A Calibrated Test of Internal Action Maps: State Signals Without Global Affine Closure

Dekun Yang

cs.AIAug 13, 2026

A Calibrated Test of Internal Action Maps: State Signals Without Global Affine Closure

Authors: Dekun Yang

Organizations: Zhejiang University

Abstract

A hidden state signal can be decodable or causally usable without supporting a reusable action map. We test whether action maps fitted without a source reach its natural post-action activation and compose. We organize the tests as an evidence lattice and validate the geometric branch on a known affine S_5 carrier: all held-source folds pass one-step, composition, inverse, decoding, and commutativity gates. Structured curvature and held-domain conjugacy raise error monotonically, but only 23/30 strongest cells flip a closure gate, bounding rather than universalizing calibration. In post-trained Qwen/Qwen3-4B, frozen final-token h28 affine maps have mean held-entity error .519, versus .398 for within-test-domain cross-fit. Seven randomized entity splits and map geometry do not support a purely entity-specific account. Earlier h4/h16 layers fit one-step transitions better, but h4 conflict-state decoding is weak and lexical controls remain unresolved. Three matched intervention datasets regenerated from one frozen checkpoint show causal effects only at h28/h36. Outcome-aware refitting improves h28 one-step error to .474 (.469 with weighting), yet no refit passes composition. Learned finite worlds likewise preserve relative algebraic signals or shared charts without held-source affine closure. Within the tested carriers, state availability, causal use, local geometry, and reusable closure are separable. The result is limited to one pretrained model, sampled final-token layers, two finite worlds, and the tested affine or diagnostic function classes.

Figures & tables

Node	Measurement	Licensed statement	Does not establish
State availability	Held-data decoding versus label controls	State information is available to the decoder	Causal use or a natural update path
Causal-use branch	Matched intervention versus wrong-state/translation controls	A state direction is locally usable by the output computation	Held-source closure or an on-manifold transition
Closure/law branch	Natural endpoints, then composition/inverse/commutativity	The tested map transfers and satisfies the reported law	Local output use or an untested carrier
Behavioral eligibility	Task success and state differentiation	Model-level interpretation is meaningful	A unique internal implementation
Branch conjunction	Causal use plus reusable closure/laws under eligibility	A causally faithful reusable operator in the tested carrier	Universality across models, layers, tokens, or function classes

Table 1: Evidence lattice. Availability enables two independent branches. Only their conjunction under behavioral eligibility licenses the strongest operator claim.

Figure 1: Evidence lattice for internal action-map claims. State availability enables independent tests of causal local use and reusable closure/laws. Behavioral eligibility guards model-level interpretation. Geometry without detected causal use and causal use without closure remain distinct partial outcomes; only their conjunction licenses a causally faithful reusable operator. Directed edges denote evidential prerequisites, not temporal emergence or mediation. The diagram contains no sampled observations.

Split	Rows/seed	Entities	State categories	Templates	Role
Train	2,500	A–D	all five	0, 1	Map fitting
Validation	500	E	all five	0, 1	Rank/ridge selection
Test	1,000	F–G	all five	0, 1	Held-entity/context evaluation

Table 2: Grounded split support. The protocol holds out entities and generated context/activation instances, not state categories or the two primary templates.

Figure 2: Calibration and sampled depth. a , Exact contextual

S_{5}

carrier results in three held-source folds; lines mark the

.50

and

.60

gates. b , Mean and empirical 95% nuisance-seed intervals over 20 seeds after averaging folds. c , Fifteen h28 action-by-dataset-seed cells for five references; cross-fit is fitted inside the test distribution, and the line is the unchanged H1 gate. d , One-step error averaged over five actions within each dataset seed versus matched intervention effects and within-seed 95% pair-bootstrap intervals. Each layer has three dataset-seed regenerations of 160 pairs from the same frozen checkpoint; colors denote layer.

Figure 3: Structured diagnostics localize the remaining alternatives. a , Median one- and two-step NRMSE under common curvature (dotted) and held-domain conjugacy (solid) across five embedding seeds and three folds. b , Three dataset-seed ratios for within/cross-entity error and same-action-across-entity/different-action-within-entity map distance. c , h4 conflict-state probe versus permuted control and h4 error after 0, 16, or 64 inserted neutral tokens. d , Seed-level h28 one-step means under train/refit and unweighted/weighted fits, alongside the frozen composition direct-map gap. U, unweighted; W, weighted. Red lines are frozen gates.

Setting	Test	Verdict	Decisive evidence
Exact carrier	Zero/support recovery	POSITIVE CONTROL RECOVERED	All formal/reproduction/support folds pass near numerical precision.
Exact carrier	Structured stress	STRUCTURED CALIBRATION LIMITED	Monotone errors, but 23/30 rather than 24/30 gate flips.
Qwen3-4B h28	Frozen H1 train-only	NOT SUPPORTED	Mean $.519$ ; 4/15 below $.50$ ; cross-fit is descriptive.
Qwen3-4B h28	Entity/context diagnostic	ACTION-DOMINANT GEOMETRY	Within/cross ratio about $.90$ ; action distance dominates.
Qwen3-4B h4	Lexical diagnostic	LEXICAL ROLE UNRESOLVED	Weak conflict probe; distance and token-only controls do not adjudicate.
Sampled layers	Paired interventions	CAUSAL DEPTH PATTERN REPLICATED	h4/h16 intervals include zero; h28/h36 positive in all datasets.

Table 3: Principal frozen and follow-up verdicts. Diagnostic labels remain separate from earlier frozen decisions.

