The Core Algorithm

Probability Matrix vs. Histogram Counting

Traditional Approach

Histogram-Based Symbol Counting

Accumulate frequency counts for each symbol
Probabilities derived from count ratios
Discrete updates on symbol occurrence
Context-dependent count tables

P(s) = count[s] / total_count

Static representation—counts persist until explicitly updated

HoloCodec Approach

Dynamic Probability Matrix

Continuous probability distribution matrix
Recalculated at each step
Gradient-based weight adjustments
Context-aware state evolution

P(s|context) = softmax(W · φ(context))

Dynamic representation—probabilities continuously shift

The Innovation

Rather than maintaining discrete frequency histograms, HoloCodec employs a continuous probability matrix that is recomputed at every encoding step. This matrix captures symbol likelihoods conditioned on the current context state.

Processing Pipeline

1. Context Encoding: Current state φ(context) → vector representation

2. Weight Matrix: W · φ(context) → raw prediction logits

3. Softmax Normalization: exp(z_i) / Σ exp(z_j) → valid probability distribution

4. Range Encoding: Map probabilities to cumulative intervals for arithmetic coding

Key Advantage

The softmax transformation ensures the output is a proper probability distribution (sums to 1.0) regardless of input scale, enabling seamless integration with range coders. This approach allows the model to adapt its predictions smoothly based on learned patterns rather than rigid count accumulation.