// Supra-Omega Resonance Theory

SORT Research

Research by Gregor Herbert Wegener · Berlin, Germany
Part of the broader independent research program at Independent Research & Systems Modeling.

A Level-0 structural assessment framework for structural consistency, projection behavior, scale coupling, and cross-domain coherence analysis.

SORT addresses structural coherence before dynamical modeling. The contribution is a reusable structural grammar for reading coherence, drift, projection behavior, boundary effects, and cross-domain diagnostic relations. The current realization uses a constructively complete 22-operator core, a global projector Ĥ, and a calibrated projection kernel κ(k). Empirical model comparison, production validation, and Level-1 replacement dynamics are outside the framework-definition layer.

Ĥ =22∏i=1Ô_i

Global structural projector formed from 22 idempotent resonance operators.

Level-0 Structural Framework Operator-Projection Architecture Structural Consistency Scale Coupling Cross-Domain Coherence SORT v7 pre-audit artifact available

// Framework Foundations

Framework Foundations

The mathematical and conceptual architecture underlying the SORT framework. An accessible overview of the structural layer, public core objects, and the relationship between SORT and established Level-1 theories — without requiring the full whitepaper.

Level-0 Structural Layer

SORT defines a structural description layer that precedes model dynamics, field equations, and empirical parametrization. It complements Level-1 theories such as GR, QFT, and ΛCDM by providing consistency conditions, projection relations, boundary behavior, and cross-domain coherence criteria for comparing and embedding model outputs without modifying their dynamics.

Pre-dynamical layer: precedes field equations and parametrization
Embedding criterion: structural coherence across Level-1 models
Non-replacement: Level-1 dynamics remain unchanged
Cross-domain reach: shared core, domain-specific reading

Public Core Objects

The public core is a constructively complete realization composed of 22 idempotent resonance operators, a global projector, and a calibrated projection kernel. These objects form the stable reference layer used by all domain modules.

{Ôᵢ}ᵢ₌₁²² Ĥ κ(k)

The global projector is the ordered structural composition of the 22 operators:

Ĥ = 22 \prod i=1 Ô i

Here ∏ denotes structured operator composition. Ĥ denotes the global structural projector.

A small set of public invariants captures the algebraic core:

Ôᵢ² = Ôᵢ Σᵢ cᵢ = 0 Ĥ² = Ĥ κ(0) = 1

22-Operator Algebra

Within the present realization, the framework is implemented through 22 idempotent resonance operators forming a closed algebraic structure. Operators are organized into balanced positive and negative subsets satisfying light-balance, with idempotency, commutator closure, and Jacobi consistency. Minimality and generator-independence remain open research questions.

Idempotency: each operator satisfies Ô² = Ô
Duality: 11 constructive and 11 reductive weights
Light-Balance: global structural neutrality Σwᵢ = 0
Commutator closure within the operator span
Jacobi identity verified for the complete algebra

Physical Interpretations of the Operators

Symbol	Interpretation
Ô₁	Origin (null state); vacuum informational substrate \|0⟩, a^k\|0⟩ = 0.
Ô₂	Projection; boundary projection by coupled kernel π_κ.
Ô₃	Entanglement; non-local coherence, e.g. \|Φ⁺⟩ = (\|00⟩ + \|11⟩)/√2.
Ô₄	Duality; wave–particle complementarity δ_wave δ_particle.
Ô₅	Timelessness; cyclic time parametrization τ(t) = artanh(t) + C.
Ô₆	Spatial structure; emergent metric γ = ∑ g_ij dxⁱ dx^j.
Ô₇	Light structure; invariant propagation λ = c dt.
Ô₈	Memory; temporal integration μ = ∫ m(t) dt.
Ô₉	Boundary layer; idempotent boundary projection β² = β, β ∂φ = ∂φ_boundary.
Ô₁₀	Self-reference; Gödel-type self-application σ = R² σ_R, σ_R φ = φ ∘ Self.
Ô₁₁	State; phase manifold Φ = e^iθ, θ ∈ ℝ/2πℤ.
Ô₁₂	Emergence; folded simplicity η = ∑ e_i · fold(θ).
Ô₁₃	Number; symmetry signature Z = ∏ z_i.
Ô₁₄	Resonance; superposition R = ∑ r_k e^iωt.
Ô₁₅	Polarity; complementary directions ρ = ρ(φ₁ − φ₂).
Ô₁₆	Entropy; ε = k ln W.
Ô₁₇	Scaling; recurrence σ = ∑ s_k(ℓ/ℓ₀).
Ô₁₈	Symmetry breaking; Σ = σ_break · Δφ.
Ô₁₉	Information; I = −∑ p_i log p_i.
Ô₂₀	Observation; B = ∫ Dψ e^iS.
Ô₂₁	Integration; J = ∏ frag_i.
Ô₂₂	Unity; cyclic closure Ω̂² = Ω̂.

Projection Kernel κ(k)

The projection kernel mediates scale-dependent structural coupling and is central to SORT's cross-scale capabilities. The Gaussian kernel is retained as a scale-dependent structural weighting and damping function. Projector idempotency must be defined and validated for a separately constructed global projector.

κ(k) = exp[ -(σ₀ L_H k)² / 2 ] σ₀ = 0.00190643 (historical MOCK v3 reference scale) L_H = 4285 Mpc κ(0) = 1

Scale-dependent damping as a structural relation
Amplification function: η(k) = κ(k) − 1
Domain invariance: same kernel form across modules

Structural Assessment Layer

SORT reads a system as a structured state, maps the observed condition through a V1–V4 diagnostic grammar, anchors the reading to an application / scenario / metric / regime structure, and connects compatible cases to an evidence interface. This layer sits before implementation-specific telemetry, scoring, weighting, production thresholds, or intervention logic.

