Causal-fate dynamics of unrealized influence

• Generated influence may be partially realized, retained as a latent consequence, transformed by later dynamics, and eventually re-enter the trajectory or disappear.Source: §1 · p. 2
• The realized state alone may not determine the future when histories have generated different latent consequences that remain unexhausted.Source: §1 · p. 2
• The studies examine a connectome-constrained neural model, dynamically updated BGP history, and a Transformer that explicitly transports and selectively realizes latent contextual influence.Source: §1 · pp. 2–3
• The neural model motivates a biological hypothesis and routing observations are compatible with future-relevant history, but neither establishes a biological mechanism or verifies the finite-transport equations.Source: §1 · pp. 2–3
• Identical visible chess configurations can differ in future admissibility because castling rights depend on prior moves and a draw claim depends on repetition history.Source: Figure 1 · p. 3

2.1 Unrealized influence becomes a dynamical state

• Generated influence need not equal the contribution admitted at a step: latent states represent unexhausted consequences that transport can transform, and augmenting the realized state can distinguish futures.The resulting balance identity is not a partition into non-negative or mutually exclusive fractions; vector contributions can cancel, and admitted influence can include transported history.Source: §2.1 · pp. 4–5

2.2 Finite transport gives latent influence a counterfactual meaning

• Finite transport makes the aggregate latent displacement the exact difference completing the realized state into the fully realized counterfactual, while overlapping channel embeddings can make its decomposition nonunique.The complete represented state evolves under the fully realized map when initialized identically to that reference. The latent state also captures history-dependent information needed when identical realized states can lead to different futures under the same realization protocol.Source: §2.2 · p. 7

2.3 Selective inter-neuronal realization sustains propagation in a connectome-constrained neural model

• In the connectome-constrained Caenorhabditis elegans model, selective realization preserved principal propagation structure with 3.51499% relative error, r > 0.99, and complete sign agreement.Local-autonomy dynamics failed to reproduce the selected propagation structure, with relative error 1. This is a model-based functional witness motivating a biological hypothesis, not evidence that unresolved influence carriers have been identified in living animals; testing that hypothesis requires direct experiments. The model uses the structure of Kunert, Shlizerman and Kutz and propagation data from Randi et al..Source: §2.3 · pp. 8–9, Figure 2 · p. 9

2.4 Ordered operational routing history retains future information beyond the current local state

• Routing history improved held-out relative log-loss over the strong current-state model by 1.97% for Logistic regression and 9.54% for HistGradientBoosting.Gains were positive in 7 of 10 and 10 of 10 held-out time blocks, respectively. Reviewed trajectories showed the unmatched front could be enhanced, consumed, or expired; chronology helped in some comparisons but had no uniform advantage over every same-content representation. These findings concern predictive history, not the additive latent state of the finite-transport construction.Source: §2.4 · pp. 10–11

2.5 Causal-fate dynamics defines a viable Transformer architecture

• The selective-realization Transformer achieved perplexity 39.9138, 3.27% above the fully realized model, while realizing 88.40% of token–layer influence opportunities.It admitted accumulated contextual influence in 26.52 of 30 layers per token on average, while every layer had at least one token realizing cross-token influence. The experiment tested functional viability, not computational efficiency; the implementation used pretrained SmolLM2-135M [19] and WikiText-2 [20].Source: §2.5 · p. 11, Figure 4 · p. 12
• Causal-fate dynamics defines a viable Transformer architecture for explicitly carrying and selectively realizing contextual influence.Source: §2.5 · p. 12

