Exact survivor density and hazard for the Type A/B sieve

Theorem

Let

Status: theorem note

Date: 2026-08-14

Claim boundary: this note applies the prime number theorem for arithmetic progressions, the divergence of reciprocal primes in the class 3 mod 4, and the Type A/B congruence characterization. It proves a density-one statement for Type A/B coverage of primes. It does not prove pointwise coverage of every prime and therefore does not solve the Erdős-Straus conjecture. No literature-priority claim is made without a separate prior-art review.

1. Finite-depth survivor states

Let

m_k=4k-1, \qquad T_k=\{-e,-4e\pmod{m_k}:e\mid k\}.

For an odd prime p, let

C_{AB}(p)=\min\{k\ge1:p\bmod m_k\in T_k\},

with C_AB(p)=infinity if no such layer exists.

For a fixed depth K, define

M_K=\operatorname{lcm}(m_1,m_2,\ldots,m_K).

Define the reduced survivor set

S_K= \left\{ a\in(\mathbb Z/M_K\mathbb Z)^\times: a\bmod m_j\notin T_j\text{ for every }1\le j\le K \right\}.

Except for the finitely many primes dividing M_K, a prime survives the first K Type A/B layers if and only if its residue modulo M_K belongs to S_K.

2. Exact finite-depth prime density theorem

By the prime number theorem for arithmetic progressions, each reduced residue class modulo fixed M_K contains asymptotically the same proportion of primes.

Therefore the relative density among primes of surviving through layer K exists and is exactly

\boxed{ \delta_K=\frac{|S_K|}{\varphi(M_K)}. }

Likewise, define the exact-depth residue set

E_k= \left\{ a\in(\mathbb Z/M_k\mathbb Z)^\times: a\bmod m_k\in T_k, \quad a\bmod m_j\notin T_j\text{ for }j<k \right\}.

Then the relative prime density of exact minimal depth k is

\boxed{ \mu_k=\frac{|E_k|}{\varphi(M_k)}. }

Since exact depth k is precisely survival through k-1 followed by failure at k,

\boxed{ \mu_k=\delta_{k-1}-\delta_k. }

Thus the minimal-depth spectrum is literally the support of the density drop:

\boxed{ k\text{ is infinitely prime-realizable}\iff\mu_k>0.}

The implication from a nonempty reduced residue class to infinitely many prime realizations is Dirichlet's theorem.

3. The exact survival-history hazard

Whenever delta_(k-1)>0, define

\boxed{ h_k=\frac{\mu_k}{\delta_{k-1}} =1-\frac{\delta_k}{\delta_{k-1}}.}

This is the exact asymptotic conditional probability that a prime hits Type A/B layer k, conditioned on having survived every earlier layer.

It is not a heuristic independence model. All arithmetic dependence, all direct shadows, and all joint shadow closure are already encoded in the finite residue set S_(k-1).

This is the correct arithmetic replacement for the earlier raw and prime-conditioned empirical hazard models.

A structural gap is exactly a zero-hazard layer:

\boxed{h_k=0\iff\mu_k=0.}

4. Exact hazard on the prime-modulus backbone

Suppose

q=m_k=4k-1

is prime.

Because q is larger than every earlier modulus m_j, it divides none of them. Hence

\gcd(q,M_{k-1})=1

and

M_k=qM_{k-1}.

Every survivor residue modulo M_(k-1) therefore has exactly q-1 reduced CRT extensions modulo M_k, one for each nonzero residue modulo q.

Exactly |T_k| of those extensions hit the target layer. Consequently

|S_k|=|S_{k-1}|\bigl(q-1-|T_k|\bigr)

and

\boxed{ h_k=\frac{|T_k|}{q-1}=\frac{|T_k|}{4k-2}.}

Using the exact trap-cardinality identity

|T_k| = 2\tau(k)-1-\mathbf 1_{4\mid k}\tau(k/4),

we obtain the closed formula

\boxed{ h_k = \frac{2\tau(k)-1-\mathbf 1_{4\mid k}\tau(k/4)}{4k-2} \qquad(4k-1\text{ prime}).}

Thus every prime-modulus backbone layer is not merely realizable. Its conditional asymptotic hazard is known exactly and is independent of the entire preceding shadow history.

