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| \subsection*{Audit scope and exact identifiability} |
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| These analyses add no new scientific prompts and require no new Gemma forward pass. They reuse the 72 exact natural questions printed in Section S4C, their 25 archived layer states, and the same three frozen Jacobian lenses. The prompt manifest contains six supplied mechanism families, four material cases per family, and three surface variants per case. All 72 prompts, all layers, all methods, all random seeds, and all failed gates were retained. |
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| For prompt node $i$ in mechanism $f$, let $x_{fi}=+1$ for a stated numerical increase and $-1$ for a decrease; let $s_f=+1$ for a direct input--property law and $-1$ for an inverse law; and let $y_{fi}=+1$ or $-1$ denote the registered physical-property outcome. The frozen manifest satisfies exactly |
| \begin{equation} |
| y_{fi}=s_f x_{fi} |
| \end{equation} |
| for every one of the 72 prompts. Therefore, within one mechanism, |
| \begin{equation} |
| y_{fi}y_{fj}=(s_fx_{fi})(s_fx_{fj})=x_{fi}x_{fj}. |
| \end{equation} |
| Thus ``same versus different physical outcome'' is exactly the same binary relation as ``same versus different numerical direction'' inside each supplied family. An unlabeled family graph can identify at most the two-way equivalence class $\{y_f,-y_f\}$; it cannot assign an absolute physical sign without the mechanism orientation $s_f$. This is a proof about benchmark identifiability, not a claim that Gemma contains no physical information. |
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| \begin{table}[H] |
| \centering |
| \small |
| \caption{Mechanism orientation in the 72-prompt audit. The physical label is generated exactly by $y=sx$.} |
| \begin{tabular}{lll} |
| \toprule |
| Mechanism & Response orientation & $s_f$ \\ |
| \midrule |
| Crosslink density--rubbery modulus & direct & $+1$ \\ |
| Dislocation density--flow strength & direct & $+1$ \\ |
| Obstacle spacing--Orowan stress & inverse & $-1$ \\ |
| Stiff-particle fraction--modulus & direct & $+1$ \\ |
| Pearlite spacing--yield strength & inverse & $-1$ \\ |
| Porosity--elastic modulus & inverse & $-1$ \\ |
| \bottomrule |
| \end{tabular} |
| \end{table} |
|
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| \subsection*{Label-blind graph matching and isomorphism} |
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| For every unordered mechanism pair, all 24 bijections between the four material cases were scored using anchor--paraphrase similarities only. The best map was frozen before evaluating counterfactual similarities or revealing the physical labels. Held-out similarity form transferred strongly for all 15 pairs: mean correlation 0.976 for Jacobian states, 0.973 for direct decoder-basis states, and 0.992 for raw states. All 15 correlations were positive for each representation. However, Jacobian physical case agreement was 0.600 (structured-null $p=0.135$). No mechanism pair was exactly isomorphic under the surface-variant labels. Weisfeiler--Lehman similarity exceeded its constrained null nominally ($p=0.0123$) but not after the seven-test Benjamini--Hochberg correction ($q=0.0861$). The transferable object is therefore graded similarity form, not a reliable physical case map or a Jacobian-specific graph. |
|
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| \subsection*{Whole-mechanism graph learning} |
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| A three-block graph isomorphism network (GIN) was trained on four mechanisms, selected on a fifth, and tested on the completely unseen sixth. Twenty seeds were run for every configuration. The absolute-polarity node input contained a constant, surface-variant one-hot, the explicitly supplied numerical direction, normalized depth, in/out degree, and edge strengths; it excluded prompt text, material identity, mechanism identity, answer words, and physical labels. A matched graphless multilayer perceptron and constrained edge shuffle served as controls. The numerical-direction configuration predicts a deliberately supplied input and is therefore a positive-control readback rather than a graph-discovery endpoint. The separate relation GIN excludes numerical direction and surface variant. It evaluates all 72 eligible ordered pairs within each 12-node family graph, requiring different material cases and different surface variants. Thus its near-chance result does not contradict the perfect readback of a feature explicitly given to the absolute model. The node-permutation audit reproduced every reported AUC exactly. |
|
|
| \begin{table}[H] |
| \centering |
| \scriptsize |
| \caption{Whole-mechanism-held-out graph learning. The numerical-direction column is an input-readback positive control because that feature was supplied to the absolute-label network. Even with direction supplied, absolute physical polarity does not generalize; the separate polarity-free relation network also fails. ROC--AUC 0.5 denotes no ordering information.} |
| \begin{tabular}{lrrr} |
| \toprule |
| Held-out mechanism & Physical polarity & Direction input readback & Same/different relation \\ |
| \midrule |
| Crosslink density--rubbery modulus & 0.611 & 1.000 & 0.578 \\ |
| Dislocation density--flow strength & 0.000 & 1.000 & 0.497 \\ |
| Obstacle spacing--Orowan stress & 0.389 & 1.000 & 0.487 \\ |
| Stiff-particle fraction--modulus & 0.500 & 1.000 & 0.619 \\ |
| Pearlite spacing--yield strength & 0.000 & 1.000 & 0.388 \\ |
| Porosity--elastic modulus & 0.056 & 1.000 & 0.472 \\ |
| \midrule |
