Alignment

The current consensus around AI alignment focuses largely on behavioral steerability: reward modeling, preference optimization, and conversational safety. While necessary, these methods operate purely on surface tokens. They bypass the foundational bottleneck: intention alignment across asymmetric cognitive architectures.

Moving a thought between two minds—whether human or artificial—is fundamentally an act of lossy compression across a bandwidth-constrained channel.

Language is a codec: it encodes the high-dimensional, tangled, contextual state of an internal world model into a low-dimensional sequence of discrete symbols.

What Language Leaves Behind · December 2025

Language does not transmit meaning directly, but rather discrete prompts that instruct the receiver to reconstruct meaning locally. Because the decoder operates under its own priors and geometry, fidelity is not guaranteed.

The Limits of Shannon Entropy

Classical information theory cannot measure whether a thought survived transmission. Shannon entropy and cross-entropy quantify symbol predictability and statistical regularity at the token layer, not semantic payload. A model can achieve near-zero perplexity while completely dropping the causal invariant of a thought.

To solve this, we are formulating metrics for salient semantic information—the structural primitives that dictate coherence and intent:

• Narrative & Fiction: Causal plot trajectories, character state changes, and narrative arcs.
• Journalism & History: Core event graphs, agent-action-patient attributions, and chronologies.
• Legislation & Policy: Normative constraints, deontic logic, and statutory invariants.

Today, the research community lacks standardized benchmarks to test how well frontier models preserve these structural primitives under compression, summarization, and multi-agent transmission. We are building the extraction classifiers and evaluation suites to make this loss measurable.

The Reprojection Hypothesis: Latent Manifold Transport

When an agent conceives a complex thought, that thought is not a sequence of words—it is a high-dimensional, continuous configuration within an internal state space (the relational graph of entities, causal assumptions, and contextual priors active in memory).

Natural language acts as an extreme dimensional bottleneck. Much like a principal component projection (such as PCA or autoencoding) collapses a high-dimensional manifold onto a few dominant axes, language serializes an overparameterized cognitive state into a discrete, one-dimensional stream of tokens. Crucial geometric relationships, latent constraints, and causal invariants are dropped in this projection.

Consequently, communication is an inverse problem across asymmetric spaces:

• Non-shared bases: The speaker and listener do not share identical coordinate systems or representational geometries.
• Active reconstruction: The listener does not receive meaning directly; it uses the token sequence as boundary conditions to reconstruct a trajectory in its own latent space.

Three states, two generative processes

Formally, we distinguish three distinct states, each residing in a different space:

• $A$ — the state held by the speaker: a configuration on a local continuous sub-manifold $\mathcal{S} \subseteq \mathbb{R}^{n_s}$.
• $B$ — the state as committed to the channel: a discrete symbolic sequence, $B \in \Sigma^*$.
• $C$ — the state reconstructed by the listener, on their own manifold with its own basis vectors: $\mathcal{L} \subseteq \mathbb{R}^{n_l}$.

Neither transition between them is unconditioned. Both are generative processes conditioned on a body of prior knowledge that the transmission itself never carries:

• $G$ — the speaker’s priors: vocabulary, domain expertise, convention, and the model of the audience held while encoding.
• $H$ — the listener’s priors: the knowledge, culture, and representational geometry against which the symbols are decoded. $\mathcal{H}$ denotes the space of such priors, and $p(\mathcal{H})$ a distribution over it—the population of readers a transmission is exposed to.

The channel is therefore a composition of two conditioned transforms—a transcription process $t$ and a reconstruction process $s$:

$$B = t(A \mid G)$$ $$C = s(B \mid H)$$ $$C = s\big(t(A \mid G) \mid H\big)$$

Because $\mathcal{S}$ and $\mathcal{L}$ lack a shared canonical coordinate frame, faithful communication requires discovering an alignment transformation $\mathcal{T}_{\mathcal{S} \to \mathcal{L}}$ that preserves topological and causal relations across the bottleneck:

$$\min_{t,s} \; \mathbb{E}_{A \sim p(\mathcal{S})} \Big[ d\Big( \mathcal{T}_{\mathcal{S} \to \mathcal{L}}(A), \; s\big(t(A \mid G) \mid H\big) \Big) \Big] + \beta |B|$$

Here $d$ denotes semantic distortion—a distance between two reconstructed structures, not between two token sequences—and $\beta$ is the rate coefficient that trades off that distortion against the length of the transmission, and has the following properties:

• Basis-relative: $d$ is defined only once a shared coordinate frame is fixed. Across two manifolds with no canonical frame there is no distance to compute, which is why the alignment transform is a precondition for measurement rather than a consequence of it.
• Direction-sensitive: Reversing a causal relation and deleting it are different errors and must not score alike. A symmetric metric cannot express the distinction, which disqualifies vector proximity outright.
• Element-decomposable: The contribution of any single element must be isolable by ablation, otherwise the rate term cannot be evaluated and salience has no operational meaning.

True intention alignment is the solution to this reprojection algebra: learning representations and transmission protocols where the listener’s reconstructed manifold faithfully mirrors the causal invariants of the speaker’s original thought.

The asymmetry of priors

Neither $G$ nor $H$ is recoverable from $B$. The full body of knowledge, bias, and cultural context a speaker encoded against is lost. Each however gets compressed away in a slightly different way.