Assessment chain

Observation \to V1 \to V2 \to V3 \to V4 \to Application \to Scenario Class \to Metric Set \to Regime Classification \to Evidence Interface

Structural interface

S_AI \to Ĵ_AI \to P̂κ(Ĵ_AI) \to R_AI(Δ)

The V1–V4 grammar reads:

V1: observed structural phenomenon
V2: structural cause or coupling
V3: structural effect space
V4: decision or utilization surface

This is a public assessment grammar, not an implementation-specific assessment engine. Operator selection, telemetry mapping, scoring, weighting, thresholds, and intervention playbooks remain outside the public protocol.

Kernel-Damping Evidence Interface

The Core-3 evidence protocol provides analysis-layer structural reproducibility evidence and reconstructs declared structural risk transitions under the inherited historical reference scale σ₀ = 0.00190643. It reports scenario-level coherence using the coefficient of variation across implied structure modes.

κᵢ = rᵢ(1) / rᵢ(0) ξᵢ = \sqrt( -2 ln κᵢ ) / σ₀ CV_S = sξ,S / ξ̄_S

Core-3 analysis-layer structural reproducibility evidence: 3 applications, 20 scenarios, 104 declared metrics, overall CV = 0.141, classification: coherent.

These values support reproducibility of declared analysis-layer calculations only. They do not claim production validation, benchmark superiority, vendor telemetry analysis, or runtime optimization.

Modular Domain Architecture

SORT decomposes into domain modules sharing a common mathematical core while providing domain-specific interpretations. All modules consume the public core API; no module modifies operator definitions.

M_SORT = M_COSMO \oplus M_AI \oplus M_CX \oplus M_QS

SORT-AI: domain architecture for advanced AI systems
SORT-CX: complex systems stability and emergence
SORT-QS: quantum systems diagnostics and coherence
SORT-COSMO: projection-level cosmological effects

Public Core vs Internal Engine

The framework distinguishes a public reference layer — operators, projector, kernel, invariants, and module signatures — from internal engine details such as full transition matrices, implementation-specific routines, and application mechanisms. This separation supports structural review without disclosing operative implementation details.

Public: definitions, invariants, kernel form, module boundaries
Internal: full matrices, implementation routines, application logic

Definitions

Compact reference for core SORT architecture terms. These definitions distinguish the frozen structural reference layer from validation runs and future execution layers.

Resonance operator. An idempotent structural operator acting on the SORT projection space.

A resonance operator is an idempotent structural operator within the SORT projection space. It defines a stable subspace or structural fragment of the framework. In the public core, the current constructive realization contains 22 such operators. The key property is idempotency, Ôᵢ² = Ôᵢ, which means repeated application preserves the structural state rather than generating an unbounded dynamical cascade.
Light-Balance. A structural neutrality condition over operator weights, not a claim about electromagnetic light.

Light-Balance denotes the structural weight-neutrality condition Σᵢ cᵢ = 0. It expresses internal balance across constructive and reductive operator contributions. The term "light" is used in the SORT lexicon as a structural and resonant metaphor, not as a claim about electromagnetic radiation. In the framework, Light-Balance functions as a consistency condition for the global operator structure.
Validation. Internal structural validation of algebraic consistency, reproducibility, and artifact integrity.

Unless otherwise stated, validation in SORT means internal structural validation. It includes idempotency, closure, Jacobi consistency, consistency residuals, deterministic reproducibility, convergence behavior, and artifact integrity. It does not automatically imply empirical prediction, production validation, benchmark superiority, or replacement of Level-1 physical or engineering models.
MOCK. Model-Operator Consistency Kernel, the frozen structural and contractual reference architecture.

MOCK v4 is the frozen structural and contractual reference architecture. It specifies the operator topology, projection consistency conditions, domain isolation rules, and public reference structure against which further validation and execution work is aligned. MOCK v4 is not a simulator, not a runtime system, and not a production execution environment. It defines how SORT is structurally organized, not what is numerically executed. Evidence releases and validation runs use MOCK v4 contracts as references but are executed outside MOCK v4.
SORT v7 Validation Runs. Controlled deterministic replay protocols; the existing workstation run is pre-audit.

SORT v7 Validation Runs refer to deterministic numerical re-execution protocols for the established SORT core under controlled single-node conditions. The existing workstation package is retained as a Pre-Audit Frozen Artifact, Superseded for Current Validation Claims, with Audited Revalidation Pending. The v7 Validation Runs reference MOCK v4 contracts, but they are not a new MOCK version, not a production runtime, and not an HPC execution environment. The first frozen package is documented in the SORT Version 7 Workstation Validation Run.
SWORD Execution Engine. Planned v8+ scalable execution layer built on top of MOCK.

SWORD, Structured Workflow for Operator-Resolved Dynamics, is the planned v8+ scalable numerical execution layer of SORT. It is intended to operationalize execution beyond controlled validation runs, including future scalable and potentially HPC-oriented workflows. SWORD is an execution layer, not an architecture definition. It builds on MOCK and does not redefine MOCK. SWORD is currently in planning; it is not yet fully published or implemented.

SORT Glossary

Selected terminology from the SORT Whitepaper v4 glossary. The lexicon replaces particle-, string-, and brane-based metaphors with operatoric and resonant terms.

Resonance carrier. Operatoric counterpart to particle-like mediation.

A resonance carrier is the SORT term for a minimal idempotent interaction between fragment operators. It replaces particle-like or bosonic mediation with a structural transfer of resonant phase between operators. The term is intended to describe coherence mediation inside an operator algebra, not the propagation of a physical particle.
Order contribution. The structural weight assigned to a fragment operator.

An order contribution is the numerical or structural weight cᵢ associated with a fragment operator. Across the full operator set, these contributions satisfy the Light-Balance condition Σᵢ cᵢ = 0. The concept replaces central-charge-like language with an algebraic neutrality condition inside the SORT operator structure.
Structural equilibrium. The closure and balance condition preserving global consistency.

Structural equilibrium denotes the condition that the operator algebra remains closed, idempotent, and balanced under admissible compositions. It is the SORT counterpart to consistency-preserving mechanisms in other mathematical frameworks, but it is expressed through operator closure, residual bounds, and light-balance neutrality rather than through geometric or field-theoretic anomaly language.
Resonant surface. An operator-defined correlation boundary, not a geometric worldsheet.

