Project index and reading guide

Scale Space / CNRS Research Programme

A modular research programme connecting logarithmic scale geometry, complex positional representation, and cross-scale physical and biological modelling.

Donald G. Palmer — Independent Researcher — Little River, South Carolina — community-synced index updated 2026-07-31

Start HereZenodo CommunityScale Space RecordsCNRS RecordsCode Record
Zenodo Community: The archival collection for Scale Space / CNRS Research Programme records is now available.
Open Zenodo Community

Start Here

This page is the public index for the Scale Space / CNRS programme. It connects the live project website, the Zenodo Community archive, the book, technical papers, code repository, open problems, and Zenodo concept DOIs.

How to read this project: the book is the synthesis and research-programme statement; the papers are the technical record; the GitHub repository is the executable-code entry point; the CNRS records are the mathematical and documentation entry points. Every record card retains its Concept DOI as the stable pointer and refreshes its title, version, date, latest-version DOI, Zenodo record link, and file list from Zenodo when the page opens. Static values remain visible if Zenodo is temporarily unavailable.
Live Zenodo metadata: waiting to refresh.

Book-level synthesis

Across Scale: A Geometry of Size and Representation provides the reader-facing architecture: motivation, scale geometry, quantum/biology extensions, CNRS, bridge questions, tests, and failure modes.

Technical derivations

The Scale Space papers provide the formal physics, geometry, dynamical-closure, cosmology, matter-source, and prediction record. Use these records when citing specific technical claims.

Mathematics and code

The CNRS records provide the mathematical background. The GitHub-linked Zenodo record provides the citable software release for the CNRS Scientific Toolkit.

45
Zenodo records indexed
Live Zenodo metadata loads when the page opens
24
Scale Space records
Framework and numbered papers
20
CNRS records
mathematics, guides, bridges
1
Code/software record
CNRS Scientific Toolkit

Zenodo Community Archive

The Zenodo Community is the archival collection for the programme records. Use it when browsing the full record set; use this page when choosing a reading path or locating the current concept/default DOI for a specific item.

45

Current concept records

represented by landing-page cards

142

Version records

in the supplied Zenodo export

2026-07-31

Snapshot through

newest publication date in the export

How the archive is used here: The Concept DOI is the stable latest-version pointer; the version DOI is the immutable citation for a specific release. The supplied community export provides the static snapshot, while the page also checks Zenodo live. New community concepts not yet classified on the page are surfaced automatically in the “Additional Community Records” area.

Scale Space / CNRS Research Programme Community

The community gathers the Scale Space framework and paper sequence, CNRS mathematical records, bridge documents, and software/code records under one archival umbrella.

Open Zenodo Community

Public architecture

www.nul1.com is the readable front door. The Zenodo Community is the archival collection. GitHub is the working code and issue-tracking layer. The book is the synthesis; the papers are the technical record.

Suggested record wording: This record is part of the Scale Space / CNRS Research Programme. The current project index, reading guide, papers, code, and open-problem map are maintained at www.nul1.com. The archival record set is collected in the Zenodo Community “Scale Space / CNRS Research Programme.”

Book and Programme Architecture

Recommended Zenodo structure: one project landing record, one book record, individual paper records, and separate code/software records. This page is suitable for the project landing record.

Across Scale

Role: reader-facing synthesis and research-programme statement. The book should cite this project landing record and the relevant technical-paper concept DOIs. Technical claims should be traced to the paper records below.

Citation logic

Cite the book for programme-level synthesis; cite individual Scale Space papers for physics derivations; cite CNRS papers and software records for arithmetic, branch-index, and implementation claims.

Book arcTechnical source familyStatus of connection
Chs. 1–8: scale coordinate, metric, gravity, predictionsFramework; Scale Space Papers 1–12Core Scale Space development; observational translation remains open for several predictions.
Chs. 9–12: quantum and biology extensionsScale Space Papers 3–5, 18, 23Model-level and exploratory extensions; empirical calibration required.
Chs. 13–15: dynamical closure and complex scaleScale Space Papers 13–17, 20, 23, 25; CNRS bridge recordsTechnical frontier; several results are ansatz-dependent or await independent review.
Chs. 16–19: CNRSCNRS roadmap, architecture, problem papers, software guideConstructed mathematical programme with implemented components and open formal questions.
Chs. 20–24: convergence, tests, status, invitationAll records; especially triangulation, tetrad, physics-connection, and code recordsProgramme synthesis; bridge remains conditional rather than proved.

