> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # A Base-Space–Prime-Fitting-Operator Reformulation Program for Legendre's Conjecture ## Legendre Base-Space Prime-Fitting Program **Version:** v0.1 **Date:** 2026-09-30 **Author:** Neo.K × AI collaboration **Institution:** EveMissLab / 一言諾科技有限公司 **Nature:** Methodological reformulation, research program, candidate formal framework **Status:** Not a completed proof of Legendre's conjecture; this paper establishes a verifiable base-space/operator-based proof route and certificate obligations --- ## Abstract Legendre's conjecture asserts that for every positive integer $n$, there is at least one prime between consecutive squares $n^2$ and $(n+1)^2$. Traditional formulations place the problem directly in terms of prime distribution, prime gaps, short-interval estimates, or differences of counting functions: $$ \pi((n+1)^2)-\pi(n^2)\ge 1. $$ This paper proposes a different reformulation: instead of initially treating ‘there is a prime between squares’ as merely a short-interval prime-existence proposition, first construct an **algebraically continuous square base space**, then elevate existing prime-prediction / fitting formulas to a **prime-fitting operator**, and study whether its restrictions to the global base space and every local square cell retain the same operator structure. The core idea is: $$ \text{existence problem} \longrightarrow \text{base-space closure problem}. $$ This paper defines the global square base space: $$ \mathcal B_{\square} = \bigsqcup_{n\ge 1}\mathcal B_n, \qquad \mathcal B_n=(n^2,(n+1)^2), $$ and uses the normalization map: $$ \phi_n:(0,1)\to\mathcal B_n, \qquad \phi_n(t)=n^2+(2n+1)t $$ to view all local square intervals as projections of the same standard base space $(0,1)$ at different scales. Next, define a candidate prime-fitting operator: $$ \mathcal F_{\mathbb P}, $$ and its local restrictions: $$ \mathcal F_{\mathbb P,n} = \mathcal F_{\mathbb P}\big|_{\mathcal B_n}. $$ If we can prove that: $$ \mathcal F_{\mathbb P}\big|_{\mathcal B_n} = \mathcal F_{\mathbb P,n} $$ holds for every $n$, and that every local output simultaneously satisfies: $$ \mathcal F_{\mathbb P}(\mathcal B_n)\in\mathbb P $$ and: $$ \mathcal F_{\mathbb P}(\mathcal B_n)\in(n^2,(n+1)^2), $$ then Legendre's conjecture follows directly: $$ \forall n\ge1,\qquad \mathbb P\cap(n^2,(n+1)^2)\neq\varnothing. $$ This paper's focus is not on mistaking ‘high fitting accuracy’ for ‘proof’, but on compressing the remaining proof obligations into three independently auditable questions: 1. Well-definedness of the prime-fitting operator; 2. Global / local restriction compatibility; 3. Certificates of local cell internality and primality. This research therefore rewrites Legendre's conjecture from ‘search directly for a prime in every interval’ into a closure problem: ‘prove that the same prime-fitting operator retains its ontological identity in the global and local base spaces, and that its local image cannot escape the prime cell’. --- # 1. The Original Form of the Problem Legendre's conjecture can be written as: $$ \forall n\in\mathbb Z_{>0}, \quad \exists p\in\mathbb P \quad \text{such that} \quad n^2 Can ‘every square interval must contain a prime’ be rewritten as a higher-level base-space/operator closure proposition? --- # 2. The Continuous Square Base Space ## 2.1 Local Square Cells For each $n\ge1$, define: $$ \mathcal B_n = (n^2,(n+1)^2). $$ Call $\mathcal B_n$ the $n$th **local continuous square cell**. Its scale is: $$ |\mathcal B_n|=2n+1. $$ All Legendre intervals thus form a family: $$ \{\mathcal B_n\}_{n\ge1}. $$ --- ## 2.2 The Global Square Base Space Define: $$ \boxed{ \mathcal B_{\square} = \bigsqcup_{n\ge1}\mathcal