A resonant surface is an informational or operator-defined correlation domain on which projection relations are evaluated. It replaces geometric surface metaphors with a structural boundary inside the projection space. The term should be understood as a correlation surface defined by the operator-kernel structure, not as a physical membrane in spacetime.
Dimensional membrane. A transition layer between projection levels or resonance domains.

A dimensional membrane is a structural transition layer linking adjacent resonance or projection domains. It replaces brane-like geometric imagery with operatoric connectivity. In SORT, such transitions are mediated by projection relations between operator subspaces rather than by extended physical objects.
Resonance chain. A finite ordered linkage of idempotent operators.

A resonance chain is a finite ordered linkage of idempotent fragment operators. It replaces operator-product or expansion-language with a stable structural chain in which repeated composition preserves coherence rather than generating uncontrolled divergence. The term emphasizes finite algebraic closure rather than infinite excitation spectra.

The full terminology is developed in the Whitepaper v4 glossary. On this website, the terms are used only as orientation for reading the public framework documents.

// Whitepaper Series & Orientation

Whitepaper Series & Orientation

The SORT whitepaper series is organized as complementary structural perspectives on the same framework. Version 4 provides exploratory depth, Version 5 provides mathematical hardening, Version 6 provides modular architecture, and the Orientation Note guides reviewers through the relation between them.

Reviewer Guide

Orientation Note

Structural guide for reviewers and expert readers. Explains how Whitepaper versions 4, 5, and 6 relate to each other, what SORT claims, what it does not claim, and how readers should navigate the whitepaper series.

Purpose: reviewer orientation; not a new theoretical claim.

The description of Version 5 as mathematically hardened reflects its historical intended role. The numerical validation layer is currently subject to corrective audit.

Read Orientation Note →

Version 4 · Conceptual Foundation

Whitepaper v4 — Exploratory Foundation

Establishes the conceptual foundation and maximal exploratory scope of SORT. Introduces the complete 22-operator set, physical interpretation context, cosmological resonance carriers, and extended derivational pathways.

Scope: conceptual depth, physical interpretation, exploratory appendices.

Open Whitepaper v4 →

Version 5 · Historical Pre-Audit

Whitepaper v5 — Mathematical Hardening

Status: Historical Pre-Audit Whitepaper. Numerical validation interpretations under corrective review.

Version 5 introduced the historical MOCK v3 numerical layer, kernel calibration, and mathematical-hardening program. Several inherited validation interpretations are currently subject to corrective audit.

Scope: algebraic rigor, kernel calibration, historical validation tolerances.

Open Whitepaper v5 →

Version 6 · Architectural Reference

Whitepaper v6 — Modular Architecture

Status: Architectural Reference. Inherited numerical validation claims under review.

Version 6 remains the principal modular and architectural consolidation document. Numerical validation claims inherited from Version 5 are being reassessed through the current corrective audit.

Scope: canonical architecture, public core, domain modules, reproducibility.

Open Whitepaper v6 →

Version 7 · In Preparation

Whitepaper v7 — Preview

Planned axiomatic consolidation of SORT as a Level-0 structural framework. Version 7 will clarify coherence axioms, formal claim boundaries, constructive completeness, workstation-scale structural validation, and the roadmap toward embedding competence, equivalence analysis, and minimality research.

Version 7 is planned as an axiomatic consolidation, not an expansion. It defines Level-0 coherence conditions, states claim boundaries explicitly, and separates structural validation from empirical model fitting. v7 does not claim empirical validation, HPC validation, minimality, structural necessity, or predictive power at the framework-definition layer.

Visual Overview

SORT v6 Structural Overview Slides

A compact visual orientation deck introducing SORT as a modular operator-projection framework for structural analysis. It summarizes the shift from dynamics to structure, the 22-operator core, idempotency, Light-Balance, resonance projection, domain modules, and the path from internal consistency to future validation.

Use as conceptual orientation. Canonical claim boundaries are defined by the Orientation Note and Whitepaper v6.

Open Overview Slides →

Framework Status

Current status: constructively complete Level-0 structural framework definition. Validation status: historical numerical validation interpretations are under corrective audit; the Version 7 workstation run is superseded for current validation claims pending audited revalidation. Open questions: minimality, generator independence, equivalence class characterization, and structural necessity.

Evaluation criteria at the framework level include internal mathematical consistency, operator and projection coherence, invariance properties, boundary behavior, reproducibility, and cross-domain consistency. Level-1 empirical prediction and replacement dynamics are not part of the framework-definition layer.

// Domain Research Modules

Research Modules

Each SORT domain module applies the same Level-0 operator-projection core to a different class of structural objects. The modules do not introduce separate theories. They define domain-specific interpretation layers for advanced AI systems, complex systems, quantum systems, and cosmological inference spaces.

All modules share the same Public Core — 22 idempotent resonance operators, the global projector Ĥ, and the projection kernel κ(k). What changes between modules is not the mathematics, but the interpretation of the structural state, the coupling surface, the evidence layer, and the domain-specific diagnostic question. The modules are Research Domains, not collections of use cases.

SORT-AI

SORT-AI is the canonical domain architecture for advanced AI systems. It reads AI deployments as structurally coupled execution systems whose relevant behavior emerges across models, runtimes, schedulers, orchestration layers, policy surfaces, memory paths, deployment boundaries, and evidence requirements. The module becomes most informative where local metrics, benchmark scores, or layer-specific monitoring remain plausible while the composed system exhibits incoherence, cost drift, auditability gaps, or emergent instability.

