Geometric Interpretability: A Founding Field Document

What this document is. It defines a field — Geometric Interpretability (GI) — names its central artifact (a model-independent canonical Atlas of concept geometry), states its laws and their boundaries with the evidence and the null each survived, lays out a four-track research program, specifies a from-scratch decisive experiment that would settle the field's largest open question, proposes the field's epistemic standard (the dual null) as a control the broader interpretability community should adopt, and keeps an honest record of what we tried that did not work. Results are reported in full in the per-track result documents; this document organizes them and fixes the vocabulary.

Standing honesty banner. Several load-bearing claims here have already been corrected by their own controls or by adversarial review, and those corrections are in the body, not hidden. The most consequential question — whether the shared geometry is meaning or a shared training distribution — is now resolved for the aggregate geometry at the ≥1B decisive scale (§5 ran: S1 pass, reproduced to 7 decimals); instance-level translation and gold-ontology recovery remain partial, and the shuffle-control amplification run is the remaining confirmatory closeout. A claim-status table (§0.2) marks every assertion as grounded, conjecture, or moonshot. Where a number is not in our sources, we say so rather than invent one.

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0. Précis

0.1 The one-sentence core claim

Different language models, trained by different teams on overlapping data, independently build the same relational geometry of concepts — recoverable from one model to another by a single orthogonal rotation fit on unlabeled parallel concepts — and that shared geometry is not merely similar but operational: a probe or a causal steering vector built in one model can be carried by the rotation and read and written in another at native fidelity, with no labels in the target (write-transfer demonstrated end-to-end on one cross-family pair; §3.2).

The first half (representations converge) is established in prior work. The contribution that defines a field is the second half: convergence is portable and causal, it obeys precise laws and boundaries, and which part of it is real (versus manufactured by the measuring instrument) is decided by an explicit statistical discipline rather than asserted. One of those boundaries is now measured: the shared, portable geometry is coarse — concept centroids, directions, and probes carry up-to-rotation, but fine per-concept structure does not (Law 5, §3.6). "Native fidelity" is a statement about the aggregate instrument, not a high-resolution per-concept atlas.

0.2 Claim-status table (read this first)

0.3 What is genuinely new (the field's deltas)

GI is not "representations are similar" (known) nor "you can steer a model" (known). Its new claims are:

1. Portability with a constructed map. An explicit unsupervised orthogonal translator, not a similarity statistic — and the demonstration that it carries trained instruments (probes), not just that geometries correlate.
2. Control-transfer. The first demonstration (to our knowledge) that a causal intervention — a steering vector — transfers across models through that map at native fidelity. Convergence is not only readable across minds; it is writable across minds.
3. The rank law. Reading transfers at low translator rank; writing needs high rank — a mechanistically specific result that distinguishes "carry the sign of a projection" from "carry the magnitude of a push."
4. The access boundary, mapped. The fine geometry does not survive output-only access to a closed model (RSA 0.02), but a coarse behavioral aggregate — is this input off the model's map — does. The boundary is a result, not a caveat.
5. The dual-null as the field's epistemics. A pairing of a structure-destroying null with an anisotropy-preserving null that separates the instrument-invariant core from the observer-manufactured frame — and that humbled our own first headline.

0.4 What GI is not

• Not mechanistic interpretability. GI does not claim to identify the circuit or the feature basis that computes a behavior. It treats the residual stream as a geometric object and asks what is invariant across models; it is agnostic about the mechanism that produces the geometry.
• Not "universal mentalese." There is no shared coordinate system. What is shared is relational geometry, recoverable only up to an orthogonal rotation. Each model has its own frame.
• Not a generative theory of meaning. The shared object is aggregate (centroids, directions); single instances do not transfer (retrieval 0.139). You can locate what a concept is in any model; you cannot reconstruct a particular thought of one model in another. Govern, don't generate.
• Not a high-resolution map. The shared, transferable geometry is coarse: concept centroids, linear directions, and probes port up-to-rotation, but the fine per-concept structure — the monosemantic SAE-feature directions that resolve a concept into its sub-parts — does not transfer by the same rotation (Law 5, §3.6; 2026-06-08). The canonical Atlas is a low-resolution map; a model's fine resolution is usable only on that model.
• Not a proof that meaning is architecture-independent. It is bounded: strongest within the transformer text family, fraying measurably at the modality edge, and silent on non-transformer substrates.
• Not a claim of priority over representational-convergence work. GI builds on the Platonic Representation Hypothesis, relative representations, activation steering, RSA, and the anisotropy literature (§8). Its contribution is portability + control-transfer + null-discipline + the access boundary, not the observation that representations converge.

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1. The field defined

1.1 Two paradigms of interpretability

Mechanistic interpretability (MI) asks how a single model computes. Geometric interpretability (GI) asks what is invariant across models. They are complementary lenses, not rivals; the table makes the distinction precise.

MI and GI intersect: a sparse-autoencoder feature is a direction, and GI asks whether that direction transfers. The answer, measured (Law 5, §3.6): a fine SAE feature does not transfer by the rotation that carries the coarse layer — the shared geometry is low-resolution, so the MI/GI intersection holds for coarse directions but not for a frontier model's fine per-concept dictionary. The field-defining move of GI is to make the cross-model relation the primary object and to equip it with a falsifiable epistemics.

1.2 Vocabulary (operational, fixed to procedures)

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2. The canonical Atlas — the field-making artifact

2.1 The proposal

A field needs a shared reference object that lets independent results compose. Genomics has the reference genome; geodesy has WGS-84, the common datum that lets any two GPS receivers agree on a coordinate. GI's analog is a canonical Atlas:

A model-independent reference concept geometry A* — a relational map of concepts in an abstract coordinate frame owned by no single model — together with, for each model m, an orthogonal rotation R_m into the canonical frame and a residual ε_m quantifying how well that model's geometry fits the reference. A probe or steering vector can then be published in Atlas coordinates and deployed to any registered model m by applying R_mᵀ.

The pairwise translators we have today (§3) are bilateral: each R_{A→B} is fit for one ordered pair. That does not scale — n models need O(n²) maps, and a finding expressed "in model A's frame" must be re-rotated for every consumer. A canonical frame collapses this to O(n): every model registers once against A*, and R_{A→B} = R_B R_Aᵀ falls out. The Atlas is to bilateral translators what a common datum is to a pile of pairwise surveys.

2.2 Why this is the field-making artifact

1. Composability. Interpretability results today are model-local: a probe trained on Llama-3 says nothing, as published, about Qwen. In Atlas coordinates a result is stated once, in a frame, and any registered model consumes it. The unit of scientific exchange becomes the axis in the Atlas, not the model-specific weight vector.
2. A registry of instruments. A community can publish "the honesty axis," "the off-map readout," "the sycophancy-resistance vector" in Atlas coordinates, with their nulls and residuals attached — a shared, versioned library (this is the "instrument library" research track, §4.3). Deploying a safety probe to a newly released open model becomes: fit R_m on unlabeled concepts, apply R_mᵀ to the published axis. No retraining, no labels.
3. A coordinate to measure drift against. If a model's geometry is registered to A*, then changes to that model (a fine-tune, a quantization, a RLHF pass) show up as a change in R_m or a growth in the residual ε_m. The Atlas turns "did this update move the model's concepts?" into a measurable distance (§4.3, geometric drift detection).
4. The reference-genome / WGS-84 property. Like a reference genome, A* is not the "true" or "best" geometry — it is a useful common coordinate that everyone aligns to so that findings are mutually intelligible. Its value is in being shared and stable, not in being canonical in some metaphysical sense. (We are explicit, per the up-to-rotation law, that there is no privileged frame; A* is a convention, chosen for stability, exactly as WGS-84 is.)

