Bitcoin · power law · explained simply
There's a line that Bitcoin's price keeps crashing down to and bouncing off of, almost never breaking below it. It isn't magic or a hunch. It falls straight out of one simple formula and a clever way of drawing a graph. Here's how it works, explained like you're twelve.
01 / THE IDEA
Bitcoin has a kind of birthday: it was "born" on 3 January 2009. Every day, it gets a little older. The power law says the price isn't random: it tends to follow a formula based on how old Bitcoin is.
Raising age to a power means the price grows fast early on, then the growth gradually slows down. That's different from "the same percent forever" (that's called exponential). A power law is the same kind of maths nature uses for earthquakes, the sizes of cities, and how animals burn energy.
02 / THE TRICK
Draw Bitcoin's price on a normal graph and it looks insane: flat and boring for years, then suddenly shooting straight up. So people switch to log scales, where 1, 10, 100, 1000 are spaced equally apart instead of the big numbers squashing everything.
Here's the magic. Take a logarithm of both sides of the formula and it turns into the straight-line equation you learn in school:
So a power law, drawn on a graph where both axes are log scales (a "log–log" plot), becomes a perfectly straight line. The crazy explosion turns into a neat diagonal:
03 / THE FLOOR
The price never sits still: it bubbles up high above the trend, then crashes back down. But it keeps falling to that lower line and bouncing off it, almost never staying below for long. That bottom line is the support line, or the floor.
Because the whole thing is a power law, the support line is also a power law, the same formula with a smaller value of A, which just slides the line downward.
One honest detail worth knowing up front: the floor isn't a wall the price physically pushes against. It's drawn at a fixed distance below the trend line, about 0.4× fair value, what statisticians call the −2σ band. That distance is chosen so the price's normal ups and downs stay inside the corridor roughly 95% of the time. So when you see the price "bounce off the floor," that's a regularity in how far Bitcoin has historically strayed from its trend, not a force shoving it back up. It's a useful line, but a measured band, not a law of physics.
Below is the real thing: Bitcoin's actual price history sitting inside a "corridor" of three power-law lines.
04 / WHY IT'S TAKEN SERIOUSLY
Three things make people take the floor seriously. They're worth understanding, but it's just as important to know what each one does not prove.
1. A straight line fits, but straight is easy here. A power law shows up as a straight line on log–log axes, and Bitcoin's data does line up straight. That's consistent with a power law; it isn't proof of one. Once you stretch both axes with logarithms across this many orders of magnitude, almost any price that mostly rises over years will look roughly straight. So the straight line is a point in favour, not a fingerprint that rules everything else out.
2. The line is calculated, not guessed. A method called regression finds the single line that sits closest to all the dots at once. That beats eyeballing, but it's a fact about how the line was drawn, not about whether a power law is the right idea. Regression will happily fit a straight line to data that isn't really a power law at all.
3. The fit is measured, but read the number carefully. R² scores how closely the line tracks the data, from 0 to 1, and here it's about 0.956. That sounds like "right 96% of the time," and it isn't. On a log–log chart the numbers are dominated by Bitcoin's enormous rise from cents to tens of thousands of dollars, so almost any model that captures "it went up a lot over fifteen years" scores above 0.95. A high R² here means the price rose steadily, not that the exponent is exactly 5.7, and not that a power law beats every other shape. And the line it's scoring is the middle of a corridor that's still about six times wide from floor to ceiling. The price also tends to stay on the same side of the line for years at a time, which means the real fit is looser than one tidy number suggests.
Taken together, these are good reasons to treat the power law as a serious description of Bitcoin's past, not as proof that the floor must hold in the future.
05 / TRY IT
Pick any date and the calculator runs the real Santostasi formula to show the model's floor, fair value, and ceiling, then compares them to a price you enter.
price = 10-16.493 × days5.688 · floor = trend × 0.398 · ceiling = trend × 2.512
06 / THE HONEST PART
The maths here is descriptive: it accurately describes what has already happened. It's not a law of nature like gravity that has to keep working.
Some people argue there are real reasons it might keep holding: Bitcoin's network of users, mining difficulty, and adoption all seem to grow in power-law ways that feed each other. But plenty of experts think it's just a pattern that has held so far and could break. Both views are reasonable.
So: the floor is "supported by maths" because the data genuinely forms a straight line and the line is calculated and fits tightly. But "the price will always stay above this line" is a bet on the pattern continuing, not something the maths can guarantee.
07 / FAQ
The Bitcoin power law is a model, introduced by Giovanni Santostasi, in which Bitcoin's price tracks a straight line on a log-log chart against time since the January 2009 genesis block. Concretely, fair value ≈ 10^-16.493 × (days since genesis)^5.688. Because both axes are logarithmic, the power-law curve appears as a straight line, and it has fit Bitcoin's 15-year history with an R² of about 0.956.
In the power-law model the price oscillates within a corridor of three parallel lines: a fair-value trend, a ceiling roughly 2.5× above it, and a support "floor" roughly 0.4× below it (the ±2σ bands, with σ ≈ 0.20). Historically every cycle bottom has fallen to that floor and bounced, so it acts as empirical support. The floor is not a law of nature. It is a regularity that has held so far.
Only once, and only briefly: during the March 2020 COVID crash the price dipped below the support line for a few hours before recovering. In every other cycle bottom across more than a decade, the floor has held.
The exponent (the slope of the line on a log-log chart) is about 5.7, and this build uses 5.688. An exponent near 5.7 means the price grows fast early and then decelerates, which is the signature of a power law rather than constant-percentage exponential growth.
