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"caption": "\\textbf{Can either readout recover an engineering term that was deliberately omitted from the description?} Before running the 50 prompts, we declared 150 scientifically appropriate one-token terms and then asked where each term ranked among Gemma's 262,144 vocabulary entries. (A) The cutoff $k$ is relaxed from rank 1 to rank 100; a higher curve means more declared terms have become strongly readable. Thin teal curves are the three independent Jacobian fits, the thick teal curve is their mean, and gray is direct unembedding of the same states. At $k=100$, the fractions are 7.2\\% and 3.3\\%, respectively. (B) Each point is one prompt's recovery area under the curve (AUC); a point above the diagonal favors Jacobian transport. Eleven prompts improve, 36 tie, and three decline. (C) The same differences are grouped at the scientifically appropriate level: gray circles are the five phrasings and diamonds are the ten family means. Positive values favor the Jacobian lens. Boundary attack and notch resistance produce the largest gains, whereas cyclic damage and particle strengthening favor direct unembedding. Across families, the mean difference is $+0.0129$, but the hierarchical 95\\% interval spans $-0.0187$ to $+0.0486$ and the one-sided family sign-flip test gives $p=0.234$. Thus the experiment finds selective, mechanism-dependent gains rather than a universal Jacobian advantage.",
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"caption": "\\textbf{Target-free family vocabularies.} Each row is one physical family and each cell is a word selected without the declared concept list. Support is the number of independent phrasings, out of five, in which the word appeared under the three-fit consensus rule; darker blue-green cells indicate stronger background-corrected support. Asterisks mark exact declared-word overlap added only after ranking. Read across a row to ask whether the words form a recognizable materials neighborhood. Boundary attack, surface oxidation, and rapid transformation show coherent mechanism or microstructure terms; high-temperature deformation and cleavage show why stable decoding can still be generic. The figure therefore measures what assembles naturally, not whether the algorithm can find a supplied answer.",
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"caption": "\\textbf{Do the freely discovered words identify the materials mechanism without a supplied answer list?} Each bar is leave-one-phrasing-out accuracy across the 50 prompts and ten balanced mechanism families. A classifier learned four phrasings per family and predicted the held-out fifth using only target-free candidate words from the corresponding readout. ``Exact prompt words removed'' deletes literal input copies; ``morphology-aware removal'' additionally deletes conservative prefix-related variants. The dotted line is 10\\% chance. The dashed line is a TF-IDF classifier that instead sees the prompt words themselves. The target-free Jacobian and direct lists remain well above chance after the strictest removal, at 58\\% and 64\\%, but the prompt baseline reaches 56\\% and direct does not differ significantly from Jacobian. Thus the discovered vocabulary contains cross-phrasing mechanism information, while the comparison rules out a lens-specific superiority claim. The complete confusion matrix and per-prompt predictions are in the Supplementary Information.",
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"caption": "\\textbf{What does the stricter target-free filter remove, and what materials vocabulary remains?} Rows A-E correspond to the five exact prompts printed in the text. Each left panel shows the original-style unrestricted leading words from lens fit 0. Each right panel removes input and output words, the frozen 214-word function list, and the globally common readout scaffold, then requires the word to occur in all three independent fits at that layer. No declared concept was supplied to either selection. The gray region marks the fixed 38-92\\% analysis band. The comparison shows that much of the visually continuous unfiltered stream is generic discourse vocabulary, while the stricter view retains localized physical neighborhoods: \\code{corrosion}/\\code{damage} (A), \\code{resilience}/\\code{protection} (B), \\code{irreversible}/\\code{mechanism} (C), and \\code{collapse}/\\code{disintegration} (D). The generic cleavage stream (E) is a useful negative case. Gaps mean that none of the displayed words passed the strict rule at that layer; they are not evidence of absent computation. Ribbon thickness is a rank-derived visualization score, not token probability, causal influence, or a literal chain of thought.",