Diagnostic Focus

AI infrastructure coupling and topology-sensitive behavior
Runtime control coherence across schedulers, orchestrators, and policy layers
Agentic system stability, retry amplification, and tool-mediated emergence
Benchmark and evaluation drift, deployment-context projection instability
Evidence surfaces, auditability, and reconstructability
SORT-Sovereign projection into strategic and regulatory decision spaces

Open Preprint →

Advanced artificial intelligence systems increasingly exhibit behaviors that are not adequately captured by component-local metrics, benchmark scores, or layer-specific monitoring. These behaviors arise across coupling surfaces, control regimes, deployment boundaries, and emergent interaction patterns. SORT-AI defines a canonical domain architecture for reading advanced AI systems as structurally coupled systems built from model execution, runtime orchestration, control policies, memory paths, serving layers, agentic workflows, deployment boundaries, and evidence requirements. The relevant analytical object is the composed system, not the isolated component. The domain is organized through Domain, Cluster, Application, and Structural Dimensions V1–V4, linking observed phenomena to structural causes, effect spaces, and decision surfaces. It currently comprises 52 applications across Coupling, Learning, Control, Emergence, and Evidence. Runtime Control Coherence (AI.04) serves as the canonical example of locally correct mechanisms producing globally incoherent behavior under scale.

SORT-CX

SORT-CX is the complex-systems domain module of SORT. It treats emergence, stability islands, collective modes, scale-dependent correlations, critical transitions, and multilayer systemic risk as structural phenomena rather than as outputs of one specific dynamical model, simulation rule, or fitted dataset. The module is designed for systems whose macrostructure cannot be reduced to microscopic update rules, but can be read through projection geometry, fixed-point stability, non-local kernels, and resonance-space drift.

Diagnostic Focus

Emergence as projection rather than time evolution
Stability landscapes and idempotent fixed points
Drift diagnostics in resonance space
Operator-dominated, kernel-dominated, and drift-dominated regimes
Pattern formation, critical transitions, and tipping points
Multilayer systemic risk and cross-scale coupling

Open Preprint →

Complex systems across physics, biology, ecology, technology, and society exhibit emergent structures that cannot be reduced to microscopic rules or simple dynamical laws. SORT-CX applies the shared SORT operator-projection framework to the structural analysis of complex systems. Emergence is formulated as a projective process rather than a temporal evolution, structural stability as idempotent fixed points under operator projection, and structural change as drift in resonance space. The module enables classification into operator-dominated, kernel-dominated, and drift-dominated regimes independent of specific differential equations, agent-based update rules, or dataset-specific modeling assumptions. SORT-CX is therefore not a competing theory of complex systems. It is a structural diagnostic overlay that becomes informative whenever local dynamical models remain valid but cannot account for stability landscapes, scale-dependent correlations, or multilayer systemic risk by themselves.

SORT-QS

SORT-QS is the quantum-systems domain module of SORT. It maps quantum channels, error sectors, stabilizer-like constraints, and operator chains into a structural resonance-space diagnostic layer. The module does not modify quantum mechanics or replace Hilbert-space dynamics. Instead, it provides a hardware-agnostic overlay for projector-based error correction, kernel-based noise filtering, and operator-chain diagnostics that complements rather than substitutes existing CPTP-level analysis.

Diagnostic Focus

Projector-based quantum error correction diagnostics
Kernel-based noise filtering in operator-valued mode space
Operator-chain drift and Jacobi defect analysis
Stabilizer-like structures and logical-subspace constraints
CPTP-aware interpretation boundaries
Hardware-agnostic verification layer

Open Preprint →

SORT-QS adapts the SORT operator-projection framework to finite-dimensional quantum devices. Coherent and incoherent error processes, noise filtering mechanisms, and diagnostic procedures are represented through idempotent resonance operators, a global projector, and a non-local kernel acting on operator space rather than configuration space. Quantum channels are mapped to structured resonance manifolds in Liouville space, and error sectors are encoded as constrained projectors satisfying algebraic closure and idempotency. Deviations from ideal projector structure are read as resonance defects, providing a structural diagnostic for error correction, noise filtering, and operator-chain analysis. The framework remains hardware-agnostic and is presented without device-specific assumptions or empirical performance claims; it is a structural overlay on standard quantum-information descriptions, not a replacement for them.

COSMO

SORT-COSMO

SORT-COSMO is the cosmology domain module of SORT. It reads cosmological tensions and anomaly classes as projection-induced structural effects in inference space, not as modifications of gravitational dynamics, field content, ΛCDM, or empirical parameter fitting. The module is intentionally non-dynamical and treats observables as structural projections of the shared operator-kernel backbone, compatible with general relativity and the standard cosmological model.

Diagnostic Focus

Scale-dependent Hubble drift
Early galaxy overdensity
Early supermassive black-hole seeds
Low-ℓ CMB modulation
Large-scale cosmic coherence
Projection-induced potential structures

Open Preprint →

SORT-COSMO provides a projection-based structural framework for analysing large-scale cosmological phenomena without modifying gravitational dynamics, introducing new fields, or performing empirical parameter fitting. Cosmological observables are treated as structural projections of a shared operator-kernel backbone composed of idempotent resonance operators, a global consistency projector, and a non-local projection kernel. Scale-dependent drift, long-range coherence, and emergent structural amplification arise as projection effects rather than as consequences of altered expansion histories or additional physical degrees of freedom. The framework is compatible with GR and ΛCDM and remains a structural diagnostic layer rather than a competing cosmological model. It addresses tensions and anomaly classes — Hubble drift, early galaxy overdensity, early SMBH seeds, low-ℓ CMB modulation, large-scale cosmic coherence, and projection-induced potential structures — as inference-space phenomena rather than dynamical revisions.

// Why Level-0 Matters

Research Highlights

These highlights summarize the scientific role of SORT as a Level-0 structural layer. SORT does not compete with dynamical or empirical models. It adds a structural diagnostic layer for projection, composition, boundary behavior, scale coupling, and global coherence.

01Level-0, Not Level-1

SORT operates before model dynamics. It does not define equations of motion, fit empirical parameters, or replace GR, QFT, ΛCDM, quantum mechanics, AI monitoring, or complex-systems simulation. It defines structural coherence conditions under which outputs of such Level-1 models can be compared, projected, and diagnostically interpreted.

02Structural, Not Dynamical

Dynamical models explain how a system evolves. SORT asks whether the resulting structural states remain coherent under projection, composition, scale coupling, and boundary transfer. This is why the same Public Core can be applied across AI systems, complex systems, quantum systems, and cosmological inference without modifying the underlying domain dynamics.

