ND0001

Beta(b,c,i,x)

Exact hygienic Gödel-beta extraction; a signature-identical alias of checked BetaAt.

Conservative notation; not a theorem, primitive, or axiom.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Hygienic expanded first-order definition

((exists ff_h_defined_beta_at. ff_h_defined_beta_at + S (x) = S ((S (i)) * c)) /\ exists ff_q_defined_beta_at. b = ff_q_defined_beta_at * S ((S (i)) * c) + (x))

The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.

Direct definition dependencies

none — first-order arithmetic only

Definitions depending on this notation

CRTPrefixLCM(b,c,l,M)CRTPrefixSolution(r,s,b,c,l,x)CRTPairwiseCoprimePrefix(b,c,l)CRTPositiveModuliPrefix(b,c,l)

Checked theorems using this definition

CR0003 · crt_positive_moduli_prefix_last_nonzeroCR0007 · crt_prefix_solution_lastCR0008 · crt_prefix_solution_successor_introCR0009 · crt_pairwise_coprime_prefix_lastCR000A · crt_positive_moduli_prefix_product_nonzeroCR000B · crt_prefix_product_common_multipleCR000C · crt_pairwise_coprime_prefix_product_is_lcmCR000F · crt_prefix_lcm_successor_introCR0012 · crt_pairwise_coprime_prefix_product_coprime_lastCR0014 · crt_prefix_solutions_pointwise_congruentCR0015 · crt_prefix_solution_transport_common_multiple

Separate complete second-wave branches: Full G011 proof · Alpha v27.