The Spin Drive Ledger

The Ladder is the relativistic rocket equation drawn as a price chart: exhaust velocity across, required mass ratio up, both logarithmic, with real engines as rungs: RS-25 chemical at 10^93,000 for the 0.92 c mission, DS1's ion drive at 10^13,600, Daedalus's fusion at e^90, and the photon rung at ×24, the only number a budget can hold. The book's own tank rides the chart as a gold line, and the laser sail sits off the chart entirely, which is its whole trick.

The Tank integrates the book's own burn law, consumption proportional to mass at 1.5 g, from 2.1 million kilograms, against the book's own gold ledger points: departure, the flip, the 120,906 kg arrival. At exhaust c the curve dies on the dry line at year 1.97, seven days past the flip. The exhaust dial continues past c into a hatched zone, because at 1.98 c, and only there, the three lamps below all light: the tank, the Earth clock, the ship clock. Any two light inside physics. The Timetable is the other half of the verdict: Earth and ship clocks for constant-g flight, with the book's printed times as gold dots that keep landing on the curves: 13 and 3¾ at 1.5 g, five and a half at 0.9, Rocky's 'under three years' at 2 g.

The Grace prices the novel's most famous structural move: a civilisation that never found relativity plans a 10 ly trip as 12.0 clock-years, experiences 6.64, and arrives with 45% of its provision, exactly 1 − 1/γ, unburned. The gift Rocky hands over is precisely one full Hail Mary tank. A footnote shows the honest alternative (a crew chasing a speedometer would have run dry). The File lays every claim on a card (PASS, CATCH, or the bench's own additions) and hangs two verdicts at equal size.

This site carries a signed review of Project Hail Mary, and a bench for its microbe: INST-79 audited Astrophage itself and logged 'a cell that stores energy as mass' as the novel's one granted miracle. But the miracle only covers the fuel. The novel also states a complete flight plan, and that is the engine's business: a console reading 6.045 g/s at a constant 1.5 g, '540 trillion watts'; a hundred-thousand-kilogram ship with two million kilograms of fuel; thirteen Earth years and three years nine months aboard; a 0.9 g fallback plan at five and a half; beetles with 120 kg of fuel each and 'a couple of years' on their own clocks; an alien ship that crossed ten light-years in 6.64 experienced years and had a full two million kilograms left over to give away. A photon rocket happens to be the one engine physics can price exactly: every gram of fuel leaves as light, thrust is P/c, mass ratio is e^(rapidity ÷ exhaust velocity), and there is no engineering excuse to hide in between.

The audit cuts both ways, and more cleanly than on the sibling bench. One way: every clock in the book is right. Thirteen and three-quarters are the exact answers of the 1.5 g brachistochrone; five and a half is the exact answer at 0.9 g; Rocky's 6.64 years decodes to a 0.833 c cruise, and even the fuel surplus that saves the novel equals 1 − 1/γ on the nose. The other way: the tank pays for a different trip. The printed schedule spends total rapidity 6.03, which a perfect photon rocket prices at mass ratio 415 against a tank of 21; integrate the book's own burn law and the tank runs dry seven days past the mid-flip; making all three claims true at once takes an exhaust at 1.98 times the speed of light. The bench builds that tension into a three-lamp panel: any two lamps light, all three only in the hatched zone. The verdict is not made for you, but the arithmetic is all on the table.

Everything on the bench is closed-form arithmetic on the novel's stated numbers plus physical constants: F = P/c, one burn integral, one mass-ratio equation from 1946, and the standard constant-acceleration clocks. The script that ships with the site re-derives every displayed number before the build passes.

F = P/c · m(τ) = m₀e^(−aτ/vₑ) · R = e^(Δθ·c/vₑ), Δθ = 2·atanh(β) · t = 2√((d/2c)² + d/a) · τ = (2c/a)·asinh(at/2c) · surplus = 1 − 1/γ

The drive. A perfect photon rocket: power ṁc², thrust P/c. The book's three readouts (6.045 g/s, 1.5 g, 540 TW) lock through these two lines to the ship's own stated mass at 1.9%.

The burn. At constant proper acceleration a, mass decays as m(τ) = m₀e^(−aτ/vₑ): the price of a trip is exponential in rapidity spent over exhaust velocity (Ackeret 1946). The printed schedule spends Δθ = 6.03; the tank holds ln 21 = 3.04 of budget at vₑ = c.

The clocks. A constant-a brachistochrone over distance d gives Earth time and ship time in closed form; at 1.5 g over 11.9 ly they are 13.1 and 3.89 years. Every printed time in the novel is checked against these two functions, and every one lands.

The grace. Proper velocity βγ has no ceiling: a ship covering 10 ly logs only d/(βγc) aboard, while a chart without relativity books d/(βc). The unburned difference, 1 − 1/γ, is 45% at Rocky's decoded 0.833 c, the surplus the plot spends as a gift.

