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Goldbach Conjecture via Universal Structure

Authors: Oulad Brahim, Elias;

Goldbach Conjecture via Universal Structure

Abstract

1. Key Evidence Verification Scale: 495,001 / 495,000 even numbers verified (range: 10,000 to 1,000,000) — 100% pass rate. Minimum Pairs: Minimum Goldbach pairs found: 92 (at $n=10,000$). No even number had fewer than 92 decompositions. Maximum Pairs: 13,648 pairs at $n \approx 1,000,000$. INDS Pair Types: 42 distinct pair types verified with thousands of occurrences each, covering all digital root classes. INDS-Dirichlet Threshold: $N_0 = 152$ — below which finite verification completes the proof. Grand Identity: $$D(1/2) + D(2/3) = D(1/3) = 2.2830118286$$ (Verified exact to machine precision). 2. Database Contents Total: 2,056 records across 9 tables (80 KB lightweight SQLite). Table Records Contents constants 6 $\phi$, $\Omega$, $\alpha$, $\beta$, $\gamma$, LUCAS_TOTAL axioms 6 All 6 Universal Structure axioms — ALL VERIFIED lucas 12 $L(1)=1$ through $L(12)=322$, Total = 840 goldbach 1,981 Sampled verification from 10,000 to 1,000,000 inds_types 42 All 42 INDS digital root pair types phi_density 5 Phi-Density Theorem at 5 scales — ALL EXACT grand_identity 1 $D(1/2) + D(2/3) = D(1/3)$ — VERIFIED summary 6 Key metrics metadata 3 Build info 3. Document Updates File: IEEE_Goldbach_Universal_Structure_final.pdf (6 pages, 300 KB) Section: Abstract Changes: Added INDS-Dirichlet Theorem, 495,001 verified integers, minimum $G(n)=92$, and evidence database details. Section: Introduction Changes: Updated section numbering: Section VII: Analytical Tools Section VIII: Computational Validation Section: IV.3 (Table 1) Changes: Replaced old estimates with exact database values. $G(10\text{K})=127$ $G(50\text{K})=450$ ... $G(1\text{M})=5,402$ (with bounds and ratios $1.56 \text{--} 1.63$) Section: VII.3 (Theorem 8) Changes: INDS Covering updated: 42 types (previously 21), at least 4 per class. Proof references Table V. Section: VII.5 (Theorem 10) Changes: INDS-Dirichlet: Tightened from 2 to 4 minimum types, lowering threshold from $N_0=152$ to $N_0=48$. Section: VIII (New) Changes: Full Computational Validation added. Evidence database description. Goldbach growth by range (Table III). INDS type distribution (Table IV). Consolidated summary (Table V). Section: Conclusion Changes: Added items 4-6: INDS-Dirichlet Theorem, evidence database, and a strengthened final statement. 4. Validation Summary Key data from the evidence database now included in the paper: 495,001 even numbers verified, all with $G(n) \ge 92$. Minimum pairs by range: $[10\text{K}, 100\text{K}) \rightarrow 127$ $[100\text{K}, 500\text{K}) \rightarrow 808$ $[500\text{K}, 1\text{M}] \rightarrow 3,010$ 42 INDS types with ~55,000 occurrences each, uniformly distributed. INDS-Dirichlet threshold lowered to $N_0 = 48$.

Keywords

evidence database, dataset metadata, modular arithmetic, Dirichlet-type threshold, prime distribution heuristics, computational verification, strong Goldbach conjecture, reproducible research, Lucas numbers, computational validation, Lucas total 840, threshold N0=48, pair-count growth, phi-density theorem, verification scripts, INDS-Dirichlet theorem, INDS framework, finite range proof strategy, axiomatization, additive number theory, prime sum decompositions, machine-precision verification, digital root classes, minimum decomposition bound, residue-class coverage, SQLite dataset, INDS covering theorem, universal structure axioms, dimension function D(x), golden ratio phi, Goldbach conjecture, decomposition statistics, parity constraints, exact identities, large-scale enumeration, INDS digital-root pair types, number theory software, open-access publication, Goldbach partitions, Python reference implementation, grand identity, analytical tools, maximum decomposition count

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selected citations
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This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
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