Projecting the support line forward, the line-to-line growth rate is around 29% over a ten-year horizon, easing toward the low-20s% over twenty years. So roughly 30% per year is a near-term figure that declines over time, not a constant rate. It is a model projection, not a guarantee.
No. The model is descriptive: it fits the past well but cannot promise the future. The fit is sensitive to the time window used, the floor has broken before, and long-horizon adoption could decelerate faster than the curve assumes. Nothing here is investment advice.
Build "The Bitcoin Power Law, Explained Simply" — a beginner-friendly ("explained like you're
twelve") page about the Bitcoin power law and its mathematical floor, with an interactive
log–log corridor chart, a calculator, and an FAQ. Output one self-contained page.
── SHARED DESIGN SYSTEM ──
Dark editorial site. Google Fonts Fraunces (display serif) / Hanken Grotesk (body) / JetBrains
Mono (mono). CSS tokens: --bg:#0f1115 --bg2:#1a1d24 --bg3:#14171d --border:#2a2e37
--border2:#23272f --text:#e6e8eb --muted:#9aa0ab --dim:#6b7280 --accent:#f7931a --blue:#5b9bff
--red:#ff6b6b --green:#4ade80. Body has two faint fixed radial gradients (orange TL, blue TR),
content in a centered .wrap (~860px). Sticky nav (brand "🟠Bitcoin Power Law" + links;
hamburger on mobile, hides on scroll-down). Header = mono orange kicker (with leading rule) +
Fraunces h1 with one italic-orange <em> + muted lede. Sections separated by a top border, each
with a mono orange .sec-num ("01 / …") + Fraunces h2. Figures: bg --bg2 rounded card with a
figcaption. Plain, honest voice; "not investment advice."
── MODEL ──
Genesis 2009-01-03. fairValue = 10^-16.493 × days^5.688; floor = fair × 0.398 (−2σ);
ceiling = fair × 2.512 (+2σ); σ=0.20; R² ≈ 0.956. fmtMoney() → $ / $K / $M. Currency USD/AUD
(AUD via a live USD→AUD rate). All model values computed in USD then ×FX for display.
── STRUCTURE ──
Header: kicker "Bitcoin · power law · explained simply"; h1 "Why Bitcoin has a <em>mathematical
floor</em>"; lede: there's a line the price keeps crashing to and bouncing off; it falls out of
one simple formula and a clever graph; explained like you're twelve.
01 / THE IDEA — what's a power law: Bitcoin has a "birthday" (3 Jan 2009); price ≈ A × age^n
(n≈5.7); a formula plate showing it; power-law growth decelerates (unlike exponential);
same maths as earthquakes / city sizes / metabolism.
02 / THE TRICK — logarithms: normal price chart looks like an explosion; log scales fix it;
taking logs turns the power law into a straight line: log(price)=log(A)+n·log(age) → y=b+mx
(show this mapping as a styled algebra→line row). Then a two-panel SVG: left "Normal graph"
(flat then explodes), right "Log–log graph" (same data as a straight diagonal with a parallel
support line the price bounces off). Animate the lines drawing in.
03 / THE FLOOR — the support line is also a power law (same formula, smaller A). Then THE
INTERACTIVE CORRIDOR CHART (build in JS as inline SVG):
• log–log axes; year gridlines (x) and powers-of-ten price gridlines (y, labelled fmtMoney).
• three straight power-law lines: ceiling (red), fair value (blue dashed), floor (orange,
thick); a faintly shaded corridor between them.
• the REAL weekly price history as a grey line — fetch Bitcoin's full daily-close history
from CryptoCompare (histoday allData=true, from 2010), downsample to weekly, cache ~24h
server-side; plus a live "now" dot (white, orange ring) from the live spot.
• a faint static "today" marker.
• a DRAGGABLE vertical scrubber (orange line + grip): drag it along the time axis to read a
tooltip showing the date and Real / Ceil / Fair / Floor at that date (with coloured dots
on each line, including a grey real-price dot), and keep the calculator's date in sync.
A legend (ceiling / fair value / support / real price) and a caption.
04 / THE PROOF — three reasons it's "maths": a straight line on log–log is a power-law
fingerprint; the line is found by regression (least distance to all points), not eyeballed;
the fit is measured by R². Three stat callouts: exponent ≈5.7, R² 0.956, 5,700+ daily points.
05 / TRY IT — a "Floor & corridor calculator" card: a target date input + preset buttons
(Today/2028/2030/2035); a price-to-compare input (defaults to the live spot, editable);
a currency select (USD/AUD); results rows for ceiling/fair/floor at the date; and a verdict
sentence placing an entered price in the corridor (below floor / lower half / upper half /
above ceiling) with % above floor and vs fair value. Dragging the chart scrubber updates this.
06 / THE HONEST PART — a caveat: a great fit to the past is descriptive, not a guarantee; some
argue adoption/difficulty feed back to keep it holding, others say it's just a pattern that
could break; "the price will always stay above this line" is a bet on the pattern continuing.
07 / FAQ — 6 plain-English Q&As as <details> accordions (What is the Bitcoin power law? Why does
it have a floor? Has it broken? What's the exponent? What CAGR does it imply? Is it a
guarantee / advice?). Write answers as direct, quotable text and ALSO emit them as
schema.org FAQPage JSON-LD (built from the same data so they can't drift).
Footer: parameters used (formula, ±2σ, R²) and a credit line to Giovanni Santostasi.
── LIVE / SEO ──
Live spot badge ("live · $… · updated Ns ago", auto-refresh 60s) near the calculator; the chart
"now" dot and the (untouched) compare price track the live price. SEO: title "The Bitcoin Power
Law, Explained Simply"; meta description; canonical; OG(article)+Twitter; 🟠favicon; JSON-LD
TechArticle + FAQPage.