The Type A/B minimal-depth spectrum

Status: active research note

Date: 2026-08-14

Claim boundary: the theorems below are elementary consequences of the Type A/B congruence system, CRT, covering congruences, and Dirichlet's theorem. The terminology and organization are part of the CENTL/FCF research program; no literature-priority claim is made here without a separate prior-art review.

1. Minimal Type A/B depth

For a prime p, define

with C_AB(p)=infinity when no Type A/B witness exists.

Write

Then C_AB(p)=k exactly when

and

The Mordell hard residue classes used here are

2. Exact candidate progression

Fix a layer k, hard class h in H, and a unit trap residue t in T_k compatible with h modulo gcd(840,m_k).

CRT gives a progression

that simultaneously enforces

Write every integer in this progression as

For each earlier layer j<k, put

The earlier hit condition induces a finite forbidden set

such that

Thus the entire minimal-depth problem for this candidate becomes a finite congruence-avoidance problem in the single parameter s.

3. Shadow closure is a covering-system problem

For a fixed (k,h,t) candidate, the family

is a finite covering system on the parameter s.

• If the union covers every integer s, the candidate is jointly shadowed by the earlier Type A/B layers.
• If some s escapes the union, the candidate survives the entire earlier shadow closure.
• Direct shadowing is the special case where one earlier layer alone covers every possible s.

This is stronger than the original pairwise shadow graph. The graph records one-layer implications; the covering system records their complete union.

Define the shadow closure of the earlier layers on this candidate by

Then

if and only if the candidate is impossible as a minimal-depth class.

4. Exact depth-realization theorem

Let

Suppose there exists s0 such that

and

Then every integer in the progression

avoids every earlier Type A/B layer and hits layer k.

Because the residue is coprime to the modulus, Dirichlet's theorem gives infinitely many primes in this progression. Consequently,

Converse

If infinitely many primes in the fixed (k,h,t) candidate have minimal depth k, then at least one of their parameter residues modulo Q is an avoiding residue coprime to LQ. Therefore the criterion above is also necessary for infinite prime realization of that fixed candidate.

This converts a question about an infinite prime population into a finite modular decision problem.

5. The hard-class depth spectrum

Define

A layer can fail to enter this spectrum for at least two immediately detectable reasons:

1. it has no prime-compatible hard-class trap candidate;
2. every admissible candidate is already forced into an earlier Type A/B layer.

The full obstruction is joint shadow closure. A layer lies in D_H exactly when at least one compatible candidate has an uncovered parameter class that is reduced modulo its total period.

The problem therefore has two dual sides:

• obstruction: earlier congruence classes cover the entire parameter line;
• realization: an uncovered reduced residue class survives, and Dirichlet populates it with infinitely many primes.

6. Structural gaps, latency gaps, and the arrival function

Finite prime sweeps can show a zero at layer k for two fundamentally different reasons.

Structural gap

A layer is a structural gap when

No sufficiently large search will ever produce a hard-class prime with exact depth k.

Latency gap

At a finite cutoff X, a layer is a latency gap when

but its first realization lies beyond X.

Define the arrival function

for k in D_H, and set lambda_H(k)=infinity for structural gaps.

This separates an arithmetic impossibility from a finite-range absence. It also gives a new inverse view of the record frontier: instead of asking how deep C_AB has become by prime size X, ask when each admissible depth first enters the prime population.

7. The k=104 anomaly resolves

The finite hard-prime sweep through 10^7 had a striking pattern. Among the zero-first-hit layers through k=109, every layer except k=104 was already explained by either absence of admissible hard-prime candidates or complete direct shadowing.

Layer 104 was the lone apparent hole.

For k=104,

Take

Since

this is a Type A trap with

CRT gives

The parameter

avoids every Type A/B layer j<104, and produces

Direct deterministic checking gives

It is in the hard class

and

The full parameter-period certificate uses

and

Moreover,

Therefore Dirichlet gives the stronger statement

The absence of depth 104 in the earlier 10^7 data was therefore a finite-range effect, not structural impossibility. The first hard-class occurrence in an exhaustive sieve through 11035249 is exactly 11035249, so

Thus the unique unexplained zero in the earlier range was a latency gap sitting just beyond the cutoff.

8. Exact Erdős-Straus witness for p=11035249

For the Type A congruence

with d=4, n=26, write

For p=11035249,

Set

The Type A solution (du,dv,duv) is

Hence

CENTL separately certifies this exact rational identity in depth-spectrum-contracts.centl.

9. Computational spectrum result through k=300

The accompanying depth_spectrum_probe.py performs exact modular checks for the hard classes. In the current experiment through k=300:

• 66 layers have no admissible hard-prime candidate;
• 39 layers have admissible candidates but every one is completely directly shadowed by an earlier layer;
• the remaining 195 layers all receive an explicit Dirichlet realization certificate with an avoiding parameter s <= 5000.

So, through k=300, every layer not already killed by the two simplest structural obstructions is explicitly certified as infinitely prime-realizable in the hard classes.

Independent high-range exploratory computation has continued this same dichotomy much farther, but the repository's reproducible certificate floor remains k<=300 until the higher-range run is packaged and independently checked.

This is a finite computational theorem-certificate result, not a proof that the same dichotomy holds for every k.

10. The theorem target hiding behind the data

The most important conjectural strengthening suggested by the calculations is:

Direct-shadow completeness conjecture. Once prime compatibility is imposed, every admissible Type A/B candidate that is not completely directly shadowed by a single earlier layer possesses an uncovered reduced parameter class and is therefore infinitely prime-realizable.

Equivalently, in this system there may be no genuinely new obstruction created only by a union of several partial shadows.

If true, this would collapse the full covering-system obstruction problem to the much simpler direct-shadow graph.

That would give a structural characterization of the hard-class minimal-depth spectrum:

This is not proved. It is now a concrete theorem target generated by the exact data.

11. Research target

The emerging object is no longer merely a list of difficult primes. It is the minimal Type A/B depth spectrum, its arrival function, and its obstruction theory.

The immediate questions are:

• Is direct-shadow completeness true for all k?
• Can every non-shadowed candidate be shown to contain a reduced avoiding class without search?
• Is there a structural characterization of the complement of D_H?
• Can the shadow graph be quotiented to an irreducible congruence sieve whose vertices are exactly the prime-realizable depths?
• What governs the growth of lambda_H(k) on the realizable spectrum?
• Are record values of C_AB(p) controlled by unusually delayed arrivals, unusually sparse residue support, or both?
• What asymptotic information about C_AB(p) follows from the density and geometry of these realizable depth classes?

A proof of direct-shadow completeness, or a counterexample exhibiting a genuinely joint-only shadow obstruction, would be a major structural step for this program.