Al-Hindara Project v7.0.0 — v7.7.2 Audit Draft & Numerical Variational Evidence

Al-Hindara Project — v7.0.0 Release

https://zenodo.org/records/23924915

https://doi.org/10.5281/zenodo.23924915

This release marks a major milestone in the Al-Hindara project, transitioning from exploratory frameworks to a formalized audit draft (v7.7.2). It introduces rigorous domain analysis for the spectral problem and presents stable numerical evidence for a spectral gap, clearly separating analytically proven results from open questions.

📄 What's Included in this Release?

1. The Consolidated Manuscript (v7.7.2 Audit Draft)

The LaTeX source (all-frames2-tex.tex) for the updated manuscript, integrating all four frameworks.

• Analytical Corrections: Establishes the formal domain inclusion (g \in \mathcal{D}(L_F)) and proves that the kernel is strictly one-dimensional ((\dim \ker L_F = 1)).
• Numerical Variational Audit: Presents a constrained B-spline Rayleigh-Ritz computation. The numerical evidence strongly supports a positive spectral gap (discrete eigenvalue (\lambda_{1,\mathrm{RR}} \approx 1.7685)), proving stability across quadrature refinements and endpoint truncations.
• Strict Evidence Classification: The §Z Audit section rigorously categorizes every claim as Proven, Numerically Verified, or Open.
• Open Problems: Explicitly formulates the remaining open questions regarding the continuous self-adjoint realization and spectral convergence.

2. Numerical Verification Code (Jupyter Notebook)

• Includes all-frames2.ipynb, containing the full Python implementation for the B-spline Rayleigh-Ritz method, piecewise Gauss-Legendre quadrature, and eigenvalue computation.

---

📌 Current Status & Remaining Work

To maintain scientific integrity, the following items are explicitly documented as Open in the manuscript:

• The continuous spectral gap: While numerically stable at (\lambda_{1,\mathrm{RR}} \approx 1.7685), proving this analytically for the continuous operator remains open.
• Compact resolvent: The compactness of the embedding (D(q_F) \hookrightarrow L^2(0, 2\pi)) requires further rigorous treatment.
• The self-adjoint domain of (L) on ((0, 2\pi)).

"This document serves to preserve a starting point, not to serve as a final verdict..."

v6.2.0 - Consolidated Manuscript (All-Frames) & Master Calculator EN

Al-Hindara Project — v6.2.0 Release

https://zenodo.org/records/23602940

https://doi.org/10.5281/zenodo.23602940

This release presents the consolidated scientific output of the Al-Hindara project, combining the four exploratory frameworks into a unified manuscript, alongside the English version of the Master Calculator.

📄 What's Included in this Release?

1. The Consolidated Manuscript (v11)

We are releasing the LaTeX source (all-frames-tex.tex) for the "Consolidated Manuscript Draft."

• Unified Structure: It integrates the Numerical Damping, Geometric Ratios, Algebraic Symmetry-Breaking, and Spectral Theory frameworks into one cohesive document.
• Clear Categorization: The manuscript precisely distinguishes between Analytically Proven identities (e.g., the Gauging Identity), Numerically Verified results (e.g., the transition ratio (R_{13/12} \approx 0.60352)), and Empirically Observed field measurements.
• Scope Statement: Includes a dedicated "Summary and Remaining Work" section and an Epilogue that outline the current achievements and the open questions.

2. Master Calculator (English Version)

• Released calculator-en.html.
• Fully translated UI, error handling, and mathematical definitions to English, maintaining complete functionality for Frames 1-4, including Python verification scripts.

📌 Current Status & Remaining Work

To ensure full reproducibility and traceability, the following tasks are scheduled for the final publication phase:

• Traceability Audit: A line-by-line verification of the Traceability Matrix against the original HTML frames is pending.
• DOI Verification: Confirming the validity and metadata of the referenced DOI records (10.5281/zenodo.23540283 and 10.5281/zenodo.23540796).
• LaTeX Compilation: Final testing of the LaTeX compilation process and PDF formatting checks.
• Spectral Analysis: The complete mathematical formulation of the self-adjoint domain and boundary conditions for the spectral operator remains an open problem for future work.

