Preregistration: A Structural Invariant Across Symbolic Systems and Its Neural Homologue in the Anterior Mid-Cingulate Cortex

Preregistration: A Structural Invariant Across Symbolic Systems and Its Neural Homologue in the Anterior Mid-Cingulate Cortex Giovanni Esposito Independent Researcher, Gold Coast, Australia December 12, 2025 Abstract This preregistration specifies two linked tests of the ψ–GIE framework: 1. A structural universality test showing that a single principal component (“ψ–PC1”) derived from simple, substrate-independent metrics (entropy, Lempel–Ziv complexity, burstiness, lag-1 autocorrelation, and length) explains a dominant share of variance across diverse symbolic systems (text, code, biological sequences, audio, neural weights). 2. A neural homology test predicting that a variance-corrected repair cost measure (∆RC) derived from the same framework will correspond to activation in the anterior mid-cingulate cortex (aMCC) during volatility and control-demand manipulations, with a characteristic asymmetry distinguishing it from generic volatility or conflict models. The central claim is not that this is the “first universal law” of structure, but that ψ–PC1 represents a unifying structural invariant that appears across (i) explicit structural metrics, (ii) learned language-model manifolds, and (iii) biological control circuits in cortex. We specify: •explicit hypotheses and thresholds; •eight falsification routes; •a 10-step replication recipe; 1 •a power and multiple-comparisons plan; •a concrete analysis protocol for aMCC, including cross-species predictions. All analyses are preregistered; results in either direction (support or refutation) will be reported. Contents 1 Overview and Core Hypotheses 4 2 The ψ–Triangulation Principle 4 3 False-Positive Immunity (FPI) Constraint 5 4 Structural Universality: Methods 6 4.1 DomainsandData .............................. 6 4.2 StructuralMetrics .............................. 6 4.2.1 Explicit Metric Definitions . . . . . . . . . . . . . . . . . . . . . . 7 4.3 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . 7 4.4 PilotWorkDisclosure............................. 8 5 Manifold Concordance: Embedding-Based Tests 8 5.1 EmbeddingModels .............................. 8 5.2 Embedding PCA and Alignment . . . . . . . . . . . . . . . . . . . . . . . 9 5.3 Null Distribution for Alignment . . . . . . . . . . . . . . . . . . . . . . . 9 6 Neural Homology: ∆RC and aMCC 10 6.1 Definition of ∆RC .............................. 10 6.2 Datasets.................................... 10 6.3 ROI Definition: aMCC vs dACC . . . . . . . . . . . . . . . . . . . . . . 11 6.4 GLM and Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 11 6.5 AsymmetryPrediction ............................ 12 7 Cross-Species Gradient 12 8 Dimensionality Collapse and Interpretation of PC1 13 9 Power Analysis and Multiple Comparisons 14 9.1 PowerAnalysis ................................ 14 9.2 Multiple Comparison Correction . . . . . . . . . . . . . . . . . . . . . . . 14 2 10 Falsification Criteria and Contingency Plans 15 10.1 Partial-Support Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . 15 10.2 Partial-Support Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . 16 11 Replication Recipe 16 12 Limitations and Boundary Conditions 17 13 Resources, Timeline, and Sharing Plan 18 14 Registration and Versioning 19 15 Conclusion 20 15.1KeyCommitments .............................. 20 15.2AnticipatedOutcomes ............................ 20 3 1 Overview and Core Hypotheses The preregistration covers two main strands: 1. Structural Universality (Symbolic Axis) Across ∼15 symbolic domains, we compute five simple structural metrics per object: Shannon entropy, Lempel–Ziv complexity, burstiness, lag-1 autocorrelation, and length. Principal component analysis (PCA) on the resulting feature matrix is expected to yield a dominant component (ψ–PC1) that: •explains ≥70% of total variance; •is clearly separated from null and anti-ψmodels; •aligns strongly with principal components computed from language-model embeddings (BERT, GPT-2, RoBERTa, MiniLM, GloVe, and—where applicable— Grok or equivalent). 2. Neural Homology (aMCC and ∆RC) We define ∆RC as a variance-corrected “repair cost” derived from volatility and structural deviation in behavioural or task-state sequences. We hypothesize that: •aMCC activation will track ∆RC more closely than dACC or generic conflict measures; •a characteristic asymmetry will appear: aMCC overshoot during rising volatility versus a more conservative response during stabilizing periods; •cross-species strength of the ∆RC–aMCC relationship will increase with hierarchical depth (rodent <monkey <human). These two strands are linked by a triangulation principle: if the same latent axis emerges in (i) explicit structural metrics, (ii) artificial language-manifold geometry, and (iii) biological control circuitry, then it is reasonable to treat it as a shared structural invariant rather than a pipeline artefact. 