3 Discussion

• When finite maps and a fully realized reference are specified in additive coordinates, the complete represented state evolves exactly under the fully realized map.This is one exact representation, not a claim that every system shares the same ontology or coordinate decomposition.Source: §3 · p. 12
• In the connectome-constrained C. elegans model, selective realization preserved the principal propagation structure that local-autonomy dynamics failed to reproduce.The result motivates a hypothesis about unresolved influence but does not identify physical carriers in living animals.Source: §3 · p. 12
• A bounded ordered cross-observer history retained short-term predictive information beyond the current local route in the tested BGP models.The analysis establishes neither a direct causal edge between collectors nor the exact additive finite-transport equations.Source: §3 · pp. 12–13
• The Transformer computation graph implements token-local evolution, transported latent contextual influence, and selective realization, establishing architectural viability rather than efficiency or improved prediction.The study does not establish reduced attention cost or computational speed.Source: §3 · p. 13
• The three studies support distinct claims: a model-motivated neural hypothesis, a routing observation compatible with future-relevant history, and an executable Transformer construction.Together they do not verify a universal physical mechanism; empirical relevance depends on the chosen state, maps, and realization rule.Source: §3 · p. 13

4.1 Rule-defined chess illustration

• Chess examples hold visible piece placement fixed while changing castling rights or repetition history, illustrating how history can alter available claims under FIDE rules.The examples follow Articles 3.8.2.1 and 9.2.Source: §4.1 · p. 13

4.2.1 Data, model and causal-fate construction

• The neural experiment combined the public C. elegans propagation atlas with connectome and synapse-sign resources [25] for a 188-neuron head network.A selective trajectory used fully coupled and local maps, maintaining realized state and latent residual while releasing selected influence; the 168 stimuli were split into 118 selection and 50 evaluation stimuli.Source: §4.2.1 · pp. 13–14

4.2.2 Propagation-function evaluation

• Propagation was evaluated on valid masked entries using relative error as the primary metric, with Pearson correlation and sign agreement also computed.The local-autonomy comparison used the fully realized propagation matrix as its target.Source: §4.2.2 · p. 14

4.3.1 Data sources and target

• The BGP analysis used Route Views Chile and RIPE RIS rrc06 updates over a fixed 24-hour window, targeting local route changes within 30 seconds.The held-out test contained 13,630 samples from 81 episodes and 14 prefixes.Source: §4.3.1 · pp. 14–15

4.3.2 Representations and online state update

• The ordered propagation-front representation augmented current route features with five newest unmatched remote updates and queue summaries, matching local events to the oldest eligible remote occurrence.Entries older than 300 seconds expired; controls tested shuffled or unordered event content and age encoding.Source: §4.3.2 · p. 15

4.3.3 Models, chronological split and uncertainty

• Logistic regression and HistGradientBoosting were evaluated with chronological episode-level splits, online-only features, and held-out log loss with block-resampled 95% intervals.Hyperparameters were selected on validation log loss and frozen before testing.Source: §4.3.3 · p. 15

4.4.1 Model and evaluation data

• The Transformer study used fixed pretrained SmolLM2-135M weights [19] on WikiText-2 [20], with 8,255 effective test prediction positions.Validation data were used only to select the realization threshold.Source: §4.4.1 · p. 16

4.4.2 Layer construction and metrics

• Across Transformer depth, the full map retained causal self-attention while the token-local map carried realized state and latent contextual influence was selectively admitted.Evaluation included next-token cross entropy, perplexity, and realization ratios; these metrics do not measure attention cost or speed.Source: §4.4.2 · p. 16

Data availability

• The C. elegans propagation data, Worm Neuro Atlas repository, connectome resources, and synapse-sign predictions [25] are publicly available.The implementation used the Worm Neuro Atlas repository at a specified commit.Source: Data availability · p. 16
• The BGP analysis used public Route Views and RIPE RIS archives, but study-generated processed tables and figure-source records lack a separate external archive.The large raw source archives are not redistributed with the manuscript.Source: Data availability · p. 16
• The Transformer experiment used public WikiText [20] and SmolLM2-135M [19] resources, while study-generated result objects are not separately archived.Source: Data availability · p. 17

Code availability

• The repository contains source code, configurations, tests, and instructions for obtaining inputs and reproducing all three experiments.Source: Code availability · p. 17
• Third-party raw data and pretrained model weights are not redistributed in the code repository.Source: Code availability · p. 17

