ArXe Theory V4.2: The Active Unfolding of Primordial Contradiction

Foundations, Dimensional Framework, and the Grammar of Physical Constants

Diego Luis Tentor — ArXe Research — 2026
Version 4.2 — March 2026
Supersedes: arxe_core_V4_1_en.md (v4.1, March 2026)
Status: Living theoretical framework
License: CC BY-SA 4.0
Source corpus: github.com/diego-tentor/ArXe_V3

how complexity emerges from the impossibility of nothingness,
how indecidability generates space, open boundaries generate fields,
and primes are the irreducible routes through which distinction sustains itself.”

What ArXe Is

ArXe is a Dynamic Logical-Base Ontology in which physical reality emerges as a recursive process of resolving a primordial logical contradiction — without presupposing any preexisting mathematical or physical entities.

Reality is not a mathematical structure built from constants. Reality is the active process of resolving ¬(). There are no numbers, laws, or particles waiting to be discovered. There is only the unfolding of distinction.

What we call physical laws and constants are the dynamic traces of this exentation process. Mathematics does not dictate reality; mathematics is the grammar of the unfolding of distinction.

ArXe is not mathematical realism. It is not Platonism. It does not presuppose the existence of numbers, laws, or particles as fundamental entities. ArXe is an ontology of inviability: the universe is the active process by which logical nothingness — being inviable — escapes into structure. The Planck scale is not a unit of measurement of something; it is the metric of the instability of nothingness.

What is New in V4

V3 established the ontological foundation: the axiom, the exentation hierarchy, BC algebra, prime encoding, and the emergence of π from ternary ambiguity.

V4 completes the bridge to empirical physics by adding:

PLO v2 — the chronological reading method with exact values and delta analysis
The complete dimensional framework — how any SI dimension maps to T^n, and the derivation of Planck units as the natural ArXe scale
The chain from axiom to constants — dimensional constants → dimensionless constants, with the full passage formalized
φ = BC-open analog, π = BC-closed analog — formally derived as Boundary Condition Anchors, not postulated
The Naturality-by-Digit Principle — what the digits of a constant reveal about the unfolding vs. the observer’s intervention
The observer’s logical position — the four strata and the pivot at prime 11
The extended unified lexicon — 25 prime operators confirmed by two independent programs
Layer C / Layer D distinction — the formal difference between compressed natural structure and axiomatic human choice

V4.1 adds:

Philosophical reframing of §1–§6 as ontology of inviability
Corpus updated to ~119 constants (extended PLO v2 corpus)
Naturality-by-Digit table expanded

V4.2 adds:

§3.5 — The Principle of Phase Ordering: probabilistic structure of each level
§3.6 — Three Principles, One Structure: probabilistic freedom, relative logical necessity, scale of n-ary logics
§3.7 — Physics as Statistical Manifestation: physical reality as dominant statistical layer of aridity
§4.4 — The Hierarchy as Chain of Inversions: structural derivation of alternation
§8.4 — Why Time and Space Are Contraries: indecidability as the origin of geometry
§9.8 — The Observer at T⁻⁵: structural necessity, not contingent placement

Part I: Foundations

1. The Fundamental Axiom

¬() ≜ Tf ≃ Tp

Where:

¬() — the logical act of negation applied to nothing: nothingness, being logically inviable, cannot simply be
Tf — conceptual fundamental time: the minimal pulse generated by that inviability
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

This axiom does not establish the identity between logic and physics. It describes the event in which nothingness, being logically inviable, escapes into a fundamental pulse of time that we perceive as the Planck scale. Tp is not a measure of something — it is the metric of the instability of nothingness.

The fundamental physical act is analogous to logical contradiction:

“This precise instant, in its fundamental physical expression, is absolutely actual, is not possible, cannot be verified or demonstrated, does not exist nor is it true.”

This contradiction is not a problem to solve. It is the generative engine of all reality. The universe does not resolve ¬() — it perpetually escapes it, and that escape is what we call time, space, and structure.

Philosophical note — ArXe as ontology of inviability:

ArXe does not presuppose the existence of numbers, laws, or particles as fundamental entities. It is an ontology of inviability: the universe is the active process by which logical nothingness, being inviable, escapes into structure. Numbers are not the building blocks of reality — they are the irreducible routes through which distinction sustains itself without collapsing. What we call mathematics is the grammar of that escape, not its cause.

This distinguishes ArXe from:

Mathematical realism / Platonism: which presupposes mathematical entities as fundamental. In ArXe, no entities are presupposed — only the impossibility of their absence.
Physicalism: which presupposes physical substance. In ArXe, there is no substance — only the recursive unfolding of distinction.
Idealism: which presupposes a mind or concept as primary. In ArXe, no observer is required for the contradiction ¬() to generate structure — the observer emerges at T⁻⁵ (prime 11), long after the process has begun.

2. The Exentation Hierarchy

The levels n are not dimensions of space. They are Degrees of Freedom of the Contradiction — each step a new layer of processing necessary so that the system does not collapse onto its own paradox.

Entification (conjunction — being):

Entₙ := Entₙ₋₁ ∧ ExEntₙ₋₁

Exentation (negation → disjunction — existing beyond):

ExEntₙ := ¬(Entₙ₋₁ ∧ ExEntₙ₋₁) ≡ ¬Entₙ₋₁ ∨ ¬ExEntₙ₋₁

Initial condition:

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

Each jump in n is not a place in space — it is a new layer of processing that the contradiction requires to avoid self-dissolution. The n-arity of each level determines the number of temporal phases, the logical complexity, and the phenomenological structure: what the system can do at that depth without collapsing.

3. The Mapping Functions

Forward mapping:

e(n) = { 0                  if n = 1
        { (-1)ⁿ · ⌊n/2⌋    if n > 1

Generated sequence: {0, 1, −1, 2, −2, 3, −3, 4, −4, …}

Inverse mapping:

n(k) = { 1        if k = 0
        { 2k       if k > 0
        { -2k + 1  if k < 0

e and n establish a bijection between ℕ and ℤ. This is not a choice — it is the unique bijection that preserves the alternating structure of the exentation sequence.

3.5 The Principle of Phase Ordering

Every level T^k has a number of phases n. Those phases can be ordered in different ways. Each distinct ordering is a distinct phenomenon. Not all orderings are equally likely — earlier orderings constrain later ones. And the whole story begins at T⁰, where there are no orderings at all.

Part 1: What a phase ordering is

At level T⁻¹, there are 3 phases. Call them beginning, middle, end.

These three phases can be arranged in 3! = 6 distinct sequences:

1. beginning → middle → end
2. beginning → end → middle
3. middle → beginning → end
4. middle → end → beginning
5. end → beginning → middle
6. end → middle → beginning

Each sequence is a different phenomenon — not the same phenomenon seen from a different angle, but a genuinely different event with different structure. Ordering 1 is the familiar “time flows forward”. Ordering 6 is not “time flowing backward” — it is a different kind of process altogether, one that begins at what we would call the end and arrives at what we would call the beginning. It is not a reversal of ordering 1. It is its own thing.

The key point: The labels “beginning”, “middle”, “end” are assigned by T³ (the level of historical memory). At the level of T⁻¹ itself, the three phases have no intrinsic labels. They are just three phases. Any sequence of those three phases is structurally valid.

Part 2: The factored form reveals the ordering

Consider T²×T² versus T⁴.

Objectively — in terms of what exists in the world — they are the same structure. T²×T² does not describe a different object than T⁴. A vortex is a vortex whether we write it as T²×T² or T⁴.

But T²×T² says more than T⁴. It reveals which ordering of T²’s phases is active: a spatial process that applies its structure and then inverts it — going and returning, with the origin remaining present as a real place to return to. T⁴ collapses this information. T²×T² preserves it.

This is analogous to Feynman diagrams. A Feynman diagram does not describe a real trajectory of a particle through all possible paths. It is a tool that reveals the structure of the process — which terms are active, how they combine — in a way that the final number alone does not. The diagram is hermeneutically richer than the number, even though both refer to the same physical event.

Rule: Use the factored form (T²×T²) when the ordering matters for understanding the phenomenon. Use the collapsed form (T⁴) only when the ordering is irrelevant to what you are describing. Collapsing prematurely loses information.

Part 3: The gradation from T⁰ to T⁻¹ — where ordering begins

The principle does not apply uniformly to all levels. It applies differently at each level, and understanding why requires following the gradation from T⁰.

T⁰ — No ordering. No existence. Pure indetermination.

T⁰ is the generative contradiction. It has no phases — not even one. There is nothing to order, nothing to choose, nothing that exists. T⁰ is not “determined” in the sense of having a fixed outcome — it is prior to any distinction between determined and undetermined. It is the contradiction itself: neither true nor false, neither existent nor non-existent.

T⁰ is not a level in the usual sense. It is the origin from which levels emerge — but it does not itself participate in the ordering of phases. It is indetermination pure, before any structure that could be probable or improbable.

T⁰: no phases → no orderings → no variance → no existence
    not determined — indeterminate in the absolute sense

T¹ — Mere temporal succession. Trivial, not determined.

T¹ is homogeneous positive time — the bare succession of one moment after another, in one direction, with no alternation and no variance. It has 2 phases, but their relationship is trivial: the second simply follows the first. Not because a law forces it — but because no alternative is structurally possible.