03Boundary Conditions of Existing Models

SORT becomes most informative where existing models remain locally valid but globally incomplete: good local metrics with system-level incoherence, stable components with unstable composition, standard dynamics with anomalous inferred observables, or valid quantum operations with structurally fragile operator chains.

04Paradoxes as Projection Signals

Paradox-like phenomena are treated not as contradictions in the underlying system, but as signals that the explanatory level is too local. In AI this appears as benchmark success with deployment drift. In cosmology it appears as inferred scale tensions. In complex systems it appears as emergence not reducible to micro-rules. In quantum systems it appears as operator-chain fragility not captured by simple performance metrics.

05Shared Core, Domain-Specific Interpretation

The same Public Core — 22 idempotent operators, the global projector Ĥ, and the projection kernel κ(k) — is used across all domains. What changes is not the mathematical substrate, but the interpretation of the structural state, the coupling surface, the evidence layer, and the diagnostic question.

06Complementary, Not Replacement Theory

SORT extends existing scientific and engineering methods by adding a structural diagnostic layer. It does not compete with local diagnostics, physical theories, simulation methods, or empirical models. It explains the layer at which composition, projection, non-local coupling, and boundary effects become visible.

Level-0 and Level-1 Are Different Questions

Layer	Question	Output	Examples
Level-1 Dynamics	How does the system evolve?	dynamics, predictions, telemetry	GR, ΛCDM, QFT, quantum mechanics, AI runtime mechanics, simulations, monitoring systems
Level-0 Structure	Is the resulting structure coherent under projection and composition?	projection, closure, drift, boundary behavior	SORT operators, global projector Ĥ, κ(k), fixed points, structural consistency

Level-0 and Level-1 ask different questions. When local dynamics, monitoring, or simulations produce outputs that are individually valid but globally difficult to reconcile, SORT provides a structural layer for reading coherence, drift, composition failure, and projection-induced behavior.

// Publications

Publications

Articles, preprints, whitepapers, and technical documents establishing the framework and its domain applications. Core Framework and Whitepapers are listed first; domain papers follow.

Source of bibliographic record: $GregorWegener.bib$ . External links open in a new tab.

// Reproducibility & MOCK Archives

Reproducibility, MOCK Archives & Validation Evidence

Reproducibility in SORT is organized through archived numerical exploration, frozen public architecture, and machine-readable catalog sources. MOCK v3 provides historical exploratory numerical evidence. MOCK v4 is the frozen structural and contractual reference architecture required to audit, reference, and extend that evidence without redefining the framework. The public assessment methodology behind these archives is described in the Structural Assessment Layer under Framework Foundations.

Research Audit Status — June 2026

An independent numerical audit has initiated a controlled corrective and revalidation process for the historical MOCK v3 numerical layer and the existing SORT Version 7 workstation validation package.

Historical publications, archived outputs, DOI records, and frozen artifacts remain preserved for provenance. The existing SORT Version 7 workstation validation run is retained as a pre-audit frozen artifact and is superseded for current validation claims pending an audited revalidation run.

The MOCK v4 structural reference architecture, the Level-0 Structural Assessment Framework, the public methodology, and separately scoped analysis-layer evidence protocols remain available under their stated limitations.

Repository Audit Status · Pre-Audit Repository Snapshot

Public Source Repository

The public SORT repository provides the version-controlled technical trace of the framework. It contains the MOCK v3 historical exploratory numerical archive, the MOCK v4 structural and contractual reference architecture, catalog materials, README files, checksum-oriented artifacts, and public documentation for external review.

Repository: $github.com/gregorwegener/SORT$
Contains $mock_v3$ and $mock_v4$ archives
Provides replay and architecture documentation
Supports external review of the public reference layer
Separates numerical exploration from architectural contracts

Open GitHub Repository →

MOCK v3 — Historical Exploratory Numerical Snapshot

MOCK v3 is the replayable historical exploratory numerical snapshot of SORT. It contains a calibrated three-layer numerical pipeline, deterministic seeding, σ₀ calibration, kernel construction, historical validation outputs, and article-level modules for exploratory cosmological observables. These outputs are treated as historical exploratory numerical evidence, not current validation evidence or final empirical proof.

The frozen MOCK v3 Gaussian kernel artifact remains numerically reconstructable from the historical Golden-Run reference value.

Previous claims that the Gaussian filtering operator itself was validated as idempotent are not treated as current validation evidence.

Version: v3.0.0-final · Seed: 117666
Three-layer pipeline: algebraic → kernel → spectral
Corrected article modules: Hubble Drift, Early Galaxies, SMBH Seeds
Legacy modules: CMB Anomalies, BAO Wiggles, Intergalactic Bridges
SHA-256 verification for generated artifacts

Selected historical exploratory outputs

σ₀ Golden-Run value 0.00190642767773082 σ₀ rounded 0.00190643 H_local_pred 72.79 km/s/Mpc δH/H₀ (SORT) 0.0800 Hubble residual -0.21 km/s/Mpc (0.15σ) Early galaxy enhancement 1.151 \to 1.528 (z=6 \to z=14) SMBH seed masses ~10⁴ - 10⁸ M☉ Legacy demonstrations CMB asymmetry \cdot BAO wiggles large-scale correlations

Workstation-scale archive. Treated as historical exploratory numerical evidence pending corrected revalidation; not current validation evidence and not a final empirical proof.

View MOCK v3 Archive →

MOCK v4 — Frozen Structural and Contractual Reference Architecture

MOCK v4 is the frozen structural and contractual reference architecture. It does not run numerical simulations and does not recompute MOCK v3 outputs. Instead, it defines the public system architecture, API contracts, domain isolation patterns, evidence structures, control semantics, and capability registry that allow prior and future numerical work to be audited against a stable framework baseline.

Architecture: final and frozen
Public Core API: operators, projector, kernel, invariants, schema
Domain modules: AI, complex systems, quantum systems, cosmology
Support layer: catalog, evidence, control, capabilities, engine hooks
Tests validate structure and contracts, not numerical correctness
MOCK v4 is the structural contract reference, not an execution engine

Evidence releases and validation runs use MOCK-v4 contracts as references, are executed outside MOCK v4, and do not change the MOCK v4 freeze status.

