisation · p. 19

B. The interface: learned encoding, fixed action

  • The six-dimensional hue interface learns its read-in but fixes A_∆ = diag(I, R(∆), R(2∆)) from measured structure, so its action is never fitted.Against learned generators, learned linear actions, and a larger generic map, the fixed action is exact in increments and composition, with read-out error no worse; the learned linear action fails on an unseen large increment.Source: B. The interface: learned encoding,… · pp. 19–20, Figure 8 · p. 21

C. What bounds the interface: the three error terms

  • The carrier-sufficiency identity bounds decoded error by read-in error, action mismatch, and target tail; only read-in error is removed by training.The mismatch error norm scales first-order with plane misalignment (log-log slope 0.99); a 12–16% tail-energy fraction corresponds to ε_tail ≈ 0.35–0.40. The bound applies to feature-space error, not read-out degrees, and measured closure limits fixed linear maps rather than this nonlinear-read-in interface.Source: C. What bounds the interface: the… · pp. 20–21

D. Generalisation on unseen shapes and parameters

  • The interface reads hue on unseen shapes at 3.4° median error by the per-cell-median criterion and on real regions at 8.0° versus 10.0° for the non-circular control.The unaligned route reaches 69.5°; failures concentrate at low saturation and value, which the interface specification treats as a degenerate domain.Source: D. Generalisation on unseen shapes and… · p. 21

E. What the structure buys

  • With sparse hue supervision, operator-based augmentation achieves 1.3° median read-out error versus 3.5° for input-space interpolation and 25.5° for real data only.The structured read-out also beats a plain probe on the dense grid and held-out sheet.Source: E. What the structure buys · p. 21
  • In matched-budget plain-CNN stage-1 controls, correct pairing wins; same-width untrained residuals do not, rank-16 and rank-64 variants land at or below it, and shuffled pairing collapses win rate.The results distinguish the operator-family capacity and pairing experiment from the interface comparison.Source: E. What the structure buys · pp. 21–22

F. Useful operations … IX. CONCLUSION

Baseline-matched operations and a measurable operating-domain rule show how transformation structure can support use, while the discussion and conclusion delimit which theoretical and empirical claims generalize. The paper connects compatible actions, measured organization, and a fixed interface, alongside explicit limits on prediction and scope.

G. Operating domain on real regions

  • On 750 pre-registered COCO regions, hue concentration separates success from failure with moderate ROC AUC 0.660; on an unaligned 120-region control, concentration alone has almost no discriminative power.The threshold is published with its calibration curve rather than as a threshold with an implied error bar.Source: G. Operating domain on real regions · p. 22, Table 16 · p. 24

F. Useful operations

  • Feature-space augmentation cuts hue read-out error by an order of magnitude versus input-space interpolation; composition reaches never-fitted parameters, detection retains a third of seen-shift AP50 at an unseen shift, and editing takes microseconds versus milliseconds.Table 15 reports each capability against ground truth and its own baseline.Source: F. Useful operations · p. 22, Table 15 · p. 24

VIII. DISCUSSION

  • The theory’s fibre condition and inherited composition law apply broadly, but the kernel-preservation and closed-form theorems concern single-region linear layers, while harmonic organization and the interface are colour-specific measurements.The discussion distinguishes these scopes explicitly and notes that carrier sufficiency holds for any carrier, action, and read-in.Source: §8.1 · pp. 24–25
  • Faithfulness scores characterize a representation–consumer pair, not a representation alone; a random orthogonal operator ranks first by projection at 15 of 42 sites, while depth reduces the reach of fixed low-parameter families, not the transformations represented.A consumer that does not read the transformed direction cannot grade an operator, and gate changes count only where the consumer reads them.Source: §8.1 · pp. 23, 25
  • Across real sites, the closed-form prediction is biased high by 0.40 (MAE 0.43 versus measured mean 0.31) and orders no better than naive baselines (Spearman 0.48; Pearson −0.44).Shared-rate extrapolation gains 14.9× on the synthetic carrier but 0.83× and 0.75× on the two real sites, so it does not transfer.Source: Figure 12 · p. 26

