# From Transient Axioms to Backfillable Bridges: Result-Induced Intermediate Proposition Reconstruction in the Case of the Riemann Hypothesis ## ——A Non-Proving Reverse Structural Design Experiment **Author: Neo.K (Theoretical Conception) / Aletheia (Collaborative Organization and Formalization)** **Version: v1.0 (Methodological Preliminary Case Draft)** **Date: 2026-07-09** --- ## Abstract This article originates from a seemingly joking artificial intelligence generation event: after being asked to "invent a new number theory to prove the Riemann Hypothesis," a large language model generated several self-created axioms with semantics such as "prime field symmetry," "dimensional folding," "zero resonance," and "critical strip closure," and quickly deduced that all non-trivial zeros lie on the critical line. This generation is obviously invalid as a rigorous mathematical proof: it contains incorrect functional equations, jumps from set symmetry to pointwise fixation, undefined terms, conclusion smuggling, and circularity risks. However, if these "axioms" are not viewed as permanent premises, but reinterpreted as **transient bridge propositions** reverse-generated from the target result, the problem fundamentally changes: the real research task is no longer to accept the axioms, but to downgrade them one by one into proof obligations, and seek existing theorems, new lemmas, operator constructions, positivity conditions, or local-global decompositions to backfill them. The purpose of this article is neither to prove the Riemann Hypothesis nor to claim that the original AI-generated axioms are correct. This article only completes the first phase of work: using the Riemann Hypothesis as a case study, it establishes an intermediate proposition system that is more precise, semantically controlled, falsifiable, backfillable, and circularity-checkable than the original generation. To avoid the two failures of "semantic over-breadth leading to candidate explosion" and "semantic over-narrowness leading to premature deletion of true paths," this article proposes a **mesoscopic semantic window**, a dual-layer architecture of **hard anchor layer—bridge proposition layer**, and a finite semantic universe centered on the completed Riemann function, the multiset of zeros, involution symmetry, test function spaces, Weil-type quadratic forms, and structured generating families. This article proposes three hard anchors and six candidate bridge propositions. The three hard anchors respectively handle: analytic and symmetric structures, the prime-zero coupling provided by explicit formulas, and the counterexample witnesses provided by positivity criteria. The six candidate bridge propositions handle: symmetric orbit observability, defect quantifiability, structured compression of negative witnesses, arithmetic decomposition on generating families, local-global compensated positivity, and closure transmission under quadratic form topologies. This article specifically points out: if a Weil-type positivity criterion is adopted, then "off-axis zeros lead to the existence of a negative witness" should no longer be masqueraded as a new axiom; the truly unknown and potentially research-valuable part is whether any negative witness can be compressed into a controllable generating family, and non-negativity established through provable arithmetic decomposition on that generating family. Finally, this article proposes a crucial judgment: the failure of the original method may not simply be due to too many candidate solutions, nor simply due to incomplete axioms, but both are co-produced by the same defect—the lack of intermediate structures that can control search convergence. When exclusion mechanisms are not formalized, function spaces are not fixed, generating families are not restricted, and continuity is not specified, every natural language axiom unfolds into a large number of mutually inequivalent formalized candidates; candidate explosion thus becomes the computational manifestation of axiom incompleteness. Therefore, this article positions this case as the first preliminary paper for the subsequent "Result-Induced Intermediate Theorem Generation Method" and "Reverse Axiom Backfilling Method," rather than a proof draft of the Riemann Hypothesis. **Keywords:** Riemann Hypothesis, reverse axiom backfilling, result induction, intermediate theorems, Weil positivity, explicit formulas, semantic width, candidate explosion, proof search, AI mathematical reasoning --- # 1. Origin of the Problem: Preserving Unformalized Structures from an Erroneous Proof ## 1.1 The Original Event Consider the following interaction mode. A user makes a clear but extreme request: > Do not use existing number theory frameworks; invent a new number theory and prove the Riemann Hypothesis. The model subsequently generated several new axioms, for example: 1. Prime field symmetry; 2. Dimensional folding; 3. Zero resonance; 4. Reciprocal conjugation; 5. Critical strip closure. It then jumped from "if $\rho=\sigma+it$ is a zero, then its symmetric position is also a zero" to: $$ \sigma=1-\sigma, $$ therefore: $$ \sigma=\frac12. $$ As a proof, this reasoning fails. The most direct reasons include: - Writing an incorrect functional equation as $\zeta(s)=\zeta(1-s)$; - Erroneously deducing that every zero must be a symmetric fixed point from the symmetry