---

“This document serves to preserve a starting point, not to serve as a final verdict...”

v6.1.0 - The Tree and the Shadow (Frame 2 - Empirical & Geometric)

🌳 v6.1.0: The Tree and the Shadow (Al-Hindara Frame 2)

https://doi.org/10.5281/zenodo.23540796

https://zenodo.org/records/23540796

This release completes the empirical and geometric companion to the Mother Paper (Frame 1). It introduces the full manuscript, interactive visual simulations, and the statistical field measurement tools that bridge the exact reciprocal Silver Ratio ((\sqrt{2}-1)) with real-world biological observations.

🚀 What's New in v6.1.0?

• Added Frame 2 Manuscript: The complete LaTeX source and compiled PDF (frame2-tex.tex & frame2_tex.pdf) for "The Tree and the Shadow: Geometric Derivation of the Reciprocal Silver Ratio (\sqrt{2}-1) and an Exploratory Statistical Analysis of Tree Crown Ratios".
• Interactive Visual Simulation (tree2.html): A dynamic web app demonstrating how a (22.5^\circ) angle precisely yields the ratio (0.4142).
• Field Measurement Tool (tree3.html): An open-source TOST (Two One-Sided Tests) statistical tool for conducting equivalence testing in the field.
• Raw Data: Included the pilot study dataset (trees-test.csv) for full reproducibility.

🔗 Related Links

• Frame 2 DOI (OSF): 10.17605/OSF.IO/YC9MB
• Frame 1 (Mother Paper): View in Releases v6.0.0
• Zenodo Concept DOI: 10.5281/zenodo.21003989
• Live Visual Simulation: tree2.html

v6.0.0 - The Numerical Framework (Frame 1 - Mother Paper)

🌌 v6.0.0: The Numerical Framework (Al-Hindara Mother Paper)

https://zenodo.org/records/23540283

https://doi.org/10.5281/zenodo.23540283

This major release introduces the theoretical foundation of the Al-Hindara Unified Theory. It marks the completion of Frame 1, which bridges the numerical framework (Riemann zeta zeros) with the geometric and empirical study (Frame 2).

🚀 What's New in v6.0.0?

• Added Frame 1 Manuscript: The complete LaTeX source and compiled PDF (frame1-tex.tex & frame1_tex.pdf) for "The Numerical Framework: A Damping Model, the Riemann Zeta Zeros, and the Proposed Integer Cutoff 21".
• Scientific Rigor: Explicitly separates the three levels of evidence: Exact Mathematical Results, Exploratory Empirical Observations, and Conjectural Modeling Proposals.
• Mathematical Correction: Included a formal sign analysis for the accumulation function (A(x)), demonstrating a minimum at (x=2) rather than a previously hypothesized maximum at (x=42).
• Reproducibility: Added a comprehensive checklist to ensure all mathematical claims can be independently verified.

🔗 Related Links

• Zenodo Concept DOI: 10.5281/zenodo.21003989
• Frame 2 (Tree & Shadow) PDF: View in repository
• Visual Simulation: tree2.html

v5.0.1 - New Rethinking of Z(x) Roots.

v5.0.1 — New Rethinking of Z(x) Roots

Second release of the Al-Jabri Field study, extending the original v5.0.0 with two independent numerical appendices.

---

📌 DOI & Record

https://zenodo.org/records/23303899

https://doi.org/10.5281/zenodo.23303899

Main DOI: 10.5281/zenodo.23285748
Zenodo record: https://zenodo.org/records/23285748
Repository: https://github.com/Jabri-web/Jabri-com

---

🔍 What's New in v5.0.1

1. Appendix C — Statistical Independence Test

A Kolmogorov–Smirnov test on the critical-line modulus
$|Z(\tfrac12+it)|$ over $t \in [14,60]$ shows:

• p-value ≈ 0.75 → no statistical correlation with the classical Riemann zeros $\gamma_n$.
• The eight zeros $R_1,\dots,R_8$ are an intrinsic feature of $Z$, not a shadow of $\zeta(s)$.
• Full code: wells/Wells.ipynb
• Figure: wells/jabri_v2_appendix.png

2. Appendix D — Asymptotic Coefficient

An analytic + numerical derivation of the large-$t$ behaviour:

$$|Z(\tfrac12+it)| \sim C,\frac{\ln t}{t^{6}}, \qquad C = 2\pi e^{-1/42} \approx 6.1353$$

• Verified numerically over $t \in [10^2, 10^5]$.
• Log-log slope: $-5.9$ (theory: $-6$).
• Coefficient: $C = 6.2$ (theory: $6.1353$, error $1.5%$).
• Full code: wells/appendix-d.ipynb
• Paper: wells/appendix-d.pdf
• Figure: wells/jabri_appendix_D.png

---

📁 Files in This Release

File	Role
wells/README.md	Package overview
wells/CITATION.cff	Machine-readable citation
wells/Wells.ipynb	Appendix C — code
wells/jabri_v2_appendix.png	Appendix C — figure
wells/appendix-d.tex	Appendix D — LaTeX source
wells/appendix-d.pdf	Appendix D — compiled PDF
wells/appendix-d.ipynb	Appendix D — code
wells/jabri_appendix_D.png	Appendix D — figure
wells/appendix-d_full.csv	100 sampled points (D)
wells/appendix-d_summary.csv	Summary statistics (D)

(The main paper jabri_5.tex/pdf/ipynb/png is unchanged from v5.0.0.)

---

🎯 Main Results (Cumulative)

1. Real-axis zeros: $w_n = 2/n$, $n \in \mathbb{N}$ — proven analytically.
2. Critical-line zeros: exactly 8 numerical zeros in $t \in (0,60]$ at 80-digit precision.
3. Independence: statistically independent from Riemann zeros ($p=0.75$).
4. Asymptotic coefficient: $C = 2\pi e^{-1/42} \approx 6.1353$.

---

⚠️ Scope Statement

This release does not claim:

• that the zeros of $Z$ coincide with $\gamma_n$ (Appendix C shows the opposite);
• that $C$ is a universal physical constant (it is a characteristic asymptotic coefficient of $Z$);
• any connection to physics, energy levels, or spacetime.

It is a self-contained numerical and analytic study of one elementary field.

---

📖 Citation

@misc{aljabri2026zeros,
  author       = {Abdulla M. N. Al-Jabri},
  title        = {On the Zeros of the Field $Z(x) = x^{-5}\ln(x)\sin(2\pi/x)e^{-x/21}$},
  year         = {2026},
  month        = {October},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.23285748},
  url          = {https://doi.org/10.5281/zenodo.23285748}
}

v5.0.0 - 5 wells in a new approach of Z(x).

v5.0.0 — 5 wells in a new approach of Zx

https://zenodo.org/records/23285748

https://doi.org/10.5281/zenodo.23285748

Release date: 10 October 2026
Author: Abdulla M. N. Al-Jabri — Sana'a, Yemen

---

Overview

This release introduces a focused study of the elementary field

Z(x) = x⁻⁵ · ln(x) · sin(2π/x) · e^(-x/21)

on the positive real axis and on the Riemann critical line
x = 1/2 + i·t.

The note is short, self-contained, and reproducible at 80-digit
precision.

---

Key results

Analytic zeros on x > 0

w_n = 2/n,    n = 1, 2, 3, ...

Proof: the factor sin(2π/x) vanishes iff 2π/x = nπ iff x = 2/n.

Numerical zeros on the critical line — t ∈ [0, 60]

Eight zeros are located, six in the unit interval (0, 1) and two
isolated outliers:

Root	t
R₁	0.0907486257
R₂	0.2196313827
R₃	0.3403140486
R₄	0.4798481018
R₅	0.6703777044
R₆	0.9957246005
R₇	1.8809618716
R₈	15.0539247442

---

Contents

File	Description
jabri_5.pdf	Compiled paper (3 pages)
jabri_5.tex	LaTeX source
jabri_5.ipynb	Reproduces the 8 numerical zeros
jabri_5.csv	Numerical output
jabri_5.png	Plot of Im[Z(1/2 + i·t)] over t ∈ [0,60]
README.md	Documentation

---

Reproduce

pip install mpmath numpy pandas matplotlib

Open jabri_5.ipynb, run all cells. Runtime ≈ 15 seconds.

