4 Filter Independence

The most significant result of this paper is the universality of the classification rule above, not any single curve shape in isolation. The source code of the python software used for testing filters is included as supplementary material.

We tested more than two hundred distinct filter definitions, including:

• Geometric filters with varying distance thresholds δ₁,δ₂,δ₃ spanning two orders of magnitude;
• Geometric filters using different hierarchy definitions (pairs, triplets, quadruplets; equilateral, isosceles, and arbitrary configurations);
• Different decoding maps D, including linear, logarithmic, and random projections from bitstrings to spatial coordinates;
• Recursive pattern-matching filters with varying substring lengths k and varying composition rules;
• Arbitrary pattern counting: selecting a fixed pattern at random — for example, 010110100010 — and counting its occurrences in S_t as a function of entropy level.

In every case, the observed shape — monotone increasing, monotone-flattening, or a genuine hump — matched the prediction of the (a,b) classification rule in Section 3 exactly. No filter produced a shape outside these three categories, and no filter in the a ≥ b regime produced a hump under any amount of additional simulation.

The filter-independence extends across representational levels. Applying the analysis to the execution trace of the host computer running the simulation — a bitstring with no direct physical interpretation as geometry — reproduces the same (a,b)-determined shape for the same pattern composition. This is the key result: the shape is an intrinsic combinatorial property of how patterns of a given composition proliferate under this process, not a product of any specific physical model, geometric embedding, or filter definition. The physical particle spectrum is, in this sense, a gauge choice overlaid on a universal informational phenomenon: different filters give different particle definitions, but the same predictable family of distributions.

We recommend that any Level-k filter intended to represent physical matter formation have its effective (a,b) composition checked explicitly against this classification before its output curve is interpreted as evidence of matter-like emergence; a hump is only obtained when the filter’s implied composition satisfies a < b.