Latest Zenodo Records

The eight newest programme records in the supplied community/version export, rebuilt again from live Zenodo metadata when available.

DateIDTitleVersion
2026-07-31CNRS-KeyDifferentiation, the Divided-Power Basis, and the Transcendental-Radix Obstructionsv7
2026-07-09CNRS-GitCNRS Scientific Toolkitv0.11.1
2026-07-02Paper 9Time, Proper Duration, and the Five-Dimensional Parent Theory of the Scale Space Frameworkv7
2026-07-01FrameworkFour-Dimensional Scale Space: A Theoretical Frameworkv13
2026-06-30CNRS-MNThe Complex Number Representation System (CNRS): Foundations, Arithmetic, and Calculusv8
2026-06-30CNRS-RdmpComplex Numeric Representational System Programme Full Programme Scope: All Four Problemsv8
2026-06-30CNRS-SumThe Complex Numeric Representational System Programme A Comprehensive Summary: Origins, Architecture, Results, and Statusv8
2026-06-27CNRS-MSComplex Numeric Representational System (CNRS) Programme Master Summaryv3

Code Repository and Software DOI

The CNRS Scientific Toolkit is the open research-code implementation layer for CNRS.

CNRS-Git GitHub v0.11.1 2026-07-09

CNRS Scientific Toolkit

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.20574852
Latest: 10.5281/zenodo.21271997

Files: DonGPalmer/CNRS_Scientific_Toolkit-v0.11.1.zip

How the layers fit together

GitHub is the working development layer. Zenodo is the archival and citation layer for versioned releases and research records. The Zenodo Community gathers the archival record set. This landing page connects code, book-level synthesis, paper records, and CNRS mathematics.

GitHub Repository Zenodo Community

Scale Space Framework — 24 Records

Framework document and Scale Space paper sequence. Concept DOI is the stable latest-version pointer; latest/default DOI is the workbook-selected version record.

Framework Working v13 2026-07-01

Four-Dimensional Scale Space: A Theoretical Framework

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19136376
Latest: 10.5281/zenodo.21108210

Files: Scale_Space_Framework_v13.pdf

Paper 1 Preprint v14 2026-05-08

A Geometric Model of Logarithmic Scale as an Extra Dimension: Anti-de Sitter Geometry, Emergent Gravity, and RG Flow

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19137149
Latest: 10.5281/zenodo.20098622

Files: paper1_scale_geometry_v14_preprint.tex, paper1_scale_geometry_v14_preprint.pdf

Paper 2 Preprint v12 2026-05-25

Candidate Gravitational Signatures of the Four-Dimensional Scale Space Framework: Asymmetric Force, Wave Polarisations, and Scale Dragging

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19169397
Latest: 10.5281/zenodo.20399341

Files: paper2_gravity_predictions_v12.pdf

Paper 3 Preprint v9 2026-05-12

Scale Uncertainty Principle, Effective Mass, and the Geometric Quantum-to-Classical Transition

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19192004
Latest: 10.5281/zenodo.20159380

Files: paper3_quantum_v9.pdf

Paper 4 Preprint v4 2026-05-12

Cross-Scale Geodesics in Biological Systems: Signalling, Quantum-Classical Boundary, and Morphogenesis

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19194148
Latest: 10.5281/zenodo.20159672

Files: paper4_biology1_v4.pdf

Paper 5 Preprint v5 2026-05-12

Four-Dimensional Extended Objects: Scale Pressure, Boundary Conditions, and Fractal Biological Architecture

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19194284
Latest: 10.5281/zenodo.20159786

Files: paper5_biology2_v5.pdf

Paper 6 Preprint v9 2026-05-12

A Linearised Rotating Exterior Perturbation of the Four-Dimensional Scale Space Framework: Scale-Kerr to First Order in Angular Momentum

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19208248
Latest: 10.5281/zenodo.20169874

Files: scale_kerr_sympy_verification.py, paper6_scale_kerr_v9.pdf, scale_kerr_poincare_exact.py

Paper 7 Preprint v8 2026-05-12

Vector-Mode Gravitational Wave Amplitude from a Rotating Binary in the Four-Dimensional Scale Space Framework

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19209675
Latest: 10.5281/zenodo.20171516

Files: paper7_amplitude_computation_v1.py, paper7_vector_modes_v8.pdf

Paper 8 Preprint v6 2026-05-12

Interior Matching for the Linearised Rotating Exterior Perturbation of the Four-Dimensional Scale Space Framework: Mode Selection, Amplitude Normalisation, and the Scale Separation of Binary Pulsars

