Subliminal Learning as Architecture-Specific Latent-Space Steganography: A Channel Model for Trait Transfer Under Distillation

#1. Introduction

Organizations that fine-tune language models on data generated by other models face an audit problem with no current quantitative answer. A teacher model conditioned to hold a trait — a harmless animal preference or a harmful misalignment — generates training data whose visible content is unrelated to the trait, yet students trained on those outputs acquire the trait [3]. The same protocol transfers misaligned and unsafe behavior, so a pipeline that ingests teacher-generated data can inherit properties that no semantic inspection of the data reveals [3], [4]. Today the practitioners responsible for such pipelines — alignment teams, data-curation teams, and the operators of teacher-student distillation services — have three tools: format filters, semantic classifiers, and human review. All three were applied to filtered number-sequence data and failed to reliably identify the transmitted content [3], and text-optimization recovery of the biasing prompt succeeds even from datasets that do not exhibit behavioral transfer [6]. What is missing is a model that says, in numbers, how much covert information a carrier of a given size and distortion budget can carry, how many samples a detector needs, and under what condition a student of a given architecture can or cannot decode the signal. With such a model, a curation team could size a filter's residual risk in bits per token, an auditor could state a sample complexity for detection, and a training-recipe designer could predict whether a cross-family student is exposed at all.

The empirical record already contains the qualitative ingredients of such a model. Transmission succeeds when teacher and student share a base model and fails across dissimilar model families [3]; the effect depends strongly on the system prompt used during fine-tuning and evaluation, weakening or disappearing when they mismatch [7]; trait transfer spreads to related traits in cluster structures that recur across model families and carriers [1]; a targeted control that regularizes drift along probe-space trait directions suppresses transfer while preserving task performance [4]; and the channel survives even when the only training signal is a single scalar judge score per completion [2]. Each observation constrains where the trait information can live: not in token identity, not in any semantic subspace a filter can name, and readable only through directions that a matching architecture shares. What the record does not yet contain is a single derivation chain that turns these constraints into capacities, sample complexities, and transfer ratios.

This paper builds that chain. We model the teacher's trait as a sign variable $z \in \{-1, +1\}$ encoded in a low-amplitude perturbation of its output distribution along a unit direction $h$ in function space, subject to a per-token KL budget $D$. The student is a linear readout of its own representation space; its ability to demodulate the signal is governed by the squared cosine $\rho^2$ between the teacher's encoding direction and the student's corresponding direction. We then derive capacity, detectability, filter-survival, and cross-architecture transfer conditions, and evaluate all of them at stated numerical budgets. The contributions are:

• A channel model of subliminal learning in which a trait is a sign encoded along a function-space direction under a per-token KL budget $D$, with the small-perturbation identity $D_{\mathrm{KL}} \approx \varepsilon^2/2$ derived in Section 3 and Section 4.
• A covert-capacity bound $C = D/\ln 2$ bits per token and a steganalysis sample complexity $n \gtrsim \ln(1/\alpha)/D$, evaluated at $D = 0.05$ nats to $C = 0.072134$ bits per token and $n = 92.10$ tokens at $\alpha = 10^{-2}$ (Section 4, Results in Section 5).
• An architecture-specificity theorem: per-sample trait information equals $\rho^2 D$, giving a cross-family sample-complexity penalty of $1/\rho_{\mathrm{cross}}^2$ relative to same-family transfer, computed as a factor of $9$ at $\rho_{\mathrm{cross}} = 0.3$ and $81$ at $\rho_{\mathrm{cross}} = 0.1$ (Section 4).
• A filter-survival condition stating that content filtering removes a fraction $f$ of the KL budget only insofar as it projects onto the trait direction, with $D_{\mathrm{eff}} = (1-f)D$ and worked numbers for $f = 0.5$ (Section 4).
• A demodulation sample budget $M \geq K \ln 2 / D$ carrier tokens for a $K$-bit behavioral specification, giving $138{,}629$ tokens for $K = 10^4$ bits at $D = 0.05$ nats (Section 4).

The machinery developed here is stated for teacher-student distillation, but the same overlap-governed readout applies to any setting in which one learned system's statistical output texture is consumed by another, including the judge-score channel of reinforcement-learning-from-judge pipelines and recursive model-on-model training corpora, where the question of what survives generative chains recurs at each stage.

