# The Standard Model and Beyond: Complete Observer-Dependent Reinterpretation of Particles

> Complete observer-dependent reinterpretation of elementary particles shows that the Standard Model describes 39 fundamental roles (not 17), distributed across two adjacent recursion levels (d=0 and d=−1), bridges, and trans-level entities. The gauge group SU(3)×SU(2)×U(1) is derived structurally from the ODTOE triad architecture. Cosmological proportions ΩΛ:ΩDM:Ωb = φ²:1:Z match Planck 2018 within 1.2σ with zero free parameters. Proton-to-electron mass ratio mp/me = 6π⁵ = 1836.12 reproduced to 0.002% accuracy. All 34 of 39 roles confirmed by PDG 2025, 2 have experimental candidates (HNL), 3 are pure ODTOE predictions.

Source: https://odtoe.org/en/articles/standard-model-beyond
Author: Anton Pankratov · Observer-Dependent Theory of Everything (ODTOE) · CC BY 4.0

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THE STANDARD MODEL AND BEYOND: COMPLETE OBSERVER-DEPENDENT REINTERPRETATION OF PARTICLES Anton S. Pankratov Independent researcher, Kazan, Russia E-mail: anton.s.pankratov@gmail.com ORCID: 0009-0002-4870-2995

ABSTRACT A complete observer-dependent reinterpretation of elementary particles within the formalism of ODTOE (Observer-Dependent Theory of Everything) is presented. It is shown that the Standard Model describes 39 fundamental roles (not 17), distributed across two adjacent recursion levels (d = 0 and d = −1), bridges, and trans-level entities. Each of the 39 roles is interpreted as a stable configuration of the field of potential states H at specific values of coherence S and dimensionality d. The gauge group SU (3) × SU (2) × U (1) is derived structurally from three independent aspects of the ODTOE triad architecture. The universal invariant 17 is obtained as a combinatorial constant of a single level of infinite recursion Φ = ι ◦ Ô. Cosmological proportions ΩΛ : ΩDM : Ωb = φ2 : 1 : Z (where Z = (π − 3)/(1 − (π − 3)φ)) match Planck 2018 data within 1.2σ with zero free parameters. The proton-to-electron mass ratio mp /me = 6π 5 = 1836.12 is reproduced to 0.002% accuracy. Recursive infinite nesting generates twelve falsifiable predictions. All 34 of 39 roles are confirmed by PDG 2025, 2 have experimental candidates (HNL), and 3 are pure ODTOE predictions. Keywords: Standard Model, ODTOE, toroidal topology, gauge group, triad architecture, φ-scaling, infinite recursion, falsifiable predictions, golden ratio, cosmological proportions.

I. INTRODUCTION: 17 PARTICLES AS 17 OBSERVER CONFIGURATIONS The Standard Model (SM) describes 17 elementary particles: 6 quarks, 6 leptons, 4 gauge bosons and the Higgs boson, as well as 3 fundamental interactions (strong, electromagnetic, weak). Gravity remains outside the SM. SM counting convention (not ODTOE). The number “17” in the SM is a projection, not a structural constant. It arises from a convention: the gluon is counted once (although there are 8), W + and W − as one type, antiparticles are not distinguished, the proton and neutron are considered “composite” rather than roles. ODTOE shows that behind this convention lie 39 fundamental roles — 17 at each of two adjacent levels (d = 0 and d = −1) + 3 bridges + 2 trans-level entities.

In the Observer-Dependent Theory of Everything (ODTOE), particles are interpreted as stable configurations of the field of potential states H at specific values of coherence S and dimensionality d. The number 17 appears in ODTOE as a structural invariant of a single level: N (d) = R × 3 + O × 3 + Ô × 8 + δΨ × 3 = 17 — this is not a coincidence with the SM particle count, but the deep reason from which the SM projection yields the same number. The three interactions are types of bonds between coherence clusters. The gap between the SM and gravity is explained by different regimes: the SM operates in the quantum regime (S < 1, stochastics active), GR describes the classical regime (S → 1, stochastics suppressed). The key ODTOE formula: R = Ô(Ψ) — reality is the result of the observation operator acting on the field of potential states. The fixed point Ψ∗ = Φ(Ψ∗ ), where Φ = ι◦ Ô, defines the self-consistent configuration. Each elementary particle is δ Ô(Ψ): a minimal configuration generated by the act of observation.

Key distinction from the SM ODTOE does not divide particles into “elementary” and “composite.” The proton is just as fundamental a role for d = 0 as the u-quark for d = −1. Both levels are shown as equal. The SM classifies the proton and neutron as “composite” (made of quarks at d = −1). But from the perspective of d = +1, the electron is also “composite” (a projection of the operator). The division into “elementary” and “composite” is an artifact of the observation perspective. This article shows that the complete set of distinguishable roles for the two-level observer window is 39 (not 17), and that all anomalies of modern physics — from dark matter to neutrino oscillations — find structural explanation in the infinite recursion Φ = ι ◦ Ô.

II. FERMIONS AND BOSONS: TORUS TOPOLOGY The fundamental distinction between fermions (matter) and bosons (forces) receives a geometric explanation in ODTOE through toroidal topology.

II.1. Fermions: spin-1/2 and the double traversal of the torus Fermions (all quarks and all leptons) have spin 1/2. Returning the wave function to its original state requires two full rotations (4π): one rotation (2π) gives ψ → −ψ. Through toroidal topology: a fermion winds around the torus twice along the minor angle θ before returning. This is analogous to a Möbius strip: one pass reverses orientation, two restore it. A torus with a “twist” = spin-1/2. The gap for two rotations: 2(π − 3). Energy: [2(π − 3)]2 = 4(π − 3)2 ≈ 0.080. This is four times greater than for a single rotation, consistent with fermions having mass.

II.2. Bosons: spin-1 and the single traversal Gauge bosons (photon, gluon, W , Z) have spin 1. One full rotation (2π) closes the wave function. A boson winds around the torus once along θ, without a twist. Gap: (π − 3). Energy: (π − 3)2 .

II.3. The Higgs boson: spin-0 and the absence of traversal The Higgs does not wind around the torus along θ. It “stands still” in toroidal space. Through ODTOE: the Higgs is a configuration without internal rotation, pure “presence” at level d. Its nonzero vacuum condensate (⟨H⟩ ̸= 0) means nonzero “presence density” on each torus. This “presence” gives mass to other particles: it slows their θ-rotation, generating inertia.

II.4. Generation mechanism Three generations of quarks and leptons reflect the three junctions of the triad selfobservation loop: Generation

Loop junction

Physical meaning

1st (u, d / e, νe ) O → Ô 2nd (c, s / µ, νµ ) Ô → R ι 3rd (t, b / τ , ντ ) R − →O

Initiation of the observation act Actualization of configuration Loop closure (return)

Leptons (Ô0 ): pure mass substitution, quantum numbers identical. Baryons (R0 , O0 ): replacement of one quark with its next-generation analog along the line d → s → b (generations of observer O−1 at level d = −1). The line u → c → t terminates: the t-quark (172.76 GeV) decays in ∼ 5×10−25 s, faster than hadronization (∼ 3 × 10−24 s), so t-baryons do not form.

