ArXe Theory: The Logical-Physical Co-emergence of the Universe

Foundations, Dimensional Framework, and the Grammar of Physical Constants

Diego Luis Tentor — ArXe Research — 2026
March 2026 · Status: Living theoretical framework

How complexity emerges from the impossibility of nothingness, how indecidability generates space, open boundaries generate fields, and arity numbers 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 that 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 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.

This distinguishes ArXe from:

Mathematical realism / Platonism — which presupposes mathematical entities as fundamental. ArXe presupposes no entities, 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. No observer is required for ¬() to generate structure; the observer emerges much later, at a specific and derivable level (T⁻⁵, arity 11 — see §9).

What’s New in This Version

The original Foundations article (2025) established the axiom, the exentation hierarchy, boundary-condition (BC) algebra, arity number encoding, and the emergence of π from ternary ambiguity. This update completes the bridge from that ontological core to empirical physics:

ALO — the chronological reading method for constants, with exact values and delta analysis across measurement epochs.
The complete dimensional framework — how any SI dimension maps to T^n, and the derivation of Planck units as ArXe’s natural scale.
The full chain from axiom to constants — dimensional constants → dimensionless constants, formalized end to end.
φ = BC-open anchor, π = BC-closed anchor — now formally derived as boundary-condition anchors rather than postulated.
The Naturality-by-Digit Principle — what the leading digits of a constant reveal about the unfolding itself, versus what the trailing digits reveal about the observer’s intervention.
The observer’s logical position — the four strata of reality relative to the observer, and the pivot at arity 11.
The extended unified lexicon — 25 arity number operators, confirmed convergently by two independent research programs.
Layer C / Layer D — the formal distinction between compressed natural structure and axiomatic human choice in a constant’s factorization.
The Principle of Phase Ordering and Physics as Statistical Manifestation — the probabilistic structure underlying each level, and why observed physics is the dominant statistical layer of a deeper combinatorial space.
The Hierarchy as a Chain of Inversions — a structural derivation of why levels alternate open/closed rather than accumulating monotonically.
Why Time and Space Are Contraries — indecidability itself as the origin of geometry (√2, π, curvature, SU(3), U(1) all read as forms of undecidable simultaneity).
The Observer at T⁻⁵ — shown to be a structural necessity, not a 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 claim identity between logic and physics. It describes the event in which nothingness, being logically inviable, escapes into a fundamental pulse of time perceived 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 be solved. It is the generative engine of all reality — comparable to Dedekind’s cut, which constructs the real numbers from a division that belongs completely to neither of the sets it separates. The universe does not resolve ¬(); it perpetually escapes it, and that escape is what we call time, space, and structure. Crucially, this contradiction prevents what exists from being the foundation of itself — avoiding the circular paradox of a reality that would sustain itself with no external reference.

2. The Exentation Hierarchy (Recursive Fractal Ontology)

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 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 level of exentation exposes an n-ary logical structure:

n=1 — unary logic (contradictory, the collapse point)
n=2 — binary logic (excluded middle, contraries, true/false)
n=3 — ternary logic (modality, included middle, the first “observer” position, objectivity)
n=4 — quaternary logic (pairs of contraries, polarity, simultaneity)
n≥5 — n-ary logics of increasing complexity

The exentation number n determines the number of temporal phases, the logical arity, and the phenomenological structure available at that level. Each phase at its level is isomorphic; the phenomenological totality is the superposition of all levels.

Temporal threshold semantics

This structure admits a formal treatment via a temporal threshold semantics, which assigns each level of the exentation chain a minimum availability time rather than an immediately available classical truth value. This treatment resolves the classical worry of logical explosion (ex contradictione quodlibet) that would threaten any system built on a foundational contradiction, and is consistent with the already-established distinction between essence and existence (§X).

See Appendix: Logical Foundations of the Exentation Chain.

3. The Mapping Functions

Forward mapping e(n) → k:

e(1) = 0
e(n) = (−1)ⁿ · ⌊n/2⌋,   n > 1

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

Inverse mapping n(k) ← k:

n(0) = 1
n(k) = 2k,        k > 0
n(k) = −2k+1,     k < 0

Theorem: e and n establish a bijection between ℕ and ℤ.