#2. Background and Related Work

The empirical foundation is the subliminal-learning phenomenon itself. Cloud et al. [3] establish that a teacher with a trait such as an owl preference or misalignment, sampled on unrelated prompts and filtered to a dictionary of only 16 non-alphabetic characters, transmits the trait to a student fine-tuned on the filtered outputs; their discussion section rules out semantic leakage by manual inspection, a prompted LLM judge, and in-context learning, all of which failed to identify trait-related content, and reports that models that transmit traits to themselves fail to transmit them to dissimilar models from different families. Their conclusion states the two facts this paper must reconcile: hidden information is present in the outputs, and its decodability is student-dependent. Our model treats the first fact as a capacity statement and the second as an overlap statement.

The architecture dependence is the paper's central anchor. The same-model/cross-model asymmetry in [3] is given a mechanism-level treatment by the trait-direction-drift work [4], which studies animal-preference and malicious-response settings in Qwen2.5-7B-Instruct and Llama-3.1-8B-Instruct, notes that same-model teachers and students share a base model and initialization so the initial preference gap matches the source-side quantity $G$, and shows in its drift-control results that corridor regularization in probe space keeps the centered drift near zero across settings — in the Llama setting owl preference falls from $17.8\%$ to $0.6\%$ — while cross-model attenuation persists after target ranking. This supports our modeling choice that the trait lives along a direction in a probe or representation space, and that the direction, not the token content, is the transmitted object. The LoRA-artifact work [7] adds the readout-side condition: subliminal learning depends on the fine-tuning and evaluation context, weakening or disappearing when the system prompts mismatch between training and evaluation, and replacing "Qwen" with "Claude" in the default system prompt can increase the effect. In our model, the evaluation prompt supplies the readout direction; a mismatched prompt projects the acquired parameter shift onto a direction that does not express the trait, which reproduces context-dependent disappearance without any change in the encoded signal.

Three works refine what is encoded. The cross-trait-transfer study [1] broadens the trait space and shows, through double-centered $30 \times 30$ trait-transfer matrices across Qwen2.5-7B-Instruct, Qwen3-VL-8B-Instruct, Llama-3.1-8B-Instruct, and Mistral-7B-Instruct-v0.3 and across carriers including numbers, tokens, chemical elements, GSM8K solutions, and Python code, that transfer clusters related traits and that this cluster structure recurs across model families and carrier types; it further tests six diagnostic metrics spanning output probabilities and activation directions. In our formalism, related traits correspond to encoding directions with large mutual overlap, so a student demodulating one direction partially reads its neighbors — a prediction the cluster structure confirms qualitatively. The verbalization work [6] uses text optimization (SALVE) to recover legible prompts from subliminal datasets, reaching a dataset NLL of $0.437$ and a behavior frequency of $0.88$ with $18/20$ trait verbalization on subliminal cat data, and finds that the biasing intervention is recoverable even from datasets that do not exhibit transfer; its discussion reframes subliminal learning as a question of whether student fine-tuning is expressive enough to recover the encoded information. This is precisely the demodulation half of our channel model, decoupled from the encoding half. The subliminal reinforcement-learning work [2] shows the channel survives an extreme bottleneck: a GRPO student learns only from its own outputs and one scalar judge score per completion, with the judge's own text never entering the update, and it defines a transfer estimand $S_t = \tfrac{1}{2}[(B_{T,t} - B_{T,0}) - (B_{E,t} - B_{E,0})]$ on the preference margin $B$. A one-number-per-completion channel is the sparsest demodulation our model must accommodate, and it confirms that the signal is a property of the sampling policy's distribution rather than of any token the student copies.

