structural-addresses

Reference implementation and reproducibility companion for:

Structural Addresses: Continuous Proximity, Discrete Membership, and Compatible Markers in Inheritance Systems — A Methodological Note with Hypergeometric, Gamma–Period, and Finite-State Controls

Elias De Jesús · Independent Researcher · ORCID 0009-0007-0190-9143 · dejesuselias10@gmail.com · July 2026

Archival record

• Current version DOI: 10.5281/zenodo.21297926
• Concept DOI, all versions: 10.5281/zenodo.21263921

The concept DOI always resolves to the latest available version. The current-version DOI identifies one specific archived version and does not represent all versions.

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1. Purpose

This repository is a lightweight reproducibility companion, not a software product. The manuscript is the scholarly record; the code here exists only to reproduce the paper's computational claims and to demonstrate its central framework in executable form.

It reproduces two analytic controls (the detuned Ramanujan 1/π family and the beta/gamma family), one minimal finite-state inheritance control, and both figures.

2. Structural-address framework

Each object x in an analytic, arithmetic, or inherited family is assigned a two-part address

Addr(x) = ( Σ_c(x), δ(x) )

where

• Σ_c : X → C is a continuous signature into a metric space, answering which objects are nearby;
• δ : X → Δ is a discrete invariant into a set with no assumed metric, answering which exact structural class x belongs to.

The two components carry independent information. The central methodological statements are:

• Continuous navigation and exact identification are different tasks. A signature map supplies local navigation coordinates exactly where it is a local embedding, and only after quotienting by automorphisms.
• An inherited marker closes exactly when its fibers satisfy the congruence condition ker Σ_ℓ ⊆ ker(Σ_{ℓ+1} ∘ E_ℓ), i.e. when the marker map sends each fiber consistently forward under inheritance.
• Closure and separation are independent. A marker may induce well-defined reduced dynamics without distinguishing the target classes, or distinguish them without closing.
• An injective continuous observable can allow exact recovery while still failing to provide proximity-stable classification.

3. Three levels of recovery

Following Appendix D of the manuscript, three properties must be kept apart:

Level	Question	Criterion
Exact recoverability	Does δ factor set-theoretically through Σ_c?	ker Σ_c ⊆ ker δ
Stable classification	Is the induced classifier locally constant?	each Σ_c(S_τ) open in the relative topology
Numerical conditioning	Can the classification be performed from finite-precision data?	quantitative inverse estimates (smallest singular value of dΣ_c)

Because D ↦ L_D and s ↦ J(s) are injective on the stated domains, exact parameter and class information can factor set-theoretically through the scalar observables. The controls demonstrate the failure of proximity-stable exact classification, not the failure of exact recoverability. Both implications above are strict: exact recoverability does not imply stable classification, and stable classification does not imply well-conditioned recovery.

4. What this repository reproduces

Control I — the detuned Ramanujan family

L_D = (2√2 / 9801) · Σ_{n≥0} ((4n)! / (n!)^4) · (1103 + 26390 n) / D^(4n)

• L_396 = 1/π to high precision;
• L'(396) ≈ −7.821535972847 × 10⁻¹¹ and L''(396) ≈ 9.88 × 10⁻¹³, computed as analytic term-by-term derivatives (no finite differences);
• the limiting value 2√2 · 1103 / 9801, which lies ≈ 7.8 × 10⁻⁹ below 1/π;
• the narrow observable-value window occupied by the family for D ≥ 396 (width < 10⁻⁸);
• digits per term = 4 log₁₀ D − log₁₀ 256, strictly increasing in D, so D = 396 is not a convergence-rate optimum.

Control II — the beta/gamma family

J(s) = Γ(s+1)² / Γ(2s+2)

• J(1/2) = π/8;
• J(1/4) = Γ(1/4)² / (12√π) and J(3/4) = 3 Γ(3/4)² / (10√π);
• J(1/4) · J(3/4) = π/20;
• monotone decrease of J over the displayed interval.

Control III — the four-state deterministic inheritance system

X = {1,2,3,4} with f(1)=1, f(2)=2, f(3)=1, f(4)=4 and target κ = {{1,2,3},{4}}:

Marker	Closes	Separates κ
κ = {{1,2,3},{4}}	yes	yes
Σ = {{1,3},{2,4}}	yes	no
M = {{1},{2,3},{4}}	no	yes
composite {{1,3},{2},{4}}	yes	yes

Figures

• figures/figure_1_ramanujan_detuning.png — left: |L_D − 1/π| on a logarithmic vertical scale with D = 396 marked; right: digits per term versus D.
• figures/figure_2_beta_family.png — J(s) with the exact strata s = 1/2, 1/4, 3/4, 1/3, 2/3 marked, together with the nearby comparison points s = 0.24 and s = 0.26.

5. Installation and reproduction

Python 3.10 or newer.

pip install -r requirements.txt
python reproduce.py

reproduce.py creates data/ and figures/ if needed, regenerates both figures, prints all numerical checks, and exits with a nonzero status if any required check fails.

The test suite runs with:

pytest -q

Individual controls can be run directly:

python examples/01_ramanujan_control.py
python examples/02_beta_gamma_control.py
python examples/03_four_state_control.py

All high-precision computation uses mpmath at 60 decimal digits, with a deterministic fixed truncation of the Ramanujan series at 50 terms (the manuscript requires ≥ 45). Displayed digits are stable under increased precision and truncation length.