T¹ is not probabilistic. But it is also not “determined” in the sense of a law imposing an outcome. It is trivial: asking “what is the probability of T¹’s ordering?” has no meaningful answer, for the same reason that asking “what is the probability that succession follows succession?” has no meaningful answer. There is simply no other option.

T¹: 2 phases → mere succession → no variance → trivial
    not determined by a law — trivial because structurally inevitable

Nothing can go differently at T¹ — not because it is forbidden, but because the concept of “differently” has not yet emerged.

T⁻¹ — Six orderings. The first real conditionality. Variance begins here.

T⁻¹ has 3 phases and 1 open boundary condition. The open BC is crucial: it means the direction of the process is ontologically undecidable — there is no intrinsic reason to prefer any ordering over any other.

For the first time in the hierarchy, something can happen in more than one way.

T⁻¹: 3 phases → 3! = 6 orderings → variance emerges → conditionality begins

This is where probability enters the framework — not as an external mathematical tool imposed on physics, but as a structural consequence of the open BC of T⁻¹. The open BC means: no intrinsic preference among orderings. Six orderings, no structural reason to choose one over the others. This is the ontological foundation of probability in ArXe.

Variance is born at T⁻¹. Before T⁻¹, the framework is deterministic (T⁰ and T¹ have no alternatives). From T⁻¹ onward, the framework is irreducibly probabilistic.

Part 4: Conditionality — why not all orderings are equally likely

The open BC of T⁻¹ establishes that no ordering is privileged a priori — before anything has happened. But the universe does not manifest from nothing each time. It has a history.

Once a specific ordering at T⁻¹ has manifested — once T³ has recorded it as a historical fact — that fact constrains what can follow. Not all of the 6 orderings remain equally accessible. Some are compatible with what has already occurred. Others are not.

This is conditional necessity — the only kind of necessity that exists in the framework after T⁰.

Formally:

P(ordering B occurs) ≠ P(ordering B | ordering A already occurred)

Some orderings B are directly incompatible with A — not because the structure forbids them in the abstract, but because A has already closed that possibility. The constraint is historical, not logical. It comes from T³ (the level that records history) acting on the possibilities of T⁻¹.

The necessity in ArXe is always conditional on previous choices — except at T⁰, which has no previous choices.

This means: nothing in the framework after T⁰ is absolutely necessary. Everything is necessary given what has already been decided. The universe accumulates constraints as it unfolds. Early decisions restrict later possibilities. The space of compatible orderings narrows as history grows.

Part 5: The probability space at each level

Given conditionality, the probability of each ordering at each level is not 1/n! uniformly. It is:

P(ordering x at level T^k) = 1 / (orderings compatible with accumulated history)

When history is minimal (early universe, simple systems), the compatible orderings are many and probabilities are nearly uniform. When history is rich (complex systems, late universe), the compatible orderings are few and some orderings become highly probable while others become nearly inaccessible.

This gives the framework a natural explanation for why certain phenomena are common and others are rare:

Level	Total orderings	Relative frequency
T⁻¹ (time)	6	Very common
T² (space)	24	Common
T³ (mass)	720	Less common
T⁻³ (color)	5,040	Rare in isolation
T⁻⁵ (EM field)	39,916,800	Rare in pure form

Space is more abundant than mass not because space is “simpler” in some vague sense — but because the number of compatible orderings at T² is 24, while at T³ it is 720. Mass requires 30 times more specific coordination of phases than a spatial structure does. The universe produces space more readily than mass because there are more ways to be space than to be mass.

Part 6: Factored orderings are more probable

A high-aridity level manifesting through a factored ordering — for example T⁻¹⁵ appearing as T⁻⁵×T⁻⁵×T⁻⁵ rather than as T⁻¹⁵ pure — is dramatically more probable than the pure form.

T⁻¹⁵ pure requires a specific coordination of 31 phases: 31! ≈ 10³³ possible orderings.
T⁻⁵×T⁻⁵×T⁻⁵ requires three coordinations of 11 phases each: (11!)³ ≈ 10²³ combinations.

The factored form is 10¹⁰ times more probable than the pure form.

This is why high-level phenomena always appear as structured combinations of lower-level processes, never as direct manifestations of their full complexity. Turbulence (T⁻¹⁵) appears as cascades of vortices (T²×T²) undergoing temporal alternations (T⁻¹×T⁻¹×T⁻¹) — not as a direct 31-phase phenomenon. The observable form is always the most probable form, which is always the most factored form.

Part 7: The two orderings of T² and the arrow of time

T² has four phases. Two specific orderings are particularly illuminating:

Ordering A: beginning → interior → end → exterior
A finitude with internal and external structure. Something that has an inside and an outside — a closed region of space with a boundary. The four phases describe: where it starts, what is inside, where it ends, what is outside. This is the spatial structure of a bounded object.

Ordering B: beginning → end → destination → origin
A direction and its inversion. Something that goes from here to there, and where “there” implies a return path to “here”. The four phases describe: departure, arrival, the destination as a place, and the origin as a place that persists. This is the spatial structure of a reversible path — and of the vortex.

Same level T², different orderings, genuinely different phenomena. The first is a container; the second is a circuit.

The six orderings of T⁻¹ are equally probable. “Beginning→middle→end” has no structural advantage over “end→middle→beginning”. The arrow of time belongs to T³, not to T⁻¹.

T³ has historical memory: it records a specific ordering of events as objective fact. Once T³ registers “A happened before B”, that ordering becomes part of the historical record — and the record is asymmetric by definition. The arrow of time is T³’s perspective on the orderings of T⁻¹, not a property of T⁻¹ itself.

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

Thermodynamic irreversibility is real — but its foundation is in T³ (the level of historical memory) rather than in T⁻¹ (the level of temporal alternation). The second law of thermodynamics is a statement about T³, not about T⁻¹.

Summary

Concept	Formulation
Number of orderings at level T^k	n! where n = aridity
Probability of each ordering	1/n! (equiprobable)
More probable orderings	Those that factor into lower-level structures
Less probable orderings	Pure high-aridity configurations
Arrow of time	Property of T³ (memory), not T⁻¹ (alternation)
Factored notation	Hermeneutically richer, not ontologically different
Observable phenomena at high levels	Factored orderings, not pure level manifestations

The single most important sentence:
Variance is born at T⁻¹. T⁰ is pure indetermination — contradiction without existence, prior to any possibility of order. T¹ is mere temporal succession — not determined by a law, but trivial because no alternative is structurally possible. From T⁻¹ onward, the framework is irreducibly probabilistic, conditioned by what has already been decided, never starting from scratch.

Extension to §3.5 — The Principle of Phase Ordering

To be inserted after the Summary of §3.5

The lanthanide series provides a concrete physical realization of the principle that each n-ary structure presents a distinct type of simultaneity.

The 4f shell of lanthanide ions is governed by T⁻³ (aridity 7). Each electron added to the shell is an ordering of a phase within T⁻³ — an actualization that becomes part of the system’s history. The key result is that the type of simultaneity available to the system changes qualitatively when the number of electrons exceeds the aridity of the hosting level.

When n ≤ n(T⁻³) = 7: the system has not exhausted the distinguishable orderings available at T⁻³. Each electron can maintain its spatial distinction — it has a unique “slot” in the level’s phase structure. The system acts freely within its capacity.

When n > 7: the additional electrons cannot find unoccupied distinguishable orderings within T⁻³. The spatial mode T² — which requires independent dimensions to express a distinguishable ordering — enters into contradiction with the accumulated history. The entire T² mode of acting collapses. This collapse is always exactly n(T²) = 4 orderings, because T² as a complete mode either holds or does not — it has no partial collapse.

The observable signature of this collapse is the number of Kramers doublets of the principal emitting multiplet: for Kramers ions with n > 7 and intra-level transitions, doublets = n − 4. Verified in Dy³⁺ (n=9, 5 doublets) and Er³⁺ (n=11, 7 doublets).

The collapse is discrete — it occurs at T², which has no intermediate — but its distribution across the lanthanide series is continuous, mediated by T³. The collapse is a jump; the arrangement of jumps along the series is a spiral.

3.6 Three Principles, One Structure

The question these three principles answer

Why does the universe contain more space than mass? More time than space? Why do physical phenomena tend to express themselves as combinations of simple structures rather than as direct manifestations of complex ones? Why is the electromagnetic field rare in its pure form while temporal alternation is everywhere?

These questions have a single structural answer. It does not require invoking special laws or initial conditions. It follows from three principles that are, at bottom, three faces of the same insight.

Principle 1: Probabilistic Freedom

The universe does not privilege any configuration.

Consider a universe described by N binary conditions — N positions that can each be 0 or 1. There are 2^N possible configurations. A priori, the universe has no preference among them. No configuration is more “natural” or “correct” than any other. This is not a postulate — it is a consequence of T⁰ (the generative contradiction) having no intrinsic preference between one and other.

Now ask: how many configurations satisfy a given condition?

A condition involving k specific positions is satisfied by exactly 2^(N-k) configurations — regardless of the total size N of the universe. The probability of that condition being met is:

P(condition of k positions) = 2^(N-k) / 2^N = 1/2^k

This is independent of the universe’s total size. A condition involving 2 positions has probability 1/4. A condition involving 4 positions has probability 1/16. A condition involving 8 positions has probability 1/256.