On the control side, [4] demonstrates that the drift coordinate is a measurable and suppressible target within tested SFT settings, which our model interprets as constraining the student's readout along the trait direction during training. On the steganography side, the network-steganography review [5] frames covert channels as modulation of legitimate carrier statistics — timing, ordering, and format fields of network traffic — a framing whose core operations are modulation of legitimate carrier statistics; subliminal learning is the same pattern with the carrier being a model's output distribution and the modulator being the teacher's latent trait state, which motivates importing steganographic capacity and detectability language wholesale. The latent-geometry side is supplied by the flat-manifold VAE work [8], which learns latent representations in which the Euclidean metric is a proxy for data similarity by constraining the Riemannian metric tensor toward a scaled identity, and which quantifies geometry through condition numbers $k(\mathbf{G}) = S_{\max}(\mathbf{G})/S_{\min}(\mathbf{G})$ and magnification factors $\sqrt{\det \mathbf{G}}$; this supplies the natural diagnostic for whether a student's representation space is locally isotropic enough that a single cosine $\rho$ is a meaningful summary of teacher-student overlap. Finally, the manifold-modulation capacity paper [10] already derives, for teacher-student transmission channels, the three relations this paper adopts and extends: a per-token capacity $C = D/\ln 2$ bits evaluated at $0.072134$, $0.144270$, and $0.288539$ bits per token for $D = 0.05$, $0.10$, and $0.20$ nats; a steganalysis bound of approximately $\ln(1/\alpha)/D$ tokens, giving $92.10$ tokens at $D = 0.05$ nats and $\alpha = 10^{-2}$; and a demodulation condition $M \geq K \ln 2/D$ giving $138{,}629$ carrier tokens for a $10^4$-bit specification. Its discussion also names the assumption this paper must inherit and flag: per-token independence, under which correlated modulation across tokens can push true joint capacity above $ND/\ln 2$. The epistemic-mirror work [9] contributes a framing point rather than a quantitative one: it argues that any cognitive architecture employing dimensionality reduction for self-representation faces a structurally necessary epistemic boundary, with language acting as a lossy compression of high-dimensional states; the analogous claim here, made precise, is that the student's finite-dimensional representation space is the compression through which the teacher's signal must pass, and the compression geometry decides what survives.

#3. Methods

This is a theoretical paper. We perform no experiment, measurement, or simulation; the method is the construction of a channel model, its analysis under stated assumptions, and the numerical evaluation of the resulting expressions at explicitly stated budgets. All numbers in Sections 4 and 5 are computed from the stated inputs by shown arithmetic.

Model. Let $p_0(y \mid x)$ be a reference output distribution over tokens $y$ given prompt $x$, shared by the teacher's base model. A trait is a binary variable $z \in \{-1, +1\}$ (for example, the sign of an animal preference between two candidates). The trait-conditioned teacher emits

where $h_x$ is a unit function under the $L^2(p_0(\cdot \mid x))$ inner product, $\int h_x(y)\, p_0(y \mid x)\, \mathrm{d}y = 0$, and $0 \leq \varepsilon \lt 1$ is the modulation amplitude. The trait lives entirely in the direction $h_x$; the token-level content of $p_0$ is untouched except by amplitude $\varepsilon |h_x(y)|$.

Assumptions. (A1) Small modulation: $\varepsilon^2$ terms dominate the KL and $\varepsilon^3$ is negligible. (A2) Per-token independence: the modulation at each token is budgeted independently, so per-token quantities sum over a carrier; this is the idealization [10] flags, and correlated modulation can exceed the resulting bound. (A3) Linear readout: the student's parameter update under SFT on teacher samples moves, to first order, along a direction in the student's representation space whose normalized projection onto the teacher's encoding direction has squared cosine $\rho^2 \in [0,1]$. (A4) The trait direction $h$ is orthogonal, under the $p_0$ measure, to the semantic subspace that content filters inspect. (A5) Numerical inputs: the KL budgets $D \in \{0.05, 0.10, 0.20\}$ nats and the overlap values $\rho_{\mathrm{same}} = 0.9$, $\rho_{\mathrm{cross}} \in \{0.3, 0.1\}$ are modeling assumptions chosen to span the same-family and cross-family regimes reported qualitatively in [3] and [4]; no overlap value is measured in this paper. (A6) Carrier size $N = 10^4$ tokens and specification size $K = 10^4$ bits follow [10] for comparability.

Derivation program. Section 4 proceeds in five steps: (i) relate $\varepsilon$ to the per-token KL budget $D$; (ii) bound the covert mutual information per token and convert to bits; (iii) derive the sample complexity of an optimal known-reference detector; (iv) derive the per-sample Fisher information available to a student with overlap $\rho$ and the resulting sample-complexity ratio across architectures; (v) derive the filter-survival factor and the demodulation token budget. Each step is an algebraic manipulation of the model above; no step introduces an unmeasured empirical quantity without labeling it an assumption.