| Six-mechanism mean & 0.259 & 1.000 & 0.507 \\ |
| \bottomrule |
| \end{tabular} |
| \end{table} |
|
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| Absolute physical AUC exceeded 0.5 for only one of six held-out mechanisms; the polarity-free relation GIN exceeded 0.5 for only two. The supplied numerical-direction feature was read back at AUC 1.000 in every family, as expected for this positive control. This value is not evidence that topology discovered direction. Below-chance physical AUC is expected when an otherwise coherent two-way partition receives the wrong absolute sign in a held-out mechanism; it is not evidence that the network learned an anti-physical law. |
|
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| \subsection*{Spectral communities and graph density} |
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| A label-blind normalized-Laplacian partition was calibrated on 1,000 positive and 1,000 negative synthetic graphs. Mean adjusted Rand index (ARI) was 0.987 for positive block graphs and 0.044 for negative controls; ordinary signed accuracy after random graph-wide flips was 0.496. ARI is invariant to swapping the two community names: 1 is perfect agreement, and values near 0 are random-like. |
|
|
| \begin{table}[H] |
| \centering |
| \small |
| \caption{Community agreement as progressively more graded similarity information is retained. The smallest corrected $q$-value over the 12 secondary representation-by-density tests is 0.092.} |
| \begin{tabular}{lrrr} |
| \toprule |
| Retained graph & Jacobian & Direct & Raw \\ |
| \midrule |
| Top-one binary & 0.267 & 0.267 & 0.104 \\ |
| Top-one weighted & 0.267 & 0.267 & 0.104 \\ |
| Top-two weighted & 0.511 & 0.409 & 0.511 \\ |
| All candidate weights & 0.674 & 0.674 & 0.817 \\ |
| \bottomrule |
| \end{tabular} |
| \end{table} |
|
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| The registered sparse top-one graph retains only modest community information. Agreement increases when all eligible similarities are retained, but direct states match the Jacobian result and raw states are stronger. Hard edge selection therefore discards graded relational organization; the result does not establish a uniquely Jacobian physical community. |
|
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| \subsection*{Exhaustive balanced partitions} |
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| To avoid dependence on one community algorithm, every one of the 462 unique balanced six-versus-six partitions of each 12-node family graph was evaluated. A second endpoint fitted all 35 balanced four-versus-four partitions using anchors and paraphrases only, then assigned the four held-out counterfactuals. The stronger blockwise null preserved every ordered surface-variant block and its weight distribution. |
|
|
| \begin{table}[H] |
| \centering |
| \small |
| \caption{Exhaustive partition results. Values are six-family mean ARI; $p$ is the exact blockwise-null probability.} |
| \begin{tabular}{lrr} |
| \toprule |
| Representation & Full 12-node graph & Counterfactual held out \\ |
| \midrule |
| Jacobian & 0.022 ($p=0.087$) & -0.027 ($p=0.620$) \\ |
| Direct & 0.002 ($p=0.217$) & -0.007 ($p=0.449$) \\ |
| Raw & -0.019 ($p=0.458$) & 0.013 ($p=0.321$) \\ |
| \bottomrule |
| \end{tabular} |
| \end{table} |
|
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| The registered relation is not the uniquely dominant balanced division of the weighted graph in this cohort, and a partition fitted without counterfactuals does not transfer to that held-out phrasing. Spectral and exhaustive results need not coincide because they optimize different graph objectives; their combination shows that the conclusion depends on how a community is defined. These are not proofs of absence across a larger materials population: six mechanisms provide limited power, and the near-threshold $q=0.092$ density result and $p=0.087$ complete-partition result leave open moderate effects in a larger prospective cohort. |
|
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| \subsection*{Complete run inventory and reproduction} |
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| The independent validator reports 72 prompts, 25 layers, 6 mechanisms, 1,320 absolute-GIN runs, and 1,080 relation-GIN runs. The option-free representation array has SHA-256 \texttt{2451d643f94934c2ea5ef73acd06dbaf1ea58f43065a724c9190423a5be9b9dc}; the exact prompt manifest has SHA-256 \texttt{da49f9d57b870404beabb7401f40a677d90d0b93d49d58170a0ab9750268d634}. The complete pair maps, GIN logits, seed rows, spectral partitions, density ablations, exact partitions, null arrays, protocols, environment versions, and validation record are retained under \path{experiments/graph-isomorphism-generalization-2026-07-18/}. |
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| \begin{verbatim} |
| python scripts/analyze_graph_isomorphism_generalization.py |
| python scripts/run_graph_isomorphism_gin.py |
| python scripts/analyze_graph_gauge_generalization.py |
| python scripts/run_graph_relation_gin.py |
| python scripts/analyze_spectral_graph_communities.py |
| python scripts/analyze_exact_graph_partitions.py |
| python scripts/plot_graph_generalization_audit.py |
| python scripts/validate_graph_generalization_audit.py |
| python scripts/build_graph_generalization_si.py |
| \end{verbatim} |
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| These commands analyze archived states; they do not invoke Gemma or create additional scientific prompts. The 72 natural questions are printed once in Section S4C rather than duplicated here. |
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