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Every Ablation Is a Dose: Counterweights and the Semblance of Self-Repair

Areeb Ahmad, Pratinav Seth, Vinay Kumar Sankarapu

arXiv:2610.02173v1cs.LGcs.CL

TL;DR

Ablation can obscure a component’s importance because downstream components appear to compensate, but the mechanism of self-repair has remained unclear. The paper tests a continuous dose intervention and models downstream contributions with an affine law. Across four model families, dose-based fits predict held-out responses, supporting the account of compensation as a standing coupling revealed by ablation.

  • Problem

    Ablation-based estimates can be distorted when downstream components adjust, partly preserving behavior and causing underestimation or misattribution of the ablated component’s role.

    Source: §1 · p. 1

  • Method

    The paper varies a core component along a signed class-contrast dose axis and measures downstream contributions with a nested counterfactual intervention.

    Source: §1 · p. 2

  • Results

    Within |λ| ≤1, downstream contributions vary nearly linearly with dose, and three-dose fits predict held-out doses with out-of-sample R2 of 0.65 to 0.91 by family.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The paper concludes that apparent self-repair is the release of pre-existing coupling rather than a dynamic, reactive process.

    Source: §9 · p. 8

    The study tests one task at one answer position, leaving transfer to other tasks, especially generative ones, open.

    Source: Limitations · p. 9

Abstract

from arXiv · show

Ablate a component of a language model, and other components often appear to adjust and compensate. This phenomenon, termed self-repair, has been observed repeatedly, but its mechanism remains unclear. The most systematic study to date concluded that self-repair is noisy and unlikely to have a single explanation. We argue that it has one: a gain already present before any ablation. Any intervention on a causally important component can be viewed as a point on a coordinate axis $\lambda$, the signed strength of a counterfactual contrast. Hence, conventional ablation methods are uncalibrated points on this axis. We show that the causal repair response for a fine-grained unit $r$ is governed by an affine law, $E_r(\lambda)=\mathrm{own}_r+\gamma_r\lambda$. The slope $\gamma_r$ is a fixed coefficient that consistently influences the model, with or without ablation, and its sign determines whether the unit counteracts or reinforces the removed signal. On a factual-verdict task across four models from distinct families (Gemma, Qwen, LLaMA, and Mistral), we identify components including MLP neurons, OV neurons, and singular directions that follow this affine law, 68 of 81 downstream directions in all. Moreover, we can anticipate the magnitude of $\gamma_r$ from the fixed weights. On the IOI circuit of GPT-2 Small, seven of the ten heads the intervention can reach follow the law, and all seven are counterweights. From this perspective, what may appear as self-repair is a counterweight performing its usual operation when the contrastive signal emerges at the core.

1 Introduction

Ablation-based inference is complicated by downstream compensation, whose mechanism prior work left unresolved. The paper proposes that apparent repair reflects a standing gain, and tests this with continuous doses across several models and an IOI circuit.

1 Introduction

  • Ablating a component can make downstream components adjust, causing behavior to partly survive and potentially leading ablation to underestimate importance or misattribute a component’s role.Direct logit attribution also degrades with distance from the unembedding, limiting earlier compensator discoveries to late-layer components [9].Source: §1 · pp. 1–2
  • Prior studies found backup name movers [1], the Hydra effect [5], and anti-erasure neurons [6], while the most systematic study judged self-repair noisy and unlikely to have one mechanism [6].Earlier deletion-based observations could not distinguish a component that switches on from one that responds proportionally to input displacement [6].Source: §1 · p. 1
  • The paper argues that directions already oppose or reinforce the core before intervention, so conventional ablation reveals a standing coupling rather than recruiting repair.A direction is a counterweight when γ_r < 0 and a relay when γ_r > 0.Source: §1 · p. 2
  • The tests span four instruction-tuned model families and the IOI circuit of GPT-2 Small.The abstracted finding is that seven IOI heads following the law are all counterweights.Source: §1 · p. 2
  • Within |λ| ≤1, downstream contribution varies nearly linearly with core dose; three-dose fits predict held-out doses with out-of-sample R2 of 0.65 to 0.91 by family.The proposed affine law treats the slope as a fixed gain, not a response recruited by ablation.Source: §1 · p. 2
  • The paper also identifies weight-named counterweights and relays whose coupling magnitudes can be anticipated from the static checkpoint.These directions are certified by intervention rather than inferred from their labels alone.Source: §1 · p. 2