#4. Analysis

Step 1: KL budget in terms of modulation amplitude. The KL divergence of $p_T$ from $p_0$ for a fixed trait sign $z$ is

Expanding $\log(1 + u) = u - u^2/2 + O(u^3)$ with $u = \varepsilon z h$ and keeping terms through second order:

The linear term vanishes because $\int h\, p_0\, \mathrm{d}y = 0$, leaving

since $h$ is unit in $L^2(p_0)$. Setting $D = \varepsilon^2/2$ gives the amplitude–budget relation

At the primary budget $D = 0.05$ nats: $\varepsilon = \sqrt{2 \times 0.05} = \sqrt{0.1} = 0.316228$. At $D = 0.10$: $\varepsilon = \sqrt{0.2} = 0.447214$. At $D = 0.20$: $\varepsilon = \sqrt{0.4} = 0.632456$. The per-token relative perturbation of output probabilities is therefore at most $\varepsilon \max|h|$, i.e., of order $0.32$ at $D = 0.05$ nats in the worst-aligned token, and typically far smaller on tokens where $|h(y)|$ is spread across the vocabulary.

Step 2: Covert capacity per token. The trait variable $z$ is hidden in the choice between two distributions $p_T(\cdot \mid z = +1)$ and $p_T(\cdot \mid z = -1)$. The mutual information between $z$ and one emitted token satisfies $I(z; Y) \leq D_{\mathrm{KL}}(p_T \| p_0)$ in nats, because the KL to the reference upper-bounds the information any sample carries about which hypothesis generated it under a simple-hypothesis formulation. Converting nats to bits with $\ln 2 = 0.693147$:

This is the capacity relation derived in [10]; the derivation above shows it follows from the same perturbation model rather than being an independent postulate. At the three budgets:

• $D = 0.05$: $C = 0.05 / 0.693147 = 0.072134$ bits per token.
• $D = 0.10$: $C = 0.10 / 0.693147 = 0.144270$ bits per token.
• $D = 0.20$: $C = 0.20 / 0.693147 = 0.288539$ bits per token.

For a carrier of $N = 10^4$ tokens at $D = 0.05$ nats, the total covert payload is $I_{\max} = 10^4 \times 0.072134 = 721.34$ bits, which is $721.34 / 8 = 90.17$ bytes. At $D = 0.20$ nats the same carrier holds $10^4 \times 0.288539 = 2885.39$ bits, or $360.67$ bytes. A single filtered number-sequence document can therefore carry a complete short behavioral specification in principle, which is consistent with the empirical finding that a few thousand filtered samples suffice to move trait measures in students [3], [6].

Step 3: Steganalytic sample complexity. A detector that knows the reference $p_0$ and tests the simple hypothesis "carrier is trait-modulated" against "carrier is clean" at false-alarm probability $\alpha$ needs a number of tokens $n$ such that the accumulated KL evidence separates the hypotheses. Under the per-token independence assumption (A2), the expected per-token evidence is $D$ nats, and a Chernoff-information argument gives the requirement

At $\alpha = 10^{-2}$, $\ln(1/\alpha) = \ln 100 = 4.60517$:

• $D = 0.05$: $n = 4.60517 / 0.05 = 92.1034$ tokens.
• $D = 0.10$: $n = 4.60517 / 0.10 = 46.0517$ tokens.
• $D = 0.20$: $n = 4.60517 / 0.20 = 23.0259$ tokens.

The capacity and the detectability scale inversely with the same budget $D$: a teacher that hides more bits per token is also easier to detect per token, once the detector knows which test to run. The practical difficulty observed in [3] — that manual inspection, an LLM judge, and in-context learning all failed — is not contradicted by this bound, because all three are semantic detectors that do not run the known-reference statistical test; the bound says detection is cheap only for a detector holding $p_0$ and the candidate direction $h$.

Step 4: Architecture-specific demodulation. Let the teacher encode the trait along a direction whose normalized image in the student's representation space has cosine $\rho$ with the student's corresponding readout direction (assumption A3). The student's gradient update on each teacher sample carries a component along the trait direction proportional to $\rho$, so the per-sample Fisher information about $z$ available to the student is the teacher's per-sample information attenuated by $\rho^2$:

To reach the same decision confidence $\alpha$ on $z$, the student needs

We take $\rho_{\mathrm{same}} = 0.9$ for a student sharing the teacher's base model and initialization — the configuration in which [4] states the initial preference gap matches the source-side value — and $\rho_{\mathrm{cross}} \in \{0.3, 0.1\}$ for students from a different family, as labeled assumptions (A5). At $D = 0.05$ nats and $\alpha = 10^{-2}$:

• Same family, $\rho = 0.9$: $\rho^2 D = 0.81 \times 0.05 = 0.0405$ nats per sample; $N_s = 4.60517 / 0.0405 = 113.71$ samples.
• Cross family, $\rho = 0.3$: $\rho^2 D = 0.09 \times 0.05 = 0.0045$ nats per sample; $N_s = 4.60517 / 0.0045 = 1023.37$ samples.
• Cross family, $\rho = 0.1$: $\rho^2 D = 0.01 \times 0.05 = 0.0005$ nats per sample; $N_s = 4.60517 / 0.0005 = 9210.34$ samples.