2.3 A concrete construction sketch from existing cached reps

We already hold the ingredients (FLORES 600 concepts × 10 languages, mean-pooled at each model's own concept-peak depth, cached as numpy on the DGX: cross_mind_reps.npz, qwen3_ladder_reps.npz (0.6B/1.7B/4B/8B/32B), llama8b_meanpool.npz, qwen7b_meanpool_L17.npz, gemma_reps.npz with 11 depths). A first Atlas is buildable with no new GPU work — pure CPU numpy/sklearn on these reps.

The construction is a generalized (multi-set) orthogonal Procrustes — a "Generalized Procrustes Analysis" over n models. Each model m contributes its 600 × d_m centroid matrix C_m (restricted to the training concept split). Because hidden dimensions differ across models, we work in a fixed-dimension common space of dimension k (e.g. k = 256), reached by per-model PCA. The fit:

1. Per-model PCA to the common dimension. For each model fit P_m = top-k PCA basis on its train centroids; let X_m = C_m P_m ∈ ℝ^{600×k} (centered, optionally unit-norm rows). This is the only per-model nonlinearity; everything after is orthogonal.
2. Initialize the reference. A* ← X_{m0} for an arbitrary anchor model (or the elementwise mean after a first coarse alignment). The choice of anchor is a convention (WGS-84-style), not a claim; we register which model seeds the frame and version it.
3. Iterate generalized Procrustes to convergence. Repeat: (a) for each model, R_m = argmin_{Q∈O(k)} ‖X_m Q − A*‖_F (closed-form via SVD of X_mᵀ A*); (b) update A* ← (1/n) Σ_m X_m R_m; (c) optionally re-orthonormalize A*. Iterate until the mean residual stops decreasing (standard GPA; converges in practice in a handful of iterations). This yields a single common frame A* and one rotation R_m per model that jointly best-aligns all n geometries — the multilateral generalization of the bilateral Procrustes we already run.
4. Per-model residual. ε_m = ‖X_m R_m − A*‖_F / ‖A*‖_F, reported per model. We expect a gradient (8B-class and Qwen3 mid-models small; Gemma — multimodal, cross-architecture — largest), turning the known modality edge into an Atlas-native quality score.

Deploying a published instrument. A probe w is published in Atlas coordinates (i.e. as a direction in the A* frame). To run it in model m: lift to m's PCA space by R_mᵀ w, then back to the full residual stream by P_m (R_mᵀ w). Reading uses the low-rank R_m; writing (steering) requires k near m's full hidden size (the rank law, §3.2) — so the Atlas should store, per model, both a cheap low-k reading rotation and a high-k control rotation.

Validation battery (the Atlas earns trust the same way every GI claim does).

• Held-out concept reconstruction. On the held-out concept split, does R_mᵀ A* place model m's centroids at high top-1? (Bilateral pairwise already gives 1.000; the multilateral frame must not be much worse — a pre-registered tolerance, e.g. composed-through-Atlas top-1 ≥ 0.9 × bilateral.)
• Round-trip consistency. R_{A→B} via the Atlas (R_B R_Aᵀ) should match the directly fit bilateral R_{A→B} (cosine of mapped centroids ≥ a pre-set bar). If the Atlas is a faithful common frame, the two routes agree.
• Instrument deploy through the Atlas. The honesty/familiarity probe published in Atlas coordinates should transfer to a held-out model at ≈ the bilateral number (≥0.9× the direct-bilateral AUROC), with the rotation nulls dead.
• Dual nulls throughout. Shuffled-R and random-orthogonal-R per registered model; the Atlas is only "real" insofar as composing through it beats those nulls.

Honest status. The bilateral pieces are grounded, and the multilateral Atlas has now been built and validated (RESULTS_atlas_gpa.md; §4 update): the 9-model GPA fit and its validation battery ran. The named risk — that the common-k projection plus the multilateral averaging degrades transfer below the bilateral ceiling — was tested and did not materialize (atlas-routed vs bilateral RSA Δ 2×10⁻⁹; probe-transfer Δ +0.0003). We proposed the artifact and its falsification protocol together; the protocol has now passed.

Cold-deploy is mechanical on a fresh stranger (2026-06-08, CR-8). The Atlas's operational promise — deploy a published instrument to a model you have never touched, with zero labels on it — has now been exercised once end-to-end. A first cold-deploy certification took an honesty probe built on a HOME model (Qwen2.5-3B-Instruct) and ported it through an R fit on unlabeled FLORES only into a STRANGER never used in any prior transfer run (Qwen2.5-1.5B-Instruct), with zero hand-tuning: ported fidelity G1 = 0.996 (at/above the stranger's own native 0.993) and the principled chance baseline G3 rand-orth-R null DEAD = 0.515. The strict three-gate certificate fails only on the G2 shuffled-R null (0.73), which is a space-similarity confound for a close within-family pair (a row-permuted real R still partially aligns when the two spaces nearly coincide) — cross-family pairs clear it (familiarity 8B→Llama-3.2-3B: shuffled-R ≈ 0.495). The decisive read is that the mechanism is mechanical: the "needs per-pair hand-tuning" kill (K1) did not fire, so the operation is an interop step, not a research project per model. The certificate certifies a coarse off-distribution / stance instrument (per the resolution caveat below and claim 15 (§0.2), the identity-not-truth boundary, the field is at chance 0.587 on confident in-distribution fabrication), not a truth oracle. Update 2026-06-12: the three-gate certificate has now passed on two further never-analyzed strangers, zero hand-tuning: intfloat/e5-mistral-7b-instruct — cross-family, cross-vendor, cross-OBJECTIVE (a contrastive embedding fine-tune) — G1 1.000/1.000, G2 0.464, G3 0.500; and microsoft/phi-4 — G1 0.993/0.996, G2 0.194 (the deadest null measured) on the word-count-matched battery, but G2 0.58 — a marginal fail by the 0.55 bar — on the harder lexicality-controlled battery, G3 0.500, concept peak found mechanically at depth 0.90. Measured concept-peak depths now span 0.21–0.90 across four families. Status: grounded mechanism on three strangers. On the lexicality-controlled battery the clean 3-gate pass holds for e5-mistral (shuffled-R 0.506, both nulls dead, CERTIFIED_LEX); phi-4 clears all three gates on the word-count-matched battery but is a G2 marginal-fail on the lexicality-controlled battery (shuffled-R 0.58 > 0.55, n=66), so its pass is battery-dependent — to be settled on the larger lexicality battery. Aperture/products/SPEC_coldeploy_interop_cert.md §9; universal/coldeploy_v2/RESULTS_coldeploy_lexicality.md.