of the zero set; - Using mathematically undefined terms; - Placing content close to the conclusion itself into "axioms"; - Failing to prove the inevitable relationship between the so-called "topological collapse" or "resonance" and the locations of the zeros. However, this article refuses to adopt another equally crude treatment: treating all generated structures as having zero value just because the proof is flawed. What is truly worth preserving is the following operation: $$ \text{Target Result} \rightsquigarrow \text{A set of candidate intermediate structures sufficient to make the result hold}. $$ Here, the symbol: $$ \rightsquigarrow $$ does not denote logical implication, but represents "generation reverse-induced by the target." In other words, what the model performed was not a valid proof, but potentially an uncontrolled **reverse structural design**. --- ## 1.2 Axioms are Not Endpoints, but Transient Nodes The original error can be written as: $$ A_1\land A_2\land\cdots\land A_n \Rightarrow P, $$ where $P$ is the target proposition, such as the Riemann Hypothesis. Traditional critiques usually stop at: $$ A_i\text{ unproven}, $$ thus the whole is invalid. But another approach is to rewrite the role of each $A_i$: $$ A_i: \text{Permanent Axiom} \quad\longrightarrow\quad A_i: \text{Transient Bridge Proposition}. $$ At this point, the research task becomes: $$ T_{i1},T_{i2},\dots,T_{im} \Rightarrow A_i, $$ ultimately hoping to form: $$ \{T_j\} \Rightarrow \{A_i\} \Rightarrow P. $$ If all $T_j$ can be proven without using $P$, then the original "cheating axioms" might be backfilled into genuine proof nodes. Therefore, the basic stance of this article is not: > Self-created axioms can prove anything. But rather: > If self-created axioms are viewed as transient intermediate nodes, they can become tools for reverse-generating proof obligations; their validity depends on whether non-circular backfilling can be subsequently completed. --- # 2. Research Boundaries: This Article is Not a Proof of the Riemann Hypothesis ## 2.1 Target Proposition Let: $$ \zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s}, \qquad \Re(s)>1, $$ and be analytically continued to a meromorphic function on the complex plane except at $s=1$. Define the completed Riemann function: $$ \xi(s) = \frac12 s(s-1)\pi^{-s/2} \Gamma\!\left(\frac{s}{2}\right)\zeta(s). $$ The Riemann Hypothesis can be stated as: $$ \forall \rho\in\mathcal Z, \qquad \Re(\rho)=\frac12, $$ where $\mathcal Z$ denotes the multiset of non-trivial zeros of $\xi$. As of the version date of this article, this problem remains unsolved. --- ## 2.2 This Article Only Performs Conditional Structural Design This article does not claim: $$ RH. $$ This article only investigates whether a set of intermediate propositions can be constructed: $$ B_1,\dots,B_m, $$ such that: $$ B_1\land\cdots\land B_m \Rightarrow RH, $$ and each $B_i$ satisfies: 1. Clear semantics; 2. Falsifiability; 3. Not directly equivalent to RH; 4. Not smuggling RH into definitions; 5. Having more than one but finitely controllable backfilling directions; 6. Attackable by existing mathematical tools. Therefore, if the results of this article hold, they are merely: $$ \text{A better proof search architecture}, $$ and not: $$ \text{A proof of the Riemann Hypothesis}. $$ --- # 3. Why the Original Five Axioms Failed ## 3.1 Semantic Explosion of "Prime Field Symmetry" Natural language: > All prime numbers constitute a global topological field. Could at least be formalized as: - A graph with primes as vertices; - A measure weighted by prime powers; - The local factor space of the Euler product; - Adèle or idèle class spaces; - The prime-side distribution in explicit formulas; - A certain spectral triple; - The local term decomposition of some kernel operator; - The fundamental group or homological data of an unknown topological space. Therefore: $$ A_{\text{prime-field}} \rightsquigarrow \left\{ A^{(1)},A^{(2)},\dots,A^{(N)} \right\}. $$ When $N$ is very large, the so-called axiom is not a candidate proposition, but an unrestricted search universe. --- ## 3.2 "Dimensional Folding" Specifies No Operator If one says: > The real part is folded to the central axis. Then one must at least answer: - Which space? - Which mapping? - Is it linear? - Is it continuous? - Is it a projection? - Is it a quotient mapping? - Is it the fixed point set of an involution? - Does it preserve zero multiplicity? - Is it compatible with the functional equation of $\xi$? Without the above content, then: $$ \text{folding} $$ is merely a metaphor, not a mathematical operation. --- ## 3.3 "Zero Resonance" Confuses Zero Values and Phases For a complex function: $$ \zeta(s)=0 $$ should not be roughly described as "modulus returning to zero plus argument returning to zero." At a zero, the argument itself requires more careful handling; substituting "dual resonance" for local zero order, analytic structure, or spectral conditions provides no valid theorem. --- ## 3.4 "Reciprocal Conjugation" Miswrites the Functional Equation The correct core symmetry should primarily be placed on the completed function: $$ \xi(s)=\xi(1-s), $$ and combined with: $$ \xi(\overline{s}) = \overline{\xi(s)}. $$ If $\rho$ is a zero, then the related symmetric positions form a closure. This is not equivalent to: $$ \rho=1-\overline{\rho}. $$ Set