III. THE COMPLETE TABLE OF 39 ROLES By the self-similarity principle: if d = −1 contains 17 roles (with 8 operator modes), then d = 0 also contains 17 roles — with 8 operator modes, not 3. The ODTOE ternary architecture: at each level d the loop contains three roles — O (observer), Ô (operator), R (observable). Each role × 3 generations. At each junction — a gap δΨ. Between levels — bridges. Across all levels — the Higgs and the photon.

III.a. LEVEL d = 0 (atomic) — ALL 17 ROLES Observable R0 and observer O0 (6 baryons)

Gen.

R0 (obs.) R0 (obs.) R0 (obs.) O0 (observer) O0 (observer) O0 (observer)

Spin Particle

Proton p Σ+ Σ+ b Neutron n Λ0 Λ0b

Quarks

m/m1

## PDG

uud uus uub udd uds udb

1.000 1.268 6.193 1.000 1.187 5.981

conf. conf. conf. conf. conf. conf.

Operator Ô0 — a network of 8 leptonic modes By self-similarity with d = −1: the operator at d = 0 is also a network (32 − 1 = 8 channels), not an arrow with 3 generations. The three “vertices” of the d = 0 loop (R0 , O0 , Ô0 ) generate 8 communication channels — just as three color vertices (r, g, b) generate 8 gluons.

Channel type

7 Ô0 fwd, 1st Ô : H → C 8 Ô0 fwd, 2nd Ô : H → C 9 Ô0 fwd, 3rd Ô : H → C 10 Ô0 rev, 1st ι: C → H 11 Ô0 rev, 2nd ι : C → H 12 Ô0 rev, 3rd ι : C → H √ 13 Ô0 diag. 1 (Ô − ι)/ 2 √ 14 Ô0 diag. 2 (Ô + ι − 2δ)/ 6

d=−1 analog g1 (r → g) g2 (g → b) g3 (b → r) g4 (g → r) g5 (b → g) g6 (r → b) g7 g8

Statu

## e− µ− τ− e+ µ+ τ+ L7 (HNL) L8 (HNL)

0.511 MeV 105.7 MeV 1776.9 MeV 0.511 MeV 105.7 MeV 1776.9 MeV ∼17 MeV? ∼keV–GeV?

cand searc

ODTOE discovery: the positron, antimuon, and antitau are not “mirrors” of the electron, but independent reverse channels of the operator network. The forward action of the operator (Ô : H → C) manifests as the electron, the reverse (ι : C → H) as the positron. This is a reformulation of the Wheeler–Feynman one-electron hypothesis in the language of ODTOE. Diagonal modes L7 , L8 : charge 0, lepton number 0, superposition of leptonantilepton pairs. Search status (March 2026): • PDG 2025 maintains a “Heavy Neutral Leptons” (HNL) section — particles with the quantum numbers predicted by ODTOE (Q = 0, L = 0, spin 1/2). • X17 (ATOMKI anomaly): hypothetical neutral particle with mass ∼17 MeV; independent confirmation from Vietnam (2024); MEG II (June 2025) weakens but does not close the hypothesis. • Sterile neutrinos ∼keV: dark matter candidate; 3.5 keV line (2014); XRISM (2025) did not confirm, KATRIN+TRISTAN (2026) continue searching. • MiniBooNE/LSND anomaly: MicroBooNE (December 2025) excluded the single light sterile neutrino model, but “heavy sterile neutrino decaying to νe + scalar” remains — this is L7 or L8 in ODTOE.

• Experiments: SHiP, DUNE, FCC-ee, PIONEER, LEGEND-1000 — all target HNL. Gaps δΨ0 (3 neutrinos) Neutrinos are one of the deepest consequences of ODTOE. The self-observation loop Φ = ι ◦ Ô is spiral (π ̸= 3, π is transcendental): each turn does not close exactly, leaving a gap δΨ. Neutrinos are the materialization of the fundamental incompleteness of the strange loop closure.

Gen.

Upper limit

Estimate

15 δΨ0 16 δΨ0 17 δΨ0

νe νµ ντ

< 1.1 eV (KATRIN) ≈ ν1 ∼ 0–0.05 eV < 0.19 MeV ≈ ν2 ∼ 0.009–0.05 eV < 18.2 MeV ≈ ν3 ∼ 0.05–0.06 eV

Neutrino properties derivable from δΨ properties: • Mass: the loop nearly closes, |δΨ| is infinitesimal. The dispersion D(η) = D0 ·(1− S) links the gap to coherence: |δΨ| ∝ (1−S), hence mν ∝ (1−S). Experimentally: Σmν < 0.12 eV. • Zero charge: δΨ belongs neither to the Ô phase (charge −1), nor to R (+1), nor to O (0 as agent). The spiral residue is orthogonal to the triad architecture. • Weak interaction: δΨ is “perpendicular” to the loop components — generated by it, but not participating in its functioning. Analogy with Gödel’s theorem: a true statement unprovable within the system. • Ubiquity: every turn of every strange loop at every level of ∞-recursion produces its own δΨ. Hence ∼ 1089 neutrinos in the visible Universe. • Left-handedness: the self-observation spiral has a definite chirality (traversal direction O → Ô → R → ι → O), and δΨ inherits this chirality. • Oscillations: the loop continues its spiral motion — the phase of δΨ shifts relative to segments. The vector δΨ rotates in the space of junctions with a frequency determined by the spectrum of Φ. Hence transitions between generations νe ↔ νµ ↔ ντ . Note on neutrino masses: from oscillations: ∆m221 ≈ 7.5 × 10−5 eV2 , |∆m232 | ≈ 2.5 × 10−3 eV2 . Under normal hierarchy (m1 < m2 < m3 ) the mass ordering matches generations. ODTOE predicts normal hierarchy (JUNO 2025+ will measure). Total d = 0: 17 roles = 6 (baryons) + 8 (leptonic modes) + 3 (neutrinos). Of these, 15 are confirmed, 2 are predicted (L7 , L8 ).

III.b. LEVEL d = −1 (subnuclear) — ALL 17 ROLES Internal structure of the proton and neutron. Contains the same three roles + gaps, but here the operator is a network, not an arrow.

Observable R−1 and observer O−1 (6 quarks)

Gen.