IX. CONCLUSION

  • The conclusion links fibre-compatible actions, metric-dependent linear defects, rectifier regime costs, and retained-block closure to measured colour organization and a fixed interface whose action is never fitted.It summarizes colour orbits as low-order and cross-shape shared, substantially inherited rather than learned, with operability eroding by depth along the readout–transformation pairing.Source: §IX · pp. 26–27
  • The conclusion bounds the claims: the closed form is not a calibrated predictor for real backbones that violate isotropy, and no tested basis supplies a shared decay rate for the dissipative family.Both limits are presented as findings about what survives compression, not as assumptions set aside.Source: §IX · p. 27

APPENDIX … Appendix B.2 general

The appendices document the scope, status, assumptions, and evidence for the paper’s claims, while Appendix B.2 develops when transformations induce or realise actions on neural representations. Its results connect fibre preservation, linear and piecewise-linear realisability, and limits on composing approximate actions.

APPENDIX; A.3 Measured claims: depth, learning and the interface:

  • Appendices A–G collect deferred claims, assumptions, proofs, protocols, full grids, methods, capability tables, and positioning, with code, seeds, and raw outputs available in the repository.Claim statuses and supporting evidence are collected in Tables A.1–A.3 and B.4; each claim states its scope, and measured claims are not presented as theorems.Source: APPENDIX · p. 27, §A.3 · p. 27

B.1 Standing assumptions:

  • The standing assumptions require exact transformed references, disjoint fit and evaluation samples, a linear full-row-rank site with invertible input action, and isotropic inputs for the exact closed form.Where an induced action is claimed, fibres must be preserved.Source: §B.1 · p. 27

B.2 The general layer (the fibre condition):

  • An induced transformation on encoded features exists if and only if the encoder’s fibres are preserved, and any such induced family inherits the transformations’ composition law.Thus, a network-level composition experiment tests the defect of a fitted family, not the truth-level composition law; the approximate fibre condition yields an induced-map defect bounded by η after quotienting.Source: §B.2 · pp. 27–28

Appendix B.2 general

  • A linear realisation exists exactly when g ker W ⊆ ker W; more generally, the closure criterion is A ker B ⊆ ker B, with optimal operator K⋆ = BAB†.The optimal residual is ∥BAPker B∥2; a verified design condition showed a 9 × 10−16 gap over 40 random selections.Source: Appendix B.2 · p. 28
  • The closed-form bound is tight when the cross term vanishes, and TF depends on more than (W, g); a verified κ sweep had maximum deviation 0.0091 and median 0.014.The two worst cells had effective rank 4 of 32.Source: Appendix B.2 · p. 28
  • For piecewise-linear sites, a fixed realisation exists exactly when the joint system over realised transition cells is consistent; gate changes create rule diversity, not the obstruction.The appendix also gives two counterexamples showing that the gate-change share is not itself the obstruction, and treats it as an empirical proxy for slack for linear consumers.Source: Appendix B.2 · p. 28
  • A closed orbit has no exact coordinate-monotone realisation, and the best fit reported L2 = 998 against 924 for monotone parameter families in a fixed basis.The theorem also gives a per-coordinate cost bound.Source: Appendix B.2 · p. 28
  • Composition error obeys En ≤ ε + LEn−1; carrier sufficiency gives a three-term L2 bound, with measured tail energy 12–16% and mismatch exponent 0.99.Hue chains were flat and heat chains monotonically decreasing; the local transporter accepts the bound’s terms deliberately, paying the composition cost measured in §5.4.Source: Appendix B.2 · p. 28

B.3 The single-region linear layer (Theorems 1 and 2):

A single-region linear realisation exists exactly when the transformation preserves the encoder’s kernel. When it does not, the obstruction comes from discarded information entering retained features, while the defect is measured in the representation-induced metric.

