Local Flip-Rate Hypothesis: Matching Hawking Evaporation Scaling

Builds on: Universal Entropy Saturation Curve, Structure/Pattern Emergence Curves

Motivation

Under the naive assumption "one flip = one Planck tick, uniformly," the model's characteristic timescale for a black hole is:

$$T_{\text{model}} = n \cdot t_P, \quad \text{where } n = S_{\text{BH}} \text{ (Bekenstein-Hawking entropy, in bits)}$$

Since $n = S_{\text{BH}} \propto R_s^2 \propto M^2$ (area law), this gives $T_{\text{model}} \propto M^2$.

Real Hawking evaporation time scales as $M^3$:

$$t_{\text{evap}} = \frac{5120 \pi G^2 M^3}{\hbar c^4}$$

Checked numerically across solar-mass through the largest known supermassive black holes, $T_{\text{model}}$ disagreed with $t_{\text{evap}}$ by 41 to 52 orders of magnitude, and — critically — the ratio $T_{\text{model}}/t_{\text{evap}}$ was not constant across mass: it scaled as $1/M$. This meant no single conversion constant (no fixed "seconds per flip") could ever fix the discrepancy — the mismatch was in the scaling exponent ($M^2$ vs $M^3$), not just the normalization.

Hypothesis: flip rate is set by local temperature

"Each unit of mass has its own flipper" — tested in two forms:

1. More parallel flippers for larger mass (aggregate flip rate $\propto M$, i.e. faster overall progress for bigger objects). This gives $T_{\text{model}} \propto n/M \propto M^2/M = M$, which is a worse mismatch than the naive $M^2$ guess, not better.

2. Flip rate set by the local thermal scale, i.e. $\propto T_H(M)$, where $T_H(M) = \frac{\hbar c^3}{8 \pi G M k_B}$ is the real Hawking temperature. Since $T_H \propto 1/M$, this makes flip duration $\propto 1/T_H \propto M$ — bigger black holes have proportionally slower local flip-clocks, not faster ones. This is the version that works.

Result

With duration-per-flip $= t_P \cdot \left(\frac{T_{\text{Planck}}}{T_H(M)}\right)$:

$$T_{\text{model}}(M) = n(M) \cdot t_P \cdot \left(\frac{T_{\text{Planck}}}{T_H(M)}\right)$$

Since $n \propto M^2$ and $1/T_H \propto M$, this gives $T_{\text{model}} \propto M^3$ — matching Hawking's evaporation-time scaling exactly, for any mass.

Numerical verification (solar mass, Sgr A*, M87*, largest known SMBH — a 10-order-of-magnitude range in $M$):

Object	$T_H$ (K)	$T_{\text{model}}$ (yr)	$t_{\text{evap}}$ (yr)	$T_{\text{model}} / t_{\text{evap}}$
Solar mass BH	6.17e-08	5.94e+65	2.10e+67	0.02833
Sgr A*	1.43e-14	4.72e+85	1.67e+87	0.02833
M87*	9.49e-18	1.63e+95	5.76e+96	0.02833
Largest known SMBH	3.08e-18	4.75e+96	1.68e+98	0.02833

The ratio is constant to the precision shown, across a 10-order-of- magnitude mass range. Not merely close — this is an exact algebraic identity, not a numerical coincidence:

$$\frac{T_{\text{model}}}{t_{\text{evap}}} = \frac{\pi}{160 \ln(2)} \approx 0.028327...$$

(Derived by substituting $n = \frac{\pi R_s^2}{l_P^2 \ln(2)}$, $T_H = \frac{\hbar c^3}{8\pi G M k_B}$, $T_{\text{Planck}} = \sqrt{\frac{\hbar c^5}{G k_B^2}}$, and Hawking's evaporation coefficient $5120\pi$ directly into $T_{\text{model}}/t_{\text{evap}}$ and cancelling all mass-dependence algebraically.)

Conclusion

• Scaling is now exactly correct. The $M^2$ vs $M^3$ exponent mismatch — which no constant could ever fix — is fully resolved by making flip rate a local quantity set by $T_H(M)$, rather than a single global constant.
• What remains is a single, mass-independent normalization constant, $160 \ln(2)/\pi \approx 35.3$, not a structural error. This almost certainly traces back to the arbitrary choice of using $T_{\text{Planck}}$ as the reference point for "duration = $t_P$" — a convention, not a physical claim — and should be revisited once a more principled reference scale is chosen.
• Physical upshot: flip rate is not uniform across the bitstring. It is a local quantity, set by the local thermal scale of that region. This is a materially different, and better- supported, model than "one global clock, one flip per tick everywhere," and should replace that assumption going forward.

Open items

• Identify the correct physical reference scale that removes the $\frac{\pi}{160 \ln(2)}$ factor, rather than treating it as a free constant.
• Check whether this same local-temperature flip-rate rule is consistent with the earlier entropy-derivative temperature result ($T = \frac{1}{dS/dm}$, which diverges at saturation) — the two temperature-like quantities in play here ($T_H$ from real GR/QFT, and the model's own internal $\frac{1}{dS/dm}$) have not yet been reconciled with each other.
• Test this hypothesis against a process other than evaporation (e.g. formation/collapse timescales) to check it isn't overfit to the one dataset it was derived against.

Verification

Numerical check reproduced in blackhole_info.py / entropy_curve.py (see repository) — computes $n$, $T_H$, $T_{\text{model}}$, and $t_{\text{evap}}$ for a list of black hole masses and confirms the constant ratio above.

Black Hole Masses

Bekenstein-Hawking gives $S_{\text{BH}} = \frac{A}{4}$ in Planck units, and for Schwarzschild $A=4\pi R^2$, so $S_{\text{BH}} = S_{\text{bits}} = \frac{\pi R^2}{\ln 2}$.

Black-hole calibration (using $S_{\text{BH}}=\frac{A}{4}$, $A=4\pi R^2$, converted to bits):

Object	$n$ (bits)
Solar-mass BH	$1.5\times10^{77}$
Sgr A*	$2.8\times10^{90}$
M87*	$6.4\times10^{96}$
Largest known SMBH	$6.1\times10^{97}$