Junction

18 R−1 (obs.) 19 R−1 (obs.) 20 R−1 (obs.) 21 O−1 (observer) 22 O−1 (observer) 23 O−1 (observer)

## O → Ô Ô → R R→O O → Ô Ô → R R→O

+2/3 +2/3 +2/3 −1/3 −1/3 −1/3

u-quark 2.16 MeV conf. c-quark 1.27 GeV conf. t-quark 172.7 GeV conf. d-quark 4.67 MeV conf. s-quark 93.4 MeV conf. b-quark 4.18 GeV conf.

u-quark as observable R−1 : charge +2/3 — positive like the proton (R0 ), but “incomplete” — a fragment of actualization at the substructural level. Lighter than the d-quark: the observable is lighter than the observer. d-quark as observer O−1 : charge −1/3 — negative like the electron (Ô0 ), but “incomplete.” Heavier than the u-quark: the observer carries greater inertia I(C), contains cognitive coherence. The observer is formalized as a triple O = (B, A, H), where B is coherence (belief), A is the attention vector, H is the horizon of accessible configurations. Coherence B unfolds through four components: B(O, C) = F w1 · E w2 · (1 − σ)w3 · Λw4

## (III.1)

where F is the focus of attention, E is emotional coherence, σ is the entropy of doubt, Λ is empirical reinforcement. t-quark — the heaviest particle (≈ 172.7 GeV, heavier than the Higgs!). Through ODTOE: this is the observable R−1 at the third (maximal) toroidal recursion level — the ultimate inertia I(C). The configuration is so “heavy” that it decays in ∼ 5 × 10−25 s — the lifetime T (C) is minimal. The t-quark mass exceeds the Higgs mass (125 GeV) because the t-quark is the ultimate actualization at the 3rd junction (loop closure), while the Higgs is a self-referential potentiality parameter. The Yukawa coupling yt ≈ 1 means in ODTOE: the 3rd junction of R−1 is in resonance with the field H. Second generation (c, s): the same architectural pair at a higher energy scale. The sharp increase in c-quark mass (≈ 1.27 GeV) compared to the u-quark (≈ 2.16 MeV) reflects increased inertia I(C) upon transition to a torus of larger radius R × φ. Operator Ô−1 — a bond network (8 gluons) Why 8 and not 3: at d = −1 the operator connects three color vertices (r, g, b) to each other. Number of channels = 32 − 1 = 8. This is a network (all pairs), not an arrow (one direction).

Channel type

d=0 analog

Ô−1 fwd, 1st Ô−1 fwd, 2nd Ô−1 fwd, 3rd

r→g g→b b→r

e− µ− τ−

gluon g1 gluon g2 gluon g3

conf. conf. conf.

Ô−1 rev, 1st Ô−1 rev, 2nd Ô−1 rev, 3rd Ô−1 diag. 1 Ô−1 diag. 2

g→r b→g r→b √ (rr̄ − gḡ)/ 2 √ (rr̄ + gḡ − 2bb̄)/ 6

## e+ µ+ τ+ L7 L8

gluon g4 gluon g5 gluon g6 gluon g7 gluon g8

√ The 9th channel (rr̄ + gḡ + bb̄)/ 3 = colorless singlet — the trace of the matrix Ô−1 . This channel is not confined (unlike the 8 gluons) because the trace is invariant under all unitary transformations: Tr(U AU −1 ) = Tr(A). The full group of the operator is U (3) = SU (3) ⊕ U (1): 8 traceless generators (gluons, SU (3)) + 1 trace generator (photon γ, U (1)). The role of the 9th channel = photon, not the Higgs. The Higgs is the substrate (field H) in which the 3 × 3 matrix unfolds; it is not a channel of the operator (see Section III.d for details). The gluon is the observation operator Ô−1 at the nucleon level. Gluon masslessness: as a pure operator at its level, it does not “sit” on the torus but mediates the connection. Confinement (impossibility of isolating a free gluon) in ODTOE: the operator does not exist outside the act of observation. The gluon is a pure process, inseparable from participants. Double origin of U (1). Electromagnetic U (1) has two roots: (a) topological — the fundamental group of the loop π1 (S 1 ) = Z (Section VI.2); (b) algebraic — the trace of the ternary operator matrix. Both roots lead to the same group U (1), explaining the uniqueness of electromagnetism. Gaps δΨ−1 — “sub-neutrinos” (3 predicted particles)

Gen.

Junction

## R−1 → O−1 O−1 → Ô−1 Ô−1 → R−1

sub-νe sub-νµ sub-ντ

predicted predicted predicted

Why not detected: D-Prot: we are d = 0 observers, and δΨ−1 “lives” entirely inside d = −1. We see neutrinos (δΨ0 ) because they are gaps of our level. Sub-neutrinos are gaps of the embedded level. Where to search: at very high energies (∼ 104 GeV and above). FCC (100 TeV) may approach. Possibly already manifesting as anomalies in gluon interactions or unexplained energy losses in deep inelastic scattering. Total d = −1: 6 quarks + 8 gluons + 3 sub-neutrinos = 17 roles

III.c. BRIDGES BETWEEN d = 0 AND d = −1 (3 bosons) Massive bosons mediating role transmutation between levels.

Function

conf. conf. conf. conf. conf.

35 Transmutation O → R 36 Transmutation R → O 37 Self-check

β −: n → p β +: p → n Coherence check

## W− W+ Z0

## 80.4 GeV 80.4 GeV 91.2 GeV

W -boson — the role transmutation operator. β − -decay (n → p + e− + ν̄e ): the observer (neutron) transmutes into the observable (proton) — potentiality transitions to actuality with generation of the operator (electron) and gap (antineutrino). The W mass (≈ 80 GeV) reflects the enormous inertia I(C) of role switching. Z-boson — the loop coherence “self-check” operator. A particle interacts but does not change its role. The Z mass (≈ 91 GeV) is slightly larger than W : coherence checking costs more than acting, requiring full “self-scanning.”

III.d. TRANS-LEVEL (all levels simultaneously) — 2 entities

Tr(Ôd ) Potentiality field H

Function

9th channel, trace Substrate, mass

Photon γ Higgs H

## 125 GeV

Photon γ = Tr(Ôd ) — the 9th channel of the ternary operator matrix. At each level d, the operator Ôd is described by a 3 × 3 matrix yielding 9 channels: 8 traceless (SU (3) generators) + 1 trace (U (1) generator). Eight confined channels = gluons; the free trace = photon. The photon exists at all levels simultaneously because the trace is invariant under all unitary transformations: Tr(U AU −1 ) = Tr(A). Three photon properties from trace properties: (a) masslessness — the trace is not bound to any vertex, acquires no inertia I(C); (b) speed c = r0 /τ0 — the photon does not “pass through” levels but is present at all simultaneously; the speed of light is the actualization front speed H → C, invariant at all levels (cd = rd /τd = r0 /τ0 = const, since rd = r0 · φd and τd = τ0 · φd ); (c) trans-levelness — the photon does not belong to a specific d because the trace is identical at all levels. Higgs H ̸= operator channel. The Higgs is the field of potential states H, the substrate in which the 3 × 3 operator matrix Ô unfolds. Not bound to a specific level d — it is one for the entire hierarchy. The Higgs mass (≈ 125 GeV) is a self-referential parameter: potentiality determining the inertia of all configurations itself possesses inertia. Fixed point: Ψ∗ = Φ(Ψ∗ ) — the field determines mass, mass determines the field. Two trans-level poles reflect two poles of the observation cycle: γ = actuality (operator, identical at all levels), H = potentiality (substrate, containing all levels).

