Special-relativistic time dilation γ=1/√(1−β²), recovered from cosmic-ray muon survival (Frisch–Smith)

A muon's mean lifetime is 2.197 µs, so at β=0.9952 it should travel only β·c·tau0 ≈ 660 m before decaying — yet muons born ~15 km up reach sea level in force. How do they survive a fall that should cost ~50 lifetimes?

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Method

Forward-model the Frisch–Smith 1963 experiment (Mt Washington, h=1907 m, β=0.9952, tau0=2.1969811 µs). A muon's decay clock ticks in PROPER time; the lab sees it dilated by γ, so survival to a detector h below is S=exp(−(h/βc)/(γ·tau0)). A SEEDED Monte-Carlo (mulberry32) of 4–5 million individual muons (proper lifetimes ~ Exp(tau0)) measures the surviving fraction S, and γ is inverted as γ=(h/βc·tau0)/(−ln S) — a recovery formula that never contains γ (only h, β, c, tau0, all loaded from the reference). Repeated across five β; a Galilean control (no dilation, decay in lab time) is run as a falsification test and cross-checked against the real summit/sea-level flux.

The law it recovers

gamma = 1 / sqrt(1 - beta^2)

Measurements, controls & cross-checks

Perturbation worst rel error

0.00082

Survival rel

0.7522

Survival gal

0.0546

Survival ratio

13.79

Control

Name

Galilean (no time dilation, gamma=1)

Recovered gamma

1.0003

Predicted sea level per hr

31

Note

ruled out — predicts ~31/hr against the measured ~408/hr

Frisch smith

Top flux per hr

563

Predicted relativistic per hr

424

Measured per hr

408

Relativistic miss

0.038

Module certification

Rung

validated + honest-module

Certified utc

2026-07-25T19:26:39Z

Gates

H-M in scripts/muon-derisk.mjs EXECUTE the shipped MuonModule.ts headless (sha-pinned, TS stripped by 26 asserted pairs): init statics + 240-muon rain arrays bit-for-bit vs replica; 600-call lockstep (dt=1/120) with both 16x240 thin-instance lane buffers bit-exact EVERY call; %8 live-cadence FROZEN-CONTENT HUD (75 writes, all payloads bit-equal — every shown number is an init static); display earned + priced; answer-free scan with planted-violation check; tamper self-tests surgical (known -> A+C-ratio only with recovery unchanged; sha -> only H; post-strip 1-ulp TAU0 -> only the doubles-pinned statics, the f32 rain and toFixed strings absorb it)

Module systematics

Shown gamma

10.212757

Oracle gamma

10.218544

Note

the module (N=4e6, static seed 1729) and the oracle gate A (N=5e6) share ONE mulberry32 stream: survivors 3008441/4e6 is an exact integer prefix of 3761158/5e6 (tail 752717/1e6). The shown-vs-oracle gap -5.79e-3 is z=-1.26 of the EXACT prefix-vs-full binomial SE 4.60e-3, and the shown gamma sits -0.56 SE from the law (12-seed ensemble at the module N: 10.21361 +/- 0.00930, pinned seed z=-0.09) — pure Monte-Carlo count noise, fully priced, no bias mechanism. The HUD discloses both the earned gamma (10.213) and the law (10.218) on adjacent lines.

Disclosed law feeds

the visual rain applies gamma_law per muon (dilated death heights; certified per-muon ratio = gamma_law to 8.7e-8 f32) and the chart guide curves are analytic exp(-(y/betac)/(gamma tau0)); the MC dots and all HUD numbers are earned. The forward model itself uses gamma (proper-time decay IS the hypothesis under test, per reduces_to); the recovery formula never does.

Overclaim fixed

9th overclaim caught by execution: the module doc-comment claimed the recovery lands 10.218 — at the module own N=4e6/seed=1729 it lands 10.213 (10.218 is the law/oracle-5M value; the HUD was already honest, printing both). Comment corrected, zero numeric change.

What it reduces to

Special-relativistic time dilation (Einstein 1905) as confirmed by the Frisch–Smith 1963 muon experiment. Validates a textbook result: the Lorentz factor γ=1/√(1−β²) is recovered to ~1e-5 from survival statistics alone, tracks the law across β, and the no-dilation (Galilean) alternative is falsified by an order of magnitude against the real measured flux. This is the lab's first special-relativity world — the SR companion to ?world=schwarzschild (general relativity). It does NOT derive time dilation from a deeper postulate; it assumes the muon decays in proper time and shows the emergent, measurable consequence matches the landmark experiment while the classical alternative cannot.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- muon (scripts/muon-derisk.mjs)

Oracle

scripts/oracles/muon.reference.json

Sources

D. H. Frisch & J. H. Smith, Am. J. Phys. 31, 342 (1963); A. Einstein, Ann. Phys. 17, 891 (1905); muon mean lifetime tau0=2.197 µs (PDG).