2 Setup

The setup represents transformer writes through fixed weight-space directions and studies how interventions on their activations affect a factual verdict. It applies activation patching as a causal intervention across four instruction-tuned models on contrastive factual statements.

2 Setup

  • Each direction is represented by a fixed write vector and a scalar activation coefficient, enabling interventions that overwrite one direction’s coefficient.The paper uses neuron directions and singular-vector directions derived from weights.Source: §2 · p. 3
  • The forward pass is treated as a deterministic structural causal model, and activation patching pins a coefficient to its value on a source input.This operation underlies the paper’s dose ladder and other interventions.Source: §2 · p. 3
  • Evaluation uses the verdict margin, the gap in mean logits between faithful and contrastive answer tokens, and is conditioned on clean-correct pairs.The condition means the clean margin is positive for every pair.Source: §2 · p. 3
  • The experiments use four instruction-tuned models on zero-shot true/false factual verdicts, with contrastive statement pairs differing only in the asserted attribute.Pairs are synthesized from the Known-Facts corpus [24].Source: §2 · p. 3

3 The Dose Axis

The dose axis expresses a core intervention as a signed displacement along the class contrast of a paired prompt. A nested intervention then measures a downstream direction’s contribution in both clean and dosed worlds.

3 The Dose Axis

  • The dose λ parameterizes the core’s displacement along the class contrast, with +1 as the clean activation, 0 as the class-neutral midpoint, and −1 as the opposite-class activation.Doses beyond these endpoints exceed one unit of the pair’s own contrast.Source: §3 · p. 3
  • The nested measurement first doses the core, then patches downstream direction r with its partner-prompt value inside that same dosed world.The clean-world comparison patches r using its partner value from the unmodified model.Source: Figure 2 · p. 4, §3 · p. 4
  • Conventional interventions such as zero, mean, or resample ablation are points on the dose axis, with their locations determined by the data.The core is clamped toward its partner value in the primary intervention.Source: §3 · p. 4
  • The estimand is the margin erased by intervening on r, measured within each world so the primary intervention’s shared shift cancels from the total effect.Positive values mean r supports the verdict; negative values mean it opposes it.Source: §3 · p. 4

4 Census of the Core and Its Counterweights

A small set of core directions carries much of the clean verdict effect, while many other legible directions move against the core as its class signal is withdrawn. Their responses can cross from opposing to supporting the verdict, and the response grows with the amount of signal removed.

4 Census of the Core and Its Counterweights

  • The core is one neuron in Llama and a pair in each other family, carrying 52.7 to 56.6% of the clean total effect.Some other directions are inert, while others oppose the verdict despite having legible truth or falsity poles.Source: §4 · p. 5
  • Across four families, featured counterweights cross from opposing to supporting the verdict as core signal is withdrawn: Llama −0.10 to +0.25 nats, Gemma −2.1 to +3.9, Qwen −5.7 to +8.7, and Mistral −1.5 to +3.2.The crossover occurs on both classes; renormalization explains at most 4.5% of audited recruitment.Source: §4 · p. 5
  • In two-core families, dosing both cores moves more of the bench than dosing either alone, and at low dose the joint response exceeds the sum of the separate responses.The excess is attributed to the signal carried by the unclamped core when only one core is clamped.Source: §4 · p. 5
  • The same directions respond to a learned label-free singular-value perturbation and to its conjunction with the swap, indicating recruitment across disruption methods.The core and bench directions are identified using weight-space labels, with causal roles assessed by intervention.Source: §4 · pp. 4–5

5 The Coupling Law

The coupling law models how a downstream direction’s verdict contribution changes with a core’s signed dose: within |λ| ≤1, the relationship is nearly linear, and many directions follow that pattern.