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"caption": "\\textbf{Do differently worded descriptions of the same mechanism remain close in the model's full state space?} (A) For each of five folds, four phrasings per family define ten centroids and the fifth phrasing is classified; the procedure is repeated at every registered layer and for every baseline representation. (B) Full-dimensional Jacobian classification peaks at 62\\% at 43.9\\% depth: 31 of 50 descriptions are correct, above the 13-of-50 threshold obtained after shuffling family labels and retaining the best of 25 layers ($p=0.0002$). (C) UMAP is only a display of those 50 best-layer states. Its axes have no physical units and no test uses them; color and two-letter code denote family, marker shape denotes phrasing, and stars are family centroids. (D) The full-dimensional confusion matrix reveals the physical specificity: boundary attack (BA), line-defect motion (LD), and notch resistance (NR) are separated across all five phrasings, while ductile failure (DU), high-temperature deformation (HT), and surface oxidation (OX) overlap. CL, CY, PS, and RT denote cleavage, cyclic damage, particle strengthening, and rapid transformation. (E) The same classifier reaches 54\\% on raw hidden states and 76\\% on a simple average of the prompt's own word embeddings. The result therefore supports statistically reliable family structure in the transported states, but it does not show that Gemma created that structure independently of the engineering vocabulary in the prompt.",
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"caption": "\\textbf{Does physical equivalence overcome deliberately stronger wording similarity?} Every triplet compares an anchor with a differently worded prompt that has the same physical answer and a near-verbatim prompt whose physical relation is reversed. Positive margin means that the same-physics pair is closer; negative margin means that wording dominates. (A) In the disjoint replication cohort, both Jacobian and direct decoder-basis states begin strongly wording-aligned and rise sharply after roughly 60\\% depth. The shaded region is the prospectively frozen 80-96\\% late window. The frozen late-minus-middle Jacobian change is $+1.402$ ($0.951$-$1.793$), positive in 24/24 triplets and 6/6 families. (B) The absolute late-window means are positive in both the discovery and disjoint cohorts, but their family-hierarchical intervals cross zero; the replication's registered primary endpoint therefore failed. (C) Four replication families end positive, obstacle spacing is approximately neutral, and porosity remains wording-aligned, demonstrating real mechanism heterogeneity. (D) A target-free comparison of decoded word sets also trends positive late, but its intervals cross zero. Teal and purple nearly coincide throughout, showing that the transition is not uniquely produced by Jacobian transport. This is a representational similarity test, not a causal intervention or a literal chain of thought.",
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"caption": "\\textbf{Where does option-free comparative graph structure appear, and where does it fail?} Every node is one complete materials question and every selected edge points to the most similar eligible different material case. AUC uses all eligible candidates rather than only the selected edge; 0.5 denotes no ordering information. (A) The same 72 scientific stems are read at three prompt boundaries. At the natural question end, no answer word or response scaffold is present and Jacobian AUC is 0.642. A checkpoint-suffixed variant shows no detectable signal, while the explicit answer-mapping condition reaches 0.816. Because suffix and token position differ across conditions, this comparison localizes prompt-boundary sensitivity rather than a unique causal effect of the added words. Direct unembedding and raw states follow the same pattern. (B) At the natural question end, solid bars show the fraction of 144 selected edges that preserve the registered binary relation and hatched bars show all-candidate AUC. The dashed line is the exact case-preserving null mean. Jacobian selected-edge precision is 0.674 ($p=0.02195$) and AUC is 0.642 ($p=0.01235$); word and character TF-IDF are at or below the null. (C) The positive result depends on supplying the governing mechanism. When candidates come from other mechanisms, AUC falls to 0.513; when raw numerical direction deliberately conflicts with physical outcome, it falls to 0.471. (D) The actual 72-node natural-question graph contains six supplied mechanism islands and 144 selected edges. Circles, squares, and triangles are anchor, paraphrase, and near-verbatim counterfactual prompts. Filled and open nodes are the two registered orientations; teal and coral edges preserve or reverse the registered binary relation. Ninety-seven edges are teal. This figure demonstrates an option-free comparative regularity; Figure~\\ref{fig:graph-identifiability} determines what that regularity can identify.",