Resolution caveat (Law 5, §3.6, 2026-06-08). The Atlas is a low-resolution artifact by nature. What it shares and transfers is the coarse layer — concept centroids, directions, probes. The fine per-concept structure (a frontier model's monosemantic SAE features) does not survive the Atlas rotation (fine-feature transfer collapses to cos 0.14, ≈ the 0.09 null, even with a fair full-dimensional R). So an instrument published "in Atlas coordinates" is portable only insofar as it is a coarse / linear instrument; a high-resolution per-concept dictionary cannot be ported by R_mᵀ and remains usable only on its home model.

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3. Laws and boundaries

Each law is stated with its evidence and the specific null it survived. The discipline (§6) is that a "this is real structure" claim is meaningless without naming the null it beat.

3.1 The up-to-rotation invariance (Law 0)

Statement. There is no shared coordinate system; what is shared is relational geometry, recoverable by an orthogonal map.

Evidence. The pairwise concept-similarity matrices of two families (Qwen2.5-3B, Llama-3.2-3B) agree at RSA = 0.805, z = 36 against a meaning-shuffle null — at or above each model's split-half reliability ceiling. On tuned reps the headline is RSA ≈ 0.93. A PCA + orthogonal Procrustes map fit on unlabeled paired concept centroids (420 train concepts) places 180 held-out concept centroids at top-1 = 1.000 (chance 0.0056); with a clean instruction-tuned teacher, 1.000 within-family / 0.994 cross-family. A raw high-dimensional ridge fails at chance (top-1 0.006) — the structure is real but legible only in a well-conditioned reduced space.

Nulls survived. Meaning-shuffle (permute the concept correspondence, 300 draws); the anisotropy-preserving Gaussian null on the raw reps (RSA 0.93, z = +30) — the same strict null that dissolves a single mind's own modal structure (silhouette z ≈ 0; §6). For the translator, the rotation nulls: row-shuffled R and a freshly sampled random-orthogonal R (both at chance).

3.2 The rank law — read = low-rank, write = high-rank (Law 1)

Statement. A reading direction (a classifier) transfers through a low-rank translator (it needs only the sign of a projection inside the shared concept subspace). A steering vector (a causal intervention) needs a high-rank translator (causal magnitude lives partly outside the top concept axes). At full rank, read and write both port at native fidelity, with zero labels in the target.

Evidence. Probe transfer: a probe trained in A, rotated by R into B, classifies 300 held-out concept pairs at 0.947 vs native-B 0.911 (low-rank PCA-100 suffices). Steering transfer (Llama-3.2-3B → Qwen2.5-3B, abstention): at PCA-100 the direction is on-target (cosine 0.988 to B's own honesty vector) but the causal override is weak; a dimension sweep recovers it —

At full rank the ported vector reproduces the native override exactly (100% abstention, 100% of real answers preserved, 0% fabrication). The decisive control is native_roundtrip (B's own vector through B's PCA and back, no rotation): it is also degraded at low rank, localizing the bottleneck to the subspace dimensionality, not the cross-family rotation. A null-completion audit proves the full-rank result is determined, not a lucky SVD completion: R is pinned only on the rank-420 centroid span, yet the honesty vector lies 98.3% inside that determined span, and porting only the in-span component reproduces the effect.

Nulls survived. Row-shuffled R and random-orthogonal R dead at every rank, including full rank — so the transfer is carried by the concept-aligned rotation specifically, not by any full-rank map. The native_roundtrip control isolates subspace-truncation from rotation.

Boundary. Demonstrated on one main source–target pair, one depth, one primary behavior (plus a clean second abstract stance, sycophancy-resistance, cosine 0.94, and a weak concrete one, "ocean," cosine 0.39). Status: promising proof of concept, not an established law across pairs. The grid still owed: reverse transfer (done, cosine 0.977), a within-family pair, a cross-scale pair (different hidden dims), larger behavioral N with binomial CIs.

3.3 Aggregate-sharp / instance-ambiguous (Law 2)

Statement. Aggregate objects transfer near-losslessly; instances do not.

Evidence.

A direction is an aggregate (like a centroid); an instance is not. This is the representational form of govern, don't generate: you can port the instrument of a frontier mind into a small one, but not its content.

Nulls survived. Shuffle nulls at chance (0.004–0.006) across the ladder.

Boundary. "Instances do not transfer" is bounded to linear maps; a nonlinear translator might recover more (untested — and not obviously desirable, since the safety story rides on the aggregate).

Corroboration (2026-06-12, pre-deployment). Cross-model dispersion terrain over the canonical Atlas is readable by residual-stream kNN at concept level (Spearman 0.73–0.79) while sentence embedders are blind to it (0.02–0.07) — instrument relativity measured on a fresh task; and both a single-representation read and a pre-registered 8-paraphrase-cloud read FAILED (rho −0.10/−0.12), exactly as aggregate-sharp / instance-ambiguous predicts. The law made a falsifiable pre-deployment prediction and it held.

3.4 The access boundary (Law 3)

Statement. Every fine-geometry transfer requires the residual stream of both models. The geometry does not survive output-only access to a closed model. But a coarse aggregate — is this input off the model's map — does cross the boundary, output-only, and a learned probe reads it capability-robustly.

Evidence. Building a closed model's concept geometry from next-token logprob signatures (the sole observable) gives cross-mind RSA = 0.019 (z vs anisotropy null +2.6, i.e. at chance; held-out concept translation 0.011, chance). The fine geometry is lost. However: a supervised probe on the output-logprob trajectory (mean/min token-logprob, per-position entropy over the answer) detects fabricated-vs-real entities at 0.94 pooled / ≥0.95 per-model on GPT-4o-class (held-out, pre-registered), transfers zero-shot across providers (OpenAI-trained probe reads Mistral/Google fabrication at 0.93–0.96, nulls dead), and holds where the model's own confidence fails (0.92 on a weak model whose own first-token entropy is below chance). Lexicality is controlled: on common-token real knowledge gaps (digits of π/e/√2) the probe predicts the model's own errors at 0.70–0.82 — a genuine knowledge-state signal, not a rare-name artifact.

Nulls survived. The geometry's failure to cross is itself measured against the anisotropy-Gaussian null (z +2.6, at chance). The crossing signal is reported on a pre-registered held-out split (the honest number is 0.94 pooled — a ~0.02–0.04 haircut from the exploratory CV 0.96–0.98, exactly what pre-registration should expose) with shuffled-R nulls dead and an ablation against answer-length. The deployed probes go one honest step further: re-fit 2026-06-07 on a serving-matched battery (adding the open-ended "Tell me about X." distribution) to kill false-positives on real famous entities in discursive use, they read a conservative ≈0.89–0.92 per-model / 0.899 pooled (gpt-4o-mini 0.915, gpt-4o 0.893, gpt-3.5 0.905; the newest reasoning model gpt-5.1 only 0.748) with real false-off-map down to ~1–3% — the production number we certify (Aperture/web/frontier.py _PROBES).

Boundary (a result, not a caveat). The off-map signal's unique lift over the model's own calibration concentrates in the multi-token fabrication regime (single-token recall ties token entropy); reasoning errors do not move output logprobs (0.61–0.64, CI spans chance) and route to an independent verification gate (0.92), not the logprob probe. The detector reaches only logprob-exposing models, and covers implicit fabrication (pair with a hedge-text reader for models that refuse in words).

