🌌 GEO — Hidden Geometry Framework

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🌌 GEO — Hidden Geometry Framework

Author: Leonel Hernan Torreblanca

Framework: GEO — Hidden Geometry / Geometría Oculta

📄 Framework / Paper Archive (OSF):
https://doi.org/10.17605/OSF.IO/YHDMZ

💻 Framework Code Archive (Zenodo):
https://doi.org/10.5281/zenodo.20225304

🧪 Cosmological MCMC Validation (GitHub):
https://github.com/LeoTorreblanca/GEO-Cosmology-MCMC

📦 Cosmological MCMC Archived Release (Zenodo):
https://doi.org/10.5281/zenodo.22103137

Overview

GEO (Hidden Geometry Framework) is an exploratory mathematical and cosmological framework investigating whether stable geometric transfer relations can reproduce selected observational structures through an internal operator architecture.

The framework explores geometric operators, transfer structures, hidden efficiency relations, cosmological applications, planetary architectures, and reproducible numerical experiments.

The central objective is not to introduce arbitrary phenomenological corrections to individual datasets, but to investigate whether a restricted geometric architecture can generate stable relations that can subsequently be confronted with observations.

This repository is the principal public repository of the GEO framework. It contains mathematical notes, validation studies, experimental results, GEO-Lens applications, numerical tests, and links to dedicated reproducibility repositories.

GEO should presently be regarded as an exploratory framework under active numerical and theoretical testing, not as an established physical theory.

1. Canonical GEO architecture

A central quantity appearing throughout the framework is the canonical partition / efficiency parameter

$$\eta=\frac{3}{5}=0.6.$$

The corresponding geometric fraction is

$$f_c=\sqrt{\eta}=0.774596669241483.$$

A complementary sector can be represented by

$$f_{\mathrm{out}}=1-f_c.$$

The framework distinguishes the canonical parameter $\eta$ from the effective GEO state $\mu$ relevant to a particular physical channel.

The canonical radial relation is

$$R=\mu^{1/3}.$$

This distinction is important.

The expression

$$R=\eta^{1/3}$$

must not be interpreted as the general GEO radial law.

The canonical law is

$$R=\mu^{1/3},$$

where $\mu$ denotes the relevant GEO efficiency or effective state.

For the specific Hubble-channel realization investigated in the cosmological analysis,

$$\mu_H=\eta=0.6.$$

Therefore, specifically in that channel,

$$R=\mu_H^{1/3}=0.843432665301749.$$

The equality $\mu_H=\eta$ is a channel realization being tested by the framework, not a replacement of the general distinction between $\mu$ and $\eta$.

2. GEO operator chain

The Hubble realization uses the GEO operator

$$\Phi=1.88961381521168.$$

The associated intensity factor is

$$\alpha=\frac{\Phi(1-\eta)}{\sqrt{2}}=0.534463497023985.$$

Using the Hubble-channel radial state,

$$R=\mu_H^{1/3},$$

the corresponding projection factor is

$$P_{\mathrm{GEO}}=1+\alpha(1-R)=1.083679525222552.$$

The local GEO realization is consequently written as

$$H_{0,\mathrm{GEO}}=P_{\mathrm{GEO}}H_{0,\mathrm{primitive}}.$$

This relation provides the bridge between the primitive cosmological expansion parameter and the locally realized GEO value used in the Hubble-channel tests.

3. GEO-Lens application to the Hubble tension

The Hubble tension provides one of the principal cosmological test cases of the GEO framework.

Historically, the GEO operator chain was evaluated using the reference primitive value

$$H_{0,\mathrm{primitive}}=67.40;\mathrm{km,s^{-1},Mpc^{-1}}.$$

Application of the canonical GEO projection gives

$$H_{0,\mathrm{GEO}}=67.40\times1.083679525222552=73.040000;\mathrm{km,s^{-1},Mpc^{-1}}.$$

This numerical reconstruction motivated the subsequent CLASS, profile-likelihood, and MCMC investigations.