B.3 The single-region linear layer (Theorems 1 and 2):

  • Among the four action blocks, C alone obstructs realisation: it moves discarded information into retained features, whereas D loses information without creating a contradiction.A is expressible, and B is invisible; D costs information while C costs consistency.Source: §B.3 · p. 28
  • Under Assumption B, Theorem 1 states that there exists ρ ∈ Rd×d such that z(gx) = ρ z(x).Source: §B.3 · p. 28
  • With an isometric encoder on the retained subspace, the defect ratio is κ = ∥C∥F / ∥A − I∥F and the optimum is ρ⋆ = A; otherwise the optimum is W-conjugate.The RMS-criterion transfer yields the defect law in the induced metric; the mean-ratio transfer differs from it by at most 0.0011 on the Table 3 sweep.Source: §B.3 · pp. 28–29

Why the metric appears. · Reading.

The defect law uses an encoder-induced metric because feature scaling affects defect size, and its calibration can depend on input covariance. The reading also distinguishes the geometry of the retained–discarded split from rank.

Why the metric appears.

  • Encoder scaling makes the induced metric W necessary for defect size, while block C alone determines whether a transformation exists and enters the defect.Omitting W assumes the encoder is an isometry on the retained subspace, which holds after whitening but not generally.Source: Why the metric appears · p. 29
  • The closed form ρ⋆=A holds exactly when the predictable–residual cross term vanishes, a condition that Assumption C guarantees.The condition is the Cov(xR, xK)-weighted cross-covariance of the predictable part and residual being zero.Source: Why the metric appears · p. 29
  • Input anisotropy shifts TF from 0.069 to 0.429 while the bottom-50% variance-direction displacement share shifts 0.486 → 0.440.The consumer-invisible subspace, not low-variance directions, is the theoretically correct discarded subspace; cross-site TF comparisons require reported or matched variance geometry, and real sites need not satisfy Assumption C.Source: Why the metric appears · pp. 29–30
  • When d^2/K is not small, the measured defect can overstate the irreducible value by as much as the O(p d^2/K) estimation term.The two worst trained-weight cells have absolute errors 0.23 and 0.07 and effective rank 4 of 32; reported scores therefore include fitted rank, fit and held-out sample counts, and a tuned-λ ceiling.Source: Why the metric appears · p. 30

Reading.

  • Rank is not the governing variable: arbitrarily large rank changes cannot make κ large, whereas a small change in split geometry can.Source: Reading · p. 30
  • For kernel-preserving g, rank 8 / 48 corresponds to 0.9999992 / 0.9999943; residual is 2.6 × 10^-5 versus displacement 34.3.Source: Reading · p. 31
  • At fixed rank, the mixing sweep moves κ from 4.5 × 10^-16 to 9.34 and TF from 1.0000 to 0.0041, with maximum 0.0091.Source: Reading · p. 31
  • Across 32 trained linear-weight cells, ρ(κ, TF) = −0.968 versus ρ(eff. rank, TF) = +0.502.The same passage reports that input anisotropy at fixed W, g moves from 0.0 to 1.0 while TF moves from 0.069 to 0.429.Source: Reading · p. 31

B.3b Carrier closure and optimal linear realisation: … B.5 Realisation and composition (Theorems 3, 4, 5):

The section characterizes when a fixed linear action can realize a transformation after encoding, across carriers and piecewise-linear networks, and states limits on monotone realization and error under composition. These results distinguish information lost by the encoder from incompatibility among the regions visited by a transformation.