invariance: $$ J(\mathcal Z)=\mathcal Z $$ does not imply pointwise fixation: $$ J(\rho)=\rho. $$ --- ## 3.5 "Critical Strip Closure" Smuggles the Target If the axiom directly states: > All zeros must fall on the axis of symmetry. Then: $$ A_5\approx RH. $$ At this point: $$ A_5\Rightarrow RH $$ does not reduce any proof burden. This exposes the first major risk of the reverse generation method: > The closer an intermediate proposition is to the target, the easier it is to semantically just rename the target. --- # 4. Dual-Layer Architecture: Hard Anchors and Candidate Bridges This article proposes: $$ \mathfrak S = \mathfrak H \cup \mathfrak B, $$ where: $$ \mathfrak H = \{H_0,H_1,H_2\} $$ is the hard anchor layer, $$ \mathfrak B = \{B_1,\dots,B_6\} $$ is the candidate bridge layer. The hard anchor layer only carries known structures or explicitly adopted existing criteria. The candidate bridge layer is: $$ \text{proof obligations}. $$ --- # 5. Domain Reduction of the Semantic Universe ## 5.1 Finite Object Classes The first version of this article only allows the following core objects: $$ \mathfrak U_{\mathrm{RH}} = \{ \xi,\, \mathcal Z,\, J,\, \mathcal H,\, Q,\, \mathcal G \}. $$ where: - $\xi$: the completed Riemann function; - $\mathcal Z$: the multiset of non-trivial zeros; - $J$: the involution mapping; - $\mathcal H$: the space of admissible test functions; - $Q$: the quadratic form induced by explicit formulas or Weil-type criteria; - $\mathcal G$: the structured generating family to be constructed. This does not mean other mathematical objects are permanently forbidden. Rather, in the first phase, all new terms must ultimately map back to: $$ \mathfrak U_{\mathrm{RH}} $$ or explicitly state the necessity of adding a seventh class of objects. --- ## 5.2 Mesoscopic Semantic Window Suppose an intermediate proposition $M_i$ has $N_i$ reasonable formalized candidates. Define the rough semantic width: $$ W_{\mathrm{sem}} = \sum_i \log N_i. $$ The rough combinatorial search space is: $$ |\Omega| \approx \prod_iN_i. $$ If: $$ W_{\mathrm{sem}}\gg1, $$ then: $$ |\Omega| $$ experiences candidate explosion. But if: $$ W_{\mathrm{sem}}\to0, $$ it may cause, due to overly strong presuppositions: $$ \Omega \cap \Omega_{\mathrm{true}} = \varnothing. $$ Therefore, the ideal state is not the minimal semantic domain, but: $$ W_{\min} < W_{\mathrm{sem}} < W_{\max}. $$ This article calls this: > **Mesoscopic Semantic Window** or: > **Mesoscopic Semantic Window**. --- # 6. Hard Anchor $H_0$: Analytic-Symmetric Structure Define: $$ J(s) = 1-\overline{s}. $$ Then: $$ J^2(s)=s. $$ For any: $$ \rho=\sigma+it, $$ we have: $$ J(\rho) = 1-\sigma+it. $$ The fixed point condition: $$ J(\rho)=\rho $$ is equivalent to: $$ \sigma=\frac12. $$ Define the off-axis defect: $$ \delta(\rho) = \left| \Re(\rho)-\frac12 \right|. $$ Therefore: $$ \delta(\rho)=0 \iff J(\rho)=\rho. $$ The hard anchor only gives: $$ J(\mathcal Z)=\mathcal Z, $$ it does not give: $$ \forall \rho\in\mathcal Z,\quad J(\rho)=\rho. $$ This distinction must be permanently preserved. --- # 7. Hard Anchor $H_1$: Prime-Zero Coupling of Explicit Formulas Riemann-Weil type explicit formulas connect the zero side with the prime/prime-power side. This article does not fix a unique normalization, but abstractly represents it as: $$ \mathcal Q_{\mathcal Z}(f) = \mathcal Q_{\mathcal P}(f) + \mathcal Q_\infty(f), \qquad f\in\mathcal H. $$ where: - $\mathcal Q_{\mathcal Z}$: the zero side; - $\mathcal Q_{\mathcal P}$: the prime or prime-power local terms; - $\mathcal Q_\infty$: the Gamma factor and infinite place terms. The important methodology here is not the specific constants of a certain formula, but: $$ \text{Zero Geometry} \leftrightarrow \text{Arithmetic Local Data}. $$ The original "prime field" semantics are thus replaced by a checkable system of local terms. --- # 8. Hard Anchor $H_2$: Positivity Criteria and Negative Witnesses When adopting a Weil-type positivity criterion on an appropriate function class, it can be abstractly written as: $$ RH \iff \forall f\in\mathcal H, \quad Q(f)\ge0. $$ Therefore: $$ \neg RH \Rightarrow \exists f\in\mathcal H, \quad Q(f)<0. $$ This article calls such an $f$ a: > **Negative Witness** denoted as: $$ w\in\mathcal W_-, $$ where: $$ \mathcal W_- = \{ f\in\mathcal H:Q(f)<0 \}. $$ There is an important correction here: **"The existence of off-axis zeros leads to the existence of a negative witness" is no longer listed as a new axiom.** If the function space and positivity criterion are fixed, this is merely the contrapositive result of an existing equivalent structure. What truly requires new methods is: 1. Can negative witnesses be structured? 2. Can they be restricted to a computable generating family? 3. Can $Q$ on the generating family be decomposed into provably positive local terms? 