---

DOI

Archived on Zenodo:
https://doi.org/10.5281/zenodo.XXXXXXX

---

Changelog

• Initial release v5.0.0.
• Fixes and refines the field study from earlier exploratory work.
• Cleaned LaTeX build, added plot, and packaged all sources.

---

License: CC BY 4.0
Sana'a, Yemen — October 2026

Jabri Zx v3.4.0 — Numerical Convergence of Z_N(44)

Summary

Numerical note on the convergence of the partial sum

Z_N(x) = -sum_{k=1}^{N} 2x / (x^2 + gamma_k^2)

where gamma_k are the imaginary parts of the non-trivial zeros of the
Riemann zeta function.

Main result: Z_N(44) converges to a finite limit

Z_infinity(44) ~ -1.00 ± 0.05

Table

N	Z_N(44)	Increment
10	-0.286869	—
30	-0.517821	-0.230952
50	-0.622073	-0.104252
100	-0.741834	-0.119761
300	~ -0.87	~ -0.13
1000	~ -0.93	~ -0.06

Tail estimate: |Z_inf - Z_N| ~ 2x (log N)^2 / N.

Files

• jabri_zx_v3.4.0.tex — the note (pdfLaTeX)
• jabri_zx_v3.4.0.pdf — compiled PDF
• jabri_zx_v3.4.0.py — reference implementation
• Z44_convergence.png — Figure 1
• z44_convergence.csv — Table 1 data
• README.md, LICENSE

Status

Numerical note. No claim about the Riemann Hypothesis.
No physical constant derived. The result is a finite limit of a
real-analytic sum.

Author

Abdulla M. Al-Jabri
ORCID: 0009-0003-3319-3822

v3.2.2 - self correction

v3.2.2 closes v3.2.1.

We did not claim a final theory. We tested a simple kernel
Z(x) = -sum 2x/(x^2+gamma_k^2) — the standard Riemann-Weil kernel —
and checked whether any functional F[Z] yields G, H0, or w without hand-inserted numbers.

Result: No.

With 30 zeros: Z(12)=-0.3286, Z'(12)=-1.78e-2, no G=6.67e-11, no H0, w_proxy(32.9)=-0.992.
The first critical point Z'≈0 moves with N. Claims JR=∞, C=1.2=Z(12), A=Z'(12)=H0 calibration, G∝1/Z' are retracted.

This release contains:

• jabri-zx-v3.2.2.tex (Technical Note)
• jabri-zx-v3.2.2.py + make_figure.py (verifiable)
• figure1.png + Z_values.csv
• RETRACTIONS.md + CITATION.cff + LICENSE MIT

We publish the negative result as is: here is what the numbers actually give, try another path.

Truth over beauty.

v2.1.0 - osf license

• Fixed LICENSE (full MIT + CC-BY-4.0 note)
• Added OSF DOI 10.17605/OSF.IO/YC9MB to footer
• Added Wiki, GitHub, OpenAIRE links
• Removed v0 issue

A-Phase closed: S_Z = 1.82 (local)
Next: B-Phase Z(x1,x2)

v2.0.0 -constraint

A-Phase complete.

V1 DOI: 10.5281/zenodo.20100622 proved Z(x).
V2 proves real-valued Z(x) yields S_Z=1.82 < 2 (Bell local bound).
This is not a failure but Bell's theorem constraint.

Next B-Phase: Non-factorizable Z(x1,x2) = Z(x1)Z(x2) e^{iPhi(x1,x2)} for entanglement up/down analogy.

Files: quantum_v2/ folder
Site: https://jabri-com.vercel.app/quantum_v2/
OSF: https://osf.io/yc9mb/