B.3b Carrier closure and optimal linear realisation:

  • A carrier action descends through encoder B exactly when it preserves the encoder’s discarded subspace: A ker B ⊆ ker B.Equivalently, a fixed linear K satisfying KB = BA exists exactly under this condition.Source: B.3b Carrier closure and optimal… · p. 31
  • For an isotropic carrier, K⋆ = BAB† is a least-squares realisation; with coordinate selection and orbit-orthogonal carriers, it equals the action projected onto retained coordinates.The residual vanishes exactly when the closure condition holds; across forty random selections, the reported gap to PAP⊤ was 9 × 10−16 and closure verdicts matched residual signs.Source: B.3b Carrier closure and optimal… · p. 31

Corollary (conjugate pairs).

  • For full continuous rotations with one two-dimensional block per frequency, closure under coordinate selection holds exactly when each block is retained or dropped whole.For a single fixed angle or repeated frequencies whose blocks may mix, the criterion concerns invariant subspaces rather than individual pairs; measured-feature results are identified as Table D.21.Source: Corollary (conjugate pairs) · p. 31

B.4 The piecewise-linear layer (Propositions C, D, E):

  • A fixed linear action realizes a piecewise-affine site on visited cells exactly when one operator satisfies ρAr = AsT and ρbr = bs across every realized region transition.With no gate-pattern changes, this reduces to ρAr = ArT and ρbr = br on the visited regions.Source: §B.4 · p. 31

Consequence.

  • Even without region crossings, visited regions can obstruct a global linear action; expanding the operator family yields at most 0.087 gain at ResNet-50 sites and 0.11 at ConvNeXt sites.The median gap over the 32-site grid is 0.005; the families separate only where both fail, at DINOv2 heat block8 (−0.40 against −0.99).Source: Consequence · p. 31
  • Gate changes create a distinct local affine rule but do not themselves determine whether one fixed operator works; consistency depends on the joint system across visited regions.Measured flip count was uncorrelated with TF (ρ = +0.49, n.s.), whereas flip depth and gate-change share each had ρ = −0.83 with TF.Source: Consequence · pp. 31–32

Status.

  • The gate-change proxy is not identical to fixed-linear slack: its consumer-visible share is informative only when the change term is unpredictable from z, and the check covers six site-algebra points.The stated guarantee is for linear consumers; no claim is made for families outside the fixed-linear class.Source: Status · p. 32

B.5 Realisation and composition (Theorems 3, 4, 5):

  • A closed non-degenerate orbit with a decreasing coordinate has no exact realization by a coordinate-monotone parameter family; under a Lipschitz truth action and uniform single-step error, composition admits an accumulation bound.A coordinate decreasing by A over intervals of length at least ℓ contributes L2 error at least A²ℓ/4; the monotone-family comparison is reported in Table 7.Source: §B.5 · p. 32

Uniformity. … C.1 Synthetic orbit construction:

The appendix specifies when the paper’s measured transformation claims hold uniformly, bounds carrier-interface error, and documents the synthetic orbits and exact transformations used for evaluation.

Uniformity.

  • Uniform guarantees require the single-step error and Lipschitz constant to hold across visited states; varying values yield pointwise recursions with product-form bounds.The three regimes are measured in the consumer’s seminorm, not inferred from the generator; independently estimating ε and L is not done here.Source: Uniformity · p. 32
  • The carrier-sufficiency setup permits any synthesis, action, and measurable read-in, and its three-term bound follows by adding and subtracting two synthesized carriers.Band-limitation and plane sharing are measurable premises of the construction but are not hypotheses of the theorem; the triangle inequality assumes no cross-term cancellation.Source: Uniformity · pp. 32–33

Three consequences.