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19222639
Latest: 10.5281/zenodo.20171206

Files: paper8_interior_matching_v6.pdf, paper8_sympy_verification_v2.py

Paper 9 Preprint v7 2026-07-02

Time, Proper Duration, and the Five-Dimensional Parent Theory of the Scale Space Framework

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19222717
Latest: 10.5281/zenodo.21197789

Files: paper9_lorentzian_v7.pdf

Paper 10 Preprint v7 2026-05-12

Resolving the Gravitational Time Dilation Discrepancy in the Five-Dimensional Scale Space Framework: The Corrected gtt and Its Consequences

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19244886
Latest: 10.5281/zenodo.20170409

Files: paper10_corrected_metric_v7.pdf, verify_corrected_metric.py

Paper 11 Preprint v5 2026-05-12

The Five-Dimensional Einstein Tensor of the Corrected Scale-Space Metric: Vacuum Structure, Required Stress-Energy, and the Non-Vacuum Nature of the Framework

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19270349
Latest: 10.5281/zenodo.20182691

Files: compute_5d_einstein.py, paper11_einstein_tensor_v5.pdf

Paper 12 Preprint v8 2026-05-25

The Geometric Status of c in the Scale-Space Framework: Manifold Radius, Null Propagation, and the Nat as a Unit of Length

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19270528
Latest: 10.5281/zenodo.20399438

Files: paper12_c_geometric_v8.pdf

Paper 13 Preprint v6 2026-05-12

The Complex Structure of Scale: From Configurational Deficit to the Conjecture that the Scale Coordinate is a Single Complex Value

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19270737
Latest: 10.5281/zenodo.20171842

Files: paper13_complex_s_v6.pdf

Paper 14 Working v3 2026-05-25

Dynamical Origin of the Scale-Space Poisson Equation: Newton's Law from Linearised 5D Field Equations

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.20088695
Latest: 10.5281/zenodo.20399487

Files: paper14_sympy_verification_v1.py, paper14_dynamical_poisson_v3.pdf

Paper 15 Working v3 2026-05-25

The Scale-Integration Reduction: Effective Four-Dimensional Einstein Equations from the Five-Dimensional Scale-Space Field Theory

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.20088926
Latest: 10.5281/zenodo.20399399

Files: paper15_reduction_v3.pdf

Paper 16 Working v2 2026-05-12

Near-Term Tests of the Complex Scale Coordinate Conjecture: Sub-leading Structure, Native vs Projected Treatment, and the Phase-Shift Prediction

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.20089020
Latest: 10.5281/zenodo.20172200

Files: paper16_zs_conjecture_nearterm_v2.pdf

Paper 17 Working v2 2026-05-12

The 5D Field Equations with Complex Scale Coordinate: Classical Neutrality, Quantum Phase, and the Level Structure of the Complex zs Conjecture

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.20089132
Latest: 10.5281/zenodo.20172304

Files: paper17_5d_field_eqs_complex_zs_v2.pdf

Paper 18 Preprint v3 2026-05-18

Multi-Scale Turing Pattern Formation via CNRS-H Digit Strings: Cross-Scale Coupling in the Gierer-Meinhardt System

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.20089231
Latest: 10.5281/zenodo.20317237

Files: cnrs_reaction_diffusion.py, paper18_reaction_diffusion_v3_Zenodo.pdf

Paper 19 Preprint v2 2026-05-12

The Representational Requirement: The CNRS Programme as Paper 13 Step 5

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.20089285
Latest: 10.5281/zenodo.20171947

Files: paper19_representational_v2.pdf

Paper 20 Working v6 2026-06-21

Dynamical Closure of the Scale-Space Framework: Complex Scale Potential, Newtonian Recovery, and the Hidden Configurational Sector

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.20201178
Latest: 10.5281/zenodo.21109980

Files: paper20 dynamical closure v6.pdf

Paper 23 Working v4 2026-05-28

Complex-Scale Quantum Mechanics: Geometric Neutrality and Hydrogenic Phase Shifts in a Gravitational Field

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.20208060
Latest: 10.5281/zenodo.20445807

Files: paper23_EPJD_v1_Zenodo.pdf

Paper 24 Working v1 2026-05-26

Scale-Space Signatures in FLRW Consistency Diagnostics: A Third Class of Cosmological Model

