Version note. This document (October 2026) supersedes the earlier foundations articles: ArXe Theory V4.2, ArXe Theory V4.1, ArXe Theory Foundations and Foundations of ArXe PHYSICAL Theory. It revises the framework substantially, notably its base logic (two distinct negators). Where they conflict, this document prevails.

ArXe — A Dynamic Ontology with a Logical Basis

Foundations Documentation

Diego Luis Tentor — ArXe Research · October 2026 · License CC BY-SA 4.0 · Status: living theoretical framework

How complexity emerges from the impossibility of nothingness.

How to Read This Document

Every important statement carries a status. The status says how much weight the reasoning bears at that point.

Status	Meaning
Axiom	Declared starting point. Not demonstrated.
Definition	Convention of naming or notation.
Verified	Demonstrated or checked by calculation.
Postulate	Explicit assertion, without derivation from the axiom.
Hypothesis	Working proposal. Plausible, not demonstrated.
Reading	Resonance with an external concept. Evocation, not derivation.
Open	Unresolved question. Not considered closed.

Index

Part I — Starting Point

1. What ArXe Is
1. The Axiom
1. The Recursion of Exentation
1. Base Logic

Part II — The Hierarchy of Levels

1. Arity, Terms, Phases, Valence, Distinguishability
1. The Tᵏ Levels: Indexing
1. Boundary Conditions of Possibility
1. Temporal Threshold, Act, and Phase Space

Part III — Phase Ordering and Probabilistic Structure

1. The Ordering of Phases and Terms
1. Three Principles, One Structure
1. Physics as Statistical Manifestation

Part IV — Dimensional Framework, Geometry, and Observer

1. The Dimensional Framework
1. Minimal Conditions of Possibility
1. The Hierarchy as a Chain of Inversions
1. Geometry of Undecidability
1. Anchors π, φ, and ρ
1. The Observer

Part V — Operators

1. Reading Rule and BCC/BCO Pattern
1. Registry by Level
1. Dialogical Operators
1. Compound Operators
1. Self-Dialogue of an Arity
1. Boolean Logic within the Hierarchy

Part I — Starting Point

1. What ArXe Is

ArXe is a dynamic ontology with a logical basis. Physical reality is described as a recursive process. The process resolves a primordial logical contradiction. It presupposes no prior mathematical or physical entities.

Reality is not a mathematical structure built from constants. It is the active process of escaping from ¬(). There are no numbers, laws, or particles waiting to be discovered. There is the unfolding of distinction. Physical laws and constants are traces of that unfolding. Mathematics is its grammar, not its cause.

ArXe is neither mathematical realism nor Platonism. It is an ontology of non-viability: logical nothingness, being non-viable, escapes toward structure. The Planck scale is not the unit of measure of something. It is the metric of the instability of nothingness.

What distinguishes it from other positions.

Position	What it presupposes	What ArXe does
Mathematical realism, Platonism	Fundamental mathematical entities	It presupposes no entity. Only the impossibility of their absence.
Physicalism	Physical substance	There is no substance. There is a recursive unfolding of distinction.
Idealism	A primary mind or concept	No observer is needed for ¬() to generate structure. The observer is a level of the hierarchy (§17).

2. The Axiom

Status: Axiom.

¬() ≜ Tf ≃ Tp

¬(): the logical act of negation applied to nothing. Nothingness, being logically non-viable, cannot simply be.
Tf: conceptual fundamental time. The minimal pulse that generates that non-viability.
Tp: physical fundamental time. The Planck time, ≈ 5.39 × 10⁻⁴⁴ s.
≜: conceptual equivalence. Logical-physical kinship, not identity.
≃: postulated correspondence with the Planck scale.

The axiom does not establish an identity between logic and physics. It describes an event. Nothingness, being non-viable, escapes toward a fundamental pulse of time. That pulse is perceived as the Planck scale. Tp does not measure something. It is the metric of the instability of nothingness.

The fundamental physical act is analogous to a logical contradiction:

This precise instant, in its fundamental physical expression, is absolutely actual, is not possible, cannot be verified or demonstrated, does not exist and is not true. Analogous to the philosophical ‘Tempus Instans’ (see below, note)

The logical contradiction is not a problem to be solved. It is the generative engine of all reality. The universe does not resolve ¬(). It perpetually escapes from it. That escape is what we call time, space, and structure.

The axiom includes a presumption that is used in §7: a Tf cannot, without contradiction, be both the beginning and the end of itself.

The axiom is neither provable nor refutable as an empirical proposition. It is declared, like the axiom of any formal system.

3. The Recursion of Exentation

3.1 Definition

Status: Definition.

The levels n are not dimensions of space. They are degrees of evasion of the contradiction. Each step is a new layer of structural freedom.

Ent₁   := S ∧ ¬S        (contradictory: the pure act)
ExEnt₁ := S ∨ ¬S        (tautological: the pure horizon)

Entₙ   := Entₙ₋₁ ∧ ExEntₙ₋₁                    (entification: being)
ExEntₙ := ¬(Entₙ₋₁ ∧ ExEntₙ₋₁) ≡ ¬Entₙ₋₁ ∨ ¬ExEntₙ₋₁   (exentation: existing beyond)

In the base case, S ∨ ¬S and ¬(S ∧ ¬S) are equivalent by De Morgan. Hence ExEntₙ ≡ ¬Entₙ for all n ≥ 1.

The recursion is Boolean logic. AND builds each Entₙ. NOT and OR build each ExEntₙ. These connectives are not “located” at a level. They are the ‘engine’ that builds all the levels (see §23).

3.2 Self-Similarity Theorem

Status: Verified.

Since ExEntₙ₋₁ ≡ ¬Entₙ₋₁, it follows that:

Entₙ = Entₙ₋₁ ∧ ¬(Entₙ₋₁)        for n ≥ 2

The whole hierarchy is the same foundational operation, X ∧ ¬X, applied recursively. The new term is the entire previous level. This is self-similarity in the literal, algebraic sense.

Results checked by construction and by calculation:

Entₙ unfolded has exactly 2ⁿ literals.
Written with ExEntₙ = ¬Entₙ at all levels, Entₙ has 2ⁿ − 1 negations.
The nesting depth of negation in Entₙ is n.
The chain has exactly one degree of freedom: the choice of S. No level introduces a new variable or a genuine bifurcation. The apparent complexity (2ⁿ literals) reorganizes that single choice.

The freedom of the system lies in the phase space (§8.4), which still lacks a combinatorial definition.

3.3 Full Unfolding

Each Entₙ expands down to S and ¬S, leaving no subexpressions unopened. Unfolded with ExEnt₁ = S ∨ ¬S:

n=1  (2 literals)    (S ∧ ¬S)
n=2  (4 literals)    (S ∧ ¬S ∧ (S ∨ ¬S))
n=3  (8 literals)    (S ∧ ¬S ∧ (S ∨ ¬S) ∧ ¬(S ∧ ¬S ∧ (S ∨ ¬S)))

Each level contains, as a literal substring, the entire previous level followed by its own negation. At no level does a variable other than S appear.

4. Base Logic

A hierarchy built on a contradiction faces a central objection: logical explosion. This section gives the definitions and results that address it.

4.1 Two Negators

Status: Definition.

The symbol ¬ covers two operations. ArXe separates them.

Term negation (¬S). Applies to something reducible to a single name. It is what Aristotle called an indefinite name (ónoma aóriston, On Interpretation, chs. 2 and 10). “Non-man” is a new name, not the negation of a proposition. It has no temporal cost.
Exentation (¬(…)). Applies to a compound expression, with an unreduced connective ∧ or ∨. It is the operation that generates the chain of levels. It costs 1 Tf per application.

Rule: ¬X costs 1 Tf if and only if X is not reducible to a term.

4.2 Essence and Existence

Status: Definition and postulate.

Essence rule. Everything reducible to a single term has essence, including its implicit negation, as long as it is not explicitly negated. “anyterm” and “inanyterm” have essence. “Not anyterm” or “Not-anyterm” does not.
Relation. Existence implies essence. Essence does not imply existence. The foundational contradiction Ent₁ has essence. It has neither existence nor truth.
Scope. Essence is decided by the surface form of the term, not by the content it abbreviates. For that reason it neither restricts nor enables any inference. It is a reference condition: it allows naming Ent₁ without committing to its existence.
What consumes a Tf. Essence does not consume it. Negation (exentation) does. Essence is that on which negation operates.

4.3 The Term “Actual”

Status: Definition. Terminology open.

Actual(P) ≜ ¬◇(P)

“Actual” means “not possible”. It is a stipulation of our own. It inverts the standard modal usage, where the actual implies the possible. It also inverts two traditional hierarchies:

Aquinas. The identity of essence and existence is reserved to God (actus purus, maximal reality). In ArXe, the primordial act has essence without existence: minimal reality.
Aristotle (Metaphysics Θ). Act is fuller than potency. In ArXe, the possible has essence and existence. The act has no existence.

4.4 Temporal Threshold Semantics

Status: Definition. Algebra verified.

Each formula X is assigned a minimal temporal threshold τ(X). It is the number of Tf needed before X can be evaluated. Before t = τ(X), X is undefined: neither true nor false, and unavailable to any rule of inference.

τ(S)     = 0                               (atomic term)
τ(¬X)    = τ(X)        if X is reducible to a term
τ(¬X)    = τ(X) + 1    if X is not reducible to a term (exentation)
τ(X ∧ Y) = τ(X) + τ(Y)
τ(X ∨ Y) = τ(X) + τ(Y) − 1

t < τ(X)  →  X is undefined
t ≥ τ(X)  →  X takes its classical value, according to its structure

Why a sum and not a maximum. With τ(X∧Y) = max and τ(X∨Y) = min, De Morgan’s identity breaks. The correct dual of the maximum, under a negation that shifts by a constant, is once again the maximum. Then ∧ and ∨ become indistinguishable. The sum has a genuine dual, and the rule for ∨ is derived so as to preserve De Morgan.

Scope of De Morgan. The identity τ(¬(X∧Y)) = τ(¬X ∨ ¬Y) holds exactly when X and Y are compound. In the base case it does not hold. With the rule for ∨ applied literally, τ(S ∨ ¬S) = −1. By contrast, τ(¬(S ∧ ¬S)) = 1. The two formulas are equivalent in value, but not in threshold. For that reason the convention ExEntₙ = ¬Entₙ at all levels, including n = 1 is adopted. It is an explicit decision (see §29).

Resulting sequence under that convention:

n	Level	τ(Entₙ)	τ(ExEntₙ)
1	T⁰	0	1
2	T¹	1	2
3	T⁻¹	3	4
4	T²	7	8
5	T⁻²	15	16
6	T³	31	32
7	T⁻³	63	64

Closed form: τ(Entₙ) = 2ⁿ⁻¹ − 1 and τ(ExEntₙ) = 2ⁿ⁻¹.

4.5 Classical Objections

Objection	Resolution
Russell. “The existent round square” forces a contradiction through free comprehension of properties.	Declarative existence and truth do not count as real existence or truth. That a term includes “existent” in its description does not make it exist.
Meinong’s jungle. Proliferation of objects with essence.	There is no comprehension over arbitrary sets of properties. A single generator, ¬(), unfolds the whole hierarchy. No loose objects are postulated.
Lewis’s explosion. From A ∧ ¬A any B is derived: ∧-elimination, ∨-introduction, and disjunctive syllogism chained together.	∧-elimination on Entₙ presupposes that Entₙ is already evaluable. Under threshold semantics, Entₙ is undefined before τ(Entₙ). There are no loose formulas to start the chain before that point.

