The debt ratio moves by (r − g) × d minus the primary balance: debt is sustainable at any level as long as nominal growth stays above the average interest rate on the stock, or the primary surplus covers the gap. Two corollaries people get wrong. The growth needed to stand still is r, not r × d — a high ratio multiplies the pain of a positive gap, it doesn’t multiply the growth requirement. And r is the average rate on the outstanding stock, which lags the marginal yield by years (Stock vs flow repricing).
The specimen (pythianism, 09/2026): “Japan has 260% debt/GDP; if the average rate hits 3% Japan must grow 7.8% just to service the debt. It grew 1.2%.” 260% × 3% = 7.8% of GDP is the interest bill, not the required growth — the stabilizing growth rate is ~3%. And the 3% is the long-end yield, not the average coupon, which is around 1% (memory), with roughly half the stock held by the BoJ whose interest income remits back. Japan’s real problem is the primary deficit plus a rising r; the number in the post inflates it by a factor of several. The same arithmetic run on the US: 124% × ~3.3% average coupon ≈ 4% of GDP in interest, against ~5% nominal growth — the gap is small but the primary deficit is ~3–4% of GDP (Deficits at full employment), which is where the ratio’s climb comes from. This is the quiet variable behind Fiscal dominance: a central bank capping r keeps the sign of r − g negative.
The same condition from the bear side of the Debasement trade: puckrin’s “can nominal GDP grow faster than the effective interest cost while the primary deficit comes down” is r − g with the primary balance made explicit — and his point that r reprices to expected inflation within a few years is why g can’t be inflated up for long (The inflation surprise only works once).
The commonest way to get the sign wrong: compare real GDP growth with nominal debt growth. fthegurus (09/2026, https://x.com/fthegurus/status/2092692555456504054) sets Bessent’s “we will grow our way out” against real GDP of +2.9/+2.8/+2.1/+1.5% (2023–26) and debt growth of +7.2/+6.9/+6.1/+7.5% — “debt is growing 2–3× faster than the economy.” Debt is a nominal quantity, so the right comparison is nominal GDP (~6.3%, ~5.3%, ~4.5% for 2023–25, memory); the gap is real but it is ~1.4×, not 2–3×, and it comes from the primary deficit, not from growth failing. The conclusion (you don’t grow out of a 6% deficit at 5% nominal growth) survives; the multiplier is inflated by the unit mismatch — a Check the stat against its own chart with no chart.