2 The ψ–Triangulation Principle A genuine invariant should appear under multiple, independent instruments. Here we require triangulation across: 1. Symbolic structure — via PCA on simple structural metrics (entropy, LZ, burstiness, lag-1 autocorrelation, length). 2. Machine manifolds — via PCA on learned sentence embeddings (BERT, GPT-2, RoBERTa, MiniLM, GloVe, and, where available, Grok or equivalent). 4 3. Biological control systems — via ∆RC-dependent activation in aMCC during volatility and control-demand tasks. Triangulation criterion: The framework is considered supported if all three modalities exhibit a dominant axis whose orientation is substantially concordant: •symbolic ψ–PC1 explains ≥70% variance and is clearly separated from null and anti-ψmodels (Sections 4–10); •embedding-PC1 in each language model aligns with ψ–PC1 with cosine ≥0.75 (Section 5); •∆RC predicts aMCC activation with effect sizes and gradients specified in Sections 6–7. 3 False-Positive Immunity (FPI) Constraint To guard against artefacts, we impose a four-layer False-Positive Immunity (FPI) constraint: 1. Null rejection (statistical): Real PC1 variance minus null PC1 variance must be ≥0.30 with p<0.00625 (Bonferroni-corrected). 2. Domain independence (robustness): Leave-one-domain-out (LODO) recomputation of ψ–PC1 and its alignment with embedding-PC1s must change variance explained and alignments by <5% (we relax earlier 2% to a more realistic 5% threshold given measurement noise). 3. Model independence (generality): Cosine alignment between ψ–PC1 and each embedding model’s PC1 must satisfy cos(ψ-PC1,Model-PC1)≥0.75 for at least four of five models (Grok, BERT, GPT-2, RoBERTa, MiniLM, GloVe) and never fall below 0.60. 4. Metric substitution stability (invariance): When substituting alternative but related metrics (e.g., BWT-based complexity for LZ, dispersion for burstiness), the recomputed ψ–PC1 must maintain cosine ≥0.70 with the original ψ–PC1. 5 4 Structural Universality: Methods 4.1 Domains and Data We will construct a dataset using the following 12 core domains (fixed a priori): 1. Whale song (audio, converted to symbolic or token representation) 2. Birdsong 3. Genomic sequences (e.g., human accelerated regions, exons) 4. Protein sequences (e.g., from ASTRAL SCOPe) 5. EEG time series segments (resting state or task) 6. Neural network weights (e.g., GPT-2 layer parameters) 7. Ancient text (pre-1900 literary or religious works) 8. Modern news text (post-1990) 9. Source code (C/Python repositories, e.g., Linux kernel excerpts) 10. Legal contracts or legislative prose 11. Poetry (diverse traditions and epochs) 12. Mathematical expository or proof-like text Additional domains (e.g., narrative fiction, social media excerpts, musical scores) may be added as exploratory extensions, but all preregistered tests will at minimum use this fixed core set. Leave-one-domain-out (LODO) analyses will always be computed with respect to these 12 core domains. Each object will be truncated to a maximum of 1 MB to standardize computational load and avoid domination by extreme lengths. 4.2 Structural Metrics For each object we compute five structural features: 1. Shannon entropy of symbol frequencies; 2. Lempel–Ziv complexity (or equivalent compressibility proxy); 3. burstiness index (based on inter-event intervals or token occurrences); 4. lag-1 autocorrelation (temporal dependence); 5. total length (number of tokens or bytes). 6 4.2.1 Explicit Metric Definitions To ensure full reproducibility, we specify the exact formulations: •Shannon entropy:H=−Pipilog2piover token frequency distribution, where piis the proportion of tokens of type i. •Lempel–Ziv complexity: We use the LZ76 variant (incremental parsing over the symbol sequence), normalized by sequence length to yield a complexity measure in [0,1]. •Burstiness: Computed using the Goh et al. (2008) definition B=σ−µ σ+µ, where µand σare the mean and standard deviation of inter-token intervals (time between successive occurrences of the same token type, or inter-event intervals for time-series data). •Lag-1 autocorrelation: Pearson correlation coefficient ρbetween the sequence {xt}and its lag-1 shift {xt−1}, where xtrepresents token identity or other appropriate quantization of the sequence. •Length: Total token count (or byte count for non-tokenized sequences). These metric definitions will remain fixed for all analyses; no alternative formulations will be introduced after data inspection. All metrics will be z-scored (standardized to mean 0, variance 1) across the full set of objects before PCA. 4.3 Principal Component Analysis We perform PCA (full SVD, e.g., via