The value $73.040000$ should therefore be understood as the canonical historical GEO reconstruction from the fixed 67.40 input.

It should not be confused with the posterior value obtained when the primitive cosmological parameters themselves are sampled against Planck/NPIPE data.

That later statistical test is documented independently in GEO-Cosmology-MCMC.

4. Cosmological MCMC validation

A dedicated numerical study now tests the GEO Hubble realization using Cobaya, CLASS, Planck likelihoods, matched control calculations, and multiple MCMC chains.

👉 Repository:
https://github.com/LeoTorreblanca/GEO-Cosmology-MCMC

📦 Archived release:
https://doi.org/10.5281/zenodo.22103137

The principal likelihood configuration includes:

Planck 2018 low-$\ell$ TT;
Planck 2018 low-$\ell$ EE;
Planck NPIPE CamSpec TTTEEE;
a local-$H_0$ likelihood for the joint comparison;
a matched $\Lambda$CDM control.

The analysis preserves the distinction between the primitive cosmological expansion parameter and the GEO local realization:

$$H_{0,\mathrm{GEO}}=P_{\mathrm{GEO}}H_{0,\mathrm{primitive}}.$$

The GEO projection factor is fixed by the canonical operator chain,

$$P_{\mathrm{GEO}}=1.083679525222552.$$

4.1 Extended GEO-29 posterior

The extended GEO-29 calculation gives

$$H_{0,\mathrm{primitive}}=67.7213\pm0.4884;\mathrm{km,s^{-1},Mpc^{-1}}.$$

After application of the canonical GEO projection,

$$H_{0,\mathrm{GEO}}=73.3882\pm0.5292;\mathrm{km,s^{-1},Mpc^{-1}}.$$

The corresponding posterior means include

$$\Omega_m=0.309465,$$

and

$$\sigma_8=0.818567.$$

The best sampled GEO-29 point gives

$$H_{0,\mathrm{primitive}}=67.833577;\mathrm{km,s^{-1},Mpc^{-1}},$$

which maps to

$$H_{0,\mathrm{GEO}}=73.509859;\mathrm{km,s^{-1},Mpc^{-1}}.$$

4.2 Matched GEO versus ΛCDM comparison

The matched best-point comparison gives, for GEO minus the $\Lambda$CDM + local-$H_0$ control,

$$\Delta\chi^2_{\mathrm{CMB}}=-3.242000,$$

$$\Delta\chi^2_{\mathrm{local},H_0}=-16.696923,$$

and

$$\Delta\chi^2_{\mathrm{joint}}=-19.938923.$$

Negative values correspond to a lower best-point chi-square for the GEO realization in this exact matched likelihood configuration.

The best sampled GEO-29 point has

$$\chi^2_{\mathrm{CMB}}=10962.919,$$

$$\chi^2_{\mathrm{local},H_0}=0.204111,$$

and

$$\chi^2_{\mathrm{joint}}=10963.123111.$$

The corresponding matched $\Lambda$CDM + local-$H_0$ control has

$$\chi^2_{\mathrm{CMB}}=10966.161,$$

$$\chi^2_{\mathrm{local},H_0}=16.901034,$$

and

$$\chi^2_{\mathrm{joint}}=10983.062034.$$

These numbers refer specifically to the likelihood, priors, model mapping, nuisance parameters, and numerical configuration documented in the dedicated MCMC repository.

They should not be interpreted as a general Bayesian evidence ratio or as proof that GEO supersedes $\Lambda$CDM.

5. Independent profile test of the canonical eta node

The canonical GEO efficiency was also tested through profile-likelihood calculations in which the corresponding geometric fraction was allowed to vary.

The wide profile gives

$$f_{c,\mathrm{best}}=0.774088414673177.$$

Since

$$\eta=f_c^2,$$

this corresponds to

$$\eta_{\mathrm{best}}=0.599212873731233.$$

The canonical GEO prediction is

$$f_{c,\mathrm{GEO}}=\sqrt{\frac{3}{5}}=0.774596669241483,$$

and

$$\eta_{\mathrm{GEO}}=0.600000000000000.$$

At the canonical node, the profile penalty is

$$\Delta\chi^2_{\mathrm{GEO}}=0.001005928553.$$

Thus, within this profile experiment, the canonical GEO node lies extremely close to the numerical likelihood minimum.

