Closure of the correspondence between arity level, recursion/paradox count, philosophical name, and full logical unfolding.
Consolidated table
| Tf | Arity | Contradiction in n steps | Paradox | Recursion | Philosophical name (EN) | ||
|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | — | First ‘cursion’ | Entia | ||
| 2 | 2 | 2 | 1 | Recursion 1 | Parity | ||
| 3 | 3 | 3 | 2 | Recursion 2 | Appearance, Perience | ||
| 4 | 4 | 4 | 3 | Recursion 3 | Ex-perience | ||
| 5 | 5 | 5 | 4 | Recursion 4 | Persistence |
Notes on this table:
- Paradox and Recursion are kept as separate counts deliberately — they are not collapsed into a single quantity, pending a case where they might diverge.
- Phases (individual unit) and Phase space (set of possible configurations) are kept as distinct concepts. The “Phase space” column is explicitly left undefined until its combinatorics are established — it is not a blank left by omission, it is an acknowledged open task.
- The philosophical names (Entia, Parity, Appearance/Perience, Ex-perience, Persistence) identify relational aspects, not objects or observable phenomena — none of them names Time, Space, Mass, or any other physical phenomenon directly. This is consistent with the logical unfolding below: at no level does any variable other than
Sappear. - On the English equivalents: “Entia” is a direct loanword already current in Anglophone philosophy (as in “entia non sunt multiplicanda…”). “Parity” is not a literal translation of “Parencia” — the Spanish wordplay (par-ente) does not survive translation — but it is the sibling term already present in the source material for the same level, and it has an exact English cognate. “Appearance” is an exact cognate of “Aparencia”. “Perience” (from clipping “ex-perience”) is a rare but attested neologism in English philosophical and speculative writing, paralleling the Spanish clipping of “ex-periencia”. “Ex-perience” and “Persistence” are exact cognates, sharing the same Latin roots as their Spanish counterparts.
Full logical unfolding, by level
Each Entₙ is shown fully expanded down to S/¬S — with no unopened subexpression left. The number of literals doubles exactly at each level (2ⁿ), which has been verified computationally, not merely derived algebraically.
n=1 (Encia / Entia) — 2 literals
(S ∧ ¬S)n=2 (Parencia / Parity) — 4 literals
(S ∧ ¬S ∧ (S ∨ ¬S))n=3 (Aparencia, Periencia / Appearance, Perience) — 8 literals
(S ∧ ¬S ∧ (S ∨ ¬S) ∧ ¬(S ∧ ¬S ∧ (S ∨ ¬S)))n=4 (Ex-Periencia / Ex-perience) — 16 literals
(S ∧ ¬S ∧ (S ∨ ¬S) ∧ ¬(S ∧ ¬S ∧ (S ∨ ¬S)) ∧ ¬(S ∧ ¬S ∧ (S ∨ ¬S) ∧ ¬(S ∧ ¬S ∧ (S ∨ ¬S))))n=5 (Persistencia / Persistence) — 32 literals
(S ∧ ¬S ∧ (S ∨ ¬S) ∧ ¬(S ∧ ¬S ∧ (S ∨ ¬S)) ∧ ¬(S ∧ ¬S ∧ (S ∨ ¬S) ∧ ¬(S ∧ ¬S ∧ (S ∨ ¬S))) ∧ ¬(S ∧ ¬S ∧ (S ∨ ¬S) ∧ ¬(S ∧ ¬S ∧ (S ∨ ¬S)) ∧ ¬(S ∧ ¬S ∧ (S ∨ ¬S) ∧ ¬(S ∧ ¬S ∧ (S ∨ ¬S)))))Underlying structural theorem
Entₙ = Entₙ₋₁ ∧ ¬(Entₙ₋₁)The entire hierarchy is the same foundational operation (X∧¬X) applied recursively, taking the entire previous level as the new term — directly visible in the unfolding above, where each level contains, as a literal substring, the entire previous level followed by its own negation.
Degrees of freedom of this chain, for any n: exactly 1 (the choice of S). No level introduces a new variable or a genuine branching point — all the apparent complexity (2ⁿ literals) is the rearrangement of a single initial choice. This leaves open the question of where the system’s real freedom enters (the “several possible states” mentioned elsewhere in the corpus): the most likely candidate is the still-undefined “phase space” in this table.