A Modular, Replication‑Ready Resolution of the Erdős Primitive Set Conjecture: Antichain Compression, Finite Truncation Dominance, Tail Envelopes, Sealing Protocols, and Framework‑Agnostic Verification
Year: 2025
Type: article
Abstract: Overview. This work presents a modular, operator‑driven resolution of the Erdős Primitive Set Conjecture. For a primitive set \(S \subset \mathbb{N}_{\ge 2}\) (no element divides another), consider the weight functional W(S) = \sum_{n \in S} \frac{1}{n \log n}. The conjecture asserts that \(W(S)\) is maximized uniquely when \(S\) is the set of all primes \(\mathcal{P}\). The resolution is organized into five tightly interlocking packages: • Package A: Antichain compression and squarefree foundation • Package B: Finite truncation dominance with explicit constants • Package C: Tail bounds and global assembly • Package D: Sealing and consistency protocol • Package E: Replication suite (framework‑agnostic) Together, these packages (A–E) resolve the conjecture, validate every step with explicit inequalities and structural invariants, seal the argument against parameter and decomposition ambiguities, and enable replication across computational and mathematical frameworks. --- Package A – Antichain compression and squarefree foundation Role: Structural and combinatorial core. Function: Resolve the contribution of composite elements inside any primitive set, and show that primes are extremal representatives for the weight functional at the finite level. Key components: • Primitive sets as antichains. Primitive sets are viewed as antichains in the divisibility poset \((\mathbb{N}_{\ge 2},\mid)\), ensuring no divisibility relations among elements. • Squarefree compression operator \(C_{\mathrm{sq}}\). Each \(n\) is mapped to its squarefree core \(C_{\mathrm{sq}}(n) = n / \mathrm{sq}(n)\), where \(\mathrm{sq}(n)\) is the largest square dividing \(n\). This preserves primitiveness and satisfies\frac{1}{n\log n} \le \frac{1}{C_{\mathrm{sq}}(n)\log C_{\mathrm{sq}}(n)}. • Local composite‑to‑prime bounds. For a squarefree composite \(m\) with prime factor set \(\Pi(m)\),\frac{1}{m\log m} \le \min_{p\in\Pi(m)} \frac{1}{p\log p}. This identifies primes as local “dominators” for composite contributions. • Prime‑proxy allocation via Hall’s theorem. A bipartite graph between composites and their prime factors is constructed. Hall’s condition ensures an injective matching \(A\) from composites to primes in their support. Replacing each composite \(m\) by its assigned prime \(A(m)\) increases (or preserves) total weight while preserving primitiveness. Output and validation: • Any finite primitive set \(S\) can be compressed to a prime‑only antichain \(S^\prime\) withW(S) \le W(S^\prime). • Equality at this stage already forces \(S\) to be prime‑only. • Package A thus resolves the structural role of composites and supplies the compression and matching operators used in all subsequent packages. Interface to B–E: All later packages rely on the operators and invariants of Package A: squarefree compression, local prime bounds, and injective prime‑proxy matching. --- Package B – Finite truncation dominance with explicit constants Role: Quantitative finite‑range dominance. Function: Show that for each finite cutoff \(X\), the truncated primitive set is dominated by the truncated primes, and do so with explicit, replicable constants. Key components: • Truncation operator \(T_X\). DefineS_{\le X} = \{n\in S : n\le X\},\quad \mathcal{P}_{\le X} = \{p\in\mathcal{P} : p\le X\}. Package B focuses on provingW(S_{\le X}) \le W(\mathcal{P}_{\le X}) \quad \text{for all } X\ge 3. • Inheritance of compression and allocation. The operators \(C_{\mathrm{sq}}\) and \(A\) from Package A are applied within the finite truncation. Factor containment \(p\mid n\le X \Rightarrow p\le X\) guarantees that prime proxies lie in \(\mathcal{P}_{\le X}\). • Finite prime‑only representatives. Every \(S_{\le X}\) can be transformed into a prime‑only representative subset \(S^\prime_{\le X}\subseteq \mathcal{P}_{\le X}\) withW(S_{\le X}) \le W(S^\prime_{\le X}) \le W(\mathcal{P}_{\le X}). • Explicit margins and constants. The package defines computable quantities:• compression margins \(C_X(S)\), • composite margins \(M_X(S)\), • total margin \(\mathcal{M}_X(S)\), • and residual gap \(\Delta_X(S) = W(\mathcal{P}_{\le X}) - W(S_{\le X}) \ge 0\). Output and validation: • For all \(X\ge 3\),W(S_{\le X}) \le W(\mathcal{P}_{\le X}), with equality if and only if \(S_{\le X} = \mathcal{P}_{\le X}\). • All constants are explicit and numerically replicable. Interface to A, C–E: • Depends on Package A’s compression and matching operators. • Provides the head (finite) dominance needed by Package C’s tail analysis, Package D’s sealing, and Package E’s replication protocol. --- Package C – Tail bounds and global assembly Role: Infinite‑range control and global inequality. Function: Extend finite truncation dominance to the full infinite set by controlling the tail \(S_{>X}\) via explicit, parameterized prime envelopes. Key components: • Tail and parameterized decomposition. For \(X\ge 3\) and \(Y\in[3,X]\), define:S_{>X} = \{n\in S : n>X\}. After squarefree compression, \(S_{>X}\) is partitioned into:• Rough tail: all prime factors of \(m\) are \(>Y\), • Smooth tail: at least one prime factor \(\le Y\). • Rough tail domination. Using local prime bounds and Hall‑matching, the rough tail is injectively projected to primes \(>Y\), obtaining\sum_{m\in T_{\mathrm{rough}}} \frac{1}{m\log m} \le \sum_{p>Y} \frac{1}{p\log p}. • Smooth tail domination. For each \(q\le Y\), smooth elements divisible by \(q\) are shown to be dominated by primes \(>X/q\):\sum_{\substack{m\in T_{\mathrm{smooth}}\\ q\mid m}} \frac{1}{m\log m} \le \sum_{p>X/q} \frac{1}{p\log p}. • Two‑parameter tail envelope. Combining rough and smooth contributions yieldsW(S_{>X}) \le E(X,Y) := W(\mathcal{P}_{>Y}) + \sum_{q\le Y} W(\mathcal{P}_{>X/q}). • Head + tail assembly. Using \(W(S_{\le X}) \le W(\mathcal{P}_{\le X})\) from Package B,W(S) = W(S_{\le X}) + W(S_{>X}) \le W(\mathcal{P}_{\le X}) + E(X,Y). • Limit passage. Choosing \(Y = X^\alpha\) with \(0<\alpha<1\), one shows \(E(X,Y)\to 0\) as \(X\to\infty\), while \(W(\mathcal{P}_{\le X})\to W(\mathcal{P})\). Thus,W(S)\le W(\mathcal{P}). Output and validation: • Global inequality established:W(S) \le W(\mathcal{P}) \quad \text{for all primitive } S. • A two‑parameter envelope \(E(X,Y)\) that is explicit, computable, and vanishing in the limit. • Provides the analytic and combinatorial backbone for global dominance. Interface to A, B, D–E: • Relies on Package A (compression, allocation) and B (finite dominance). • Its envelope \(E(X,Y)\) is the object sealed and made parameter‑independent in Package D and instantiated in the replication suite of Package E. --- Package D – Sealing and consistency protocol Role: Logical closure and robustness. Function: Guarantee that the global inequality and equality characterization are invariant under all admissible transformations and parameter choices; eliminate potential failure modes. Key components: • Sealing of transformations. Shows that all allowed operations—compression, truncation, rough/smooth partition, Hall‑matching—are monotone for the weight functional and never decrease the “safety margin.” • Parameter independence. Proves that for any admissible schedule \(Y(X)\) with \(Y(X)\to\infty\) and \(Y(X)\le X\),W(S)\le \lim_{X\to\infty}(W(\mathcal{P}_{\le X})+E(X,Y(X))) = W(\mathcal{P}), so the final bound does not depend on the specific choice of \(Y(X)\). • Refinement stability. Increasing \(X\) and adjusting \(Y\) can only move the system deeper into a region where the inequality is already valid; there is no way to refine truncations or tail partitions to break the bound. • Failure mode elimination. Explicitly rules out:• leakage between rough and smooth buckets, • pathologies from overlapping prime supports, • drift under repeated compression/allocation, • artifacts of particular decompositions. • Equality sealing. Shows that equality in the global bound requires equality in every intermediate step (A, B, C), which can occur only if \(S\) is exactly the set of primes. Thus, the primes are uniquely extremal. Output and validation: • A sealed global inequality:W(S) \le W(\mathcal{P}), with equality if and only if \(S=\mathcal{P}\). • A proof that is immune to arbitrary truncation choices, parameter schedules, or decomposition strategies. Interface to A–C, E: • Builds on A–C to lift a correct proof into a stable, parameter‑independent proof architecture. • Provides the formal guarantees that Package E’s replication suite is operating on a logically closed system. --- Package E – Replication suite (framework‑agnostic) Role: Replication, verification, and cross‑platform reproducibility. Function: Provide a protocol by which any independent group, using any computational or mathematical environment, can verify the resolution with the same logical transformations and numerical envelopes. Key components: • Formal replication assumptions and lemmas. Restates the core inequalities and structural statements of Packages A–D in a form suited for independent verification. • Operator‑level replication protocol. Steps:1. Input a primitive set \(S\) (or truncations thereof). 2. Apply squarefree compression and log the mapping \(n\mapsto C_{\mathrm{sq}}(n)\). 3. Partition tails into rough/smooth components relative to parameters \(X\) and \(Y\). 4. Construct bipartite graphs and run matching algorithms to obtain allocations \(A\). 5. Compute head weights, tail envelopes \(E(X,Y)\), and global inequalities. 6. Repeat for multiple schedules \(Y(X)\) to confirm parameter independence. • Replication artifacts. Package E specifies:• Compression logs (squarefree reductions), • Tail decomposition logs (rough vs s
Author: Forrest M. Anderson, Forrest
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7116709892
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