module module high

`IndisputableMonolith.Mathematics.BirchTateStructure`

This module defines a simplified algebraic structure for K₂ of the ring of integers to support Birch-Tate relations inside the Recognition Science mathematics layer. Number theorists working on RS derivations of zeta invariants or BSD components would cite it when linking K-theory to the phi-ladder. The module is purely definitional, importing the RS time quantum and the M-005 BSD scaffold to establish the required objects without internal proofs.

claimThe module introduces the structure $K_2(R)$ for the ring of integers $R = O_K$ of a number field $K$, together with auxiliary maps such as the w2-invariant and connections to zeta values at negative integers.

background

Recognition Science places this module in the mathematics domain after the fundamental time quantum τ₀ = 1 tick (from Constants) and the M-005 Birch-Swinnerton-Dyer scaffold. The module supplies the K-theoretic ingredient needed to relate the order of K₂(O_K) to special values of the Dedekind zeta function, consistent with the phi-ladder and eight-tick octave already fixed in the upstream chain. Sibling declarations inside the same file (K2RingOfIntegers, w2Invariant, ZetaValue) are the concrete objects that realize this link.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the algebraic K-theory foundation that feeds the BirchTateConjecture, birch_tate_for_Q, and zeta_phi_orbits declarations. It completes the number-theoretic half of the M-005 BSD derivation path, allowing later steps to connect K₂ orders to the RS-native constants and the phi-fixed-point structure.

scope and limits

Does not prove the Birch-Tate conjecture itself.
Does not compute explicit orders of K₂ groups.
Does not derive physical predictions or mass formulas.
Limits attention to simplified structures, omitting full Quillen K-theory machinery.

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Mathematics.BirchSwinnertonDyerStructure module_import

declarations in this module (21)

structure K2RingOfIntegers L45
structure ZetaValue L50
def BirchTateConjecture L55
def IsTotallyReal L60
def w2Invariant L64
theorem birch_tate_for_Q L70
theorem birch_tate_quadratic L77
theorem lictenbaum_connection L83
theorem k2_phi_paths L94
theorem zeta_phi_orbits L103
theorem birch_tate_path_orbit_duality L110
theorem abelian_phi_product L117
theorem w2_phi_euler_characteristic L124
theorem has_bsd_structure L131
def birch_tate_from_ledger L133
theorem birch_tate_structure_chain L135
theorem birch_tate_implies_bsd L137
structure Resolution L143
theorem birch_tate_abelian_proven L152
theorem birch_tate_rs_prediction L159
theorem birch_tate_summary L167