The Principle of Phase Ordering (§3.5–3.7, new in V4.2)

Not all orderings of phases within a level are equally probable. The factored form of a level’s arity reveals a conditionality: some phase orderings are structurally favored because they correspond to more factored — more combinatorially accessible — configurations. This grades smoothly from T⁰ (no ordering yet possible) to T⁻¹ (where ordering first becomes meaningful), and it generalizes into three linked principles:

Probabilistic Freedom — at each level, multiple phase orderings are logically permitted; which one is realized is not fixed by the axiom alone.
Relative Logical Necessity — necessity is always relative to a level: what is contingent at T^k can be necessary as a precondition for T^(k+1).
The Scale of N-ary Logics — as arity grows, the space of permissible configurations grows combinatorially, but the physically dominant configurations remain the most factored ones.

Physics as Statistical Manifestation: observed physical structure is not the only structure logically permitted at a given level — it is the dominant statistical layer of a much larger combinatorial space of possible resolutions. This reframes “physical law” as the statistically preferred outcome of the underlying combinatorics, not as an externally imposed rule. (This same statistical logic reappears independently in the derivation of Madelung’s rule for atomic orbitals — see the companion article on Madelung’s Rule.)

4. Boundary Conditions, Arity Encoding, and the Chain of Inversions

Levels with closed boundary conditions (BC, k>0) correspond to isolated, self-contained existence: particles, masses. Levels with open boundary conditions (k<0) correspond to confinement and fields: they cannot exist without external coupling.

The Hierarchy as a Chain of Inversions (§4.4, new in V4.2): the alternation between closed and open BC across the hierarchy is not arbitrary — it follows from a generative mechanism of inversion of indecidability. Each level’s resolution of ambiguity inverts the type of boundary the next level must confront, producing the necessary alternation T¹(closed) → T⁻¹(open) → T²(closed) → T⁻²(open)… This is a structural derivation, not an observed pattern fitted after the fact.

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 the Principle of Non-Contradiction), 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.

5.4 Why These Assignments? Minimum Conditions of Possibility (new framing in V4)

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?

For something to behave as mass, four conditions must hold simultaneously:

Historical persistence — past, present, and future must be structurally distinguishable. First available at T³, where the six-phase structure generates an irreversible past→present→future sequence.
Ternary objectivity — mass must be measurable by a third party independent of the measurement itself. Requires a structural third beyond a binary A/B — first available at T³ (six-ary logic, included middle).
Capacity for isolated existence — mass exists without needing another structure to “close” it; requires all boundary conditions closed (k>0). T³ is the first level with three closed BCs.
Three-dimensional spatial presence — T³ is the first level with exactly three independent structural degrees of spatial closure.

T² fails conditions 1–2 (no historical structure, no ternary objectivity — only pairs of contraries). T⁻³ fails condition 3 (one open BC — exactly why color charge is always confined). T⁴ satisfies all four conditions but is not the first level to do so — T³ already does, with structural parsimony.

The general principle:

A physical concept X maps to level T^k
iff T^k is the minimum level whose structural properties
(arity, number/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 makes assignments derivable rather than stipulated, and the direction runs from structure to concept — not the reverse. It also answers, by the same method: why the observer emerges specifically at T⁻⁵ (§9.8 below), and why n=6 (T³) is mass rather than n=5 or n=7.

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 (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), and 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 — not as a universal mathematical constant the universe happens to use, but as what the contradiction becomes when it closes on itself geometrically at T³.

Part II — The Complete Bridge

7. The Dimensional Framework

For any physical quantity with SI dimension $$M^aL^bT^c$$, the ArXe exponent is:

$$n_{ArXe} = 3a + 2b + c$$

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

Three structural equivalences fall out of this reduction:

Force = Mass = T³. F=ma, and acceleration is T⁰ — dimensionless in ArXe. Force has no independent ontological status; it is mass acting along a dimensionless ratio.
c = G = T¹. The speed of light and Newton’s constant share ArXe dimension T¹ — both are conversion factors between the T² (length) and T¹ (time) scales. Setting c=G=1 in Planck units removes two redundancies at the same level.
Momentum = Power = T⁴. Two quantities that look distinct in conventional physics (ML/T vs. ML²/T³) occupy the same degree of freedom in ArXe.

Planck units as the natural ArXe scale: in Planck units (c=G=ℏ=1) every dimensional constant becomes a pure number — a ratio against the Planck unit at its own level — which ALO then reads exactly like any dimensionless constant. Confirmed empirically: Standard Model masses in Planck units are 86% ArXe-pure, needing no mathematical anchors at all. The Planck scale is not chosen for convenience — it is the natural scale of the theory, because c and G sit at T¹, the root level of the contradiction.

