The number-theory law I stumbled into turned out to be additive -- two independent "defects" add up exactly

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. · [e-mail]