Reproducibility statement

• External resources used in the studies are publicly available from their original providers.The manuscript uses saved result tables for its numbers and figures; study-generated result objects are not deposited in a separate public archive.Source: Reproducibility statement · p. 17
• The neural experiment fixes its source revision, connectome, kernel-window rule, and readout, then selects realization policy on 118 stimuli and evaluates on 50.The stimulus split is not independent validation of the calibrated neural substrate.Source: Reproducibility statement · p. 17
• The BGP study fixes its collectors, 24-hour UTC window, chronological episode-level split, and evaluation protocol; the Transformer study fixes its model revision and WikiText configuration.Source code and execution instructions are maintained in the repository identified in the Code availability statement.Source: Reproducibility statement · p. 17

Predictive-state necessity

• If two admissible histories share a realized state but yield different future trajectories under a common protocol, the realized state is not predictively sufficient.An exact predictive representation must retain history-dependent information that cannot be recovered from the realized state alone.Source: Predictive-state necessity · p. 22
• The missing predictive distinction need not have a unique coordinate system, but causal-fate dynamics constrains its latent coordinate through the same finite dynamics as the fully realized reference.An invertible coordinate change can relabel the information but cannot eliminate the need to distinguish histories with the same present and different futures.Source: Predictive-state necessity · p. 22

Aggregate uniqueness and channel-level representation freedom

• With fixed fully realized maps, local maps, inputs, initial state, and realized trajectory, the compatible aggregate latent displacement is unique, while channel-level latent state is unique only if B is injective.If B is not injective, states differing by an element of its null space yield the same aggregate displacement; channel claims then require additional constraints or independent observations.Source: Aggregate uniqueness and channel-level… · p. 23

Public data and fixed propagation object

• The fixed propagation object is a pairwise stimulus–response matrix from the public atlas, with the neural model using atlas-associated connectivity and synapse-sign resources.The model includes 188 head neurons and both directed chemical synapses and electrical gap junctions.Source: Public data and fixed propagation object · p. 24

Connectome preprocessing

• Chemical and gap-junction connectivity matrices are restricted to the 188 head neurons, transformed with log(1 + w), and normalized by incoming row sums.Gap-junction counts are symmetrized before normalization, and chemical reversal potentials are assigned by synapse sign.Source: Connectome preprocessing · p. 24

Neural model and full/local decomposition

• The local neural map retains membrane, gate, and stimulus terms while withholding cross-neuronal chemical and gap-junction currents from the full map.The state contains membrane voltage and synaptic gate variables for each of the 188 neurons.Source: Neural model and full/local… · p. 25

Selective-realization protocol

• Selective realization transports latent remainder and candidate non-local influence through the finite-displacement recurrence, using a 0.1-s interval and a fixed realization threshold.The production integrator uses RK4 with a 0.0125-s step; numerical checks matched the implemented recurrence to recorded precision.Source: Selective-realization protocol · pp. 25–26

Readout, calibration and stimulus-stratified evaluation

• The readout is calibrated against the frozen empirical propagation object, then realization settings are selected on 118 stimuli and evaluated on the remaining 50.The operating point is selected among candidates meeting prespecified propagation-quality criteria, choosing the lowest qualifying realization frequency.Source: Readout, calibration and… · pp. 26–27

Public data and MRT reconstruction

• The study reconstructs BGP announcements and withdrawals from RouteViews Chile and RIPE RIS rrc06 over a fixed 24-hour UTC window.Events are ordered by timestamp and record order, and convergence episodes are defined by gaps no greater than 300 seconds.Source: Public data and MRT reconstruction · p. 27

Prefix and peer selection

• From 119,612 shared active prefixes, the study retains 50 by a prespecified event-count ranking and selects one anchor peer per collector before evaluation.The selected peers are AS27678 at RouteViews Chile and AS25152 at rrc06, each covering 29 selected prefixes.Source: Prefix and peer selection · p. 28

Unmatched-front state and feature definitions

• The unmatched queue retains remote updates not consumed by a corresponding local event, removing events older than 300 seconds and clearing queues at episode gaps.A local event consumes the oldest unmatched event with the same fingerprint; the five-slot view shows the newest surviving entries, while summaries cover all retained entries.Source: Unmatched-front state and feature… · pp. 28–29