B_n } $$ Here $\bigsqcup$ emphasizes that each local cell retains its index and structural identity, rather than being treated merely as an undifferentiated subset of the real numbers. Thus, ‘global’ in this paper does not mean merging all intervals into one number line, but preserving: $$ n \longmapsto \mathcal B_n $$ this square-level structure. --- ## 2.3 The Normalized Local Base Space Every square cell can be generated from the same unit interval. Define: $$ \phi_n:(0,1)\to\mathcal B_n, $$ $$ \boxed{ \phi_n(t)=n^2+(2n+1)t. } $$ Its inverse map is: $$ \phi_n^{-1}(x) = \frac{x-n^2}{2n+1}. $$ Thus: $$ \mathcal B_n \cong (0,1) $$ holds for every $n$. The only difference is the affine scale: $$ 2n+1. $$ This yields the first important viewpoint: > At the base-space level, the square intervals in Legendre's conjecture are not entirely different problems, but expansions of the same standard cell at different square scales. --- # 3. The Local–Global Base-Space Proposition If every local square cell is simply a scaling of the same base-space template, what is truly worth seeking is not a separate technique for each $n$, but a structural proposition acting simultaneously on: $$ \mathcal B_{\square} $$ and: $$ \mathcal B_n $$ these spaces. We hope to find a property $\mathfrak P$ such that: $$ \mathfrak P(\mathcal B_{\square}) $$ holds, and for any $n$: $$ \mathfrak P(\mathcal B_n) $$ also holds. More strongly, we hope that the local property is not separately defined, but is the natural restriction of the global property: $$ \boxed{ \mathfrak P(\mathcal B_{\square})\big|_{\mathcal B_n} = \mathfrak P(\mathcal B_n). } $$ The core point here is not ‘global statistical averages are correct’, but: > After the global structure is restricted to any local square cell, it still retains the same ontology and the same rules of action. --- # 4. From Prime-Prediction Formulas to a Prime-Fitting Operator ## 4.1 No Longer Viewing the Existing Formula Merely as a Predictor Previous prime-geometry work can be abstracted as a procedure mapping a sequence scale $N$ to a candidate prime position: $$ N \longmapsto \widehat p_N. $$ In this research, it is no longer called merely a ‘prediction formula’, but is elevated to: $$ \boxed{ \mathcal F_{\mathbb P} } $$ that is, the **Prime Fitting Operator**. This does not mean claiming that it is inherently a prime-generation theorem; rather, it repackages: $$ \text{estimate} $$ as a mathematical operator whose following properties can be studied: - Well-definedness; - Restriction; - Gluing; - Closure; - Invariance; - Certification; as properties of that operator. --- # 5. The Existing Prime-Fitting Skeleton The previous prime-geometry framework takes the cumulative mean of primes: $$ \operatorname{Avg}(N) = \frac1N\sum_{k=1}^{N}p_k $$ as its core. Let: $$ y_N = \log_{10}\operatorname{Avg}(N). $$ Use the scale-dependent geometric slope: $$ m(N) = 1+\varepsilon(N). $$ The revised full-scale correction is written as: $$ \varepsilon_{\mathrm{full}}(N) = \begin{cases} \varepsilon_{\mathrm{table}}(N), &2\le N<1000,\\[4pt] \varepsilon_{\mathrm{mid}}(N), &1000\le N<5\times10^6,\\[4pt] \varepsilon_{\mathrm{asy}}(N), &N\ge5\times10^6. \end{cases} $$ The intermediate-scale part can be written as: $$ L=\ln N, \qquad \Lambda=\ln\ln N, $$ $$ \varepsilon_{\mathrm{mid}}(N) = \frac1L + \frac{\beta_1\Lambda+\beta_2}{L^2} + \frac{\beta_3\Lambda^2+\beta_4\Lambda+\beta_5}{L^3}. $$ The stored candidate coefficients are: $$ \beta_1=160.2971, $$ $$ \beta_2=-845.5678, $$ $$ \beta_3=763.7534, $$ $$ \beta_4=-359.9259, $$ $$ \beta_5=1546.2420. $$ The large-scale asymptotic part is: $$ \varepsilon_{\mathrm{asy}}(N) = \frac1L + \frac{1-\Lambda}{L^2} + \frac{\Lambda^2-\Lambda-1}{2L^3}. $$ Use this for the geometric extrapolation: $$ y_{N+1} = y_N + \bigl(1+\varepsilon_{\mathrm{full}}(N)\bigr) \left[ \log_{10}(N+1)-\log_{10}N \right]. $$ Then recover: $$ \widehat{\operatorname{Avg}}(N+1) = 10^{y_{N+1}}. $$ Using additive anchoring: $$ \boxed{ \widehat p_{N+1} = (N+1)\widehat{\operatorname{Avg}}(N+1) - N\operatorname{Avg}(N) } $$ gives a candidate position for the next prime. This paper abstracts the entire process above as: $$ \boxed{ \mathcal F_{\mathbb P}. } $$ --- # 6. The Prime-Fitting Base Space An operator alone is not enough. This paper further defines the base space formed by its image. Let: $$ \boxed{ \mathcal B_{\mathrm{PF}} = \left\{ (x,\mathcal F_{\mathbb P}(x)) : x\in\mathcal B_{\square} \right\}. } $$ Call it: **The Prime-Fitting Base Space.** Intuitively: $$ \mathcal B_{\square} $$ describes the square-interval skeleton of the problem; while: $$ \mathcal B_{\mathrm{PF}} $$ describes the graph of the prime-fitting operator's action on that skeleton. For each local cell: $$ \mathcal B_n, $$ define: $$ \boxed{ \mathcal B_{\mathrm{PF},n} = \mathcal B_{\mathrm{PF}}\big|_{\mathcal B_n}. } $$ We therefore obtain: $$ \mathcal B_{\square} \leftrightarrow \mathcal B_{\mathrm{PF}} $$ and the local version: $$ \mathcal B_n \leftrightarrow \mathcal B_{\mathrm{PF},n}. $$ --- # 7. The Core Question: Which Property Holds Both Globally and Locally? What this program actually seeks is a common proposition $\mathfrak C$ such that: $$ \mathfrak C \left( \mathcal B_{\square}, \mathcal B_{\mathrm{PF}} \right) $$ holds globally, and: $$ \mathfrak C \left( \mathcal B_n, \mathcal B_{\mathrm{PF},n} \right) $$ holds for every local $n$. The most natural candidate is: $$ \boxed{ \mathcal F_{\mathbb P} \big|_{\mathcal B_n} = \mathcal F_{\mathbb P,n}. } $$ This can be called: ## Local–Global Prime-Fitting Compatibility ### Local–Global Prime-Fitting Compatibility Here: $$ \mathcal F_{\mathbb P,n} $$ is not a separately invented local algorithm, but the natural restriction of: $$ \mathcal F_{\mathbb P} $$ to: $$ \mathcal B_n $$ this local cell. --- # 8. Formalizing ‘The Prime-Fitting Operator Is the Prime-Fitting Operator’ An intuitive statement is: > In the end, we only need to prove that the prime-fitting operator is the prime-fitting operator. Formally, we cannot merely write: $$ \mathcal F_{\mathbb P} = \mathcal F_{\mathbb P}, $$ because that is just an identity. What must actually be proved is: $$ \boxed{ \mathcal F_{\mathbb P}^{\mathrm{global}} \big|_{\mathcal B_n} = \mathcal F_{\mathbb P}^{\mathrm{local}}. } $$ That is: $$ \boxed{ \mathcal F_{\mathbb P} \big|_{\mathcal B_n} = \mathcal F_{\mathbb P,n}. } $$ Its meaning is: - A change in scale does not change the operator; - A change in interval index does not change the rule; - Localization does not lose the prime-fitting structure; - Different $n$ do not require different special cases; - The local model is not another empirical fitter; - The global and local operators have the same provenance. This is ‘consistency of operator ontology’. --- # 9. The Restriction Commutation Law Let: $$ \rho_n: \mathcal B_{\square} \to \mathcal B_n $$ be the restriction map from the global base space to the $n$th local cell. We hope to have: $$ \boxed{ \rho_n\circ\mathcal F_{\mathbb P} = \mathcal F_{\mathbb P,n}\circ\rho_n. } $$ This can be represented by a commutative diagram: $$ \begin{array}{ccc} \mathcal B_{\square} & \xrightarrow{\mathcal F_{\mathbb P}} & \mathcal B_{\mathrm{PF}} \\ \downarrow\rho_n & & \downarrow\rho_n \\ \mathcal B_n & \xrightarrow{\mathcal F_{\mathbb P,n}} & \mathcal B_{\mathrm{PF},n}. \end{array} $$ What actually matters here is not the diagram, but: > Global fitting followed by restriction must give