4. Can positivity be stably transmitted to the closure? --- # 9. Candidate Bridge $B_1$: Symmetric Orbit Observability For: $$ \rho\in\mathcal Z, $$ define the complete symmetric orbit: $$ \mathcal O(\rho) = \{ \rho,\, \overline{\rho},\, 1-\rho,\, 1-\overline{\rho} \}. $$ Candidate proposition $B_1$: > If $\delta(\rho)>0$, then there exists some admissible test $f\in\mathcal H$ such that this off-axis orbit produces a non-zero distinguishable contribution to $Q$ or its zero-side decomposition. Formalization template: $$ \delta(\rho)>0 \Rightarrow \exists f\in\mathcal H: \mathfrak D_f(\mathcal O(\rho))\neq0. $$ where $\mathfrak D_f$ is not allowed to be defined arbitrarily, but must be derived from: $$ Q,\quad \mathcal Q_{\mathcal Z}, \quad\text{or explicit formulas}. $$ ### Significance $B_1$ does not claim that off-axis zeros inevitably lead to a contradiction. It only excludes: $$ \text{Off-axis orbits being completely invisible to all admissible tests}. $$ ### Falsifiability Conditions If there exists: $$ \rho\in\mathcal Z, \quad \delta(\rho)>0, $$ and: $$ \forall f\in\mathcal H, \quad \mathfrak D_f(\mathcal O(\rho))=0, $$ then $B_1$ fails. --- # 10. Candidate Bridge $B_2$: Defect Quantifiability Candidate proposition $B_2$: There exists a defect functional naturally derived from established objects: $$ \mathfrak E: \mathcal Z/J \rightarrow \mathbb R_{\ge0}, $$ satisfying: $$ \mathfrak E([\rho])=0 \iff \delta(\rho)=0. $$ where: $$ [\rho] $$ denotes the quotient class of the $J$-symmetric orbit. ### Difference from the Original "Dimensional Folding" The original statement was: > All points are forcibly folded to the central axis. The new proposition only requires: > Centrally fixed orbits and off-axis orbits can be distinguished by a natural invariant. Therefore: $$ \mathfrak E>0 $$ does not automatically lead to a contradiction. It merely establishes a measurable defect. ### Candidate Implementations Allowed candidates include: - Spectral defects; - Some quadratic form distance; - Involution non-fixation degree; - Kernel operator asymmetry measure; - Orbit contribution differences derived from explicit formulas. But each class must explicitly provide: $$ \mathfrak E([\rho]). $$ --- # 11. Candidate Bridge $B_3$: Structured Compression of Negative Witnesses This is the first truly core new bridge of this article. Hard anchor $H_2$ only gives: $$ \neg RH \Rightarrow \exists w\in\mathcal H: Q(w)<0. $$ But $\mathcal H$ might be too large, and negative witnesses might be too irregular. Candidate proposition $B_3$: There exists a structured generating family: $$ \mathcal G = \bigcup_{m\ge1}\mathcal G_m \subseteq \mathcal H, $$ such that any negative witness can be compressed into a negative witness of finite complexity or within a controllable limit. Strong version: $$ Q(w)<0 \Rightarrow \exists m,\ \exists g\in\mathcal G_m: Q(g)<0. $$ Weak version: $$ Q(w)<0 \Rightarrow \exists \{g_n\}\subset\operatorname{span}(\mathcal G) $$ such that: $$ g_n\to w $$ and: $$ Q(g_n)\to Q(w)<0. $$ Therefore, there exists a sufficiently large $n$: $$ Q(g_n)<0. $$ ### Why It Is Important If $B_3$ holds, counterexample search does not need to traverse the entire $\mathcal H$. It is compressed to: $$ \mathcal G. $$ Therefore: $$ \text{Infinite Function Space Search} \longrightarrow \text{Structured Generative Search}. $$ --- # 12. Candidate Bridge $B_4$: Arithmetic Decomposability on Generating Families Candidate proposition $B_4$: For each: $$ g\in\mathcal G_m, $$ there exists an explicit decomposition: $$ Q(g) = L_\infty(g) + \sum_{p\le P(m)}L_p(g) + R_m(g), $$ or more generally: $$ Q(g) = L_\infty(g) + \sum_pL_p(g) + R(g), $$ where: - $L_\infty(g)$: infinite place / analytic term; - $L_p(g)$: local arithmetic term for prime $p$; - $R_m(g)$: truncation, interaction, or approximation remainder term. Requirements: 1. Each term is explicitly defined; 2. The mode of series convergence is explicit; 3. $R_m$ has a controllable bound; 4. The decomposition does not use RH; 5. The parameters and local terms of $g$ are computable or estimable. ### Difference from the Original "Prime Field" It no longer says: > Primes form a global topological field. But rather requires: $$ \text{Global Quadratic Form} = \text{Infinite Term} + \text{Prime Local Terms} + \text{Controllable Remainder}. $$ --- # 13. Candidate Bridge $B_5$: Local-Global Compensated Positivity Candidate proposition $B_5$: For all: $$ g\in\mathcal G, $$ it is provable that: $$ L_\infty(g) + \sum_pL_p(g) + R(g) \ge0. $$ That is: $$ Q(g)\ge0. $$ But to avoid smuggling in the entirety of RH, $B_5$ must satisfy: - It is only proven for $\mathcal G$; - The definition of $\mathcal G$ is not conditioned on $Q(g)\ge0$; - It must not be defined as: $$ \mathcal G = \{g:Q(g)\ge0\}; $$ - Positivity must come from independent local estimates, pairwise cancellation, operator positivity, convexity, monotonicity, or other provable mechanisms. ### Core Problem The original model stated: > Off-axis zeros will cause the prime field to collapse. This article replaces it with a truly attackable problem: > Does there exist a natural generating family such that negative local contributions in its explicit formula decomposition are necessarily compensated by other places? Namely: $$ \sum_{p\in\mathcal N(g)}|L_p^-(g)| \le L_\infty^+(g) + \sum_{p\in\mathcal P(g)}L_p^+(g) + R^+(g). $$ This is the verifiable "global compatibility." --- # 14. Candidate Bridge $B_6$: Closure Transmission under Quadratic Form Topology Candidate proposition $B_6$: If: $$ g_n\to f $$ in an appropriate topology, then: $$ Q(g_n)\to Q(f). $$ More precisely, it requires: $$ \overline{\operatorname{span}(\mathcal G)}^{\,\tau_Q} = \mathcal H, $$ where $\tau_Q$ is a topology sufficient to guarantee that $Q$ is continuous or lower semi-continuous. A strong form can be adopted: $$ g_n\xrightarrow{\tau_Q}f \Rightarrow Q(g_n)\to Q(f), $$ or a weak form: $$ g_n\xrightarrow{\tau_Q}f \Rightarrow Q(f) \ge \liminf_{n\to\infty}Q(g_n). $$ If for all $n$: $$ Q(g_n)\ge0, $$ then the weak form is sufficient to deduce: $$ Q(f)\ge0. $$ ### Difference from the Original "Critical Strip Closure" The original statement directly confined the zeros to the critical line. This article changes it to: > Can positivity be transmitted along a specified topology from a structured generating family to the complete test space? This is a genuine functional analysis problem. --- # 15. Conditional Derivation Skeleton Assume: $$ H_0,H_1,H_2 $$ hold, and the candidate bridges: $$ B_3,B_4,B_5,B_6 $$ are all completely backfilled. From $B_5$: $$ \forall g\in\mathcal G, \quad Q(g)\ge0. $$ From $B_6$: $$ \forall f\in\mathcal H, \quad \exists g_n\in\operatorname{span}(\mathcal G): g_n\xrightarrow{\tau_Q}f. $$ Then by the continuity or lower semi-continuity of $Q$: $$ Q(f)\ge0. $$ Therefore: $$ \forall f\in\mathcal H, \quad Q(f)\ge0. $$ From hard anchor $H_2$: $$ RH. $$ Alternatively, using a proof by contradiction chain: Assume: $$ \neg RH. $$ From $H_2$: $$ \exists w\in\mathcal H: Q(w)<0. $$ From $B_3$: $$ \exists g\in\mathcal G $$ or there exists a sequence $g_n$, ultimately satisfying: $$ Q(g)<0. $$ But from $B_4+B_5$: $$ \forall g\in\mathcal G, \quad Q(g)\ge0. $$ Contradiction. Hence: $$ RH. $$ --- # 16. Why This is Still Not a Proof Because at least the following parts have not yet been completed: $$ B_3,\quad B_4,\quad B_5,\quad B_6. $$ Any one of these could be: - False; - Valid only on an overly small $\mathcal G$; - Equivalent to RH; - Bearing a proof burden no lower than RH; - Incompatible with other bridges. Therefore, this article only establishes: $$ (B_3\land B_4\land B_5\land B_6) \Rightarrow RH. $$ More precisely, if $H_2$ adopts a known equivalent criterion, then this article is designing: $$ \text{A candidate backfilling path towards that positivity criterion}. $$ --- # 17. Non-Circularity Check of Intermediate Propositions ## 17.1 Direct Equivalence Risk If: $$ B_i\iff RH, $$ then: $$ B_i $$ is merely a renaming. Therefore, it is required that each bridge is accompanied by: $$ \operatorname{Dep}(B_i), $$ indicating its proof dependencies. If: $$ RH\in\operatorname{Dep}(B_i), $$ then the backfilling is invalid. --- ## 17.2 Implicit Usage Risk Even if the paper does not write: $$ RH, $$ it might use estimates that only hold under RH. Therefore, a dependency graph must be established: $$ \mathcal D = (V,E), $$ where: - $V$: theorems, lemmas, estimates, definitions; - $E$: proof dependencies. Requirement: $$ P\notin\operatorname{Anc}(B_i), $$ where $P=RH$. --- ## 17.3 Definition Smuggling Risk Prohibited: $$ \mathcal G = \{g\in\mathcal H:Q(g)\ge0\}. $$ Because in this case, $B_5$ becomes true by definition. Similarly prohibited: $$ \mathfrak E(\rho)=0 \iff \Re(\rho)=\frac12 $$ if $\mathfrak E$ merely repackages the right side without an independent construction. --- # 18. Candidate Explosion and Axiom Incompleteness May Be the Same Problem The original reflection proposed two possibilities: 1. Too many candidate solutions; 2. The original axioms are incomplete. This article proposes: > The two might not be competing explanations, but two facets of the same defect. Consider the original statement: > Off-axis zeros cause topological collapse. If it does not specify: - Space; - Topology; - Defect measure; - Test functions; - Acting operators; - Observables; then the formalized candidates might be: $$ \Omega = \{ \text{Homological Change}, \text{Index Change}, \text{Non-real Spectrum}, \text{Positivity Destruction}, \text{Kernel Singularity}, \text{Metric Degeneration}, \dots \}. $$ That is: $$ \text{Missing Nodes} \Rightarrow \text{Formalization Bifurcation}. $$ Therefore: $$ \boxed{ \text{Axiom Incompleteness} \Longrightarrow \text{Candidate Explosion} } $$ may hold in many cases. Conversely, candidate explosion makes it impossible for researchers to judge which missing node truly needs to be filled, therefore: $$ \boxed{ \text{Candidate Explosion} \Longrightarrow \text{Incompleteness is Hard to Locate} } $$ forming a feedback loop: $$ \text{Missing Nodes} \rightarrow \text{Increased Bifurcation} \rightarrow \text{Search Loses Focus} \rightarrow \text{Missing Nodes Harder to Identify}. $$ --- # 19. Design Principles for Semantic Domain Reduction ## Principle 1: Fix Parent Objects, Do Not Fix a Unique Technique For example, fix: $$ Q,\mathcal H,\mathcal G $$ the three classes of roles. But do not pre-specify that: $$ \mathcal G $$ must be a certain type of basis. This preserves multiple candidates. --- ## Principle 2: Fix Quantifiers Natural language: > Off-axis zeros will be seen. Changed to: $$ \forall \rho\in\mathcal Z, \quad \delta(\rho)>0 \Rightarrow \exists f\in\mathcal H: \mathfrak D_f(\rho)\neq0. $$ Once quantifiers are fixed, the search domain shrinks drastically. --- ## Principle 3: Fix Failure Witnesses Every proposition must answer: > What observation would make it fail? For example, the failure witness for $B_5$ is: $$ \exists g\in\mathcal G: Q(g)<0. $$ --- ## Principle 4: Allow Modular Replacement If a certain decomposition for $B_4$ fails, it does not mean the entire framework dies. One can: $$ B_4^{(a)} \rightarrow B_4^{(b)} \rightarrow B_4^{(c)}. $$ But each version must still occupy the same semantic role: > Arithmetic decomposability on generating