ArXe Theory V4.1: The Active Unfolding of Primordial Contradiction

Foundations, Dimensional Framework, and the Grammar of Physical Constants

Diego Luis Tentor — ArXe Research — 2026
Version 4.1 — March 2026
Supersedes: arxe_core_V4.md (v4.0, March 2026)
Status: Living theoretical framework
License: CC BY-SA 4.0
Source corpus: github.com/diego-tentor/ArXe_V3

“We do not prove. We show:
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

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.

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.

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

A careful reader will notice that the table in 5.3 assigns Mass to T³, Length to T², and Time to T¹ — and may reasonably ask: why these assignments and not others?

This question deserves a precise answer, because the entire framework rests on it.

The wrong question and the right question

The wrong question is: “Why did ArXe assign mass to T³?”

This phrasing implies an arbitrary decision — as if someone looked at the hierarchy and said “let’s put mass here.” If that were the case, the assignments would be post-hoc labels, not structural results, and the framework would be circular.

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 then 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 is not satisfied at T¹ (where time is homogeneous positive flow — one direction, no distinction between before and after as a record) or at T² (where simultaneity is the dominant structure). It 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” — what the framework calls ternary objectivity: 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). At T² (four-ary, pairs of contraries), there is no structural third — only pairs. At T⁻¹ (three-phase, ternary logic) the third exists but the level has an open BC and cannot exist in isolation.

Condition 3: Capacity for isolated existence
Mass can exist without requiring another structure to “close” it. A mass does not need to be coupled to something else to be — it simply is. Formally, this requires all boundary conditions to be closed (k > 0). T³ is the first level with k = 3 > 0 and three closed BCs, satisfying this condition. All negative levels (T⁻¹, T⁻², T⁻³…) have at least one open BC and cannot exist in isolation.

Condition 4: Three-dimensional spatial presence
Mass occupies space in three dimensions. It requires that the three basic spatial conditions (extension, reversibility, depth) all be simultaneously active. 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: At T², there is historical persistence in a limited sense (space persists), but there is no past-present-future structure. Ternary objectivity is also 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 the right complexity (n=7, closest to T³’s n=6) but has one open BC. It cannot exist in isolation — it requires coupling. This is exactly why color charge (T⁻³) is always confined: its open BC means it structurally cannot exist without a partner to close it.

T⁴ satisfies all four conditions — but it is not the first level that does so. T⁴ has four closed BCs, historical persistence, ternary objectivity, and three-dimensional presence. It satisfies the minimum conditions for mass and more. But T³ already satisfies all four with exactly three closed BCs — no more, no less. By the structural parsimony principle, T³ is the minimum level at which mass becomes possible. T⁴ and beyond can have mass (they inherit the conditions), but mass does not require them.

The general principle

This reasoning generalizes to all level assignments:

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.

This principle has three consequences:

1. Assignments are derivable, not stipulated.
They are not labels chosen to make the framework look good. They follow from asking which structural conditions each physical concept requires and finding the minimum level that provides them.

2. Absence is as informative as presence.
If a concept cannot be assigned to any level — because no level satisfies its minimum conditions — this is a structural result of the framework, not a gap to be filled by adding a new level. Conversely, if a level has no known physical concept assigned to it, this does not mean the level is empty: it means we have not yet identified the physical concept whose minimum conditions it satisfies.

3. The framework is not circular.
The charge of circularity would be valid if T³ were defined as “the level of mass.” It is not. T³ is defined by its structural properties: six-ary logic, three closed BCs, zero open BCs, recursive structure generated by three exentations from T⁰. Mass is then identified as the physical concept whose minimum conditions of possibility are exactly those structural properties. The direction of determination runs from structure to concept, not from concept to structure.

Application to the open questions

This criterion directly addresses several questions that the framework leaves implicit:

Q: Why does the observer emerge at T⁻⁵ (prime 11)?

The minimum conditions for observation are: (a) the capacity to register a state without being modified by the registration — which requires a gauge freedom, i.e., an open BC that allows choosing a reference without affecting the measured system; (b) sufficient internal complexity to encode the registered state — which requires at least 11 phases (n=11); (c) the ability to project onto spatial structure without exhausting it — which requires four closed BCs alongside the one open BC. T⁻⁵ is the first level satisfying all three. The observer does not emerge at T⁻⁵ because we placed it there — it emerges there because that is the minimum level where something can register without being fully determined by what it registers.

Q: Why is n=6 mass and not n=5 or n=7?

n=5 corresponds to T⁻² (k=-2), which has one open BC — it cannot exist in isolation. It fails condition 3. n=7 corresponds to T⁻³ (k=-3), which also has one open BC — same failure. n=6 is T³ (k=3), the first even-arity positive level with three closed BCs and zero open BCs. It is not that n=6 is mass. It is that n=6 is the first arity value that makes isolated, historically persistent, ternarily objective, three-dimensionally present existence structurally possible — and mass is what we call things that have exactly those properties.

Q: What determines the jump between levels?