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"caption": "\\textbf{What can the option-free graph identify?} (A) A numerical increase raises the property for a direct law but lowers it for an inverse law. In this prompt set, physical polarity is exactly $y=sx$; within one mechanism, the same/different physical relation collapses algebraically to the same/different numerical relation. (B) A graph isomorphism network is trained on four mechanisms, selected on a fifth, and tested on an unseen sixth. The purple AUC of 1.0 is a positive-control input readback: numerical direction $x$ was deliberately supplied to the absolute-label network, so this curve does not show graph discovery. The actual test asks whether graph structure can combine that supplied $x$ with an unseen law to infer physical polarity $y$; it fails, as does a separate same/different relation network that receives neither $x$ nor surface-variant identity. Points are held-out mechanisms and connecting lines guide the eye. (C) Community agreement rises when graded similarities are retained instead of only the strongest edge, showing that sparsification discards structure. Direct and especially raw states are at least as strong as the Jacobian states, so this organization is not lens-specific. (D) Exhaustively evaluating every balanced partition shows that the registered physical split is not the uniquely preferred division, including when counterfactual phrasing is held out. Bars are means, dots are mechanisms, and intervals show the corresponding blockwise null distribution. Together, the panels support transferable comparative geometry while rejecting a universal, Jacobian-specific physical-sign graph.",
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"caption": "\\hlyellow{\\textbf{Constitutive orientation persists without a supplied equation across Gemma scale.} Every law contributes eight matched up-minus-down state contrasts along a model-specific direction fitted only on the same earlier development laws. (A) The matched 4B prompt-format comparison. Open markers supplied the equation and prescribed sign composition; filled markers used the corresponding equation-free questions. Both use the original 4B neutral scale, and black segments are medians. (B) The complete 60-law distributions for every available condition. Each waterfall independently orders all laws from most inverse to most direct; horizontal bars begin at that condition's calibrated neutral zero. The 4B supplied and equation-free facets are the matched format pair. The 12B and 31B facets contain equation-free tests only. Each facet has its own horizontal robust-$z$ scale, so signs and category ordering are comparable across facets but absolute bar lengths are not common hidden-state units. Purple, gray, and teal denote inverse, neutral, and direct relations. (C) Direct-inverse AUC for the hidden-state direction, output-logit change, and lexical controls across the same four available conditions. Hidden and output separation survives scaffold removal and reaches 1.000 at 12B and 31B, whereas word and character controls remain near chance. (D) The three registered pairwise AUCs, neutral-cut directional accuracy, and exact generated-answer accuracy for the equation-free cohort. Both larger checkpoints orient 40 of 40 directional laws and reach ceiling on all three relational AUCs; only 31B converts that representation into nearly perfect generated answers. The comparison supports a scale-robust ordinal constitutive orientation, not an invariant law-by-law magnitude, unfamiliar-law generalization, or a literal reasoning trace.}",
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"caption": "\\textbf{Can an automated interpreter recognize a materials mechanism from discovered words alone?} (A) Twenty candidate sets (ten mechanism families from each of two readouts) were shuffled and stripped of the original prompt, method identity, and answer key. The automated interpreter selected one of ten family labels for each set in five order-randomized passes; it did not solve the original engineering question. (B) Accuracy stays far above the 10\\% chance level on every pass. (C) Majority-vote correctness for every family and readout: both columns are correct for nine families and both miss cleavage. Thus both naturally assembled vocabularies carry recognizable materials meaning, with no Jacobian-specific advantage in this secondary test (paired difference 0, $p=1.0$). (D) Each point is one Jacobian family. The horizontal coordinate is predeclared exact-term recovery; the vertical coordinate is the fraction of automated passes assigning the target-free words correctly. Cleavage is weak on both measures, but most families are semantically identified even when exact targets are rarely top-ranked, producing only a weak relationship ($\\rho=0.254$). Controlled recovery and target-free recognition therefore measure complementary properties. The five passes are repeated judgments from one automated system, not independent scientific expert (hence this figure is limited to reproducible machine-based semantic validation).",
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"caption": "\\textbf{Two magnified examples of the controlled-rank measurement.} The panels print the exact prompts, predeclared absent words, and one-token continuations. (A-C) The notch-resistance example is an earlier development item excluded from all held-out inference. Panel B plots \\code{toughness}'s full-vocabulary rank at every sampled layer for three independent Jacobian fits and direct unembedding; upward is stronger because rank 1 is best. The registered 38-92\\% band is shaded. Panel C reduces each word to its best rank within that band, showing that the localized rank-2 event is specific to \\code{toughness}. (D-F) The boundary-attack example is one of the 50 held-out prompts and was already counted in Figure~\\ref{fig:controlled}; it is not a new test. All three lenses sustain \\code{corrosion} at rank 1 over two sampled layers, while direct unembedding reaches rank 111. Panel F shows that \\code{sensitization} and \\code{boundary} do not receive the same advantage. Thus Figure~\\ref{fig:controlled} asks how often declared concepts are recovered across the frozen population, whereas this figure shows the layer location, fit-to-fit agreement, duration, and word specificity that its aggregate statistic necessarily discards. Neither trajectory is a literal chain of thought.",