families. --- ## Principle 5: Prohibit Infinite Free Addition of Terms If a new object $X$ is added, one must explain: $$ X $$ which gap, unexpressible by existing bridges, it solves. --- # 20. Mapping Between the Original Five Axioms and the New System | Original Statement | Original Problem | New Correspondence | |---|---|---| | Prime field symmetry | "Field" undefined | $H_1+B_4$: Explicit formulas and local term decomposition | | Dimensional folding | No operator, no space | $B_2$: Defect functional | | Zero resonance | Chaotic local zero structure | $B_1$: Orbit observability | | Reciprocal conjugation | Functional equation miswritten | $H_0$: Completed function and involution | | Critical strip closure | Nearly smuggling RH | $B_6$: Closure transmission of positivity | | "Topological collapse" | No exclusion mechanism | $H_2+B_3+B_5$: Negative witness compression and generating family positivity | It can be seen that the new system is not simply replacing five sentences with five prettier sentences. Instead, it dismantles the original semantics into: $$ \text{Known Parts} + \text{Truly Unknown Parts} + \text{Transmission Parts} + \text{Exclusion Parts}. $$ --- # 21. The Most Important New Judgment: What is Truly Unknown is Not "Whether Negative Witnesses Exist" If an appropriate Weil-type criterion is adopted, then: $$ \neg RH \Rightarrow \exists f\in\mathcal H: Q(f)<0 $$ is already provided by the equivalence relation. Therefore, further listing: > Off-axis zeros necessarily produce negative witnesses as a new axiom has limited research value. The true difficulty might be: $$ \boxed{ \mathcal W_- \neq\varnothing \Rightarrow \mathcal W_-\cap\mathcal G\neq\varnothing } $$ That is: > Can any negativity be seen within some structured, decomposable, and computable generating family? If the answer is yes, the counterexample space is compressed. Coupled with: $$ \forall g\in\mathcal G, \quad Q(g)\ge0, $$ this forms a genuine exclusion mechanism. --- # 22. Research Plan: How to Backfill Item by Item ## 22.1 Backfilling Directions for $B_1$ - Localized tests in explicit formulas; - Interpolation functions for specific orbits; - Paley–Wiener type function spaces; - Separability of kernel functions on the zero side. Target: $$ \delta(\rho)>0 \Rightarrow \exists f: \mathfrak D_f(\rho)\neq0. $$ --- ## 22.2 Backfilling Directions for $B_2$ - Fixed point defects of the involution $J$; - Natural distances on the quotient space; - Spectral offsets; - Positivity defects of kernel matrices. The target is not to prove RH, but to construct: $$ \mathfrak E([\rho]). $$ --- ## 22.3 Backfilling Directions for $B_3$ - Dense subspaces; - Countable bases; - Reproducing kernel expansions; - Band-limited test families; - Finite-parameter bump functions; - Mellin/Fourier controllable families. Core problem: $$ Q(w)<0 \Rightarrow \exists g\in\mathcal G: Q(g)<0? $$ --- ## 22.4 Backfilling Directions for $B_4$ - Explicit formulas; - Mellin transforms; - Fourier transforms; - Prime-power local terms; - Trace formula type decompositions. --- ## 22.5 Backfilling Directions for $B_5$ - Pairwise cancellation; - Local positive definite kernels; - Operator positivity; - Schur complements; - Convexity; - Chains of inequalities; - Scale recurrences. --- ## 22.6 Backfilling Directions for $B_6$ - Choosing a $Q$-norm; - Closable quadratic forms; - Lower semi-continuity; - Friedrichs extension type mechanisms; - Reproducing kernel Hilbert space closures. --- # 23. Failure Modes ## 23.1 Generating Family Too Small If: $$ \overline{\operatorname{span}(\mathcal G)}^{\,\tau_Q} \neq \mathcal H, $$ then generating family positivity is insufficient to deduce global positivity. --- ## 23.2 Generating Family Too Large If $\mathcal G$ is almost equal to $\mathcal H$, then: $$ B_5 $$ might be as difficult as RH. --- ## 23.3 $Q$ is Discontinuous Even if: $$ g_n\to f, $$ it does not guarantee: $$ Q(g_n)\to Q(f). $$ In this case, "dense" is not enough. --- ## 23.4 Remainder Uncontrollable If: $$ R_m(g) $$ has an uncontrollable sign and magnitude, the local positivity decomposition fails. --- ## 23.5 Candidate Modules Mutually Incompatible It is possible that: $$ B_3^{(a)} $$ requires a certain function class, but: $$ B_4^{(b)} $$ only holds in another function class. Therefore, one must check: $$ \operatorname{Dom}(B_3) \cap \operatorname{Dom}(B_4) \cap \operatorname{Dom}(B_5) \neq\varnothing. $$ --- # 24. Possible Roles of Computational Experiments This article does not advocate proving RH with finite computations. What computation can do is: ## 24.1 Stress Testing Candidate Generating Families Randomly or systematically generate: $$ g\in\mathcal G_m $$ and compute the approximation: $$ Q(g). $$ Search for: $$ Q(g)<0. $$ If found, it falsifies that version of $B_5$. --- ## 24.2 Semantic Width Estimation Record the number of candidates for each bridge: $$ N_i. $$ Estimate: $$ |\Omega| \approx \prod_iN_i. $$ --- ## 24.3 Backfill Graph Search Construct a directed hypergraph: $$ \mathfrak G_{\mathrm{proof}} = (V,E), $$ where: - Nodes: propositions; - Hyperedges: a set of premises deducing a conclusion. Search for: $$ \{T_j\} \Rightarrow B_i. $$ --- # 25. The Correct Role of Artificial Intelligence in This Method The case in this article specifically shows that large language models may simultaneously possess two opposing tendencies: ## 25.1 Generativity