  • The interface error bound separates read-in error, action mismatch, and target tail; only read-in error is removed by training.Mismatch is first order in action difference, while squared error is second order in plane misalignment; the measured error-norm slope is 0.99. A 12–16% tail energy fraction corresponds to ε_tail ≈ 0.35–0.40, not 0.12–0.16.Source: Three consequences · p. 33

B.6 Status of every statement:

  • Appendix B.6 fixes the definitions behind measured numbers and identifies the producing scripts and result files for each subsection.Source: §B.6 · p. 33

C.1 Synthetic orbit construction:

  • Synthetic orbits use jittered single-shape 224 × 224 images, with four training shapes, two held-out shapes, and hue holdouts at 20° and 260°.Centres, radii, and orientations are jittered by ±12 px, [70, 92] px, and uniformly, respectively; colour is written through HSV→RGB.Source: §C.1 · p. 33
  • 1.83 × 10−7 relative error confirms heat-semigroup composition, while a truncated Gaussian kernel departs by 22.1%.The exact reference is spectral multiplication for heat and re-rendered recolouring for hue.Source: §C.1 · p. 33
  • Hue rotation is exact in HSV-like coordinates, and heat uses the exact discrete semigroup Bt = exp(−tL) via per-channel DCT spectral multiplication.Heat uses t = σ^2/2 and site rescaling t_site = (σ^2/2)(H/224)^2.Source: §C.1 · p. 33

C.2 Sites and backbones: … D.2 Depth grid (hue and heat):

The merged sections specify the sites, readouts, fitting and evaluation protocols, then report real-region, harmonic-readout, and depth-grid findings. They distinguish measured results from unreachable interventions and controls.

C.2 Sites and backbones:

  • The orbit readout uses pose-averaged GAP vectors after per-shape DC removal, while the depth-grid readout applies one matrix at each spatial location or patch token.Source: §C.2 · p. 33

C.3 Depth-grid protocol:

  • Interventions change logits by exactly 0 despite input-side displacement of 0.188–0.623; the σ = 1.0 grid is single-seed, while the heat pilot sweeps σ ∈ {0.5, 1.0, 2.0} on CIFAR plain CNN.Source: §C.3 · p. 33

C.4 The three tasks and their holdouts:

  • Prediction, transport, and composition are evaluated with distinct holdouts, including unseen shapes or states, disjoint image-index fit and held-out blocks, and controls beside each path.Composition varies the decomposition while holding the total fixed; never-fitted hue and heat composites are read without refitting.Source: §C.4 · p. 33

C.5 Fitting protocol:

  • Operator fits use ridge-regularised least squares by default (λ = 10−3), but stage-1 fitting budgets differ across arms, so no cross-arm comparison is drawn from that ranking.Large dimension relative to row count can limit estimation; at ResNet-50 heat depth 4, d²/K = 428.0 (d = 2048, 9800 fitted rows).Source: §C.5 · p. 33

C.6 Power gate and baselines:

  • A site is powered when the consumer-level effect size is at least 0.05; sites the transformation does not move for the consumer are excluded rather than scored.Mandatory baselines include the no-op, trivial depth rule, random operator, and a shuffled-pair control.Source: §C.6 · p. 33

C.7 Real-region protocol:

  • 8.0° median hue error was achieved on the held-out high-concentration COCO regions (m ≥ 0.7, n = 69); the +40° control route gave 10.0°.On the test half, the aligned route separated high- from low-concentration regions by 0.238 (0.770 versus 0.532); the ground-truth-box and raw routes had AUCs of 0.660/0.662/0.579. In a disjoint 120-region control, concentration alone was near chance (AUC 0.523 at threshold 10, 0.480 at 20).Source: §C.7 · pp. 34–35

D.1 Harmonic spectra grid:

  • 0.71° zero-shot hue readout was achieved at the ResNet-50 mid layer, on the same order as the dedicated code head on stage 3; CLIP was weakest at its final CLS.Source: §D.1 · p. 35

D.2 Depth grid (hue and heat):

  • Ridge and conv-res transfer generally declined with depth across the reported hue and heat grids, while random-control trends varied by backbone.The reported Spearman trends include ridge and conv-res values as low as −0.97; some Procrustes and random trends were weaker or positive. The grids distinguish three-seed hue and heat σ = 2 blocks from the single-seed σ = 1 control.Source: §D.2 · pp. 35–36

D.3 Readout × transformation crossed grid: … D.7 Cross-cutting note:

Across the crossed grids, transformation–readout outcomes varied with depth, composition, and boundary conditions; attribute probes improved substantially from null baselines, while consumer-visible gate-change share tracked transformation fit more closely than flip fraction.