Cross-configuration calculations give a median preferred value

$$\eta_{\mathrm{median}}=0.6.$$

The mean canonical-node penalty across the tested configurations is

$$\left\langle\Delta\chi^2_{\mathrm{GEO}}\right\rangle=0.089354084305,$$

with a maximum tested penalty of

$$\Delta\chi^2_{\mathrm{GEO,max}}=0.267647200015.$$

These configurations are not all statistically independent.

Consequently, these calculations establish compatibility and cross-configuration numerical stability of the canonical $\eta=3/5$ node within the tested setup.

They do not independently establish $\eta=3/5$ as a measured universal constant of nature.

6. MCMC convergence and numerical stability

The dedicated GEO cosmological study contains matched short-chain and extended-chain calculations.

The principal extended GEO-29 calculation used four MPI chains with 30,000 stored rows per chain:

$$4\times30,000=120,000$$

stored chain rows.

The final recorded convergence diagnostic was

$$R-1=0.017309752619.$$

A stricter pre-specified target was

$$R-1<0.01.$$

That strict stopping criterion was not formally reached before the sample cap.

For this reason, GEO-29 is reported as an extended, numerically stable / near-converged MCMC calculation, rather than as a chain that formally satisfies the stricter $R-1<0.01$ criterion.

The shorter GEO-28B and extended GEO-29 runs nevertheless give closely consistent posterior results.

For GEO-28B,

$$H_{0,\mathrm{GEO}}=73.3998\pm0.4998;\mathrm{km,s^{-1},Mpc^{-1}}.$$

For GEO-29,

$$H_{0,\mathrm{GEO}}=73.3882\pm0.5292;\mathrm{km,s^{-1},Mpc^{-1}}.$$

This agreement provides an internal numerical stability check while the exact convergence diagnostic remains explicitly reported.

7. Reproducibility package

The cosmological MCMC repository contains the publication-oriented reproducibility package used for the current GEO cosmological validation.

It includes:

matched GEO and $\Lambda$CDM Cobaya configurations;
Planck/NPIPE likelihood configuration;
MCMC checkpoints;
proposal covariance matrices;
convergence histories;
posterior summaries;
best-point comparisons;
machine-readable CSV tables;
publication-quality figures;
profile-likelihood validation of $\eta$;
cross-configuration tests;
software environment freeze;
Python dependency freeze;
CLASS source hashes;
GEO-modified source hashes;
SHA256 chain manifests;
consistency-audit scripts;
reproducibility documentation.

The frozen publication audit reports:

AUDIT PASSED
All frozen numerical quantities are internally consistent.

The archived release is available at:

https://doi.org/10.5281/zenodo.22103137

8. Technical validation kit — CLASS

The public implementation of GEO-related cosmological calculations for CLASS v3.x is available separately.

👉 GEO Launch Kit:
https://github.com/LeoTorreblanca/GEO_Launch_Kit

The GEO Launch Kit contains source modifications and diagnostic scripts used in the development and reproducibility analysis of GEO-Lens calculations.

It is released under the MIT License for independent inspection, testing, reproduction, and scientific discussion.

CLASS-GEO-Lens

Public implementation:

https://github.com/LeoTorreblanca/CLASS-GEO-Lens

Archived release:

https://doi.org/10.5281/zenodo.20529415

CLASS-GEO-Lens should be distinguished from the dedicated GEO-Cosmology-MCMC statistical validation repository.

The former provides a public implementation environment for the GEO cosmological mapping.

The latter contains the dedicated matched MCMC and profile-likelihood analysis.

9. Scientific documentation

The mathematical development of the GEO Hubble realization is documented through the technical sheets included in this repository.

These documents describe the mathematical and historical development of the framework.