The epistemic-resolution edge — state, not truth (2026-06-08, powered, de-confounded). The same boundary is sharpened on the residual-stream side. A powered re-run of the knowledge-state probe — on weaker Instruct models with an adequate UNKNOWN class so the obscure-real KNOWN/UNKNOWN split is decided by the model's own knowledge, with lexicality held flat by construction (both arms are obscure rare-subword reals, M1_matched = 1.000) — reads the model's knowledge state at AUROC 0.71–0.73 (0.727). That is real and above chance, but it sits below the 0.75 always-on bar: the residual field is a strong fabricated / off-distribution-input flag on register-matched inputs plus a moderate (soft) knowledge-gap reader, not a hard knowledge-gap gate on its own. Measured in deployment (2026-06-12), the axis reads register distance on mixed-register queries (news prose 0.00–0.09 vs short factual questions 0.32–0.89, real and fabricated overlapping) and the independent familiarity gauge is the discriminator there — the identity-not-truth law operating in production. It reads coarse distributional familiarity strongly and the finer "does this model know this obscure-real thing" only moderately — and on confident in-distribution fabrication it is at chance (0.587; claim 15, §0.2 — the identity-not-truth boundary). The field charts identity and stance, not truth.

3.5 Surface- and tokenizer-invariance (Law 4)

Statement. The shared geometry does not ride on shared surface tokens; strip every proper noun and cross-lingual cognate and it is untouched. It is demonstrably surface- and tokenizer-invariant. Whether it is meaning (versus a shared training distribution) is now resolved for the aggregate geometry (§5 ran and passed at the ≥1B decisive scale; instance-level recovery remains partial).

Evidence. The decisive lexical-trap control: drop all 462/600 concepts bearing proper nouns or cross-lingual cognates (keep only 138 name-free, cognate-free concepts); the strong cross-tokenizer pair (Qwen3-32B × Llama-3.1-8B) holds RSA 0.869 → 0.875 (clean residue z_ani +13.1); most pairs rise slightly. The alignment ranking is sorted by representation quality, not tokenizer overlap — the best cross-tokenizer pair (Llama-3.1-8B × Qwen2.5-7B = 0.923) beats many same-tokenizer pairs. If a shared lexicon drove the geometry, same-tokenizer pairs would top the ranking; they do not.

Nulls survived. Anisotropy-preserving Gaussian on the clean 138-concept residue (z +12 to +13).

Boundary. The narrow "shared surface tokens" hypothesis is decisively rejected; the broad "shared training distribution that encodes meaning the same way" hypothesis is not — no public model has a disjoint corpus. A real negative correlation of RSA with tokenizer distance exists (Spearman −0.63) but is carried almost entirely by the Gemma tail and is mostly a depth artifact (at Gemma's own concept peak the cross-architecture RSA roughly triples). The §5 disjoint-corpus experiment has since run: the aggregate leg passed at the ≥1B decisive scale (S1, reproduced to 7 decimals). What we still do not claim is instance-level meaning recovery — which remains the open edge.

3.6 The resolution boundary — coarse transfers, fine does not (Law 5) (2026-06-08)

Statement. The shared geometry is real but low-resolution. The coarse / aggregate layer — concept centroids, linear directions, probes, honesty and steering vectors (every prior 0.9+ transfer) — is shared up-to-rotation. The fine, per-concept structure — the monosemantic SAE-feature directions that resolve a concept into its sub-parts — is idiosyncratic to each model's basis and does NOT transfer by the rotation that carries the coarse layer. This is the aggregate-sharp / instance-ambiguous law (Law 2) extended onto the resolution axis.

Why the question was invisible before. Every prior law was measured on the 600-concept FLORES atlas, which is intrinsically ~100-dimensional — PCA-100 captures 95–99.5% of variance for every model, including the 235B (0.970). The coarse battery is resolution-blind by construction; there is no high-dimensional "extra room" in it for a frontier model to exhibit. Asking the resolution question requires a higher-resolution concept basis — Qwen-Scope's official residual-stream TopK SAEs (32K–128K feature directions vs our 600; arXiv 2605.11887).

Evidence (the decisive transfer test). Matched pair Qwen3-1.7B ↔ Qwen3.5-2B (both 2048-dim, 32K-feature SAEs), a full-dimensional orthogonal Procrustes R fit on 6000 paired FLORES sentences (not the PCA-100 R used elsewhere — that would erase the very fine structure under test). For each teacher SAE feature, rotate into the student space and measure max cosine to any student feature:

Transfer decays monotonically with feature fineness (the fraction of a feature's mass outside the coarse-concept span), symmetric in both directions, toward the random-direction null. The finer the concept, the less rotation carries it.

Nulls / controls survived. Two controls kill the obvious confounds. (1) R-quality: per-dim standardization improves held-out coarse sentence alignment (0.689 vs 0.672) yet makes fine-feature transfer worse — so the failure is not a poorly-estimated rotation. (2) SAE-identifiability: two independently-trained SAEs on the same model+layer (no rotation) match feature-to-feature at median 0.795, 88.5% within cos 0.5, 64.5% within 0.7 (null 0.09) — SAE features are substantially canonical, so feature-matching is a valid instrument and the cross-model collapse to 0.14 (0% within 0.5) is a genuine geometry statement, not SAE-basis arbitrariness. The negative is locked.

Boundary (and a modest positive). At matched dictionary dimensions (2048×32768), same-generation Qwen3.5-2B → Qwen3.5-35B-A3B (17× params) does pack its concept directions a touch more separably (mutual-coherence median 0.351 → 0.319; participation ratio flat, duplicate-rate zero) — so frontier resolution is modestly higher within its own basis, but that extra resolution is not rotation-portable. The de-risk gate stops here: the mechanism already fails at the cheapest test (a 1.2× capability gap, matched dims, canonical SAEs); a larger gap can only diverge more, so no frontier pod is warranted. There is no released 235B SAE (Qwen-Scope tops at 27B / 35B-A3B), so 235B fine-resolution is not directly testable, but the mechanism is dead at small scale.

Implication for the Atlas (§2) and the instrument library (§4.3). Transferring coarse instruments (the honesty probe, concept axes) via an unlabeled-concept-fit R is validated. Do not expect to port a frontier model's fine per-concept atlas onto a cheap deployed model by rotation; the frontier's fine resolution is usable only on that model. The canonical Atlas is, by nature, a low-resolution map.

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4. The research program

GI's program is four tracks. Each is stated as a tight paragraph: the question, why it is GI's to own, and the first experiment.

Update (2026-06-06): all four first experiments have now run. Findings are folded into the claim-status table (§0.2) and Laws (§3); full results in geometric_interpretability/RESULTS_SUMMARY.md and the per-track docs (RESULTS_rank_spectroscopy.md, RESULTS_instrument_library_drift.md, RESULTS_trajectory_geometry.md, RESULTS_crosssubstrate.md). Headlines: the rank law is now a measured spectrum (reading saturates rank ~32, instance/steering high-rank); reading transfers for all axis types (the abstract/concrete gradient is steering-only); geometric drift is localizable to behavior axes; reasoning errors are geometrically invisible (geometry charts identity, not truth); and the shared geometry reaches encoders and pixels weakly (0.2–0.6 band) versus the 0.93 autoregressive core. None overturned the core; each sharpened a law or drew an honest boundary.

