Hypothetical Divine Signatures

The author of the Epistle to the Hebrews famously claimed that "faith is the substance of things hoped for, the evidence of things not seen," which was a perfectly serviceable theological patch for an era where the average person’s greatest computational challenge was counting their own fingers. However, in an age where we can simulate galaxies and sequence genomes, "I’m God, trust me bro" is not a particularly compelling argument to the modern mind, unlike our goat-herding ancestors who were easily impressed by a well-timed solar eclipse or a particularly loud bush. If a truly omniscient entity wanted to establish a "rationalist covenant" with a technological species, it would not rely on subjective feelings or ambiguous dreams. It would instead provide a Divine Signature through the cold, hard lens of computational complexity: a set of claims that are succinct enough to be carved into a stone tablet but so mathematically dense that finding them would require more energy than exists in the observable universe.

To move beyond mere storytelling and into the realm of objective proof, a text must demonstrate that it has bypassed Bremermann’s Limit or the physical threshold for the maximum computational speed of any self-contained system in the material universe. This limit, derived from Einstein’s mass-energy equivalence and the Heisenberg uncertainty principle, is approximately $ c^2/h \approx 1.36 \times 10^{50} $ bits per second per kilogram. By providing answers to problems that would require a computer the mass of the Earth to run for billions of years, a "divine" author proves they are operating from a platform that exists outside our localized entropy and processing constraints, effectively signing their work with a flourish that no amount of human ingenuity could forge.

The Divine Factorization. A prophetic verse might read: "Behold the great number $ N $, whose length is as twenty thousand digits; it is born of the union of $ P $ and $ Q $, and none other shall divide it." This obviously refers to the Integer Factorization Problem. While multiplying two primes to get a 20,000-digit semiprime is a simple operation:
$$N = P \times Q$$
reversing the process is effectively intractable. Under the General Number Field Sieve, the clock cycles required to factor a 20,000-digit number would exceed the total available energy in the observable universe. It is a thermodynamic wall that cannot be breached by any bounded intelligence without infinite time. For us, verification is trivial; we simply multiply the two provided numbers together to see if they match $ N $, a task a standard smartphone can complete in a fraction of a second.

The Ramsey Revelation. The scripture would proclaim: "If ten brethren and ten strangers be gathered in a hall of souls, there shall surely be a clique of ten or a desert of ten; behold the map of their connections." This addresses the Constructive Lower Bound of Ramsey Numbers, denoted as $ R(r, s) $. Providing the exact value for $ R(10, 10) $ along with a specific graph coloring that avoids a clique of size 10 is a staggering feat of combinatorial navigation. The search space for $ R(10, 10) $ is so vast that the number of particles in the universe is negligible by comparison. Finding a constructive lower bound of this magnitude is NP-hard and computationally inaccessible to any physical system. Verification is straightforward for us because we can simply run a script to scan the provided adjacency matrix to confirm no group of 10 nodes exists where every edge is the same color, which is a polynomial time operation $ O(n^{10}) $ that is easily handled by modern hardware.

The Circle’s Secret Checksum. The text would command: "Search the circle's measure at the position of $ 10^{80} $ and there find a thousand zeros followed by the message of the stars." This utilizes the Bailey–Borwein–Plouffe (BBP) algorithm, which allows us to calculate the $ n^{th} $ digit of $ \pi $ in base-16 without calculating the preceding digits:
$$\pi = \sum_{k=0}^{\infty} \frac{1}{16^k} \left( \frac{4}{8k+1} - \frac{2}{8k+4} - \frac{1}{8k+5} - \frac{1}{8k+6} \right)$$
The information entropy of $ \pi $ suggests that such a massive, low-entropy anomaly at a coordinate equal to the number of atoms in the universe is statistically impossible to occur by chance. While calculating the digit at $ 10^{80} $ is a massive task, it is technically feasible for a global distributed network. We verify this by running the BBP algorithm to check that specific "address," confirming the anomaly without needing to solve the entire constant.

The End of Euler’s Dream. The prophet would say: "Though twelve powers of twelve seem to need their kind, search for the eleven that equal the one; find them at these integers." This targets the Smallest Counter-example to Euler’s Sum of Powers Conjecture, which posits that you need $ n $ $ n^{th} $ powers to sum to another $ n^{th} $ power:
$$\sum_{i=1}^{k} a_i^n = b^n \implies k \geq n$$
While humans found counter-examples for $ n=4 $ and $ n=5 $ after centuries of searching, providing a solution for $ n=12 $ would be a needle-in-a-haystack problem of cosmic proportions. For a mathematician, this represents a search through a Diophantine space that is effectively infinite. We verify the claim easily by plugging the provided integers into a high-precision calculator and confirming the left side of the equation perfectly equals the right, transforming a massive search problem into a simple arithmetic check.