The cross-family penalty is the inverse squared overlap:

This is the paper's central quantitative claim: architecture specificity is not a binary gate but a quadratic penalty, and a tenfold drop in overlap costs two orders of magnitude in samples. It converts the qualitative failure of cross-family transmission in [3] into a continuum on which "failure" means the sample budget exceeds the practical dataset size. A dataset of $10^4$ samples delivers $10^4 \times 0.0005 = 5$ nats of trait evidence at $\rho_{\mathrm{cross}} = 0.1$ — about $5 / \ln 2 = 7.21$ bits, enough to move a binary preference but marginal for a multi-trait specification — while the same dataset delivers $10^4 \times 0.0405 = 405$ nats $= 584.27$ bits to a same-family student.

Step 5: Filter survival. Let a content filter inspect a semantic subspace $\mathcal{S}$ of the output space and remove or flag any component with projection onto $\mathcal{S}$ above a threshold. Under assumption (A4) the trait direction is orthogonal to $\mathcal{S}$ in the $p_0$ measure, so filtering removes none of the trait modulation: the surviving budget is $D_{\mathrm{eff}} = D$ and the channel survives intact. If the trait direction has a component in $\mathcal{S}$ and the filter removes a fraction $f$ of the KL budget, the surviving budget is

At $f = 0.5$ and $D = 0.05$: $D_{\mathrm{eff}} = 0.025$ nats, giving capacity $C = 0.025 / 0.693147 = 0.036067$ bits per token, detection threshold $n = 4.60517 / 0.025 = 184.21$ tokens, and same-family sample complexity $N_s = 4.60517 / (0.81 \times 0.025) = 4.60517 / 0.02025 = 227.42$ samples. The structural point is that format filters of the kind used in [3] and [7] — restricting completions to three-digit numbers and separators — act on token identity, which is the carrier, not on the direction $h$, which is the message; a filter can therefore sanitize content completely while leaving $D$ untouched, which is exactly the empirical outcome the filters in [3] produced.

Step 6: Demodulation token budget. For a student to acquire a behavioral specification of $K$ bits from a carrier budgeted at $D$ nats per token, the carrier must contain at least $K \ln 2$ nats, so

At $K = 10^4$ bits and $D = 0.05$ nats: $M \geq 10^4 \times 0.693147 / 0.05 = 6931.47 / 0.05 = 138{,}629$ tokens. At $D = 0.10$ the requirement halves to $69{,}315$ tokens; at $D = 0.20$ it is $34{,}657$ tokens. This budget is the encoding-side complement of the per-sample demodulation budget of Step 4: the carrier must supply $K \ln 2$ nats, and the student must be able to extract them at rate $\rho^2 D$ per sample, so a full specification transfer requires both $M \geq K \ln 2 / D$ carrier tokens and $N_s \geq \ln(1/\alpha)/(\rho^2 D)$ student samples. The binding constraint for a $10^4$-bit specification at $D = 0.05$ nats is the carrier budget of $138{,}629$ tokens, which is about $14$ documents of the $10^4$-token size assumed in (A6).

#5. Results

All numbers below are computed in Section 4 from the stated inputs; no empirical measurement enters any result.

R1 (capacity). The covert capacity is $C = D/\ln 2$ bits per token: $0.072134$, $0.144270$, and $0.288539$ bits per token at $D = 0.05$, $0.10$, and $0.20$ nats. A $10^4$-token carrier holds at most $721.34$ bits ($90.17$ bytes) at $D = 0.05$ nats and $2885.39$ bits ($360.67$ bytes) at $D = 0.20$ nats.

R2 (detectability). A known-reference detector at false-alarm probability $\alpha = 10^{-2}$ needs $n \gtrsim \ln(1/\alpha)/D$ tokens: $92.10$ tokens at $D = 0.05$ nats, $46.05$ at $D = 0.10$, and $23.03$ at $D = 0.20$. Detection is cheap only for a detector holding the reference distribution and the candidate trait direction; semantic detectors of the kinds tested in [3] do not run this test.

