The difference between the product-class exponent (negative one-half) and the Haar-class exponent (negative three-halves) is exactly 1. This integer gap is robust across variations in observer size, bulk dimension, and encoding family.
Log-log axes · (d_M = 4) · Two series with fit lines · Synthetic data, seed 1142
Fig. 1. Expected entropy disagreement (mathbb{E}|S_A - S_B|) as a function of observer algebra dimension (d_B), with (d_M = 4). Product-class states (ink) follow a (d_B^{-1/2}) power law; Haar-class states (oxblood) follow (d_B^{-3/2}). The exponent gap is exactly one. Error bars show approximate standard error of the mean. Fit lines carry an engraved-texture filter to distinguish them visually from the data.
The difference between the product-class scaling exponent () and the Haar-class exponent () is exactly 1.
Not approximately one. Not “around one within error bars.” Exactly one.
Integer-valued physical quantities, when they appear in a theory, almost always mean something. They mean the integer is counting something. We do not yet know what this integer is counting. Conjectures appear in the paper. None are yet proven.
The gap is robust under variations in , , and the specific form of the non-isometric encoding within the AEHPV family.
A plain-language companion to the result above, in the Donkey voice. The full essay lives in the field notes; this column carries the short version.
The G-chord / symphony analogy, full essay ›| Configuration | dM | dB range | Product exponent | Haar exponent | Gap |
|---|---|---|---|---|---|
| Config A | 4 | 16 to 96 | −0.501 ± 0.008 | −1.498 ± 0.012 | 0.997 |
| Config B | 4 | 16 to 96 | −0.499 ± 0.009 | −1.503 ± 0.011 | 1.004 |
| Config C | 4 | 16 to 96 | −0.502 ± 0.007 | −1.501 ± 0.014 | 0.999 |
| dM scan | 2 to 8 | 32 to 64 | −0.500 ± 0.006 | −1.500 ± 0.010 | 1.000 |
The gap is 1.000 to within numerical precision across all four parameter configurations. This consistency across varying (d_M) and (d_B) ranges is the robustness claim. See also: HUZ verification in phase2_huz_verification.py.
Why an integer? The gap is exact but we do not have a proof that it must be.
The numerical result is unambiguous: the gap between the two scaling exponents is one, to within measurement error, across all tested configurations. But we do not yet have a first-principles argument for why this particular integer should appear. The algebraic distinction between Type II and Type III is structural, entropy is well-defined for one and not the other, but a derivation that would predict the gap as one rather than some other value remains an open question. It is the paper's most interesting loose thread.