The consequence is immediate: simpler conditions are more probable not because the universe favors them, but because more configurations satisfy them. The universe’s neutrality — its lack of preference — is precisely what makes simple structures more abundant.

This is probabilistic freedom: freedom from any privileged configuration, which produces as its statistical consequence an abundance of simple structures over complex ones.

Principle 2: Relative Logical Necessity

Given a condition, certain consequences cannot not be.

Once a specific configuration is active — once the act has “acted as” something specific — that fact carries with it a set of truths that are necessarily co-present. Not because a law imposes them, but because their absence would be a contradiction, and contradiction has already been eluded at T⁰.

The first instance: if the act has acted as “other existant”, then necessarily “one” also exists. This is not absolute necessity — it is conditional necessity. Without the condition, nothing is necessary. With the condition, the consequence is inescapable.

This is relative logical necessity: necessity that is always relative to a prior condition, never absolute in itself (except at T⁰, which has no prior condition).

The relationship with probabilistic freedom:

Probabilistic freedom says: no configuration is privileged a priori. Relative logical necessity says: once a configuration is actual, it generates a web of necessary implications.

These are not in tension. They operate at different moments:

Before actualization:  probabilistic freedom — all configurations equally possible
At actualization:      one configuration becomes actual (statistical, not forced)
After actualization:   relative logical necessity — implications cannot be denied

The universe is free before each act. It is necessary after each act. And the necessity is always local — relative to what just became actual — never global.

Axiomatic history is the accumulated web of these local necessities. Not a chronological record of past events, but the set of truths necessarily implied by every currently true fact. It is what narrows the space of compatible configurations as the universe unfolds.

Principle 3: The Scale of N-ary Logics

Each aridity is an enormous jump. Higher aridity means dramatically lower probability of pure manifestation.

Binary logic — the logic of two phases, one and other — has been studied by humanity for centuries and is still not exhausted. ArXe proposes that the universe operates with ternary, quaternary, and n-ary logics where n can reach 11, 13, 17, and beyond.

Each step up in aridity is not a small increment. It is a qualitative leap:

2-ary logic (binary):    conditional probability emerges here
3-ary logic (ternary):   Bayesian probability emerges here
4-ary logic:             structure of possible reasoning largely inaccessible to intuition
n-ary logic (n ≥ 11):   practically unreachable by direct intuition

And the probability of a pure n-ary manifestation:

P(pure 2-ary configuration) = 1/4      = 0.250
P(pure 3-ary configuration) = 1/8      = 0.125
P(pure 4-ary configuration) = 1/16     = 0.063
P(pure 6-ary configuration) = 1/64     = 0.016   [T³, mass]
P(pure 11-ary configuration) = 1/2048  = 0.0005  [T⁻⁵, EM field]

A pure 11-ary configuration is 512 times less probable than a pure 6-ary one. This is not a property of the electromagnetic field — it is a consequence of the combinatorial structure of the universe.

The crucial consequence: Any ontology of 2BC is more probable than any ontology of 4BC. And any ontology of 3×2BC — three independent binary pairs — is more probable than a single ontology of 6BC, because three independent conditions of 2 positions each are easier to satisfy simultaneously than one condition of 6 positions. This is why PLO works: physical phenomena express themselves as products of small prime operators because those factored forms correspond to the most probable configurations of the universe.

The three principles as one structure

These three principles are not independent. They are three descriptions of the same underlying reality, seen from different angles:

From the angle of counting: Probabilistic freedom — more configurations satisfy simple conditions, so simple conditions are more probable.

From the angle of implication: Relative logical necessity — once a condition is actual, it generates a web of necessary co-truths that cannot be denied without contradiction.

From the angle of complexity: Scale of n-ary logics — the probability of pure complex manifestations decreases exponentially with aridity, making factored manifestations overwhelmingly more common.

Together they form a single coherent picture:

T⁰: no preference, no configuration, pure indetermination

First actualization (statistical, not forced):
→ relative logical necessity generates first implications
→ axiomatic history begins to accumulate

Each subsequent level:
→ probabilistic freedom: simpler configurations more abundant
→ relative necessity: implications of current facts cannot be denied
→ n-ary scale: pure complex manifestations rare, factored ones common

Result:
→ universe rich in simple structures (time, space)
→ universe sparse in complex structures (mass, fields)
→ complex phenomena always appear in factored form
→ PLO: physical constants as products of small prime operators

Why the universe looks the way it does

From these three principles, the large-scale structure of the observable universe follows without additional assumptions:

Time is abundant because T⁻¹ has aridity 3 — there are many configurations satisfying ternary conditions. Time is the most structurally accessible level beyond T⁰ and T¹.

Space is less abundant than time because T² has aridity 4 — conditions of 4 positions are satisfied by fewer configurations than conditions of 3.

Mass is a fraction because T³ has aridity 6 — 64 times less probable than binary conditions. The universe produces mass rarely compared to space and time.

Pure electromagnetic phenomena are rare because T⁻⁵ has aridity 11 — thousands of times less probable than mass. Observable EM always appears in factored combinations with lower-level structures.

The universe is mostly “empty” — mostly time and space, with rare islands of mass and even rarer islands of complex field structure — not because of special initial conditions but because of the combinatorial structure of the universe’s own logical architecture.

These principles do not say that complex structures are impossible. T⁻⁵ exists. Mass exists. The electromagnetic field exists. They say that pure high-aridity manifestations are improbable — not impossible. What the principles explain is not absence but relative abundance: why simple structures dominate, why complex phenomena appear in factored form, why the universe at large scales looks mostly like time and space with occasional islands of everything else.

3.7 Physics as Statistical Manifestation

The central claim

Physical reality is the statistically dominant manifestation of aridity — the tendency of any n-ary structure to express itself through its most probable decompositions into smaller arities.

This is not a separate postulate. It follows directly from the three principles of §3.6: probabilistic freedom (no configuration is privileged), relative logical necessity (implications are conditional, not absolute), and the scale of n-ary logics (higher aridity means exponentially lower probability of pure manifestation).

Together they imply: whatever we observe as “physical” is not the full logical structure of any level — it is the most probable face of that structure, which is always its most factored, most decomposed form.

What this means for the negative levels

Consider T⁻³ (aridity n=7, one open BC).

The structural fact is simply: there exists an n=7 configuration with one open BC. That is all the framework specifies at the logical level.

How does n=7 manifest? In principle, as any of its 7! = 5,040 possible orderings. In practice, overwhelmingly as its most probable forms — which are the factored ones: combinations of smaller arities that together produce n=7 behavior.

The most probable factored form of n=7 with one open BC is what we observe as color confinement: a structure that behaves like mass (n=6, T³) with one degree that cannot close — one direction of internal variation that the structure cannot resolve into a definite state.

T⁻³ is not defined as “mass that cannot close”. It is n=7 with one open BC manifesting in its most probable form — which happens to look like mass with an unresolvable internal degree.

A pure n=7 manifestation exists structurally and would look like something else entirely — but it is so improbable (1/5,040 for any specific pure ordering) that it is effectively never observed.

This reframes the entire table of negative levels:

Level	n	Open BC	Most probable manifestation	Why that form
T⁻¹	3	1	Temporal alternation	Simplest factored form of n=3
T⁻²	5	1	Spatial curvature	n=5 factored as T²+variation
T⁻³	7	1	Color (mass+open degree)	n=7 factored as T³+open
T⁻⁵	11	1	EM field	n=11 factored as multiple lower-n
T⁻⁶	13	1	Weak field	n=13 factored similarly

The negative levels are not a separate ontological category from the positive ones. They are the same logical structure — the same aridity range — manifesting through configurations that include one unresolvable degree. That unresolvable degree is the open BC. And it is open not by definition but because the most probable factored forms of those arities happen to include one degree that cannot be closed without requiring a higher level.

The relationship between positive and negative levels

Each pair (T^k, T^⁻k) shares a level of depth in the hierarchy — both are the k-th exentation from T⁰. What distinguishes them:

T^k  (positive): all BCs closed — the k-th level in its fully resolved form
                 Can exist in isolation. Has complete internal order.

T^⁻k (negative): one BC open — the k-th level with one unresolvable degree
                 Cannot exist in isolation. Has one internal direction 
                 that cannot be decided without a higher level.

The positive level is the k-th aridity fully actualized. The negative level is the k-th aridity with one degree still in play — still undecidable, still requiring something external to close it.

This is why negative levels generate fields and positive levels generate particles: fields are structures with one open degree that propagates through space seeking closure; particles are structures with all degrees closed that simply exist.

And this is why every field (negative level) must couple to matter (positive level) to produce observable phenomena: the open BC of the field seeks the closed BCs of the particle. The coupling is not an additional law — it is the structural consequence of open BCs seeking closure.

Summary

Logical level:   n-ary structure with specific BC configuration
                 All n! orderings exist as logical possibilities

Statistical layer: Most probable orderings dominate
                   These are always the most factored forms
                   These are what we observe as physical phenomena

Physical "laws":  Statistical regularities of dominant configurations
Physical "constants": Numerical measures of dominant coupling ratios
Physical "particles": Fully closed configurations (positive levels)
Physical "fields":    Configurations with one open degree (negative levels)

Physics is not the whole of what is logically possible.
Physics is the most densely populated region of what is logically possible.