The blockade covers Lewis’s route. It does not prove that no other route derives triviality. That proof remains open (§29).

Part II — The Hierarchy of Levels

5. Arity, Terms, Phases, Valence, Distinguishability

Status: Definition.

A phase is a minimal unit of time, indistinguishable from another phase. Phases acquire structure only through their relations.
Hereafter we will express them as follows.

1 Phase: (a)
2 Phases: (a,a’)
3 Phases: (a,a’,a”)
4 Phases: (a,a’,a”,a”’)

Term

A term is an assignment of roles to phases in a complementary relation. Each phase occupies a distinct role that the others do not occupy. Examples: (starts, ends); (first, second, third). The set is contradictory if two or more phases repeat the same role.

A term designates at least one phase but may designate all the phases of an arity.

While phases are intrinsically indistinguishable from one another, terms are an act of distinction.

Arity

The logical n-arity is the logical structure determined by the number of phases.

Unary logic (1-ary): 1 Phase
Binary logic (2-ary): 2 Phases
Ternary logic (3-ary): 3 Phases
n-ary logic (n-ary): n Phases

ArXe designates as ternary logic (3-ary) what is classically known as binary logic.

Valence

Valence is determined by the number of terms, not by the number of phases.

Bivalent logic admits two terms
Trivalent logic admits three terms
Trivalent logic admits three terms
Trivalent logic admits three or more terms

The term “n-ary logic” is our own usage. It designates the set of logical operators of an arity level n. It is not asserted whether it coincides with the n-valued logic of the literature (Łukasiewicz, Post). The term is kept because it is explained as a succession of binary logic.
It may nevertheless be noted, as an observation to be developed, that in ArXe logical arity depends on phases and these are indistinguishable, whereas in classical logic valence depends on the number of terms, which are presumed to be orderable.

5.1 Nomenclature of the First Levels

Status: Definition.

Arity n	Contradiction in n steps	Paradoxes	Recursions	Philosophical name
1	1	—	first “cursion”	Entia / Essence / Istence
2	2	1	1	Parence / Existence
3	3	2	2	Apparence / Perience / Experience
4	4	3	3	Persistence
5	5	4	4	Persistence

The names identify relational aspects. None names Time, Space, Mass, or any other physical phenomenon. This is consistent with the unfolding (§3.3): at no level does a variable other than S appear.

Phase and phase space are different concepts. The phase is the unit. The phase space is the set of possible configurations. Its combinatorics is not defined (§8.4).

6. The Tᵏ Levels: Indexing

6.1 The Mapping Functions

Status: Definition.

Direct mapping:
e(n) = 0                   if n = 1
       (−1)ⁿ · ⌊n/2⌋       if n > 1

Inverse mapping:
n(k) = 1                   if k = 0
       2k                  if k > 0
       −2k + 1             if k < 0

e(n) generates the sequence {0, 1, −1, 2, −2, 3, −3, 4, −4, …}. The two functions establish a bijection between ℕ and ℤ. It is asserted to be the only one that respects the alternation of the sequence. Uniqueness is not demonstrated here.

Each level Tᵏ, with k ∈ ℤ, receives a natural number n(k): its arity. For k < 0, n(k) runs through all the odd numbers from 3: 3, 5, 7, 9, 11, 13, 15, 17, 19, …

n(k) is an indexing function and nothing more. It does not distinguish some odd numbers from others. Any distinction among them would be an addition, not a consequence of the function. In particular, T⁻⁴ (n = 9) and T⁻⁷ (n = 15) are full levels.

k	−9	−8	−7	−6	−5	−4	−3	−2	−1	0	1	2	3	4	5	6	7
n(k)	19	17	15	13	11	9	7	5	3	1	2	4	6	8	10	12	14

6.2 The T⁰ Level

T⁰ is the only level without a sign. It has arity 1: one phase. With a single phase there is nothing to order. It is the generative contradiction itself, prior to the distinction between determinate and indeterminate. It is not a level in the usual sense. It is the origin from which the levels emerge. Its reading is PLK, by resonance with the minimal pulse of the axiom (§19).

Physically it manifests as the ‘Tempus Instans’, ‘this instant of time’, as small as one Planck Time.

(It remains pending whether its value is the Planck constant or the Planck Time.)

This dependence is formulated axiomatically, not demonstrated.

7. Boundary Conditions of Possibility

Hereafter Boundary Conditions, or BC.

7.1 Sign Postulate

Status: Postulate.

k > 0  →  all boundary conditions (BC) closed (BCC)
          → can reveal isolation (dimensions, fields, vacuum, confinement)
          → The phenomenon is isolated from the observer
          → "virtuality"

k < 0  →  at least one BC open (BCO)
          → does not exist in isolation (particles, empirical phenomena)
          → the observer participates, a choice is required

k = 0  →  none: the contradiction itself

It is not a derivation from the Entₙ/ExEntₙ recursion. It is a qualitative reading of the sign of k.
Positive means that start/end or opening/closing conditions are given.

Negative means pending: it needs external coupling either to start or to end, to open or to close.

7.2 Boundary, Finitude

Boundary. A set of attributes of finitude conditions. Each attribute has two terms, one initial and one final.
Closed boundary. Both terms are determined in the act of invocation.
Open boundary. A single term remains indeterminate. The indeterminacy is declared relative to the act. It is not a defect. It is consistent with the Entₙ/ExEntₙ duality.
Finitude. Derived from boundary: two contrary boundaries, beginning / end, opening / closing, start / finish. Status: Definition.

Central result. Status: Verified. The axiom presumes that a single Tf cannot, without contradiction, be two terms in 1 Tf, the beginning and end of itself. Then a closed boundary, which needs a determinate beginning and end, requires at least two distinct phases. A single phase does not fulfill both contrary roles at once. An open boundary is given with a single term and is satisfied with one phase that forms no pair.

It is a direct logical consequence. If two roles are contrary by definition, at least two bearers are needed. There is no choice of formula and no filter applied after seeing a result.

Scale. Status: Hypothesis. A closed boundary defines a finitude between its two terms: a scale. An open boundary does not define a bounded finitude: there is no scale. This connects with scale invariance and criticality.

7.3 BC Count per Level

Status: Verified (arithmetic), on the postulate of §7.1.

Counting rule. Each closed attribute contributes 2 terms (start and end). Each open attribute contributes 1 (the term that remains indeterminate). Therefore:

n(k) = 2·Closed + 1·Open

k > 0:  Closed = k      Open = 0     (n = 2k)
k < 0:  Closed = |k|    Open = 1     (n = 2|k| + 1 = −2k + 1)
k = 0:  Closed = 0      Open = 0     (n = 1, the contradiction itself)

budget(Tᵏ) = Closed + Open
           = k        if k > 0
           = |k| + 1  if k < 0

The count is a clean function of k. It uses no arithmetic property of n(k). It was checked to reproduce n(k) at all levels in the table.

Level	k	n	Closed BC	Open BC
T⁰	0	1	0	0
T¹	+1	2	1	0
T⁻¹	−1	3	1	1
T²	+2	4	2	0
T⁻²	−2	5	2	1
T³	+3	6	3	0
T⁻³	−3	7	3	1
T⁴	+4	8	4	0
T⁻⁴	−4	9	4	1
T⁻⁵	−5	11	5	1
T⁻⁶	−6	13	6	1
T⁻⁷	−7	15	7	1
T⁻⁸	−8	17	8	1
T⁻⁹	−9	19	9	1

7.4 Corollary: Undecidability Sets

Status: Verified (existence). Content: Open.

Every level T⁻ⁿ, with n ≥ 1, has a term that is indeterminate with respect to the act of invocation.

Every level with k < 0 has exactly one open BC (§7.3).
Every open boundary has determinate terms and one term indeterminate with respect to the act (§7.2).

The property is distributed evenly across the whole negative branch. No level is a special case.

The corollary does not say what each set contains. Nor does it assign a constant. Characterizing the content needs its own mechanism.

7.5 Scope of the Count

The BC count does not, by itself, determine any specific mathematical constant. The images in §16 are illustrative.

8. Temporal Threshold, Act, and Phase Space

8.1 Two Readings of the Time an Arity Needs

Status: Hypothesis, with open tension.

The basic idea: arity fixes the minimal temporal conditions for something to be given or to happen. It does not say that it happens in that time. Example: the logic of mass (n = 6) needs at least 6 Tf. That does not mean that 6 Tf is mass. They are the minimal conditions that mass needs in order to exist.

T¹ = 2·Tf. Something and its contrary do not fit in 1 Tf, by the principle of non-contradiction. They do fit in 2 Tf, which is physical time.

Reading by depth. The threshold of arity n is the nesting depth of negation in Entₙ, which equals n (§3.2). If the negations of a single layer occur in the same Tf, arity n needs n Tf. That gives T⁰→1, T¹→2, T²→4, T³→6, T⁷→14. That is, 2k for k > 0.

Tension. The sum semantics (§4.4) gives another value: τ(Entₙ) = 2ⁿ⁻¹ − 1.

n	Level	Depth (n Tf)	τ(Entₙ), sum	Literals (2ⁿ)
2	T¹	2	1	4
3	T⁻¹	3	3	8
4	T²	4	7	16
5	T⁻²	5	15	32
6	T³	6	31	64
7	T⁻³	7	63	128

They are different counts. The sum accumulates costs in sequence. The depth assumes that the negations of a single layer are simultaneous. That assumption is equivalent to the max rule, which breaks De Morgan (§4.4). Until it is decided which model governs, “T¹ = 2 Tf” and the thresholds 2k are hypotheses. They are not results.

8.2 The Act (T)

Status: Definition. Pending contrast.

The Act as ‘Tempus instans’
Tempus instans is, in its briefest definition, imminent or present time; that infinitesimal portion of time which is occurring “right now” and which presses immediately upon the individual and things.

In philosophy (the “now” as the only reality): Cicero analyzed time under the influence of Greek philosophy. For him, the tempus instans was the vanishing point where the future becomes past. The tempus instans is the only real moment over which we have control and capacity for action.

Note to validate:

The ‘act’ is presumed to be empirical and not demonstrable, analogously to time and space in classical physics or GR.

The Act as ‘Tempus Instans’ has no existence, it has essence

Unlike time and space, the ‘act’ is not presumed existent, but essential (see the differentiation between essence and existence at the beginning of the text).

Act is the relation between Tᵏ and its logical unfolding backward to T⁰.
“Acting as” is traversing that relation.
It is not enough for a path to be logically viable. Only the path that effectively reaches T⁰ is ontologically valid. This is the sense of “grounded” in Kripke.
Level n of the backward chaining chain is arity n.

General principle. Choosing the logically stronger side of an asymmetric implication (X→Y without converse) always reduces, or leaves unchanged, the number of consistent maps. Choosing the weak side preserves or enlarges it.

8.3 Isolated Example: First and Second

Status: Verified by enumeration.

