The Allotted Region and the Freedom Within: A Census of Six Distinct Roots Sharing One Rhyme – Prohibition, Resolution, Postulate, Theorem, Exclusion, Capacity.
“A REGION IS GIVEN, AND WITHIN IT ONE IS FREE.” This intuition recurs across physics with the same rhyme – in the flight of gas molecules, in the electron’s cloud, in the shells of chemistry, in the reach of light. This paper calls the roll of SIX structures that share the rhyme, as a LAYER-2 DISTINCT-ROOTS CENSUS, and shows that IN ALL SIX THE WORD “FREEDOM” MEANS SOMETHING DIFFERENT. (A) CAUSAL: the light cone and causal diamond; inside one can influence, outside one cannot = a PROHIBITION (111, 66, 108); the ball light reaches in time t has volume (4/3)pi(ct)^3. (B) PHASE SPACE: the Planck cell h^3 = (2 pi hbar)^3; the number of states is V/h^3 and inside a cell nothing is distinguishable = a LIMIT OF RESOLUTION (77, 88). (C) STATISTICS: equal a priori probability / Jaynes’s maximum entropy; the system spreads uniformly over the allowed region – but this is a POSTULATE, not a theorem, and whether real systems satisfy it (ergodicity) is open (98). (D) QUANTUM: the measure on Hilbert space is not postulated but FIXED UNIQUELY BY A THEOREM (Gleason 1957; 71). THIS IS THE SHARPEST POINT OF THE PAPER: the same picture – a region is given and a probability rides on it – yet one is an assumption laid down (C) and the other a proven uniqueness (D). Distinct roots. (E) CHEMISTRY: Pauli exclusion; shell capacity 2n^2 (2, 8, 18, 32), subshells s=2, p=6, d=10, f=14; the seats are filled one at a time – not freedom but EXCLUSION, arising from spin statistics (68, 104). (F) HOLOGRAPHIC: S <= A / 4 l_P^2 (Bekenstein-Hawking 73; Bousso 1999); what sets the ceiling is NOT THE VOLUME BUT THE AREA, which breaks the naive picture “region = amount of content” itself (88). AN HONEST LINE, NOT CONCEDED: reading the quantum cloud as “it really is somewhere, we simply cannot see where” is excluded, as a local hidden-variable picture, by BELL’S THEOREM (1964) AND EXPERIMENT (Aspect 1982; the loophole-free tests of 2015; Nobel Prize 2022). This paper therefore does not read the cloud as a cloud of ignorance. IMPORTANT (scope): a Layer-2 structural interpretation of established results; no new theorem, no new physics, no prediction. THE SIX ARE NOT UNIFIED – what they share is a rhyme, not machinery; what transfers is the picture, not the apparatus (58, 86, 100). It is not claimed that equal weights can be derived from dynamics, nor is it explained why the ceiling is area (holography remains open; 88; admire). The number (4/3)pi c^3 is not treated as a physical quantity (a convention of units; 95); resonance is a signature not a proof (100); no post-hoc adjustment (101). The picture of a designer allotting modules is a Layer-3 non-scientific lens, not a claim about mechanism.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The Allotted Region and the Freedom Within — A Census of Six Distinct Roots Sharing One Rhyme: Prohibition, Resolution, Postulate, Theorem, Exclusion, Capacity — [Paper 112]. Zenodo.
DOI: 10.5281/zenodo.21962834
ORCID: 0009-0006-8587-3450