The dedicated MCMC repository should be used for the current statistical cosmological validation.

10. GEO experimental test series

The GEO framework developed through a sequence of numerical and geometric tests.

PRUEBA 1 — SOP emergence

Initial exploration of:

growth suppression;
stable $S_8$ regions;
emergence of an effective geometric fraction;
complementary-sector interpretation.

PRUEBA 2 — Geometric node analysis

Analysis of preferred geometric regions including:

$3/4$;
$\sqrt{3/5}$;
$\pi/4$;
effective geometric bands.

PRUEBA 3 — Architectural transfer structure

Investigation of:

geometric partition;
active/complementary transfer;
efficiency structure;
architectural consistency.

PRUEBA 4 — Prediction law

Exploration of the radial prediction relation

$$R=\mu^{1/3}$$

and its transfer consistency.

The relevant state is $\mu$.

The historical expression $R=\eta^{1/3}$ should not be treated as the general GEO radial identity.

PRUEBA 5 — Observational consistency

Exploratory consistency analysis involving:

$S_8$;
effective suppression;
$E_G$;
geometric prediction stability;
cross-observable behavior.

These exploratory stages motivated the later dedicated profile and MCMC analyses.

11. Multi-planetary empirical validation

The GEO framework has also been explored outside the primary cosmological Hubble application.

A dedicated empirical study analyzes orbital architectures in multi-planetary systems using data derived from the NASA Exoplanet Archive.

👉 GEO Exoplanets Validation:
https://github.com/LeoTorreblanca/GEO-Exoplanets-Validation

The study investigates geometric transition statistics, orbital spacing, node structure, and possible architecture-dependent discontinuities.

Reported exploratory results include:

analysis of 2,099 multi-planetary systems with $N\geq3$;
a reported 98.09% coherence rate under the adopted transition definition;
changes in transition frequency around higher-multiplicity systems;
concentration of selected internal transition locations.

These results belong to a separate empirical application of GEO and should not be treated as statistically independent evidence for the cosmological Hubble realization without an explicit joint statistical model.

12. Spanish technical reconstruction

A dedicated Spanish-language technical reconstruction of the framework is maintained separately.

👉 GEO — Geometría Oculta ESP:
https://github.com/LeoTorreblanca/GEO-Geometria-Oculta-ESP

This repository provides a cleaner Spanish-language presentation of the framework, its geometric architecture, and its experimental development.

13. Current scientific status

The current public GEO program now contains several distinct levels of evidence and development.

Mathematical framework

GEO proposes a constrained internal geometric architecture involving partition, projection, complementary sectors, effective states, and geometric transfer operators.

Historical Hubble reconstruction

The canonical Hubble operator chain maps

$$67.40\longrightarrow73.040000;\mathrm{km,s^{-1},Mpc^{-1}}.$$

This is the original fixed-input GEO reconstruction.

Profile-likelihood validation

When the relevant efficiency parameter is allowed to vary in the profile experiment,

$$\eta_{\mathrm{best}}=0.599212873731233,$$

very close to the canonical prediction

$$\eta_{\mathrm{GEO}}=0.6.$$

Cosmological MCMC validation

The Planck/NPIPE MCMC calculation independently samples the primitive cosmological scale and obtains

$$H_{0,\mathrm{primitive}}=67.7213\pm0.4884;\mathrm{km,s^{-1},Mpc^{-1}},$$

which the fixed GEO operator maps to

$$H_{0,\mathrm{GEO}}=73.3882\pm0.5292;\mathrm{km,s^{-1},Mpc^{-1}}.$$

Within the exact matched joint likelihood experiment, the best sampled comparison gives

$$\Delta\chi^2_{\mathrm{joint}}=-19.938923.$$

These are stronger numerical tests than the original fixed-input reconstruction because the primitive cosmological parameter is sampled within an explicit likelihood analysis.

They remain tests of the proposed GEO realization rather than proof of the framework as a fundamental physical theory.