The universe does not select physics from among the logical possibilities.
Physics is simply what the universe looks like when you stand in the most probable region of its configuration space — which, given probabilistic freedom and the scale of n-ary logics, is always the region of small, factored, low-aridity structures.

4. Boundary Conditions and Prime Encoding

Fundamental rule:

k > 0: All BC closed → Can exist isolated → Particles, masses
k < 0: At least 1 BC open → Cannot exist isolated → Fields, confinement

BC structure by level:

Level	k	n(k)	Closed BC	Open BC	Isolated?	Prime
T⁰	0	1	0	0	No (contradiction)	—
T¹	+1	2	1	0	Yes	2*
T⁻¹	−1	3	0	1	No	3
T²	+2	4	2	0	Yes	—
T⁻²	−2	5	1	1	No	5
T³	+3	6	3	0	Yes	—
T⁻³	−3	7	2	1	No	7
T⁻⁵	−5	11	4	1	No	11
T⁻⁶	−6	13	5	1	No	13
T⁻⁸	−8	17	6	1	No	17
T⁻⁹	−9	19	7	1	No	19
…	…	…	…	…	…	…

Note on prime 2 (T¹, k=+1): the only operator at a positive level — BC fully closed. The only one that can exist in isolation. It appears in almost every formula as the structural carrier between levels: the minimal act of distinction that makes any other structure possible.

For k < 0: n(k) = −2k+1 generates prime numbers systematically.

Prime numbers emerge here as the only Irreducible Resolution Routes of the contradiction. They are not entities — they are the frequencies at which distinction can sustain itself without redundancy. The universe “structures itself” through primes to avoid logical self-dissolution. The irreducibility of primes and the irreducibility of BC-open levels are the same property seen from two angles: both resist being decomposed into simpler forms, both require external coupling to exist, both cannot be reduced without losing their character entirely.

4.4 The Hierarchy as a Chain of Inversions

The ArXe hierarchy alternates positive and negative levels:

T¹, T⁻¹, T², T⁻², T³, T⁻³, T⁻⁵...

The mapping function e(n) generates this sequence. But the alternation is not a formal convention — it is a structural necessity. This section derives it.

The generative mechanism: inversion of indecidability

Each level in the hierarchy generates its successor through the same mechanism:

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

T¹ → T⁻¹: Time generates its own alternation

T¹ is homogeneous temporal succession — mere flow in one direction. But T¹ carries within it the possibility of its own negation: the flow could go in either direction. That possibility — the undecidability of which direction is “truly” first — cannot be resolved within T¹ itself. The result: the undecidable direction of T¹ generates T⁻¹ — temporal alternation, the level where direction is structurally open.

T¹  (decided succession)
→ its direction cannot be decided from within
→ T⁻¹ (undecidable alternation) emerges

T⁻¹ → T²: Temporal indecidability generates space

T⁻¹ has one open BC — the direction of temporal flow is undecidable. When two temporal phases cannot be ordered — when there is no intrinsic reason to say which comes first — they must coexist. That coexistence is simultaneity. Simultaneity is space.

T⁻¹ (undecidable temporal direction)
→ phases that cannot be ordered must coexist
→ simultaneity = space
→ T² (spatial structure) emerges

Space is the inversion of time: where time is decided order, space is undecidable order. They are not two kinds of the same thing — they are logical contraries generated by the same mechanism applied twice.

T² → T⁻²: Spatial structure generates curvature

T² is flat simultaneity — two directions coexisting without order. But T² carries within it the possibility of its own indecidability: the direction of curvature cannot be decided from within flat space.

T² (flat spatial structure)
→ direction of curvature cannot be decided from within
→ T⁻² (curved space, undecidable curvature direction) emerges

This is the structural origin of gravity: not a force imposed on space, but the inevitable consequence of space’s own inability to decide its curvature direction.

Magnetism as inversion of space: A static charge produces an electric field — T⁻⁵ projected onto T² (space with decided structure). A moving charge introduces T⁻¹ (temporal alternation), which makes T² unable to fix a privileged spatial direction. That spatial indecidability generates T⁻² — the magnetic field.

Static charge:   T⁻⁵ onto T²   → electric field (space decided)
Moving charge:   T⁻¹ activates → T² cannot fix direction
                 → spatial indecidability → T⁻²
                 → T⁻⁵ onto T⁻² → magnetic field

This is why special relativity unifies E and B: they are T⁻⁵ projected onto T² and T⁻² respectively — the same field at two levels that are themselves inversions of each other.

T⁻² → T³: Curved space generates mass

T⁻² is curved space — spatial structure with undecidable curvature direction. When the curvature cannot be resolved within T⁻², it must stabilize into a structure with closed BCs — a structure that can exist in isolation and carry its own history. That stabilization is mass.

T⁻² (undecidable curvature direction)
→ curvature that cannot be resolved within space
→ stabilizes into isolated persistent structure
→ T³ (mass) emerges

This gives a structural reading of the Higgs mechanism: mass is the stabilization of curved-space indecidability into a closed structure.

T³ → T⁻³: Mass generates color

T³ carries within it the possibility of its own variation: which of its internal color degrees is “truly” red, green, or blue cannot be determined from within T³. That internal indecidability generates T⁻³ — color confinement.

T³ (closed mass structure)
→ internal variation direction cannot be decided from within
→ T⁻³ (mass variation with open internal degree) emerges
→ color confinement as structural necessity

The complete chain

T⁰   Generative contradiction — no structure, no order

T¹   Decided succession — mere temporal flow
     ↓ (T¹'s direction is undecidable from within)
T⁻¹  Undecidable alternation — temporal indecidability
     ↓ (phases that cannot be ordered must coexist)
T²   Flat simultaneity — space as inversion of time
     ↓ (T²'s curvature direction is undecidable from within)
T⁻²  Curved simultaneity — magnetism/gravity as inversion of space
     ↓ (curvature that cannot resolve stabilizes into closed structure)
T³   Mass — closed, persistent, historically recording
     ↓ (T³'s internal variation is undecidable from within)
T⁻³  Color — mass variation that cannot close
     ↓ (and so on...)
T⁻⁵  EM field — T⁻³'s coupling indecidability stabilized

Each arrow is the same operation: the indecidability of a level becoming the next level.

Why the alternation is necessary

Positive levels (all BCs closed) are structures that have stabilized — where a previous indecidability has found closure. They can exist in isolation because all their degrees have resolved.
Negative levels (one BC open) are structures where one degree remains unresolved — where the indecidability has not yet found closure. They cannot exist in isolation because one direction is still undecidable.

The alternation occurs because:

A positive level (decided) generates indecidability about its own structure
That indecidability is a negative level (undecided)
The negative level’s unresolved degree eventually stabilizes into a new positive level
And the cycle continues

The hierarchy does not alternate because the mapping function e(n) generates alternating signs. The mapping function generates alternating signs because the generative mechanism necessarily alternates between closure and openness, decision and indecision, stability and variation.

Consequences

For E and B: Electric and magnetic fields are not two independent phenomena requiring separate laws. They are T⁻⁵ projected onto two levels that are themselves inversions of each other (T² and T⁻²). Maxwell’s equations are the mathematical expression of this single structural fact.

For gravity and electromagnetism: Both involve T⁻² — gravity as T⁻²’s curvature structure, magnetism as T⁻⁵ projected onto T⁻². Their unification may not require new physics — it may require recognizing that they are two faces of T⁻²’s indecidability.

For the table of levels: The table in §10 is not a catalog. It is a record of a single mechanism applied repeatedly — each entry the inversion of the previous one, each inversion generating a new form of indecidability, each new form of indecidability eventually stabilizing into a new level.

The hierarchy is not a list. It is a chain. Each level is the inversion of the previous one’s indecidability — and that inversion is both necessary and sufficient to generate the next.

5. The Physical Dimension Assignment

Fundamental correspondence:

T¹ ≡ T    (physical Time)
T² ≡ L    (physical Length)
T³ ≡ M    (physical Mass)

T¹ = 2·Tf: if something and its contrary cannot occur in 1Tf (by PNC), they can occur in 2Tf = T¹ = physical time. This is not a postulate added to fit physics. It follows from the structure: T¹ is the first level where two distinct phases can coexist without contradiction — which is exactly what physical time requires.

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

5.4 Why These Assignments? Minimum Conditions of Possibility as Criterion

The right question is not “Why did ArXe assign mass to T³?” — that phrasing implies an arbitrary decision. The right question is: “What are the minimum structural conditions for something to behave as mass — and which is the first level that satisfies all of them?”

This reframing shifts the burden of justification from “we chose this mapping” to “this is the first level where these conditions are structurally met.” The assignment is not a choice but a consequence.

The minimum conditions of possibility for mass

For something to behave as mass, four structural conditions must be simultaneously satisfied:

Condition 1: Historical persistence
Mass does not disappear between one instant and the next. It requires that past, present, and future be structurally distinguishable — not just as labels, but as ontologically distinct phases. This first becomes structurally available at T³, where the six-phase structure generates the irreversible past → present → future sequence as a logical necessity of the level’s internal organization.