## III.e. FINAL SUMMARY Level

Roles

Count

Details

d=0 d = −1 Bridges Trans-level TOTAL

R0 × 3, O0 × 3, Ô0 × 8, δΨ0 × 3 R−1 × 3, O−1 × 3, Ô−1 × 8, δΨ−1 × 3 W +, W −, Z 0 γ, H

− − − + + p, Σ+ , Σ+ b , n, Λ , Λb , e , µ , τ , e , µ , u, c, t, d, s, b, 8 gluons, 3 sub-ν Transmutation Universal 34 conf. + 2 cand. + 3 pred.

UNIVERSAL INVARIANT: 17 The number of roles at EACH recursion level:

## N (d) = R × 3 + O × 3 + Ô × (32 − 1) + δΨ × 3 = 3 + 3 + 8 + 3 = 17

## (III.2)

This is not “the number of elementary particles” — it is the structural constant of a single level of infinite recursion Φ = ι ◦ Ô. The SM obtained the same number 17 for a different reason — as a counting convention collapsing 39 roles of the two-level window: antileptons “hidden” in leptons, 8 gluons collapsed into “1 type,” baryons classified as “composite,” diagonal modes (L7 , L8 ) and sub-neutrinos not anticipated. The coincidence of two different “17”s (N (d) = 3 + 3 + 8 + 3 vs. NSM = 3 × 2 × 2 + 4 + 1) is not accidental but reflects the SM convention unconsciously reproducing the structural invariant of a single level.

22 “extra” roles — where they are What is hidden Proton p, neutron n Σ+ , Σ+ b (2nd and 3rd gen. proton) Λ0 , Λ0b (2nd and 3rd gen. neutron) e+ , µ+ , τ + (reverse channels) L7 , L8 (diagonal modes) 7 “additional” gluons 3 sub-neutrinos (δΨ−1 )

Count

Why not in SM “17”

“Composite” “Composite” “Composite” “Antiparticles” No SM analog “One type” Beyond D-Prot horizon

conf. conf. conf. conf. cand. conf. pred.

17 conf., 2 cand., 3 pred.

IV. FOUR INTERACTIONS THROUGH ODTOE IV.1. Strong interaction: internal coherence of the triad The strongest of all forces. In ODTOE — coherence S → 1 inside the nucleon, binding the triad architecture at level d = −1. Carrier: gluon (operator Ô−1 ). Confinement: the loop does not break because the operator does not exist outside the act.

IV.2. Electromagnetic interaction: the R–Ô bond The bond between observable and operator at the atomic level d = 0. Carrier: photon γ = Tr(Ôd ), the 9th channel of the ternary matrix (Section III.d). The fine-structure constant: α−1 = π(4π 2 + π + 1) ≈ 137.036

## (IV.1)

A self-referential formula containing only π and integers, reflecting the closed nature of the loop. The approximation α−1 ≈ 360/φ2 = 137.51 (99.7% accuracy) is the zeroth order; the full formula is the exact result. Speed of light c = r0 /τ0 — a geometric identity of the φ-torus, not an empirical constant. At each level d, the minimal radius rd = r0 · φd and elementary duration τd = τ0 · φd grow synchronously, so cd = rd /τd = r0 /τ0 = const for any d. The speed c is not the photon speed but the actualization front speed H → C: in one tick τ0 the loop Φ actualizes exactly one configurational volume r0 . The ultimacy of c follows from the discreteness of the observation act.

IV.3. Weak interaction: role transmutation The process of switching loop components: observer ↔ observable. Carriers: W ± , Z 0 . Massiveness means high restructuring inertia. The weak interaction generates neutrinos (gap δΨ) and allows changing the particle “generation.”

IV.4. Gravity: beyond the Standard Model The SM does not include gravity. ODTOE explains: the SM describes the regime S < 1 (quantum); gravity arises at S → 1 (classical). Two limiting cases of one theory. Spacetime curvature in GR corresponds to the potential gradient ∇U (C). Gravity is not a “fifth force” but what the self-observation loop looks like at S → 1. Unification does not require “quantizing gravity”; it requires recognizing that both descriptions are projections of a single cycle Φ onto different coherence regimes.

## V. RECURSION 3-6-9: UNIVERSE

## FROM QUARKS TO THE

The particle structure reproduces the 3-6-9 pattern at level d = −1: 3 (the observer looks): 3 quarks in a nucleon. Triad architecture at the subatomic level. 6 (the result returns): 6 quarks total (3 pairs ×2 = forward + reverse loop traversal). 6 forward/reverse leptonic modes (e− , µ− , τ − , e+ , µ+ , τ + — same logic; the full network Ô0 = 8 channels with two diagonal L7 , L8 ).

9 (the cycle becomes self-aware): nucleon = Ψ∗ — a fixed point, self-consistent configuration containing the entire triad architecture. After 9 — return to 1 of the next level. The nucleon (Ψ∗−1 ) becomes an element of the atom (Ψ∗0 ), the atom an element of the molecule (Ψ∗+1 ). An infinite spiral 3 → 6 → 9 → 3 → 6 → 9 at each level.

VI. DERIVATION OF THE GAUGE GROUP SU (3) × SU (2) × U (1) FROM ODTOE AXIOMATICS VI.1. Problem statement The SM gauge group SU (3)×SU (2)×U (1) is postulated in the standard approach based on experimental data. ODTOE shows that this specific group is derived structurally from three independent aspects of the triad architecture, where the observation operator Ô, the field of potential states Ψ ∈ H, and the self-observation cycle Φ = ι ◦ Ô are primary.

VI.2. U (1): phase invariance of the strange loop Initial construction. The strange loop Φ : H → H is topologically equivalent to the circle S 1 . The fundamental group π1 (S 1 ) = Z directly generates the group U (1) ∼ = S 1. Derivation. The observable configuration R does not depend on the absolute phase of Ψ: Ô(eiθ Ψ) = Ô(Ψ)

for all θ ∈ [0, 2π)

## (VI.1)

This condition is global U (1)-invariance. Upon localization θ → θ(x), the differential structure of Ô requires a compensating field (the standard gauge argument), generating the electromagnetic potential Aµ . Physical meaning. U (1) is the phase rotation group inside a single torus (θrotation). Charge q ∈ Z is the winding number around S 1 . Charge discreteness follows from the integrality of π1 (S 1 ) elements. Correspondence. constant:

U (1) governs electromagnetic interaction.

α−1 = π(4π 2 + π + 1) ≈ 137.036 — a self-referential formula containing only π and integers.