D.3 Readout × transformation crossed grid:

  • For hue→hue, 3/4 backbones declined with depth (median ρ = −0.80); heat→hf declined in 4/4 (median ρ = −0.60), while hue→hf and heat→hue had median ρ values of +0.00 and +0.30.The other depth trends were hue→hf decline in 2/4 backbones and heat→hue decline in 1/4.Source: §D.3 · p. 36

D.4 Composition grids:

  • The composite power gate had effect size 0.881, top-1 flip 0.795, and accuracy 0.915→0.225; its random-orthogonal control TF rose from −0.76 to −0.17.Source: §D.4 · p. 37
  • Heat two-step versus four-step composition had ridge value 0.173 ± 0.070; the global linear family agreed most tightly between paths but transported worst.This is described as the consistency–fidelity split.Source: §D.4 · p. 37

D.5 Attribute and perimeter grids:

  • Null-to-real probe error changed from 88.10→18.89 for hue, 0.370→0.070 for saturation, 0.368→0.044 for value, and 0.980→0.256 for quantity.The fully clipped-bin false-confidence control had mean absolute report change exactly 0.0 (n=24) for both attributes; the Table 6 headroom ≥1 row reports realised report change 0.174 and read-out error 0.0143.Source: §D.5 · p. 38

D.6 Boundary grids:

  • Across the three boundary-grid sets, regression targets were near zero (CV R2: −0.018, +0.010, −0.095), so the control bounds ranking power rather than calibrated error.Source: §D.6 · p. 38

D.7 Cross-cutting note:

  • On ResNet-50, consumer-visible crossing share rose from 0.220 (hue) and 0.252 (heat) at layer 2 to 0.627 and 0.516 at layer 4, while gate flip fraction fell from 0.091 and 0.094 to 0.064 and 0.051.Source: §D.7 · p. 38
  • Across six (site, algebra) points, consumer-visible gate-change share correlated with TF at Spearman −0.886, versus +0.486 for flip fraction (p = 0.33, n.s.).The raw gate-change share had Spearman −0.829 against TF.Source: §D.7 · p. 38

D.8 Realisation-error and metric diagnostics: · D.9 Direct test of the fibre condition: · Box E1. Local phase-conditioned transporter (§5.3):

The appendix separates local transport performance from the fixed carrier action: the local transporter fits a phase-conditioned generator, while the carrier action is fixed by measured structure and only its read-in is learned. It also specifies diagnostics for realisation error, fibre closure, and metric distortion.

D.9 Direct test of the fibre condition:

  • D.9’s implementation specifications state the hyperparameters and identify what is fitted versus not fitted for four construction and measurement components.The passage gives no component-specific settings or outcomes.Source: §D.9 · p. 38

Box E1. Local phase-conditioned transporter (§5.3):

  • The local transporter fits nine 40° segment generators in a 24-dimensional PCA subspace, then smooths them with a third-order Fourier field whose phases are shared across shapes.The Fourier fit reaches R2 = 0.8672; inference applies the fixed field using the read phase, without refitting on a test shape or starting state.Source: Box E1. Local phase-conditioned… · pp. 38–39
  • Across five sites, the local construction beats the copy baseline by 1.55–15.94× while the global family reaches only 0.08–0.84×, and its full-turn closure error is 0.0037 on stage 3 and 0.0062 on ResNet-18 layer 2.The comparison uses four training shapes and two holdout shapes.Source: Box E1. Local phase-conditioned… · p. 39
  • The carrier action is fixed as diag(I2, R(∆), R(2∆)), holding two scalar channels fixed while rotating the fundamental and second harmonic, so only the read-in is learned.The construction learns correspondence but not the action law; unlike the local transporter, it never fits an action.Source: Box E1. Local phase-conditioned… · p. 40
  • The carrier head’s measured per-(s, v) orbit amplitude anchors its radial scale and phase gauge, and the six-channel head is evaluated zero-shot on two unseen shapes.Where semantic labels are unavailable, the channels are additionally anchored to label-free saturation-weighted hue, saturation, and value statistics.Source: Box E1. Local phase-conditioned… · p. 40