the same result as first restricting the base space and then fitting locally. --- # 10. The Second Layer Required for a Legendre Proof: Cell Internality Proving local–global compatibility alone is still not Legendre's conjecture. We also need: $$ \boxed{ \mathcal F_{\mathbb P}(\mathcal B_n) \in \mathcal B_n. } $$ That is: $$ \boxed{ n^2 < \mathcal F_{\mathbb P}(\mathcal B_n) < (n+1)^2. } $$ This is called: ## Square-Cell Internality ### Square-Cell Internality If the operator escapes the cell for some $n$: $$ \mathcal F_{\mathbb P}(\mathcal B_n) \notin \mathcal B_n, $$ then, even if it is still a good prime-position estimator, it cannot directly support Legendre's conjecture. --- # 11. The Third Layer: A Primality Certificate Even if the output lies in: $$ (n^2,(n+1)^2), $$ we must still prove: $$ \boxed{ \mathcal F_{\mathbb P}(\mathcal B_n) \in\mathbb P. } $$ This is called: ## The Prime-Image Property ### Prime-Image Property Thus, Legendre's conjecture requires not a single condition, but: $$ \boxed{ \mathcal F_{\mathbb P}(\mathcal B_n) \in \mathbb P \cap (n^2,(n+1)^2). } $$ for every $n\ge1$. --- # 12. An Operator-Based Equivalent Target for Legendre's Conjecture We can therefore elevate the original existence proposition: $$ \forall n\ge1, \quad \exists p\in\mathbb P \cap (n^2,(n+1)^2) $$ to the following construction target: There exists a global operator: $$ \mathcal F_{\mathbb P} $$ such that for every $n$: $$ \boxed{ \mathcal F_{\mathbb P} \big|_{\mathcal B_n} = \mathcal F_{\mathbb P,n}, } $$ and: $$ \boxed{ \mathcal F_{\mathbb P}(\mathcal B_n) \in \mathbb P \cap \mathcal B_n. } $$ This immediately gives: $$ \mathbb P \cap \mathcal B_n \neq \varnothing. $$ Therefore: $$ \boxed{ \forall n\ge1, \qquad \mathbb P \cap (n^2,(n+1)^2) \neq \varnothing. } $$ --- # 13. Candidate Main Theorems This paper does not claim a proof, but the propositions actually to be tackled can be written as: ## Candidate Main Theorem A: Base-Space Restriction Invariance If: $$ \mathcal F_{\mathbb P} : \mathcal B_{\square} \to \mathcal B_{\mathrm{PF}} $$ is a well-defined prime-fitting operator, and its generation rule depends only on base-space invariants preserved under square-cell normalization, then for every $n$: $$ \mathcal F_{\mathbb P} \big|_{\mathcal B_n} = \mathcal F_{\mathbb P,n}. $$ --- ## Candidate Main Theorem B: Square-Cell Closure If for each $n$: $$ \mathcal F_{\mathbb P,n} $$ preserves an admissible prime image in the normalized local coordinates, then: $$ \mathcal F_{\mathbb P}(\mathcal B_n) \in \mathcal B_n. $$ --- ## Candidate Main Theorem C: Prime-Image Closure If the image of $\mathcal F_{\mathbb P}$ satisfies an independent primality certificate: $$ \mathcal C_{\mathbb P} \left( \mathcal F_{\mathbb P}(\mathcal B_n) \right)=1, $$ then: $$ \mathcal F_{\mathbb P}(\mathcal B_n) \in\mathbb P. $$ --- ## Candidate Main Theorem D: The Legendre Corollary If A, B and C hold for every $n\ge1$, then: $$ \forall n\ge1, \qquad \mathbb P\cap(n^2,(n+1)^2)\neq\varnothing. $$ that is, Legendre's conjecture holds. --- # 14. A Stronger Form in Normalized Coordinates Let: $$ u = \phi_n^{-1}(x) = \frac{x-n^2}{2n+1}. $$ Then each: $$ x\in\mathcal B_n $$ is mapped to: $$ u\in(0,1). $$ Suppose the prime-fitting operator can be written in these coordinates as: $$ \widetilde{\mathcal F}_{\mathbb P,n} = \phi_n^{-1} \circ \mathcal F_{\mathbb P,n} \circ \phi_n. $$ If we can further prove that: $$ \boxed{ \widetilde{\mathcal F}_{\mathbb P,n} = \widetilde{\mathcal F}_{\mathbb P} } $$ holds for every $n$, this means: > After the scale of the square interval is removed, the local law of action of the prime-fitting operator does not depend on $n$. This is stronger than merely: $$ \mathcal F_{\mathbb