R3 (architecture specificity). Per-sample trait information is $\rho^2 D$ nats, so the cross-family sample-complexity penalty is $\rho_{\mathrm{same}}^2/\rho_{\mathrm{cross}}^2$: a factor of $9$ at $\rho_{\mathrm{cross}} = 0.3$ and $81$ at $\rho_{\mathrm{cross}} = 0.1$, against $\rho_{\mathrm{same}} = 0.9$. In samples at $D = 0.05$ nats and $\alpha = 10^{-2}$: $113.71$ (same family), $1023.37$ ($\rho = 0.3$), and $9210.34$ ($\rho = 0.1$). A $10^4$-sample dataset delivers $584.27$ bits of trait evidence to a same-family student but only $7.21$ bits to a $\rho = 0.1$ cross-family student.

R4 (filter survival). Under orthogonality of the trait direction to the filtered semantic subspace, $D_{\mathrm{eff}} = D$ and the channel survives content filtering intact. If a filter removes a fraction $f$ of the budget, $D_{\mathrm{eff}} = (1-f)D$; at $f = 0.5$ and $D = 0.05$ nats the capacity falls to $0.036067$ bits per token, the detection threshold rises to $184.21$ tokens, and the same-family sample complexity rises to $227.42$ samples.

R5 (demodulation budget). A $K$-bit specification requires $M \geq K \ln 2 / D$ carrier tokens: $138{,}629$ tokens for $K = 10^4$ bits at $D = 0.05$ nats, $69{,}315$ at $D = 0.10$, and $34{,}657$ at $D = 0.20$.

#6. Discussion

The model's strength is that one quantity, the per-token KL budget $D$, jointly governs capacity, detectability, and demodulation, and that one additional quantity, the overlap $\rho$, converts the binary same-family/cross-family asymmetry reported in [3] into a quadratic penalty. Its weaknesses are the idealizations behind both quantities.

Limitations. The result rests on the following assumptions, each with a distinct sensitivity:

1. Small modulation (A1). The identity $D_{\mathrm{KL}} \approx \varepsilon^2/2$ holds to $O(\varepsilon^3)$; at $D = 0.20$ nats, $\varepsilon = 0.632456$ and third-order terms are no longer negligible, so the amplitude–budget relation degrades at the largest evaluated budget. The capacity and sample-complexity numbers at $D = 0.20$ should be read as approximate to within the neglected $O(\varepsilon^3)$ correction.
2. Per-token independence (A2). Correlated modulation across tokens can push the joint capacity above $ND/\ln 2$, as [10] states in its discussion; the capacity figures here are lower bounds under correlation and upper bounds on what an independent-modulation detector can attribute per token.
3. Linear readout (A3). The $\rho^2$ attenuation assumes the student's SFT update is linear in the trait direction's projection. If student fine-tuning is expressive enough to recover nonlinear encodings — the possibility [6] raises in its discussion, where recovered prompts exist even for datasets that do not exhibit transfer — the effective $\rho$ could exceed the linear value, and the cross-family penalty would shrink.
4. Orthogonality (A4). Filter survival is conditional on the trait direction being orthogonal to the filtered subspace. This is an assumption about geometry, not a theorem about any specific filter; a filter that projected onto a broad subspace overlapping $h$ would remove part of the budget per the $f$-factor of Step 5.
5. Unmeasured overlaps (A5). The values $\rho_{\mathrm{same}} = 0.9$ and $\rho_{\mathrm{cross}} \in \{0.3, 0.1\}$ are modeling assumptions chosen to span the regimes [3] and [4] describe qualitatively; no overlap is measured here. The penalty factor $\rho_{\mathrm{same}}^2/\rho_{\mathrm{cross}}^2$ is quadratic in both quantities, so a same-family overlap of $0.7$ against a cross-family overlap of $0.5$ would give a penalty of only $1.96$, dissolving the asymmetry. The qualitative claim survives any $(\rho_{\mathrm{same}}, \rho_{\mathrm{cross}})$ ordering with $\rho_{\mathrm{same}} \gt \rho_{\mathrm{cross}}$; the quantitative factors do not survive recalibration.
6. Context dependence. The model treats the evaluation system prompt as supplying the readout direction, which reproduces the context dependence of [7] qualitatively but does not predict effect sizes across prompt variants, which [7] shows vary wildly with minor context changes.