14. Present limitations and next tests

Earlier versions of the GEO documentation identified a full cosmological MCMC analysis as an outstanding validation step.

That step has now been performed.

The dedicated GEO-Cosmology-MCMC repository provides the current Planck/NPIPE MCMC validation and its reproducibility package.

The principal remaining limitations are therefore no longer the absence of an MCMC analysis.

They are:

Stricter chain convergence

The extended run reached

$$R-1=0.017309752619,$$

but not the stricter target

$$R-1<0.01.$$
Independent cosmological datasets

The GEO mapping should be confronted with additional independent BAO, supernova, growth, weak-lensing, and other cosmological likelihood combinations.
Alternative local-$H_0$ likelihoods

The sensitivity of the result to different local distance-ladder determinations should be quantified.
Bayesian model comparison

The reported $\Delta\chi^2$ comparison is not a Bayesian evidence calculation.

Bayesian evidence, information criteria where appropriate, and explicit treatment of model complexity remain future tests.
Out-of-sample prediction

Additional observables should be predicted before being included in parameter estimation.
Physical derivation of channel realization

The identification

$$\mu_H=\eta$$

remains a physical hypothesis of the Hubble-channel realization and should be derived or independently tested beyond its present numerical performance.
Independent replication

External reproduction of the complete pipeline remains essential.
Broader theoretical embedding

The relation between the GEO operator architecture and established relativistic field equations, perturbation theory, conservation principles, and fundamental dynamics requires further formal development.

These limitations define the next stage of the GEO research program.

15. Interpretation policy

The numerical results in this repository and its associated validation repositories should be interpreted according to the exact experiment that produced them.

In particular:

$\eta=0.6$ is the canonical GEO value;
the profile calculations show compatibility with that value but do not independently establish a new universal constant;
$R=\mu^{1/3}$ is the canonical radial law;
$\mu_H=\eta$ is the specific Hubble-channel realization tested here;
$H_0=73.040000$ is the historical fixed-input GEO reconstruction;
$H_{0,\mathrm{GEO}}=73.3882\pm0.5292$ is the extended MCMC posterior result;
$\Delta\chi^2_{\mathrm{joint}}=-19.938923$ refers to the exact matched likelihood comparison documented in GEO-Cosmology-MCMC;
the reported $\Delta\chi^2$ is not a Bayesian evidence ratio;
the longest MCMC run is near-converged / numerically stable under the reported diagnostic, but did not satisfy the stricter $R-1<0.01$ target.

This separation is maintained to make the framework falsifiable, auditable, and reproducible.

16. Repository ecosystem

The public GEO research program is distributed across dedicated repositories so that theoretical development, implementation, and numerical validation can be inspected separately.

Main framework

GEO — Hidden Geometry Framework

https://github.com/LeoTorreblanca/GEO-hidden-geometry-framework

Framework archive:

https://doi.org/10.5281/zenodo.20225304

Cosmological MCMC validation

GEO Cosmology MCMC Validation

https://github.com/LeoTorreblanca/GEO-Cosmology-MCMC

Archived release:

https://doi.org/10.5281/zenodo.22103137

CLASS implementation

CLASS-GEO-Lens

https://github.com/LeoTorreblanca/CLASS-GEO-Lens

Archived release:

https://doi.org/10.5281/zenodo.20529415

17. Official channels

18. Repository structure

scripts/  -> Reproducible GEO scripts
figures/  -> Generated plots and visual outputs
results/  -> Numerical outputs and console logs
pdf/      -> Individual technical reports
paper/    -> GEO manuscript and preprint versions
docs/     -> Mathematical and technical documentation

Dedicated large-scale cosmological MCMC products are maintained in GEO-Cosmology-MCMC rather than duplicated in this repository.

19. Reproducibility

The GEO project follows a public reproducibility-oriented structure.

The main framework contains the mathematical and experimental architecture.

Dedicated validation repositories contain the corresponding numerical implementations, configurations, results, and diagnostic products.