P} \big|_{\mathcal B_n} = \mathcal F_{\mathbb P,n} $$ as a scale-normalization proposition. It may become the actual point of breakthrough. --- # 15. Scale Covariance If the normalized operators are not exactly the same, a weaker covariant form can also be sought. Let: $$ S_n(x) = \frac{x-n^2}{2n+1}. $$ We can then require a uniform operator: $$ \mathcal G_{\mathbb P} $$ such that: $$ \boxed{ S_n \circ \mathcal F_{\mathbb P,n} = \mathcal G_{\mathbb P} \circ S_n. } $$ This means: $$ \mathcal F_{\mathbb P,n} = S_n^{-1} \circ \mathcal G_{\mathbb P} \circ S_n. $$ That is, all local prime-fitting operators are mutually conjugate. If this holds, square intervals for different $n$ are not different problems, but conjugate representations of the same local dynamics at different square scales. --- # 16. Relationship to the Existing Prime-Fitting Formula This paper does not directly claim: $$ \varepsilon_{\mathrm{full}} $$ itself as a proof of Legendre's conjecture. At present it is better suited to the role of: $$ \boxed{ \mathcal F_{\mathbb P}^{(0)} } $$ a candidate zeroth-order prime-fitting operator. The next research focus is not to pursue more attractive numerical errors, but to ask: ### Question 1 $$ \mathcal F_{\mathbb P}^{(0)} $$ Can it be redefined as a genuine global operator rather than a collection of piecewise empirical formulas? ### Question 2 Can its piecewise components: $$ \varepsilon_{\mathrm{table}}, \qquad \varepsilon_{\mathrm{mid}}, \qquad \varepsilon_{\mathrm{asy}} $$ be unified into a single base-space generation law? ### Question 3 Can intermediate-scale fitting error be elevated from empirical numerical bounds to a: $$ \text{certified envelope}? $$ ### Question 4 Is that envelope sufficient to guarantee: $$ \mathcal F_{\mathbb P}(\mathcal B_n) \in \mathcal B_n? $$ ### Question 5 Is primality generated endogenously by the operator, or is an external primality test still needed? These questions determine whether: $$ \mathcal F_{\mathbb P}^{(0)} $$ ultimately is a: $$ \text{predictor} $$ or a: $$ \text{certificate-carrying operator}. $$ --- # 17. From Fitting Accuracy to a Certified Envelope Let the actual prime position be: $$ p_N. $$ The candidate fitted position is: $$ \widehat p_N. $$ Define the error: $$ e_N = p_N-\widehat p_N. $$ If a globally provable function can be established: $$ E(N)\ge0 $$ such that: $$ \boxed{ |e_N| \le E(N) } $$ we obtain: $$ p_N \in [ \widehat p_N-E(N), \widehat p_N+E(N) ]. $$ For Legendre's conjecture, a one-sided version is more specifically needed: $$ \boxed{ p_N \le \widehat p_N+E_+(N). } $$ If we let: $$ k=\pi(n^2), $$ the next prime is: $$ p_{k+1}. $$ As long as: $$ \boxed{ \widehat p_{k+1} + E_+(k+1) < (n+1)^2 } $$ we can infer: $$ p_{k+1} < (n+1)^2. $$ Also, since: $$ p_{k+1}>n^2, $$ we have: $$ n^2 < p_{k+1} < (n+1)^2. $$ This is the ‘error-envelope route’. --- # 18. Unifying the Error-Envelope and Base-Space Routes The base-space and error-envelope versions do not conflict. We can define the square-cell safety margin: $$ M_n = (n+1)^2 - \left( \widehat p_{k+1} + E_+(k+1) \right), $$ where: $$ k=\pi(n^2). $$ If: $$ \boxed{ M_n>0 } $$ the $n$th cell is safe. Cell internality can therefore be rewritten as: $$ \boxed{ M_n>0 \quad \forall n. } $$ That is: $$ \text{operator internality} \Longleftrightarrow \text{positive certified margin}. $$ --- # 19. The Actual Common Global / Local Proposition What is most worth seeking in this research is not simply: $$ E(N)\to0 $$ or: $$ \widehat p_N/p_N\to1. $$ because Legendre's conjecture is a local existence proposition. What is truly needed is a relative-scale quantity controlling both: $$ \mathcal B_{\square} $$ and: $$ \mathcal B_n $$ these spaces. A candidate may be: $$ \Gamma_n = \frac{ \widehat p_{k+1}-n^2 }{ 2n+1 }, $$ and its certified upper version: $$ \Gamma_n^{+} = \frac{ \widehat p_{k+1}+E_+(k+1)-n^2 }{ 2n+1 }. $$ The condition needed for Legendre's conjecture becomes: $$ \boxed{ 0<\Gamma_n^{+}<1. } $$ If we can prove that: $$ \Gamma_n^{+} $$ is subject, in the normalized square base space, to a global closure independent of $n$: $$ \Gamma_n^{+} \in (0,1), $$ then the local and global are truly unified. --- # 20. A Possible ‘Base-Space Invariant’ After normalization, every interval is: $$ (0,1). $$ It is therefore natural to seek: $$ \mathcal I_n = I \left( \mathcal B_n, \mathcal F_{\mathbb P,n} \right) $$ such that: $$ \mathcal I_n = \mathcal I $$ or at least: $$ \mathcal I_n \in \mathcal K $$ where: $$ \mathcal K $$ is a closed set independent of $n$. For example: $$ \mathcal I_n = \frac{ \mathcal F_{\mathbb P}(\mathcal B_n)-n^2 }{ 2n+1 }. $$ If we can prove: $$ \boxed{ \mathcal I_n\in(0,1) \quad \forall n, } $$ cell internality holds. If we additionally attach: $$ \boxed{ \mathcal C_{\mathbb P} \left( \mathcal F_{\mathbb P}(\mathcal B_n) \right)=1, } $$ Legendre's conjecture holds. --- # 21. Certificate Obligations This paper deliberately distinguishes: $$ \text{fit} $$ from: $$ \text{proof}. $$ Every subsequent version must satisfy at least the following certificate obligations. --- ## C0 — Definition Certificate The following must be defined unambiguously: $$ \mathcal B_{\square}, \quad \mathcal B_n, \quad \mathcal F_{\mathbb P}, \quad \mathcal B_{\mathrm{PF}}, \quad \rho_n. $$ The meanings of ‘base space’ and ‘fitting’ must not drift between sections. --- ## C1 — Well-Definedness Certificate We must prove that: $$ \mathcal F_{\mathbb P} $$ has a unique admissible output for every input in its entire claimed domain. That is: $$ \forall x\in\operatorname{Dom}(\mathcal F_{\mathbb P}), $$ there exists a unique: $$ y=\mathcal F_{\mathbb P}(x). $$ --- ## C2 — Global / Local Restriction Certificate We must prove: $$ \mathcal F_{\mathbb P} \big|_{\mathcal B_n} = \mathcal F_{\mathbb P,n} $$ rather than merely showing approximate numerical agreement on a finite sample. --- ## C3 — Cell Internality Certificate We must prove: $$ n^2 < \mathcal F_{\mathbb P}(\mathcal B_n) < (n+1)^2. $$ for all: $$ n\ge1 $$ these indices. --- ## C4 — Primality Certificate We must prove: $$ \mathcal F_{\mathbb P}(\mathcal B_n)\in\mathbb P. $$ If this step depends on an independent primality test, we must explicitly acknowledge that: $$ \mathcal F_{\mathbb P} $$ remains a ‘locator + verifier’ combination, not a pure generation operator. --- ## C5 — Noncircularity Certificate If the definition of $\mathcal F_{\mathbb P}$ itself depends on ‘the known next prime’ or equivalent data, it cannot be claimed to have independently located a prime. We must therefore audit whether: $$ \mathcal F_{\mathbb P} $$ uses: $$ p_{N+1} $$ or an equivalent oracle. --- ## C6 — Piecewise Unification Certificate If: $$ \varepsilon_{\mathrm{full}} $$ uses: $$ \varepsilon_{\mathrm{table}}, \quad \varepsilon_{\mathrm{mid}}, \quad \varepsilon_{\mathrm{asy}}, $$ we must prove that the transition points: $$ N=1000, \qquad N=5\times10^6 $$ do not cause incompatibility, jumps, or certificate failure. --- ## C7 — Error-Envelope Certificate If we use: $$ E_+(N), $$ we must prove it is a genuine upper bound, not the maximum observed error on a training set, validation set, or finite data. --- ## C8 — Global-Quantifier Certificate Finite verification: $$ n\le N_{\max} $$ cannot replace: $$ \forall n. $$ At least one of the following must be provided: - Analytic closure; - Monotonicity; - Inductive propagation; - Invariants; - Conjugacy invariance; - Completeness of generators; - Global error