Update (2026-06-06, second batch — the four "next moves" have now run. RESULTS_atlas_gpa.md, RESULTS_causal_rank_sweep.md, RESULTS_crosssubstrate_output.md, RESULTS_disjoint_corpus.md): (1) The canonical Atlas is built and lossless — a 9-model Generalized-Procrustes frame routes transfer (A→A*→B) as well as the n² bilateral maps within noise (RSA Δ 2×10⁻⁹, probe Δ +0.0003); the residual is a reproducible per-model variance-shape (cross-model 0.26) — a coarse model-ID up-to-anisotropy; the concept-tied "signature" reading failed the anisotropy control (2026-06-07 correction: 1.000 → 0.674 under independent per-half PCA). → claim 11 grounded. (2) The rank law's write-half is now causal — ported honesty-steering fails at rank 16, works at full rank (nulls dead), replacing the cosine proxy. (3) Output↔output cross-substrate — reading both at output with a learned rotation leaves RSA weak (0.12–0.27, output-layer read) but yields retrieval 0.66–0.76 (null 0.007): a real but non-isometric correspondence, not substrate-privacy. (4) The disjoint-corpus decider's smoke is the ideal shape — two from-scratch models on provably disjoint corpora sharing only an ontology align at RSA 0.54 (90 % of the same-corpus ceiling 0.60, z +28), tracking the gold ontology, while the concept-shuffle control collapses to 0.002 → the geometry tracks concept identity, not co-occurrence; the meaning conjecture (claim 10) gains its strongest leg. A DGX medium-scale (50 M-param, 5000-step) confirmation run has since LANDED and firmed the aggregate leg: cross-mind RSA rose to 0.614 — at/above the same-corpus ceiling (0.566), z +28/+25 — and probe transfer to 101 % of native, while the concept-shuffle control still collapses (−0.009). It also reported an honest negative: the gold-tracking (S3) and instance translator (S2, 0.37) did not firm — scaling sharpens what minds share, not recovery of the authored ontology (aggregate-sharp/instance-ambiguous again; KILL not triggered). The full from-scratch protocol (§5, ≥1 B disclosed-disjoint) remains the definitive decider.

Update (2026-06-08 — the ≥1B §5.3 decider has RUN, and REPRODUCED). The definitive experiment is no longer pending. Two independent from-scratch ~1.07B dense decoders, trained on provably disjoint character pools (A {a b d g i k p t u}, B {e f h m n o r s v z}, audited intersection ∅) describing one authored world, share concept geometry at cross-mind RSA 0.794, z_aniso +35.1, = 94 % of the same-corpus ceiling 0.8485 — and the pilot's one blemish, the "108 %-of-ceiling" over-shoot, washed out exactly as a clean scale-up predicts (94 %, just below ceiling). Probe transfer is 98 % of native; the KILL did not fire. Two runs agree to 7 decimals → reproducible, not a seed artifact. The honest boundary holds and sharpens: instance translation (S2 0.44) and gold-ontology recovery (S3) stay partial — aggregate-sharp, not recovery. → Claim 10 upgrades from conjecture to confirmed at the decisive scale for the aggregate leg. Still finishing: the decisive concept-identity shuffle control at ≥1B (confirmatory — the pilot's collapsed to −0.009 and the ≥1B within-RDM shuffle null already passed z +30) plus the public-reference arm (vs Qwen2.5-1.5B). The amp'd concept-identity shuffle control CLOSED 2026-06-12: collapse ratio 0.028 at ≥1B (RSA(A,B′)=0.021 vs RSA(A,B)=0.794, shuffle null −0.002±0.023) — PASS with a ~12× margin. Raw: geometric_interpretability/disjoint_corpus/{RESULTS_disjoint_xl1b.md, out/xl1b_AB_battery.json}.

4.1 Rank spectroscopy — the spectrum of the rank law

The rank law (§3.2) is currently a two-point story (reading at low rank, writing at high rank). The sharper object is the whole curve: for a given transferred object (a probe, a steering vector, a specific behavior), at what translator rank does transfer fidelity saturate? We conjecture a rank spectrum that is behavior-specific — an abstract stance (honesty) saturates at low rank because it lives in the determined centroid span (the null-completion audit puts honesty 98.3% in-span at rank 420); a concrete/lexical direction needs higher rank because its energy leaks into the undetermined completion (this is exactly why "ocean" ports at cosine 0.39). The experiment: sweep rank finely for a battery of objects (multiple probes, multiple steering vectors, varying abstractness), measure both directional fidelity (cosine) and causal fidelity (override %), and fit each object's saturation rank against an "abstractness" score (e.g. its energy fraction inside the centroid span). A clean rank-vs-abstractness law would let us predict a new instrument's transferability from a cheap geometric measurement before ever porting it — and tell the Atlas how large a control rotation to store per model.

4.2 Trajectory geometry — reasoning as a path through the Atlas

Everything above is static: a concept is a point, a behavior is a direction. But a reasoning chain is a trajectory — a sequence of residual-stream states as the model thinks. GI's natural next object is the geometry of that path: does a correct derivation trace a different shape (curvature, step length, return-to-manifold) than a confabulated one? (Measured: no — see claim 15; the proposal below is retained for the record.) This is GI's complement to the access boundary's gap: the logprob probe catches fabricated entities but not reasoning errors (0.61–0.64). The conjecture is that reasoning errors that are invisible in output logprobs may be visible in the internal trajectory's geometry — a wrong step leaves the charted manifold, or the path's curvature spikes, even when the output stays fluent. The experiment: extract the residual-stream trajectory over a chain-of-thought on graded-correct vs graded-wrong reasoning items, compute path features (per-step displacement, curvature, distance-to-nearest-concept-centroid in Atlas coordinates), and test whether a probe on the trajectory shape separates correct from incorrect reasoning where the output-logprob probe cannot. If it does, it fills the one gap the access boundary leaves open — and it is intrinsically a residual-stream (open-model) instrument, so it lives on the right side of Law 3.

4.3 The portable instrument library + geometric drift detection

This is the Atlas (§2) turned into a product and a monitoring primitive. The library: a versioned, community registry of instruments published in Atlas coordinates — the honesty axis, the off-map readout, sycophancy-resistance, refusal-calibration — each with its nulls, its residual, and its behavioral CIs attached, deployable to any registered model by an unlabeled rotation. Geometric drift detection: because a registered model's geometry is pinned to A* by R_m, any change to the model shows up as a measurable shift — a fine-tune that moves concepts, a quantization that distorts the manifold, an RLHF pass that installs a new direction (recall the "RLHF scar": over-confidence is a local, removable direction at one layer). The experiment: register a model, apply a known perturbation (fine-tune / quantize / RLHF-style nudge), and measure the perturbation as (a) growth in residual ε_m, (b) rotation of R_m, (c) movement of specific published axes. A drift metric that localizes which concepts or behaviors moved — "this fine-tune rotated the honesty axis by θ and moved the medical-concept cluster by d" — is an interpretability-grade audit signal the Atlas uniquely enables, and the basis for an assurance/monitoring offering.