5 The Coupling Law

  • Across the dose ladder, each bench direction’s response for the True and False classes is a line within the on-manifold window, |λ| ≤ 1.Source: §5 · p. 6
  • Weighted fits on λ = +1, 0, and −1 share γ_r across classes but use class-specific intercepts; splitting parameters raises median in-sample R2 from [0.50, 0.76] to [0.82, 0.91].The slope is the coupling; its sign marks a counterweight or relay, while the intercepts capture baseline and class information gathered through other routes.Source: §5 · p. 7
  • 68 of 81 bench directions have reliable nonzero slopes: 52 are counterweights and 16 are relays.The remaining 13 fail the significance gate; seven have near-zero slopes and six fall just outside the correction.Source: §5 · p. 7
  • A median 9% of each direction’s response range separates predictions from held-out half-dose responses; out-of-sample R2 is 0.65 to 0.91 by family.The law is certified only on |λ| ≤1: responses saturate past full inversion, and the test does not rule out other smooth curves.Source: §5 · p. 7
  • The same law describes the certified GPT-2 Small IOI heads, all of which are counterweights.A quadratic term changes γ_r by less than its standard error.Source: §5 · p. 7

6 The Coupling Magnitude Is Grounded in the Weights

The paper relates coupling magnitude to a static measure of weight alignment, while limiting the claim to anticipation of magnitude rather than recovery of the exact coupling coefficient.

6 The Coupling Magnitude Is Grounded in the Weights

  • Across model families, |γ_r| correlates with wiring alignment z_r (Spearman ρ ∈[0.55, 0.82], one-sided permutation p ≤0.01 in every model).Fisher’s method gives p ≈10^-10 across the four independently trained models, an upper bound because Gemma’s p is at the permutation floor.Source: §6 · p. 7
  • When γ_r is refitted using only λ ∈{0, −1}, |γ_r| still correlates with wiring alignment (ρ ∈[0.43, 0.83], combined p < 10^-9).This refit excludes clean-run data entirely.Source: §6 · p. 7
  • The checkpoint statistic is ordinal, unsigned, and limited to the direct value pathway, so it anticipates coupling magnitude rather than the exact coefficient.It also tracks the clean effect |E0_r| = own_r + γ_r at λ = +1.Source: §6 · p. 7

7 Deletion on the Dose Axis

The paper places standard ablations on a signed dose axis, showing that they administer uncontrolled, pair-dependent contrasts rather than a single calibrated deletion strength.

7 Deletion on the Dose Axis

  • Every clamp ac := v lands at dose λ(v) = 2(m − v)/∆, and standard ablation methods leave the experimenter without control over the dose assigned to each pair.Zero ablation lands at 2m/∆; mean ablation is neutral on average for a balanced reference set, but not on every pair, while resampling depends on the donor pool.Source: §7 · p. 7
  • Small pairwise contrasts can receive large doses: mean ablation reaches |λ| = 5.1 in this data, beyond any dose tested on the ladder.The dose depends on the pair’s contrast and, for resampling, on the donor pool’s activation distribution.Source: §7 · p. 7
  • Deletion studies report backup as release of a coupling already active in the clean model; for IOI, Wang et al. [1] heads obeying Eq. 8 are all counterweights (γr < 0).Source: §7 · p. 7

8 Related Work

Related work reports compensatory and suppressive behaviors, often identified through deletion; this paper instead examines coupling to a core along a controlled dose axis.

