Type-one primes through 101

$\forall p\le101$, the primes dividing no $a(n)$ are exactly ${3,11,17,23,29,41,47,53,59,83,89,101}$, with the term index $n$ unbounded. Here $a(n)$ and $b(n)$ are the integer coordinates of $\prod_{k=0}^n(1+k\sqrt{-3})$. Preregistration #18634 maps to D5.S3.Arith.Congruence.BalaA363411TypeOne.result (Proved).

This is an explicit OEIS conjecture in the source family documented in bala3634-valuations.md.

The source and literature screen recorded in #18634 found no proof of this exact assertion in the searched scope; completeness is ASSUMED-UNVERIFIED.

BalaA363411TypeOne.nonzero_of_block extends each nonzero residue block to all indices by strong induction. BalaA363411.real_one handles the prime $3$; listed_data and unlisted_data certify the exact finite prime list. The primary dossier gives the four issue-to-result mappings.

An index or prime within the displayed domain violating the assertion would refute it.

The kernel-checked declaration is D5.S3.Arith.Congruence.BalaA363411TypeOne.result. The axiom closure of every public declaration is contained in {propext, Classical.choice, Quot.sound}.

| Item | Status | Evidence | | --- | --- | --- | | $\forall p\le101$, the primes dividing no $a(n)$ are exactly ${3,11,17,23,29,41,47,53,59,83,89,101}$, with the term index $n$ unbounded. | proved | Kernel-checked BalaA363411TypeOne.result. |

What the settlement shows

The conclusion holds throughout its stated domain. The primary dossier gives the shared mechanism and distinguishes the finite prime bound from the unbounded term indices. Broader classifications remain open there.

Literature completeness and priority are ASSUMED-UNVERIFIED. Four-slot escape registration is unfinished in #21978; no declared_validated status is asserted.