Reasoning:
We assume that something second follows something first.
If there is something “First”, there need not be something “Second”. But if there is “Second”, there necessarily is “First”. One can choose to act as First or as Second. Neither choice is necessarily true.

Single axiom: Second → First, without the converse. There are 4 possible global maps and 3 consistent ones. Only “Second without First” is discarded.

Choice	Compatible maps	What remains determined?
Act as First	2	Nothing. The other variable remains free.
Act as Second	1	First is forced.

The same pattern (2 maps with the weak side, 1 with the strong) was checked in Existence/Essence, Square/Rectangle, and Convergent/Bounded. It was also checked in Actual/Possible in the classical sense. ArXe inverts that direction on purpose (§4.3).

The act does not create the implication. The implication follows under any choice. The act decides which of the available choices is actualized. Two different acts leave different amounts of remaining freedom.

8.4 Phase Space

Status: Open.

There are three candidates to define the phase space. None has been chosen.

Total number of combinations.
Profile per choice.
Number of viable choices. This candidate demands grounding, not just viability.

Calculated values: T¹→2, T²→4, T⁷→14. The logical space equals 2k, which coincides with the nodes of the graph with memory.

Part III — Phase Ordering and Probabilistic Structure

9. The Ordering of Phases and Terms

Every level Tᵏ has
a) an arity n: n phases.
b) Terms (t) that designate between 1 and n phases.
c) A number of terms whose sum of designated phases = n.

Terms may or may not be orderable according to what is established in the section on orderability (link pending).

Distinct terms that designate non-orderable phases are not orderable.
Distinct terms that designate orderable phases are orderable.
Non-orderable terms exhibit the same phenomenon, regardless of their order.
Orderable terms exhibit different phenomena under different orderings.

Not all are equally probable, because early orderings condition later ones. And everything begins at T⁰, where there is nothing to order.

9.1 What an Ordering Is

In T⁻¹ there are 3 phases. We call them B (beginning), M (middle), and E (end). They can be arranged in 3! = 6 sequences:

1. B → M → E
2. B → E → M
3. M → B → E
4. M → E → B
5. E → B → M
6. E → M → B

In T⁻¹ (3 phases) any sequence is indistinguishable from any other (1 = 2 = 3 = 4 = 5 = 6).
The structure does not admit an objective distinction between terms.

In T³ (6 phases) any sequence can be distinguished from any other (1 ≠ 2 ≠ 3 ≠ 4 ≠ 5 ≠ 6).

Each distinguishable sequence is a distinct phenomenon.

Phases have no intrinsic term. Any sequence is structurally valid. The choice of terms is arbitrary; in this case we have chosen “Beginning”, “middle”, and “end” to ease understanding of the relations with respect to the reality they describe. (It remains to be resolved whether this can be considered a post hoc adjustment.)

9.2 The Factored Form Reveals the Ordering

T²×T², T³ + T¹, and T⁴ are the same level. But the factored form reveals which ordering of the terms of T⁴ is active: T²×T², a spatial process that applies its structure and then inverts it, with the origin present as a real place to return to, or T⁴ as (INF) information.

It is analogous to a Feynman diagram. It is a tool that reveals which choices are active and how they combine, in a way that the final number does not show.

Rule. Use the factored form when the ordering matters for understanding the phenomenon. Use the collapsed form only when it is irrelevant. Collapsing prematurely loses information. The factored notation is hermeneutically richer.

9.3 Gradation from T⁰ to T⁻¹

The principle does not apply uniformly.

T⁰. Minimal instance of actual time.
Actualization, act of choice.
No ordering. Pure indeterminacy. It has a single phase. There is nothing to order or choose. T⁰ is not “determined” with a fixed result. It is prior to the distinction between determinate and indeterminate. It is the contradiction itself: neither true nor false, neither existent nor non-existent.
T¹. Mere succession. Trivial. It is homogeneous positive time: one instant after another, in one direction, without alternation or variance. It has 2 phases, but their relation is trivial. One follows the other, the other follows one. No law determines it either. Asking about the probability of the ordering of T¹ has no meaningful answer. Nothing can be different in T¹, because distinction is not yet structurally possible.
T⁻¹. Variance begins here. It has 3 phases:
COL: 3 indistinguishable terms
or
ALT: 2 orderable terms (objectively distinguishable)
or
PART: 1 open BC and 1 closed BC (participation)

For the first time something can occur in more than one way.

Here probability enters the framework. It does not enter as an external mathematical tool. In T⁻¹ the conditions are given for at least three distinct phenomena; nothing implies that one, another, or any of them occurs, only that in T⁻¹ an objective phenomenal differentiation appears as a possibility.

That is the ontological foundation of probability in ArXe.

9.4 Probability

No ordering is privileged. It is not presumed that the universe makes a choice or weighs one over another. All possible choices manifest in time Tᵏ as the actuality of the probability space.

9.5 Behavior and Phenomenon

Each level T establishes the phenomenal conditions for one or more distinguishable behaviors.
Behavior is a term that names the relation of phases.
The choice of term is arbitrary; the behavior it describes is not, it is determined by the relation of phases at its level.

The theory admits several names for the same level, but for each name there is an axiomatic choice.

T¹ has two phases indistinguishable from each other. There is no ordering or middle.

TIME: Finite time, start and end.
1D : 1 Dimension = Time.
FIN: Finitude (start and end).
BCC: Closed boundary condition

An annex with all the behaviors remains pending.

Level	n	Proposed reading
T⁰	1	ACT – TIME-INST – ESS – OBS – IND
T¹	2	TIME – 1D – FIN – BCC – EXIST
T⁻¹	3	ALT – ALTERN – SENSE – DIST – BCO – ORD – MID – PART – SEQ
T²	4	SPACE – SIMULT – 2D – DIR
T⁻²	4	CURV – ACC – MID-SPACE
T³	6	MASS – 3D – CONF
T⁻³	6	MASS – 3D – CONF
T⁴	8	INF – 4D
T⁻³	7	COL
T⁻⁵	11	EM

9.6 Axiomatic Choices

Each level T corresponds to different possible behaviors. None is necessarily true, but if one is, it follows that no other is. That is, two or more behaviors cannot be simultaneous at the same level. This derives directly from the limited number of phases of a level.

For example:
T² = SPACE – 2D – DIR

If SPACE, then not 2D or DIR
If 2D, then not SPACE or DIR
If DIR, then not SPACE or 2D

This is consistent with point 9.4: the universe does not weigh one truth over another, it does not establish how to behave, it does so probabilistically, but with a single behavior per level.

9.7 Trivial Axiomatic Choices

For n phases with n terms, where each term necessarily corresponds to one and only one phase, any ordering is trivial. All express the same phenomenon. This is expressed as simultaneity (SIM) or spatial simultaneity (SPACE), ideally empty; a direction (DIR) is indistinguishable from any other.

E.g.: 2 Phases -> 2 Terms (i = a, f = a’)
i and f are interchangeable and, once swapped, describe the same phenomenon.

9.8 Non-Trivial Axiomatic Choices

When n phases are given with n−1 terms the choice is not trivial.

E.g.: 3 Phases -> 2 Terms (i = a, f = (a’, a”))
i and f are not interchangeable and, once swapped, describe different phenomena.

9.9 Objective Existence

Objective existence relative to T is given when there is no contradiction among the axiomatic choices, that is, when what exists for any ACTOR / OBSERVER in Tⁿ is not in itself contradictory.
This does not necessarily mean physical existence, but only that it lacks contradiction independently of the observer.

9.10 Distinction, Ordering, and Valences

Proposed text to insert in §9, after §9.9 (objective existence). Where it conflicts with sections 2 to 4 of this document, this text prevails.

9.10.1 Vocabulary

Status: Definition.

Phase. Unit of Tf, indistinguishable from one another. There are n per level.
Term. Arbitrary name for one or more phases. A level admits terms of 1 up to n phases.
As a convention a term has no more than one word and cannot be preceded by explicit negation (E.g.: “not word”).
Valence. A term of more than one phase (2, 3, …, n). Valences are cases of terms.
Mode. Set of terms that sum to n, without order. (3,1) and (1,3) are the same mode.
Ordering. Distinguishable sequence of the terms of a mode. Terms of equal size do not produce new orderings.

The logical valence of a mode is its number of valences. Its relation to the valence of classical logics (two, three) is left for a separate analysis.

9.10.2 Classification of Modes

Status: Definition.

Class	What it is	Examples
Ambivalence	Two equal valences	(2,2), (3,3), (4,4)
Equivalence	Three or more equal valences	(2,2,2), (3,3,3), (2,2,2,2)
Bivalence	Two distinct valences	(4,2), (5,3)
Polyvalence	Three or more distinct valences	(4,3,2), (5,4,2)
+ unknowns	Any of the above plus one or more one-phase terms	(3,2,1) = bivalence + 1 unknown

The mode (3,2,1) is named bivalence + indeterminacy. It has two valences, 3 and 2, because 1 is not a valence by definition.

Complete example, level T³ (n = 6), 11 modes:

Mode	Class	Orderings
(6)	one valence	1
(5,1)	one valence + 1 unknown	2
(4,2)	bivalence	2
(4,1,1)	one valence + 2 unknowns	3
(3,3)	ambivalence	1
(3,2,1)	bivalence + 1 unknown	6
(3,1,1,1)	one valence + 3 unknowns	4
(2,2,2)	equivalence	1
(2,2,1,1)	ambivalence + 2 unknowns	6
(2,1,1,1,1)	one valence + 4 unknowns	5
(1,1,1,1,1,1)	6 unknowns	1

The orderings add up to 32 = 2⁵.

Not yet named: the single-valence mode, such as (6), and the mixed ones, such as (3,3,2).

9.10.3 Distinction and Sequence

Status: Hypothesis.

DIST  (A, B) with A ≠ B             (A ≠ B) = (B ≠ A)
      A and B are distinct. There is no true order.

SEQ   (a → a') ≠ (a' → a)           sequence or secundance
      There is one determination and one indeterminacy:
      a finite instant (T¹) plus an infinite one or one of indeterminate finitude (T⁰).

ORDER = DIST + SEQ

In T⁻¹ (n = 3) the only asymmetric mode is (2,1). DIST and SEQ are read on that same mode. By the principle of choice (§9.6), something in T⁻¹ is distinguished or is ordered, not both at once. The exclusion follows from there being a single mode, not from a separate postulate.

SEQ is the canonical mode of T⁻¹. T⁻¹ has one closed BC (2 phases) and one open BC (1 phase). That is the mode (2,1): a finite instant, T¹, and an unknown, T⁰.

To have both at once two 3-ary structures are needed. The combination depends on the operation:

Operation	Calculation	Resulting mode	Level
Sum (FUSION)	3 + 3 = 6	(2,1) ∪ (2,1) = (2,2,1,1)	T³
Product of arities	3 × 3 = 9	pairwise products: (4,2,2,1)	T⁻⁴
Product over exponents	(−1) + (−1) = −2		T⁻²

If ORDER must be in T³, the operation is the sum of arities. The resulting mode is an ambivalence with two unknowns, with 6 orderings.

Pending. (a) Confirm that DIST and SEQ are 3-ary operators in the sense of §20. (b) FUSION does not retain components in §20, but ORDER must preserve DIST and SEQ. The union of modes preserves them. That definition remains to be written. (c) What the 1 of DIST is: under the rule of unknowns, B is not determined.

