Math Check

Dataset: BTC-Solved-Unsolved.txt — all 160 Bitcoin Challenge puzzles. Method: steps 1–4 below (gap/ratio profiling, bit-level & masking distribution, predictive model fit, actionable strategy) plus the five advanced non-linear/structural checks in their own section further down (bit-position bias matrix, spectral periodicity, cross-key algebraic relations, nibble/byte entropy, Benford's Law), executed in your browser at load time from btc-puzzles.json, and reproduced independently by math_check.py (pure standard library). No figure on this page is typed in by hand.

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Every solved row is re-derived end-to-end: private key → compressed public key → HASH160 → P2PKH address, and must equal the address published with the puzzle (—). A single mistyped hex digit would break that link, so the statistical work below runs on provably intact keys. Puzzle ranges are [2^(n-1), 2^n − 1] where n = bit length of the key.

all solved keys sorted by puzzle number and private-key value

normalised position inside each range   p = (k − 2^(n−1)) / 2^(n−1)

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The deliverable script performs all four steps from a terminal — no GPU, no network, no dependencies — and prints the honest sub-space for any unsolved target. It prefers the raw BTC-Solved-Unsolved.txt when present and falls back to btc-puzzles.json.

loading script…

what zero-GPU analysis can and cannot do

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Steps 1–4 use linear and geometric summaries; these five checks attack the 83 solved private keys non-linearly and structurally — bit-position bias matrix, spectral / multi-lag periodicity, cross-key algebraic relations, nibble & byte entropy, and Benford's Law. Each check is computed live in your browser from btc-puzzles.json and mirrored line-for-line by advanced() in math_check.py. Thresholds are family-wise α = 0.01 (Bonferroni across scanned candidates); hit counts are judged against their exact null expectation with a Poisson tail. No figure below is precomputed.

check 1 — bit-position bias matrix (positional probability)

relative bit-probability matrix — 64 buckets over r = p/(n−1)

absolute free-bit positions with ≥ 40 keys — static pinning scan

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check 2 — spectral & frequency domain (value series u = k/2^n − 0.5 · concatenated bit stream)

value-series multi-lag autocorrelation — lags 1…60

Fisher periodogram — top 5 of 41 ordinates

bit-stream autocorrelation — max |r| per 128-lag block

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check 3 — cross-key algebraic multipliers & relations (all pairs)

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check 4 — suffix, nibble & byte entropy profiling (LSB-first)

nibble positions j = 0 (least significant) … 33 — keys contributing, entropy, χ²(15)

byte positions b = 0 (least significant) … 31 — zero share and entropy

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check 5 — Benford's law compliance (decimal & hexadecimal)

decimal leading digit vs Benford log10(1 + 1/d)

hex leading digit vs Benford log16(1 + 1/d)

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The checks above summarise keys as numbers; these five examine their bit and digit structure directly — consecutive-key XOR gaps, greatest-common-divisor profiling of the gaps, the full Hamming-weight (popcount) distribution against B(n, ½), base-10 conversion artifacts, and a forbidden-residue matrix across the small primes 3…31. Every figure is computed live in your browser from btc-puzzles.json and mirrored line-for-line by lowlevel() in math_check.py. Thresholds are family-wise α = 0.01 with exact null expectations (Bonferroni across scanned candidates, Poisson tails for hit and empty-class counts). Nothing below is precomputed.

check 6 — bitwise XOR gap analysis (K_i+1 ⊕ K_i)

consecutive XOR gaps — popcount against the exact per-gap expectation

LFSR signature scan — Berlekamp-Massey linear complexity per XOR bit plane

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check 7 — GCD gap profiling (ΔK = K_i+1 − K_i)

divisibility of the gaps by m — observed vs GAPS/m

largest neighbouring-gap gcds

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check 8 — full Hamming weight (popcount) distribution shape

relative popcount r = pop/n in ten bands — observed vs binomial expectation

largest standardised popcounts

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check 9 — decimal (base-10) conversion artifacts

last decimal digit — keys and consecutive gaps

base-10 pattern tests — observed vs null expectation

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check 10 — forbidden residues across the small primes 3 … 31

residue matrix per modulus — χ², empty classes, and the lowest bucket

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The sections above treat the keys one at a time; these five ask whether the solved private keys as a set behave like independent uniform draws. Check 11 measures every pairwise Hamming distance H(K_i, K_j) and tests the whole matrix against resampled datasets. Check 12 runs the NIST SP 800-22 runs test and an LZ78 parse per key and on the aggregated 3850-bit stream, looking for the compressible repeated substrings a passphrase-derived (brainwallet) key would leave behind. Check 13 computes Shannon mutual information between bit positions — the quartile MSB-32 × LSB-32 block and all 1953 position pairs — to expose non-linear dependency. Check 14 pushes each key through the Collatz map and Rule 30 and compares trajectory length, odd-step share, parity runs and 1-density against simulations of the same bit length. Check 15 maps each key into the secp256k1 field as (K³ + 7) mod p and tests quadratic residuosity and the uniformity of the field image.

Every figure is computed live in your browser from btc-puzzles.json and mirrored line-for-line by deep() in math_check.py. Thresholds are family-wise α = 0.01 with Bonferroni across the candidates each test scans, and every Monte-Carlo null uses one deterministic xorshift32 stream with identical seeds on both sides. Where a null is discrete and small-n the null is enumerated exactly (LZ78 below 21 bits) or the inference is carried by an aggregate test rather than a per-key z. Nothing below is precomputed.

check 11 — pairwise binary Hamming distance matrix (all N(N−1)/2 pairs)

most extreme pairs — Hamming distance against the binomial model

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check 12 — NIST SP 800-22 runs & LZ78 Lempel-Ziv complexity

exact per-key LZ78 null — full enumeration of every n-bit key (n ≤ 20)

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check 13 — bit-segment Shannon mutual information (non-linear links)

ten strongest position pairs — bias-corrected MI and G²

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check 14 — Collatz trajectory & Rule-30 state-space dynamics

aggregate dynamics tests — the inference, since per-key Monte-Carlo z's are descriptive

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check 15 — secp256k1 field geometry, (K³ + 7) mod p

field-geometry sub-tests — observed against the length-matched null

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