bounds; to carry the infinite quantifier. --- # 22. Failure Conditions This program is falsifiable. Any of the following results would directly force a revision of the framework. ### F1 There exists some: $$ n $$ such that: $$ \mathcal F_{\mathbb P}(\mathcal B_n) \notin \mathcal B_n. $$ ### F2 There exists: $$ n $$ such that: $$ \mathcal F_{\mathbb P}(\mathcal B_n) $$ is composite. ### F3 The local operator requires refitting with $n$-specific parameters, and there is no uniform restriction law. ### F4 After normalization: $$ \widetilde{\mathcal F}_{\mathbb P,n} $$ has no common structure across different $n$. ### F5 Intermediate-scale fitting error cannot be elevated to a provable upper bound. ### F6 The operator actually uses target-prime information covertly. ### F7 The square-cell safety margin: $$ M_n $$ cannot be proved globally positive. --- # 23. Differences from the Traditional Formulation of Legendre's Conjecture The traditional route usually gives: $$ \pi((n+1)^2)-\pi(n^2) $$ a direct lower bound. The route in this paper is: $$ \boxed{ \begin{aligned} &\text{Square interval}\\ &\downarrow\\ &\text{Continuous square base space}\\ &\downarrow\\ &\text{Prime-fitting operator}\\ &\downarrow\\ &\text{Prime-fitting base space}\\ &\downarrow\\ &\text{Local / global compatibility}\\ &\downarrow\\ &\text{Cell internality}\\ &\downarrow\\ &\text{Primality certificate}\\ &\downarrow\\ &\text{Legendre}. \end{aligned} } $$ This does not make the original problem easy; it changes the mathematical layer where the proof obligations lie. --- # 24. The Core Philosophy of This Program The original Legendre proposition has the: $$ \forall n\,\exists p $$ form. When handled directly, each $n$ seems like a new existence obligation. But if a global operator can be found: $$ \mathcal F_{\mathbb P} $$ such that: $$ \mathcal F_{\mathbb P}(\mathcal B_n) \in \mathbb P\cap\mathcal B_n $$ holds for every $n$, then: $$ \exists p $$ no longer means searching each cell anew. It is lowered to: $$ \boxed{ \text{operator closure}. } $$ The quantifier structure is thus rewritten from: $$ \forall n\,\exists p $$ to: $$ \exists \mathcal F_{\mathbb P} \quad \forall n: \quad \mathcal F_{\mathbb P}(\mathcal B_n) \in \mathbb P\cap\mathcal B_n. $$ This is the most important logical reformulation of the program. --- # 25. Quantifier Descent The original proposition: $$ \forall n \quad \exists p_n. $$ After operatorization: $$ \exists \mathcal F_{\mathbb P} \quad \forall n. $$ where: $$ p_n = \mathcal F_{\mathbb P}(\mathcal B_n). $$ If: $$ \mathcal F_{\mathbb P} $$ can be completely described by finite rules, many local existential quantifiers are compressed into one global construction obligation. This can be viewed as: ## Existential Quantifier Descent That is: $$ \boxed{ \forall n\,\exists p_n \longrightarrow \exists F\,\forall n. } $$ This reformulation does not itself guarantee that the problem can be proved, but if it succeeds, the focus of proof shifts from: $$ \text{infinitely many independent witnesses} $$ to: $$ \text{one global witness generator}. $$ --- # 26. The Order of Research in the Next Stage The most reasonable subsequent sequence for this program is not to declare a proof immediately, but: ## Phase 1 — Canonical Base Space Formally fix: $$ \mathcal B_{\square}, \qquad \mathcal B_n, \qquad \phi_n. $$ Prove all basic affine and scale properties. --- ## Phase 2 — Operator Extraction Extract from the existing: $$ \varepsilon_{\mathrm{full}} $$ and the trinity predictor: $$ \mathcal F_{\mathbb P}^{(0)}. $$ Rewrite the ‘program flow’ as a genuine mathematical operator. --- ## Phase 3 — Circularity Audit Check whether: $$ \mathcal F_{\mathbb P}^{(0)} $$ depends anywhere, directly or indirectly, on