The three fundamental physical limits on computation — bit storage (Bekenstein), erasure energy at temperature (Landauer), and operations from total energy (Lloyd) — applied first to any abstract system, then to a black hole ergosphere.
R only affects the Bekenstein bit-storage bound — that's a different axis (bits, not ops/s) and isn't on the ladder at all.
T and P drive Landauer; E drives Lloyd. The Panel 1 arrow follows whichever of those two is smaller, so when Lloyd is the binding limit, moving T or P does nothing visible.
In Panel 2, the arrow uses Lloyd's bound on M·c² as the ultimate ceiling. Spin a and field B change the BZ power and horizon radius (used by the on-screen rows) but don't change the rest-mass energy, so they don't move this arrow.
Even the sliders that do drive the arrows can produce small visible movement because the bar spans 80 decades — a 2-decade change in Landauer (e.g. T from 2.7 K to 300 K) is only ~2.5% of the bar width.
Bekenstein bound: S ≤ 2π k_B R E / (ℏ c). Maximum entropy that fits inside a sphere of radius R holding total energy E. Convert to bits via N_bits = S/(k_B ln 2). This is a *storage* limit — how many distinguishable states the system can hold.
Landauer limit: E_min = k_B T ln 2 per irreversible bit erasure. At available power P and temperature T, max erasures per second is P / (k_B T ln 2). This is a *throughput* limit imposed by thermodynamics — you can't erase faster than this without the heat bath complaining.
Lloyd limit: f_max = 2 E / (π ℏ). Maximum operations per second a system with total energy E can perform. This is the *quantum-mechanical* throughput ceiling, derived from the Margolus–Levitin theorem (a quantum state takes ≥ πℏ/(2E) to reach an orthogonal state).
The three limits answer different questions but are often quoted together. A real system is bounded by whichever is most restrictive for its question. For ops/sec, that's typically Landauer at room temperature (because k_B T ln 2 is much larger than ℏ-derived bounds) and Lloyd at low temperatures (because there's no thermal floor).
Panel 2 evaluates the same three limits at the ergosphere of a spinning IMBH. The key observation: at CMB temperature (because Hawking radiation is negligible for IMBH masses), Landauer's k_B T floor drops by ~2 orders of magnitude vs. room-temperature Earth, and Lloyd's bound on M·c² rest-mass energy is enormous. The compute density per kg of mass-energy is many orders of magnitude higher than anything achievable on a planetary surface. This is the quantitative foundation of the Macro Transcension Hypothesis's "compress inward" claim.
It doesn't compute "subjective ops/sec" with time-dilation factored in — that lives in the Gravitational Time Dilation tool, which is cross-linked. It doesn't address how a civilisation would actually engineer a computational substrate from spacetime curvature; that's the ✦ engineering-fiction tier. It uses split-monopole BZ for the ergospheric power input (κ ≈ 0.044) — see the BZ/Kardashev tool for that calculation in detail.
🔬 The three limits are derived from established physics and have been experimentally probed in many regimes. ✦ Their application to a civilisational compute substrate at a black hole horizon is engineering fiction.
Landauer's k_B T ln 2 at room temperature (300 K) is 2.87×10⁻²¹ J/bit — a thermodynamic floor on irreversible erasure. The principle was directly measured in Bérut et al. 2012 (Nature 483:187) using a single colloidal particle in a double-well optical trap; the measured energy per bit erasure agreed with the Landauer prediction to within experimental error. Modern silicon is still ~10⁹ to 10¹⁰ times above this floor: a 5 nm node CMOS switch dissipates ~10⁻¹² J/op; the most efficient deployed supercomputer (Frontier, on the Green500 list) achieves about 65 GFLOPS/W = 1.5×10⁻¹¹ J/FLOP. Bitcoin SHA-256 on the best 2024 ASICs (Bitmain Antminer S21) runs ~10⁻¹⁰ J/hash. Reversible-computing demos exist (DNA-based, optical) but operate at femtohertz speeds — Landauer is the floor for the irreversible compute most real machines do.
Lloyd 2000 (Nature 406:1047) computed the maximum operations per second a 1 kg, 1 litre system could perform: 5.4×10⁵⁰ ops/s. Such a system contains 10³¹ bits of storage and would last about 10⁻¹⁹ s before collapsing into a black hole — at which point the compute substrate IS the gravitational field rather than the engineered hardware. The Margolus–Levitin bound (1998) gives 6×10³³ ops/s per joule, so 1 J of energy supports six nonillion operations per second. The modern AI training run is rough orders of magnitude below this: GPT-4 training (~50 GWh ≈ 1.8×10¹⁴ J) at Margolus–Levitin would support 10⁴⁸ ops/s, vs the ~10²² ops/s actually achieved — a factor of 10²⁶ inefficiency relative to the quantum limit.
The Bekenstein bound applied to a 1 kg / 1 L sphere holding M·c² rest energy gives about ~1.6×10⁴² bits (S_max = 2πk_B R E/ħc; R≈0.062 m, E=Mc²=9×10¹⁶ J). For comparison, the entire 2024 worldwide installed storage is roughly 10²³ bits (~10 ZB) — so we're ~9 orders of magnitude short of Bekenstein for one kilogram of matter. The Sun has Bekenstein storage of ~6×10⁷⁷ bits. The observable universe, applying the bound to its energy content, comes out to about 10¹²⁰ bits (Lloyd 2002, Phys Rev Lett 88:237901) — a number that appears in cosmological information-theoretic arguments and in the Bousso entropy bound generalisations.
Lloyd's bound applied to M·c² for various black holes: 1 M☉ stellar-mass hole → 1.08×10⁸¹ ops/s. Häberle 8,200 M☉ IMBH (this tool's default) → 8.85×10⁸⁴ ops/s. Sgr A* (4.3×10⁶ M☉) → 4.6×10⁸⁷ ops/s. M87* (6.5×10⁹ M☉) → 7×10⁹⁰ ops/s. The ergosphere ops/s/kg ratio is the same constant 2c²/(πℏ) ≈ 5.4×10⁵⁰ regardless of mass — it's the per-kilogram Lloyd ceiling. The interesting thing about an IMBH isn't the per-kg rate (a kilogram of any matter has the same ceiling); it's that the IMBH packages 10³⁴+ kilograms into a single self-organising compute substrate that needs no engineered cooling or power infrastructure.
The Hawking temperature scales as 1/M, so for IMBH and supermassive holes it's vanishingly small: Häberle 8,200 M☉: T_H = 7.5×10⁻¹² K, ~12 orders of magnitude below CMB. Sgr A*: T_H = 1.5×10⁻¹⁴ K. These holes are net absorbers of CMB photons today, not net emitters — they grow (very slowly) rather than evaporate, until the CMB cools below T_H in the cosmologically distant future. For all of Panel 2's parameter range, treating T = T_CMB ≈ 2.7 K is correct.