Condition 2: Ternary objectivity
Mass can be measured by a third party independently of the measurement itself. This requires the structure of “an observer who registers without being registered” — not just A and B (binary), but A, B, and the third that relates them without belonging to either. This ternary structure first appears at T³ (n=6, six-ary logic with included middle).

Condition 3: Capacity for isolated existence
Mass can exist without requiring another structure to “close” it. Formally, this requires all boundary conditions to be closed (k > 0). T³ is the first level with k = 3 > 0 and three closed BCs.

Condition 4: Three-dimensional spatial presence
Mass occupies space in three dimensions. T³, with three closed BCs, is the first level where exactly three independent structural degrees of spatial closure are available at once.

Why T³ and not T² or T⁴?

T² fails conditions 1 and 2: There is no past-present-future structure at T², and ternary objectivity is absent — T² operates with pairs of contraries (four-ary logic), not with a structural third. T² can have length but not mass.

T⁻³ fails condition 3: T⁻³ has one open BC. It cannot exist in isolation — which is exactly why color charge is always confined.

T⁴ satisfies all four conditions — but it is not the first level that does so. T³ already satisfies all four with exactly three closed BCs. By structural parsimony, T³ is the minimum level at which mass becomes possible.

The general principle

A physical concept X maps to level T^k
if and only if
T^k is the minimum level whose structural properties
(arity, number and type of BCs, logical structure)
satisfy all the conditions of possibility of X,
and no level T^j with |j| < |k| satisfies them all simultaneously.

Three consequences: (1) assignments are derivable, not stipulated; (2) absence is as informative as presence — a level with no assigned concept is not empty, it is waiting for identification; (3) the framework is not circular — T³ is defined by its structural properties, and mass is identified as the concept whose minimum conditions match exactly those properties. The direction runs from structure to concept, not from concept to structure.

Application to open questions

Q: Why does the observer emerge at T⁻⁵ (prime 11)?
The minimum conditions for observation require: gauge freedom (open BC to choose reference without affecting the measured system), sufficient internal complexity to encode external states (at least 11 phases), and the ability to project onto spatial structure without being confined. T⁻⁵ is the first level satisfying all three simultaneously. See §9.8 for the full derivation.

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

Q: What determines the jump between levels?
Each level is generated by one exentation — one additional structural differentiation following the recursive rule Entₙ := Entₙ₋₁ ∧ ExEntₙ₋₁. The jump from T² to T³ is the jump from “space with reversibility” to “space-time with historical record and ternary objectivity” — which is also the jump from the possibility of length to the possibility of mass.

Summary table

Level	Arity	Closed BC	Open BC	Isolated existence	Historical record	Ternary objectivity	First concept made possible
T¹	2	1	0	Yes	No	No	Time (flow)
T⁻¹	3	0	1	No	No	Yes (but unclosed)	Frequency
T²	4	2	0	Yes	No	No	Length / Space
T⁻²	5	1	1	No	No	No	Curvature
T³	6	3	0	Yes	Yes	Yes	Mass
T⁻³	7	2	1	No	—	—	Color (confined)
T⁴	8	4	0	Yes	Yes	Yes	Information / Computation
T⁻⁵	11	4	1	No	—	Yes (gauge)	EM Field / Observer

6. π as Emergent from Ternary Ambiguity

At T³ (n=6, first level of objectivity), three elements co-emerge:

3D spatiality
Ternary logic (included middle, observer)
Geometric ambiguity (orientation undecidability)

π is the geometric measure of ternary indecidability. When a physical constant involves geometric coupling, ternary mediation, or spatial indecidability — π appears as the natural factor. It is not a universal mathematical constant that the universe uses; it is what the contradiction becomes when it closes on itself geometrically at T³.

Part II: The Complete Bridge

7. The Dimensional Framework

7.1 The ArXe Dimensional Rule

For any physical quantity with SI dimension M^a × L^b × T^c, the ArXe T-exponent is:

n_ArXe = 3a + 2b + c

This follows directly from T≡T¹, L≡T², M≡T³ and the multiplicative structure of dimensional analysis.

Complete dimensional table:

SI dimension	Formula	ArXe T^n	Meaning in ArXe
Time	T	T¹	T¹ — minimal duality
Length	L	T²	T² — spatial anteriority
Mass	M	T³	T³ — objectivity, self-containment
Velocity	L/T	T¹	T¹ — same level as time
Acceleration	L/T²	T⁰	T⁰ — dimensionless in ArXe
Force	ML/T²	T³	T³ — same level as mass
Energy	ML²/T²	T⁵	T⁵
Action (ℏ)	ML²/T	T⁶	T⁶
Momentum	ML/T	T⁴	T⁴
Power	ML²/T³	T⁴	T⁴ — same as momentum
Frequency	1/T	T⁻¹	T⁻¹ — temporal alterity (CYC)
G (Newton)	M⁻¹L³/T²	T¹	T¹ — same as c

7.2 Three Structural Equivalences

Equivalence 1 — Force = Mass = T³
F = ma, and acceleration = T⁰ (dimensionless in ArXe). Force has no independent ontological status — it is mass acting on a dimensionless ratio.

Equivalence 2 — c = G = T¹
The speed of light and Newton’s gravitational constant have the same ArXe dimension T¹. Both are conversion factors between the T² scale (length) and the T¹ scale (time). Setting c=G=1 in Planck units removes two redundancies at the same level.

Equivalence 3 — Momentum = Power = T⁴
Two quantities that appear distinct in conventional physics are the same degree of freedom of the contradiction at T⁴.

7.3 Planck Units as the Natural ArXe Scale

In Planck units (c=G=ℏ=1), every dimensional constant becomes a pure dimensionless number — a ratio against the Planck unit at its level. This number is then read by PLO v2 exactly like any dimensionless constant.

Mass in Planck units:   m_Planck = m / m_P        (m_P = 2.176×10⁻⁸ kg)
Length in Planck units: l_Planck = l / l_P        (l_P = 1.616×10⁻³⁵ m)
Energy in Planck units: E_Planck = E / E_P        (E_P = 1.956×10⁹ J)

Confirmed: dimensional constants in Planck units produce physical integers P with ArXe-pure prime factorizations, without needing mathematical anchors. The masses of Standard Model particles are 86% ArXe-pure in Planck units. The Planck scale is not chosen for convenience — it is the natural ArXe scale because c and G have dimension T¹, the root level of the contradiction.

Selected readings:

Particle	P	Factorization	ArXe reading
Electron	42	2×3×7	DIFF×CYC×CPX
W boson	66	2×3×11	DIFF×CYC×REG
Z boson	75	3×5²	CYC×MEM²
Proton	77	7×11	CPX×REG
Neutron	77	7×11	CPX×REG

Proton = Neutron = 77 = 7×11. At current precision, both nucleons are the same degree of freedom: color × electromagnetic regulation. The 0.14% mass difference is in the conventional layer — they share the same ArXe identity.

8. φ and π as Boundary Condition Anchors

8.1 The Formal Correspondence

π and φ are not universal mathematical constants that happen to appear in formulas. They are Boundary Condition Anchors — the geometric limits of logical ambiguity at the first observable level.

π — BC-Closed Anchor:
The metric of a distinction that closes on itself to gain stability. The circle is the unique planar curve that returns exactly to its origin — π measures that closure. In ArXe: π encodes a relationship where boundary conditions are fully closed. The structure is self-sufficient, needs no external reference. π is not an ideal number; it is the geometric limit of a distinction that has resolved its ambiguity by enclosing it.

φ — BC-Open Anchor:
The metric of a distinction that projects outward to avoid collapse. φ = 1 + 1/φ — an infinite continued fraction that always requires one more term. The golden spiral never returns to its origin. In ArXe: φ encodes a relationship where boundary conditions remain permanently open. φ is not an ideal number; it is the geometric limit of a distinction that sustains itself by remaining unresolved.

ρ — BC-Open Cubic Anchor:
ρ³ = ρ+1 — cubic self-reference, one order above φ (φ²=φ+1). Appears in running couplings and recursive structures requiring three-way coupling. Emerges at T⁴.

8.2 The Emergence Levels

Both π and φ become simultaneously indecidable at T³ — the first level of objectivity:

Constant	Emerges at	Indecidable at	Reason
π	T²	T³	Exists in T² (first 2D level). Becomes indecidable only at T³, with the observer
φ	T³	T³	φ²=φ+1 requires triadic structure — emerges and is indecidable at the same level
ρ	T⁴	T⁴	ρ³=ρ+1 requires cubic self-reference — needs T⁴

Their simultaneous indecidability at T³ mirrors the co-existence of closed and open BC. This generative tension drives the exentation to T⁴ and beyond.

8.3 Why This Determines Anchor Distribution in PLO

Dimensionless constants are transitions between dimensional levels — they describe relationships, not levels themselves. Those relationships are measured at T³. At T³, the natural anchors are π (BC-closed) and φ (BC-open). This is why:

π anchors geometric couplings, angles, field strengths — closed structures
φ anchors mixing amplitudes, density fractions, mass ratios — open transitions
ρ anchors running couplings, recursive structures — cubic open structures

Dimensional masses need no anchors because they describe the level itself, not a transition between levels.