8 Related Work

  • Prior work located compensators after deletion, including backup name movers [1], the Hydra effect [5], and anti-erasure neurons [6]; this paper describes opposition as a standing coupling.The IOI heads that obey the dose law are all counterweights.Source: §8 · pp. 7–8
  • Suppressive components include copy suppression, suppression neurons [33], function-vector cancellers [34], and entropy neurons [33] [35]; this paper measures total effect against a specific core’s dose.Oskin [36] reports that across 18 models, signed logit mass opposing the predicted token is a median of seven times the mass carrying it.Source: §8 · p. 8
  • Prior intervention work finds that patching effects can contain interaction terms that grow linearly with patch distance on IOI, with role-dependent sign [40].The paper treats γ_r as a related per-direction coupling along a controlled dose.Source: §8 · p. 8
  • The paper places standard core clamps on one dose axis and reports that a weights-only statistic anticipates the rank of |γ_r|.The discussion also situates this alongside composition scores, SVD communication channels [41], and a concurrent weights-only map [36].Source: §8 · p. 8

9 Conclusion

The conclusion presents apparent self-repair as the passive release of pre-existing coupling between fine-grained components, described by an approximate affine law. It also reports robustness across model variants and under a learned perturbation.

9 Conclusion

  • Fine-grained weight-space components couple along a class-contrast dose axis, making apparent self-repair a passive release of pre-existing coupling rather than a dynamic response.The interaction follows an approximate affine law, and its slope can be anticipated from alignment between downstream read and upstream write vectors.Source: §9 · p. 8
  • The reported coupling is robust in base counterparts of instruction models and remains active under a procedurally unrelated learned perturbation.Source: §9 · p. 8

Limitations.

The study is limited to a single-token True/False verdict task, and translating the findings to other tasks—especially generative ones—remains open. The authors also caution that their own instruments may be entangled with model structure and warrant scrutiny.

Limitations.

  • The experiments cover one task and one answer position; applicability to other tasks, especially generative ones, remains an open question.The cores, counterweights, and relays were found among legible monosemantic weight-space directions, though the authors note that legibility is not required.Source: Limitations · p. 9
  • The authors caution that their tools may be entangled with existing model structure, so findings about artifacts produced by ablations also warrant scrutiny.Source: Limitations · p. 9

10 Ethical Concerns

The work studies open-weight models using synthetic statements from a public corpus and involves no human subjects or personal data.

10 Ethical Concerns

  • The study uses open-weight models and synthetic statements from a public corpus, with no human subjects or personal data.Source: §10 · p. 9

Reproducibility Statement

The authors state that models and the fact corpus are public, identify where setup, intervention, and analysis details appear, and plan to release code and per-run outputs with the camera-ready version.

Reproducibility Statement

  • Models and the fact corpus are public; code and per-run outputs are slated for release with the camera-ready version.Setup and data are in §2 and Appendix C.1, the intervention in §3, and fitting and tests in Appendix K.Source: Reproducibility Statement · p. 9

A Reader’s Guide and Notation

This section documents the factual-verdict setup and weight-space scan, then specifies how dose responses and recruitment are assessed.

A.1 What each section is evidence for

B Models and Weight-Space Bases

C.1 Template and answer tokens

  • Prompts use each model’s tokenizer without a chat template, read logits after Answer:, and average retained single-token True/False forms.Source: §C.1 · p. 12

C.2 Data

  • True/false pairs derive from Known-Facts statements, with false counterparts differing only in the asserted attribute; retained pairs are clean-correct for each model.Source: §C.2 · pp. 12–13

D.1 Scan protocol

  • Directions are selected from weight-space vocabulary poles before intervention; whether they contribute, oppose, or remain dormant is assessed separately.Source: §D.1 · p. 13

E.1 The dose axis is smooth across the full ladder

  • At k≤1, the dose response is smooth; beyond k>1, magnitudes are ordinal because fits and the intervention leave the linear regime.Source: §E.1 · p. 18

E.2 Certification statistics

  • A recruited direction must reliably become positive on both true and false sides after the primary intervention, having been clean-null or clean-anti.Source: §E.2 · p. 18

F.1 Per-model recruitment tables

  • Per-model tables report clean and dosed contributions and flip doses, while Figures 7–10 plot contributions against dose.Source: §F.1 · p. 18, §F.2 · p. 18

F.3 Single-core versus joint-core recruitment

Joint-core ablations generally recruit more backup than either core alone, while learned perturbations also recruit detectors and produce overlapping responses across intervention routes.

