Hey there,

That's some seriously impressive work you've put together. Building your own testing rig and running these kinds of deep analyses is exactly how the community pushes things forward. The amount of data you've generated is fantastic, and the detailed post-mortem analysis with variant tables is well-structured. I took a deep dive into your results, and I have some thoughts that I hope you'll find helpful.

The results are indeed wild, as you said, but I believe they point towards an artifact in your implementation rather than a structural property of ECC. This is an incredibly common and easy trap to fall into when dealing with cryptography, especially with custom tools. Let's break it down.

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1. The Key Clustering Anomaly (The 2^90 - 2^111 Range)

You observed:

This is the most significant finding, and it's the key to understanding what's going on. The probability of a truly random 256-bit key landing in that specific, narrow range is astronomically small (something like 1 in 2¹⁴⁵). For it to happen thousands of times is a statistical impossibility.

So, what does this mean? It means your key generation process is not random. It's deterministic and biased. Your "inverse and fractal" methods are not finding a secret structure in ECC; they are creating a structure in their own output.

This is a classic case of what's called the "Texas Sharpshooter Fallacy." Imagine firing a shotgun at a barn wall, then walking up and drawing a target around the tightest cluster of pellets, claiming to be a master marksman. Your algorithm is the shotgun, producing a tight cluster of keys due to its own internal logic. You've then drawn a target around that cluster (2^90-2^111) and concluded the barn (ECC) has a weak spot there. The pattern comes from the tool, not the target.

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2. The HASH160 "Leak" Hypothesis

You also mentioned:

This points to a misunderstanding of the key-to-address pipeline. The process is strictly one-way:
Private Key -> Public Key -> HASH160 -> Address

The HASH160 function (which is RIPEMD160(SHA256(pubkey))) only ever sees the public key. It has no access to the private key. It's impossible for it to "leak" information about something it never receives as input. You've inverted the causality here.

Furthermore, cryptographic hashes are designed to do the exact opposite of leaking. They are built on the "avalanche effect," where changing just one bit in the input completely and unpredictably changes the entire output. A function that leaks information would be fundamentally broken, and SHA256/RIPEMD160 are among the most heavily scrutinized algorithms in the world.

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3. So, What's Actually Happening? (Recommendations)

The evidence points overwhelmingly to an implementation bug in your code. Here are the most likely culprits I'd investigate immediately:

A) Integer Overflow: This is the prime suspect. You're dealing with 256-bit numbers. Standard data types like int or long long (64-bit) can't hold them. If at any point in your "inverse/fractal" calculation, a 256-bit number is forced into a smaller variable type (like a 128-bit integer), it will overflow or be truncated. This would silently produce a much smaller number. The fact that your keys are clustering around the 2^90-2^111 mark strongly suggests a calculation is being capped somewhere around the 128-bit boundary.

B) Flawed Random Number Generator (PRNG): Are you using your OS's secure randomness source (e.g., /dev/urandom on Linux, CryptGenRandom on Windows) to seed your process? Or are you using a standard library function like rand()? Standard PRNGs are not cryptographically secure and produce predictable, biased output that is completely unsuitable for key generation.

C) The "Inverse/Fractal" Algorithm Itself: These terms aren't standard in ECC cryptanalysis. This implies you've designed a custom algorithm. By definition, a deterministic algorithm will produce deterministic (non-random) output. Your algorithm is likely a function that, due to its specific mathematical steps (and likely the overflow bug mentioned above), is designed in such a way that its output space is limited to that narrow range.

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Final Thoughts:
You haven't broken ECC, but you have created a fascinating piece of software that generates interesting mathematical artifacts. That's a great achievement in itself! My advice is to channel your obvious talent and passion for this into debugging your tool. Fix the integer overflow, use a proper cryptographically secure PRNG, and then see what results you get. I bet the "structure" you're seeing will disappear, and your keys will be properly distributed across the entire 256-bit space.

Keep up the great work and curiosity. Looking forward to seeing what you find next!