THM0003canonicalv1Iterator Ladder Saturation
The iterator ladder has exactly 4 levels (0-3); higher hyper-operations lose canonical injectivity because K6 prevents a 5th orbit channel.
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Iterator Ladder Saturation
The iterator ladder has exactly 4 levels (0-3); higher hyper-operations lose canonical injectivity because K6 prevents a 5th orbit channel.
Iterator Ladder Saturation
Summary
The iterator ladder has exactly 4 levels (0-3); higher hyper-operations lose canonical injectivity because K6 prevents a 5th orbit channel.
Statement
%
\label{thm:ladder-saturation}
The iterator ladder $\mathcal{L} = (L_0, L_1, L_2, L_3)$
has exactly four levels.
Higher hyper-operations (pentation, hexation, \ldots)
are definable but lose canonical injectivity.
The ladder saturates at level~3 (tetration).
Proof / Justification
Levels $L_0$--$L_3$ each consume one orbit channel
(Observation~\ref{obs:channel-consumption}).
After $L_3$, all four channels are consumed.
By Lemma~\ref{lem:pentation-non-inj},
no level-4 operation can achieve canonical injectivity.
The same argument blocks all higher levels.
Source Context
- Registry source:
book-01.jsonlline 18 - Manuscript source:
2nd-edition/book-i-categorical-foundations/02_mainmatter/part02/ch08-iterator-ladder.texlines 261-269
Lean / Formalization Notes
- Formalization:
formalized - Module:
TauLib.BookI.Orbit.Ladder - Name:
Tau.Orbit.ladder_saturation
Dependencies
- Canonical: I.D06, I.D03, I.K6
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
I.T02iterator-ladder-saturationthm:ladder-saturationRelease lines
corpus_v3_workingcorpus_v2Relations
Formalized by (1)
Appears in (1)
Downstream uses (computed) (3)
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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