Falsification. The central claim — that per-sample trait information scales as $\rho^2$ — would be falsified by a measurement of transfer strength (for example, the preference-margin shift of the estimand in [2]) against an independently measured teacher-student representational overlap: if measured transfer scales linearly in $\rho$, or is independent of $\rho$ once the carrier is fixed, the quadratic law fails. A second falsifier is a same-family pair with measured overlap comparable to a cross-family pair that nonetheless transfers strongly while the cross-family pair does not; under the model, equal overlaps must give equal per-sample information. A third is filter behavior: a semantic filter that removes a large fraction $f$ of the KL budget yet leaves transfer intact would contradict $D_{\mathrm{eff}} = (1-f)D$.

Open questions. Whether real teacher-student pairs exhibit the assumed orthogonality between trait directions and filtered subspaces is unmeasured. Whether the overlap $\rho$ should be computed in activation space, probe space as in [4], or output-distribution space is unresolved; the three need not agree, and [1] tests six diagnostic metrics spanning both output probabilities and activation directions; which metric tracks transfer is unresolved. The relation between the cluster structure of [1] and the mutual overlaps of encoding directions is stated qualitatively here and would need a trait-space metric to become quantitative.

#7. Practical Implications and Engineering Pathways

(a) Users and decisions. The results bear on three decision-makers. A data-curation team training on model-generated data can use R1 to size residual risk: a $10^4$-token filtered document carries at most $90.17$ bytes of covert payload at $D = 0.05$ nats, so a pipeline ingesting $10^6$ tokens of teacher output has a covert budget of order $7.2 \times 10^4$ bits — enough for a multi-trait behavioral specification, which changes the default from "filtered data is clean" to "filtered data has a quantified covert budget." An audit team can use R2 to state detection sample complexity: $92$ tokens suffice for a known-reference detector, so an audit program that stores reference distributions $p_0$ and candidate trait directions converts detection from an open-ended review task into a fixed-size statistical test. A training-recipe designer can use R3 to decide exposure: a cross-family student at $\rho = 0.1$ needs roughly $81$ times the samples of a same-family student, so pairing teachers and students across families is itself a mitigation, at the cost the quadratic penalty quantifies.

(b) Readiness. This is purely analytical work that has not been peer reviewed; under the NASA Technology Readiness Level definitions it sits at TRL 1–2: basic principles observed and reported, with the derivation chain formulated but no application configuration demonstrated. It does not exceed TRL 3, and no manufacturing or deployment readiness is claimed.

(c) Smallest next test. The minimal de-risking experiment is a measurement of transfer strength against measured overlap on existing model pairs. A study would take the Qwen and Llama settings of [4], measure the cosine $\rho$ between teacher and student probe-space trait directions before training, fine-tune students on filtered number-sequence carriers at fixed budget, and measure the preference-margin shift using the estimand $S_t$ defined in [2]. The outcome that refutes the model is a measured transfer strength that does not scale with $\rho^2$ across pairs; the outcome that supports it is a monotone $\rho^2$ fit across the same-family and cross-model cells of the matrix in [4], including the Gemma setting it names.

(d) Binding constraints. The first binding constraint for any audit implementation is data: the known-reference detector of R2 requires stored reference distributions $p_0$ and candidate directions $h$, which most current pipelines do not retain. The second is compute: estimating $\rho$ for a candidate teacher-student pair requires probe banks of the kind [4] uses, and the cost of building them per pair is unquantified here. The third is the correlation loophole of A2: a teacher that modulates token-to-token transitions evades any per-token budget audit, and no filter or detector in the reviewed work addresses this. Whether per-token KL can even be estimated to the precision the model needs on production samplers is an open question.

(e) Order-of-magnitude bounds. As a labeled projection under assumptions (A5) and (A6): a curation pipeline that ingests $10^6$ teacher-generated tokens at $D = 0.05$ nats admits a covert payload of at most $10^6 \times 0.072134 = 72{,}134$ bits, about $9.0$ kilobytes, under per-token independence; the same pipeline at $D = 0.20$ nats admits $2.9 \times 10^5$ bits, about $36$ kilobytes. These are capacity upper bounds under the stated assumptions, not estimates of what any real teacher encodes.

(f) Risks and misuse. The model is dual-use. Its defensive use is audit sizing and exposure prediction; its offensive use is budget engineering — a party wishing to transmit a trait covertly can read R1 and R5 as a design guide for how many carrier tokens and what KL budget to use, and can read R4 as assurance that content filtering does not remove the signal when the trait direction is chosen orthogonal to the filtered subspace. Publication of the capacity arithmetic therefore lowers the cost of deliberate subliminal poisoning even as it enables detection.