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"caption": "Lexical-adversarial physical-equivalence results. Margin is centered cosine(anchor, physical paraphrase) minus cosine(anchor, near-verbatim physical counterfactual). Positive means that physics outranks wording. Intervals use the frozen two-stage family/triplet bootstrap.",
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"caption": "Post hoc answer-scaffold audit and prospectively frozen arbitrary-code negative control. The scaffold comparison reuses stored states and was defined after both source experiments were inspected. Its first four rows are positive-minus-negative decoder-logit differences, equivalently pairwise log-odds units; they are not the centered-cosine margins reported in the preceding table.",
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"caption": "Complete mechanism inventory for the two lexical-adversarial cohorts. Each row contains four material systems, each rendered as anchor, physical paraphrase, and lexical counterfactual. ``Positive/negative'' are the exact one-token scientific outcomes used only for clean-answer and answer-scaffold audits; the full-state similarity endpoint does not search for these words.",
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"caption": "\\textbf{Post hoc signed-relation graph across different material cases.} A source prompt selects one nearest eligible target in each other surface-variant group. Same material cases are prohibited. Precision is the fraction of 144 directed edges whose source and target share the registered binary relation. The mechanism family is supplied when candidates are defined. The later exact audit shows that this within-family physical relation is identical to same/different numerical direction, so these values measure comparative structure rather than independent identification of a constitutive-law sign.",
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"caption": "\\textbf{Graph-identifiability audit summary.} All endpoints reuse the same 72 natural questions and archived states; no new prompt or Gemma forward pass was added.",
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"caption": "\\textbf{Post hoc cross-phrasing mechanism graph from the original 50-prompt suite.} Every prompt selects one nearest neighbor in each other phrasing fold; 10\\% is balanced mechanism chance. This graph asks whether differently worded descriptions from the same family remain neighbors. It is complementary to, and less lexically controlled than, the signed-relation graph.",
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"caption": "\\textbf{Independent lens fits give nearly identical controlled ranks.} Each cell is the Spearman correlation across all 150 held-out prompt--concept pairs. The three Jacobian fits use independent 1,000-record WikiText samples yet agree at $\\rho=0.9978$--$0.9987$. Direct unembedding is much less correlated with their ordering ($\\rho\\approx0.21$). This figure establishes repeatability of the fitted measurement, not that its decoded terms are causal or universally more accurate.",
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"caption": "\\textbf{Sample semantic stream.} The exact prompt was: ``After substantial plastic elongation, a metal bar separated along a surface filled with microscopic pits around second-phase particles. Neighboring pits had joined into larger depressions.'' The one-token continuation was ``The.'' The ribbons were selected from the unrestricted leading decoded tokens at each layer, so the plotting code was not given a word list. The scientifically meaningful term \\texttt{coalescence} appears naturally among those tokens, alongside generic discourse words that reveal the readout's noise. However, \\texttt{coalescence} had been declared separately as a tracked concept when this earlier development prompt was designed. This is therefore an illustrative one-lens development display, not an unanticipated held-out discovery and not a literal chain of thought.",
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"caption": "\\textbf{Complete target-free cross-phrasing classification.} (A) Accuracy for Jacobian and direct candidate lists before and after removing exact or morphologically related prompt words. The dotted line is balanced chance; the dashed line is a classifier that sees TF--IDF prompt words. The strict-filter result remains above chance for both readouts but does not favor the Jacobian lens. (B) Prompt-level confusion matrix for the strict Jacobian configuration. Each row contains five independently written prompts. Boundary attack and surface oxidation are perfectly identified, whereas cleavage, ductile failure, high-temperature deformation, and notch resistance share vocabulary with neighboring mechanisms. The complete 50-row predictions, family accuracies, macro-F1 scores, paired correctness, and permutation outputs are stored in \\texttt{experiments/candidate-nonsteering-2026-07-16/}.",