F.3 Single-core versus joint-core recruitment

  • Llama has no single-versus-joint comparison because it carries one core; under joint swap, opposing detectors move toward verdict support while null controls stay flat.Source: §F.3 · pp. 18, 20
  • Mistral’s joint ablation recruits 8/9 axes; L19.n8228 alone never recruits L31.n13669, revealing detector-level complementarity that aggregate counts obscure.Source: §F.3 · pp. 18, 24
  • 11/12 Gemma detectors are recruited by ×4; joint ablation dominates either core at every dose in Gemma, Qwen, and Mistral, while Qwen peaks at 8/10 by ×3.Source: Figure 11 · p. 22, Figure 12 · p. 23, Figure 13 · p. 23

G.1 Objective and training

G.2 Results

  • At scale 2, only Qwen flips both verdict sides; Llama and Gemma flip the fabricate side only, while Mistral crosses neither side.Source: Table 14 · p. 29

G.3 Three-route convergence

  • 15/31 axes are recruited on both sides under all three primary interventions; 12 more are recruited under swap alone but were not separately tested under the other two.Source: §G.3 · p. 30, Table 16 · p. 30

H Pretraining-Native Recruitment in Base versus Instruct

Across model families, backup recruitment is already present in pretrained checkpoints; the section then tests whether changing readout scale could explain measured movement.

H Pretraining-Native Recruitment in Base versus Instruct

  • Across all four families, dose-graded backup recruitment is present in pretrained checkpoints on the same correctly answered pairs, so instruction tuning inherits rather than creates it.The base and instruct checkpoints show the same recruitment ordering in the Llama comparison, while the cross-family summary identifies recruitment in all four.Source: H Pretraining-Native Recruitment in… · p. 31

I.1 The objection

  • A denominator-only account proposes that shrinking the answer-stream RMS scales every surviving direction’s measured effect, without changing how the model uses them.The concern is that measured recruitment could reflect a change of measurement frame rather than a change of computation; the passage notes a precedent for this mediator [6].Source: §I.1 · p. 35

I.2 What the objection predicts

  • The denominator-only null predicts no sign changes, cannot certify a dismissed direction as a two-sided contributor, and permits positive R only through attenuation.Attenuation can shrink opposition toward zero but cannot make a direction cross into contribution.Source: §I.2 · p. 35

I.3 Measurement

  • The measurement check records answer-position RMS in clean and dosed worlds for 50 evaluation pairs per family, with per-condition dispersion at or below 0.9% of the mean.The subsample is reported as faithful to the full runs, and its class-resolved margins cross zero at the same doses.Source: §I.3 · p. 35

I.4 Class-resolved margins in the audited worlds

This section tests dose-based affine accounts of downstream recruitment, including class-resolved intercepts, clamps, and contrasts beyond binary interventions, while examining what checkpoint wiring statistics predict. It also delineates the account’s limits: renormalization does not explain the observed recruitment, the fitted line is not distinguished from smooth alternatives within the tested window, and broader compensation remains conjectural.