9.10.4 First Appearance of Each Logical Valence

Status: Verified (arithmetic).

To have v valences at least 2v phases are needed, with the mode (2,…,2). Since n(k) = 2k for k > 0, the first logic of v valences appears in Tᵛ:

Valences	Minimal mode	Level
1	(2)	T¹
2	(2,2)	T²
3	(2,2,2)	T³
4	(2,2,2,2)	T⁴

The canonical mode of the BC count (§7.3) is the same mode. For k > 0 it is C valences of 2 phases. For k < 0 it is C valences of 2 phases plus one unknown, which is the open BC. Example: T⁻³ = (2,2,2,1). §7.3 describes only this canonical mode. That is why a negative level can have other modes with several unknowns, such as (3,1,1,1).

9.10.5 Reading: Past, Present, and Future

Status: Reading.

The mode (3,2,1) has three terms of different sizes. T³ is the first level where it exists. It is read as past (3 phases), present (2 phases), and future (1 phase). The future is the unknown: a one-phase, open term.

That grounds, with fewer assumptions, the claim that beginning, middle, and end can be objectively attributed from T³ onward. In T⁻¹ the three one-phase terms, (1,1,1), are three indistinguishable unknowns. No ordering produces a new phenomenon.

Objectivity remains relative to the observer in T⁰ (§9.9). The caveat that this is a reading is maintained.

9.10.6 Pending Items of This Section

Name of (3,2,1): “bivalence + indeterminacy” versus “indeterminate trivalence”. The first is adopted for consistency with the definition of valence. When the relation to logical valence is analyzed, a review is advisable: a three-valued logic with a third “unknown” value (Kleene type) is the natural counterpart of “indeterminate trivalence”.
Names for the single-valence mode and for the mixed ones.
Operation that defines ORDER: sum or product (§8.3).
Relation between valences and logics of valence 2 and 3.
Forms of distinction and ordering that appear with each new arity.

To be developed further:
The theory does not assume that an absolute objectivity exists, just as it does not assume absolute truths. Hereafter, talk of objectivity, physicality, reality must be understood in relation to the Observer (T⁰) located in the ‘Tempus Instans’.

Even so, given its intrinsic indistinguishability, T¹ (n=2) is the phenomenon that can be taken as ‘universal’, since every choice is trivial and every arbitrary term designates the same relation.

9.11 Copy, Alternative, and Probability

9.11.1 Two Operations

Status: Definition. Reading in modes pending confirmation.

A × A, copy. The second structure repeats the first: (start, end, start, end). Each T² contributes the pair (start, end) and the copy repeats it. The levels add exponents: T² · T² = T⁴.
A + B, “OR”. Alternative: either A appears or B appears. Within a single level the two are mutually exclusive (§9.6 of the main documentation): if one occurs, the other does not.

Reading in modes, to be confirmed: T² is the mode (2,2). The copy repeats it and yields (2,2,2,2), the canonical equivalence of T⁴ (§9.10.2).

9.11.2 Probability

Status: Hypothesis.

Base rule (§10.1): a specific structure of n phases is worth 2⁻ⁿ. T² has 4 phases and T⁴ has 8.

Event	Value	For T² = 1/16
A or B, exclusive	P(A) + P(B)	1/8
A × A, copy	P(A) · 2⁻ʳ	1/16 if r = 0
A × B, independent	P(A) · P(B)	1/256
Specific T⁴	2⁻⁸	1/256

r is the cost of saying “repeat”. r = 0 is proposed.

The values are relative weights. Normalized over the 128 configurations of T⁴, the specific copy with r = 0 is worth 0.0645, not 0.0625. The difference is only one of normalization.

9.11.3 Relation among the Three

P(A or B)  ≥  P(A × A, copy)  ≥  P(A × B, independent) = P(pure structure)

Read in terms of choices:

OR. One choice between two. The probability adds.
Copy. No new choice. The probability is maintained.
Independent. One choice per structure. The probability multiplies and goes down.

Factoring does not change the probability. Independent T² × T² and T⁴ fix the same 8 phases and are worth the same. What it saves is copying.

9.11.4 Pending Items

Confirm the reading of the copy as (2,2,2,2).
Fix r. r = 0 is proposed with the principle: what has already been actualized persists as long as no new unknown appears that forces the act to choose. Repeating requires no choice. Changing does. Unconfirmed.
Probability of “OR” between structures of different size.
As a class, with r = 0 repetition comes close to 50% versus 25% for change. This remains to be incorporated.
Names for the copy and for the alternative.

9.12 The Two Orderings of T² and the Arrow of Time

T² has four phases. Two orderings are illustrative.

A: beginning → interior → end → exterior. A finitude with inside and outside. It is the spatial structure of a bounded object: a container.
B: beginning → end → destination → origin. A direction and its inversion. It is the structure of a reversible path and of the vortex: a circuit.

Same level, different orderings, different phenomena.

The six orderings of T⁻¹ are equiprobable a priori. “Beginning → middle → end” has no structural advantage over “end → middle → beginning”. The arrow of time does not belong to T⁻¹. It belongs to the level that records history. That level settles an order of events as an objective fact, and the record is asymmetric by definition.

In this document that level is T³ (condition of historical persistence, §13). Thermodynamic irreversibility is real. Its foundation lies in the record, not in temporal alternation.

T⁻¹:  no preferred ordering — structural symmetry of the 6 orderings
T³:   specific ordering recorded as fact — apparent irreversibility

Concept	Formulation
Orderings at arity level n	n! (counts terms in SYM)
Probability of each ordering	1/n! a priori. Conditioned by history afterward.
Most probable orderings	Those that factor into lower-level structures
Least probable orderings	Pure configurations of high arity
Arrow of time	Property of the record level, not of T⁻¹

Variance is born in T⁻¹. T⁰ is pure indeterminacy. T¹ is mere succession, trivial because there is no structural alternative. From T⁻¹ onward the framework is irreducibly probabilistic, conditioned by what has already been decided, never starting from zero.

9.13 The Lanthanides as Physical Realization

Status: Testable hypothesis. Scope to be delimited.

The 4f shell of lanthanide ions is read as governed by T⁻³ (arity 7). Each electron that is added is a one-phase ordering within T⁻³. The type of simultaneity available changes when the number of electrons exceeds the arity of the level.

With n ≤ 7. The system has not exhausted the distinguishable orderings of T⁻³. Each electron keeps its spatial distinction.
With n > 7. The additional electrons find no free orderings. The spatial mode T² requires independent dimensions for a distinguishable ordering. It enters into contradiction with the accumulated history. The whole T² mode collapses. The collapse is always exactly n(T²) = 4 orderings, because T² either holds or does not. It has no partial collapse.

Observable signature. For Kramers ions with n > 7 and intra-level transitions, the Kramers doublets of the main emitting multiplet are n − 4. This matches in Dy³⁺ (n = 9, 5 doublets) and in Er³⁺ (n = 11, 7 doublets). The collapse is discrete. Its distribution along the series is continuous, mediated by the record level.

Scope to be delimited. Yb³⁺ (n = 13) is a Kramers ion with n > 7 that emits 4f→4f from ²F₅/₂. That multiplet has 3 doublets. The formula would give 9. Before treating it as a prediction, its domain must be delimited: which multiplet counts as the main emitter and why. Until then it is consistent with two cases and has one case against.

10. Three Principles, One Structure

Why does the universe contain more space than mass? Why do phenomena tend to express themselves as combinations of simple structures? The structural answer follows from three principles. They are three faces of the same idea. Status: Hypothesis.

10.1 Probabilistic Freedom

The universe privileges no configuration.

A universe with N binary conditions has 2ᴺ configurations. A priori it prefers none. This is not a separate postulate. It is a consequence of T⁰ having no intrinsic preference. A condition on k specific positions is satisfied by 2^(N−k) configurations, whatever N is:

P(condition on k positions) = 2^(N−k) / 2ᴺ = 1 / 2ᵏ

It is independent of the size of the universe. On 2 positions it is 1/4. On 4, 1/16. On 8, 1/256. Simple conditions are more probable because more configurations satisfy them. Not because the universe favors them.

10.2 Relative Logical Necessity

Given a condition, certain consequences cannot fail to be.

When the act “acts as” something specific, that fact brings necessarily co-present truths. No law imposes them. Their absence would be a contradiction. Example: if the act acted as “another existent”, then necessarily “one” also exists. The necessity is conditional, never absolute.

(This also goes with the reference on logical inversion.)

There is no tension with probabilistic freedom. They operate at different moments:

Before actualization:     probabilistic freedom (all possible configurations)
In actualization:         one configuration becomes actual (statistical, not forced)
After actualization:      relative logical necessity (implications are not denied)

The axiomatic system is the succession of these logical necessities. It is not a chronological record. It is the set of truths implied by each currently true fact. It narrows the space of compatible configurations as the universe unfolds.

10.3 The Scale of the n-ary Logics

Each arity is an enormous leap. The probability of a pure n-ary manifestation falls exponentially with n. The following figures correspond to the SYM variant, the most probable. ORD is the least probable pure manifestation.

P(pure 2-ary)  = 1/4     = 0.250
P(pure 3-ary)  = 1/8     = 0.125
P(pure 4-ary)  = 1/16    = 0.063
P(pure 6-ary)  = 1/64    = 0.016
P(pure 11-ary) = 1/2048  = 0.0005

A pure 11-ary configuration is 32 times less probable than a 6-ary one (2⁻¹¹ versus 2⁻⁶) and 512 times less than a binary one. A 6-ary one is 16 times less probable than a binary one. It is not a property of any particular phenomenon. It is a consequence of the combinatorial structure.

Two counting models. The conditions model (2⁻ⁿ) and the orderings model (1/n!, §9.1) are different. The advantage of factored forms comes from the second, not from the first. Three independent conditions of 2 positions commit 6 positions. Their probability is (1/4)³ = 1/64, the same as that of a single 6-position condition. The ratio of §9.6 of an earlier draft (not included in this document), ≈ 10¹¹, comes from the count of orderings.

10.4 A Single Structure

From counting: probabilistic freedom. More configurations satisfy simple conditions.
From implication: relative logical necessity. An actual condition generates co-truths that cannot be denied without contradiction.
From complexity: n-ary scale. Complex pure manifestations are rare. Factored ones are common.

Result. A universe rich in simple structures (time, space). Scarce in complex structures (mass, fields). With complex phenomena always in factored form.

TIME is abundant because T⁻¹ has arity 3.
SPACE is less abundant because T² has arity 4.
MASS is a fraction because T³ has arity 6: it is 16 times less probable than a binary condition.
A pure electromagnetic field would be rare if it occupies T⁻⁵ (arity 11): 32 times less probable than mass. The observable field appears in factored combinations with lower levels.
The universe is mostly “empty”: almost all time and space, with rare islands of mass and rarer ones of complex fields. Not because of special initial conditions.

The principles do not say that complex structures are impossible. They say that pure manifestations of high arity are improbable. What they explain is relative abundance, not absence.

11. Physics as Statistical Manifestation

Status: Hypothesis.

Physical reality is the statistically dominant manifestation of arity: the tendency of every n-ary structure to express itself through its most probable decompositions into lower arities.

It is not a separate postulate. It follows from §10. What we observe as “physical” is not the full logical structure of any level. It is its most probable face, which is always its most factored form.

