bit concept

Definition	The unit of a budget. A direction quantized with b bits is stored at one of two to the power b levels. Chapter 0 section 0.7.
Example	Four bits give 16 levels; a fifth bit gives 32, halving the step and quartering the squared error.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `bit`, kind concept.
Status	measures [replicated]. Corrections: none recorded.
Defining equation	Book equation 0.13.
Assumptions and scope	The unit of a budget. A direction quantized with b bits is stored at one of two to the b levels, so one more bit doubles the levels, halves a uniform quantizer’s step, and quarters the squared error of the high-rate model, and bits add across directions since the levels multiply. Every flip comparison is at matched bits, since a code that spends more bits is a different budget and not a better reader.
Prior art	none recorded
Evidence	`geometric-observation/claims/LEDGER.md:69`, `lean/DataMiningAsObservation/Bit.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

Each bit doubles the levels and halves the step.

Equation

Book equation 0.13.

\[D(b)=\sum_i s_i v_i\,4^{-b_i},\qquad b_i=\max\!\Big(0,\ \tfrac12\log_2\frac{s_i v_i}{\theta}\Big),\qquad \sum_i b_i=B.\]

Book equation 4.2.

\[D(b)=\sum_i s_i\sigma_i^{2}\,2^{-2b_i},\qquad s_i=v_i^{\top}P_C\,v_i,\qquad b_i^{\star}=\max\!\Big(0,\ \tfrac12\log_2\frac{s_i\sigma_i^{2}}{\theta}\Big),\qquad \sum_i b_i^{\star}=B,\]

Conditions

The unit of a budget. A direction quantized with b bits is stored at one of two to the b levels, so one more bit doubles the levels, halves a uniform quantizer’s step, and quarters the squared error of the high-rate model, and bits add across directions since the levels multiply.
Every flip comparison is at matched bits, since a code that spends more bits is a different budget and not a better reader.

Conditions are curated in entries.toml rather than read from a record.

Ledger

measures. GO-7 [replicated]. A stored description’s description rate and its conditional Landauer reset content are operationally separate resources: the same finite-\(n\) code index needing \(\hat R\approx0.67\) bits/symbol to describe is fully recoverable from retained … geometric-observation/claims/LEDGER.md:69.

First stated

Chapter 0 section 0.7 of Data Mining as Observation, with the program’s matched-bits comparisons in Volume 14 and turboquant-pro.

Measurements

none

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/Bit.lean, theorems levels_succ, levels_add, step_succ, sqError_succ, sqError_antitone, levels_zero, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation primer S, chapters 0, 1, 2, 4, 6, 10, 11, 12, 13, 14.

budget; water-filling; the flip; Landauer's principle.

Status

Generated 2026-09-10 by encyclopedia/generate.py; book at observation-data-mining f3914f0; the commit of every record is listed in the encyclopedia’s provenance.

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