Each level is generated from the previous by one exentation — one additional structural differentiation. The jump is not arbitrary: it follows the recursive rule Entₙ := Entₙ₋₁ ∧ ExEntₙ₋₁. What changes at each jump is the number of available phases (arity), the configuration of open and closed BCs, and consequently the structural capacities of that level. The jump from T² to T³ is the jump from “space with reversibility” to “space-time with historical record and ternary objectivity” — which is why it is also the jump from the possibility of length to the possibility of mass.

Summary

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

The table is not a list of assignments. It is a record of which structural thresholds each level crosses — and which physical concepts become possible at each threshold for the first time.

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. Force is not a separate degree of freedom of the contradiction; it is T³ in motion.

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. This is the geometric expression of BC-closed levels (k > 0). 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. The structure requires external coupling to exist. This is the geometric expression of BC-open levels (k < 0). 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.

9. PLO v2 — The Reading Method

9.1 What PLO v2 Is

Physical constants are the chronometric traces of the exentation process. They are not independent facts about the universe — they are the imprints left by the process of distinction unfolding through specific degrees of freedom.

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, Not a Placement

Why the Observer at T⁻⁵ — Necessary or Contingent?

The question

The framework assigns the electromagnetic field — and with it, the observational capacity of physical systems — to T⁻⁵ (prime 11, four closed BCs, one open BC). A natural question follows: is this contingent? Could the observer have emerged at a different level? Or is there a structural argument that makes T⁻⁵ the necessary minimum for anything to function as an observer?

This question has two parts that must be answered separately:

Why T⁻⁵ and not some other level?
Is this necessity absolute or conditional?

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 at least four structural conditions:

Condition O1: Registration capacity
The observer must be able to 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 — a minimum number of distinguishable internal phases.

Condition O2: Non-exhaustion of what is observed
The act of registration must not consume or fully determine the observed system. An observer that collapses what it observes into itself is not an observer — it is an absorber. This requires that the observer have a degree of freedom that is not fixed by the observation — a gauge freedom. Formally: at least one open BC, which allows the observer to choose its reference without that choice affecting the measured system.

Condition O3: Projection onto spatial structure
The observer must be able to project the registered state onto the spatial level (T²) where physical phenomena unfold. Without this capacity, registration remains internal and cannot be communicated or compared. This requires enough closed BCs to couple stably with T².

Condition O4: Not being mass itself
If the observer were a positive level (k > 0, all BCs closed), it would be a massive structure — it would interact gravitationally and electromagnetically with what it observes in a way that makes the observation inseparable from the interaction. The observer must be a field-like structure (k < 0, at least one open BC) to retain the gauge freedom of condition O2.

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

Let us check each negative level in sequence:

T⁻¹ (prime 3, 0 closed, 1 open):
Fails O1 — zero closed BCs means no stable internal structure to encode external states. It has gauge freedom (open BC) but nothing to write on. Cannot observe.

T⁻² (prime 5, 1 closed, 1 open):
Fails O1 — one closed BC provides minimal structure, but insufficient phases (n=5) to encode the complexity of a physical state in T². Has gauge freedom but cannot project onto T² with enough resolution.

T⁻³ (prime 7, 2 closed, 1 open):
Partially satisfies O1 (7 phases, more internal structure) and O2 (open BC). But fails O3 — T⁻³ is the level of color confinement. Its coupling to T² is mediated through T³ (mass), not direct. It cannot project independently onto spatial structure — it is always confined within hadrons. Cannot observe directly.

T⁻⁴: structural gap
n(-4) = 9 = 3², not prime. As established elsewhere, levels whose arity is not prime do not generate irreducible ontological operators — they are composite structures, not independent levels. 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 (2 closed BCs) without being confined
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 — the physical structure that T⁻⁵ describes — 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, k < 0) to not be an object itself.

Is this necessary or contingent?

The necessity here is conditional, not absolute:

What is absolutely necessary: If something functions as an observer in a universe with the level structure of ArXe, it must operate at T⁻⁵ or higher. No level below T⁻⁵ satisfies all four conditions. This is a structural theorem, not a contingent fact.

What is not absolutely necessary: That observers exist at all. The framework does not require that T⁻⁵ structures couple to T³ structures in ways that produce observers. It only says that if such coupling occurs, it necessarily occurs at T⁻⁵ or above.

The conditional form:

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 also explains why observation in quantum mechanics has always been associated with electromagnetic interaction — not as a postulate but as a structural consequence. Measurement is always ultimately an electromagnetic process because T⁻⁵ is the minimum level where registration without collapse is structurally possible.

The hierarchy of observers

This reasoning also implies a hierarchy: levels above T⁻⁵ can also function as observers, with increasing complexity.

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. An observer sensitive to identity, not just presence.
T⁻⁸ (Hyperspace, prime 17): Registers states that require hyperspatial structure. Speculative at this stage.

Each level up adds observational capacity — more phases, more closed BCs, more internal structure to encode — 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 not coincidence — it 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 V4.1 — March 2026
Diego Luis Tentor — ArXe Research
License: CC BY-SA 4.0

“We do not prove. We show — how complexity emerges from the impossibility of nothingness.”

ArXe Theory V4.1:Foundations, Dimensional Framework, and the Grammar of Physical Constants