The coupling

## (VI.2)

VI.3. SU (2): double torus traversal and the spinor bundle Initial construction. Fermions require a double traversal of the torus along θ: 2π gives ψ → −ψ, only 4π returns ψ → ψ. The spinor field on the torus is described as a section of a bundle with structure group SU (2) — the double cover of SO(3). The double covering precisely corresponds to the double torus traversal. In the triad architecture O, Ô, R, transitions between components form doublets: the pair (O, R) is connected by the operator Ô, which switches roles. This switching is an operation in a two-dimensional role space isomorphic to the fundamental representation of SU (2). Physical meaning. SU (2) is the role transmutation group. Weak isospin is “up/down” in the pair (O, R). W ± perform the switching O ↔ R (charged currents); Z 0 checks without switching (neutral current). Why SU (2) and not SO(3)? Because fermions require a double traversal. Describing half-integer spin requires a double cover, and SU (2) is the universal cover of SO(3). W and Z masses. In ODTOE — incompatibility of full role symmetry with specific actualization Ψ∗ (the fixed point fixes a specific role distribution, breaking full SU (2)symmetry).

VI.4. SU (3): triad architecture at level d = −1 Initial construction. At level d = −1, the triad architecture is reproduced: u-quark (R−1 ), d-quark (O−1 ), gluon (Ô−1 ). Three colors (r, g, b) are a manifestation of the triadicity. Derivation. Three loop components realize three “color” states. The group of unitary transformations in three-dimensional complex space is U (3) = SU (3) ⊕ U (1). It contains 32 = 9 generators: 8 traceless (gluons g1 –g8 , SU (3) generators) + 1 trace generator (photon γ, U (1) generator). Eight gluons are confined √ (traceless, not invariant under basis change); the 9th channel (trace, (rr̄ + gḡ + bb̄)/ 3) is free — this is the photon, not an additional gluon. The trace is invariant: Tr(U AU −1 ) = Tr(A), so the 9th channel carries no color charge and is not confined. Confinement. The closure requirement on Φ at d = −1 means the observable configuration = “colorless” (color singlet). A hadron = a closed loop = Ψ∗ at level d = −1. Confinement affects the 8 traceless channels; the trace (photon) is free by definition. Why SU (3) for the strong interaction, not U (3)? The full operator group is U (3), but it decomposes: U (3) = SU (3) ⊕ U (1). The strong interaction is described by the SU (3) part (confined channels). The remaining U (1) part (trace = photon) describes the electromagnetic interaction. Thus, U (1) in the SM gauge group has a double origin: (a) topological — π1 (S 1 ) = Z (Section VI.2) and (b) algebraic — the trace of the ternary matrix Ô. Both roots lead to the same U (1).

VI.5. Why the product SU (3) × SU (2) × U (1), not a sum The three factors act on different aspects of the loop and commute: • U (1) governs the absolute phase of θ-rotation (inside the torus) • SU (2) governs role switching O ↔ R (loop architecture) • SU (3) governs the internal triad structure at d = −1 (color) Phase does not depend on who is observer and who is observable. Role switching does not depend on color. Color does not depend on absolute phase. The three symmetries are orthogonal — the group is a direct product.

VI.6. Two derivations of the number 17: SM projection vs. ODTOE structure SM projection (convention). The SM obtains 17 by collapsing the full picture: NSM = 3 × 2 × 2 + 4 + 1 = 17

## (VI.3a)

This is not a structural constant, but a counting convention. ODTOE structural invariant. At each level d, the ternary loop O → Ô → R → O contains: • R × 3 generations = 3 (observable: p/Σ+ /Σ+ b at d = 0, or u/c/t at d = −1) • O × 3 generations = 3 (observer: n/Λ0 /Λ0b at d = 0, or d/s/b at d = −1) • Ô × (32 − 1) channels = 8 (operator network: 8 leptonic modes at d = 0, or 8 gluons at d = −1) • δΨ × 3 gaps = 3 (νe /νµ /ντ at d = 0, or sub-νe /sub-νµ /sub-ντ at d = −1)

N (d) = 3 + 3 + 8 + 3 = 17

for any d ∈ Z

## (VI.3b)

VI.7. Full distribution of 39 roles in the d = 0 observer window Level

R×3

O×3

Ô × 8

δΨ × 3

Total

d=0 d = −1 Bridges Trans-level

p, Σ+ , Σ+ b u, c, t

n, Λ0 , Λ0b d, s, b

e − , µ− , τ − , e + , µ+ , τ + , L 7 , L 8 g1 –g8 W +, W −, Z 0 γ, H

νe , νµ , ντ sub-νe , sub-νµ , sub-ντ

## TOTAL

VI.8. Why exactly 3 generations, not 2 or 4? The triad architecture has exactly three junctions: O → Ô, Ô → R, R → O. Each junction generates one generation. Two junctions yield an open chain (no loop). Four junctions are impossible in triangular architecture (would require a fourth component, but the observation act is triadic: π > 3, not π > 4). Three junctions are the only number compatible with a closed minimal loop. The number 3 is a property of horizontal topology (junctions at one level), while infinite recursion is a property of vertical structure (levels d). Infinity goes inward, not sideways. Confirmation: the Z 0 decay width gives Nν = 2.9840 ± 0.0082 — exactly three light neutrinos [14].

VI.9. Electroweak unification SU (2) × U (1) → U (1)em At high energies (T ≫ mW ) the triad architecture is fully symmetric: all three components are equal. The group SU (2) × U (1) is fully realized. At low energies (T ≪ mW ) the fixed point Ψ∗ fixes a specific role distribution. Potentiality H “crystallizes” into a vacuum condensate ⟨H⟩ ̸= 0. Only U (1)em remains. Three generators acquire mass (W + , W − , Z 0 ), one remains massless (photon) [19]. Through ODTOE: spontaneous symmetry breaking is not “breakage” but actualization. The transition from full potentiality (all roles equal) to a specific configuration (roles fixed) is the act of observation Ô(Ψ) = R.

VI.10. Grand unification and gravity The three factors do not “unify” into a simple group because they describe three orthogonal aspects of the loop: phase (inside the torus), role (loop architecture), position in the substructural triad (embedded level). The unifying structure is the cycle Φ itself, not a group. Gravity is the limiting regime S → 1, where stochastics are suppressed and the loop appears as smooth geometry. Unifying QM and gravity does not require quantizing gravity or adding a graviton; it requires recognizing that both descriptions are projections of a single Φ onto different coherence regimes S. Remark. The established correspondence U (1) ↔ phase invariance, SU (2) ↔ double traversal, SU (3) ↔ triad architecture is a structural analogy. A rigorous derivation of gauge symmetry requires constructing a bundle with connection — a task beyond the scope of this work.

