module module moderate

`IndisputableMonolith.Mathematics.BirchTateStructure`

The BirchTateStructure module supplies simplified structural definitions for K₂ of the ring of integers inside the Recognition Science mathematics layer. It rests on the time quantum from Constants and the BSD scaffold from the imported module. Number theorists examining Birch-Tate links within RS would reference these objects. The module contains only definitions and no proofs.

claimDefines a simplified structure for the group $K_2(\mathcal{O}_K)$ where $\mathcal{O}_K$ denotes the ring of integers of a number field $K$.

background

Recognition Science imports the fundamental time quantum $\tau_0 = 1$ tick from the Constants module. The upstream BirchSwinnertonDyerStructure module formalizes a structural RS scaffold for BSD derivation components under label M-005. The present module introduces K₂ of the ring of integers as a simplified structure to support further number-theoretic constructions such as zeta values and the Birch-Tate conjecture.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

Supplies the K₂ ring-of-integers structures that feed the sibling declarations BirchTateConjecture, ZetaValue, and w2Invariant. It completes the number-theoretic portion of the M-005 BSD scaffold.

scope and limits

Does not prove any value of K₂ or the Birch-Tate conjecture.
Does not connect to the J-function, phi-ladder, or forcing chain T0-T8.
Does not compute K₂ for concrete fields or link to physical constants.
Does not address full K-theory or Lichtenbaum conjectures.

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