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Type A/B shadow ancestry: formal working lemmas

Synthesis · hosted from the CENTL repository

Research library · Synthesis

Synthesis

This note separates proved elementary structure from computational conjecture targets for WS-CAND-003.

Source in the repository

This note separates proved elementary structure from computational conjecture targets for WS-CAND-003.

Definitions

For k >= 1, write

m_k=4k-1,

and

T_k=\{-d,-4d\pmod{m_k}:d\mid k\}.

Let

H=\{1,121,169,289,361,529\}\pmod{840}.

A hard-class-compatible candidate at layer k is a pair (h,t) with h in H, t in T_k, gcd(t,m_k)=1, and

t\equiv h\pmod{\gcd(840,m_k)}.

It defines the CRT progression

x\equiv h\pmod{840},\qquad x\equiv t\pmod{m_k}.

A candidate is directly shadowed by layer j<k when every integer in that CRT progression lies in T_j modulo m_j.

Lemma 1: modulus ancestry

For integers 1 <= j < k,

m_j\mid m_k

if and only if there exists an integer s >= 1 such that

\boxed{m_k=(4s+1)m_j}

and equivalently

\boxed{k=(4s+1)j-s.}

Proof

If m_j | m_k, let q=m_k/m_j. Since both moduli are congruent to 3 mod 4,

3q\equiv3\pmod4,

so q == 1 mod 4. Write q=4s+1. Then

4k-1=(4s+1)(4j-1)

and expansion gives

k=(4s+1)j-s.

The converse follows by substituting that expression for k. QED.

For fixed s, the identity

4((4s+1)j-s)-1=(4s+1)(4j-1)

is a univariate polynomial identity in j. The automated harness asks CENTL to certify the fixed-quotient identity for every ancestry quotient actually observed in the finite shadow graph.

The first nontrivial ancestry family is s=1, hence

\boxed{k=5j-1,\qquad m_k=5m_j.}

Lemma 2: direct shadow collapses to residue reduction along ancestry

Assume j<k and m_j | m_k. Let (h,t) be an admissible candidate at layer k. Then (h,t) is directly shadowed by j if and only if

\boxed{t\bmod m_j\in T_j.}

Proof

Every integer in the current candidate progression satisfies

x\equiv t\pmod{m_k}.

Since m_j | m_k, this forces

x\equiv t\pmod{m_j}

for every member of the progression. Therefore every member is captured by layer j exactly when that one forced residue belongs to T_j. QED.

This removes the need for a general CRT fibre computation on modulus-ancestry edges. The general fibre criterion is still needed for shadow edges where m_j does not divide m_k.

Corollary: exact full-shadow test on an ancestry edge

If m_j | m_k, then layer k is completely directly shadowed by j on the hard-prime population exactly when every admissible candidate (h,t) at layer k satisfies

t\bmod m_j\in T_j.

This is a finite divisor-generated residue condition. It is the natural starting point for classifying infinite shadow families.

Lemma 3: first-hit primes certify global non-union-shadowing

Suppose a prime p has

C_{AB}(p)=k.

Set

h=p\bmod840,\qquad t=p\bmod m_k.

Then the current candidate class (h,t) is not covered by the union of all earlier Type A/B trap layers.

Proof

The prime p itself belongs to the current CRT class. By the definition of C_AB(p)=k, it lies in no trap T_j for j<k. Hence the union of earlier layers cannot cover the entire current class. QED.

This is stronger than direct_novel: the latter excludes only one-layer containment, whereas a first-hit prime excludes collective coverage by all previous layers at once.

For the current finite record,

C_{AB}(9658489)=2622,

so its (169,10449) candidate class at modulus 10487 has an explicit globally non-union-shadowed witness.

Theorem target A: classify k=5j-1

The automation observes all three possibilities along the first ancestry family:

  • complete direct shadow;
  • partial direct shadow;
  • no direct shadow from the immediate ancestor.

The research target is a necessary-and-sufficient condition on j for

T_{5j-1}^{\mathrm{admissible}}\bmod(4j-1)\subseteq T_j.

A finite list of observed examples is not an infinite-family theorem. The automated ancestry-candidate-families.json exists to discover candidate conditions that can then be proved separately.

Theorem target B: irredundant core

For candidates without an explicit first-hit prime, determine whether several earlier layers can jointly cover the current CRT progression even though no one earlier layer covers it. The exact target is collective union shadowing.

The checked-in automation currently makes only two rigorous union-level statements:

  1. a completely directly shadowed class is certainly union-shadowed;
  2. a class containing a verified first-hit prime is certainly not union-shadowed.

All classes between those two categories remain open to stronger exact covering analysis.

Theorem target C: growth and distribution of C_AB

The existing unboundedness theorem rules out any universal constant ceiling. Useful next questions are therefore growth, record frequency, hard-class distribution, and whether the irredundant ancestry structure yields nontrivial upper bounds for large families of primes.