Dreams as Sheaf Probes: The Transmission Gear Between Cache Identity and Context Identity

Sleep acts as a compiler that integrates moment-to-moment identity updates into a coherent life narrative. During waking, the brain generates local identity fragments (diffs); during sleep, it merges them into a global self-model. Dreams are the thermodynamic byproduct of this process—not meaningful messages but visible residue of compatibility checks performed with relaxed safeguards. Dream content targets the weakest parts of the identity structure, explaining their vividness, lack of epistemic value, and discontinuity. The framework yields testable predictions about dream content bias, post-learning replay, and sleep disorders such as night terrors, insomnia, fatal familial insomnia, and lucid dreaming.

Study at a glance

Abstract

Four published instruments describe identity and consciousness at different temporal resolu- tions: the First-in Cache Identity (FCI) formalizes the self as a rolling buffer with anti-aliasing and monotone fictionalization at the micro-scale (seconds to minutes); the Sheaf-of-Contexts Identity (SCI) formalizes the self as a global section glued from context-local sections with ˇ Cech obstruction at the macro-scale (months to lifetime); the Individuals as Invariants frame- work provides the phenomenal field Φ and the CHIA composite-head architecture; and the Sleep-as-GLUE monograph (Monograph 98) establishes sleep as the canonical GLUE step in the Intercalation Lemma applied to the consciousness mezzanine. What has been missing is the transmission gear: the explicit mechanism that moves data from the FCI cache (RAM) to the SCI sheaf (map). We prove that sleep is that compiler. The wake–sleep cycle implements a discrete deployment schedule in which waking generates FCI local sections (the diffs) and sleep runs the GLUE operation that integrates them into the SCI global section (the merge). Dreams are the thermodynamic exhaust of this compilation—not hallucinations, not messages, but the visible residue of sheaf-compatibility audits conducted at blur β <β⋆ with guards deliberately relaxed. Dream content is Murphy-scheduled: it targets high-Ob/Lc overlaps where the identity sheaf is weakest. Dream phenomenology—vividness without epistemic worth, confidence without fidelity, discontinuity without detection—follows from FCI’s monotone fictionalization theorem operating at extreme parameters during the GLUE phase. We derive falsifiable predictions for dream content bias, vividness–blur coupling, post-learning replay specificity, and pathological modes (night terrors as cohomological write failures, insomnia as residual catastrophe, fatal familial insomnia as mezzanine evaporation, lucid dreaming as partial guard restoration). The synthesis completes the three-scale invariant theory of identity: FCI at the micro-scale, sleep at the meso-scale, SCI at the macro-scale, unified by the Intercalation Lemma and metered by the Murphy horiz