OCS / Tools / Spin Economics End to End
Five-stage cascade: tidal capture rate → spin-up time → BZ power → Bekenstein-Landauer compute limit → time-dilation clock speedup. From stellar capture to Kardashev-II compute ops/sec — the complete Phase 1 engineering chain.
The IMBH grows by tidally capturing stars that wander within the tidal disruption radius. In OC's current core (stellar density ~10³ M☉/pc³), the estimated capture rate is ~10⁻³ M☉/yr. This is the natural accretion rate; Phase 2 feeding would increase it by 2–3 orders of magnitude via active stellar deflection. Rate estimate is order-of-magnitude; exact value depends on loss-cone dynamics and stellar mass function.
Accreting from the ISCO gradually increases the spin parameter. The mass fraction needed scales as 1 − √(r_ISCO(a★)/r_ISCO(a₀)). This stage converts the accretion rate from Stage 1 into a time. M and Ṁ are inherited from Stage 1, but may be overridden for scenario exploration.
At the target spin, the BZ mechanism extracts rotational energy via the magnetic field threading the horizon. Power scales steeply with spin: P ∝ a²/(1+√(1−a²))². M_final and a★ are inherited from Stage 2.
The Bekenstein-Landauer-Lloyd limit sets the maximum number of bit operations per second for a given power P and temperature T. Running computronium at temperature T, the maximum ops/sec is P/(k_B × T × ln 2). Lloyd's absolute limit (2E/πℏ per second) is independent of temperature. P_BZ is inherited from Stage 3.
A computer at the ISCO runs in a time-dilated frame. Relative to a distant observer, gravitational time dilation at the ISCO provides a "clock speedup" — more subjective compute time per cosmic second. This is the ultimate advantage of ISCO-proximate computation. M_final and a★ are inherited from Stage 2.
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