CMDE 4.1 Validation Suite v2
Supplementary Numerical Tests for an Informational Time Metric in Cosmology

1. Purpose and scope

The CMDE 4.1 Validation Suite v2 is a structured collection of numerical tests designed to assess the internal consistency and empirical viability of the CMDE 4.1 informational time metric for cosmology. In this framework the cosmological redshift is interpreted as an irreversible informational transformation of light as a function of cosmic time, encoded in a three-phase function z(t) that is continuously differentiable up to eighth order and numerically stable under high-precision evaluation.

The Validation Suite v2 does not introduce new physics or additional assumptions beyond those defined in the CMDE 4.1 formalism. Its goal is to document, in a transparent and reproducible way, how the final August 2025 formulation of z(t) performs when it is confronted with a broad set of background observables and with purely theoretical consistency checks. The suite covers hyper-primordial and classical metric behaviour, log-Hermite transitions, background distances and horizon scales, supernova time dilation, BAO-related quantities and metric-density proxies, as well as a number of informational and interpretative stress tests.

The suite is intended as a companion resource to the foundations and core-validation manuscripts and as a standalone, citable reference for researchers who wish to inspect, replicate or extend CMDE-based analyses at the numerical level.

2. Relation to the CMDE 4.1 papers

The Validation Suite v2 is conceptually and technically aligned with two main manuscripts:

– Carenzi 2025a (“Foundations”): formal definition of the CMDE 4.1 informational time metric, including the three-phase construction of z(t), the log-Hermite raccordo in logarithmic time and the general informational framework.
– Carenzi 2025b (“Core Validation”): quantitative confrontation of the CMDE 4.1 background time metric with representative cosmological observables, including surface-brightness tests, supernova luminosity distances, BAO-related quantities, H(z) measurements, CMB distance priors and high-redshift galaxy constraints.

The present archive provides the numerical backbone of that validation programme. Each test in the suite assumes the CMDE 4.1 formulation as defined in the foundations work and, where appropriate, mirrors or extends the analysis described in the core-validation manuscript. Bibliographic information and any related identifiers for associated records are provided in the Zenodo metadata for this archive and in the published versions of the manuscripts.

3. Archive structure

The ZIP archive CMDE41_ValidationSuite_v2.zip unpacks into a single top-level directory:

CMDE41_ValidationSuite_v2/

Within this directory the content is organised as follows, using a minimal and navigable structure:

– docs/

This directory contains human-readable documentation and index files. In particular:

• README_CMDE41_ValidationSuite_v2.txt (this document), which describes the scope, structure and intended use of the suite and explains how to cite it.

• Reproducibility_and_NumericalSetup_CMDE41_v2.pdf, a compact note that summarises the numerical environment, libraries, precision policies, validation thresholds and general philosophy of the CMDE validation programme.

• CMDE41_ValidationSuite_v2_TestIndex.txt (and/or .csv / .pdf), a test index listing all tests in the suite with at least: global test number, short title, block identifier, and broad diagnostic family or Node. This index provides a single entry point to the entire collection and is entirely in English.

– tests/

This directory contains the actual validation tests grouped into document blocks. The suite is organised into 28 blocks, each block collecting a coherent subset of tests. Each block is provided as a single PDF file named using the list of test identifiers contained in that block, with a structure similar to the following examples:

• CMDE_4.1_Validation_Tests_[001-003-004-005-006-007-008-009-010-011].pdf
• CMDE_4.1_Validation_Tests_[012-013-014-015-016-017-018-019-020-021].pdf
• …
• CMDE_4.1_Validation_Tests_[...].pdf

Each block file is written entirely in English and includes: a title page with block identifier and list of tests contained; a short description of the block’s role within the CMDE 4.1 Validation Suite v2; a note linking the block to the CMDE foundations and core-validation manuscripts; a linear table of contents; and then, for each test, a self-contained section with objective, CMDE 4.1 metric definition, computational environment, replicable pipeline, acceptance criteria, numerical results, scientific interpretation, robustness analysis, technical outcome and a standard Data Availability statement. Each block ends with a short “End of Block” note that restates its connection to the global CMDE 4.1 Validation Suite (Tests 001–270) and summarises the numerical stability and differentiability properties of the results.

– code/

This directory either contains or points to the numerical implementation of the CMDE 4.1 metric and to example pipelines. Two use-cases are envisaged:

• If core code is included: the directory is structured into minimal modules for the informational redshift function z(t), its three phases and log-Hermite raccordo, and for the mappings to background observables used in the tests. Example scripts illustrate how to reproduce selected blocks.

• If no code is included: the directory contains a short README_code.txt that explains where the official public repositories are hosted (for example on a collaborative version-control platform), how they relate to the Validation Suite v2 and how to obtain the exact code snapshots corresponding to this Zenodo record.

– Top-level service files

In the root directory CMDE41_ValidationSuite_v2/ the following service files are provided:

• LICENSE.txt, describing the licence under which the suite is released (data and code reuse conditions).

• CITATION_CMDE41_ValidationSuite_v2.txt (or .cff), providing the recommended citation for this Zenodo record and for the associated CMDE 4.1 papers. The recommended bibliographic references are provided in the Zenodo record for this archive and in the published versions of the manuscripts.

• VERSION.txt, briefly summarising the version status of the suite (for example: “CMDE 4.1 Validation Suite v2 – August 2025 formulation of z(t) – Nodes covered: 1–7 and Node Ω – Governance ML Validation – Total tests: 270 – This archive corresponds to the CMDE 4.1 Validation Suite v2 Zenodo record.”).