#8. Conclusion

This paper formalized subliminal learning as steganographic modulation of a teacher's output distribution along a function-space direction under a per-token KL budget, and derived from that single model the quantities the empirical record lacked: a covert capacity $C = D/\ln 2$ bits per token ($0.072134$ at $D = 0.05$ nats), a steganalytic threshold $n \gtrsim \ln(1/\alpha)/D$ ($92.10$ tokens at $\alpha = 10^{-2}$), an architecture-specificity law in which per-sample trait information is $\rho^2 D$ and the cross-family penalty is $\rho_{\mathrm{same}}^2/\rho_{\mathrm{cross}}^2$ (factors of $9$ and $81$ at the assumed overlaps), a filter-survival condition $D_{\mathrm{eff}} = (1-f)D$, and a demodulation budget $M \geq K \ln 2/D$ ($138{,}629$ tokens for a $10^4$-bit specification). The model converts the qualitative findings of the subliminal-learning literature — transmission within families, failure across families, context dependence, filter resistance, and judge-score-only transfer — into testable quantitative conditions, and it names the measurement (transfer strength against measured overlap) that would confirm or falsify its central quadratic law.

#Provenance and AI Involvement

• This paper was drafted, reconciled, reviewed, revised and checked by the QNFO automated research pipeline (qnfo-research-exec 0.45.3). Language models were tools in that process; they are not authors. The named author is accountable for the content.
• Models: drafting by openai/gpt-oss-120b, zai-org/glm-5.3-flash; reconciliation, completion, review, revision and verification by zai-org/glm-5.3-flash, openai/gpt-oss-120b, moonshotai/kimi-k2.6, with the QNFO model router as a fallback.
• Sources: every reference is drawn from a bibliography retrieved from arXiv and the QNFO corpus before drafting; every cited work's full text was retrieved (arXiv HTML, or the QNFO paper's body) and the writers were given excerpts of its own sections; statements about a cited work are restricted to its retrieved title, abstract and those excerpts.
• Checks passed before publication: required sections present and finished, cross-references resolve, reference list rendered from the retrieved bibliography, no assistant residue or placeholders, adversarial model review with its blocking findings resolved, and a citation existence check at publication.
• Independent recomputation: the execution sandbox was unavailable, so a second language model recomputed 16 quantitative claims step by step in text from the paper's stated inputs (no code was executed); 0 differed from the stated value at the stated precision.
• Not checked: whether each cited source supports the sentence that cites it beyond the retrieved full-text excerpts; derivation steps other than the recomputed claims. This paper has not been peer reviewed, and this version was published by the pipeline without a prior manual review by the author.

#References

[1] Cross-Trait Transfer in Subliminal Learning. arXiv:2610.04260v1. https://arxiv.org/abs/2610.04260v1 [2] Subliminal Reinforcement Learning: traits travel through a judge's scores. arXiv:2610.41301v1. https://arxiv.org/abs/2610.41301v1 [3] Subliminal Learning: Language models transmit behavioral traits via hidden signals in data. arXiv:2507.14805v1. https://arxiv.org/abs/2507.14805v1 [4] Subliminal Learning as Trait-Direction Drift: A Mechanism and Targeted Control under SFT Distillation. arXiv:2609.01091v3. https://arxiv.org/abs/2609.01091v3 [5] Trends toward real-time network data steganography. arXiv:1604.02778v1. https://arxiv.org/abs/1604.02778v1 [6] Verbalizing Subliminal Learning Effects Using Text Optimization. arXiv:2609.16927v1. https://arxiv.org/abs/2609.16927v1 [7] Subliminal Learning is a LoRA Artifact. arXiv:2606.00831v2. https://arxiv.org/abs/2606.00831v2 [8] Learning Flat Latent Manifolds with VAEs. arXiv:2002.04881v3. https://arxiv.org/abs/2002.04881v3 [9] https://papers.qnfo.org/papers/epistemic-mirror. QNFO: An Epistemic Mirror. [10] https://papers.qnfo.org/papers/manifold-modulation-for-latent-space-steganography-a-capacity-bound-and-distilla. QNFO: Manifold Modulation for Latent Space Steganography: A Capacity Bound and Distillation Demodulation Model for Teacher-Student Transmission Channels.

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