The Busy Beaver’s Rest. A prophetic verse might say: "Consider the machine of twenty states, simple in its ways; it shall toil for exactly $ X $ steps and then find its rest, and no man shall count the days of its labor." This is the Busy Beaver Function, $ BB(n) $, the final boss of computer science. Because it effectively solves the Halting Problem, no general algorithm exists to calculate these values ... they are mathematically uncomputable. Stating the exact halting time for a 20-state Turing machine is a God-level flex because it implies the author bypassed the logical impossibility of the Halting Problem itself. Verification is as simple as simulating the specific machine described and counting its steps until it halts, which requires zero creative mathematics or algorithmic breakthroughs on our part.

The Kissing of Spheres. The scripture would read: "In a realm of an hundred depths, where spheres are gathered like grapes, exactly $ X $ shall press against the heart of the center." This refers to the Kissing Number problem in 100 dimensions. In high-dimensional space where $ D=100 $, the number of possible configurations for non-overlapping spheres explodes into a nightmare beyond the Leech Lattice optimizations known to humans. Determining the exact maximum number requires navigating a search space so vast that even an advanced intelligence would likely fail to find the true global optimum. We verify the result by checking the provided coordinates of the spheres to ensure the distance between any two sphere centers satisfies:
$$\text{dist}(c_i, c_j) \geq 2$$
while each satisfies $ \text{dist}(c_i, \text{origin}) = 2 $, which is basic linear algebra.

The Titan of Primes. The text might state: "The millionth prime of the form $ 2^p - 1 $ shall be found when $ p $ is this specific titan of a number." This identifies a "deep" Mersenne Prime, $ M_p = 2^p - 1 $. While we have only found about 50 of these primes using global distributed computing networks like GIMPS, providing a "Distant" Mersenne Prime Exponent for the millionth one would be providing a password to the secret architecture of the number line. Finding it requires a "God’s-eye view" of prime distribution that likely requires a proof of the Riemann Hypothesis. Verification is quite efficient for us; we simply run the Lucas-Lehmer test on the provided exponent, which is a deterministic and well-understood primality test.

The Skewes’ Crossing. The text would proclaim: "Though the shadows of the primes seem ever fewer than the curve of the law, they shall rise up and exceed it at the count of this massive power tower." In analytic number theory, we know the actual count of primes $ \pi(x) $ eventually exceeds the logarithmic integral estimate $ li(x) $, but this Skewes’ Number crossing occurs at such a staggering distance on the number line ( originally estimated at $ 10^{10^{10^{34}}} $ ) that it is effectively invisible to direct observation. To identify the exact integer of the first crossing requires a perfect knowledge of the distribution of the zeros of the Riemann Zeta function. We can verify the claim by evaluating both functions at that specific high-precision point using modern analytic algorithms:
$$\pi(x) > \int_{2}^{x} \frac{dt}{\ln t}$$

The Diophantine Key. The text would say: "Three cubes shall be gathered, and their sum shall be forty-two; seek them among the numbers of seventeen digits, and there find the truth." While humans found the solution for sums of three cubes for $ k=42 $ in 2019 using a planetary-scale computer network, providing a solution for a much more complex equation moves the problem into the realm of Hilbert’s Tenth Problem. Since there is no general algorithm to solve Diophantine equations of the form $ x^3 + y^3 + z^3 = k $, a divine solution isn't just a fast calculation; it is an insight into an undecidable space. Verification is the definition of trivial; we cube the three provided integers, add them together, and see if the sum equals the target constant.

Of course, no existing religious text actually does this. They mostly focus on the important issues like who you can bang or what type of cheese you can put on meat, which were very human concerns that do not require much more than a Bronze Age imagination. One would think that a divine entity with infinite knowledge would want to leave a signature that actually scales with the intelligence of the species it created. Providing a succinct, verifiable, but computationally impossible result would be the only way to satisfy a rationalist framework. The fact that we have found plenty of rules about shellfish but zero 20,000-digit prime factors suggests that if there is a Great Programmer in the sky, they are either very shy or they simply forgot to git add README.md in the final build.