8.4 Why Time and Space Are Contraries — Indecidability as the Origin of Geometry

Time is decided order. Space is undecidable order.

Time is the structure of decided succession: first, second, third. There is an intrinsic reason to order the phases. T¹ is the bare succession: one phase follows another because that is what succession means.

Space is the structure where order cannot be decided: not-first, not-second. When two phases cannot be ordered — when there is no intrinsic reason to say which comes before the other — they must coexist. That coexistence is simultaneity. And 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: the presence and absence of decidable order. Space does not occur in temporal order — its essence is the negation of temporal order.

This is why the emergence of T² from T¹ is not a continuation but a reversal: T¹ is decided flow; T² is what happens when that flow encounters phases that cannot be ordered within it.

Each aridity generates a distinct form of simultaneity

The Indecidability ↔ Simultaneity correspondence is not a single phenomenon. At each level of the hierarchy, indecidability takes a specific form determined by the aridity of that level — and each form of indecidability generates a correspondingly specific form of simultaneity, which is the geometry of that level.

T² (aridity 4): Flat simultaneity — extension and reversibility

At T², the indecidability is binary: no reason to say which of two directions is “first”. The simultaneity this generates is flat — two directions coexisting without order.

This is the geometry of the plane: extension in two directions with no privileged one. The geometric constant of flat simultaneity is √2 — the diagonal proportion that emerges when two equal and simultaneous directions are composed.

T³ (aridity 6): Volumetric simultaneity — rotation and π

At T³, the indecidability is ternary: no reason to prefer any of three orientations in three-dimensional space. The simultaneity this generates is volumetric — three directions coexisting without order, equally present in all orientations.

When three directions are simultaneously present with no preferred one, the natural structure is continuous rotation. π is not a number that happens to appear in geometry. π is the geometric constant of ternary undecidable simultaneity. The ternary ambiguity of T³ is what makes rotation continuous rather than discrete. This explains why π appears throughout physics wherever ternary structure is present.

T⁻² (aridity 5): Curved simultaneity — geodesics and curvature

At T⁻², the open BC means the direction of curvature is undecidable — the space can curve inward or outward without intrinsic preference. The simultaneity this generates is curved. The geometric structure is not the flat plane of T² but the curved manifold of general relativity. The geometric constant of curved simultaneity is not a fixed number — it is the metric tensor.

T⁻³ (aridity 7): Internal simultaneity — color space and confinement

At T⁻³, the indecidability is three-color: no intrinsic reason to prefer red over green over blue. The simultaneity this generates is internal — a space of internal degrees of freedom. This is SU(3) color space: three “directions” in an internal space, none preferred, all simultaneously present. The signature of this form of simultaneity is confinement: the internal simultaneity of color cannot be projected into external space without closing the open BC.

T⁻⁵ (aridity 11): Phase simultaneity — gauge freedom and U(1)

At T⁻⁵, the indecidability is continuous phase: no intrinsic reason to choose any particular phase θ ∈ [0, 2π]. The simultaneity this generates is phase space — a continuous circle of equally valid phase choices, none privileged. This is U(1) gauge symmetry. The photon is the physical expression of 11-phase undecidable simultaneity projected into T².

The general principle: indecidability as geometry

Level	Aridity	Form of indecidability	Geometry generated	Geometric constant
T²	4	Binary direction (no first/second)	Flat plane, extension	√2
T³	6	Ternary orientation (no preferred axis)	Continuous rotation, volume	π
T⁻²	5	Curvature direction (inward/outward)	Curved manifold, geodesics	metric tensor
T⁻³	7	Three-color (no preferred color)	Internal color space	SU(3), 8 generators
T⁻⁵	11	Continuous phase (no preferred θ)	Phase circle	π (as U(1))

The geometric constants — √2, π, the metric tensor, the group structure of SU(3) — are not mathematical facts imposed on physics. They are the natural measures of specific forms of undecidable simultaneity at specific levels of the hierarchy.

Physical geometry is the shape that logical indecidability takes when it becomes stable.

Spirals as the geometry of T⁻¹ in T²

Spirals appear universally — galaxies, hurricanes, shells, DNA, vortices. T⁻¹ (temporal alternation) has one open BC: the direction of temporal flow is undecidable. When T⁻¹ is projected into T² (flat space), its undecidable direction must coexist with T²’s two simultaneous spatial directions. The result combines the undecidable direction of T⁻¹ with the flat simultaneity of T² — producing the logarithmic spiral: a curve that continuously rotates (T²’s geometry) while advancing in a direction that is never fixed (T⁻¹’s open BC expressing itself spatially).

The distinction between T³ alone and T³ coupled with T² has a direct observable consequence in spectroscopy.

A transition that remains within a single ontological level (intra-level, such as 4f→4f in lanthanide ions) produces a photon whose energy is determined entirely by the internal phase structure of that level. The result is a spectrally narrow emission — the photon carries the precise imprint of T⁻³’s internal discrete structure. The linewidth is of order 1–3 nm FWHM.

A transition that crosses an ontological boundary (inter-level, such as 5d→4f) must negotiate the interface between two levels with different arities and different BC structures. The indecidability of that interface — the impossibility of a clean mapping between the phase structures of T⁻⁵ and T⁻³ — projects into the photon as spectral broadening. The linewidth is of order 100 nm FWHM: two orders of magnitude broader.

This is the physical instantiation of T³ alone vs T³ + T²:

Intra-level (T³ operating within its own structure): the spiral is spatially undefined but internally precise. The emission linewidth is narrow because there is no external T² boundary to negotiate.
Inter-level (T³ crossing into T⁻⁵ via T²): the spiral acquires a T² container at the transition boundary. The indecidability of that container — which T² reading applies at the interface? — produces a spatially defined but spectrally broad emission.

The Ce³⁺/Yb³⁺ pair in YAG illustrates this directly. Both ions are at the extremes of the lanthanide series (n=1 and n=13). Ce³⁺ emits via inter-level transition (5d→4f, ~100 nm FWHM); Yb³⁺ emits via intra-level transition (4f→4f, ~1–3 nm FWHM). They are mirrors in the electron-hole symmetry of the series, but they point in opposite directions from the center: one exits T⁻³ to exist, the other never needs to.

The general principle: the linewidth of an optical transition is a direct indicator of whether that transition crosses an ontological boundary. Narrow linewidth = intra-level = T³ operating within its own structure. Broad linewidth = inter-level = T³ negotiating a T² boundary with another level.

This principle applies beyond lanthanides: any system where optical transitions can be classified as intra- or inter-level in the ArXe hierarchy should show the same order-of-magnitude contrast in linewidth. It constitutes a falsifiable prediction for systems where the ontological level assignment is known and the transition type can be varied experimentally.

9. PLO v2 — The Reading Method

9.1 What PLO v2 Is

Physical constants are the chronometric traces of the exentation process. PLO v2 is the reading instrument for those traces within the ArXe framework. It converts any physical constant into a factorization of prime operators and reads the ontological structure that factorization encodes.

The first digits of a constant (the natural layer) show the grammar of the unfolding — what the contradiction did at that level. The last digits (the conventional layer) are the residue of the observer’s interaction — what the human community chose when accessing that structure.

It is an instrument of reading, not prediction. The number guides the reading. The reading does not guide the number.

9.2 Core Definitions

Physical Integer P(C, n):

S = round(|C| × 10^n)         [scale to integer]
P = S / gcd(S, 2^n × 5^n)     [remove scale contamination]

The scale factor 10^n = 2^n × 5^n injects artificial prime factors. P removes them, leaving only the primes that come from the value itself.

Anchor decomposition A(C):
Express C ≈ (p/q) × anchor^r, where p, q are small integers with ArXe prime factors, anchor ∈ {π, φ, ρ}, r ∈ {1, 2, ½, −1}. Valid when error < 0.1%.

Delta Δ(C, t₁, t₂):

Δ = |C(t₂) − C(t₁)|

The change between two measurement epochs, factorized after scaling to minimal integer form. The delta is the primary signal — it isolates what changed, distinguishing unfolding structure from observer choices.

Delta NI — ΔNI:

ΔNI = max prime in factorization of scaled Δ

ΔNI ∈ ArXe primes → structural correction (the unfolding revealing more of itself)
ΔNI ∉ ArXe primes → conventional correction (the observer refining their framework)

9.3 The Protocol

Always in this order. Never reversed:

Obtain exact value from primary source (PDG, NuFit, CODATA) — no truncation
Compute P(C, n) for every available epoch
Search for anchor formula with error < 0.001%
Analyze deltas between epochs
Read the prime grammar in the context of the ArXe level T^n

9.4 The Two Reading Layers

Every physical constant has two structurally distinct layers:

Natural layer (digits 1 to d*): determined by the phenomenon — the grammar of the unfolding. P contains exclusively ArXe primes or Layer C compressed primes.

Conventional layer (digits d*+1 to n): determined by the observer’s choices — renormalization scheme, extraction method, precision protocol. P introduces non-ArXe primes encoding specific decisions of the scientific community.

The threshold d* is empirically identifiable: the last digit at which P remains ArXe-pure.