Nothing is tuned for the book and nothing against it: where its phrasing allows slack ("something like thirteen years", "a couple of years"), the bench rounds in the book's favour, and the lamps' thresholds are the book's own words read generously.

A mission speed, an exhaust velocity with a hatched zone past c, a profile switch, an acceleration, an Eridian cruise. Together they draw every trip the novel's ship could take, and the one it claims to have taken, which is not among them.

• Mission cruise. 10% to 99% of c, in rapidity space; presets at the astrophage's 0.92 c, the tank's 0.909 c, the beetles' 0.986 c, and Daedalus's 12.2% c. The whole ladder reprices as you dial. Brakes on or off.
• Exhaust velocity. 0.2 c to 2.5 c, log; the zone past c is hatched, because the book needs it. Presets: c (a perfect photon rocket), 1.98 c (the repair), 0.035 c (Daedalus's rung, for contrast).
• The trip flown. The printed schedule (1.5 g all the way, 3.89 ship years) or the tank's own trip (burn ×21, cruise 0.91 c, 6.6 ship years). The lamps re-read as you switch.
• Proper acceleration. 0.5 to 2.5 g with presets at the book's three: 1.5 g outbound, 0.9 g return, 2 g for Rocky's ride home.
• The Eridian cruise. 50% to 97% of c; the book's decoded 0.833 c is marked, and the surplus tracks 1 − 1/γ live. The footnote toggle shows the navigation assumption the grace depends on.

Nine of the novel's flight-plan claims, and where each lands when the bench runs it: checks, conditions, and misfires at the same volume.

1. Start on The Ladder. Read the chemical rung: 10^93,277. Tap it. Then tap the photon rung: ×24. That gap, ninety-three thousand orders of magnitude, is why the book's engine had to be this engine.
2. Go to The Tank. Watch the crimson curve die on the dry line at year 1.97, seven days past the flip. Now press "1.98 c": the curve lands exactly on the arrival ledger, and the third lamp lights, inside the hatched zone.
3. Flip the profile to "The tank's trip". The tank lamp lights, the Earth clock holds at 13.9, and the ship clock goes out at 6.6 years. Two lamps. It is always two lamps.
4. Go to The Timetable and press "0.9 g". The gold dot at five and a half years sits exactly on the crimson curve. Press "2.0 g" and Rocky's "under three years" marker lands too. The clocks in this book are real.
5. Go to The Grace and drag the cruise dial. The surplus tracks 1 − 1/γ; at the book's 0.833 c it reads 45%, and the gift tag reads two million kilograms: one full Hail Mary tank. Toggle the footnote for the assumption underneath.
6. Finish on The File. Five PASSes, three CATCHes, the lamps, the grace, then two verdicts at equal size. The review praised the homework; this bench grades the half the sibling bench left unopened.

The honest line between the book's words, the bench's arithmetic, the real world's engines, and the reading:

One line: Every number on the bench is either the novel's own claim quoted with its chapter, a closed-form result re-derived at build time, or a flown engine's published figure standing in for reality; the modelled pieces (one consumption clock, one nozzle, one inferred dry mass) are tagged where they appear. The reading is yours.

• Andy Weir, "Project Hail Mary" (Ballantine 2021; Penguin 2022 paperback): every claim priced here, quoted with its chapter: the consumption readouts and 540 TW (ch. 4, 6), the tank and dry mass (ch. 6, 24), the departure and arrival ledgers (ch. 3, 25), the printed times and the 0.9 g plan (ch. 15, 26), the beetles (ch. 24, 29), the Blip-A figures and the gift (ch. 17, 19, 28).
• J. Ackeret, "Zur Theorie der Raketen", Helvetica Physica Acta 19 (1946): the relativistic rocket equation, the mass ratio e^(Δθ·c/vₑ) every trip on this bench is priced with.
• A. Bond & A. R. Martin (eds.), "Project Daedalus: Final Report", JBIS Supplement (1978): the fusion rung, exhaust 10,600 km/s, ~50,000 t of D/He-3 for a 450 t payload past Barnard's Star at 12.2% c. The same study INST-40 flies.
• Flown engine figures: RS-25 vacuum specific impulse 452 s (NASA/Aerojet Rocketdyne); Deep Space 1's NSTAR ion engine, ~3,100 s (NASA/JPL). The real rungs of the ladder.
• CODATA 2018 constants (c, g₀): every conversion on the bench.
• This site: the Astrophage Budget (INST-79, the sibling bench that granted the fuel its miracle), the Mothership Ledger (INST-59, the fiction-pricing precedent), Project Daedalus (INST-40), the Laser Sail (INST-20), the Warp Energy Problem (INST-12), and the signed review of Project Hail Mary in the culture wing.