The blind spot

How coarse the probe is, measured rather than argued. The fraction of states below a cap that share their headroom with another state falls at every cap tried — 39.2% below 1500, 26.8% below 192000 — while the probe's resolution — distinct readings as a fraction of states — improves only from 67.2% to 75.3%. The shape is close to c/log x and not quite it: the fraction times ln(cap) creeps 2.87 → 3.26 rather than holding, so the decay runs a shade slower than 1/log. That form vanishes, but a slow approach to a small positive limit fits the same points, and nothing here decides between them. Either way the collisions are no small-number artifact: more than a quarter of the states below 192000 still share a reading with another.

The largest blind class — the states sharing one headroom value — below 6000 is V = 24, holding 178 states, 101 of them prime, and its prime part is characterized exactly. A prime q > 5 lies in it iff no prime outside {2, 3, q} has its predecessor dividing q−1 and 3 does not divide q−1 — equivalently W(q−1) = 24q. The safe primes from r = 5 up, q = 2r+1 with r prime, are a proper sub-family of that and not the whole of it — the two smallest safe primes, q = 5 and 7, fall outside the class: 81 of the class's 101 primes below 6000 are safe and 20 are not.

Scope. The census is measured over states below the stated caps, six in all; the characterization of the prime part is a rule, proved, and the class's 77 composite members lie outside it.

verifiers: explore_headroom.py, explore_blind_spot_infinite.py

The blind class V = 24, the largest below 6000, is infinite, and no conjecture enters: the primes q with V(q) = 24 number ≫ x/log²x below x, and the non-safe members — 239, 443, 647 leading, (q−1)/2 composite — are ≫ x/log²x by themselves (theorem). The route is the characterization rewritten in divisor terms: with q−1 = 2m, the primes q > 5 with m odd and prime to 3 are the eligible primes, and the class is those of them with 2e+1 composite for every divisor e of m with 1 < e < m — a divisor with 2e+1 prime being a bad divisor — an odd cofactor with a finite list of compositeness demands on it, which is what a sieve counts. The linear sieve's lower bound, positive once the level of distribution exceeds the square of the sifting range, at the level the Bombieri–Vinogradov theorem supplies — primes in arithmetic progressions equidistributed on average to modulus x^1/2 less logarithms — gives ≫ x/log²x primes q ≤ x with (q−1)/2 free of prime factors below x^1/10; the ones among them that some prime 2e+1 cuts out number O(x log log x/log³x), by an upper-bound sieve on the three numbers e, 2e+1 and q at once; and prescribing m a prime factor in [x^1/10, x^1/4] leaves m composite but for O(x^1/4) of them, at the same order. The safe primes were one route to the class and never the class itself: the Sophie Germain question — whether infinitely many primes r have 2r+1 prime — is untouched, the theorem's constants separating nothing between the safe and the non-safe parts. Below 10⁷ the class holds 71,419 primes, 30,655 safe and 40,764 not, the non-safe share rising 0.29, 0.41, 0.51, 0.57 across the decades from 10⁴; and the fraction the proof sends to zero — (q−1)/2 free of prime factors below the sifting range, yet some 2e+1 prime — reads 0.57 at range 5 and 0.19 at 47 there, so that limit is asymptotic and not a small-scale effect.