VII. MASS HIERARCHY AND φ-SCALING VII.1. Four numbers defining reality Particle masses are not random. In the ODTOE toroidal model, the scale is set by the ratio R/r = φ (golden ratio), ensuring maximum stability by the KAM theorem. Transition between generations is a φ-jump to the next torus. Four numbers define all of reality: • π — the shape of the turn (spirality) • φ — the step of the spiral (scaling) • (π − 3)2 — the energy grain per revolution • d — the observer horizon (dimensionality)

VII.2. Key ratio: mp /me = 6π 5 mp /me = 1836.15 ≈ 6π 5 = 1836.12 (accuracy 0.002%!)

## (VII.1)

This is the ratio of the observable R0 mass to the operator Ô0 mass. The number 6 = 3! = the number of permutations of three loop vertices. π 5 = five powers of “spirality” (one per recursion level in the visibility window). The full four-layer self-referential formula gives µ = 1836.15267304 (nine correct significant digits, discrepancy with CODATA: 3.9 × 10−7 ) [10].

VII.3. φ-scaling between generations Group Ô0 (leptons) R−1 (u-quarks) O−1 (d-quarks) R0 (proton) O0 (neutron)

m1 → m2

≈ φn

m2 → m3

≈ φn

m1 → m3

≈ φn

206.8 20.0 1.27 1.19

φ11 φ13 φ6 ∼ φ0.5 ∼ φ0.4

3.9 12.9 11.5

16.8 44.8 4.89 5.04

φ6 φ10 φ8 φ3 φ3

6.3 10.6 4.7 15.3 18.9

79981 6.19 5.98

φ17 φ23 φ14 φ4 φ4

2.7 10.7 14.6

Key pattern: the φ power for m1 → m3 = (power for m1 → m2 ) + (power for m2 → m3 ). For leptons: 11 + 6 = 17 = the ODTOE invariant! For u-quarks: 13 + 10 = 23 = 17 + 6. For d-quarks: 6 + 8 = 14 = 17 − 3. Sum R−1 + O−1 = 23 + 14 = 37 ≈ 39 − 2 (all roles minus γ and H). The operator Ô0 “traverses” exactly 17 steps — the full set of roles of one level.

VII.4. Inter-group ratios — toroidal sectors Ratio

Value

logφ

mp /me mW /mp mH /mp mH /mW mτ /ms mp /md

1836.15 φ15.6 85.7 φ9.3 133.3 φ10.2 1.56 φ0.9 19.0 φ6.1 200.9 φ11.0

Interpretation R0 /Ô0 = 6π 5 bridge/observable Higgs/observable δ = 3.8% 3rd gen. lepton / 2nd gen. quark d=0 baryon / d=−1 quark

VII.5. PDG “bonus”: an unexpected match + Scheme p → Σ+ → Σ+ c : m(Σc )/m(p) = 2.614 ≈ φ = 2.618 with 0.2% accuracy!

Similarly: m(Ξ0c )/m(n) = 2.629 ≈ φ2 = 2.618 with 0.4% accuracy. This is an alternative generational ladder (d → s → c instead of d → s → b), with nearly perfect φ2 matching.

VIII. SUMMARY TABLE: ALL 39 ROLES THROUGH ODTOE VIII.1. How the SM sees its “17 elementary” (projection) The SM selects from 39 roles only those it considers “elementary,” collapsing the rest:

Particle (SM)

SM view

ODTOE view

Torus topology

u-quark

Quark, +2/3

R−1 (1st gen.)

d-quark

Quark, −1/3

O−1 (1st gen.)

c-quark

Quark, +2/3

R−1 (2nd gen.)

s-quark

Quark, −1/3

O−1 (2nd gen.)

t-quark

Quark, +2/3

R−1 (3rd gen.)

b-quark

Quark, −1/3

O−1 (3rd gen.)

Gluon g e− µ−

1 boson (8 colors) Ô−1 — 8 channels Lepton, −1 Ô0 fwd (1st) Lepton, −1 Ô0 fwd (2nd)

Double traversal, 1st Double traversal, 1st Double traversal, 2nd torus Double traversal, 2nd torus Double traversal, 3rd Double traversal, 3rd Single traversal Double traversal Double traversal, 2nd torus

νe νµ ντ 14 0/ − 1 Photon γ 15 0/ − 1 W ± 16 0/ − 1 Z 0 all Higgs H

Lepton, −1

Ô0 fwd (3rd)

Neutrino Neutrino Neutrino Boson, EM Boson, weak Boson, weak Scalar

δΨ0 (O → Ô) δΨ0 (Ô → R) δΨ0 (R → O) Tr(Ôd ), 9th channel Transmutation O ↔ R Loop self-check Field H: potentiality

Double traversal, 3rd Spiral residue Spiral residue Spiral residue Trans-level Single traversal Single traversal No traversal (spin 0)

VIII.2. What the SM hides: 22 “missing” roles

Why SM does not count

18–23

p, Σ+ , Σ+ b , n, Λ , Λb

## R0 × 3 + O0 × 3

24–26

e + , µ+ , τ +

Reverse channels of Ô0

27–28 29–35 36–38

L7 , L8 g2 . . . g 8 sub-νe , sub-νµ , sub-ντ

Diagonal channels of Ô0 (HNL) 7 additional channels of Ô−1

0/ − 1

“Composite” (from quarks) “Antiparticles” (mirrors) Not anticipated “One type” of gluon Beyond D-Prot horizon “One type” W ±

Bridge O → R

W − (separate)

Analogs across recursion levels

d=+1 (mol.)

d=0 (at.)

d=−1 (nucl.) d=−2 (sub-q.)

## R O Ô

Molecule Solvent Chemical bond

Proton p+ Neutron n0 Electron e−

u-quark d-quark Gluon g

Sub-u Sub-d Sub-gluon

The electron = the gluon of the next octave. The electron binds atoms into molecules just as the gluon binds quarks into nucleons. Quarks = leptons of the previous octave. In the sub-SM, quarks play the role of free operators analogous to electrons.

IX. COMPLETE PDG → ODTOE MAP: EVERYTHING NOT IN THE 39 ROLES IX.a. Mesons (∼200+ in PDG) — “bond fragments” Mesons (q q̄) are NOT roles of the ternary loop. They are “fragments” of the gluon string (bond Ô−1 ). When a collider breaks the loop, quarks reassemble not only into baryons (qqq = loop) but also into mesons (q q̄ = bond fragment).

IX.b. Vector mesons — “bridges WITHIN d = −1” Just as W ± /Z 0 are bridges between levels, vector mesons (J P = 1− ) are bridges within the quark loop.

IX.c. Exotic hadrons — “role molecules” Exotic hadrons add no new roles — they are combinations of existing ones: pentaquarks (baryon + meson), tetraquarks (meson + meson).

IX.d. Resonances — “excited roles” Hundreds of resonances in the PDG (N ∗ , ∆, Σ∗ , Ξ∗ , Ω∗ ) are the same 39 roles with added rotational/vibrational energy. Not new roles, but excited states of existing ones.