For the cosmological MCMC study, the archived reproducibility package contains:

source and configuration hashes;
environment information;
dependency freeze;
chain manifests;
MCMC diagnostics;
covariance matrices;
profile-likelihood source tables;
publication figures;
numerical result tables;
consistency auditing.

The archived cosmological validation release is:

Torreblanca, Leonel (2026). GEO Cosmology MCMC Validation. Zenodo.

https://doi.org/10.5281/zenodo.22103137

20. Citation

When referring to the general GEO framework, use the principal framework archive and associated paper record.

📄 Framework / Paper:

https://doi.org/10.17605/OSF.IO/YHDMZ

💻 Framework Code:

https://doi.org/10.5281/zenodo.20225304

When referring specifically to the cosmological MCMC result, cite:

Torreblanca, Leonel (2026). GEO Cosmology MCMC Validation. Version 1.0.0. Zenodo.

https://doi.org/10.5281/zenodo.22103137

When referring specifically to the CLASS-GEO-Lens implementation, use:

https://doi.org/10.5281/zenodo.20529415

21. License

Unless otherwise indicated for third-party materials or external datasets, the public GEO software and repository materials are released under the MIT License.

See the corresponding LICENSE files in each repository.

External cosmological likelihoods and observational datasets retain their original licenses, citations, and distribution conditions.

22. Scientific scope

GEO remains an open exploratory research framework.

The current results establish that the proposed geometric architecture has generated specific, falsifiable numerical relations that can be tested with standard cosmological inference tools.

The current cosmological analysis shows that:

$$\eta_{\mathrm{best}}\approx0.59921$$

is numerically close to the canonical

$$\eta_{\mathrm{GEO}}=0.6,$$

and that the Planck/NPIPE primitive posterior

$$H_{0,\mathrm{primitive}}\approx67.72$$

is mapped by the fixed GEO operator to

$$H_{0,\mathrm{GEO}}\approx73.39;\mathrm{km,s^{-1},Mpc^{-1}}.$$

Within the exact matched likelihood configuration tested so far, the best sampled GEO realization also gives a lower joint chi-square than the corresponding $\Lambda$CDM + local-$H_0$ control.

These results warrant further independent testing.

They do not remove the need for stricter convergence, independent datasets, Bayesian model comparison, theoretical derivation, and external replication.

Author

Leonel Hernan Torreblanca

GEO — Hidden Geometry Framework

2026

Name		Name	Last commit message	Last commit date
class_modification_guide		class_modification_guide
data		data
docs		docs
figures		figures
paper		paper
pdf		pdf
results		results
scripts		scripts
.gitignore		.gitignore
LICENSE		LICENSE
README.md		README.md
index.html		index.html
paper.md		paper.md
release_notes.md		release_notes.md
requirements.txt		requirements.txt

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🌌 GEO — Hidden Geometry Framework

Overview

1. Canonical GEO architecture

2. GEO operator chain

3. GEO-Lens application to the Hubble tension

4. Cosmological MCMC validation

4.1 Extended GEO-29 posterior

4.2 Matched GEO versus ΛCDM comparison

5. Independent profile test of the canonical eta node

6. MCMC convergence and numerical stability

7. Reproducibility package

8. Technical validation kit — CLASS

CLASS-GEO-Lens

9. Scientific documentation

10. GEO experimental test series

PRUEBA 1 — SOP emergence

PRUEBA 2 — Geometric node analysis

PRUEBA 3 — Architectural transfer structure

PRUEBA 4 — Prediction law

PRUEBA 5 — Observational consistency

11. Multi-planetary empirical validation

12. Spanish technical reconstruction

13. Current scientific status

Mathematical framework

Historical Hubble reconstruction

Profile-likelihood validation

Cosmological MCMC validation

14. Present limitations and next tests

15. Interpretation policy

16. Repository ecosystem

Main framework

Cosmological MCMC validation

CLASS implementation

GEO Launch Kit

Exoplanet validation

Spanish technical reconstruction

17. Official channels

18. Repository structure

19. Reproducibility

20. Citation

21. License

22. Scientific scope

Author

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