The bound is a floor and not the order (theorem): at λ = 2 and at every even λ alike the class numbers ≫ x/(log x·log log x) below x, the constant depending on λ, so N·log²x/x grows without bound and no conjecture enters — the same sieve with its sifting range lowered from x^1/10 to z = (log x)^C, which leaves ≫ x/(log x·log z) cofactors free of prime factors below z, the members some prime 2e+1 then cuts out bounded by a union over the divisor e, and that union is where the one factor log log x is paid: it sums the mean number of bad divisors over such a cofactor, which grows like log log x/log z where the typical number is bounded, so z must be a power of log x for the mean to stay small and those cofactors then number x/(log x·log log x). The census reads the count N of class primes below x as N·log²x/x 1.35, 1.50, 1.67, 1.85 across those decades and N/(x/(log x·log log x)) flat — 0.325, 0.319, 0.318, 0.320 — so the proved bound fits the counts as an order, and the order is nonetheless x/log x (theorem): the class is a fixed share of the primes, N ~ δ·π(x) with δ > 0, π(x) counting the primes below x, at λ = 2 and at every even λ. The route runs through the Bernoulli numbers: the denominator of B_2k is the product of the primes p with (p−1) | 2k (von Staudt–Clausen), so for q ≥ 11 the class is exactly the primes with denom(B_q−1) = 6q, q = 5 and 7 alone having the denominator without the headroom — 4 | q−1 lifting the wall's 2-part to 16 at q = 5 and 3 | q−1 its 3-part to 9 at q = 7, prime powers a denominator made of primes never sees. Erdős and Wagstaff proved the integers m with denom(B_2m) = 6 to have positive density, the integers with a divisor p−1 > T being a share at most ε(T) → 0 of all; the transfer to the primes q = 2m+1 runs their argument with every sieve step supplied by Pollack's Shiu bound over a sifted set, so the primes q with (p−1) | (q−1) for some prime p > T, p ≠ q, are in the limit a share at most ε(T) of the primes; and positivity is Harris's inequality — the exponent of each prime in q−1 is independent across primes by Dirichlet, avoiding a divisor p−1 is a decreasing event in those exponents, so the finitely many conditions below T keep a positive density and the tail costs at most the factor 1 − ε(T). The theorem withholds δ: every truncation at a finite T is a ceiling on it, and no census estimates it. The bad-divisor heuristic written as a model — the bad-divisor events taken independent across the divisors of m, each priced at the sieve's chance that 2e+1 is prime, of order 1/log e — predicts the census within 0.3% at every decade (observation), and read beyond the census on random integers m drawn with their factorizations and weighted and sieved to the residue law of a shifted prime, an instrument good to 3%, it keeps the model's value of N/(x/(log x·log log x)) between 0.29 and 0.37 through 10⁵⁰ while the class's share of the eligible primes falls 0.42 → 0.31: the flat census is the model's own prediction. The model reads the approach to δ and not δ; the slow decay like (log x)^−0.16 its sample could not separate from a positive proportion is excluded by the theorem.

And the same sieve runs at every even λ, V = 24 being the case λ = 2 (theorem): for every even λ the primes q ≡ 1 (mod λ) with V(q) = W(λ) — the blind class of λ's own wall — number ≫ x/log²x below x, the constant depending on λ, and those with (q−1)/λ composite are ≫ x/log²x by themselves, the cofactor m = (q−1)/λ now asked to be odd, greater than 1, prime to every prime p with p−1 dividing λ, and to have e·f+1 composite for every even divisor e of λ and every divisor f > 1 of m, the pair (e, f) = (λ, m) excepted — an odd e costs nothing, e·f+1 being even. Each odd such p is a prime the cofactor must avoid — a factor 1 − 1/p on the constant when p divides λ, 1 − 1/(p−1) otherwise — and each even divisor of λ one more demand on a composite cofactor: at 10⁷, N·φ(λ)·log²x/x — φ(λ) counting the residues prime to λ — reads 1.855 at λ = 2 and 0.423 at λ = 60, which has eight such p, seven of them odd, and the composite-cofactor share 0.571 at λ = 2 against 0.000 at λ = 48 and 60.

Scope. The class's infinitude, the order bound, the order x/log x and the lift to every even λ are theorems, resting on the census's proved characterization of the prime part; the counts below 10⁷ check proved statements rather than supporting them; the model's match to the census is an observation and its reading past 10⁷ is a sampled model's and no measurement; the density δ is proved positive and not estimated.

verifiers: explore_blind_spot_infinite.py, explore_silent_set.py, explore_blind_share.py, explore_blind_model.py, explore_blind_bernoulli.py