X. MULTI-LEVEL MAP: ANOMALIES AS SHADOWS OF OTHER LEVELS X.a. D-Prot: the visibility window of the d = 0 observer The d = 0 observer sees levels with attenuation S(ρd ) ∝ φ−|∆d| . Entanglement is maximal at our level and decreases by a factor of φ ≈ 1.618 at each next level. The current table (39 roles) covers only d = 0 and d = −1. Physics anomalies are shadows of roles from other levels leaking through D-Prot.

X.b. Self-similarity: 17 roles at EACH level By ∞-recursion Φ = ι ◦ Ô, each level d contains the same ternary loop with N (d) = 17 roles. Roles of one level become components of another.

X.c. Classification of anomalies by source level 1. Missing roles of our level d = 0 (L7 , L8 → explain MiniBooNE, X17). Roles that the SM missed because it does not know about the network structure of the operator (8 channels instead of 3). 2. Shadows of neighboring-level roles (d=+1: graviton, dark photon, WIMP; d=−2: axion, quark substructure). Belong to their levels (17 each), we see “blurred projections” through D-Prot. 3. Trans-level effects (dark energy = H pressure, cosmological proportions = φtorus geometry). Not particles but properties of recursive architecture. Full counting formula: in the d = 0 observer window — 39 full roles plus “ghost” contributions: 17 × φ−1 ≈ 10.5 (from d=+1), 17 × φ−2 ≈ 6.5 (from d=−2 and d=+2), and so on. Total ∼84 effective roles in the full D-Prot window.

XI. ENERGY PROPORTIONS: FORMULAS FROM π AND φ (zero free parameters) XI.a. Cosmological proportions — three sectors of the φ-torus The φ-torus with R/r = φ (the most irrational number, KAM-stable) generates three sectors: Sector

Dynamics

I: R-dynamics Rotation along R II: r-dynamics Rotation along r III: gap Spiral slit IV: gap2 Gap within gap

Inertia

Physics

Observed fraction

## ∝ R 2 = φ2 ∝ r2 = 1 Z (π − 3)2

Between levels Dark energy ΩΛ Within a level Dark matter ΩDM Matter in slit Baryonic matter Ωb 2nd-order gap Neutrinos Ων

Parameter Z — geometric series of spiral gaps: Z=

π−3 = 0.18367 . . . 1 − (π − 3) φ

## (XI.1)

Contributions by order: k = 1: 77.1%, k = 2: 17.7%, k = 3: 4.0%, k ≥ 4: 1.2%.

XI.b. Comparison with Planck 2018 (3-component model) ΩΛ : ΩDM : Ωb = φ2 : 1 : Z, normalization Σ = φ2 + 1 + Z = 3.8017. Parameter

## ODTOE

Planck 2018

Deviation

ΩΛ (dark energy) ΩDM (dark matter)

68.86% 26.30%

68.89% 26.07%

0.56% 0.20%

0.05σ 1.17σ

Ωb (baryons)

4.83%

4.90%

0.06%

1.06σ

All three matches within 1.2σ. Zero free parameters — only π and φ. [20]

XI.c. Self-referential correction (Φ = ι ◦ Ô) The baryonic fraction “observes itself” (strange loop): x = (Z + εx)/(K + Z + εx), ε = (π − 3)2 , K = φ2 + 1. Quadratic equation: εx2 + x(K + Z − ε) − Z = 0

## (XI.2)

Result: Ωb (s.r.) = 4.856% (σ = 0.67 from Planck), improvement by 0.39σ.

XI.d. 4-component model (with neutrinos) ΩΛ : ΩDM : Ωb : Ων = φ2 : 1 : Z : (π − 3)2

## (XI.3)

Ων = (π − 3)2 /Σ4 = 0.52% (Planck: < 0.3% at Σmν < 0.12 eV — consistent in order of magnitude). Neutrinos = 2nd-order gap of the toroidal spiral (δΨ ∝ (π − 3)2 ).

XI.e. Two types of formulas Type 1. Between levels (cosmological proportions): φ2 : 1 : Z : (π − 3)2 — a property of the torus AS A WHOLE. Determines the fractions of dark energy, dark matter, baryons, neutrinos. Type 2. Between generations (φ-scaling of masses): m(gen. n+1)/m(gen. n) ≈ φk — a property of RECURSION. The power k depends on the group (role) and junction number. Cosmological proportions are not applicable to mass distributions within a level (mp ≈ mn , not mp /mn = φ2 ). But φ-scaling is not applicable between levels (dark energy/matter are not a “generation” of baryons). The two types of formulas reflect two types of rotation on the φ-torus: along the major radius R (between levels) and along the minor radius r (within a level).

## XII. FALSIFIABLE NESTING

## PREDICTIONS

## FROM

## INFINITE

The ODTOE recursive self-similarity principle asserts: each proton contains an internal triad architecture, and this architecture is reproduced at all scales. The fixed

point Ψ∗ = Φ(Ψ∗ ) defines a self-consistent configuration linking all levels. This structure generates twelve falsifiable predictions.

XII.1. P1: Inter-scale atom/nucleus correlations Electron capture (e− +p → n+νe ) is a confirmed case of inter-level interaction. ODTOE predicts systematic correlations beyond QED. Test: precision measurement of β-decay rates in different electronic states (neutral atom vs. fully ionized).

XII.2. P2: φ-scaling of entanglement entropy The von Neumann entropy S(ρd ) ∝ φ−|d−d0 | [3]. In systems with self-similar structure (quasicrystals, Fibonacci lattice), entanglement entropy between scales should obey this law. Test: simulation on fractal lattices; measurement of correlations in quasicrystals.

XII.3. P3: Nonlocal electron correlations through Ô unity All electrons are projections of a single operator Ô. Indistinguishability is a consequence of operator identity. ODTOE predicts nonzero (though small) correlations between distant electrons without prior entanglement. Test: comparison of spin correlations in separated atoms.

XII.4. P4: Baryonic asymmetry from spirality The closed cycle length (π ≈ 3.14159) is incommensurable with the triad architecture (3 components). The increment π − 3 ≈ 0.14159 creates a systematic asymmetry Ô ̸= ι. The transcendence of π guarantees the asymmetry will not vanish. Test: analytical derivation of η ≈ 6 × 10−10 through powers of (π − 3).

XII.5. P5: Topological prohibition of the 4th generation Exactly 3 generations at each level d — period. A triangle has no fourth vertex. Infinity goes inward (nested triads), not sideways (additional junctions). Ngenerations (d) = 3

Nlevels = ∞

for any d ∈ Z

(topological invariant)

(S = 1 is unreachable → recursion does not terminate)

## (XII.3)

## (XII.4)

Confirmation: Nν = 2.984 ± 0.008 [14]. Discovery of a 4th generation (not substructure) would falsify the triad architecture.