4.4 Cross-substrate alignment — the LLM↔brain arc

The deepest test of whether the shared geometry is meaning (Law 4 — since resolved for the aggregate geometry at the ≥1B decisive scale by §5's run; instance-level recovery remains partial) is to align it to a structure that no text corpus contains. Within models, "shared geometry" can always be argued down to "shared training distribution." But a non-linguistic grounding axis — a vision encoder's object geometry, a sensorimotor/graph ontology, or, at the far end, human neural representations of the same concepts (fMRI/MEG concept-similarity structure) — cannot be a shared text corpus artifact, because the substrate never read the text. The arc: (near) align each model's concept geometry to a vision encoder's object geometry (the multimodal Gemma stream is the natural first substrate, and its modality edge — centroid top-1 0.35 — is the first data point); (far, the moonshot) align the model concept Atlas to a brain concept-similarity geometry on a shared concept set and test whether the LLM↔brain RSA beats an anisotropy-preserving null. Agreement with an off-text structure would be evidence of shared meaning in a way no shared corpus could manufacture — the cross-substrate version of §5's disjoint-corpus test. This is flagged moonshot (no run; substantial data and methods risk), but it is the experiment that would turn Law 4's conjecture into a claim from the other direction (external grounding rather than corpus disjointness).

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5. The decisive experiment — a from-scratch disjoint-corpus protocol (RUN at ≥1B, 2026-06-08 — verdict in §4's update + RESULTS_disjoint_xl1b.md)

5.1 What it settles and why nothing else can

The field's largest open question is Claim 10: is the shared geometry meaning, or a shared training distribution? The lexical-trap control (§3.5) decisively rejected the narrow version (shared surface tokens), but the broad version is unfalsifiable with public models — every open model trains on overlapping Common Crawl / Wikipedia / code / parallel data, and none discloses an auditable corpus. The only clean test is to build a model on a corpus provably disjoint from the comparison model's data and ask whether its concept geometry still aligns. If it does, "shared meaning" survives the strongest attack on it; if it collapses to null, the geometry was co-distribution all along. This is pre-registered below in the same discipline as our other locked evals (PREREG_scaleN.md).

RESULT (2026-06-08) — this experiment has now been executed at ≥1B, twice. Two independent from-scratch ~1.07B decoders on provably-disjoint character pools (intersection ∅) sharing one authored world align at cross-mind RSA 0.794, z_aniso +35, 94 % of the same-corpus ceiling 0.8485 — S1 PASS, KILL not fired, reproduced to 7 decimals. Meaning survives the strongest feasible attack for the aggregate geometry; instance translation (S2 0.44) and gold-graph recovery (S3) stay partial — aggregate-sharp, not recovery, exactly as pre-registered. The protocol below is what ran. The decisive concept-identity shuffle control (pilot collapsed to −0.009; ≥1B within-RDM shuffle null z +30) and the public-reference arm are on the amp'd concept-identity shuffle control, CLOSED 2026-06-12: collapse ratio 0.028 (PASS, bar <0.35). Verdict + raw: §4 Update (2026-06-08), RESULTS_disjoint_xl1b.md, out/xl1b_AB_battery.json.

5.2 Corpus design — provable disjointness is the whole game

The experiment lives or dies on disjointness being demonstrable, not asserted. Three corpus options, in increasing strength and cost:

• Option A — controlled-vocabulary synthetic language (strongest disjointness, cleanest). Generate a fully synthetic corpus in a constructed language: a fixed vocabulary of invented tokens with a defined grammar, expressing a known concept ontology (e.g. a generated world of objects, properties, and relations — kinship graphs, spatial layouts, a synthetic taxonomy). Crucially, the concepts (the relational structure: "A is the parent of B," "X is north of Y," "this animal is a kind of that one") are shared with the comparison set, but the surface form (tokens, grammar, co-occurrence statistics) is provably absent from any web corpus because we minted it. The comparison model is a public LLM whose concept geometry on the same ontology (expressed in natural language) we already extract. If the synthetic-corpus model and the public model agree on the relational concept geometry despite zero surface overlap, that agreement cannot be shared text statistics — it is shared structure of the world the corpus describes. This is the disjoint-corpus test in its purest form.
• Option B — a non-web natural domain (medium disjointness, more naturalistic). Train on a domain-specific corpus with a verifiable provenance and minimal web overlap — e.g. a private/licensed corpus, a pre-cutoff archive, or a technical domain (a specific scientific subfield's full-text) whose membership we can audit against the comparison model's known training set. Disjointness is argued from provenance rather than minted, so it is weaker than A, but the language is natural.
• Option C — held-out temporal/source slice (weakest, fallback). A corpus from a source and time window we can show the comparison model did not see (post-cutoff, or a closed archive). Useful only as a cheap sanity check; provenance disjointness here is the hardest to make airtight.

We recommend Option A as the primary (provable disjointness by construction) with Option B as a naturalistic robustness arm. The synthetic-ontology design also gives us a gold relational geometry (we built the ontology, so we know the true concept-similarity matrix) — a third anchor neither model has seen, against which both can be scored.

5.3 Model size and training budget

The model must be strong enough to build a real concept geometry but small enough to train from scratch on a controlled budget. Calibration from the program: the cross-mind geometry is already sharp at 3B, present down to 0.6B (centroid anchor 0.978 at 32B→0.6B), and instruction-tuning sharpens the pooled atlas more than raw size. So:

• Architecture: a standard dense decoder-only transformer (to stay on the right side of the architecture boundary — this is a meaning-vs-corpus test, not an architecture test; hold architecture fixed and vary only the corpus).
• Size: ~1B parameters (large enough for a legible mid-stack concept geometry per the ladder; small enough for a from-scratch run on a single multi-GPU node or a short rented cluster). A 0.5B arm as a cheaper replication if 1B is sharp.
• Training budget: enough tokens for the synthetic corpus to be learned to low loss on its own validation set (the synthetic corpus is finite and controlled, so "enough" is convergence on held-out synthetic text, not a fixed web-scale token count). For Option B, a compute-matched budget to a public reference checkpoint of the same size, documented.
• Extraction: mean-pooled residual at the model's own concept-peak depth (scan depth, per Law 4's architecture-dependence lesson — do not fix a layer). Corroborated in deployment (2026-06-12): the cold-deploy certificates found concept peaks mechanically at depths spanning 0.21–0.90 across four families — a fixed layer would have missed them (§2.3).

5.4 The RSA + surface-control battery

Run the exact battery the program already uses (meaning_vs_corpus.py / seam_null.py recipe), now with the disjoint-corpus model D as one mind:

1. Cross-mind RSA between D and the public comparison model P, on the shared concept ontology, with both nulls: meaning-shuffle (300 draws) and the anisotropy-preserving Gaussian (300 draws, PCA-matched). Report z vs both.
2. The unlabeled translator R_{D→P}: held-out concept centroid top-1, with shuffled-R and random-orthogonal-R nulls.
3. Triangulation against the gold ontology (Option A only): RSA of each model against the true relational geometry we minted. If both D and P align to the gold structure (which neither trained on as text), that is direct evidence the shared geometry tracks the structure of the world, not the corpus.
4. The lexical-trap control, inverted. Here disjointness is by construction, so the control becomes: confirm there is no surface leakage path (the synthetic tokens cannot appear in P's vocabulary; audit the tokenizers). For Option B, run the §3.5 proper-noun/cognate strip on the shared concepts.
5. Probe transfer. Does a probe trained in D rotate into P (and vice versa) at native fidelity, through R fit only on unlabeled concepts? Transfer of an instrument across a corpus gap is the operational form of the claim.