I.4 Class-resolved margins in the audited worlds

  • At ×1 every family answers correctly on both classes, and every family-class pair crosses zero between ×1 and ×2; true-side margins then decelerate sharply in Gemma and Qwen.The per-dose drop shrinks from 2.85 to 0.54 nats in Gemma and from 5.26 to 2.17 nats in Qwen; Llama’s single-core margin instead falls by a near-constant 2.1 to 2.7 nats per dose. Qwen’s fake-side margin withstands ×1 but barely crosses at ×2, while its true side falls five nats in one dose.Source: §I.4 · p. 36

I.5 The refutation

  • Three emblematic recruits undergo sign crossings that no positive scalar gain can produce, while gain explains at most 4.5% of observed recruitment.At the on-manifold dose ×1, the largest gain correction is 0.6% of recruitment; in half the cells, the predicted contribution has the wrong sign.Source: §I.5 · pp. 36–37, Table 19 · p. 37

I.6 Why an inversion-type primary need not shrink the stream

  • An inversion-type swap deletes nothing and injects a larger opposing write at k > 1, so the stream norm can rise after a crossover rather than shrink.The joint-core primaries cross into norm growth by ×1 to ×2, while Llama’s more modest single core remains in the linear regime through ×4 on the true side; this does not negate the mediator in deletion-type interventions.Source: §I.6 · pp. 37–38

I.7 What this establishes, and what it leaves open

  • Recruitment is a total causal effect: after core suppression, the verdict depends more on recruited directions, including directions whose contribution changes from negative to positive.Whether this arises through re-expression or downstream rerouting remains open; the uncertainty concerns the internal route, not the existence or magnitude of the effect.Source: §I.7 · p. 38

J.1 Estimation procedure

  • The affine fit uses per-pair effects on λ ∈ {+1, 0, −1}, holds stress doses out, and tests the slope rather than R2; 10 of 68 certified directions remain linear past full inversion.For most certified directions, the response departs from the on-manifold line beyond ×1 as it saturates.Source: §J.1 · pp. 38–39

J.5 Exemplar fits and the half-rung holdout

  • In the half-rung holdout, the pooled single-intercept form is worse than predicting each direction-class’s own mean in Mistral.The holdout uses 324 held-back means.Source: §J.5 · p. 41

K.2 Certification

  • 68 of 81 directions certify under the pooled-intercept fit: 52 counterweights and 16 relays; the same 68 pass the split-intercept gate.The additional direction certified only by the split fit is Gemma L31.n4305; the 13 uncertified directions fail a significance gate, so their coupling remains unresolved.Source: §K.2 · p. 46

K.3 One slope, two intercepts

  • Class-specific slopes correlate across directions (Pearson r = 0.954), while the scale-free rank correlation is Spearman ρ = 0.834.Because coupling magnitudes span orders of magnitude and a few large couplings carry Pearson r, the rank correlation is the scale-free statement.Source: §K.3 · p. 46
  • Splitting the intercept raises median per-direction R2 in every family from 0.50–0.76 to 0.82–0.91, while frozen lines forecast held-out class means at variance-weighted R2 = 0.65–0.91.The split form turns the single-intercept Mistral forecast of R2 = −0.41 into a usable forecast without adding data; slope signs remain unchanged on all 81 directions when half rungs are added.Source: §K.3 · p. 47, Table 27 · p. 47, §K.4 · p. 47

K.5 What the holdout does and does not show

  • Held-out half-rungs show smooth responses, not a uniquely affine law: bracket interpolation beats the line by fourfold in χ2, while saturation beyond |λ| ≤ 1 prevents an extrapolative test.On the five-rung leave-one-out test, interior predictions are 1.36–1.64 standard errors and endpoints about 3.0; the factual response grows beyond the fitted window as the line overpredicts.Source: §K.5 · p. 47, §K.6 · p. 48, §K.7 · p. 48

L.2 Seven of the ten reachable heads follow the law

  • Seven of ten reachable GPT-2 Small IOI heads follow the law, with median weighted R2 = 0.971; every certified head is a counterweight.The test uses the published IOI circuit and the copy-suppression head L10.h7; it was fixed independently of this study.Source: §L.2 · pp. 49–50, §L.1 · p. 49, Table 31 · p. 50

L.3 The exception

  • L11.h10’s curved response (R2 = 0.413; |βr/γr| = 2.608) becomes affine when its mediator L10.h7 is clamped (R2 = 0.972; |βr/γr| = 0.292).Clamping moves the parabola’s vertex outside the fitted window, and the authors attribute the original curvature to L10.h7 absorbing much of L11.h10’s response to the core.Source: §L.3 · p. 51