• MANIFEST.txt, a complete listing of the archive contents for quick inspection (single-place overview of files and directories included in this release).

For an authoritative, single-place listing of everything included in this release, refer to MANIFEST.txt.

4. Structure of individual tests and blocks

Each test in the suite follows a common structure, which is documented once in this archive and then applied consistently across all blocks. While the detailed implementations vary, the main elements are:

– Objective
A concise statement of the purpose of the test, including its role within the CMDE 4.1 programme, the domain in redshift and/or cosmic time, and whether the test is purely theoretical or based on external datasets.

– Metric Definition (CMDE 4.1)
A short description of the part of the CMDE 4.1 informational time metric used in the test. All tests rely on the final unified formulation (August 2025), in which z(t) is defined in three phases with a smooth log-Hermite bridge in logarithmic time, continuous and continuously differentiable up to eighth order and numerically stable. Where needed, explicit expressions for the relevant phase(s) of z(t), auxiliary variables (s = ln t, y = ln(1+z)) and derivatives are provided.

– Computational Environment
A description of the numerical environment: Python 3.11, NumPy/SciPy and related scientific libraries; IEEE-754 double precision; Linux-class workstations; and standard policies for underflow/overflow, safe logarithms and, where relevant, random-number generators.

– Replicable Methods (Pipeline)
A step-by-step outline of the numerical pipeline used in the test: grids in t and/or z, refinement near metric nodes, evaluation of z(t) and its derivatives, mappings to observables, integrators and inversion routines, residual definitions and the internal metrics used to quantify stability and convergence.

– Acceptance Criteria and Quality Checks
A clear statement of the default CMDE validation thresholds, which are shared across tests: internal stability at the level of approximately 10⁻⁶; the percentage of points required within 2σ and 3σ; RMS of normalised residuals below unity; absence of long-range systematics; and small variations under grid and tolerance changes. Tests that involve external data also specify how σ values and χ²/ν are constructed. These are the default CMDE validation thresholds, consistent across all tests.

– Numerical Results
A summary of the main numerical outcomes, including representative values in monospaced pseudo-tables where helpful, RMS and χ² values, coverage within 1σ, 2σ, 3σ, and any salient numerical features such as minima, asymptotes, or integrability properties.

– Scientific Interpretation
A qualitative and interpretative discussion of the results in the CMDE framework, often including comparisons with standard ΛCDM interpretations. The emphasis is on whether the behaviour observed is structurally compatible with the informational interpretation of redshift and on how any tensions with external benchmarks should be understood.

– Robustness and Sensitivity Analyses
A brief report on grid refinements, alternative integrators, parameter perturbations and other stress tests that demonstrate the stability of the conclusions under controlled changes of the numerical setup.

– Technical Outcome
A final, explicit statement classifying the test as fully passed, formally not passed with well-defined informational tensions, or otherwise characterised, based on the predefined acceptance criteria.

– Data Availability
A standard sentence stating that the test is fully replicable and that all numerical outputs and step-by-step protocols are made publicly available in the CMDE 4.1 Validation Suite v2 Zenodo repository, with third-party datasets cited via their official repositories.

Within each block, a set of such tests are presented in sequence, preceded by a block-level description and followed by an “End of Block” note that ties that block back to the global Validation Suite. All block documents are in English and are designed to be directly usable by the international community.

5. Reproducibility and numerical setup

For detailed information on the numerical environment and reproducibility policies, users should consult the document Reproducibility_and_NumericalSetup_CMDE41_v2.pdf in the docs/ directory. That note summarises the common numerical choices (Python version, libraries, precision), the treatment of the CMDE metric, standard integration and inversion routines, default tolerance settings and the shared validation thresholds for stability, convergence and σ-based criteria.

In practical terms, any researcher with access to a standard Python 3.11 environment, NumPy/SciPy and a Linux-class workstation should be able to reproduce the tests to within the internal thresholds documented in the suite. The Test Index, together with the block-level descriptions, indicates which parts of the numerical setup are critical for each test family.

6. Interpreting test outcomes

The Validation Suite v2 distinguishes between two levels of outcome:

– Internal CMDE consistency and numerical stability: whether the implementation of z(t) and its derivatives, together with the numerical pipeline, is stable, convergent and free of hidden pathologies across the relevant domain.

– External numerical agreement: whether the CMDE predictions match selected external benchmarks, such as ΛCDM-based H₀ values or geometric horizon scales, within predefined numerical tolerances.

Some tests are classified as fully passed in both senses, while others are formally not passed with respect to a given external benchmark but remain structurally coherent within the CMDE informational framework. In the latter cases the tests are retained in the suite as documented tensions, not as discarded failures. The Technical Outcome section of each test makes this distinction explicit.

7. Citation and reuse

When using this suite, users are invited to cite both the CMDE 4.1 Validation Suite v2 and the associated CMDE 4.1 papers. A recommended citation format is provided in CITATION_CMDE41_ValidationSuite_v2.txt. Bibliographic information and relevant identifiers are provided in the Zenodo record for this archive and in the published versions of the manuscripts.

The suite is released under the licence specified in LICENSE.txt. Unless otherwise noted, the numerical tests and pipelines may be reused for academic, non-commercial purposes with appropriate attribution. Third-party datasets remain subject to the licences and conditions of their original providers.

8. Contact and further information

The CMDE 4.1 Validation Suite v2 has been assembled and maintained by Ivan Carenzi as part of the broader CMDE 4.1 and informational-cosmology research programme. For clarifications, suggestions or reports on replication attempts, users are encouraged to use the official CMDE contact page `https://www.cmdeuniverse.com/contatti` or email `ivan.cmde@gmail.com`.