sklearn.decomposition.PCA) on the N×5matrix (objects ×metrics): •Extract eigenvalues λ1, λ2, . . . , λ5and eigenvectors v1,v2,...,v5. •Define ψ–PC1 as the first principal component eigenvector v1. •Record the proportion of variance explained by PC1: λ1/Piλi. We then construct: •anull model by shuffling metric vectors across objects (permuting rows of the feature matrix independently for each column) and recomputing PCA (e.g., n= 1000 permutations); 7 •an anti-ψmodel by constructing synthetic objects designed to break the structural relationships (e.g., heavily randomized or adversarial sequences with enforced anticorrelations between metrics) and recomputing PCA. 4.4 Pilot Work Disclosure Pilot work. Non-preregistered pilot analyses using 10–15 domains and 3 embedding models (BERT, MiniLM, GloVe) yielded PC1 variance in the range 0.75–0.85 and alignment cosines of 0.80–0.84. These values motivated the current thresholds of 0.70 (variance) and 0.75 (alignment). All pilot code and results will be archived separately from the preregistered analyses and clearly labeled as exploratory. 5 Manifold Concordance: Embedding-Based Tests 5.1 Embedding Models For each object we compute embeddings using multiple models: •BERT-base-uncased (bert-base-uncased) •GPT-2 small (gpt2) •RoBERTa-base (roberta-base) •MiniLM (all-MiniLM-L6-v2) •GloVe 300-dimensional vectors (glove-wiki-gigaword-300), averaged over tokens for text-like domains •Where available, a Grok-like or similar high-capacity model’s internal manifold Exact model checkpoints will be drawn from HuggingFace Transformers (version 4.x; exact version and commit hash to be recorded in the analysis repository). For text-based objects, we mean-pool token embeddings (or use the CLS token for BERT-like models, following best practices per model). For non-text domains (e.g., audio, genomic sequences), we convert to text-like representations or use appropriate tokenization schemes before embedding. All embedding dimensions will be column-wise z-scored across objects prior to PCA (standardized to mean 0, variance 1). 8 5.2 Embedding PCA and Alignment For each model: 1. Build an N×dembedding matrix, where Nis the number of objects and dis the embedding dimensionality. 2. Perform PCA on the embedding matrix and extract the first principal component (Model-PC1). 3. Compute the cosine similarity between ψ–PC1 (from structural metrics) and ModelPC1. Specifically, we project all objects onto each PC to obtain score vectors, then compute the cosine similarity between these score vectors: cosine(ψ-PC1,Model-PC1) = sψ·smodel ∥sψ∥ ∥smodel∥, where sψand smodel are the vectors of PC scores for all objects. 4. Perform leave-one-domain-out (LODO) tests: remove all objects from one domain, recompute both ψ–PC1 and Model-PC1, and recalculate alignment cosines. Primary alignment thresholds: •Full-data cosine: ≥0.75 for at least four of the five primary models (BERT, GPT-2, RoBERTa, MiniLM, GloVe), with no model falling below 0.60; •LODO mean cosine: within 5% of full-data cosine for each model (i.e., relative drop <0.05). 5.3 Null Distribution for Alignment We generate a null distribution for the alignment by: •shuffling object labels (permuting the association between metric-based scores and embedding-based scores); •recomputing the cosine between ψ–PC1 and Model-PC1 under these permutations (n= 1000); •estimating an empirical p-value as the fraction of permutations whose cosine exceeds or equals the observed cosine. We require pnull <0.00625 (Bonferroni-corrected for 8 primary tests) for the alignment to be considered statistically significant. 9 10.2 Partial-Support Scenarios If some but not all criteria are met, interpretations will be graded: •Strong structural universality but weak neural support:ψ−−PC1maybearobustsymbolicinvariantwithoutastraightforwardaMCChomologue.Thiswouldsuggestthestructuralaxisisrealbutitsmappingtospecificneuralcircuitsrequiresrefinement. •Neural support but weak manifold concordance: RC∆RCRCmaybeameaningfulbiologicalquantityeveniftheproposedsymbolicaxisisincomplete.ThiswouldsuggestaMCCtracksacontrol− relevantsignalthatisonlypartiallycapturedbyourstructuralmetrics. •Mixed results across models: Suggests architectureor dataset-specific effects. This would prompt refinement of the universality claim to a more cautious framing (e.g., transformer-based language models” rather than all language models”). •Strong PC1 but weak alignment: If PC1 is dominant but doesn’t align with embeddings, this suggests our structural metrics capture a real axis that is orthogonal to learned linguistic representations—still interesting but requiring reinterpretation. All outcomes (full support, partial support, or refutation) will be reported transparently, along with effect sizes, confidence intervals, and detailed discussion of boundary conditions. 