Within-sample content and order controls

• Within-sample controls preserve each sample’s event content while separately testing slot encoding and age-free chronological content order.The age-free comparisons are model- and representation-dependent, not estimator-independent tests of conditional independence; revisions followed inspection of an earlier analysis and were not a prospective independent replication.Source: Within-sample content and order controls · pp. 29–30

Model specification

• Logistic regression and HGB used training-fitted preprocessing and validation-selected hyperparameters; test results did not select the representation or correction protocol.AS-path hash-bucket codes entered the numerical branch, but their spacing did not measure path similarity.Source: Model specification · p. 30

Chronological split and uncertainty

• The analysis retained the 24-hour window and chronological 60/20/20 split, excluded boundary-crossing episodes, and used only pre-sample features to predict later route changes.Paired bootstrap intervals resampled ten consecutive 30-minute held-out blocks and condition on the fitted models and held-out window.Source: Chronological split and uncertainty · p. 30

Held-out comparisons and ablations

• 1.97% for Logistic and 9.54% for HGB were the corrected five-event front’s relative gains over the strong current-state model.Their 95% intervals were [0.23%, 3.37%] and [7.18%, 11.45%], respectively.Source: Held-out comparisons and ablations · p. 30, Table S5 · p. 31
• Without per-event ages, chronological content gained 2.58% and 1.20% over shuffled content; the primary HGB comparison with canonical inventory crossed zero.Against the canonical inventory, the corresponding gains were 0.96% and -0.32%; order-specific evidence depended on model and seed.Source: Held-out comparisons and ablations · p. 31

Time-block, collector and prefix heterogeneity

• Five-event-front gains were positive in 7 of ten time blocks, 1 of two collector directions, and 9 of 14 prefixes for Logistic; HGB counts were 10, 2, and 11.The reported group results include small and negative-gain groups.Source: Time-block, collector and prefix… · p. 32

Seed robustness and alternative horizon

• The front and age-free content comparisons were refit under three seeds, while the 120-second horizon and history-summary representation remained secondary analyses.These analyses were not replacement criteria for selecting the headline result.Source: Seed robustness and alternative horizon · p. 33, Table S10 · p. 34, Table S11 · p. 34

Matched observations and queue transitions

• The retained tolerance-matching procedure found 100 pairs with different 30-second outcomes, but none matched every checked current-state coordinate exactly.The outcome-selected pairs were not exact controlled-state interventions; queue examples describe constructed-state changes, not physical causes of later route changes.Source: Matched observations and queue… · pp. 33–35, Table S13 · p. 35

Model and evaluation data

• The experiment used public pretrained SmolLM2-135M weights without fine-tuning and evaluated next-token prediction on contiguous WikiText-2 raw-text blocks.The validation set contained 33 blocks and the test set 65 blocks.Source: Model and evaluation data · p. 36

Full and token-local layer maps

• The full layer map is the original causal-attention layer, while the token-local map reuses its parameters but restricts attention to token-local computation.Final logits are computed from the final realized state.Source: Full and token-local layer maps · p. 36

Finite-transport recurrence and token-level realization

• Each token’s contextual influence is released in full when its score exceeds the frozen threshold; otherwise it remains latent, with tokens gated independently.The terminal latent state is not forced to release.Source: Finite-transport recurrence and… · pp. 36–37

Validation selection and test evaluation

• The realization threshold was selected using validation perplexity and fixed before test evaluation; neither test data nor weight updates were used for selection.The rule chose the lowest non-trivial realization ratio within 1.06 times fully realized validation perplexity.Source: Validation selection and test evaluation · p. 37, Figure S2 · p. 37

Evaluation metrics

• Perplexity is the exponential of mean next-token cross entropy, while the realization ratio counts released token-layer entries among valid entries.Mean active layers per token equals the realization ratio times 30.Source: Evaluation metrics · p. 38

Code availability

• Source code for the three experiments is maintained at the paper’s stated GitHub repository.The passage also provides reproduction instructions there.Source: Code availability · p. 38
• Third-party inputs are obtained from their original providers rather than redistributed with the code.Source: Code availability · p. 38
• The repository is identified as the location for the experiments’ reproduction instructions.Source: Code availability · p. 38