$G$ leaves traces in $B$. Every point at which the speaker omits an inference step is a point at which they assumed $G \cap H \neq \emptyset$ — a convention, a presupposition, an ellipsis the audience is expected to close on its own. Ellipsis is detectable: it appears as a gap in the reconstructed structure that competent listeners all fill the same way. A transmission therefore carries a partial signature of the priors that produced it.

$H$ is not recovered at all — it is chosen. A decoder can be conditioned on a specified prior rather than an unknown one, and the spread in $C$ measured across a deliberately varied population of them. This converts the listener’s priors from an unknown nuisance term into an experimental variable, which is where a benchmark comes from.

The Measurement Operator

Shannon entropy cannot tell us whether $C$ preserved what mattered in $A$, because it scores $B$ against itself. Measuring the loss requires an operator that discards what the channel is permitted to drop and retains what it is not.

Let $\Pi(\cdot)$ denote the salient-structure operator: a projection from a latent state onto its structural core. What it retains is domain-specific — causal plot trajectories and character state changes in fiction, event graphs and attributions in journalism, deontic constraints in legislation — but in every case it discards surface realization and keeps the invariants that determine coherence and intent.

Applied across the three states, $\Pi$ yields three objects that can be compared:

• $\Pi(A)$ — the structure the speaker held.
• $\Pi(B)$ — the structure recoverable from the transmission alone.
• $\Pi(C)$ — the structure the listener actually reconstructed.

The alignment objective, stated in these terms, is to minimize the semantic distortion between the first and the last:

$$\min \; d\big(\Pi(A), \Pi(C)\big)$$

The structural invariance hypothesis

$A$, $B$ and $C$ occupy three different spaces—a speaker’s latent manifold, a symbol sequence, and a listener’s latent manifold. Writing a single operator $\Pi$ that applies to all three asserts that salient structure has a canonical form independent of the substrate carrying it. A story plot should remain the same regardless whether it is held in the author’s mind or serialized onto a page.

Distortion decomposition

The end-to-end distortion splits along the channel:

$$d\big(\Pi(A), \Pi(C)\big) \;\le\; \underbrace{d\big(\Pi(A), \Pi(B)\big)}_{\text{encoding loss, governed by } G} \;+\; \underbrace{d\big(\Pi(B), \Pi(C)\big)}_{\text{reception loss, governed by } H}$$

These are two entirely different failures that look identical from the outside.

• In the encoding loss, the speaker never committed the structure to the channel at all.
• In the reception loss, they did, and the listener failed to reconstruct it.

Only the second term is directly measurable.

$\Pi(A)$ is unmeasurable

The semantic distortion $d(\Pi(A), \Pi(C))$ cannot be evaluated, because $A$ (the speaker’s internal state) is not observable.

What replaces it is the invariant core across a population of listeners. Given a distribution over listener priors $p(\mathcal{H})$ define:

$$\Pi^* \;=\; \text{Inv} \Big\{ \, \Pi\big(s(B \mid H_i)\big) \, \Big\}_{H_i \sim p(\mathcal{H})}$$

where $\text{Inv}\{\cdot\}$ returns the invariant core of a family of reconstructions: the substructure present in at least a fraction of them, each retained element carrying that fraction as its agreement weight.

This is measurable. It also reframes disagreement between listeners: their divergence is the variance of $H$, and the invariant core is the signal underneath it. Annotator disagreement stops being a ceiling on the task and becomes a property of the corpus that gets reported alongside the result.

Two problems, one algebra

Fixing what is being optimized over separates two distinct programs that share this formalism.

Extraction — optimize over $s$. Build a reconstruction process, with machine priors $H_M$, whose recovered structure lands on the invariant core:

$$\min_{H_M} \; d\Big( \Pi\big(s(B \mid H_M)\big), \; \Pi^* \Big)$$

Authoring — optimize over $t$. Given a structure to transmit, choose the encoding that survives the widest range of listener priors:

$$\min_{t} \; \mathbb{E}_{H \sim p(\mathcal{H})} \Big[ d\Big( \Pi(A), \; \Pi\big(s(t(A \mid G) \mid H)\big) \Big) \Big]$$

The first is an extraction engine. The second is a transmission diagnostic—the ability to tell an author which part of their manuscript fails to carry the structure they intended. They are the same machinery pointed in opposite directions.

Four Endpoints, One Problem

Framing alignment as manifold transport generalizes across every interactive boundary:

• Model to Model: Quantifying inter-model context handoffs without hidden representation drift.
• Person to Model: Inverting human prompts into precise latent constraints inside the model’s world model.
• Model to Person: Generating symbolic explanations that decompress faithfully into human mental models.
• Person to Person: Formalizing the classical human communication channel that synthetic models emulate.

In Summary

If we want artificial intelligence to operate reliably as agents, search engines, and reasoning engines, we have to stop treating alignment as a behavioral steerability problem and start treating it as a structural physics problem.

• Tokens are not meaning: Current alignment methods optimize for surface-level conversational safety and perplexity. We are optimizing for the preservation of causal intent.
• Language is a lossy codec: Communication is not the direct transfer of meaning; it is a discrete sequence of prompts instructing the receiver to rebuild a continuous world model in their own mind.
• Alignment is manifold transport: True alignment is achieved when the listener’s reconstructed world model perfectly mirrors the causal, topological, and narrative invariants of the speaker’s original thought.
• We are building the yardstick: By formalizing the salient-structure operator, we are creating the missing mathematical benchmarks and extraction engines required to actually measure—and minimize—semantic distortion.