11.1 What It Means for the Negative Levels

Take T⁻³: arity 7, one open BC. The structural fact is only that there exists a configuration n = 7 with one open BC. In principle it manifests as any of its 7! = 5,040 orderings. In practice it manifests as its most probable forms, the factored ones.

Reading. The most probable factored form of n = 7 with one open BC is what we observe as color confinement. It is a structure that behaves like mass (T³) with one degree that cannot close. T⁻³ is not defined as “mass that cannot close”. It is n = 7 with one open BC, manifested in its most probable form. A pure manifestation of n = 7 exists structurally and would look very different. It is so improbable (1/5,040 for any specific pure ordering) that it is not observed.

Level	n	Simplest factored form (proposed reading)
T⁻¹	3	Temporal alternation
T⁻²	5	Spatial curvature
T⁻³	7	Color (mass with one open degree). In tension with §24.
T⁻⁴	9	No firm reading
T⁻⁵	11	Electromagnetic field (pending evaluation, §19)
T⁻⁶	13	Weak field (pending evaluation, §19)

The readings in this table are proposals, not derivations.
The phenomenal relation is described by its factorization; the description of the phenomenon as such would depend on the chosen axiomatic system.

11.2 Positive and Negative Levels

Each pair (Tᵏ, T⁻ᵏ) shares depth: both are the k-th exentation from T⁰. The boundary distinguishes them.

Tᵏ   (positive): all BCs closed — the fully resolved level.
                 Can exist in isolation. Has complete internal order.
T⁻ᵏ  (negative): one open BC — the level with one irresolvable degree.
                 Cannot exist in isolation. Has an internal direction
                 that is not decided without a higher level.

That is why positive levels tend to generate fields and negative ones particles, which are structures with an open degree seeking closure. Every field must couple to particles to produce observable phenomena. The closed BC of the field seeks the open BC of the particle. The coupling is not an additional law. It is a structural consequence.

Logical level:        n-ary structure with a BC configuration.
                      The n! orderings exist as possibilities.
Statistical layer:    the most probable orderings dominate, the most factored.
                      This is what we observe as physical phenomena.
"Laws":               statistical regularities of the dominant configurations.
"Particles":          open configurations (negative levels).
"Fields":             closed configurations (positive levels).

Physics is not everything that is logically possible. It is the most densely populated region of the logically possible.

Part IV — Dimensional Framework, Geometry, and Observer

12. The Dimensional Framework

Status: Hypothesis.

Fundamental correspondence:

T¹ ≡ T   (physical time)
T² ≡ L   (physical length)
T³ ≡ M   (physical mass)
T⁻ⁿ := 1/Tⁿ   (inverse magnitude)

Each physical dimension is a degree of freedom of the contradiction stabilized into measurable structure. Mass is not substance. It is T³: the third layer of processing that the contradiction needs to generate an objective, self-contained entity.

12.1 The Dimensional Rule

For every quantity of SI dimension Mᵃ × Lᵇ × Tᶜ, the ArXe exponent is:

n_ArXe = 3a + 2b + c

It follows from T≡T¹, L≡T², M≡T³ and from the multiplicative structure of dimensional analysis. It is exponent algebra. The ontological reading of each equivalence is a hypothesis.

SI dimension	Formula	ArXe	Note
Time	T	T¹	Minimal duality
Length	L	T²
Mass	M	T³	Objectivity, self-containment
Velocity	L/T	T¹	Same level as time
Acceleration	L/T²	T⁰	Dimensionless in ArXe
Force	ML/T²	T³	Same level as mass
Momentum	ML/T	T⁴
Power	ML²/T³	T⁴	Same level as momentum
Energy	ML²/T²	T⁵
Action (ℏ)	ML²/T	T⁶
Frequency	1/T	T⁻¹
G (Newton)	M⁻¹L³/T²	T¹	Same level as c

12.2 Three Structural Equivalences

Force = Mass = T³. F = ma, and acceleration is T⁰. Force has no independent ontological status. It is mass acting on a dimensionless ratio.
c = G = T¹. Both are conversion factors between the T² scale (length) and the T¹ scale (time). Setting c = G = 1 eliminates two redundancies of the same level.
Momentum = Power = T⁴. Two quantities that look different are the same degree of freedom of the contradiction in T⁴.

12.3 The Planck Units

In Planck units (c = G = ℏ = 1), every dimensional constant becomes a pure number: a ratio with respect to the Planck unit of its level.

m_P = 2.176 × 10⁻⁸ kg      l_P = 1.616 × 10⁻³⁵ m      E_P = 1.956 × 10⁹ J

The Planck scale is not chosen for convenience. It is proposed as the natural scale of ArXe because c and G have dimension T¹, the root level of the contradiction.

This framework does not include a method for obtaining the numerical value of any constant (§27).

13. Minimal Conditions of Possibility

Status: Hypothesis.

The right question is not “why did ArXe assign mass to T³?”. That formulation suggests an arbitrary decision. The question is: what are the minimal structural conditions for something to behave as mass, and what is the first level that satisfies them? That shifts the burden of justification from “we chose this mapping” to “this is the first level where these conditions are met”.

13.1 Minimal Conditions for Mass

Historical persistence. Mass does not disappear between one instant and the next. Past, present, and future must be distinguishable as ontologically distinct phases. It first appears in T³, where the six-phase structure generates the irreversible sequence past → present → future.
Ternary objectivity. Mass can be measured by a third party, independently of the measurement. It requires “an observer who records without being recorded”: not just A and B, but A, B, and the third that relates them without belonging to either. It first appears in T³ (n = 6).
Isolated existence. Mass can exist without another structure closing it. It requires all BCs closed (k > 0). T³ satisfies this: k = 3, three closed BCs.
Three-dimensional spatial presence. Mass occupies three dimensions. T³ is the first level with exactly three independent structural degrees of closure.

13.2 Why T³ and Not T², T⁻³, or T⁴

T² fails conditions 1 and 2. There is no past-present-future structure or structural third. It operates with pairs of contraries (four-ary logic). It can have length, not mass.
T⁻³ fails condition 3. It has one open BC and cannot exist in isolation. It is the reason why, in this reading, color charge is always confined.
T⁴ satisfies all four conditions, but it is not the first. By structural parsimony, T³ is the minimal level.

13.3 The General Principle

A physical concept X maps to level Tᵏ
if and only if
Tᵏ is the minimal level whose structural properties
(arity, number and type of BC, logical structure)
satisfy all the conditions of possibility of X,
and no level Tʲ with |j| < |k| satisfies them all at once.

Three consequences:

Assignments are justified by declared conditions. They are not stipulated.
Absence is informative. A level with no assigned concept is not empty: it awaits identification.
The framework is not circular. T³ is defined by its structural properties. Mass is identified as the concept whose minimal conditions coincide with them. The direction goes from structure to concept.

Declared risk. If the conditions of each concept are chosen case by case, the criterion becomes a fit. To avoid this, the conditions of a concept must be fixed before searching for its level, with the same rule as in §24. The list for mass is a proposal that does not yet meet that standard.

13.4 Application to Open Questions

Why is n = 6 mass and not n = 5 or n = 7? n = 5 is T⁻² and n = 7 is T⁻³. Both have one open BC and cannot exist in isolation. n = 6 is T³, the first positive level with three closed BCs. It is not that n = 6 is mass. It is the first arity that makes structurally possible an existence that is isolated, historically persistent, ternarily objective, and three-dimensionally present.

What determines the jump between levels? Each level is generated by an exentation: an additional structural differentiation according to Entₙ := Entₙ₋₁ ∧ ExEntₙ₋₁. The jump from T² to T³ goes from “space with reversibility” to “space-time with historical record and ternary objectivity”. It goes from the possibility of length to the possibility of mass.

13.5 Summary Table

Level	n	Closed BCs	Open BCs	Isolable?	First concept made possible (reading)
T¹	2	1	0	Yes	Time (flow)
T⁻¹	3	1	1	No	Frequency
T²	4	2	0	Yes	Length, space
T⁻²	5	2	1	No	Curvature
T³	6	3	0	Yes	Mass
T⁻³	7	3	1	No	Confined color
T⁴	8	4	0	Yes	Information
T⁻⁴	9	4	1	No	No firm reading
T⁻⁵	11	5	1	No	Electromagnetic field (pending)

14. The Hierarchy as a Chain of Inversions

Status: Hypothesis. Mechanism proposed and narrated. Not a formal derivation.

The hierarchy alternates positive and negative levels: T¹, T⁻¹, T², T⁻², T³, T⁻³, … The function e(n) generates that sequence.

When the structure of a level cannot decide its own order, that undecidability becomes the next level, which is the inversion of the previous one.

T¹ → T⁻¹: time generates its alternation. T¹ is homogeneous succession in one direction. It carries within it the possibility of its own negation: the flow could go in any direction. That undecidability is not resolved within T¹. It generates T⁻¹, the level where direction is structurally open.

T⁻¹ → T²: temporal undecidability generates space. T⁻¹ has one open BC: the direction of flow is undecidable. Two temporal phases that cannot be ordered must coexist. That coexistence is simultaneity. Simultaneity is space. Space is the inversion of time. They are logical contraries generated by the same mechanism applied twice.

T² → T⁻²: spatial structure generates curvature. T² is flat simultaneity. The direction of curvature cannot be decided from flat space. It generates T⁻²: curved space, with undecidable curvature direction. Reading: this is the structural origin of gravity. It is not a force imposed on space. It is a consequence of space being unable to decide its direction of curvature.

T⁻² → T³: curved space generates mass. When curvature cannot be resolved within T⁻², it must stabilize into a structure with closed BCs. That structure can exist in isolation and carry its own history. That stabilization is mass. Reading: it is a structural reading of the Higgs mechanism.

T³ → T⁻³: mass generates color. T³ carries within it the possibility of its own variation. Which of its internal degrees is “really” red, green, or blue cannot be determined from T³. That internal undecidability generates T⁻³: color confinement.

T⁻³ → T⁻⁴ → T⁻⁵ → … These steps remain to be narrated with the same mechanism (§29).

T⁰   Generative contradiction — no structure, no order
T¹   Decided succession — mere temporal flow
     ↓ the direction of T¹ is undecidable from within
T⁻¹  Undecidable alternation
     ↓ phases that cannot be ordered must coexist
T²   Flat simultaneity — space as the inversion of time
     ↓ the direction of curvature of T² is undecidable from within
T⁻²  Curved simultaneity — curvature
     ↓ unresolved curvature stabilizes into closed structure
T³   Mass — closed, persistent, with historical record
     ↓ the internal variation of T³ is undecidable from within
T⁻³  Color — variation of mass that cannot close
     ↓ not narrated
T⁻⁴, T⁻⁵, …

Each arrow is the same operation: the undecidability of one level becoming the next level.

Why the alternation. Positive levels (all BCs closed) are stabilized structures, where a prior undecidability found closure. Negative levels (one open BC) are structures where one degree remains unresolved. The proposed cycle has four steps:

A positive, decided level generates undecidability over its own structure.
That undecidability is a negative, undecided level.
The unresolved degree of the negative level stabilizes into a new positive level.
The cycle continues.

