Landauer's principle result

Definition	Erasing one bit costs at least Boltzmann's constant times the absolute temperature times the natural log of two. Equation 0.27. Also Landauer, reset.
Example	Erasing one bit at 300 kelvin costs at least 2.87 times 10 to the minus 21 joules.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `landauers-principle`, kind result.
Status	measures [replicated] ×3. Corrections: none recorded.
Defining equation	Book equation 0.27.
Assumptions and scope	Erasing one bit costs at least Boltzmann’s constant times the absolute temperature times the natural logarithm of two, 2.87 times ten to the minus twenty-one joules at 300 kelvin. The reset bound of chapter 13 charges that price per bit of the record’s conditional entropy given what the consumer keeps. Conditioning cannot raise entropy, so the consumer-relative bound is never above the consumer-free one, and a record that keeps nothing about the erased part costs nothing to reset. The entropies themselves are the ledger’s measurements, two source families and six codebooks.
Prior art	none recorded
Evidence	`geometric-observation/claims/LEDGER.md:69`, `geometric-observation/claims/LEDGER.md:71`, `geometric-observation/claims/LEDGER.md:73`, `geometric-observation/claims/LEDGER.md`, `geometric-observation/chapters/ch07_cost.md`, `geometric-observation/experiments/GO2-gradient-curvature-NOTES.md:1-40`, `geometric-observation/experiments/GO3-certificate-vacuity-v3-NOTES.md:1-60`, `geometric-observation/chapters/ch10_the_blind_probe.md:34-52`, `geometric-observation/chapters/ch10_the_blind_probe.md:100-118`, `geometric-observation/experiments/GO-landauer-gaussian-secondsettings-NOTES.md`, `geometric-observation/prereg/GO-P-2026-077-kv-consumer-relative.md`, `geometric-observation/results/GO13-kvaw2-governed.json`, `lean/DataMiningAsObservation/Landauer.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

Erasing a bit costs at least kT ln 2 of energy.

Equation

Book equation 0.27.

\[E_{\min}=k_B\,T\ln 2\ \approx\ 2.87\times10^{-21}\ \text{J at }300\ \text{K}.\]

Book equation 13.3.

\[W_{\mathrm{reset}}\ \ge\ k_BT\ln2\ \cdot\ H(M\mid S),\qquad H(M\mid S)\le H(M).\]

Conditions

Erasing one bit costs at least Boltzmann’s constant times the absolute temperature times the natural logarithm of two, 2.87 times ten to the minus twenty-one joules at 300 kelvin. The reset bound of chapter 13 charges that price per bit of the record’s conditional entropy given what the consumer keeps.
Conditioning cannot raise entropy, so the consumer-relative bound is never above the consumer-free one, and a record that keeps nothing about the erased part costs nothing to reset. The entropies themselves are the ledger’s measurements, two source families and six codebooks.

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.
measures. GO-8 [replicated]. On two independent source families (binary Markov; Gaussian AR(1)), a fixed stored record’s operational reset threshold rises with the age of the retained side information exactly as the staleness–work complement prices it: same record, … geometric-observation/claims/LEDGER.md:71.
measures. GO-9 [replicated]. Coordinated reset is operationally cheaper than independent reset by the records’ shared-structure information: with two consumer records sharing a component, recovering either record’s bin residual with the other record intact lowers … geometric-observation/claims/LEDGER.md:73.

First stated

Landauer, irreversibility and heat generation in the computing process, 1961, as chapter 0 section 0.13 states it, and its consumer-relative form in Volume 14 and the rate-work paper of ledger rows GO-7 to GO-9.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 4 section 4.2	GO-6, d 8 and r 4, output at or below surrogate at or below reconstruction at every rate, about 500 times, gap 0.41 to 0.005, isotropic control collapses	`geometric-observation/claims/LEDGER.md` row GO-6; `geometric-observation/chapters/ch07_cost.md`
chapter 4 section 4.4	GO-4 budget inversion, fixed m 10 rises, matched m 121, 126, 159 collapses, 3 seeds	`geometric-observation/claims/LEDGER.md` row GO-4
chapter 6 section 6.2	planted probe, overlap 0.936 vs 0.059, twelve of twelve, reconstruction 0.40, five of five	`geometric-observation/claims/LEDGER.md` row GO-1
chapter 7 section 7.2	curvature reader, identical reconstruction 0.298, flip twelve of twelve both, projected variance twelve of twelve, reconstruction Spearman negative 0.20, identity curvature tie	`geometric-observation/experiments/GO2-gradient-curvature-NOTES.md:1-40`; `geometric-observation/claims/LEDGER.md` row GO-B
chapter 7 section 7.2	real logistic model, exact Hessian, anti 300 of 300, flip 82 of 300, coupling diagnosis, bound not refutation	`geometric-observation/claims/LEDGER.md` row GO-B-optim-D4
chapter 9 section 9.3	the margin certificate, mu crit as the expected maximum of N minus 1 standard normals, rho, death at 0.948 within 6 percent, Spearman 0.991 vs 0.873, fourteen corpora, six gates, the v1 to v3 path, the standing correction	`geometric-observation/experiments/GO3-certificate-vacuity-v3-NOTES.md:1-60`; `geometric-observation/claims/LEDGER.md` row GO-3
chapter 11 section 11.7	GO-1 overlap 0.936 vs 0.059, flip 12 of 12, reconstruction 0.40	`geometric-observation/claims/LEDGER.md` row GO-1; `geometric-observation/chapters/ch10_the_blind_probe.md:34-52`
chapter 12 section 12.2	recall 0.999 vs 0.592, the derived death point within 6 percent across fourteen corpora	`openvector-bench/README.md:60-96`; `geometric-observation/claims/LEDGER.md` row GO-3
chapter 12 section 12.3	uncompressed 0.79, centred 0.84	`geometric-observation/claims/LEDGER.md` row GO-B-legal
chapter 12 section 12.3	041 non-oracle, frozen LSA TF-IDF to SVD 100 train-only, AUROC 0.975 vs 0.910, flip tied, magnitude overshot, partial	`geometric-observation/chapters/ch10_the_blind_probe.md:100-118`; `geometric-observation/claims/LEDGER.md` row GO-B-blind 041
chapter 13 section 13.5	GO-7, 0.67 bits per symbol, bin rate 0.26 equals 0.39, error 1.00 without side information, two families, six codebooks	`geometric-observation/claims/LEDGER.md` row GO-7
chapter 13 section 13.5	GO-8, 0.10 to 0.55 across ages 0 to 64, flip probability 0.05, 1 percent to 100 percent at age 32, Gaussian pass 5 of 5, the control-statistic caveat	`geometric-observation/claims/LEDGER.md` row GO-8; `geometric-observation/experiments/GO-landauer-gaussian-secondsettings-NOTES.md`
chapter 13 section 13.6	attempt three, windows 1024, 256, 32, uncertainty 0.982 to 0.892, 5 of 6, 0.4375 with SE 0.070 at 5 percent keep vs 0.30, 97 percent eviction, oracle-miss 0.370 vs 0.25, V4 0.078 vs 0.0625, contrast 0.359 with SE 0.068, n 64, seed 20260812, 89 duty cycles	`geometric-observation/claims/LEDGER.md` row GO-2/GO-12/GO-13 operational; `geometric-observation/prereg/GO-P-2026-077-kv-consumer-relative.md`; `geometric-observation/results/GO13-kvaw2-governed.json`

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/Landauer.lean, theorems eMin_300, resetBound_le, resetBound_zero, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation chapters 13.

budget; certificate; read operator; coherence time.

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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