Event horizon inspired hash assessment¶
5 October 2026. Michael proposed using the inability to escape an event horizon as the basis for a one-way hash, initially considering a black-hole/white-hole pair. The bounded research goal was to assess that connection, specify a testable mapping, and record the result. This assessment does not implement a black hole, white hole, or secure new hash.
Goal and security question¶
Desired future construction: a deterministic mapping from arbitrary message bytes to one fixed-size digest, with efficient computation and verification, and computationally difficult preimage, second-preimage, and collision searches. The current 64-bit output is an experimental size, not a SHA-256-equivalent collision-security target.
The event-horizon analogy suggests one-way behavior, but an obstruction to future-directed physical signals is different from the complexity of solving an equation. A person evaluating the equation on a computer does not need to physically move a signal outward through a horizon.
Formula from the E0 notes¶
The QuantumChat handoff uses Schwarzschild spacetime in ingoing Painleve–Gullstrand coordinates. For dimensionless radius R=r/r_s and time U=cT/r_s, the locally outgoing radial null branch is:
v = dR/dU = 1 - 1/sqrt(R)
At R<1, v<0. At R=1, v=0. At R>1, v>0. This is a coordinate velocity; it is not the locally measured speed of light. The source notes and Boonserm, Ngampitipan and Visser distinguish regular horizon-penetrating coordinates from misleading coordinate singularities.
As a scalar mapping from R to v, it has the explicit inverse:
R = 1 / (1-v)^2, for v<1
An exact rational check used square radii:
R |
v |
Recovered R |
|---|---|---|
1/4 |
-1 |
1/4 |
4/9 |
-1/2 |
4/9 |
1 |
0 |
1 |
4 |
1/2 |
4 |
This inverse recovers an input parameter. It does not describe escaping a black hole or reverse a physical signal’s future direction. The scalar equation therefore supplies no hard inversion problem. This finding is limited to the mapping tested, not all possible constructions inspired by gravity.
A concrete scalar hash candidate and its failure¶
To test whether truncating that mapping helps, this investigation introduced the following deliberately simple candidate. It is an assistant-proposed experiment, not a formula from Michael’s papers.
For a nonnegative integer input n, choose an interior radius approaching the horizon:
R(n) = (n+1)/(n+2)
-v(R(n)) = sqrt(1+1/(n+1)) - 1
h_b(n) = floor(2^b * (sqrt(1+1/(n+1)) - 1))
The result fits within b bits because the parenthesized quantity is between zero and sqrt(2)-1. This only defines the integer-input core; an arbitrary-message implementation would additionally require injective message-to-integer encoding. The core already fails.
For x>0, sqrt(1+x)-1 = x/(sqrt(1+x)+1) < x/2. Consequently:
n >= 2^(b-1) implies h_b(n) = 0
This is an analytical collision family. It also makes the all-zero proof-of-work target trivial for those inputs. Rounding discards the very small differences near the horizon.
A Decimal calculation with 100 decimal digits confirmed:
Output width |
Input n |
Output |
|---|---|---|
8 |
128 |
0 |
8 |
129 |
0 |
64 |
9223372036854775808 |
0 |
64 |
9223372036854775809 |
0 |
Reproduction:
from decimal import Decimal, localcontext
with localcontext() as ctx:
ctx.prec = 100
for bits in (8, 64):
for n in (2**(bits-1), 2**(bits-1)+1):
value = int(Decimal(2)**bits *
((1 + Decimal(1)/Decimal(n+1)).sqrt() - 1))
assert value == 0
print(bits, n, value)
The proof, rather than numerical precision, establishes the collision family. A production reference implementation would need a fully specified exact or correctly rounded arithmetic rule.
What hashing the formula would mean¶
One may feed a serialized formula, simulation state, or trajectory into an established hash. That produces a fingerprint whose security depends on the hash, not on the physical inability to escape. One may also design a new mixer inspired by the dynamics, but its arithmetic and attacks must be investigated independently.
A white-hole-themed reverse routine that efficiently recovers the original message from the public digest would defeat the hash’s intended preimage resistance. If reversal requires a secret key, the proposal instead needs an encryption specification and an adequate ciphertext space. Reversing a full retained state also differs from reversing a compressed digest.
Falsifiable next experiment¶
Any further candidate should specify message encoding, state width, exact update rule, number of rounds, boundary handling, and digest extraction. Hold a known hash as the control and use the same input corpus and target difficulty. First seek algebraic inverses, monotonic-search shortcuts, saturation regions, and explicit collisions. Only candidates that survive those checks merit broader avalanche and performance measurements. Increasing digest width alone will not fix the demonstrated scalar collapse.
Outcome: the event-horizon analogy supplies a useful question, but the direct scalar map is easily inverted and the specified finite-output projection has trivial collisions. The assessment goal is complete; developing and validating a different custom hash remains a separate research task.