The model can rapidly propose: - New intermediate concepts; - Non-standard mappings; - Cross-domain analogies; - Candidate invariants; - Reverse sufficient conditions. --- ## 25.2 Self-Flattening When the model switches to "strict error-correction" mode, it might delete: $$ \text{Erroneous Proofs} $$ along with: $$ \text{Researchable Structures within Erroneous Proofs} $$ Therefore, a three-valued evaluation is needed: $$ \{\text{valid},\text{invalid},\text{structurally-interesting}\}. $$ Instead of a binary: $$ \{\text{correct},\text{garbage}\}. $$ --- # 26. Preliminary Methodological Propositions This article proposes the following non-theorematic propositions. ## Proposition A: Transient Axiom Downgrading Principle If a set of artificially generated premises: $$ A_1,\dots,A_n $$ is sufficient to deduce target $P$, they should not be directly accepted as axioms; they should first be rewritten as: $$ A_i\in\mathcal O_{\mathrm{proof}}, $$ where $\mathcal O_{\mathrm{proof}}$ is the set of proof obligations. --- ## Proposition B: Missing Node-Candidate Explosion Coupling If an intermediate proposition lacks an object domain, quantifiers, failure witnesses, and action mechanisms, its number of formalized candidates will grow non-trivially. Abstractly written as: $$ I(M)\downarrow \Rightarrow N(M)\uparrow, $$ where: - $I(M)$: proposition completeness; - $N(M)$: number of reasonable candidates. --- ## Proposition C: Mesoscopic Semantic Optimality There exist certain problem classes for which the most effective reverse generation is neither completely free natural language generation nor single-path formalization, but: $$ W_{\min} < W_{\mathrm{sem}} < W_{\max}. $$ --- ## Proposition D: Reverse Coupling of Generative Causality and Proof Causality The result-induced process: $$ P \rightsquigarrow M \rightsquigarrow T $$ and the proof process: $$ T \rightarrow M \rightarrow P $$ form a reverse coupling structurally. This is not an equivalence in the categorical sense or the functional inverse mapping sense, but a bidirectional search structure in proof engineering. --- # 27. Relationship with Existing RH Equivalent Criteria The Riemann Hypothesis already has a massive number of equivalent restatements. For example, Li-type criteria connect RH with the non-negativity of a certain sequence; Weil-type criteria connect RH with positivity on specific test function classes. These results prove that: $$ P \iff Q $$ the "change of representation" itself is completely legitimate. But the method in this article differs from simple equivalent rewriting. This article hopes that after: $$ P \rightsquigarrow \{B_i\} $$ one then searches for: $$ \{T_j\} \Rightarrow \{B_i\} \Rightarrow P. $$ Its value lies not in recreating an equivalent proposition as difficult as $P$, but in: $$ \max_i C(B_i) < C(P) $$ or at least: $$ \text{Search Branches Can Be Modularized}. $$ --- # 28. Why Choose RH as a "Preliminary Case" Rather Than Formal Method Validation RH is not an ideal first target for method validation. Reasons: 1. The target remains unsolved; 2. It is impossible to know whether failure stems from the method or the problem itself; 3. It is already surrounded by a massive amount of deep theory, making it easy to inadvertently restate existing routes; 4. There are many equivalent criteria, easily leading to "changing names without reducing difficulty"; 5. The candidate space is enormous. Therefore, this article's use of RH is strictly limited to: > Repairing a historical thought experiment to establish a methodological preliminary case. Formal validation should choose targets that are: - Propositions known to be true; - Historically difficult; - Having multiple proof paths; - Capable of hiding standard proofs; - Allowing ex-post comparison between generated intermediate propositions and historical proofs. This part will be handled in a subsequent independent paper. --- # 29. Discussion: How the Failure of That Year Should Be Re-understood The original attempter later personally used this line of thought and failed to complete the RH proof. This article believes that there are at least four possibilities for this failure: ## 29.1 Path Does Not Exist The original generation was merely an accidental language pattern. --- ## 29.2 Path Exists But Bridges Are Wrong Some $B_i$ are false. --- ## 29.3 Path Exists But Bridges Are Incomplete The true structure is: $$ B_1,\dots,B_6, G_1,\dots,G_r \Rightarrow RH. $$ Missing $G_j$ causes the search to be unable to close. --- ## 29.4 Candidate Space Explosion Each natural language node has a massive number of formalized versions: $$ B_i^{(1)},\dots,B_i^{(N_i)}. $$ Therefore, the total space is: $$ |\Omega| = \prod_iN_i. $$ --- This article leans towards proposing a fifth, comprehensive explanation: $$ \boxed{ \text{Incomplete Bridges} + \text{Lack of Convergence Criteria} \Rightarrow \text{Candidate Explosion} } $$ That is: > Too many candidates and incomplete axioms might be dual manifestations of the same underlying problem. --- # 30. Conclusion This article has not proved the Riemann Hypothesis. This article has also not proved that the five original AI-generated axioms are true. What this article has accomplished is something else: Reconstructing an erroneous, joking "new number theory proof" laden with empty-shell terminology into an intermediate proposition system that can be