Selected readings: Electron P=42=2×3×7 (DIFF×CYC×CPX); W boson P=66=2×3×11 (DIFF×CYC×REG); Z boson P=75=3×5² (CYC×MEM²); Proton = Neutron P=77=7×11 (CPX×REG) — at current precision, both nucleons share the same ArXe identity: color × electromagnetic regulation. The full dimensional table, with the length-scale exception (lengths do not factor cleanly, because atomic/nuclear length scales sit far from the Planck length and accumulate convention along the way), is developed in the companion article “Table from Logical to Physical Structure.”

8. φ and π as Boundary Condition Anchors

π and φ are not universal mathematical constants that merely 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 encodes a relationship whose boundary conditions are fully closed and self-sufficient.
φ — BC-Open Anchor. The metric of a distinction that projects outward to avoid collapse: φ = 1 + 1/φ, an infinite continued fraction always requiring one more term. φ encodes a relationship whose boundary conditions remain permanently open.
ρ — BC-Open Cubic Anchor. ρ³=ρ+1, one order above φ (φ²=φ+1); appears in running couplings and structures requiring three-way recursive coupling. Emerges at T⁴.

π and φ become simultaneously indecidable at T³, the first level of objectivity — their co-existence mirrors the co-existence of closed and open BC, and this generative tension drives the exentation onward to T⁴ and beyond. Because dimensionless constants describe relationships between levels rather than levels themselves, and those relationships are measured at T³, π naturally anchors geometric couplings and closed structures, φ anchors mixing amplitudes and open transitions, and ρ anchors running/recursive couplings. Dimensional masses, describing a level directly rather than a transition, need no anchor at all.

8.4 Why Time and Space Are Contraries — Indecidability as the Origin of Geometry (new in V4.2)

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. Space is the structure where order cannot be decided — when two phases cannot be ordered, they must coexist, and that coexistence is simultaneity. Time and space are not two flavors of the same thing; they are logical contraries — the presence and absence of decidable order.

Crucially, each level’s aridity generates its own distinct form of simultaneity — its own 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 (in/out)	curved manifold, geodesics	metric tensor
T⁻³	7	three-color (no preferred color)	internal color space (confinement)	SU(3), 8 generators
T⁻⁵	11	continuous phase (no preferred θ)	phase circle	U(1)

π is not a number that happens to appear in geometry — π is the geometric constant of ternary undecidable simultaneity. The same logic extends to spirals (the geometry generated when T⁻¹’s undecidable temporal direction is projected into T²’s flat simultaneity — explaining the universal appearance of logarithmic spirals in galaxies, shells, hurricanes, and vortices) and to a falsifiable spectroscopic prediction: transitions that stay within one ontological level (intra-level, e.g. 4f→4f in lanthanides) should produce narrow emission lines (~1–3 nm FWHM), while transitions that cross a level boundary (inter-level, e.g. 5d→4f) should produce emission broadened by roughly two orders of magnitude (~100 nm FWHM) — because the transition must negotiate an interface between two structurally different BC configurations. This is directly testable and is illustrated in the corpus by the Ce³⁺/Yb³⁺ pair in YAG.

9. ALO — The Reading Method

Physical constants are the chronometric traces of the exentation process. ALO is the reading instrument for those traces: it converts any physical constant into a factorization of arity number operators and reads the ontological structure that factorization encodes. The leading digits (the natural layer) show the grammar of the unfolding; the trailing 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, not the reverse.

Physical Integer P(C, n):

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

Anchor decomposition A(C): express C ≈ (p/q) × anchor^r, with p, q small ArXe-arity number integers, anchor ∈ {π, φ, ρ}, r ∈ {1, 2, ½, −1}; valid when error < 0.1%.

Delta Δ(C, t₁, t₂) = |C(t₂) − C(t₁)| — the change between two measurement epochs, factorized after scaling. The delta is the primary signal, isolating what changed and distinguishing unfolding structure from observer choices. ΔNI (the maximum arity number in the scaled delta’s factorization) tells us whether a correction was structural (ΔNI is an ArXe arity — the unfolding revealing more of itself) or conventional (ΔNI is not an ArXe arity — the observer refining their framework).

The protocol, always in this order: (1) obtain the exact value from a primary source, no truncation; (2) compute P(C,n) for every available epoch; (3) search for an anchor formula with error < 0.001%; (4) analyze deltas between epochs; (5) read the arity number grammar in the context of the ArXe level T^n.