D.8 Realisation-error and metric diagnostics:

  • The realisation-error diagnostic separates within-cell error from the cost of sharing one operator across realised transitions, while the metric diagnostic compares feature-distance distortion in the site metric and consumer response.The table captions describe these diagnostics but provide no numerical values in the supplied passages.Source: Table D.20 · p. 41, Table D.22 · p. 41

D.9 Direct test of the fibre condition:

  • The direct fibre-condition test compares the known-input-hue carrier with estimated synthesis, measuring reconstruction tail, closure defect, and held-out global-linear-map error, also with the second-harmonic coordinate omitted.The supplied table caption specifies the measured quantities but does not report their values.Source: Table D.21 · p. 41

Box E1. Local phase-conditioned transporter (§5.3):

  • The appendix cautions that the reported runtime ratio is not a controlled head-to-head benchmark because the artefact omits its device and later verification runs are CPU-only.Source: Box E1. Local phase-conditioned… · p. 41

Box E3. Consumer heads: · Box E4. Hardware and runtime: · G.1 Where this paper does not compete:

These appendix sections specify how consumer heads evaluate transformations, qualify runtime and result interpretation, and delimit the paper’s claims. They distinguish the measured fixed-action interface from architecture design, perceptual theory, and universal claims about networks.

Box E3. Consumer heads:

  • The consumer is either a frozen pretrained classifier head for label-free logit comparisons or a shared ridge probe for hue, saturation, and value read-outs.The hue read-out in §7.4 uses the Box E2 code head, not a ridge probe; the YOLO arm reads a 64-dimensional stage 3 GAP feature.Source: Box E3. Consumer heads · p. 41
  • At ∆= 90◦, detector F1 is 0.008 for null, 0.778 for Procrustes, 0.905 for ridge, 0.965 for conv-res, and 0.000 for random orthogonal.The YOLO11n hue-bin detector was trained on eight hue classes with shape randomised across classes; AP50 tracks F1 throughout.Source: Box E3. Consumer heads · p. 41

Box E4. Hardware and runtime:

  • The runtime ratio compares an already-extracted code shift with pixel re-rendering and re-forwarding, and its device mismatch means it is not a controlled benchmark.The pixel route used an RTX 2060; the region-level artifact does not record its device, and later verification runs were CPU-only.Source: Box E4. Hardware and runtime · p. 41
  • The low-effect arms (0.141 and 0.013) are underpowered because of effect-size collapse, so their results do not show representational deficiency..Source: Box E4. Hardware and runtime · p. 42
  • The code auxiliary’s clean-accuracy edge is within seed noise, while its OOD ordering reflects invariance built during training, not the interface.Source: Box E4. Hardware and runtime · p. 42
  • At 90◦, never-fitted generator extrapolation reaches AP50 0.3654 and a fitted-30◦/60◦ chain 0.511, versus 0.8052 for an operator fitted directly at 90◦.The random orthogonal operator has projection coefficients 0.32–0.46 but detection F1 0.000 at every shift.Source: Box E4. Hardware and runtime · p. 42

G.1 Where this paper does not compete:

  • The paper proposes no new layer, loss, or training scheme; its interface reads existing features and fixes the action form by measurement rather than learning it.Hue rotation and diffusion are exact re-renderable references whose physical validity is assumed; perceptual appearance and universal claims about all networks are out of scope.Source: §G.1 · p. 42

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