11 Replication Recipe A minimal reproduction of the structural universality result can be achieved in approximately 30–60 minutes with standard tools: 1. Choose 10–20 diverse symbolic domains and collect at least one object per domain (examples: news articles, code files, genomic sequences, audio transcriptions). 2. Truncate each object to 1≤11MBandconverttoatokenorbytesequence. 3. Compute the five structural metrics for each object: •Shannon entropy: H=ipilog2piH = -Pipilog2piH=ipilog2pi •Lempel–Ziv complexity (LZ76, normalized) •Burstiness: B=()/(+)B = (σ−µ)/(σ+µ)B= ()/(+)forinter−tokenintervals •Lag-1 autocorrelation (Pearson ρontokensequence) •Length (token count) 4. zz z-score each metric across all objects. 5. Run PCA using standard libraries (e.g., sklearn.decomposition.PCA in Python) and extract PC1; record variance explained. 16 6. Compute embeddings with at least one standard model (e.g., BERT-base via HuggingFace Transformers): •Mean-pool token embeddings (or use CLS token) •zz z-score embedding dimensions across objects 7. Run PCA on embeddings and extract embedding-PC1. 8. Compute cosine similarity between ψ− −PC1andembedding −PC1scorevectors. 9. Generate null distributions by shuffling metric values across objects (1000 permutations) and repeat steps 5–8. 10. Compare real vs null: report PC1 variance, alignment cosines, and empirical pp p-values. Required software: Python 3.8+, numpy,scipy,scikit-learn, HuggingFace transformers. No proprietary tools required. Total compute time on a standard laptop: <2< 2 <2 hours for 50∼5050objects. Reference implementation: Analysis code will be released in a public GitHub repository with a tagged release corresponding to this preregistration, including a Jupyter notebook demonstrating the 10-step recipe on synthetic data. 12 Limitations and Boundary Conditions •Sample size and domain choice: Initial tests will use 12∼1212coredomainswithmultipleobjectsperdomain.Whilesufficienttotestthecorehypothesis, broadercoverage(50+ domains, includingnon−Westernlanguages, non−humananimalcommunicationsystems, anddiverseAIarchitectures)wouldbettercharacterizetheboundariesoftheinvariant. •Embedding models: Results may depend on specific training corpora and architectures. We mitigate this by testing multiple, diverse models (BERT, GPT-2, RoBERTa, MiniLM, GloVe); however, future systems (e.g., multimodal models, non-transformer architectures) may behave differently. The claim is restricted to contemporary language models trained on large text corpora” rather than all possible AI systems.” •Neural data: aMCC analyses depend on existing datasets with appropriate tasks (volatility, control-demand). ROI definitions and noise properties vary across studies. We therefore treat the neural strand as supporting and complementary rather than singularly decisive. Cross-species comparisons are particularly challenging given differences in task design, recording methods, and anatomical homologies. •Scope of universality: Statements are restricted to symbolic systems under predictive and resource constraints—systems that must balance accuracy and cost during learning or adaptation. We do not claim universality across: 17 –Non-symbolic biological systems (e.g., bacterial chemotaxis) –AI systems not trained on prediction tasks (e.g., pure reinforcement learners without world models) –Static, non-adaptive corpora (e.g., purely random sequences) •Causal interpretation: This preregistration establishes correlational structure (alignment across modalities) but does not test causal mechanisms. Future work could use interventional designs (e.g., training language models with manipulated structural properties, or using TMS/lesion studies to perturb aMCC function). •Alternative axes: We focus on PC1 as the dominant axis, but secondary PCs may capture additional meaningful structure (e.g., domain-specific constraints, genre effects). Future analyses should explore higher-order components and their potential biological or computational significance. These limitations will be explicitly discussed in any resulting publication, along with suggestions for follow-up studies to address them. 13 Resources, Timeline, and Sharing Plan Computational Resources •Hardware: Standard laptop or workstation (no GPU required for structural metrics; GPU optional for embedding computation to speed up processing). •Estimated total compute time: 2∼22−−444hoursforstructuralandmanifoldanalyseswith50− −100objects. •Software: Python 3.8+, numpy,scipy,scikit-learn, HuggingFace transformers, and standard fMRI analysis tools (SPM, FSL, or Nilearn) for neural analyses. Data Sources •Structural analyses: Publicly available text corpora (Project Gutenberg, arXiv, GitHub), biological sequence databases (NCBI, UniProt), audio datasets (e.g., Watkins Marine Mammal Sound Database). •Neural analyses: Public neuroimaging datasets from OpenNeuro or similar repositories, with appropriate task designs (volatility manipulation, control-demand paradigms). •All data sources and accession numbers will be listed in the final analysis report. 