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 "scientific secular 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, 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. The "soft" version of this limit considers a computer the mass of the Earth running for billions of years; the "strong" version considers harnessing every atom and cubic meter of space in the observable universe until heat death. By providing answers to problems that exceed even the strong limit, 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 ingenuity or effort could forge.

Personally, I hold no supernatural beliefs, and I am serenely indifferent to those who do. But a good hypothesis must be falsifiable, and intellectual honesty requires stating what evidence would change my mind. A verified divine signature of this kind would do exactly that. I would naturally default to material explanations until they were exhausted, but the exercise demonstrates that materialism is not unfalsifiable dogma; one can articulate the precise circumstances under which it would fall.

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. Twenty thousand digits is not even that much text; it is considerably shorter than the "begots" documenting forty generations of Levantine shepherds. While multiplying two primes to get a 20,000-digit semiprime is a simple operation, reversing the process is effectively intractable. For example, RSA-129 was published in 1977 with a one hundred million dollar prize and remained unfactored for 17 years:
$$114381625757888867669235779976146612010218296721242362562561842935706935245733897830597123563958705058989075147599290026879543541$$
$$= 3490529510847650949147849619903898133417764638493387843990820577$$
$$\times \ 32769132993266709549961988190834461413177642967992942539798288533$$
At only 129 digits, this took a massive distributed computing effort in 1994. A 20,000-digit semiprime is in another universe entirely. 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) $. The known diagonal Ramsey numbers are:
$$R(3,3) = 6, \quad R(4,4) = 18, \quad R(5,5) = 43\text{--}48, \quad R(6,6) = 102\text{--}165, \quad \ldots$$
We cannot even pin down $ R(5, 5) $ to an exact value; it is famously "impossible" to compute because the combinations grow too fast. Providing the exact value for $ R(10, 10) $ along with a specific graph coloring that avoids a clique of size 10 is succinct to state but requires navigating a search space that is thermodynamically inaccessible to our universe. To put this in perspective: the number of ways to two-color the edges of a graph with 1,000 nodes is $ 2^{499,500} $, while the number of atoms in the observable universe is roughly $ 2^{266} $. The search space is mathematically larger than any physical computer could ever process. We verify one graph; the author had to search all of them. Finding a specific "needle" graph in this haystack is a rational proof of super-universal processing power. 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:
$$\pi = \sum_{k=0}^{\infty} \frac{1}{16^k} \left( \frac{4}{8k+1} - \frac{2}{8k+4} - \frac{1}{8k+5} - \frac{1}{8k+6} \right)$$
We cannot currently scan $ \pi $ out to $ 10^{80} $ to find interesting patterns, and 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. If the text successfully predicts a thousand zeros at a specific, distant coordinate, it implies the author did not "scan" for it but rather created the fundamental constants of mathematics itself, or possesses infinite foreknowledge of the structure of irrational numbers. This was, incidentally, a plot point in Carl Sagan's novel Contact, where a message hidden deep in the digits of $ \pi $ serves as a signature from the architects of the universe. 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. This is essentially 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. Humanity has, with heroic effort, managed to prove values only for the smallest machines:
$$BB(1) = 1, \quad BB(2) = 6, \quad BB(3) = 21, \quad BB(4) = 107, \quad BB(5) = 47176870, \quad \ldots$$
Beyond five states, the function explodes beyond comprehension; $ BB(6) $ is known to exceed $ 10 \uparrow\uparrow 15 $, a tower of 15 tens. 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: how many non-overlapping spheres can touch a central sphere of the same size? In three dimensions, the answer is 12, which is easy enough to visualize over a drink. In 8 dimensions, the answer is 240. In 24 dimensions, it is 196,560, thanks to the elegant structure of the Leech Lattice. But beyond 24 dimensions, we lack these lattice optimizations that make the math manageable. In 100 dimensions, the symmetry is so complex that our best supercomputers can only give us a vague range. The number of possible configurations for non-overlapping spheres explodes into a combinatorial nightmare, and determining the exact maximum requires navigating a high-dimensional search space so vast that even an advanced intelligence utilizing quantum annealing would likely settle on a local maximum rather than the true global optimum. Providing the exact integer would reveal a perfect mastery of high-dimensional space, the kind of insight that suggests the author is comfortable navigating 100-dimensional manifolds like they were a simple game of marbles. 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:
$$p_5 = 13, \quad p_{10} = 89, \quad p_{20} = 4423, \quad p_{30} = 132049, \quad p_{40} = 20996011, \quad p_{51} = 82589933, \quad \ldots$$
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) $ (defined as $ \int_{2}^{x} \frac{dt}{\ln t} $, the "expected" number of primes up to $ 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. This number is so deep into the number line that the observable universe is not large enough to write down its digits in standard form; you must use power towers. An entity pointing to a specific exception to a universal rule at such a distance is demonstrating a "God's eye view" of the entire number line simultaneously. 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 $ \pi(x) $ using the Meissel-Lehmer algorithm and confirming that at the provided crossing point:
$$\pi(x) > li(x) \quad \text{for the first time}$$

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:
$$42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3$$
providing a solution for a much more complex equation moves the problem into the realm of Hilbert's Tenth Problem. Because this problem is generally undecidable, there is no "method" to solve it; you cannot write a program to solve all Diophantine equations. Finding a solution to a sufficiently complex one implies the ability to look through the infinite set of integers and "pick out" the needle. A divine solution transcends mere calculation; it is an insight into an undecidable space, suggesting the author is unbound by the Halting Problem. 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 really important issues of existence 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 and infinite (computational) power 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 scientific secular 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.