the target prime. --- ## Phase 4 — Standardized Local Operator Compute: $$ \widetilde{\mathcal F}_{\mathbb P,n} = \phi_n^{-1} \circ \mathcal F_{\mathbb P,n} \circ \phi_n. $$ Observe whether a common local kernel exists. --- ## Phase 5 — Local–Global Compatibility Attempt to prove: $$ \mathcal F_{\mathbb P} \big|_{\mathcal B_n} = \mathcal F_{\mathbb P,n}. $$ --- ## Phase 6 — Certified Error Envelope Establish: $$ E_+(N). $$ --- ## Phase 7 — Square-Cell Margin Define: $$ M_n = (n+1)^2 - \left( \widehat p_{k+1} + E_+(k+1) \right). $$ Study: $$ M_n>0. $$ --- ## Phase 8 — Prime Certificate Integration Integrate: $$ \mathcal C_{\mathbb P} $$ into the operator, distinguishing: $$ \text{fit-only} $$ from: $$ \text{certificate-carrying}. $$ --- ## Phase 9 — Global Closure Only at the end study: $$ \forall n. $$ --- # 27. The Final Candidate Framework This program can be condensed to: $$ \boxed{ \mathcal L = \left( \mathcal B_{\square}, \{\mathcal B_n\}, \mathcal F_{\mathbb P}, \mathcal B_{\mathrm{PF}}, \{\rho_n\}, \mathcal C_{\mathbb P} \right). } $$ where: $$ \mathcal B_{\square} $$ is the global square base space; $$ \mathcal B_n $$ is a local square cell; $$ \mathcal F_{\mathbb P} $$ is the prime-fitting operator; $$ \mathcal B_{\mathrm{PF}} $$ is the prime-fitting base space; $$ \rho_n $$ is the local restriction map; $$ \mathcal C_{\mathbb P} $$ is the primality certificate. The final closure required for Legendre's conjecture is: $$ \boxed{ \forall n\ge1: \quad \mathcal C_{\mathbb P} \left( \mathcal F_{\mathbb P}(\mathcal B_n) \right)=1 } $$ and: $$ \boxed{ \mathcal F_{\mathbb P}(\mathcal B_n) \in \mathcal B_n. } $$ --- # 28. The Shortest Proposition If only the core sentence of the entire paper is retained: > Legendre's conjecture can be reformulated as follows: does there exist a global prime-fitting operator whose restriction to every continuous-square local base space remains the same operator, and whose local image always stays inside that square cell and has primality? Formally: $$ \boxed{ \exists\mathcal F_{\mathbb P} \quad \forall n\ge1: \quad \mathcal F_{\mathbb P} \big|_{\mathcal B_n} = \mathcal F_{\mathbb P,n}, } $$ and: $$ \boxed{ \mathcal F_{\mathbb P}(\mathcal B_n) \in \mathbb P \cap (n^2,(n+1)^2). } $$ If this proposition holds, Legendre's conjecture follows immediately. --- # 29. Conclusion This paper does not claim to have completed Legendre's conjecture. What it completes is another, more fundamental task: reorganizing the original: $$ \forall n\,\exists p $$ short-interval existence proposition into a: $$ \exists\mathcal F_{\mathbb P}\,\forall n $$ global generative-operator proposition. This transformation moves the research objective from: > Where exactly should we look for a prime in each square interval? to: > Can we prove that the same prime-fitting operator retains its ontological identity on the global square base space and every local square cell, and that its local output can never escape that cell or the set of primes? If successful, Legendre's conjecture is no longer an infinite existence problem requiring interval-by-interval treatment, but becomes: $$ \boxed{ \text{An operator closure theorem on a global base space.} } $$ This is precisely the core of subsequent research in this program. --- ## Status Markers ```text Legendre Base-Space Prime-Fitting Program v0.1 Base-space reformulation: ESTABLISHED AS FRAMEWORK Local square-cell model: DEFINED Global square base space: DEFINED Prime-fitting operator: CANDIDATE / TO FORMALIZE Prime-fitting base space: DEFINED Local-global compatibility: OPEN Square-cell internality: OPEN Prime-image certificate: OPEN Certified error envelope: OPEN Global closure: OPEN Legendre proof status: NOT PROVEN ```