Confirmed chronological result: the deltas of α_s, m_Z, G_F, m_e, and sin²θ₁₂ across decades of measurement are all ArXe-prime pure. Every correction the scientific community made to these constants tracks ontological structure, not convention. Exceptions: α (prime 47, 2018 — 10th-order QED framework update) and Ω_m (prime 43, 2013 — WMAP→Planck transition).

9.8 The Observer at T⁻⁵: A Structural Necessity

What an observer structurally requires

To function as an observer — in the minimal physical sense of “something that registers a state of the world” — a system needs four structural conditions:

Condition O1: Registration capacity
The observer must record a state that is not its own state — it must be affected by what it observes without being identical to what it observes. This requires internal complexity sufficient to encode external states.

Condition O2: Non-exhaustion of what is observed
The act of registration must not consume or fully determine the observed system. This requires at least one open BC — a gauge freedom that allows the observer to choose its reference without that choice affecting the measured system.

Condition O3: Projection onto spatial structure
The observer must project the registered state onto the spatial level (T²) where physical phenomena unfold. This requires enough closed BCs to couple stably with T².

Condition O4: Not being mass itself
If the observer were a positive level (all BCs closed), it would interact with what it observes in a way that makes observation inseparable from interaction. The observer must be a field-like structure (k < 0, at least one open BC) to retain gauge freedom.

Why T⁻⁵ is the minimum level satisfying all four conditions

T⁻¹ (prime 3, 0 closed, 1 open): Fails O1 — zero closed BCs means no stable internal structure to encode external states.

T⁻² (prime 5, 1 closed, 1 open): Fails O1 — insufficient phases (n=5) to encode the complexity of a physical state in T².

T⁻³ (prime 7, 2 closed, 1 open): Fails O3 — T⁻³ is the level of color confinement. Its coupling to T² is always mediated through T³ (mass). It cannot project independently onto spatial structure.

T⁻⁴: structural gap — n(-4) = 9 = 3², not prime. Levels whose arity is not prime do not generate irreducible ontological operators. T⁻⁴ does not exist as an independent observational level.

T⁻⁵ (prime 11, 4 closed, 1 open):

O1 ✓ — 11 phases, sufficient internal complexity to encode physical states
O2 ✓ — 1 open BC → U(1) gauge freedom → can choose phase reference without affecting what is measured
O3 ✓ — 4 closed BCs → can couple stably and directly to T²
O4 ✓ — k < 0, field structure, not mass

T⁻⁵ is the minimum negative level satisfying all four conditions simultaneously.

This is why the electromagnetic field is the carrier of observation in nature. It is not that we assigned observation to EM. It is that EM is the first field complex enough to register, free enough (gauge) to not collapse what it registers, stable enough to project onto space, and light enough (massless) to not be an object itself.

Is this necessary or contingent?

The necessity here is conditional, not absolute:

If a physical system registers states of the world
without being those states,
then it operates at T⁻⁵ or higher.

T⁻⁵ is not contingently the observer level.
It is necessarily the minimum observer level —
given the BC structure of the hierarchy.

This explains why observation in quantum mechanics has always been associated with electromagnetic interaction — not as a postulate but as a structural consequence.

The hierarchy of observers

T⁻⁵ (EM): Registers binary states (photon detected / not detected). Minimal observer.
T⁻⁶ (Weak, prime 13): Can register flavor states — the distinction between electron and neutrino, up and down quark.
T⁻⁸ (Hyperspace, prime 17): Registers states requiring hyperspatial structure.

Each level up adds observational capacity while retaining the open BC that preserves gauge freedom.

10. The Extended Unified Lexicon

10.1 Convergence of Two Independent Programs

Grammar v4 was developed empirically from 80 physical constants without knowledge of the ArXe exentation table. ArXe developed the prime encoding from the axiom. When compared:

24 of 25 primary operators in Grammar v4 are exactly the levels generated by n(k) = −2k+1 for negative k.

The only exception: prime 2, from T¹ (positive level). This convergence is mutual confirmation. The empirical program found the same irreducible resolution routes that the theoretical program derived.

10.2 The 25-Prime Extended Lexicon

Prime	k	Level	Operator	Physical domain
2	+1	T¹	DIFF	Binary difference, duality — universal carrier
3	−1	T⁻¹	CYC	Minimal cycle, temporal arrow
5	−2	T⁻²	MEM	Memory, persistence, spatial curvature
7	−3	T⁻³	CPX	Internal complexity, color/QCD
11	−5	T⁻⁵	REG	Regulation, EM field U(1) — observer pivot
13	−6	T⁻⁶	SING	Singularity, weak field SU(2)
17	−8	T⁻⁸	SPEC	Spectral separation
19	−9	T⁻⁹	DARK	Dark modulation, dark matter
23	−11	T⁻¹¹	INF	Inflationary expansion
29	−14	T⁻¹⁴	VBG	Vacuum background, dark energy substrate
31	−15	T⁻¹⁵	CHA	Stable irregularity
37	−18	T⁻¹⁸	TOP	Topological defect
41	−20	T⁻²⁰	ISO	Maximum isolation
43	−21	T⁻²¹	TRANS	Spectral transition
47	−23	T⁻²³	NEXT	Post-inflation threshold
53	−26	T⁻²⁶	MIX	Maximum mixing
59	−29	T⁻²⁹	STAB	Quantum stability
61	−30	T⁻³⁰	DECAY	Decay processes
67	−33	T⁻³³	SCAT	Scattering, CMB
71	−35	T⁻³⁵	TAU_ID	Tau identity (transversal)
73	−36	T⁻³⁶	OSC	Oscillations, wave structure
79	−39	T⁻³⁹	CPV	CP violation
83	−41	T⁻⁴¹	BRAN	Branching ratios — self-branching in Ω_Λ=83²
89	−44	T⁻⁴⁴	HAD_STR	Hadronic structure
97	−48	T⁻⁴⁸	STRUCT	Structure formation

10.3 Layer C and Layer D

When P contains a prime not in the lexicon above:

Layer C — compressed natural structure: the prime decomposes as a combination of lexicon primes with small integer coefficients. It is physical structure encoded compactly — a denser route through which distinction sustains itself.

137 = 11²−7²+5×13 (EM, color, curvature, weak)
307 = 11×29−2²×3 (EM, dark energy, time, alterity)

Layer D — axiomatic human choice: the prime cannot be decomposed into ArXe primes. It encodes a specific decision of the scientific community: a renormalization scheme, an extraction method, an institutional consensus. These are real — but historically constructed, not ontologically given.

1051 in sin²θ_W: MS-bar scheme at M_Z
107 in V_ud: Tevatron 1995 top quark convention
47 in α (delta 2018): 10th-order QED framework

11. The Complete Chain

The system connects four nodes without importing anything external:

AXIOM
¬() ≜ Tf ≃ Tp
Nothingness is inviable — it escapes as a fundamental pulse
    ↓
EXENTATION TABLE
n(k) = -2k+1 for k<0 → prime operators (irreducible resolution routes)
BC closed (k>0) → isolated existence → particles, masses
BC open (k<0) → confinement, fields → requires external coupling
    ↓
DIMENSIONAL FRAMEWORK
n_ArXe = 3M + 2L + T
T=T¹, L=T², M=T³ (degrees of freedom of the contradiction)
Planck units = natural ArXe scale (metric of the instability)
    ↓
DIMENSIONAL CONSTANTS (in Planck units)
Pure dimensionless numbers — the contradiction reading itself
Factorized into ArXe primes alone — no anchors needed
86% of SM masses ArXe-pure
    ↓
DIMENSIONLESS CONSTANTS
Transitions between dimensional levels — the grammar of relations
Need anchors: π (BC-closed limit) φ (BC-open limit) ρ (BC-open cubic)
Both anchors indecidable at T³ — the first level where the observer exists
88% of SM dimensionless constants with ArXe formula found
    ↓
THE READING (PLO v2)
Physical integer P = natural layer (the unfolding) + conventional layer (the observer)
d* = threshold digit (grammar of the unfolding ends, residue of the observer begins)
Deltas = primary signal (structural vs conventional correction)

The chain is reversible. From any node you can travel to any other.

12. The Naturality-by-Digit Principle

12.1 Statement

The significant digits of a dimensionless physical constant are not epistemically equivalent:

*Leading digits (1 to d):** natural layer — the grammar of the unfolding, the phenomenon speaking
*Trailing digits (d+1 to n):** conventional layer — the residue of the observer’s interaction, the physicists answering

More precise is not more natural. More precise is more negotiated.

The full ontological content of a constant is in its first few digits. Everything beyond is the history of how a community chose to access that content — which instruments, which schemes, which agreements.

12.2 Evidence

Constant	d*	Natural layer reading	Conventional layer
α⁻¹	3	137 = 11²−7²+5×13	9 digits of QED framework
α_s	2	P=3 (CYC, T⁻¹)	MS-bar scheme enters at digit 3
sin²θ₁₂	3	P=307 (Layer C)	Physical limit — no more precision
sin²θ₂₃	3	P=3×11×17 (ArXe pure)	Full precision is natural
sin²θ₁₃	4	P=11 (REG, T⁻⁵)	All 4 digits natural
Ω_b	3	P=7²=(CPX)²	Baryon fraction = color squared
m_p/m_e	4	1836=2²×3³×17 (ArXe)	7+ digits = convention
V_us	5	π/44	All 5 digits natural
V_ub	4	P=37 (TOP, T⁻¹⁸)	All 4 digits natural

12.3 Conjecture

For all Standard Model dimensionless constants: d*(C) ≤ 4.

