DEF0204canonicalv1E₁ as Gluing Principle
Self-enrichment (E₁) is precisely the statement that local Hartogs bulk projections glue globally. Morphisms between local patches carry the same split-complex bipolar structure as the patches themselves. Physics IS E₁ — the phenomenon of local spatial structures gluing into a globally coherent space.
Payload
E₁ as Gluing Principle
Self-enrichment (E₁) is precisely the statement that local Hartogs bulk projections glue globally. Morphisms between local patches carry the same split-complex bipolar structure as the patches themselves. Physics IS E₁ — the phenomenon of local spatial structures gluing into a globally coherent space.
E₁ as Gluing Principle
Summary
Self-enrichment (E₁) is precisely the statement that local Hartogs bulk projections glue globally. Morphisms between local patches carry the same split-complex bipolar structure as the patches themselves. Physics IS E₁ — the phenomenon of local spatial structures gluing into a globally coherent space.
Statement
%
\label{def:e1-gluing}
At layer~$E_0$ (Books~I--II),
$\tau$ possesses objects with internal split-complex structure
--- the algebraic lemniscate, the bipolar spectral decomposition,
the full holomorphic apparatus ---
but the morphisms between objects carry only ordinary
($\mathrm{Set}$-enriched) structure.
Local tori exist. Local Hartogs extensions exist.
But there is no guarantee that morphisms
between different local bulks respect the bipolar structure of the objects.
The enrichment functor $\mathcal{F}_{E_1}$
promotes $\mathrm{Hom}$ spaces
from sets to \emph{split-complex modules}:
at $E_1$, the morphisms between local bulks
carry the same $\chi_+/\chi_-$ bipolar decomposition as the objects.
This enrichment forces local bulk projections to be globally compatible ---
the transition maps $\phi_{xy}$
are morphisms in a split-complex-enriched category,
and the enrichment constraints leave no room for the transition maps
to break the spectral, regularity, or discreteness guarantees.
Proof / Justification
This item is definitional. No manuscript proof is required.
Source Context
- Registry source:
book-03.jsonlline 4 - Manuscript source:
2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part00/ch02-the-eight-guarantees.texlines 258-281
Lean / Formalization Notes
- Formalization:
formalized - Module:
TauLib.BookIII.Prologue.HartogsBulk - Name:
e1_gluing_check
Dependencies
- Canonical: III.D01, III.D02
Related Results
Generated by later projection phases.
Related Publications
Generated by later projection phases.
Revision Notes
- 2026-04-24: Initial pilot migration.
Identifiers
Aliases & legacy IDs
III.D03e-as-gluing-principledef:e1-gluingRelease lines
corpus_v3_workingcorpus_v2Relations
Formalized by (2)
Appears in (1)
Downstream uses (computed) (4)
Items in the corpus that reference this one via load-bearing relations. Computed from the full corpus-v3 graph at build time.
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