Four-Dimensional Scale Space: A Theoretical Framework
Concept DOI: 10.5281/zenodo.19136376Files: Scale_Space_Framework_v10.pdf
A modular research programme connecting logarithmic scale geometry, complex positional representation, and cross-scale physical and biological modelling.
This page is the public index for the Scale Space / CNRS programme. It is designed to make the book, technical papers, the GitHub code repository, open problems, and Zenodo concept DOIs mutually discoverable.
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.
The Scale Space papers provide the formal physics, geometry, dynamical-closure, and prediction record. Use these records when citing specific technical claims.
The CNRS records provide the mathematical background. The GitHub repository provides the current executable CNRS Scientific Toolkit: code, tests, examples, issues, and future software releases.
The recommended Zenodo structure is: one project landing record, one book record, individual paper records, and separate code/software records. This page is suitable for the project landing record.
Role: reader-facing synthesis and research-programme statement. The book should cite the project landing record and the relevant technical-paper concept DOIs. Technical claims should be traced to the paper records below.
Cite the book for the programme-level synthesis. Cite individual Scale Space papers for physics derivations. Cite CNRS papers and software documents for arithmetic, branch-index, and implementation claims.
| Book arc | Technical source family | Status of connection |
|---|---|---|
| Chs. 1–8: scale coordinate, metric, gravity, predictions | Framework; Scale Space Papers 1–12 | Core Scale Space development; observational translation still open for several predictions. |
| Chs. 9–12: quantum and biology extensions | Scale Space Papers 3–5, 18, 23 | Model-level and exploratory extension; empirical calibration required. |
| Chs. 13–15: dynamical closure and complex scale | Scale Space Papers 13–17, 20, 23; CNRS bridge records | Technical frontier; several results are ansatz-dependent or await independent review. |
| Chs. 16–19: CNRS | CNRS roadmap, architecture, problem papers, software guide | Constructed mathematical programme with implemented components and open formal questions. |
| Chs. 20–24: convergence, tests, status, invitation | All records; especially triangulation, tetrad, physics-connection records | Programme synthesis; bridge remains conditional rather than proved. |
These peer-reviewed publications precede or accompany the larger Zenodo programme record.
Palmer, D. G. (2025). Advances in Theoretical and Computational Physics, 8(4), 01–09.
Palmer, D. G. (2025). Space Science Journal, 2(4), 01–05.
The CNRS Scientific Toolkit is the current open research-code implementation of the CNRS programme.
The GitHub repository provides executable code, tests, examples, and workflow demonstrations for CNRS-A arithmetic, CNRS-float, Gaussian rational representation, CNRS-H calculus, CNRS-H ODE solvers, branch-aware complex-state helpers, observation maps, scale-law utilities, and three-workflow comparison examples.
GitHub is the working code and testing layer. Zenodo remains the archival and citation layer for programme documents and versioned research outputs. This landing page connects the code repository, book-level synthesis, paper records, and CNRS mathematical records.
The following sections are generated from the Concept DOI CSV inventories. The newest addition is Scale Space Paper 24, now included in the Scale Space records.
Framework document and Scale Space paper sequence. Concept DOIs resolve to the latest Zenodo version.
Files: Scale_Space_Framework_v10.pdf
Files: paper1_scale_geometry_v14_preprint.pdf
Files: paper2_gravity_predictions_v11.pdf
Files: paper3_quantum_v9.pdf
Files: paper4_biology1_v4.pdf
Files: paper5_biology2_v5.pdf
Files: paper6_scale_kerr_v9.pdf, scale_kerr_poincare_exact.py, scale_kerr_sympy_verification.py
Files: paper7_vector_modes_v8.pdf, paper7_amplitude_computation_v1.py
Files: paper8_interior_matching_v6.pdf, paper8_sympy_verification_v2.py
Files: paper9_lorentzian_v6.pdf
Files: paper10_corrected_metric_v7.pdf, verify_corrected_metric.py
Files: paper11_einstein_tensor_v5.pdf, compute_5d_einstein.py
Files: paper12_c_geometric_v7.pdf
Files: paper13_complex_s_v6.pdf
Files: paper14_dynamical_poisson_v2.pdf, paper14_sympy_verification_v1.py
Files: paper15_reduction_v2.pdf
Files: paper16_zs_conjecture_nearterm_v2.pdf
Files: paper17_5d_field_eqs_complex_zs_v2.pdf
Files: paper18_reaction_diffusion_v2.pdf, cnrs_reaction_diffusion.py
Files: paper19_representational_v2.pdf
Files: paper20_dynamical_closure_v2.pdf
Files: paper23_complex_scale_hydrogen_v3.pdf
Files: paper24_flrw_diagnostics_v1.pdf, step_b_computation.py, step_b_final_results.csv, step_b_final_diagnostics.pdf, step c results.csv
CNRS mathematics, architecture, guides, and Scale Space bridge documents. Concept DOIs resolve to the latest Zenodo version.
Files: CNRS_comprehensive_summary_v6.pdf
Files: CNRS_programme_roadmap_v7.pdf
Files: CNRS_master_summary_v2.pdf
Files: CNRS_combined_architecture_v4.pdf
Files: CNRS_problem1_base_definition_v6.pdf
Files: CNRS_problem1_beta_golden_v3.pdf
Files: CNRS_problem2_capstone_v8.pdf
Files: CNRS_branch_index_incorporation_v2.pdf
Files: CNRS_layers12_standalone_v4.pdf
Files: CNRS_problem3_arithmetic_closure_v4.pdf
Files: CNRS_multiplication_transducer_v2.pdf
Files: CNRS_schmidt_framework_mapping_v2.pdf
Files: CNRS_user_guide_v1.pdf, cnrs_demo.py
Files: CNRS_problem4_partial_completeness_v2.pdf
Files: CNRS_physics_connection_v2.pdf
Files: CNRS_tetrad_subleading_v2.pdf
Files: CNRS_triangulation_unification_v2.pdf
Files: CNRS_multiscale_diffusion_v1.pdf, cnrs_reaction_diffusion.py
The project is designed to be engaged modularly. These are the specialist entry points most likely to advance the programme.
GR/dynamical-closure review, strong-field dynamics, rotating-body treatment, compact-object observable models, and Bianchi consistency checks.
e-base theorem, block classification, metric completeness, Layer 2 algebra, complex-base number systems, and independent proof review.
Compact-object spectroscopy, optical-clock residual modelling, SANS/SAXS calibration of g(s), and biological signal-reach test design.