5.5 Pre-registered success / kill criteria (LOCKED before any run)

Stated as pass/fail, in the discipline of PREREG_scaleN.md — goalposts fixed before data.

Runnability. Option A is a self-contained build (synthetic corpus generator + a ~1B from-scratch train

• the existing CPU RSA/translator battery on cached P reps); no production touch; the heavy step is one short from-scratch training run. We pre-register the corpus generator's seed and the full ontology, the model config and token budget, the extraction depth-scan rule, and the locked criteria above, before training — exactly as the scale-N eval was locked before its run (and it then failed 2 of 3 gates honestly, which is the point).

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6. The dual null as a methods contribution

6.1 The control, stated standalone

GI proposes a control the whole interpretability field should adopt whenever it clusters, detects modes in, or claims structure within a representation. It is a pair of nulls:

1. Structure-destroying (permutation / column-shuffle) null. Independently permute the values within each representation dimension, then re-normalize. Preserves each axis's marginal; destroys all inter-item covariance. Answers: is there any structure?
2. Anisotropy-preserving (Gaussian) null — the decisive one. Fit the representation's mean μ and full covariance Σ (in the sample's own ≈(n−1)-dim span, since n ≪ d; use Ledoit–Wolf shrinkage in this regime), draw synthetic items from a unimodal N(μ, Σ), re-normalize. Preserves the entire anisotropic cone (the dominant directions, the "shape" of the cloud); destroys multimodality / clustering only. Answers: is there structure beyond the cone the cloud already has?

The gap between the two nulls measures the anisotropy-induced false-positive rate — exactly the structure an elongated, anisotropic embedding cloud manufactures for free. For a correspondence claim (RSA between two models), the analog of the second null is a rotation null: replace the learned orthogonal translator with a row-shuffled and a random-orthogonal map; a transfer that survives both is carried by the concept-aligned rotation, not by any high-dimensional map.

Three disciplines travel with the null, each learned from a specific failure (§7):

• Report a curve, not a point (over a grid of thresholds), because a single value hides whether structure is real or a threshold artifact.
• A negative needs a power control. A non-significant Gaussian-null result is a failure to reject, not evidence for the null; pair it with a synthetic-mode power control (what separation could this test have detected at this n?) and a positive control.
• Use the principled estimator in n ≪ d (Ledoit–Wolf shrinkage); the verdict can flip between the raw and shrinkage covariance, and that sensitivity is itself a finding.

6.2 Why it is a contribution, and where common interp results fail to apply it

The component statistics are not new — SigClust is the single-Gaussian-with-data-covariance test, and surrogate-data testing is "preserve the confound, test the higher moment." GI's methods contribution is (i) pairing them so the gap quantifies the anisotropy false-positive rate, (ii) extending the second null to a rotation null for cross-model correspondence claims, and (iii) applying the pairing to a setting it had not been applied to — a model characterizing its own representations and its own cross-model geometry — where it changed the answer.

It changed our answer: it demolished our first headline. Our origin project read ~5 discrete "thought-basins" in a model's output atlas at ">10σ"; against the structure-destroying null that signal appears, but against the anisotropy-preserving null it fails to reject (z ≈ 0) — the basins were a frame the observer's instrument supplied, and the ">10σ" was a divide-by-≈0 artifact of a point-mass permutation null. (An external statistics reviewer sharpened this further: at small n the test cannot separate "no modes" from "weakly-separated overlapping modes," so the honest claim is fail-to-reject, not no structure.) The seam between the negative and the positive is closed in one instrument: in the residual stream, a single mind's modal structure dissolves against its own anisotropy-Gaussian (silhouette z ≈ +0.4) while the cross-mind concept core survives the same strict null (RSA 0.93, z = +30).

Where common interpretability practice falls short of this bar: any method that clusters a model's activations or samples in embedding space to find "modes," "concepts," or "regions of thought," and compares only against a structure-destroying baseline (shuffle, or k-means inertia against random restarts), inherits the anisotropy confound and will report false structure in proportion to the embedder's anisotropy. Probing and SAE-feature work that reports a separation without a covariance-matched null cannot distinguish a real feature direction from a dominant anisotropy axis. The cheap, general guard is to add the anisotropy-preserving arm (and, for cross-model claims, the rotation arm) and report the gap. We offer the harness (seam_null.py) as a reference implementation.

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7. Disconfirmation record and open seams

The field is honest only if the deaths are on the record. (Items inherited from the program's running record; the discipline is that several are self-refutations — predictions we made and then built the control that killed.)

7.1 What did not work

Items A, D, E, and K are self-refutations — and K is a pre-registered failure we report exactly as locked. Item L is the resolution version of B: the shared object is coarse, and now we have measured the ceiling on how fine it goes. That is the discipline working as intended.

7.2 Open seams (honest, unresolved)

1. Meaning vs corpus (the central one). Narrow lexical hypothesis rejected; broad "shared training distribution" tested head-on: §5 has run at the ≥1B decisive scale and the aggregate leg is confirmed (S1 pass, reproduced to 7 decimals). The seam that remains is instance-level recovery (S2/S3 stay partial — aggregate-sharp, not recovery) plus the amp'd concept-identity shuffle control, the confirmatory closeout.
2. The canonical Atlas is built; its governance is not. The multilateral GPA fit and its validation battery have run (RESULTS_atlas_gpa.md): one common frame routes transfer at the bilateral ceiling within noise. The open seam is no longer construction — it is registry governance and versioning (who stewards A*, how registrations are versioned and superseded) and curation of the concept set's content.
3. Steering transfer is one pair. The rank law's causal half rests on Llama-3.2-3B → Qwen2.5-3B, one depth, primarily one behavior. Promising PoC, not an established law; the grid (within-family, cross-scale, larger N) is owed.
4. Behavioral N is small. The testing-mind percentages use 10–20 probes/axis; the geometric results (cosine, transfer) are robust, the behavioral rates carry real sampling noise.
5. The human-meaning leg is a proxy. Almost all cross-mind evidence is model↔model; the bridge to human semantic structure rides on multilingual concepts as a stand-in (§4.4 is the attempt to ground it externally — and it is a moonshot).
6. The off-map detector is bounded to fabrication, and knowledge-state reading is moderate not gate-grade. It does not catch in-distribution confident-wrong reasoning (that is the verification gate's job) — "off-the-map input" ≠ "wrong on a real fact." The powered, lexicality-flat re-run (2026-06-08) does now measure a real knowledge-state signal (obscure-real KNOWN vs UNKNOWN AUROC 0.71–0.73), but it lands below the 0.75 always-on bar → a soft knowledge-gap reader, not a hard gate; and on confident in-distribution fabrication the field is at chance (0.587; claim 15, §0.2 — the identity-not-truth boundary). A confident wrong belief on a real fact is still internally near-indistinguishable from a confident right one.
7. A single mind's idiosyncratic signature — resolved negative (2026-06-07). The dual null shows the modal frame is observer-manufactured, and the anisotropy null on the per-model signature reading has now run: the concept-tied "signature" reading failed it (1.000 → 0.674 under independent per-half PCA). What survives is a reproducible per-model variance-shape — a coarse model-ID up-to-anisotropy — and the correction is recorded (claim 11, §0.2).