Candidate carrier	One-step	Composition	Causal use	Current status
Final-token residual h4	pass	not supported	not detected	Low-dimensional geometry; lexical role unresolved
Final-token residual h16	pass	not supported	not detected	Layer-local affine geometry only
Final-token residual h28	fit-scope sensitive	not supported	replicated positive	Causally usable signal without global affine closure
Final-token residual h36	not supported	not tested	replicated positive	Causal-use branch only
Last-16 static token embeddings	not supported	not tested	not tested	Does not explain h4 by itself
Other tokens/cross-layer spans	not tested	not tested	not tested	Live alternative

Table 4: Carrier coverage. Pass and not supported refer only to the listed test and frozen representation.

Figure S1: Frozen thresholds expose different degrees of boundary sensitivity. a , Number of 15 h28 action-by-seed cells below each H1 threshold. The red line is the frozen $.50$ gate; all cells first pass at $.58$ . b , H2 verdict over direct-map-discrepancy and probe-gain thresholds while other gates stay fixed; the cross is the frozen $(.50,.20)$ point. c , number of Phase 7 h4 seed cells passing the order sub-gate or full H2 as the minimum $d_{z}$ varies. Relaxing $d_{z}$ alone never changes joint H2 because direct and probe gates still fail. No panel changes an original decision.

Figure S2: State differentiation exposes failure of the preregistered H5 composite. a , Algebra violation and world incoherence across 99 nonzero checkpoints; partial Spearman $\rho=.091$ with trajectory-cluster 95% CI $[-.035,.249]$ , and the task gate fails. b , One trajectory shows H1 error rising as collapsed state representations differentiate. c , Signed margins use $(.5-\mathrm{H1})/.5$ , $(.6-\mathrm{H2})/.6$ , $(\mathrm{AUROC}-.8)/.8$ , and $(.5-\mathrm{H4})/.5$ . H3 remains diagnostic and cannot override absolute gates.

Figure S3: Early-layer one-step geometry does not satisfy the frozen composition construct. Raw cell text reports endpoint error, direct-map gap, order $d_{z}$ , probe gain, and joint H2 status. Color is signed normalized distance from the frozen sub-gate. h28 rows are frozen compatibility comparators. No layer-seed cell passes joint H2.

Figure S4: Scalar scale control preserves the sampled-depth ordering, while early representations are low dimensional. a , Mean residual norm. b , Mean action displacement norm. c , Covariance participation ratio. d , One-step error after train-fitted scalar RMS normalization. Points are three activation datasets; black bars are seed means.

Hidden-State Probing

Hidden-state probing reads model activations to study what they represent and whether those signals support monitoring or control.

See all 47 papers

A Pre-Specified Construction-Confirmation Test of Operation-Level Causal Transfer Across Finite Isomorphic Symbolic Domains
Aug 16, 2026Xinyi ShanSymbolic Reasoning in Language Models Causal Reasoning in Language Models
Independent Researcher
Algebraic Consistency Alone Does Not Certify Temporal Structure in Latent Action Models
Sep 20, 2026Di Wen, Ruodi Zhang, Kailun Yang +1Latent Action Learning Latent Dynamics Modeling
Karlsruhe Institute of Technology, Karlsruhe 76131, Germany · Hunan University, Changsha 410012, China
Erased, Rerouted, or Rescaled? Post-Training and the Causal Quotient of a Language Model's Belief State
Oct 4, 2026Weihan Li, Tianshi Zheng, Junhao Wu +1Belief Updating in LLMs RL Post-Training
The University of Tokyo · The Hong Kong University of Science and Technology · RWTH Aachen University +1

Aug 16, 2026J/K move · Enter open · S save

A Pre-Specified Construction-Confirmation Test of Operation-Level Causal Transfer Across Finite Isomorphic Symbolic Domains

Xinyi Shan

Independent Researcher

Symbolic Reasoning in Language Models Causal Reasoning in Language Models Causal Interventions in Language Models Cross-Domain Transfer Learning

Open paper AskPDF arXiv

Behavioral accuracy, linear decodability, and successful activation interventions do not by themselves show that a model carries an operation-level structure from one symbolic domain to another. We ask a narrower question in finite isomorphic state spaces: if the hidden-state difference between two operations is estimated separately for each source input, does adding that difference to a mapped recipient input move the model toward the corresponding recipient answer? The design compares this input-specific intervention with wrong-operation, norm-matched random, and no-op controls, and separates candidate construction from an independently isolated confirmation split. On a frozen Qwen2.5-7B-Instruct model at layers 20--21, one route--domain--operation candidate from a family pre-specified and frozen before confirmation access, transparent | integer_mod16--letters16 | successor->predecessor, passed both PyVene splits; its confirmation intersection--union p-value was 0.000198 and its 36-family Holm-adjusted p-value was 0.006943. A subsequent NNsight 0.7.0 experiment, pre-specified and frozen before its confirmation access, tested only this selected prompt route, without candidate or layer reselection. It reproduced all 12 confirmation effect estimates, confidence intervals, and exact sign-flip p-values numerically; its 36-family Holm-adjusted p-value was 0.007141. The result is therefore limited to one prompt route and one candidate, replicated across two intervention implementations on one model revision and one layer interval. It does not establish cross-model generalization, full-family backend independence, domain-general transfer, or algebraic invariance.

Abstract

Hidden-State Probing

Explore similar work

A Pre-Specified Construction-Confirmation Test of Operation-Level Causal Transfer Across Finite Isomorphic Symbolic Domains

Algebraic Consistency Alone Does Not Certify Temporal Structure in Latent Action Models

Erased, Rerouted, or Rescaled? Post-Training and the Causal Quotient of a Language Model's Belief State