module module moderate

`IndisputableMonolith.CrossDomain.AttentionSpace`

The CrossDomain.AttentionSpace module shows that the complexity ceiling gap45 leaves exactly 5 overflow slots. Researchers working on attention networks within Recognition Science cite it when bounding cross-domain complexity ceilings. The module organizes its content as definitions for AttentionNetwork and TickPhase together with lemmas on counts and surjectivity.

claimThe module defines the attention space whose main objects are the network $AttentionNetwork$, phase $TickPhase$, state $AttentionState$, and the ceiling relation $gap_{45}$ that admits exactly five overflow slots.

background

The module sits inside the CrossDomain domain and imports only Mathlib. It introduces AttentionNetwork, TickPhase, AttentionState along with the associated count functions networkCount, tickCount, attentionStateCount. Related lemmas include tick_eq_twoPowD, gap45, overflow_eq_D, attention_fits_under_gap, attention_plus_overflow_eq_gap and network_surj. The local setting is the cross-domain portion of the Recognition framework whose forcing chain supplies the eight-tick octave and three spatial dimensions.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the attention-space layer that feeds higher-level cross-domain results in the Recognition Science framework. It directly supports the complexity-ceiling analysis tied to the eight-tick octave (T7) and D = 3 (T8) steps of the unified forcing chain.

scope and limits

Does not treat attention outside the cross-domain setting.
Does not compute explicit numerical values for the phi-ladder.
Does not address cases with more than three spatial dimensions.
Does not supply simulation code or empirical checks.

declarations in this module (15)

inductive AttentionNetwork L22
inductive TickPhase L26
theorem networkCount L30
theorem tickCount L31
theorem tick_eq_twoPowD L32
abbrev AttentionState L34
theorem attentionStateCount L36
def gap45 L40
theorem overflow_eq_D L41
theorem attention_fits_under_gap L44
theorem attention_plus_overflow_eq_gap L47
theorem network_surj L51
theorem tick_surj L55
structure AttentionSpaceCert L59
def attentionSpaceCert L67