Prediction

The tower predicts primes it has never met. Build a sub-ring from a few primes. Compute its Euler characteristic. The answer names primes the sub-ring has never seen. This is the tower's prime lens at its sharpest: the sieve, run as algebra, seeing ahead of its own range.

Formula: for a sub-ring built from m primes, −χ = N × (m−1 − Σ 1/pi). For two primes: (p−1)(q−1) − 1. For example, {3, 5} → Z/15 gives −χ = 2 × 4 − 1 = 7 — the next prime, which the sub-ring does not contain.

Explorer

Select any two or more of the seven primes (k = 7 demo rung).

Predictions through the tower

Each rung predicts a different number of primes. The count grows as the tower climbs. Two root patterns generate all 2-prime predictions at k ≤ 8: the twin offset {2, p} → p−2, and the 3-root {3, p} → 2p−3. Higher roots enter at k = 9: {5, 7} → 23 is the first (the “other” column).

The factorization rule

New predictions at rung k come from exactly two sources (proved, verified k = 3–24):

(A) Twin term: {2, p_k} → p_k−2. Fires when p_k and p_k−2 are a twin prime pair.

(B) Factorization term: each way to write (p_k + 1) as (p_i−1)(p_j−1) with p_i, p_j smaller tower primes contributes one prediction.

“Rich” rungs: (p_k + 1) highly composite (e.g. p_k = 71, 72 = 2³·3², gains 3 new predictions). “Barren” rungs: (p_k + 1) lacks usable factorizations (e.g. p_k = 29, 30 = 2·3·5, gains 0).

Census to 5×10⁷: 20% of all rungs are rich (599,875 of 3,001,134; every one has p ≡ 3 mod 4), and rich rungs outnumber twin rungs 2.5× with the ratio growing — the factorization term dominates the long-run prediction supply. The primality gate tightens as the tower climbs: the fraction of sub-rings (two or more primes — smaller ones have −χ = −1) whose −χ is prime falls from 82% at k = 4 to 13% at k = 20 and 1.5% at k = 128 — plain 1/ln thinning of ever-bigger integers (−χ is provably odd and coprime to every prime of its own sub-ring, and a prime-density model conditioned on its small divisors tracks the census throughout) — while the absolute count of prime-−χ sub-rings still nearly doubles per rung.

Examples at k = 7

Four 2-prime sub-rings at k = 7 each name a Cunningham-chain prime the sub-ring is missing. Each naming is an algebraic identity, not a numerical coincidence: write the partner and the target as chain-linear expressions in the seed (x for the chain seeded at 2, y for the chain seeded at 3), and the prediction becomes a polynomial whose unique positive root is the seed value.

Every polynomial factors over the chain-uniqueness root (x−2) or (y−3), and the cofactor is itself a chain element. The discriminants are perfect squares of tower values: 25, 49, 25, 36. The fifth k = 7 prediction, {2, 13} → 11, is the twin offset 13−2.

The landscape

How many sub-rings at k = 7 have −χ built entirely from the seven primes?

The naming criterion

One congruence decides every naming (proved, 960/960 cases checked). An absent prime s divides −χ of a sub-ring built from m primes if and only if

1/p₁ + … + 1/p_m ≡ m − 1  (mod s)

where 1/p_i is the inverse of p_i mod s. (Since gcd(N, s) = 1, divide −χ = N(m−1) − Σ N/p_i by N mod s and the criterion falls out.) Two consequences:

2-invisibility (proved). The prime 2 is never named by any sub-ring that doesn't contain it. For odd primes, each 1/p_i = 1 (mod 2), so the sum is m (mod 2) — never m−1.

3-dominance (proved). The prime 3 is named by 51% of 3-prime sub-rings at k = 8, against a 33% baseline — small primes lean toward 2 (mod 3) (the Chebyshev prime-race bias), and the criterion rewards that imbalance. The excess fades as the tower grows (Dirichlet equidistribution).

Naming is governed by prime size alone: across k = 3–50, naming statistics show zero correlation with the tower's dynamical structure (transparency); the naming fraction for an absent prime s converges to 1/s (computed at k = 25).

How far the tower sees

The prediction horizon — how far ahead the tower's sub-rings can name primes — grows super-exponentially (computed, k = 3–12):

k=3:  horizon/p_k = 5.8   k=6:  8,445   k=10:  505M   k=12:  245B

Mid-size sub-rings (~k/2 primes) are the most productive predictors. The full ring's −χ is too large (rarely prime); single primes are too small.

Why predictions miss

The criterion turns “does some sub-ring name s” into a subset-sum problem in Z/s: s is named exactly when some subset of the k inverse values hits its required target. Misses are coverage failures, and they are rare and well understood.

The 41 story. At k = 7, fourteen of the fifteen primes after 17 are predicted — only 41 is missed. The seven inverse values {21, 14, 33, 6, 15, 19, 29} in Z/41 reach only 37 of the 41 residues, and every required target sits in the gap. That is the entire mystery. One more prime breaks the deadlock: at k = 8, −χ({17, 19}) = 16 × 18 − 1 = 287 = 7 × 41.

Misses are accidents, not structure. A random model — P(miss) ≈ (1 − 1/s)^2k−1 — matches observation near-exactly: over the primes up to 10 p_k, the expected miss rate is 22.6% vs 21.9% observed at k = 7, and 9.0% vs 8.8% at k = 8. The misses behave like coupon-collector gaps. (Even the one striking exception proved random: the Cunningham chain {41, 83, 167} all missed at k = 7 — a ~1% event under the model — but across 187 such chain pairs the misses show no chain correlation at all.)

Coverage threshold (pattern, 26 primes tested). Full subset-sum coverage of Z/s arrives at about k ≈ 1.36 log₂ s tower primes — the inverse values behave pseudo-randomly in Z/s: full coverage arrives with ~14× fewer elements than the classical Erdős–Ginzburg–Ziv bound needs to guarantee even a single zero-sum subsequence. Naming needs even less than full coverage: on average a prime is first named at k ≈ 7.2, before coverage completes at k ≈ 9.6.

Prediction density falls smoothly with distance (computed, k ≤ 11):

Density tracks the 2^k growth in sub-ring count. No clustering, no special misses — the field just thins with distance.