floor, ceiling concept

Definition	A floor is a lower bound a quantity cannot fall below and a ceiling an upper bound it cannot exceed, each proved or measured and named as such. Chapter 0 section 0.9 and chapter 13.
Example	The refresh floor was 4 slots at 10 hertz, and the Poisson ceiling at 10 expected retrievals per row is about 20.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `floor-ceiling`, kind concept.
Status	refutes or corrects [missed]. Corrections: 1 item(s), see below.
Defining equation	Book equation 0.17.
Assumptions and scope	A floor is a lower bound a quantity cannot fall below, and a ceiling an upper bound it cannot exceed, each proved or measured and named as such. The refresh floor is the shortest renewal that keeps a certificate within its error, the Poisson ceiling is the count chance allows, and the Youden ceiling bounds the best F1. A floor that was fitted as a fraction of the coherence time was refuted, and the omission floor that was first read as unbounded was corrected to a floor, so each is a measured object with a record.
Prior art	none recorded
Evidence	`geometric-observation/claims/LEDGER.md:106`, `geometric-observation/chapters/ch07_cost.md`, `geometric-observation/claims/LEDGER.md`, `lean/DataMiningAsObservation/CoherenceTime.lean`, `lean/DataMiningAsObservation/PoissonCeiling.lean`, `lean/DataMiningAsObservation/YoudenF1.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

A lower bound a quantity cannot fall below, an upper bound it cannot exceed.

Equation

Book equation 0.17.

\[\Pr[X\ge c]=1-\sum_{i<c}e^{-\mu}\frac{\mu^{i}}{i!},\qquad c^{\star}=\max\{c:\ n\Pr[X\ge c]\ge 1\},\qquad \mu=\frac{n_q\,k}{n}.\]

Book equation 5.4.

\[\begin{gathered} F_1^{\max}\ \le\ \sup_{t\in[J,\,1]}\ \frac{2t\pi}{t\pi+\pi+(t-J)(1-\pi)},\qquad J=\max_{\tau}\big(\mathrm{TPR}-\mathrm{FPR}\big),\qquad \pi=\text{prevalence}, \\ J\le 2A-1\ \text{when the ROC curve is concave, and not in general.} \end{gathered}\]

Book equation 13.2.

\[\begin{gathered} T_{\mathrm{coh}}=\frac{0.423}{f_D}, \\ \text{refuted fit:}\ \ \phi\approx0.177\,T_{\mathrm{coh}}\ (R^{2}=0.915), \qquad \text{sealed:}\ \ \phi\le 6\ \text{TTI at }10\ \text{Hz},\ \ 1\ \text{TTI at}\ \ge50\ \text{Hz}. \end{gathered}\]

Conditions

A floor is a lower bound a quantity cannot fall below, and a ceiling an upper bound it cannot exceed, each proved or measured and named as such. The refresh floor is the shortest renewal that keeps a certificate within its error, the Poisson ceiling is the count chance allows, and the Youden ceiling bounds the best F1.
A floor that was fitted as a fraction of the coherence time was refuted, and the omission floor that was first read as unbounded was corrected to a floor, so each is a measured object with a record.

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

Ledger

refutes or corrects. NEG-13 → resolved [missed]. (GO-P-2026-026, prospective, real LLM) Appendix-E’s omission floor is a rate-irreducible downstream wall on trained Llama read operators. geometric-observation/claims/LEDGER.md:106.

First stated

Chapter 0 section 0.9 and chapter 13 section 13.4 of Data Mining as Observation, with the omission floor in geometric-observation/chapters/ch07_cost.md and the Youden ceiling in theory-radar/paper/astar_paper.tex:94-110.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 4 section 4.2	commission tax of order half r log M over m, omission floor, unbounded corrected to floor, naive projector 30 percent over, VI-6	`geometric-observation/chapters/ch07_cost.md` section on mismatch; `geometric-observation/claims/LEDGER.md` VI-6
chapter 4 section 4.2	floor measured on 16 of 16 heads, naive assumption off 20 to 60 percent, about 100000 times, 0.10 nats against about 6e-7, first gate missed 0 of 16, NEG-13	`geometric-observation/chapters/ch07_cost.md`; `geometric-observation/claims/LEDGER.md` GO-P-2026-027 and NEG-13

Failures and corrections

NEG-13 → resolved, [missed]. (GO-P-2026-026, prospective, real LLM) Appendix-E’s omission floor is a rate-irreducible downstream wall on trained Llama read operators. geometric-observation/claims/LEDGER.md:106.

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/CoherenceTime.lean, theorems clarke_to_three_decimals, examples, refuted_law_exceeds_sealed, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

lean/DataMiningAsObservation/PoissonCeiling.lean, theorems mass_nonneg, tail_antitone, tail_zero, tail_le_one, expectedAtLeast_antitone, example_mean, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

lean/DataMiningAsObservation/YoudenF1.lean, theorems f1_eq, f1_le_of_youden, auroc_eq, youden_eq, youden_exceeds_auroc_form, 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, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14.

refresh floor; Poisson ceiling; Youden index; budget cliff; bar.

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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