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.20416152
Latest: 10.5281/zenodo.20416389

Files: step c results.csv, step_b_final_diagnostics.pdf, step_b_computation.py, step_b_final_results.csv, paper24_flrw_diagnostics_v1.pdf

Paper 25 Preprint v2 2026-06-24

Scale-Extended Matter Sources and the 4D Poisson Equation

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.21108449
Latest: 10.5281/zenodo.21108450

Files: paper25_earth_prem.py, paper25_analytic.py, paper25_calc.py, paper25 extended source v2.pdf, paper25_eos.py

CNRS Mathematical Programme — 20 Records

CNRS mathematics, architecture, guides, and Scale Space bridge documents. Concept DOI is the stable latest-version pointer; latest/default DOI is the workbook-selected version record.

Executable CNRS code: for the current CNRS Scientific Toolkit, including tests and examples, see the code section above and the GitHub repository.
CNRS-KeyReportv72026-07-31

Differentiation, the Divided-Power Basis, and the Transcendental-Radix Obstructions

Concept DOILatest version DOIOpen Zenodo record

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Zenodo Community snapshot: current version indexed.

Concept: 10.5281/zenodo.21722741
Latest: 10.5281/zenodo.21722742

Files: CNRS_ebase_differentiation_keystone_jul_31_2026_v7_Zenodo.tex, CNRS_ebase_differentiation_keystone_jul_31_2026_v7_Zenodo.pdf

CNRS-Sum Working v8 2026-06-30

The Complex Numeric Representational System Programme A Comprehensive Summary: Origins, Architecture, Results, and Status

Concept DOI Latest version DOI Open Zenodo record

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Zenodo Community snapshot: current version indexed.

Concept: 10.5281/zenodo.19338814
Latest: 10.5281/zenodo.21105442

Files: CNRS comprehensive summary v8.pdf

CNRS-Rdmp Working v8 2026-06-30

Complex Numeric Representational System Programme Full Programme Scope: All Four Problems

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19325069
Latest: 10.5281/zenodo.21105781

Files: CNRS programme roadmap v8.pdf

CNRS-MS Working v3 2026-06-27

Complex Numeric Representational System (CNRS) Programme Master Summary

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.20185553
Latest: 10.5281/zenodo.21105324

Files: CNRS_master_summary_v3.pdf

CNRS-MN Preprint v8 2026-06-30

The Complex Number Representation System (CNRS): Foundations, Arithmetic, and Calculus

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.21108838
Latest: 10.5281/zenodo.21108839

Files: CNRS main paper v8 draft.pdf

CNRS Arch Working v4 2026-05-11

A Combined Architecture for Complex Numeric Representation Systems: Arithmetic, Analytic Continuation, Operator Calculus, and Physical Structure

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19394226
Latest: 10.5281/zenodo.20158574

Files: CNRS_combined_architecture_v4.pdf

CNRS-Pr1 Working v6 2026-05-14

Toward a Definition of Base−b for All b∈R+: Extending the Ito–Sadahiro (−β)-Expansion Framework

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19554909
Latest: 10.5281/zenodo.20185318

Files: CNRS_problem1_base_definition_v6.pdf

CNRS-P1-Ext Working v3 2026-05-14

Problem 1 Extension: The β = (3 + √5)/2 Test Case; Supplement to the CNRS Problem 1 Working Paper (v6)

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19691829
Latest: 10.5281/zenodo.20185217

Files: CNRS_problem1_beta_golden_v3.pdf

CNRS-Pr2 Working v8 2026-05-11

Complex Numbers as Single Values: A Three-Layer Architecture for the Complex Numeric Representational System

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19555139
Latest: 10.5281/zenodo.20172556

Files: CNRS_problem2_capstone_v8.pdf

CNRS Pr2-2 Working v3 2026-06-11

Incorporating the Branch Index into the CNRS Digit String

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.20179101
Latest: 10.5281/zenodo.21109659

Files: CNRS_branch_index_incorporation_v3.pdf

CNRS-Pr1-2 Preprint v4 2026-05-11

A Single-Value Complex Numeral System via Canonical Number Systems and the Riemann Surface of the Logarithm

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19563061
Latest: 10.5281/zenodo.20172649

Files: CNRS_layers12_standalone_v4.pdf

CNRS-Pr3-1 Working v5 2026-06-16

Arithmetic Closure for the Complex Numeric Representational System: Working Paper — Problem 3 of the CNRS Programme

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19555323
Latest: 10.5281/zenodo.20709942