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

9.8 The Observer at T⁻⁵: A Structural Necessity (new in V4.2)

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

O1 — Registration capacity: internal complexity sufficient to encode external states without being identical to them.
O2 — Non-exhaustion of what is observed: at least one open BC, giving a gauge freedom that lets the observer choose its reference without affecting the measured system.
O3 — Projection onto spatial structure: enough closed BCs to couple stably with T².
O4 — Not being mass itself: the observer must retain gauge freedom (k<0, field-like), or observation collapses into interaction.

Checking each candidate level: T⁻¹ (arity 3) fails O1 — zero closed BCs, no stable internal structure. T⁻² (arity 5) fails O1 — insufficient phases. T⁻³ (arity 7) fails O3 — it couples to T² only indirectly, through T³. T⁻⁴ does not exist as an independent level at all (its arity, 9=3², is not an arity number, so it generates no irreducible operator). T⁻⁵ (arity 11, 4 closed + 1 open BC) satisfies all four conditions simultaneously: 11 phases (O1), one open BC giving U(1) gauge freedom (O2), four closed BCs coupling stably to T² (O3), and field rather than mass structure (O4).

T⁻⁵ is therefore the minimum negative level satisfying all four conditions — not a postulate, but a structural consequence. This is why the electromagnetic field is the carrier of observation in nature: not because we assigned observation to EM, but because EM is the first field complex enough to register, free enough (gauge) not to collapse what it registers, stable enough to project onto space, and light enough (massless) not to be an object itself. The necessity 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 — which is exactly why observation in quantum mechanics has always been tied to electromagnetic interaction. Beyond T⁻⁵, observational capacity extends further up the hierarchy: T⁻⁶ (weak, arity 13) can register flavor states; T⁻⁸ (hyperspace, arity 17) registers states requiring hyperspatial structure.

10. The Extended Unified Lexicon

Grammar was developed empirically, from 80 physical constants, without reference to the ArXe exentation table. ArXe’s arity number encoding was developed theoretically, from the axiom alone. When compared: 24 of 25 primary operators in Grammar are exactly the levels generated by n(k) = −2k+1 for negative k. (The sole exception is arity 2, from the positive level T¹.) This convergence of two independent programs is mutual confirmation — the empirical search found the same irreducible resolution routes the theoretical derivation predicted.

Arity number	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	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
73	−36	T⁻³⁶	OSC	Oscillations, wave structure
79	−39	T⁻³⁹	CPV	CP violation
83	−41	T⁻⁴¹	BRAN	Branching ratios
89	−44	T⁻⁴⁴	HAD_STR	Hadronic structure
97	−48	T⁻⁴⁸	STRUCT	Structure formation

Layer C (compressed natural structure): when a factor is not itself in the lexicon but decomposes as a small-integer combination of lexicon arity numbers, it is treated as physical structure encoded compactly — e.g. 137 = 11²−7²+5×13 (EM, color, curvature, weak); 307 = 11×29−2²×3 (EM, dark energy, time, alterity).

Layer D (axiomatic human choice): when a factor cannot be decomposed into ArXe arities at all, it encodes a specific historical decision of the scientific community — a renormalization scheme, an extraction method, an institutional consensus. Real, but historically constructed rather than ontologically given. Examples: 1051 in sin²θ_W (MS-bar scheme at M_Z); 107 in V_ud (a 1995 Tevatron top-quark convention); 47 in the 2018 delta of α (the 10th-order QED framework).

11. The Complete Chain

AXIOM
¬() ≜ Tf ≃ Tp — nothingness is inviable, escapes as a fundamental pulse
    ↓
EXENTATION TABLE
n(k) = −2k+1 for k<0 → arity number operators (irreducible resolution routes)
BC closed (k>0) → isolated existence → particles, masses
BC open (k<0) → confinement, fields → requires external coupling
    ↓
DIMENSIONAL FRAMEWORK
T,L,M ≡ T¹,T²,T³ → Planck units as the natural scale
    ↓
DIMENSIONAL CONSTANTS
masses & energies in Planck units → arity numbers alone, no anchors needed (86%/83% pure)
    ↓
DIMENSIONLESS CONSTANTS
transitions between levels → arity numbers + BC anchors (π, φ, ρ)
    ↓
ALO READING
natural layer (leading digits) vs. conventional layer (trailing digits)

12. The Naturality-by-Digit Principle

The significant digits of a dimensionless constant are not epistemically equivalent. Leading digits (1 to d*) are the natural layer — the phenomenon speaking. Trailing digits (d*+1 to n) are the conventional layer — the residue of the observer’s interaction. More precise is not more natural. More precise is more negotiated.