5. Density-one Type A/B coverage of primes

For every backbone depth with q=4k-1>7 prime, k>=3 and

|T_k|\ge3.

Non-backbone layers can only remove additional survivors. Therefore

\delta_K \le \prod_{\substack{q\le4K-1\\q\equiv3\pmod4\\q>7}} \left(1-\frac{3}{q-1}\right).

The reciprocal sum over primes q congruent 3 mod 4 diverges. Hence this product tends to zero and

\boxed{\lim_{K\to\infty}\delta_K=0.}

Let

\mathcal E=\{p\text{ prime}:C_{AB}(p)=\infty\}.

For every fixed K, the set E is contained in the primes surviving through K, whose relative density is delta_K. Therefore

\overline d_{\mathbb P}(\mathcal E)\le\delta_K

for every K. Sending K to infinity gives

\boxed{ overline d_{\mathbb P}(\mathcal E)=0. }

Equivalently:

\boxed{\text{A relative density-one set of primes has a Type A or Type B solution.}}

This is a pointwise weaker statement than López's conjecture that every prime has Type A or B, but it is unconditional.

6. A quantitative decay in depth

The Mertens theorem for primes in arithmetic progressions gives

\sum_{\substack{q\le x\\q\equiv3\pmod4}}\frac1q = \frac12\log\log x+O(1).

Using log(1-u)<=-u, the prime-modulus backbone alone yields

\log\delta_K \le -3 \sum_{\substack{q\le4K-1\\q\equiv3\pmod4\\q>7}} \frac1{q-1} = -\frac32\log\log K+O(1).

Hence

\boxed{ \delta_K\ll(\log K)^{-3/2}. }

This is a conservative bound because it uses only three trap residues at each prime-modulus layer and ignores every composite-modulus layer as well as the extra divisor structure in T_k.

7. Hard-class version

The same argument remains valid after conditioning on any fixed reduced residue class modulo a modulus coprime to the backbone prime q.

In particular, for the six Mordell hard classes modulo 840, every backbone prime q>7 is coprime to 840. Therefore the exact backbone hazard formula survives unchanged inside each hard class.

Consequently the set of hard-class primes not covered by Type A/B also has relative density zero within the hard-prime population.

This dovetails with the hard-class depth spectrum and arrival-function framework: structural gaps are zero-density drops, realizable depths are positive drops, and the prime-modulus backbone supplies infinitely many explicit positive drops.

8. Why this changes the research picture

The Type A/B system can now be viewed as an exact arithmetic survival process:

1=\delta_0\ge\delta_1\ge\delta_2\ge\cdots\to0,

with exact mass function

\mu_k=\delta_{k-1}-\delta_k

and exact conditional hazard

h_k=\mu_k/\delta_{k-1}.

The shadow graph describes dependencies between layers. The shadow closure describes finite covering of survivor states. The depth spectrum is the support of mu_k. The arrival function records the first prime occupying each positive-mass depth. The prime-modulus backbone gives an infinite set of layers where the hazard is closed-form and independent of the past.

This unifies what previously looked like separate empirical phenomena.

9. The next theorem targets

The strongest immediate targets are:

compute or bound h_k at composite moduli using the shadow-closure quotient rather than raw residue density;
determine whether direct-shadow completeness is true, which would characterize every zero-hazard layer by a one-edge shadow obstruction;
exploit the exact backbone recurrence together with average divisor information for k=(q+1)/4 to improve the conservative (log K)^(-3/2) survivor bound;
study whether the exact hazard sequence has a tractable Euler-product or multiplicative approximation after quotienting shadow dependencies;
relate the depth-tail delta_K to the arrival function lambda(k) and the observed record frontier.

The remaining universal question is still pointwise: density zero does not rule out an infinite sparse exceptional set, and proving that the exceptional set is empty remains equivalent to establishing Type A/B coverage for every prime.