Time Is Space

Complexity theory carves the world into time and space. Here I prove the distinction is an artefact of description, not of computation. I define a formalism in which representations are physically embodied and finite. Time is sequentially realised distinction. Space is simultaneously realised distinction. Both count one underlying resource I call tone. Space is a chord of simultaneous tones. Time is an arpeggio of sequential tones. I prove two embodied recoding results. If a present abstraction layer keeps every visible tick of a finite timeline fragment separately available, its layer space is at least the elapsed layer time. If it keeps every active program at every tick separately available, its layer space is at least the cumulative layer space. Both bounds become equalities when those embodied distinctions are exactly the present encoding. A bounded Turing-machine reconstruction places the machine inside the environment and recovers classical machine time and used-space as instances of the same resource. An exact block-product theorem further shows that program-count simplicity and weakness, meaning the count of compatible completions, coincide only when representational blocks are uniform. Physical machines rarely satisfy this condition. Standard complexity classes are not affected. Time and space are not carved into computation. You carved them in when you chose a vocabulary.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-07-30
DOI
https://doi.org/10.5281/zenodo.21710852
Primary Topic
Computability, Logic, AI Algorithms
Type
preprint
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preprint

Time Is Space

Michael Timothy Bennett
Zenodo (CERN European Organization for Nuclear Research)
Computability, Logic, AI Algorithms
preprint

Time Is Space

Michael Timothy Bennett
preprint en

Abstract

Complexity theory carves the world into time and space. Here I prove the distinction is an artefact of description, not of computation. I define a formalism in which representations are physically embodied and finite. Time is sequentially realised distinction. Space is simultaneously realised distinction. Both count one underlying resource I call tone. Space is a chord of simultaneous tones. Time is an arpeggio of sequential tones. I prove two embodied recoding results. If a present abstraction layer keeps every visible tick of a finite timeline fragment separately available, its layer space is at least the elapsed layer time. If it keeps every active program at every tick separately available, its layer space is at least the cumulative layer space. Both bounds become equalities when those embodied distinctions are exactly the present encoding. A bounded Turing-machine reconstruction places the machine inside the environment and recovers classical machine time and used-space as instances of the same resource. An exact block-product theorem further shows that program-count simplicity and weakness, meaning the count of compatible completions, coincide only when representational blocks are uniform. Physical machines rarely satisfy this condition. Standard complexity classes are not affected. Time and space are not carved into computation. You carved them in when you chose a vocabulary.

Zenodo (CERN European Organization for Nuclear Research)
Australian National University (AU)
Computability, Logic, AI Algorithms
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