View MOCK v4 Reference →

Validation Boundary

MOCK v3, MOCK v4, SORT v7 Validation Runs, and the planned SWORD execution layer refer to different levels of the research program. Keeping these levels separate is part of the claim discipline of SORT.

Layer	Role	Claim Level
MOCK v3	Historical exploratory numerical snapshot	Historical exploratory numerical evidence; not current validation evidence
MOCK v4	Frozen structural and contractual reference architecture	Structural reference baseline
SORT v7 Validation Runs	Deterministic re-execution of the established core	Pre-audit artifact; superseded pending audited revalidation
SWORD v8+	Planned scalable execution layer	Future execution layer, not a MOCK version

MOCK v3 records the historical exploratory numerical calibration phase. Its Gaussian kernel is retained as a scale-dependent structural weighting and damping function, not as current projector-idempotency evidence. MOCK v4 is cited as the frozen structural and contractual reference architecture.

Data Availability Statement

Public framework components, reproducibility materials, and catalog sources are available through the archived whitepaper artifacts, Zenodo records, and the public SORT repository. The current machine-readable application catalog is $catalog.public.json$ , Version 6.2.

SORT v6 Framework: DOI 10.5281/zenodo.18094128
MOCK v4 Reference: DOI 10.5281/zenodo.18050207
Source Repository: github.com/gregorwegener/SORT
Public Application Catalog: $catalog.public.json$ , v6.2, 107 applications

// Replay Map

Computational Artifacts & Replay Map

MOCK v3 exposes the replay structure of the historical exploratory numerical snapshot. The artifacts below show how configuration, operator definitions, kernel calibration, layered diagnostics, and phenomenon-level observables are organized for reproducible execution. Each item can be expanded for additional scientific context.

Configuration & Core Inputs

These files define the deterministic numerical environment of MOCK v3: lattice geometry, seed, operator definitions, α-parameters, kernel settings, and shared execution functions.

⚙

05_config.yaml Lattice N=128, L=160 Mpc, deterministic seed 117666.

Global configuration for the MOCK v3 replay environment. It defines the workstation-scale lattice (N=128, L=160 Mpc, dx=1.25 Mpc), Hubble reference values (H₀_CMB=67.4, H_local_obs=73.0, L_Hubble=4285 Mpc), projection parameters (β=0.5, k_CMB=0.001, k_local=0.05), the operator count (22), kernel settings, numerical tolerances, output paths, and validation settings. This file anchors reproducibility: the same configuration should generate the same Layer I–III outputs when used with the archived scripts.

⊕

06_operators.json 22 resonance operators, 11 positive and 11 negative, light-balanced.

Public operator definition file for the MOCK v3 numerical archive. It encodes the 22 resonance operators used in the exploratory calculation, including their symbolic identifiers and weights cᵢ = ±0.090909… The operator set is organized as 11 constructive and 11 reductive contributions satisfying the Light-Balance condition Σᵢ cᵢ = 0 exactly. This file links the algebraic layer to the numerical replay environment.

params_alpha_v3.json Alpha-grid for Layer III semi-spectral evolution.

Parameter grid for the reduced Layer III semi-spectral evolution: a linear α-scan from 0.0 to 1.0 with 33 grid points (step 0.03125). The α-parameters define the scan surface used to evaluate energy-series behavior, convergence tendencies, and drift response under controlled replay conditions. This is not a production optimization layer; it is part of the exploratory numerical surface of MOCK v3.

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mockv3_engine.py Core engine for σ₀ calibration, kernel construction, historical diagnostics, and observables.

Shared numerical engine for the MOCK v3 archive. It implements the common routines used by Layer II and the phenomenon modules: σ₀ calibration, Gaussian scale-dependent transfer kernel construction κ(k) = exp[−(σ₀ L_H k)²/2], amplification function η(k) = κ(k) − 1, historical diagnostic metrics, and observable generation. The engine is part of the replayable historical exploratory archive and should not be described as a production runtime.

Three-Layer Pipeline

The pipeline moves from algebraic consistency to kernel calibration and then to reduced spectral evolution. It is retained as a historical exploratory replay surface, not as current validation evidence or production execution.

Layer I — Algebraic Diagnostics Idempotency, Light-Balance, operator count, and structural sanity checks.

Layer I records the algebraic entry conditions of the MOCK v3 environment. It checks that the declared operator count matches the operator file (22), evaluates the Light-Balance residual ε_LB = 0.0, records the deterministic seed 117666, and writes structured diagnostics including layer1_metrics.json · layer1_table2.csvIts purpose is to document that the historical replay begins from a structurally consistent operator basis.

Layer II — Kernel Calibration σ₀ calibration, κ(k), η(k), M_layer2.npy, historical diagnostics.

Layer II constructs the Gaussian scale-dependent damping / weighting / transfer kernel. It calibrates σ₀ from the historical Golden-Run value (0.00190642767773082; rounded 0.00190643), builds the kernel array M_layer2, evaluates the projection correction η(k), records historical diagnostic metrics including Light-Balance consistency (ε_LB = 0.0) and drift consistency (ε_drift ≈ 1.47×10⁻⁶), and writes layer2_metrics.json · M_layer2.npyThe Gaussian kernel is retained as a scale-dependent structural weighting and damping function. Projector idempotency must be defined and validated for a separately constructed global projector.

III

Layer III — Spectral Evolution Reduced semi-spectral evolution, energy series, convergence and drift diagnostics.

Layer III runs the reduced semi-spectral evolution using the calibrated kernel and α-grid. It generates the energy series across 16 steps, tracks E_min, E_max, and E_final, and records convergence and drift behavior. The layer is used as a controlled replay surface for observing convergence and drift in the exploratory environment.layer3_metrics.json · layer3_energy_series.csv

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Checksum Manifest SHA-256 artifact manifest for replay integrity.