XII.6. P6: Quark substructure at E ≫ 104 GeV At energies above ∼ 104 GeV, quark substructure will be discovered. These are not preons (finite number of levels) but reproduction of the same loop architecture at a deeper scale. Substructural objects will have fractional charges (±1/9, ±2/9), three “sub-colors,” and binding through sub-gluons. Current LHC data: Λ ≥ 30 TeV (PDG 2024) [13]. ∞-recursion predicts substructure at scales Rq ∼ Rnucleon × φ−n .

XII.7. P7: Inter-generation masses ∝ φn × [1 + k(π − 3)2 ] m(τ )/m(e) ≈ 3477 ≈ φ16.92

## (XII.8)

m(µ)/m(e) ≈ 206.77 ≈ φ11.04

## (XII.7)

m(τ )/m(µ) ≈ 16.82 ≈ φ5.88

## (XII.6)

The exponents are not exact integers — they reflect the spiral gap (π − 3)2 at each transition. Test: if at least three of six ratios have integer n (to accuracy < 0.1), this is a statistically significant confirmation.

XII.8. P8: Scale dependence of Planck’s constant h̄ may prove to be an effective parameter depending on observation level: h̄ = h̄(d, S). Test: compare h̄ via the Josephson effect (d ≈ 0) and the Kibble balance (d ≈ 2). Discrepancy > 10−8 = evidence.

XII.9. P9: Spatial dipole trend of α Correlation of ∆α/α with baryonic density ρb along the line of sight. Webb et al. (2011) already detect a dipole trend ∆α/α ∼ 10−5 [2].

XII.10. P10: Normal neutrino mass hierarchy The junction R → O (τ -neutrino) closes the full cycle and contains the maximum gap → m1 < m2 < m3 [16]. Test: JUNO, DUNE, Hyper-Kamiokande.

XII.11. P11: Nuclear resonance width Γ/E ≈ (π − 3)2 ≈ 2% The observation grain at d = −1 determines the minimum relative uncertainty. Test: analysis of ENDF/EXFOR databases.

XII.12. P12: φ-scaling of cosmological structures The cluster hierarchy (atom → molecule → . . . → galaxy cluster) reproduces the triad architecture at each level. Test: large-scale structure statistics.

XII.13. Additional falsifiable predictions from the 39-role table F1. Generational structure of baryons: Σ+ and Σ+ b exhibit discrete transitions (weak decay, flavor change) like leptonic generations e → µ → τ — already confirmed. F2. φ4 -law: m(3rd gen.)/m(1st gen.) ≈ φ4 for baryons. F3. φ2 -law: m(Σ+ c )/m(p) = 2.614 ≈ φ with 0.2% accuracy.

F4. Gluon hierarchy: 8 gluons exhibit internal structure at high energies. F5. Sub-neutrinos: upon opening d = −2, δΨ−1 will be discovered. F6. The number 39: the complete set of roles for a two-level window. Of these, 34 are confirmed, 5 are predictions. F7. The invariant 17: each recursion level has exactly 17 roles.

XII.14. Summary table of predictions

Prediction

Verification method

## P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12

Inter-scale correlations S(ρd ) ∝ φ−|d−d0 | Correlations of unentangled e− η = f (π, φ) without parameters Exactly 3 generations ∀d Quark substructure Masses ∝ φn × [1 + k(π − 3)2 ] δh̄/h̄ depends on scale ∆α/α correlates with ρb Normal ν hierarchy Γ/E ≈ (π − 3)2 ≈ 2% φ-scaling of structures

β-decay in different states Partial Fractal lattices Precision measurements Analytical derivation Nν = 2.984 ± 0.008 Retrodiction Scattering cross-sections Mass ratio analysis Partial Josephson vs. Kibble Quasar spectroscopy Indirect JUNO, DUNE ENDF/EXFOR databases Large-scale structure

XII.15. What would refute ∞-recursion (a) Rigorous proof of the existence of point-like (structureless) objects without an internal triad — a “bottom” of recursion. (b) Discovery of a 4th generation at the same level d (not substructure). (c) Inter-scale entanglement completely excluded — |Ψ∗ ⟩ strictly separable. (d) Nuclear resonance widths systematically lack (π − 3)2 .

(e) h̄ turns out to be an absolutely exact constant to 10−12 precision.

XIII. CAVEATS AND OPEN QUESTIONS The presented derivation is structural in character: the ODTOE axiomatics contains three topological mechanisms generating SU (3) × SU (2) × U (1), a combinatorial invariant of 17 roles per recursion level, and the complete picture of 39 roles of the two-level window with twelve falsifiable predictions. Open tasks for transitioning from a structural to a rigorous mathematical derivation: (a) Rigorous proof that π1 (S 1 ) = Z generates precisely a U (1)-gauge field from the self-consistency condition Ψ∗ = Φ(Ψ∗ ). (b) Derivation of SU (2)-spinor structure from the toroidal bundle as a theorem. (c) Rigorous derivation of SU (3) from the triad architecture at d = −1 with proof excluding SO(3) and U (3). (d) Derivation of quantum numbers (spin, isospin, hypercharge, color) from observer components O = (B, A, H) and the four coherence components. (e) Quantitative connection of PMNS and CKM matrix angles with loop junction geometry. (f) Derivation of exact masses of all 17 (or 39) particles from π, φ, and (π − 3)2 . (g) Extension to the Higgs boson mass: connection of mH ≈ 125 GeV with structural parameters. (h) Proof of decomposition uniqueness: exactly three factors from minimality of the triad architecture. (i) Rigorous definition of the scaling operator Σd and proof of existence of selfsimilar fixed points. (j) Analytical derivation of baryonic asymmetry η from (π − 3) and structural parameters. (k) Refinement of the neutrino cosmological proportion: Ων (ODTOE) = 0.52% vs. Planck < 0.3% — requires either correction of the 4-component model or revision of the upper limit on Σmν . (l) Determination of exact masses of L7 and L8 from structural parameters. (m) Quantitative description of role reassignment upon observer window shift.

XIV. MAIN THESIS The Standard Model is not the definitive catalog of reality, but a single octave on an infinite keyboard: 39 stable configurations of the unified self-observation cycle Φ = ι ◦ Ô at levels d = 0 and d = −1.

The gauge group SU (3) × SU (2) × U (1) is not a postulate but a consequence of the triad loop topology. The number 17 is a combinatorial invariant reproduced at each of an infinite number of recursion levels. Cosmological proportions ΩΛ : ΩDM : Ωb = φ2 : 1 : Z are a direct consequence of φ-torus geometry [20]. The ratio mp /me = 6π 5 is a manifestation of fivefold spirality [10]. The total number of configuration types: 17 × ∞. Infinite recursive nesting is not a metaphor but a falsifiable structure with concrete predictions.

CONFLICT OF INTEREST The author declares no conflict of interest.

FUNDING This research was conducted without external funding.

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ODTOE author: Anton Pankratov Analysis and unification: March 2026