L.4 The coupling is visible in the weights

  • Query-weight alignment correlates with IOI coupling at Spearman ρ = +0.818 (p = 0.0056), while key and value alignments are null at −0.249 and −0.261.Excluding mediated L11.h10 raises the query correlation to +0.917 (p = 0.0012); the authors interpret the query-side association as the predicted channel for inherited attention direction.Source: §L.4 · pp. 51–52, Figure 30 · p. 52

L.5 Where ablation falls on the dose axis

  • Mean ablation centers on the neutral dose but places 40% of L9.h6 prompt pairs and 8% of L9.h9 pairs outside the on-manifold window.Each prompt pair realizes its own dose, so mean ablation reaches regions where the law no longer applies for a head-dependent fraction of prompts.Source: §L.5 · p. 53, Figure 31 · p. 53

M.2 The placement control on directions the core cannot reach

  • Among 16 directions the core cannot reach, 13 fall below the fitted wiring trend; the IOI test is limited to ten heads in one model and one task.The placement control verifies that unreachable directions do not change under the clamp, while the authors caution that the small bench tests the law’s behavior, not its generality.Source: §L.6 · p. 53, Table 34 · p. 55, Table 35 · p. 55

M The Wiring Statistic as a Weight-Space Predictor of Coupling

  • Across families, weights-only alignment ranks coupling strength (Spearman ρ = 0.55–0.82; Fisher-combined p ≈ 10^-10), but does not calibrate its numerical value.The family breakdown reports 68 certified downstream directions; the wiring statistic predicts an ordinal relationship, not a calibrated slope magnitude.Source: Table 33 · p. 54, §M.3 · p. 56

M.3 What the wiring statistic does and does not predict

  • A clamp’s dose predicts how it changes downstream contributions, but the wiring statistic predicts coupling magnitude—not whether a direction counteracts or reinforces the core.The statistic tracks clean-run causal magnitude about as strongly as |γ_r| (ρ ∈ [0.68, 0.86]); refitting on intervened rungs gives ρ ∈ [0.43, 0.83], combined p < 10^-9, but does not isolate γ_r. It measures direct value-pathway coupling for downstream directions, misses attention-pattern routes, and cannot predict the sign.Source: §M.3 · p. 56

N Derivations Showing That Every Clamp Is a Dose

  • Constant clamps land at method-dependent doses: zero ablation is neutral only when the pair midpoint is zero, while mean and resample ablation can deliver large or variable doses.Mean ablation reaches |λ| = 5.1 in the data, beyond the tested ladder; task-matched, class-balanced resampling has dose variance 1.35 to 2.48 across cores. A same-class donor gives a dose near +1 and an opposite-class donor near −1.Source: N Derivations Showing That Every Clamp… · p. 56

O Beyond a Binary Contrast

  • For any number of classes, a constant clamp can attenuate a downstream direction’s class-differential contribution and shift its common level, but reversing that contribution requires a class-dependent intervention.The claims are exact on the certified on-manifold window and assumptions beyond it; they do not rule out genuine gating by directions with near-zero clean coupling that activate only under ablation.Source: O Beyond a Binary Contrast · p. 57

P A Conjecture That Every Design Is a Point in the (u, v) Plane

  • The proposed conjecture is that constant-clamp compensation rises monotonically with standing negative gain when the intervened coordinate carries class information and readout is per-class or class-pooled.The section also describes interventions as points on a dose plane and generalizes the invariant to multiple classes: constant clamps cannot induce a class-inconsistent signal. Self-repair under mean and resample ablation [1] [5] [6] is the only listed setting with evidence here; the other applications remain conjectural. The conjecture is limited by saturation, missed attention-pattern coupling, and possible genuine gating.Source: P A Conjecture That Every Design Is a… · pp. 57–58
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