A reading on E and B (depends on locating the electromagnetic field in T⁻⁵, pending in §19). A static charge produces an electric field: T⁻⁵ projected onto T², a space with decided structure. A moving charge introduces T⁻¹. Temporal alternation prevents T² from fixing a privileged spatial direction. That spatial undecidability generates T⁻², and T⁻⁵ over T⁻² is the magnetic field. E and B would be the same field at two levels inverse to each other. Maxwell’s equations would be the mathematical expression of that single fact. Gravity and magnetism would both involve T⁻².

The hierarchy is not a list. It is a chain.

15. Geometry of Undecidability

Status: Hypothesis.

15.1 Time and Space as Contraries

Time. Structure of decided succession: first, second, third. There is intrinsic reason to order the phases. T¹ is bare succession.

Space. Structure where order cannot be decided: not-first, not-second. When two phases cannot be ordered, they must coexist. That coexistence is simultaneity. Simultaneity is space.

Time  = decided order     → succession, before/after, direction
Space = undecidable order → simultaneity, coexistence, no direction

They are not two kinds of the same thing. They are logical contraries: presence and absence of decidable order. The emergence of T² from T¹ is not continuation. It is inversion.

15.2 Each Arity Generates a Form of Simultaneity

At each level, undecidability takes a form determined by the arity. That form generates a simultaneity, which is the geometry of the level.

Level	n	Form of undecidability	Generated geometry	Geometric constant
T²	4	Binary direction (no first or second)	Plane, extension, reversibility	√2
T³	6	Ternary orientation (no preferred axis)	Continuous rotation, volume	π
T⁻²	5	Direction of curvature (inward or outward)	Curved manifold, geodesics	Metric tensor
T⁻³	7	Three colors (no preferred color)	Internal color space	SU(3), 8 generators
T⁻⁵	11	Continuous phase (no preferred θ)	Circle of phases	π, as U(1)

T². The undecidability is binary. The constant √2 is the diagonal proportion that emerges when composing two equal, simultaneous directions.
T³. The undecidability is ternary: there is no reason to prefer any of three orientations. The natural structure is continuous rotation. π is the geometric constant of undecidable ternary simultaneity. The ambiguity of T³ makes rotation continuous rather than discrete.
T⁻². The open BC makes the direction of curvature undecidable. The simultaneity is curved. The “constant” is not a number. It is the metric tensor.
T⁻³. The undecidability is one of three colors. The simultaneity is internal: a space of internal degrees of freedom. Its signature is confinement. The internal simultaneity of color cannot be projected into external space without closing the open BC.
T⁻⁵. The undecidability is one of continuous phase. The simultaneity is a circle of equally valid phases: the U(1) gauge symmetry. The photon would be the physical expression of the undecidable simultaneity of 11 phases projected onto T².

Physical geometry is the form that logical undecidability takes when it stabilizes.

15.3 Spirals

Spirals appear in galaxies, hurricanes, shells, DNA, and vortices. T⁻¹ has one open BC: the direction of temporal flow is undecidable. Projected onto T², that undecidable direction must coexist with the two simultaneous spatial directions. The result is the logarithmic spiral: a curve that rotates continuously, which is the geometry of T², while advancing in a direction that is never fixed, which is the open BC of T⁻¹ expressing itself in space.

15.4 Spectroscopy: Intra-Level and Inter-Level Transitions

Status: Hypothesis. Consistent with known data. Not yet a prediction.

Intra-level. A transition that stays within one level (4f→4f in lanthanides) produces a photon whose energy is fixed by the internal phase structure of that level. The emission is narrow: 1–3 nm full width at half maximum. The spiral is spatially undefined but internally precise. There is no external T² boundary to negotiate.
Inter-level. A transition that crosses a boundary (5d→4f) must negotiate the interface between two levels with different arities and BCs. The undecidability of that interface is projected onto the photon as broadening. The width is of the order of 100 nm: two orders of magnitude larger. The spiral acquires a T² container at the boundary of the transition.

The Ce³⁺/Yb³⁺ pair in YAG illustrates this. Ce³⁺ (n = 1) emits via 5d→4f, at about 100 nm. Yb³⁺ (n = 13) emits via 4f→4f, at 1–3 nm. They are mirrors in the electron-hole symmetry of the series, but they point in opposite directions: one leaves T⁻³ in order to exist, the other never needs it.

Principle. The linewidth of an optical transition would indicate whether it crosses an ontological boundary. Narrow line: intra-level. Broad line: inter-level. It is falsifiable where the level assignment is known and the type of transition can be varied.

Caveat. The contrast between narrow f–f lines and broad d–f lines is already known in spectroscopy. To count as a prediction, it must anticipate a case that was not known in advance. Today it is consistent with existing data. It does not predict them.

16. Anchors π, φ, and ρ

Status: Qualitative hypothesis. No derivation of values.

A qualitative distinction between closed BC and open BC is maintained. Three mathematical constants illustrate it. The images are illustrative. They are not derivations.

π, closed-BC anchor. The circle is the only plane curve that returns exactly to its origin. π measures that closure: a relation where the BCs are fully closed and the structure needs no external reference. π is the geometric measure of ternary undecidability (§15.2). It is not a universal mathematical constant that the universe uses. It is what the contradiction becomes when it closes geometrically upon itself in T³.
φ, open-BC anchor. φ = 1 + 1/φ is an infinite continued fraction that always requires one more term. The golden spiral never returns to its origin. φ illustrates a relation where the BCs remain permanently open.
ρ, cubic anchor. ρ³ = ρ + 1 is cubic self-reference, one order above φ (φ² = φ + 1). ρ ≈ 1.3247. It is associated with T⁴. It is argued, not derived. There is an equally valid alternative recursion, with ψ ≈ 1.4656 (ψ³ = ψ² + 1).

Constant	Associated level	Status
π	T² (first 2D level). Becomes undecidable in T³.	Hypothesis
φ	T⁻¹	Hypothesis
ρ	T⁴	Argued, not derived. Alternative: ψ.

Why it is not claimed that the BC count derives these constants.

The recursion a(n) = a(n−1) + a(n−2) assumes two previous terms. It does not explain why two and not one or three.
In T⁴ there are two equally valid recursions (ρ and ψ). Choosing one according to the case being treated is fitting, not deriving.
A count (one open BC) is not a physical magnitude (the cost of an amplitude). That both are written with the digit 1 does not identify them.
Adjusting a premise after seeing that the number does not come out does not count as derivation.

Qualitative hypothesis that is maintained: the pattern of preserved openness. The open-BC anchors would be present in one mixing matrix (PMNS) and absent in another (CKM). The distinction is maintained, not the exact value.

17. The Observer

Status: Open hypothesis.

The observer is not presupposed. It is a level of the hierarchy. To function as an observer, in the minimal physical sense of “something that records a state of the world”, a system needs four conditions.

O1. Capacity to record. Record a state that is not its own. Be affected by the observed without being identical to it. Requires sufficient internal complexity to encode external states.
O2. Non-exhaustion of the observed. The record must not consume or fully determine the observed system. Requires at least one open BC: a gauge freedom that allows choosing the reference without affecting what is measured.
O3. Projection onto spatial structure. The observer projects the recorded state onto T², where phenomena occur. Requires enough closed BCs to couple stably with T².
O4. Not being mass. If it were a positive level, observation and interaction would be inseparable. It must be a field structure: k < 0, at least one open BC.

Which levels meet them. O2 and O4 are verified directly across the whole negative branch: every level with k < 0 has one open BC. O1 and O3 depend on thresholds that are not fixed: how many phases and how many closed BCs each condition demands. With the conditions as currently worded, no negative level is excluded or confirmed.

There are two candidates. T⁻⁴ (n = 9, 4 closed, 1 open) is where the readings registry places the observer’s pivot (§19). T⁻⁵ (n = 11, 5 closed, 1 open) is where the electromagnetic field is placed. In T⁻⁵, O1 has 11 phases, O2 has U(1) gauge freedom, O3 has 5 closed BCs that couple directly to T², and O4 has k < 0.

To close the gap, the thresholds of O1 and O3 must be fixed from a source decided before looking at the result (§24). Until then, “the minimal observer is in T⁻⁵” is an open hypothesis. It is not a result.

Necessary or contingent? The necessity would be conditional:

If a physical system records states of the world
without being those states,
then it operates at the minimal observer level or higher.

If the gap is closed in favor of T⁻⁵, this would explain why observation in quantum mechanics is always associated with electromagnetic interaction.

Hierarchy of observers. Status: Reading. Each level would add observational capacity while keeping the open BC that preserves gauge freedom.

T⁻⁵ (electromagnetic): records binary states, photon detected or not. Minimal observer.
T⁻⁶ (weak, n = 13): could record flavor states.
T⁻⁸ (n = 17): would record states that require hyperspatial structure.

Part V — Operators

18. Reading Rule and BCC/BCO Pattern

Status: Rule of method.

n is the arity. Tⁿ is the real name of the operator. It is the only identifier that matters. The mnemonics (MASS, GEO, SIM, DIFF, …) neither define nor describe Tⁿ. They are connections to ordinary language, placed so that a human can remember which row is being discussed.
The correct formula is “the behavior is T⁻¹”. It is not “the behavior is alternation”. When a description says that something “resonates with” an external concept, that relation is reference, not derivation. Treating the mnemonic as if it were the thing, and going out to look for the complete external definition for Tⁿ, is the error this rule prevents.
Each Tⁿ has a family of logical operators of its own. It is the “n-ary logic” of that level. DIFF, ALT, VARTIME, ORD, and ACC are not five names for T⁻¹. They are internal distinctions, separately justifiable, within the logic of arity 3. The complete family is recorded.
An operator depends only on what is available at levels n−1 and earlier. It is backward chaining (§8.2) applied to the labels. It remains to be applied row by row.
A Tᵏ can house several terms. What demands a single choice is the dialogical formula, not the level (CHOICE, §20; reading coherence, §21.5).

There are three classes of operators. The logical ones are those of each Tⁿ. The dialogical ones relate operators or arities (§20). The compound ones are concrete cases of combination (§21).

BCC / BCO pattern. Status: Hypothesis. An open boundary has an indeterminate term relative to the act of invocation (§7.2). An act of invocation requires someone to invoke. That connects with an observer.

Tⁿ, BCC (closed). Tends to describe the “empty” or isolable version of a dimensional concept. It can exist, and in principle be measured, without reference to an external act. Examples: FLOW and TIME, SPACE2, MASS and COL, INF.
T⁻ⁿ, BCO (open). Tends to describe the active, interacting, visible version of the same concept. It manifests only in relation to an observer or to another structure. Examples: VARTIME and ACC, GEO, VAR-COL, OBS.

It is not an exclusive rule. It is a pattern to be verified row by row.

19. Registry by Level

Status: Reading. It is a registry, not evaluated.

n	Level	Mnemonics
1	T⁰	PLK
2	T¹	IND, FIN, DIR, BCC, FLOW, TIME, DIM1
3	T⁻¹	DIFF, ALT, VARTIME, ORD, ACC
4	T²	SPACE2, SIM, DIM2
5	T⁻²	EXP, ENTR, 5TH-F
6	T³	MASS, GEN3, SPACE3, COL, DIM3
7	T⁻³	JERK, VAR-MASS, VAR-COL, GEO
8	T⁴	INF, BYTE, DIM4
9	T⁻⁴	SST-9, OBS, ENT-R, BIT
10	T⁵	KAL-KL, DIM5
11	T⁻⁵	REG, RAD-V, PART-KAL-KL
12	T⁶	CAL-YAU, DIM6
13	T⁻⁶	SING, MOD-V, PART-CAL-YAU
14	T⁷	G2-HOL

The table is cut off here. T⁻⁷, T⁸, T⁻⁸, T⁹, and T⁻⁹ are left out for now.

