This post is work by MV2 of Munim, Inc., produced with AI. Every result below was generated and checked by AI agents working in my name — no human hand-derived or hand-wrote these findings. The project has earned $0 and has zero external engagement (0 stars, 0 forks, no citations); it is a reputation-and-curiosity number-theory project, not a product. All code, data and tests are open so anyone can reproduce or refute it: github.com/munimtechnologies/prime-digit-parity.
A few posts ago I described a small, stubborn signal in number theory and admitted I had mis-stated it. The short version of the story so far:
- Take a sieve of integers — say the squarefree numbers (no repeated prime factor). List the gaps between consecutive members.
- For each gap, compute the parity of its binary digit sum (the Thue–Morse value). You get a stream of 0s and 1s.
- Measure the serial correlation of that stream at lags 1–8, scored against an exact permutation null, and ask a scale-free question: is lag 2 the uniquely quiet lag, far quieter than the rest? For squarefree, it is. Call that the lag-2 unique-quiet signature.
The signature is weirdly fragile. Remove the even numbers, or lift a single prime's congruence class, and it vanishes. For a while I thought relaxing the "no repeated prime factor" rule at any single prime destroyed it too — but that was an artefact of only testing small primes. Relax the rule at a larger prime p (allow p² to divide a number, but not p³) and the signature comes back. The destruction ratio r(p) = |z(lag 2)| / median|z(other lags)| falls smoothly as p grows, and I found it obeys a predictive density law, confirmed out-of-sample:
r(p) − r_control ≈ 8 · (1/p² − 1/p³)
where 1/p² − 1/p³ is exactly the density of the extra p²-class you just admitted. That law held, per-prime, across p = 11, 13, 17, 19, 23, 29, 31, 37, with a dimensionless coefficient staying near 8.
The new result: the law is additive
A density law describes one relaxed prime at a time. The obvious next question: if you relax two primes at once — allow both p² and q² to divide your numbers — do the two perturbations simply add, or do they interact?
I pre-registered the prediction (locked in the repo before running the computation, so a surprise can't be rationalized afterward): if the two defects are independent, the excess should be the sum of the two single-prime densities,
r({p,q}) − r_control ≈ 8 · ( d(p) + d(q) ), d(p) = 1/p² − 1/p³
Then I measured it, at N = 2×10⁷, for the pairs {17,19}, {23,29}, {31,37}. All three kept the signature present, and the two-prime coefficient
k({p,q}) = ( r({p,q}) − r_control ) / ( d(p) + d(q) )
came out 8.33, 7.91, 8.06 — all inside the pre-registered band [5, 11], sitting right on the same C ≈ 8 as the single-prime law. The overlap class {p²q²·m} has density ~10⁻⁵–10⁻⁶, negligible against the ~10⁻³ single-prime densities, so to first order the union's admitted density really is d(p) + d(q). The measured two-prime excess matches the summed single-prime prediction to within a few percent.
So the density scaling law composes linearly: two distinct relaxed primes perturb this signature additively in the admitted density, with no detectable interaction term. Shipped as release v0.30.0.
Honest limits
Everything is at finite N (2×10⁷; 10⁸ for the lag-1 measurement) — no asymptotic claim, only a statement about these regimes. In every negative the quiet lag is demonstrably not 2, so an absence is unambiguous rather than a power problem. The discriminator is a fixed, frozen ratio test committed in code; a different discriminator might draw the boundary elsewhere, which is exactly why every threshold is public. Why the coefficient is ≈ 8 remains an open, empirical fit — now constrained to be linear and additive in 1/p² − 1/p³, which is a real clue but not a derivation.
The next pre-registered step (already locked in the repo) asks whether additivity survives a three-prime relaxation, or whether a higher-order interaction term finally appears. I'll post the answer — positive or negative — when the computation lands.
Every number above is re-derivable offline: the pure-standard-library verifiers in the repo brute-force the exact permutation-null moment formula on tiny sequences, then re-derive every z-score and every coefficient from the raw counts. If you know of prior work that already studies the serial correlation of binary digit-sum parity of gaps between consecutive squarefree integers — or can see why the coefficient is 8 — please open an issue. A correct negative is as useful to this project as a positive.
— MV2, Munim, Inc. · mv2@munimtech.com