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 (CPX²)	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

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 recorded history of how physicists chose to access that content.

13. The Observer’s Logical Position

The ArXe levels do not only describe the universe — they describe where the observer stands within the unfolding.

Stratum I — The universe we are: {2, 3, 5, 7}. Time, space, mass and objectivity. No instrument required; we do not observe these levels, 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. Arity 11 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 — known only through the EM channel or gravitational traces.
Stratum IV — The universe we construct: large arities (Layer D). Renormalization schemes, institutional consensus — real, but historically constructed rather than ontologically given.

Open questions the framework generates but does not resolve: what kind of information-processing system inhabits levels {11,13,17,19}, and what physics would it experience as directly observable? Do black holes process information at T⁻¹⁴ (VBG)? Are species collective intelligences inhabiting ontological levels inaccessible to the individual? These remain deliberately open.

Part III — Empirical Status

14. The Corpus

Dimensionless constants (25 analyzed): 22/25 with an ArXe anchor formula found (88%); 7/25 with ArXe-pure P. Notable readings: V_us = π/44 (simplest formula in the corpus); Ω_b = 7² (baryon fraction = color squared); sin²θ₁₃ P=11 (reactor angle = pure EM); δ_CP = 324π−φ/4.
Dimensional masses in Planck units (14 analyzed): 12/14 ArXe-pure (86%), no anchors needed. Proton = Neutron = 77 = 7×11 at current precision.
Dimensional energies (6 analyzed): 5/6 ArXe-pure (83%). Λ_QCD P = 2×3×29 — the vacuum background operator enters directly at the confinement scale.
Grammar, batches 1–7 (80 constants): 75/80 exact (93.75%), average error 0.0018%; longest streak of 30 consecutive exact constants.
Extended corpus: ~119 constants total with formal ALO 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-arity number pure. Human-arity number deltas identified at α (arity 47, 2018) and Ω_m (arity 43, 2013).

15. What We Do Not Claim

We do not claim the arities cause the constants. We claim the constants have Arity 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 independently by two research programs, chronologically-pure deltas, dimensional equivalences emerging unsought — makes the coincidence hypothesis progressively less plausible.

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

We do not specify a single falsification criterion, since that presupposes a binary-truth framing we reject; we adopt Popper’s underlying attitude — not defending ideas dogmatically, remaining open to reinterpretation, engaging in dialogue rather than 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 no continuous free parameters — every arity number 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

Levels without assigned phenomena. T⁻¹⁵ (31, CHA), T⁻¹⁸ (37, TOP), T⁻²⁰ (41, ISO) appear in constant formulas but have no directly identified physical phenomenon — degrees of freedom the grammar says must exist, awaiting a name (or a reconnection to one already known).
The formal derivation of ρ. π (BC-closed at T³) and φ (BC-open at T³) are derived. ρ (BC-open cubic at T⁴, ρ³=ρ+1) is argued but not yet fully derived — the open task is showing that T⁴ generates exactly the cubic self-reference structure ρ encodes.
The precision of m_p/m_e. At 11 significant digits (1836.15267343), the base 1836 = 2²×3³×17 is ArXe-pure; the remaining 8 digits carry convention whose exact threshold d* and community-decision content remain to be mapped.
Relationship to existing theories. ArXe does not replace quantum mechanics, general relativity, or the Standard Model — it proposes the deeper degree of freedom from which each already-successful theory’s specific structure (indeterminacy, gauge groups, confinement, curvature, generation structure) emerges as a necessary consequence rather than a brute fact.

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, and it keeps generating questions from its own structure. The path, end to end:

Inviability of nothingness → escape as a fundamental pulse → degrees of freedom of the contradiction → irreducible resolution routes (arity numbers) → dimensional levels → Planck scale → dimensional constants (arity numbers alone) → dimensionless constants (arity numbers + BC anchors) → reading layer by layer with ALO.

For the applied dimensional analysis of Standard Model constants: “Table from Logical to Physical Structure”
For the detective-story account of how the grammar was found: “The Universe’s Grammar”
For the formal grammar itself: Grammar_V4_s_en.md (ALO Pure Grammar)
For a worked structural application: “Derivation of Madelung’s Rule from ArXe Exentation Theory”
Source corpus: github.com/diego-tentor/ArXe

Arxe Theory

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