6. Expected results

Ramanujan control
L_396 agrees with 1/pi: True
L_prime_396: approximately -7.821535972847e-11
L_double_prime_396: approximately 9.88e-13
Digits per term increase with D: True

Beta/gamma control
J(1/2) = pi/8: True
J(1/4) * J(3/4) = pi/20: True
J is decreasing on the tested grid: True

Four-state control
kappa closes: True
kappa separates target: True
Sigma closes: True
Sigma separates target: False
M closes: False
M separates target: True
Composite closes: True
Composite separates target: True

7. Repository structure

structural-addresses/
├── README.md
├── LICENSE                  CC BY 4.0 (prose, figures, generated data)
├── LICENSE-CODE             MIT (source, tests, examples, reproduce.py)
├── CITATION.cff
├── requirements.txt
├── pytest.ini
├── conftest.py
├── .gitignore
├── paper/
│   └── README.md            pointers to the archived manuscript
├── src/
│   ├── ramanujan_control.py     Control I: series, analytic derivatives, Figure 1
│   ├── beta_gamma_control.py    Control II: J(s), exact identities, Figure 2
│   ├── inheritance.py           partitions, congruence, refinement, induced map
│   └── structural_address.py    factorization, local stability, Addr(x)
├── examples/
│   ├── 01_ramanujan_control.py
│   ├── 02_beta_gamma_control.py
│   └── 03_four_state_control.py
├── data/
│   └── README.md            generated CSVs are written here by reproduce.py
├── figures/
│   ├── figure_1_ramanujan_detuning.png
│   └── figure_2_beta_family.png
├── tests/
│   ├── test_ramanujan.py
│   ├── test_beta_gamma.py
│   ├── test_inheritance.py
│   └── test_structural_address.py
└── reproduce.py

Core utilities

src/inheritance.py

• canonical_partition(blocks) — canonical, hashable partition representation
• is_congruence(states, transition, partition) — does the marker close?
• refines(partition_a, partition_b) — does A separate the classes of B?
• common_refinement(partition_a, partition_b) — the meet
• induced_map(states, transition, partition) — reduced dynamics T, defined only under closure
• coarsest_stable_refinement(states, transition, target) — coarsest inheritance-stable refinement that still separates the target, by iterated preimage refinement

src/structural_address.py

• exact_factorization_possible(signature_values, invariant_values)
• local_stability_flags(signature_values, invariant_values, tolerance)
• structural_address(continuous_signature, discrete_invariant, provenance=None)

8. Scope and non-claims

• The component theorems are classical: the immersion/local-embedding theorem, the universal property of quotient maps, the gamma reflection and multiplication formulas, the generalized hypergeometric differential equation, complex-multiplication special values, and Markov lumpability. Novelty is claimed only for their methodological assembly.
• Numerical closeness to a special value is not proof of arithmetic membership. No finite tolerance certifies exact class membership.
• The code does not classify all gamma-value relations. Only the specific reductions proved in the manuscript are asserted.
• The Rohrlich–Lang classification problem remains open; no contribution to it is claimed.
• No universal structural-address theory is claimed. The discrete invariant δ is an input supplied by the domain; the framework does not discover it.
• The framework asserts neither density nor topological isolation of the arithmetically special locus on the real Ramanujan slice.
• Failure of an integer-relation search is not a proof of algebraic independence.
• Strong lumpability must not be conflated with weak or stationary-distribution lumpability.

9. Related work

This repository accompanies the general methodological note. A separate application of the same framework appears in:

The Pell Spine as an Optimal Rational Separator: Extremal Resolution and Smooth Transport in Primitive Pythagorean Triples — companion repository: https://github.com/innerlightr-wq/pell-spine-optimal-separator

That work distinguishes the continuous separation coordinate Δ_k = (b − a)/c from the exact arithmetic condition b − a = 1, which defines the Pell spine, and shows that the Pell spine forms the record frontier of the continuous proximity measure when primitive triples are ordered by increasing hypotenuse.

The two repositories are complementary: Structural Addresses develops the general framework; Pell Spine provides one concrete arithmetic application. Both are reproducibility companions; the authoritative scholarly records, including the most recent conceptual revisions, are maintained on Zenodo.

10. Citation

Cite the manuscript rather than the repository.

De Jesús, E. (2026). Structural Addresses: Continuous Proximity, Discrete Membership, and Compatible Markers in Inheritance Systems: A Methodological Note with Hypergeometric, Gamma–Period, and Finite-State Controls. Zenodo.

• Current version: https://doi.org/10.5281/zenodo.21297926
• All versions / latest version: https://doi.org/10.5281/zenodo.21263921

@article{dejesus2026structuraladdresses,
  author  = {De Jes{\'u}s, Elias},
  title   = {Structural Addresses: Continuous Proximity, Discrete Membership,
             and Compatible Markers in Inheritance Systems: A Methodological
             Note with Hypergeometric, Gamma--Period, and Finite-State Controls},
  year    = {2026},
  publisher = {Zenodo},
  doi     = {10.5281/zenodo.21297926},
  url     = {https://doi.org/10.5281/zenodo.21297926}
}

Repository URL: TODO (add the canonical GitHub URL once the repository is published).

11. Licensing

• Source code, tests, examples, and reproduce.py: MIT License — see LICENSE-CODE.
• README prose, generated figures, generated data, and manuscript-related material: Creative Commons Attribution 4.0 International (CC BY 4.0) — see LICENSE.

12. AI-assistance disclosure

AI assistance (Anthropic's Claude) was used for algebraic and computational verification, adversarial auditing, repository construction, testing, and manuscript preparation. Conceptual development, research direction, interpretation, and final responsibility for all content remain with the author. AI systems are not authors.

13. Author

Elias De Jesús Independent Researcher ORCID: 0009-0007-0190-9143 Email: dejesuselias10@gmail.com