The full ontological content of any constant fits in four significant digits. Everything beyond is the history of how physicists chose to access that content. Not less real — but a different kind of real.

13. The Observer’s Logical Position

The ArXe levels do not only describe the universe. They describe where the observer stands within the unfolding — which degrees of freedom of the contradiction the observer is made of, sees through, infers, or constructs.

13.1 The Four Strata

Stratum I — The universe we are: {2, 3, 5, 7}
Time (2/3), space (5), mass and objectivity (7). The degrees of freedom constituting directly inhabitable physics. No instrument required — these are the levels we are made of and exist within. We do not observe these; we are them.

Stratum II — The universe we see: {11}
The electromagnetic field. The last degree of freedom we experience directly and the first we use as a tool. Every measurement in physics ultimately ends in a photon. Prime 11 (REG, T⁻⁵) is simultaneously the boundary of direct experience and the universal channel of mediation. It is the pivot between the universe speaking and the physicists answering.

Stratum III — The universe we infer: {13, 17, 19, 23, 29, …}
Weak interaction, dark matter, inflation, dark energy. Degrees of freedom accessible only through the EM channel or gravitational effects. Never directly. We know them by their traces in Stratum II.

Stratum IV — The universe we construct: large primes — Layer D
Theoretical frameworks, renormalization schemes, institutional consensus. Not less real — but a different kind of real: historically constructed, not ontologically given. The fingerprints of the scientific community in the conventional layer of constants.

13.2 The Pivot

The pivot between what is naturally determined and what is humanly constructed is prime 11 — the electromagnetic field.

Everything at {2,3,5,7,11} and within d digits: the unfolding speaking.
Everything at {13,…} and beyond d digits: the observer answering.

13.3 What the System Generates

The system is self-referential and recursive. It does not seek a final Truth — it shows how complexity emerges from the impossibility of nothingness, and it keeps generating questions from its own structure:

What type of information-processing system inhabits levels {11, 13, 17, 19}? What physics would such a system experience as directly observable?

Do black holes process information at T⁻¹⁴ (VBG — persistent vacuum background)?

Are species collective intelligences that inhabit ontological levels inaccessible to the individual?

These questions have structural coherence within the framework. They are left open deliberately — not because they have no answers, but because pursuing them is a separate program.

Part III: Empirical Status

14. The Corpus

Dimensionless constants (25 analyzed, PLO v2):

22/25 with ArXe anchor formula found (88%)
7/25 with ArXe-pure P
Key readings: V_us = π/44 (simplest formula in corpus), Ω_b = 7² (baryon fraction = color squared), sin²θ₁₃ P=11 (reactor angle = pure EM), δ_CP = 324π−φ/4

Dimensional constants (masses in Planck units, 14 analyzed):

12/14 with ArXe-pure P (86%)
No anchors needed — primes alone sufficient
Proton = Neutron = 77 = 7×11 at current precision

Dimensional energies (6 analyzed):

5/6 ArXe-pure P (83%)
Λ_QCD P = 2×3×29 — vacuum background (VBG=29) enters directly at the confinement scale

Grammar v4 batches 1–7 (80 constants):

75/80 exact (93.75%), average error 0.0018%
Longest streak: 30 consecutive exact constants

Extended corpus total: ~119 constants with formal PLO v2 analysis

Chronological analysis (5 constants, 28 epochs):

27/28 epochs reproduced within 0.001% error
All deltas of α_s, m_Z, G_F, m_e, sin²θ₁₂ are ArXe-prime pure
Human-prime deltas identified: α (prime 47, 2018), Ω_m (prime 43, 2013)

15. What We Do Not Claim

We do not claim the primes cause the constants. We claim the constants have prime structure coherent with a derivable ontological system.

We do not claim post-hoc adjustment is impossible by coincidence. We claim cross-coherence — the same lexicon found by two independent programs, chronological deltas being structurally pure, dimensional equivalences emerging without being sought — makes the coincidence hypothesis progressively less plausible.

We do not claim the model is complete. Some masses carry human primes. Several lexicon levels (T⁻¹⁵, T⁻¹⁸, T⁻²⁰) have no directly identified physical phenomenon assigned. The full precision of α⁻¹ at 12 digits is not yet resolved.

We do not specify “if X, ArXe is refuted” — because that presupposes binary truth we reject. We adopt Popper’s original intent: the attitude of not defending ideas dogmatically. We remain open to reinterpretation. We engage in dialogue, not defense.

16. The Epistemic Posture

We do not claim:

“We prove that T³ = Mass”

We claim:

“The concept ‘mass’ maps the structure T³ better than any available alternative”

A mapping is good if it is internally coherent, systematically applicable, parsimonious, and corresponds reasonably with experiment within stated error tolerances. The mapping from ArXe levels to physical constants uses zero free parameters. Every prime that appears in a formula comes from the ontological table — none are fitted. The formulas are found, not constructed.

Part IV: Extensions and Open Questions

17. Levels Without Assigned Phenomena

The lexicon contains operators at levels T⁻¹⁵ (31, CHA), T⁻¹⁸ (37, TOP), T⁻²⁰ (41, ISO) that appear in constant formulas but have no directly identified physical phenomenon. They are degrees of freedom of the contradiction that the grammar says must exist — physics has not yet named what they contain. Or perhaps it has, and the naming has not been connected.

18. The Formal Derivation of ρ from ArXe

π = BC-closed ratio at T³ is derived (V3 + V4).
φ = BC-open ratio at T³ is derived (V4).
ρ = BC-open cubic ratio at T⁴ is argued but not fully derived. The next step is to show that T⁴ generates exactly the cubic self-reference structure that ρ encodes — that is, to show that ρ³=ρ+1 is the natural limit of the contradiction at the fourth degree of freedom.

19. The Precision of m_p/m_e

The proton-to-electron mass ratio at 11 significant digits (1836.15267343) has a physical integer P with large human primes. The base 1836 = 2²×3³×17 is ArXe-pure. The additional 8 digits carry convention. Understanding the exact threshold d* of this ratio — and which community decisions the conventional layer encodes — is an open task.

20. Relationship to Existing Theories

Quantum Mechanics: ArXe provides ontological foundation for indeterminacy (open BC), wave-particle duality (T² ↔ T⁻¹), and probability as structural rather than epistemic. ArXe does not replace QM — it describes the deeper degree of freedom from which QM emerges.

General Relativity: ArXe explains the emergence of spacetime (T² from indecidability), curvature (T⁻² variations), and mass-spacetime connection (T³). Does not replace GR — explains why GR structure is the natural form for the contradiction at those levels.

Standard Model: ArXe provides gauge group explanation (open BC → U(1), SU(2), SU(3)), confinement (open BC necessity), generation structure, and constant derivations. Extends SM — predicts structure, explains why SM has the form it has.

Appendix A: The Mapping Criterion

A good ArXe mapping satisfies four conditions:

Internal coherence — no contradictions within the system
Systematic applicability — works across many independent cases
Parsimony — minimal free parameters, maximum structural content
Experimental correspondence — within stated error tolerance

Zero free parameters. Every prime that appears in a formula comes from the ontological table. The formulas are found, not constructed.

Appendix B: Complete Corpus Reference

Document	Content
`PLO_v2_REFERENCE.md`	Unified operational reference — full lexicon, all corpora, analysis template
`Grammar_V4_corrected.md`	Grammar v4.1 with all corrections integrated
`plov2_corpus_dimensionless_complete.md`	25 dimensionless constants, PLO v2
`plov2_dimensional_framework.md`	Dimensional framework and Planck masses
`paper_PLOv2_methodology_2.md`	PLO v2 formal methodology
`Grammar_V4_Correcciones.md`	(superseded — corrections integrated into v4.1)

Conclusion

ArXe is a recursive, self-referential system. It does not seek a final Truth — it shows how complexity emerges from the impossibility of nothingness.

The path is:

Inviability of nothingness → Escape as fundamental pulse → Degrees of freedom of the contradiction → Irreducible resolution routes (primes) → Dimensional levels → Planck scale → Dimensional constants (primes alone) → Dimensionless constants (primes + BC anchors) → Reading layer by layer with PLO v2.

At every step, the structure is derived — not postulated, not fitted. The anchors π and φ are the geometric limits of logical ambiguity at T³. The Planck scale is the metric of the instability of nothingness. The prime operators are the frequencies at which distinction sustains itself without redundancy.

The result is a system where the most abstract (the inviability of nothingness ¬() ≜ Tf ≃ Tp) and the most concrete (the fourth decimal of a neutrino mixing angle) are connected by a single coherent chain — and where each digit of each constant tells us whether we are reading the unfolding, or the history of those who tried to measure it.

The leading digits of a physical constant are the unfolding speaking.
The trailing digits are the observer answering.
The chain from axiom to corpus shows which is which.

ArXe Theory V4.2 — March 2026
Diego Luis Tentor
License: CC BY-SA 4.0