18 Timeline •Structural universality and manifold concordance analyses: 1∼11−−222weeks(includingdatacollection, preprocessing, analysis, androbustnesschecks). •Neural analyses (aMCC): Dependent on dataset identification and preprocessing; estimated 22 2–44 4 weeks. •Total timeline from preregistration to completed analysis: 4∼44 − −888weeks. Code and Data Sharing •All analysis code will be released in a public version-controlled repository (GitHub) with a tagged release (e.g., v1.0-preregistered) corresponding to this preregistered version. •A container specification (Docker or Conda environment file) will be provided to facilitate exact reproducibility of the computational environment. •Derived datasets (feature matrices, PC scores, embeddings) will be shared via Zenodo or Open Science Framework (OSF) where licensing permits, with appropriate citations to original data sources. •Raw neuroimaging data will not be redistributed but will be referenced via original dataset DOIs/accession numbers with preprocessing scripts provided. •Any deviations from the preregistered analysis plan will be clearly documented in a “Transparency Appendix” in the final report. 14 Registration and Versioning This preregistration will be deposited in two locations: 1. Open Science Framework (OSF): As a time-stamped preregistration, locked to prevent modification. 2. Zenodo: As a citable preprint with a permanent DOI. Version history: •v1.0 (December 12, 2025): Initial preregistration (this document). •Future versions (if any) will be clearly labeled and will note all changes from the preregistered plan. Any post-hoc analyses or exploratory findings will be clearly distinguished from preregistered confirmatory tests in the final report. 19 15 Conclusion This preregistration specifies a rigorous, falsifiable program for testing whether a structurally defined principal component, ψ−−PC1, functionsasaunifying structural invariantacrosssymbolicsystems, artificiallanguagemanifolds, andbiologicalcontrolcircuits. 15.1 Key Commitments We commit to: •Eight independent falsification routes, any one of which could reject the framework; •Strong constraints against false positives via the four-layer FPI framework (null rejection, domain independence, model independence, metric stability); •Explicit thresholds for variance explained (0.70), alignment (0.75), and effect sizes (Cohen’s d0.5d ≥0.5d0.5forasymmetry); •Fully specified methods: Exact metric formulas, ROI coordinates, RC∆RCRCdefinition, andmodelversions; •Graded interpretations: Clear contingency plans for partial support or refutation; •A concrete replication recipe requiring only standard open-source tools; •Transparent reporting of all results, including effect sizes, confidence intervals, and deviations from the preregistered plan. 15.2 Anticipated Outcomes If the hypotheses are supported:ψ−−PC1willbeastrongcandidateforaconvergent structural solutionemergingwheneversystemscompresscomplexsymbolicdynamicsunderpredictiveandresourceconstraints.Thiswouldsuggestadeepconnectionbetween : information-theoretic properties of sequences (entropy, complexity, burstiness), learned representations in artificial neural networks, biological control mechanisms in mammalian cortex. Such a result would motivate further theoretical work on the underlying optimization principles and experimental work to test causal predictions. If the hypotheses are rejected: The resulting refutation will yield valuable constraints on how far structural invariants can generalize across biology, culture, and machine intelligence. Specific failure modes (e.g., strong structural axis but no neural correspondence, or strong embedding alignment but domain-specific PC1) would guide refinements to the theoretical framework and suggest productive directions for future research. In 20 either case, this preregistration ensures that the evidence will be interpretable, the methods will be reproducible, and the conclusions will be appropriately calibrated to the strength of the empirical findings. Key References •Goh, K.-I., Barabási, A.-L. (2008). Burstiness and memory in complex systems. EPL (Europhysics Letters), 81(4), 48002. •Holroyd, C. B., Yeung, N. (2012). Motivation of extended behaviors by anterior cingulate cortex. Trends in Cognitive Sciences, 16(2), 122–128. •Shenhav, A., Botvinick, M. M., Cohen, J. D. (2013). The expected value of control: An integrative theory of anterior cingulate cortex function. Neuron, 79(2), 217–240. •Lempel, A., Ziv, J. (1976). On the complexity of finite sequences. IEEE Transactions on Information Theory, 22(1), 75–81. 21