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"caption": "\\textbf{The frozen expert ontologies do not align significantly with Jacobian family geometry.} At every layer, pairwise distances among the ten family centroids are compared with (A) four broad response classes and (B) a multi-attribute materials ontology. Curves show Jacobian-transported and raw states; horizontal lines show target-layer and prompt-embedding baselines. The best multi-attribute Jacobian correlation is $\\rho=0.148$ at 58.5\\% depth, but the maximum-over-layers corrected value is $p=0.9695$. Prompt embeddings align more strongly at $\\rho=0.453$. The result does not support a claim that the lens recovered this expert ontology.",
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"caption": "\\textbf{No universal entity--process--property depth ordering is detected.} Predeclared single-token concepts were assigned before calculation to physical entity/state, process/mechanism, or response/property roles. The panels show the distribution of each concept's strongest registered depth under Jacobian and direct readouts. The largest Jacobian separation among role medians is 8.2 percentage points and does not exceed a family-blocked role-label null ($p=0.1988$). Missing sustained events are retained separately rather than assigned an artificial late depth.",
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"caption": "\\textbf{The broad physical-equivalence hypothesis fails in the first frozen cohort.} Each triplet contains an anchor, a physically equivalent paraphrase whose wording and units change, and a near-verbatim counterfactual whose physical answer reverses. Positive margin means the anchor is closer to the physical paraphrase than to the lexical counterfactual. The registered 38--92\\% band mean is negative under both readouts (panel A), and no family has a positive mean (panel B): wording dominates over the broad window. Panel C shows the unregistered late rise that motivated, but does not itself confirm, a new disjoint cohort with its late window fixed before execution. This retained negative result is why the manuscript claims a temporal shift rather than general physics-first geometry.",
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"caption": "\\textbf{Explicit semantic answer scaffolding strongly amplifies the relation contrast.} The audit reuses stored states from the disjoint cohort and was specified only after both lexical-adversarial studies were inspected. (A) Before the answer choices, the late pre-choice state weakly separates the physically equivalent pair from the counterfactual under both readouts; after the prompt presents ordinary semantic choices such as \\texttt{higher}/\\texttt{lower}, the final-state contrast is much larger. Lines connect the same triplets and open symbols are cohort means. (B) The after-minus-before gain is positive in every mechanism family. (C) Ordinary semantic answer pairs are selected more reliably than the arbitrary A/B codes used in the separate falsification, both overall (87.5\\% versus 62.5\\%) and in most families; the dashed line is 50\\% forced-pair chance. Because position, suffix, and answer-word exposure change together, this post hoc comparison does not identify which factor causes the gain. It does show why the late physical-equivalence transition cannot be interpreted as an option-free constitutive variable.",
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"caption": "\\textbf{The arbitrary A/B answer-code falsification is inconclusive because its manipulation check fails.} The prospective protocol replaced meaningful answer words with arbitrary \\texttt{A}/\\texttt{B} labels and reversed their mapping across matched prompt variants. (A) Before the mapping appears, the physical-equivalence separation remains small and heterogeneous. (B) After the mapping, the required A/B contrast is also unstable across layers; the orange line is after mapping and gray is the pre-mapping reference. Shading in A and B is the frozen late window. (C) Family means do not show a consistent transition from pre-code physics separation (circles) to post-mapping A/B separation (squares). (D) Clean forced-pair code accuracy is only 62.5\\% overall, with 0\\% for obstacle spacing; the dashed line is 50\\% chance. Only 5 of 24 triplets formed a complete anchor--counterfactual--remapped pattern, and neither code token was the global top continuation in any of 72 prompts. The frozen geometric endpoint is therefore not interpretable as successful answer-independent binding. Retaining this failure prevents the semantic-scaffold result from being presented as if arbitrary output codes had already reproduced it.",
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