The manifest records file sizes and SHA-256 checksums for the core replay files (configuration, operator definitions, α-grid, engine, layer scripts). Its purpose is artifact integrity: it allows external readers to verify that the replay scripts and configuration files have not changed between publication, review, and re-execution.07_manifest.json

Cosmological Phenomena & Observables

These modules evaluate selected cosmological phenomena and anomaly classes using the Layer I–III outputs. They do not recalibrate the framework. They generate phenomenon-specific exploratory outputs and figures.

Hubble Drift Scale-dependent H₀ drift as projection signature across CMB and local regimes.

Evaluates the corrected Hubble drift definition δH/H₀ = (H_local − H_CMB) / H_CMB. MOCK v3 records H_CMB = 67.4 km/s/Mpc, H_local_pred = 72.79 km/s/Mpc, H_local_obs = 73.0 km/s/Mpc, δH/H₀ = 0.08000, and a residual against the observation of approximately −0.21 km/s/Mpc (0.15σ). This is presented as exploratory numerical evidence and a projection signature, not as final empirical proof.11_article1_hubble_drift.py

Early Galaxy Enhancement Kernel-based high-redshift enhancement factors for early galaxy abundance.

Evaluates a kernel-based enhancement function for high-redshift structure formation. MOCK v3 records enhancement(z) = 1 + |η(k_eff(z), σ₀)| with values rising from approximately 1.151 at z=6 to 1.528 at z=14, using k_eff(z) = 0.1 · (1+z)/10 Mpc⁻¹. The module is framed as a structural projection reading of the early galaxy abundance tension, not as a completed publication claim.12_article2_early_galaxies.py

Early SMBH Seed Scaling Projection-based seed-mass scaling for early supermassive black-hole candidates.

Evaluates an exploratory projection-based SMBH seed mass surface using η₀ = 10⁻⁶ and δ_proj = 0.3. MOCK v3 records seed-mass ranges approximately from 10⁴ to 10⁸ M☉ across z=7, 10, 15, and 20, with comoving smoothing scales σ_com from ~1.02 Mpc at z=7 down to ~0.39 Mpc at z=20. The ranges are consistent with direct-collapse black-hole scenario scales. This remains a structural numerical exploration, not a final astrophysical model.13_article3_smbh_seeds.py

CMB Low-ℓ Modulation Legacy structural demonstration of hemispheric asymmetry and low-k modulation.

Legacy module evaluating a low-k / low-ℓ structural modulation surface, including hemispheric asymmetry. MOCK v3 records a hemispheric asymmetry of approximately 0.79% from a binned power-spectrum split (P_plus ≈ 0.426, P_minus ≈ 0.419). This is retained as a legacy structural demonstration and not framed as a final CMB model.14_article4_cmb_anomalies.py

Dark-Baryon Oscillator Legacy BAO wiggle demonstration through drift-coupled response.

Legacy module exploring a dark-baryon oscillator surrogate surface and BAO-like wiggle residuals. It evaluates power-spectrum samples across an 80-point k-grid and extracts the residual oscillatory component against a smoothed reference. The module is kept as a structural demonstration of oscillatory response under kernel coupling, not as a finalized physical dark-sector model.15_article5_dark_baryon_oscillator.py

Intergalactic Bridges Legacy large-scale correlation demonstration for filamentary bridge-like structures.

Legacy module evaluating large-scale correlation structure, including the radial two-point correlation ξ(r) on a 60-point r-grid, with a focus on the long-range tail ξ(r > 100 Mpc). It is presented as a replayable structural demonstration of long-range coherence signatures, not as a finalized observational claim.16_article6_intergalactic_bridges.py

Phenomenon modules consume the Layer I–III outputs. They do not recalibrate the framework. They generate phenomenon-specific exploratory outputs and figures for selected cosmological anomaly classes.

Key Formula Surface

Selected public formulas from the MOCK v3 replay archive. These formulas define the observable surface used by the phenomenon modules without exposing proprietary execution logic.

δH/H₀ = (H_local - H_CMB) / H_CMB η(k; σ₀) = exp[ -(σ₀ L_H k)² / 2 ] - 1 H_eff(k) = H_bare \cdot exp[ -(σ₀ L_H k)² / 2 ] enhancement(z) = 1 + | η(k_eff(z), σ₀) | M_seed = η_BH \cdot (4π / 3G) \cdot Φ \cdot σ

These formulas are shown as the public observable surface of MOCK v3. They should be read as part of a replayable exploratory archive, not as final empirical proof.

Replay Orientation

MOCK v3 can be replayed by installing the Python dependencies, running Layer I, Layer II, and Layer III, and then executing the phenomenon modules. The detailed commands are kept in the repository README rather than repeated on this landing page.

Install Python dependencies
Run algebraic diagnostics (Layer I)
Run historical kernel calibration diagnostics (Layer II)
Run spectral evolution (Layer III)
Execute phenomenon modules
Verify generated checksums and manifests

Open Replay Guide →

// Validation Runs

SORT Version 7 Workstation Validation Run

The SORT Version 7 Workstation Validation Run is retained as a historical pre-audit frozen artifact. It records the original deterministic Level-0 structural validation package and is superseded for current validation claims pending audited revalidation.

Pre-Audit Frozen Artifact

A deterministic Level-0 structural validation sequence for the declared SORT Version 7 operator, projection, kernel, fixed-point, drift, stability, workstation execution, and artifact-freeze chain. Status: Pre-Audit Frozen Artifact; Superseded for Current Validation Claims; Audited Revalidation Pending. Zenodo DOI: 10.5281/zenodo.20634212.

