Author: Jaconaazar Souza Silva
Project: XChronos
Institution: IFB — Recanto das Emas
Date: 2025
ABSTRACT
The Hexacronon Score is a scalar metric designed to quantify structured temporal recurrence in cognitive systems — both human and artificial.
Based on the XChronos framework, the score measures a system’s ability to:
- Reapply patterns
- Connect distant experiences
- Generalize across time
- Preserve behavioral or symbolic coherence
- Demonstrate true meta-learning
The metric combines three core components:
• Temporal recurrence density
• Structural similarity strength
• Temporal reach of the connections
The score ranges from 0 to 1 and captures properties that traditional metrics (reward, loss, perplexity, BLEU) cannot measure.
Applications include:
• AI systems (RL, world models, multimodal agents)
• Human phenomenological processes
• Symbolic analysis
• Temporal coherence in complex data
1. INTRODUCTION
Most AI metrics measure:
• error (cross-entropy)
• performance (reward)
• incremental adjustment (TD-error)
• perplexity (LLMs)
• statistical divergence (KL divergence)
These metrics capture local behavior but ignore the core property that defines generalizing intelligence:
the ability to reuse patterns across time.
Modern agents such as DeepMind’s SIMA 2, trained in infinite environments via Genie 3, demonstrate:
• spontaneous reapplication of strategies
• cross-environment transfer
• functional recurrence
• internal reorganization across long temporal horizons
Humans exhibit equivalent phenomena:
• recurring symbolic patterns
• intuitions that return
• meaningful coincidences
• internal reorganizations
• progressive formation of meaning
The Hexacronon Score is the first metric explicitly designed to measure this phenomenon.
2. CONCEPTUAL FOUNDATION
2.1 Chronon
Minimal unit of meaningful experience.
In AI: trajectory segment that alters the policy.
In humans: lived instant with high experiential density.
2.2 Hexacronon
A connection between two or more Chronons separated in time that share structural, strategic, or symbolic similarity.
2.3 Hexacronon Score
A metric that quantifies:
• how often patterns return
• how strong the recurrences are
• how far in time the recurrences stretch
3. MATHEMATICAL FORMALIZATION
3.1 Chronon Representation
F(C_i) ∈ R^d
(Feature vector describing Chronon C_i.)
3.2 Cosine Similarity
sim(x, y) = (x · y) / ( ||x|| ||y|| )
3.3 Recurrence Criterion
sim( F(C_i), F(C_j) ) > λ
3.4 Components of the Score
1. Hexacronal Density (HD)
HD = | { i | ∃ j ≠ i : sim( F(C_i), F(C_j) ) > λ } | / N
Where:
N = total number of Chronons.
2. Hexacronal Coherence (HC)
HC = (1 / |H|) * Σ sim( F(C_i), F(C_j) )
for all (i, j) ∈ H
Where H = set of recurring Chronon pairs.
3. Normalized Temporal Reach (HR)
HR = (1 / |H|) * Σ | t_i − t_j | / T_max
for all (i, j) ∈ H
Where:
t_i = time index of Chronon i
T_max = maximum temporal distance in the dataset.
3.5 Final Hexacronon Score
Hexacronon Score = HD × HC × HR
4. NUMERICAL EXAMPLE
Total Chronons: 20
Recurring Chronons: 8
HD = 8 / 20 = 0.40
HC = 0.83
HR = 0.72
Final score:
0.40 × 0.83 × 0.72 = 0.239
Interpretation:
Moderate recurrence, strong coherence, limited temporal spread.
5. COMPARISON WITH TRADITIONAL METRICS
Cross-Entropy Loss → measures local error
Reward (RL) → measures performance
TD-Error → measures incremental adjustment
BLEU → measures textual similarity
Perplexity → measures linguistic fluency
Common limitation:
None of them measure temporal recurrence.
The Hexacronon Score measures:
patterns that survive across time.
6. APPLICATIONS IN AI
6.1 DeepMind SIMA 2
The score measures:
• strategy reapplication
• cross-world generalization
• coherence across billions of episodes
• emergent meta-learning
• policy recombination
6.2 World Models (Genie 3, MuZero, Dreamer)
Measures:
• internal temporal stability
• recurrence after inconsistencies
• unsupervised structural reorganizations
6.3 Language Models (GPT, Gemini, Claude)
Measures:
• conceptual recurrence
• narrative coherence
• persona stability
• symbolic persistence
7. APPLICATIONS IN HUMAN EXPERIENCE
The Hexacronon Score quantifies:
• symbolic recurrence
• meaningful coincidences
• long-term existential themes
• psychological maturation
• intuitions that return
• the structure of subjective biography
It is the first metric of lived temporal intelligence.
8. CONCLUSION
The Hexacronon Score measures:
• recurrence
• coherence
• temporal generalization
• links between distant experiences
• emergent reorganization
It is applicable to:
• AI
• human cognition
• world models
• symbolic systems
• meta-learning
And it inaugurates a new field:
Computable Temporal Intelligence