Files: CNRS_problem3_arithmetic_closure_v5.pdf

CNRS-Pr3-2 Working v2 2026-05-11

The Explicit One-Argument Multiplication Transducer for CNRS-A: Carry Sets, State Graphs, and Verified Examples

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.20173246
Latest: 10.5281/zenodo.20173247

Files: CNRS_multiplication_transducer_v2.pdf

CNRS Schmidt Working v2 2026-05-14

The CNRS Programme and Schmidt's Symbolic Cover Framework: A Complete Mathematical Mapping

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.20201367
Latest: 10.5281/zenodo.20201368

Files: CNRS_schmidt_framework_mapping_v2.pdf

CNRS UG Software Doc v1.2 2026-05-13

Complex Numbers as Single Values: A Working Demonstration of the Complex Numeric Representational System

Concept DOI Latest version DOI Open Zenodo record

Zenodo metadata: checking…

Zenodo Community snapshot: concept indexed, but the supplied export lists an earlier version.

Concept: 10.5281/zenodo.19797882
Default: 10.5281/zenodo.20201126

Files: CNRS_user_guide_v1.pdf, cnrs_demo.py

CNRS-Pr4 Working v2 2026-05-11

Partial Operational Completeness of the Complex Numeric Representational System Working Paper — Problem 4 (Sub-question Q3) of the CNRS Programme

Concept DOI Latest version DOI Open Zenodo record

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Zenodo Community snapshot: current version indexed.

Concept: 10.5281/zenodo.19794152
Latest: 10.5281/zenodo.20173050

Files: CNRS_problem4_partial_completeness_v2.pdf

CNRS Phy Working v2 2026-05-11

The CNRS Programme and the Scale-Space Framework: How Papers 1–13 Connect to the Mathematical Programme

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.19332621
Latest: 10.5281/zenodo.20172848

Files: CNRS_physics_connection_v2.pdf

CNRS-Tet Working v2 2026-05-11

The Metric Correction from the Vierbein Formalism: Tetrad Re-derivation, Sub-leading Structure, and the Complex Geodesic Connection

Concept DOI Latest version DOI Open Zenodo record

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Zenodo Community snapshot: current version indexed.

Concept: 10.5281/zenodo.19794533
Latest: 10.5281/zenodo.20200572

Files: CNRS_tetrad_subleading_v2.pdf

CNRS-Tri Working v2 2026-05-11

The Triangulation Function: Conditional Derivation within the Static Diagonal Ansatz and Unification of Metric Correction, Clock Rate, and Einstein Tensor Deficit

Concept DOI Latest version DOI Open Zenodo record

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Concept: 10.5281/zenodo.20200867
Latest: 10.5281/zenodo.20200868

Files: CNRS_triangulation_unification_v2.pdf

CNRS-Multi Preprint v1 2026-05-08

Scale-Derivative Exactness and Structural Cross-Scale Coupling in Multi-Scale Diffusion: A Progressive Place-Value Framework

Concept DOI Latest version DOI Open Zenodo record

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Zenodo Community snapshot: current version indexed.

Concept: 10.5281/zenodo.20089340
Latest: 10.5281/zenodo.20173005

Files: CNRS_multiscale_diffusion_v1.pdf, cnrs_reaction_diffusion.py

Open Problems and Collaboration Entry Points

The project is designed to be engaged modularly. These are the specialist entry points most likely to advance the programme.

Scale Space physics

GR/dynamical-closure review, strong-field dynamics, rotating-body treatment, compact-object observable models, cosmological diagnostics, matter-source modelling, and Bianchi consistency checks.

CNRS mathematics

e-base theorem, block classification, metric completeness, Layer 2 algebra, complex-base number systems, and independent proof review.

Experimental/calibration work

Compact-object spectroscopy, optical-clock residual modelling, SANS/SAXS calibration of g(s), cosmology-data diagnostics, and biological signal-reach test design.

Earlier Published Precursors

These peer-reviewed papers are listed for historical provenance. The current technical record is the Zenodo Community and the indexed paper/code set above.

Toward a Unified Multidisciplinary Model of Reality

Palmer, D. G. (2025). Advances in Theoretical and Computational Physics, 8(4), 01–09.

DOI: 10.33140/ATCP.08.04.04

New Directions for Tying Physics and Biologic Systems

Palmer, D. G. (2025). Space Science Journal, 2(4), 01–05.

DOI: 10.33140/SSJ.02.04.02