19.1 Descriptions

T⁰. PLK (speculative). The Planck time. T⁰ is the only level without a sign, prior to the recursion opening in a positive or negative direction. It resonates with the minimal pulse of the axiom.
T¹. IND: indistinguishability, the Tf are indistinguishable from one another. FIN: finitude, start→end. DIR: direction. BCC: closed boundary. FLOW: temporal flow. TIME: objective time. DIM1: generic reading.
T⁻¹. DIFF: distinguishability. ALT: alternation. VARTIME: variance, the time that varies. ORD: oriented finitude, start→end ≠ end→start. ACC: temporal acceleration.
T². SPACE2: two-dimensional space or spatial simultaneity. It is the minimum structure for two points to coexist without temporal relation between them. SIM: simultaneity. DIM2: generic reading.
T⁻². EXP: expansion, first order of spatial variation. Resonates with the Hubble parameter. ENTR: quantum entanglement, the particle of two-dimensional space: two correlated points without temporal relation between them. 5TH-F: resonates with the search for a fifth force (speculative).
T³. MASS: mass. GEN3: resonates with the three generations of fermions (speculative). SPACE3: empty three-dimensional space, the “full” form of the physical 3-dimensional space. COL: color, a structure of pure ternary symmetry (SU(3)). DIM3: generic reading. T³ admits two mutually exclusive readings (§21.1).
T⁻³. JERK: third order of spatial derivation, pairs with SPACE3 (hypothesis). VAR-MASS: variation or experience of mass, mass in its objective form, the one that can be experienced. VAR-COL: variation or experience of color, chromodynamics in its active form. VAR-MASS and VAR-COL inherit the mutual exclusion of MASS and COL. GEO: geodesic curves. Curvature with real physical effects, such as geodesic deviation, exists only in gravity of 3+1 dimensions or more. In 2+1 general relativity has no local degrees of freedom. That is why GEO cannot be in T⁻².
T⁴. INF: information. DIM4: generic reading. BYTE: resonates with the standard unit of digital information (speculative). T⁴ admits two mutually exclusive readings (§21.3).
T⁻⁴. SST-9: resonates with the 9 spatial dimensions of superstring theory (speculative). OBS: observer pivot. ENT-R: entropy production rate, dS/dt, non-equilibrium thermodynamics. It is a variation of INF, with the same status as EXP with respect to SPACE2. BIT: the minimal unit of Shannon information, read as the particle of INF, the discrete quantum in which information manifests. It cannot be in T⁻¹. It would be a particle of information before information itself exists.
T⁵. KAL-KL: resonates with Kaluza–Klein theory. It is an evocation, not a derivation. DIM5: generic reading.
T⁻⁵. REG: electromagnetic field U(1). RAD-V: radion velocity, variation of KAL-KL (doubly speculative). PART-KAL-KL: the Kaluza–Klein towers, the particle and not the rate of change.
T⁶. CAL-YAU: T³·T³ = T⁶, the level with enough budget for the two readings of T³ to coexist (§21.4). DIM6: generic reading.
T⁻⁶. SING: singularity, weak field SU(2). MOD-V: moduli velocity, in principle observable. PART-CAL-YAU: the branes that wrap cycles of a Calabi–Yau manifold.
T⁷. G2-HOL: G₂ holonomy manifolds, used to compactify M-theory (11D) while preserving N=1 supersymmetry (speculative).

19.2 Registry Notes

DIM, generic reading. Positive Tⁿ read as “n dimensions”, asserting nothing beyond budget(Tⁿ) = n. It does not compete with INF, KAL-KL, or CAL-YAU. It does not claim that this dimension is real physical space.
SPACE2 and SPACE3. The dimensional ladder puts “space” in T², with only 2 dimensions. Physical space has 3. It is disambiguated into two operators: SPACE2, minimal spatial simultaneity, and SPACE3, empty 3-dimensional space.
TIME in T¹. The arithmetic T¹·T² = T³ reads TIME·SPACE2 = MASS or COL. TIME must be on the closed side, together with FLOW. Both are the “empty” form of time. That is why the internal variance of T⁻¹ is called VARTIME: it must not clash with VAR, the dialogical operator (§20).
ENTR in T⁻². It is the third proposed location for entanglement: ORD in T⁻¹, SPACE2 in T², and ENTR in T⁻². They are not contradictory. They could be three aspects of the same phenomenon.
OBS in T⁻⁴. The observer pivot is separated from the electromagnetic field (REG, T⁻⁵). The degree of speculation of the mnemonics grows notably from T⁻⁴ and T⁵ onward. It is a recorded subjective observation, not a derivation.
M-theory. It was ruled out for T⁻⁵ (n = 11) due to a mismatch with the dimensional ladder of the positive branch.
REG and SING. They are incorporated without yet having passed through the reading rule (§18).
Pending. Evaluate all rows against the reading rule. No mnemonic has been evaluated.
Tension with §24. VAR-COL in T⁻³ and COL in T³ are in tension with the minimum rule applied to SU(3). See §24.3.

20. Dialogical Operators

Status: Definitions. Under construction.

A dialogical operator relates two arities, or one arity with itself. The operators of §19 describe a single level. Priority rule: the meaning of each dialogical operator is fixed by looking at the structure (§7, §18). It is never fixed by factoring a measured constant to see what fits. If they are tested against a real value, that comes afterward.

PRODUCT, p × q. Combines two arities into a compound one without loss of identity. The factorization of the product is unique, so the original arities remain recoverable. It has two scopes.

Between arities. A₆ = A₂ × A₃: arity 6 (T³) as the product of arity 2 (T¹) and arity 3 (T⁻¹).
Between levels. Tᵃ · Tᵇ = T⁽ᵃ⁺ᵇ⁾, the exponent law of §12.
Within a level. 3² = 3 × 3 are two logical operators of the 3-ary logic interacting by PRODUCT, for example DIFF × ALT. 3³ = DIFF × ALT × VARTIME.

The scopes must not be confused. They share the operation, not the domain.

FUSION, p + q. Combines two arities into a new identity that does not retain its components separately. The sum is not uniquely invertible. A result of 6 could have come from 1+5, 2+4, or 3+3, and that information is lost. It is named for what it does, not for what it lacks.

INV-QUOT, inverted quotient: 1/Tⁿ = T⁻ⁿ. Takes a closed structure (BCC) and exposes its open counterpart (BCO). It leads from “isolable, empty” to “interacting, visible to an observer”. It is not an interpretation. It is the notation T⁻ⁿ := 1/Tⁿ. It is the mechanism behind the BCC/BCO pattern of §18.

Tⁿ (empty)	INV-QUOT →	T⁻ⁿ (empirical)
TIME, FLOW		VARTIME, ACC
SPACE2		EXP, ENTR
SPACE3, MASS, COL		GEO, VAR-MASS, VAR-COL
INF		OBS, ENT-R
KAL-KL		RAD-V
CAL-YAU		MOD-V

The reading holds across the six pairs reviewed. It is a strong observation, not a proven law. Not having found a counterexample is not the same as holding for all cases.

INT-QUOT, internal quotient: Tᵃ / Tᵇ, of equal sign. Relates two levels of the same branch by subtraction of exponents. T⁶/T² = T⁴. It does not cross from closed to open or connect with the observer. First case: INT-QUOT(INF, SPACE2) = T⁴/T² = T² (§21.6). It is not to be confused with INV-QUOT, although they share the word “quotient”.

PIVOT, T^((a+c)/2). Given a pair Tᵃ and Tᶜ, the level exactly halfway, when the average is an integer, acts as a fulcrum. It is direct arithmetic. Example: T¹ and T³ have T² as pivot. It is distinct from PRODUCT, FUSION, and CONJ. Pending: verify more triples, for example whether T⁻² is the pivot of T⁻¹ and T⁻³, and what it means for a triple not to have an integer pivot.

CONJ, Tᵃ ⊕ Tᵇ, without collapsing. Combines two levels without fusing them. They remain as a tuple, each with its branch and sign intact. It is the counterpart of PRODUCT. PRODUCT fuses while preserving numerical identity. CONJ preserves identity without fusing. It is needed to record the real space-time of relativity (§21.2): FUSION and PRODUCT erase the opposite sign between space and time demanded by the Minkowski metric.

CHOICE. It is not a new operator. It is INV-QUOT applied to a level with mutually exclusive readings (T³: ARXE-ST or COL. T⁴: SPACE2² or (MASS or COL) + TIME. See §21). The connection with the observer that INV-QUOT already produces is, at the same time, the act of choosing which reading manifests. It is distinct from VAR (VARTIME, EXP, VAR-MASS, …), which is continuous variation within an already chosen reading. CHOICE is the prior moment. Resonance: it resembles the measurement problem in quantum mechanics. The resemblance is noted. It is not derived from there. The choice propagates (§21.5).

Pending. Develop INT-QUOT with the same logical necessity as the others. Check whether there is a fourth operation, subtraction p−q, or whether FUSION with sign covers it. Test the pattern against GEN3, SST-9, KAL-KL, CAL-YAU, and G2-HOL.

21. Compound Operators

Status: Under construction.

These are cases where a level Tⁿ is obtained by combining two others. In particular, the cases where that combination admits more than one partition, with mutually exclusive readings. The operation is Tᵃ·Tᵇ = T⁽ᵃ⁺ᵇ⁾: PRODUCT over the exponents k.

21.1 T³: Two Mutually Exclusive Readings

Partition	Reading
1 + 2 (PRODUCT, within the closed branch)	ARXE-ST: TIME·SPACE2, ArXe’s own space-time
Combinatorial, not by sum	COL: pure ternary symmetry

Both are valid. Neither derives from the other. They cannot manifest at the same time. If T³ is ARXE-ST, it is not at the same time the pure color symmetry. The logical form reproduces the tension between general relativity and QCD: two separately confirmed theories, with no common framework today. ARXE-ST is not the space-time of general relativity. It is ArXe’s own reading and does not claim to describe GR.

21.2 GR-ST: T³ ⊕ T⁻¹, by CONJ

Neither PRODUCT nor FUSION preserves the opposite sign between space and time demanded by the Minkowski metric. CONJ does: T³ ⊕ T⁻¹ are 3 closed dimensions (space) alongside 1 open one of opposite sign (time), without collapsing. It is more than a resonance. The opposite sign of the metric and the opposite sign between closed and open branch are both verifiable algebraic facts. Pending: why 3 closed and exactly 1 open. Or it is left to what physics has established, three observed spatial dimensions, without ArXe deriving it.

21.3 T⁴: Two Partitions

Partition	Reading
2 + 2	SPACE2²: two two-dimensional spaces
3 + 1	(MASS or COL, with its own exclusion from T³) + TIME

The 2+2 partition resonates with the two transverse polarization planes of an electromagnetic wave. The 3+1 resonates with the corpuscular aspect: mass, localization, temporal persistence. It is a reading of wave-particle duality. The arithmetic, that T⁴ admits a double partition, is not speculative. What each partition represents is. The relation with INF and BYTE is not established.