Referenced public layers

MOCK v4 frozen structural and contractual reference architecture
Public Analysis Layer
SORT-AI Core-3 Kernel-Damping Evidence Release v1
SORT Version 7 Workstation Validation Run

View GitHub validation run → Read explainer → Open Zenodo archive →

Artifact Summary

Validation path	$validation_runs/sort_version_7_workstation_validation/$
Frozen package	$SORT_Version_7_Workstation_Validation.zip$
Zenodo DOI	10.5281/zenodo.20634212
Validation scope	Historical Level-0 structural validation run
Current evidential status	Superseded pending audited revalidation
Execution context	Lenovo ThinkStation P3 Ultra workstation
Operating system	Windows
CPU	Intel Core i9-13900K
Python	3.13.5
Logical threads	32
RAM	approximately 64 GB
Package status	Pre-Audit Frozen Artifact
Package size	289337 bytes
SHA-256	$2fc5e68551f70ac25e8970e51204de80c03d593003aab8123091366cac8df505$

Public Layer Workflow

The historical validation run sits inside the public layer chain. MOCK v4 defines the frozen structural and contractual reference architecture, evidence releases provide analysis-layer structural reproducibility evidence, and validation runs document deterministic structural artifacts.

SORT \to MOCK v4 \to Public Analysis Layer \to Evidence Protocols \to Validation Runs \to Future Execution Layers

MOCK v4 is the frozen structural and contractual reference architecture. The Public Analysis Layer explains how observations become structurally assessable cases. Evidence releases provide analysis-layer structural reproducibility evidence. Historical validation runs document deterministic structural artifacts and package-level reproducibility evidence under their stated status. Future execution layers, including SWORD, remain downstream and are not implemented in the public repository.

Phase Summary

The historical validation run followed the sequence environment → operators → kernel → global projector → fixed-point structure → drift and stability → workstation execution → artifact freeze. The result is a frozen, hashed, archived package documenting the original pre-audit workstation run.

The pass states below are preserved as the recorded outputs of the original pre-audit run. They do not represent the current validation status of SORT. A corrected validation protocol and independently replayed artifact package are in preparation.

Phase	Gate	Scope	Status
Phase 0 — Setup and Reproducibility	Gate 0	Environment capture, seed policy, run manifest	passed
Phase 1 — Operator Integrity	Gate 1	22-operator registry, idempotency, balance, Jacobi checks	passed
Phase 2 — Projection Kernel	Gate 2	Kernel definition, normalization, projection idempotency	passed
Phase 3 — Global Projector	Gate 3	Global projector construction, closure, composition checks	passed
Phase 4 — Fixed-Point Structure	Gate 4	Iterative projection behavior, convergence, norm tracking	passed
Phase 5 — Drift and Stability	Gate 5	Synthetic drift diagnostics and perturbation response	passed
Phase 6 — Workstation Scaling	Gate 6	Runtime, memory, safe-gate grid/thread execution	passed / included
Phase 7 — Artifact Freeze	Gate 7	Audit, inventory, hashes, reproducibility manifest, ZIP freeze	passed

What the historical run recorded

Under the declared finite workstation representation, the original pre-audit run recorded that the 22 declared operators passed implemented integrity checks, the declared Gaussian kernel was normalized as a scale-dependent interface, the global projector interface was constructed under the tested protocol, the projection apparatus exhibited controlled fixed-point behavior, the drift metric distinguished controlled structural deviations, perturbed states returned under the tested stability protocol, and the resulting artifacts were inventoried, hashed, packaged, frozen, cited, and archived.

What this does not establish

The validation run does not establish that SORT is empirically correct, that SORT is a physical theory, that SORT replaces GR, QFT, ΛCDM, AI systems theory, control theory, or any other Level-1 theory. It does not prove minimality, necessity, uniqueness, σ₀ universality, production readiness, runtime improvement, SWORD execution, or ASDV execution.

Historical recorded run status: passed. Current evidential status: superseded pending revalidation. The correct interpretation is a preserved pre-audit artifact, not current empirical proof, production validation, minimality proof, or structural necessity proof.

Relationship to MOCK v4

MOCK v4 is the frozen structural and contractual reference architecture. The SORT Version 7 Workstation Validation Run does not replace MOCK v4 and does not define a new MOCK version. It is a historical deterministic validation sequence that used MOCK-v4 contracts as a reference and is now retained as a pre-audit frozen artifact. Future SWORD work remains conceptually downstream and is not part of this validation run.

Validation Artifacts

Main SORT repository: github.com/gregorwegener/SORT
Validation run root: validation_runs/…/sort_version_7_workstation_validation
Explainer: SORT_Version_7_Workstation_Validation_Explainer.md
Phase 7 freeze outputs: phase_7_artifact_freeze/outputs
Zenodo archive: 10.5281/zenodo.20634212

Citation

Wegener, G. H. (2026). gregorwegener/SORT: SORT Version 7 Workstation Validation Run — Frozen Artifact Package (sort-v7-workstation-validation-v1.0.0). Zenodo. https://doi.org/10.5281/zenodo.20634212

@software{wegener_2026_sort_v7_workstation_validation, author = {Wegener, Gregor H.}, title = {{gregorwegener/SORT: SORT Version 7 Workstation Validation Run — Frozen Artifact Package}}, year = {2026}, publisher = {Zenodo}, version = {sort-v7-workstation-validation-v1.0.0}, doi = {10.5281/zenodo.20634212}, url = {https://doi.org/10.5281/zenodo.20634212} }

// Application Catalog

Application Catalog

The SORT Application Catalog is a public diagnostic ontology of recurring structural problem forms. It currently contains 107 structural diagnostic perspectives across SORT-AI, SORT-CX, SORT-QS, SORT-Sovereign, and SORT-COSMO, organized along three orthogonal diagnostic axes: Domain, Cluster A–E, and Structural Dimensions V1–V4. Applications are diagnostic perspectives, not products, implementations, benchmarks, or software modules.

107

Total Apps

Domains

A–E

Clusters

Technical Apps

Sovereign Apps

Cosmology Apps

Applications are structural diagnostic perspectives, not implementations. The technical domains (SORT-AI, SORT-CX, SORT-QS) contain 91 applications across clusters A–E. The Sovereign meta-domain contains 5 applications restricted to clusters A, C, and E. SORT-COSMO contains 11 IP-free cosmology applications. The source of record is $catalog.public.json$ .

Open Application Catalog → Download Catalog Source

// Contact

Contact

Scientific correspondence and collaboration inquiries.

Gregor Herbert Wegener

Berlin, Germany

gregor.wegener@independent-research-systems-modeling.com