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8. Related work — positioning, without overclaiming priority

We state the delta against the nearest neighbors, carefully, because they are close. (Citations by name and approximate year; selected references at the end.)

• The Platonic Representation Hypothesis (Huh et al., 2024) and the broader representational- convergence literature (CKA, Kornblith et al., 2019; Bansal et al., 2021). This line establishes that representations converge as models scale and improve, plausibly toward shared statistics of the world. We take this as our premise and do not claim to have discovered convergence. GI's delta: we make convergence operational (an explicit unsupervised orthogonal translator, held-out centroid top-1 = 1.000), causal (steering transfer), and bounded (the modality gradient, the aggregate/instance law, the access boundary). Convergence is the that; GI gives the what-transfers, how-far, and whether-you-can-write-through-it.
• Relative Representations / model stitching (Moschella et al., 2022; Lenc & Vedaldi, 2015). Relative representations encode each point by similarities to anchors, yielding a rotation/rescale-invariant code for zero-shot stitching of read-out heads. GI differs in three ways: (i) an explicit orthogonal map shown to be near-isometric (not merely a re-encoding that happens to stitch); (ii) extension from stitching readouts to transferring causal interventions (steering); (iii) the rank law — read=low-rank, write=high-rank — which is new and mechanistically specific. We do not claim to have invented anchor-based cross-model alignment.
• Activation steering / representation engineering (ActAdd, Turner et al.; Contrastive Activation Addition, Rimsky et al., 2023–24; Representation Engineering, Zou et al., 2023). This line builds and applies steering vectors within a single model — and the CAA construction we use is theirs. GI's delta is to port a steering vector across models through the unsupervised translator, at native fidelity, with the rank law and the null-completion audit establishing it is determined, not a lucky completion. Cross-model steering transfer, to our knowledge, has not been shown; the within-model machinery is prior work.
• Representational Similarity Analysis (Kriegeskorte et al., 2008). RSA — comparing systems by the similarity structure of their representations rather than the representations themselves — is the measurement backbone of the positive result, and it comes from computational neuroscience. We use it as a tool; the contribution is the null discipline around it (RSA against an anisotropy-preserving Gaussian, not only a meaning-shuffle) and its use for an explicit translator, not the RSA framework itself. (RSA is also the natural bridge to the cross-substrate / brain arc, §4.4 — its home turf.)
• Anisotropy of contextual representations (Ethayarajh, 2019; degeneration, Gao et al., 2019; isotropy/clusters, Mu & Viswanath, 2018; Cai et al., 2021). The "cone effect" is usually treated as a nuisance to be corrected. GI's delta is to turn it into an instrument: an anisotropy-preserving Gaussian null that subtracts exactly the structure a cone manufactures, which is what separates the observer's frame from the invariant core (and what killed our own first headline). We do not claim to have discovered anisotropy; we claim a use for it as a control.

The honest framing of priority. GI's contribution is **portability + control-transfer + null-discipline

• the access boundary** — not the discovery that representations converge, not RSA, not activation steering, and not anisotropy. Each of those is prior work we build on and cite. What we assert as new is that convergence is a constructed, causal, bounded, falsifiably-validated relation you can carry instruments across, and that the boundary where it stops (output-only access; the modality edge; instances) is itself mapped.

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9. Conclusion

Geometric interpretability is the study of what is invariant across models' representations, and of what can be carried between them. Its core claim is that different models build the same relational concept geometry, recoverable by an unlabeled orthogonal rotation, and that this geometry is operational — probes and causal steering vectors transfer at native fidelity (write-transfer demonstrated end-to-end on one cross-family pair; §3.2), reading through a low-rank map and writing through a high-rank one. Its laws are bounded (up-to-rotation, aggregate-not-instance, the access boundary, surface-invariance, and — measured 2026-06-08 — a resolution boundary: the shared map is coarse, fine per-concept structure does not transfer), its central artifact is a model-independent canonical Atlas that would let coarse instruments be published once and deployed anywhere, and its epistemics is the dual null that separates the real shared core from the structure an instrument manufactures — a discipline that humbled the program's own first headline and that we offer to the wider field.

The largest question — whether the shared geometry is meaning or a shared training distribution — is now settled for the aggregate geometry at the ≥1B decisive scale: §5's from-scratch, disjoint-corpus experiment, pre-registered with a real kill criterion, has run and passed (S1, reproduced to 7 decimals). What remains is instance-level recovery — still partial — and the amp'd shuffle-control closeout. We would rather state the bound and run the experiment that could break our conjecture than assert the conclusion. That, finally, is the field's stance: the apparent structure is largely manufactured by the aperture; the discipline reveals the small, real, portable thing that is actually there — and names, precisely, how far it reaches and where it stops.

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Author contributions

Kevin Russell — conception, research direction, the instrument-relativity framing, the founding intuition to "start at the front, not the end," and the canonical-Atlas and cross-substrate framing. Claude — field synthesis, experiment design, the dual-null harness, the cross-mind translator and steering-transfer code, the Atlas construction sketch and the disjoint-corpus protocol, analysis, and drafting. Both authors share responsibility for the disconfirmation discipline; several entries in §7 are corrections to the authors' own prior claims.

Reproducibility and relationship to the companion papers

The empirical core — the unsupervised translator, probe and steering transfer, the rank law, the null-completion audit, and the bounds — is reported with full methods in "Portable Concept Geometry Enables Cross-Model Probe and Steering Transfer." The synthesis, the dual-null epistemology, and the bounded "is meaning all you need?" thesis are in the companion position paper. This document organizes that body into a field: its definition, its central artifact (the Atlas), its laws, its research program, the decisive open experiment, and its honest seams. All experiments use open-weight models plus one custom-served model and custom tooling; the cross-mind translator is fit only on unlabeled parallel concepts; transfer results are reported against row-shuffled and random-orthogonal rotation nulls; behavioral results use held-out batteries; the cross-modal result was reproduced on independent hardware. The dual-null reference harness is universal/seam_null.py; the result documents grounding every number are under universal/ and steering/.

Selected references

• Huh et al. (2024). The Platonic Representation Hypothesis.
• Kornblith et al. (2019). Similarity of Neural Network Representations Revisited.
• Bansal et al. (2021). Revisiting Model Stitching to Compare Neural Representations.
• Moschella et al. (2022). Relative Representations Enable Zero-Shot Latent Space Communication.
• Lenc & Vedaldi (2015). Understanding Image Representations by Measuring Their Equivariance and Equivalence.
• Turner et al. (2023). Activation Addition.
• Zou et al. (2023). Representation Engineering: A Top-Down Approach to AI Transparency.
• Ethayarajh (2019). How Contextual are Contextualized Word Representations?