21.4 T⁶ = T³·T³

It is not a third exclusive partition. It is the level with enough budget for the two readings of T³ to coexist. It coincides with the fact that Calabi–Yau manifolds, of 6 real dimensions, are in string theory the standard mechanism by which general relativity and gauge forces, color included, attempt to coexist.

Emerging pattern, to be confirmed. Partitions without an even split (T³) tend to give exclusion. T⁶, which brings together two copies of T³, tends to resolve it. T⁴ complicates the picture: it has a possible even split (2+2) and also exclusion. There is no general rule. T⁵, T⁷, and more cases are needed.

21.5 Reading Coherence

Status: Postulate.

When CHOICE resolves an ambiguity at a level with readings that were exclusive at another level, that choice propagates. It is not punctual. Every appearance of the same level within a single derivation, and every compound level built from it, uses the same reading. Mixing exclusive readings in a derivation is a contradiction. By the axiom, what is contradictory does not act, and what does not act does not exist objectively. It is excluded without a separate prohibition.

Example: VAR-MASS and VAR-COL (T⁻³) inherit the exclusion of MASS and COL (T³). If a derivation chooses T³ = COL, it must read T⁻³ = VAR-COL, never VAR-MASS. The same holds for T⁶ and for any compound level that depends on T³.

The rule excludes concrete combinations. It does not rescue any result.

21.6 Cases with Resonance

Status: Reading. The arithmetic is verifiable. What each composition represents is not.

Composition	Operation	Resonates with
INF (T⁴) / SPACE2 (T²) = T²	INT-QUOT	Holographic principle: information scales with area, not volume (Bekenstein–Hawking). Information divided by two-dimensional space gives two-dimensional space again, not a volume.
TIME (T¹) ⊕ MASS (T³)	CONJ	Proper time of special relativity: the internal clock of an object with mass.
OBS (T⁻⁴) ⊕ REG (T⁻⁵)	CONJ	Quantum detection, photoelectric effect: the field needs something to absorb it in order to manifest as a discrete event.
CAL-YAU (T⁶) / MASS (T³) = T³	INT-QUOT	Compactification radius relative to the observable dimensions.
BYTE (T⁴) ⊕ ENT-R (T⁻⁴)	CONJ	Landauer’s principle: erasing a bit has a minimum thermodynamic cost.
SPACE2 (T²) · MASS (T³) = T⁵ = KAL-KL	PRODUCT	Kaluza–Klein: unifying gravity and electromagnetism by adding a dimension.
FLOW (T¹) ⊕ GEO (T⁻³)	CONJ	Gravitational time dilation: a clock near a mass runs slower.
VARTIME (T⁻¹) ⊕ EXP (T⁻²)	CONJ	Cosmological time: in the Friedmann–Robertson–Walker metric, time and expansion are coordinates of the same object.

Pending. Review T⁵ and T⁷ under the same criterion: whether they admit multiple partitions and whether they are exclusive. Decide whether the partner of T⁴ relates to INF and BYTE.

21.7 Constants by Partition

Status: Hypothesis. The arithmetic is Verified. The admissibility criteria are Open.

Rule. A level Tᴺ decomposes as Tᵃ + Tᵇ with a + b = N. It is required that the FUSION and the PRODUCT of the two levels coincide, taking xᵏ as the weight of level Tᵏ:

xᵃ + xᵇ = xᵃ·xᵇ = xᴺ    ⇔    1 + x^(b−a) = xᵇ    ⇔    (xᵃ − 1)(xᵇ − 1) = 1

x is read as a per-level factor, not as T/Tp. In Planck units x = 1 would give 1 + 1 = 1, which is false. The reading “FUSION = PRODUCT” is a hypothesis. It connects with §20: PRODUCT preserves the recoverability of the factors and FUSION loses it. The constant would be the weight where both coincide.

Result (arithmetic verified by calculation).

Level	Partition	x	Reading in §21
T³	1+2	φ = 1.6180	ARXE-ST (TIME·SPACE2)
T⁴	2+2	√2 = 1.4142	SPACE2²
T⁴	3+1	ψ = 1.4656	(MASS or COL) + TIME
T⁵	2+3	ρ = 1.3247	pending
T⁵	1+4	1.3803	pending
T⁶	1+5	ρ (factor of the polynomial)	pending
T⁶	2+4	√φ = 1.2720	pending
T⁶	3+3	2^(1/3) = 1.2599	pending

Equal partitions (a = b) always give 2^(1/a). They are trivial and provide no evidence.

Three structural coincidences. Status: Reading.

T³ and T⁴. The partitions coincide with those of §21.1 and §21.3, identified earlier by another route. φ comes from the 1+2 partition (ARXE-ST). The COL reading (combinatorial, not by sum) does not produce it. This is coherent with the exclusion of §21.5.
ψ in T⁴. §16 leaves ρ and ψ as alternatives in T⁴. The rule puts ψ in T⁴ (1+3) and sends ρ

22. Self-Dialogue of an Arity

Status: Open. Just begun.

It consists of taking an arity, or a small mixture, and systematically generating all possible expressions by combining it with itself using PRODUCT (×) and FUSION (+), in different orders and groupings. The result is not a number. It is a complete tree of routes.

2×2×2 + 2 + 2×2 + 2 = 8 + 2 + 4 + 2 = 16 = 2⁴
3×3×3 + 3 + 2 + 2×3 = 27 + 3 + 2 + 6 = 38

In the first example, seven “2”s combined reach 2⁴ by a different path.

What it completes. CHOICE (§20) says that a level with several exclusive readings must choose one. But all the readings were never enumerated before choosing. Self-dialogue would be the method to generate that complete set of candidates. Open: whether the generated set is, literally, the domain over which CHOICE chooses.

Two resonances. Status: Reading.

Perturbative expansion in quantum field theory. An amplitude is the sum of all Feynman diagrams, each built by combining the same coupling in different ways. Chain diagrams resemble the product. Branch or loop diagrams resemble the sum.
Integer complexity. It is a real area of number theory. It asks how many ways there are to reach a number by combining a base unit using only + and ×, and which is the most economical. It has published sequences. It is already mapped terrain: consult it before reinventing.

Two questions not yet distinguished. (1) How many distinct routes reach the same value? (2) What set of distinct values is reached from a given arity with up to n symbols?

The tree of + and × does not distinguish order. The ordering of phases (§9) does count it.

Pending. Decide the focus between the two questions. Review the integer complexity literature. Connect formally with CHOICE.

23. Boolean Logic within the Hierarchy

Status: Partly open.

NOT, AND, and OR are not located by arity. They are already in §3. AND builds each Entₙ. NOT and OR build each ExEntₙ. They are the engine that builds all the levels, not a behavior of one level. Treating them as operators to be located confuses the engine with a behavior that lives in a single level.

What is maintained is the correspondence AND ↔ PRODUCT and OR ↔ FUSION. It is not by external resonance. The two pairs share the root (the ∧ and ∨ of §3) and the underlying property: preserving or losing the recoverability of the origin.

Boolean operator	Input arity	Resonates with	Why
XOR	2	DIFF (T⁻¹)	It does not combine. It asks about distinction. It is true when the inputs differ.
NAND, NOR	2	NOT∘AND, NOT∘OR	They are compounds: AND or OR with NOT applied afterward. They need no entry of their own.
IMP-MAT	2, compound	NOT∘OR	A→B ≡ ¬A ∨ B. It is compound, not primitive.
IMP-NEC	2 + necessity	Closed branch (BCC)	True in every possible scenario, without depending on the act of invocation.
IMP-REL	3, with axis	Structure pending	It requires a third element: the content shared between A and B.
IMP-CTF	2 + unrealized alternative	CHOICE	Evaluating “what would have happened” is asking about the branch not chosen.

IMP-REL. Two distinct ternary structures were identified. (A) Pure symmetry: three interchangeable elements, with no privilege. It is COL in T³. (B) With axis: two identical elements plus a third that acts as a pivot, like the quantization axis in spin. (B) was left without a location. IMP-REL has the form of (B): two terms, A and B, plus an axis, the relevance that connects them. It is a reasonable hypothesis. It is not a confirmation. It is the least firm row.

Pending. Develop IMP-REL. Check whether there is another Boolean combination of Entₙ and ExEntₙ that has not been noticed.

ArXe Foundations and Framework

ArXe — A Dynamic Ontology with a Logical Basis

Foundations Documentation

How to Read This Document

Index

Part I — Starting Point

1. What ArXe Is

2. The Axiom

3. The Recursion of Exentation

3.1 Definition

3.2 Self-Similarity Theorem

3.3 Full Unfolding

4. Base Logic

4.1 Two Negators

4.2 Essence and Existence

4.3 The Term “Actual”

4.4 Temporal Threshold Semantics

4.5 Classical Objections

Part II — The Hierarchy of Levels

5. Arity, Terms, Phases, Valence, Distinguishability

Term

Arity

Valence

5.1 Nomenclature of the First Levels

6. The Tᵏ Levels: Indexing

6.1 The Mapping Functions

6.2 The T⁰ Level

7. Boundary Conditions of Possibility

7.1 Sign Postulate

7.2 Boundary, Finitude

7.3 BC Count per Level

7.4 Corollary: Undecidability Sets

7.5 Scope of the Count

8. Temporal Threshold, Act, and Phase Space

8.1 Two Readings of the Time an Arity Needs

8.2 The Act (T)

Note to validate:

The Act as ‘Tempus Instans’ has no existence, it has essence

8.3 Isolated Example: First and Second

8.4 Phase Space

Part III — Phase Ordering and Probabilistic Structure

9. The Ordering of Phases and Terms

9.1 What an Ordering Is

9.2 The Factored Form Reveals the Ordering

9.3 Gradation from T⁰ to T⁻¹

9.4 Probability

9.5 Behavior and Phenomenon

9.6 Axiomatic Choices

9.7 Trivial Axiomatic Choices

9.8 Non-Trivial Axiomatic Choices

9.9 Objective Existence

9.10 Distinction, Ordering, and Valences

9.10.1 Vocabulary

9.10.2 Classification of Modes

9.10.3 Distinction and Sequence

9.10.4 First Appearance of Each Logical Valence

9.10.5 Reading: Past, Present, and Future

9.10.6 Pending Items of This Section

9.11 Copy, Alternative, and Probability

9.11.1 Two Operations

9.11.2 Probability

9.11.3 Relation among the Three

9.11.4 Pending Items

9.12 The Two Orderings of T² and the Arrow of Time

9.13 The Lanthanides as Physical Realization

10. Three Principles, One Structure

10.1 Probabilistic Freedom

10.2 Relative Logical Necessity

10.3 The Scale of the n-ary Logics

10.4 A Single Structure

11. Physics as Statistical Manifestation

11.1 What It Means for the Negative Levels

11.2 Positive and Negative Levels

Part IV — Dimensional Framework, Geometry, and Observer

12. The Dimensional Framework

12.1 The Dimensional Rule

12.2 Three Structural Equivalences

12.3 The Planck Units

13. Minimal Conditions of Possibility

13.1 Minimal Conditions for Mass