genuinely academically critiqued. The core transformation is: $$ \text{Self-Created Axioms} \longrightarrow \text{Transient Bridge Propositions} \longrightarrow \text{Proof Obligations}. $$ This article establishes: $$ \mathfrak H = \{H_0,H_1,H_2\}, $$ and: $$ \mathfrak B = \{B_1,\dots,B_6\}. $$ Where the center truly worthy of further research is no longer: > Does symmetry directly force the zeros to be located on the central axis? But rather: > If a positivity criterion has already transformed RH into non-negativity over the entire test space, does there exist a structured generating family that can simultaneously satisfy negative witness compression, arithmetic decomposability, local-global compensated positivity, and closure transmission? Abstractly speaking: $$ \mathcal W_-\neq\varnothing \Rightarrow \mathcal W_-\cap\mathcal G\neq\varnothing, $$ but on the other hand: $$ \forall g\in\mathcal G, \quad Q(g)\ge0. $$ If both can be independently proven, a genuine exclusion mechanism is formed. Finally, this article preserves a more general methodological insight: $$ P \rightsquigarrow M \rightsquigarrow T $$ is the result-induced generation direction, while: $$ T \rightarrow M \rightarrow P $$ is the proof direction. The reverse coupling of the two may constitute a new AI-assisted proof search framework. But this proposition has not yet been proven in this article. Therefore, the next step should not be to continue obsessing over RH, but to abstract the method and perform blind test validation on targets that are "known to be true, with non-trivial proofs." --- # Appendix A: Minimal Formalized Summary ## A.1 Target $$ P:=RH. $$ ## A.2 Hard Anchors $$ H_0: J(\mathcal Z)=\mathcal Z. $$ $$ H_1: \mathcal Q_{\mathcal Z}(f) = \mathcal Q_{\mathcal P}(f) + \mathcal Q_\infty(f). $$ $$ H_2: RH \iff \forall f\in\mathcal H,\ Q(f)\ge0. $$ ## A.3 Candidate Bridges $$ B_1: \delta(\rho)>0 \Rightarrow \exists f,\ \mathfrak D_f(\mathcal O(\rho))\neq0. $$ $$ B_2: \exists\mathfrak E,\quad \mathfrak E([\rho])=0 \iff \delta(\rho)=0. $$ $$ B_3: Q(w)<0 \Rightarrow \exists g\in\mathcal G,\ Q(g)<0 $$ or its closure version. $$ B_4: Q(g) = L_\infty(g) + \sum_pL_p(g) + R(g). $$ $$ B_5: \forall g\in\mathcal G,\quad Q(g)\ge0. $$ $$ B_6: \overline{\operatorname{span}(\mathcal G)}^{\,\tau_Q} = \mathcal H $$ and $Q$ is continuous or lower semi-continuous with respect to $\tau_Q$. ## A.4 Conditional Chain $$ B_5+B_6 \Rightarrow \forall f\in\mathcal H,\ Q(f)\ge0 \Rightarrow RH. $$ Or: $$ \neg RH \overset{H_2}{\Rightarrow} \exists w,\ Q(w)<0 \overset{B_3}{\Rightarrow} \exists g\in\mathcal G,\ Q(g)<0 \overset{B_5}{\Rightarrow} \bot. $$ --- # Appendix B: Candidate Bridge Evaluation Table | Proposition | Known Status | Possibly Equivalent to RH | Falsifiability | Candidate Width | Priority | |---|---:|---:|---:|---:|---:| | $B_1$ Orbit Observability | Unfixed | Medium | High | Medium | Medium | | $B_2$ Defect Quantifiability | Unfixed | Low to Medium | Medium | High | Low | | $B_3$ Negative Witness Compression | Unknown | High | High | Medium | Very High | | $B_4$ Arithmetic Decomposition | Partial structure known | Medium | High | Medium | High | | $B_5$ Generating Family Positivity | Unknown | Very High | High | Low to Medium | Very High | | $B_6$ Closure Transmission | Depends on choice | Medium | High | Medium | Very High | --- # Appendix C: Research Integrity Statement 1. This article is not a proof of the Riemann Hypothesis. 2. This article does not claim that $B_1$ through $B_6$ are true. 3. This article does not claim that the original AI-generated "new number theory" is valid. 4. This article only proposes a reconstructed conditional research architecture. 5. If any bridge is found to be equivalent to RH in the future, it should be publicly marked as not having reduced the proof burden. 6. If any bridge is found to be false in the future, the failed version should be preserved as a methodological counterexample. 7. No computational verification can substitute for a rigorous proof over an infinite range. --- # References [1] E. Bombieri, *Problems of the Millennium: The Riemann Hypothesis*, Clay Mathematics Institute official problem description. [2] J.-F. Burnol, *The Explicit Formula in Simple Terms*, arXiv:math/9810169. [3] A. Connes, *Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function*, arXiv:math/9811068; later published in Selecta Mathematica. [4] X.-J. Li, *The Positivity of a Sequence of Numbers and the Riemann Hypothesis*, Journal of Number Theory 65 (1997), 325–333. [5] J. B. Conrey and X.-J. Li, *A Note on Some Positivity Conditions Related to Zeta- and L-functions*, arXiv:math/9812166. [6] A. Connes, *Weil Positivity and Trace Formula, the Archimedean Place*, arXiv:2006.13771. [7] A. Connes et al., *The Riemann Hypothesis: Past, Present and a Letter Through Time*, arXiv:2602.04022, 2026. (As recent review background, not as evidence for any new theorem in this article.) --- # Version Notes This article is the first in a three-part sequence: 1. **This article:** Reconstruction of candidate intermediate axioms in the case of the Riemann Hypothesis; 2. **Subsequent second article:** General methodology of the result-induced intermediate theorem generation method and reverse axiom backfilling method; 3. **Subsequent third article:** Blind test target experimental design on known but non-trivial propositions.