> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # Global Equivalence Reconstruction and Scaling: A World-Model Methodology from Floating Geometry to Arithmetic Representatives **Global Equivalence Reconstruction and Scaling: A World-Model Methodology from Floating Geometry to Arithmetic Representatives** Version: v0.1 Nature: Methodology / proposition / conjecture paper Research status: Research Program / Proposition-Conjecture Stage Primary benchmark: Perfect Cuboid / Perfect Rectangular Box Original language: Traditional Chinese Canonical source of the original: UTF-8 Markdown --- ## Abstract This study proposes a geometry–computation route distinct from the traditional approach of “searching for integer solutions directly in arithmetic space”: first treat a candidate geometric object as a complete finite world, then study global equivalence transformations acting on the entire world, exact arithmetic reconstruction of floating worlds, and methods for extracting provable mathematical propositions from computational procedures in reverse. The central question is not “how to round each floating-point number separately to an integer,” but: > Given a geometric world jointly constituted by a container, center, objects, distances, relations, and constraints, is there a global algorithm that reconstructs its numerical representation as an exact arithmetic representative while preserving the world's equivalence class, and then obtains a canonical integer representative? Perfect Cuboid is chosen as the first benchmark world because its geometric structure is simple and its constraints explicit, yet its arithmetic closure is extremely strict. If this method can establish a verifiable exact reconstruction pipeline for Perfect Cuboid or nearby floating candidates, the same method may extend to triangles, polyhedra, distance geometry, rigid framework, lattice geometry, and other finite constrained geometric worlds. This paper does not claim that the existence of a Perfect Cuboid has been proved, nor that every floating candidate must correspond to an exact rational solution. Its aim is to establish a global research framework that is computable, falsifiable, certificate-producing, and capable of being translated from algorithms into mathematical theorems. --- # 1. Restating the problem: from “numbers” to “worlds” Traditional formulations usually represent a geometric object as a collection of numbers: $$ L=(L_1,L_2,\ldots,L_n). $$ But this representation can split originally coupled geometric relations into local variables and invite erroneous operations, such as: $$ L_i\mapsto \operatorname{round}(\lambda L_i). $$ Such componentwise scaling and rounding generally do not preserve the original exact constraint, and therefore cannot be regarded as valid world-equivalence transformations. This paper instead takes a finite geometric world as the basic object. Define $$ W= (\mathcal C,c,\mathcal O,\mathcal R,\mathcal M,\Phi), $$ where: - $\mathcal C$ is the container; - $c$ is the chosen center or gauge origin; - $\mathcal O$ is the set of geometric objects; - $\mathcal R$ consists of incidence, adjacency, dependency, and other relations; - $\mathcal M$ consists of metric / geometric attributes; - $\Phi$ is the constraint system that must hold simultaneously. Numerical values are thus no longer independent quantities, but projections of the same world state. --- # 2. Global equivalence scaling ## 2.1 Center-based global scaling For any $\lambda>0$, define global scaling about $c$: $$ T_{\lambda,c}(x) = c+\lambda(x-c). $$ If every coordinate and geometric object in a world is acted on by the same $T_{\lambda,c}$, first-order lengths satisfy: $$ L' = \lambda L. $$ Areas satisfy: $$ A' = \lambda^2A. $$ Volumes satisfy: $$ V' = \lambda^3V. $$ More generally, if quantity $q$ has scale weight $w(q)$, then: $$ q' = \lambda^{w(q)}q. $$ --- ## 2.2 World equivalence Define worlds $W$ and $W'$ to be equivalent under scaling, written: $$ W\sim_{\mathrm{scale}}W', $$ if there exist $\lambda>0$ and a valid center $c$ such that $W'$ is obtained by applying $T_{\lambda,c}$ to $W$, and the following data are preserved: 1. incidence structure; 2. topology / combinatorial structure; 3. orientation class, where required by the problem; 4. angle relations; 5. dimensionless ratios; 6. normalized constraint relations; 7. all exact relations defined by constraints homogeneous of the same degree. The basic research object is therefore no longer a single numerical representation $W$, but its equivalence class: $$ [W] = \{W':W'\sim_{\mathrm{scale}}W\}. $$ --- # 3. Proposition One: global preservation of weighted-homogeneous constraints Consider the constraint: $$ F(q_1,\ldots,q_n)=0. $$ If every term of $F$ has the same total weight $d$ under the scale action, then: $$ F(\lambda^{w_1}q_1,\ldots,\lambda^{w_n}q_n) = \lambda^dF(q_1,\ldots,q_n). $$ Thus, if: $$ F(q_1,\ldots,q_n)=0, $$ then for every $\lambda\neq0$: $$ F(\lambda^{w_1}q_1,\ldots,\lambda^{w_n}q_n)=0. $$ ### Proposition 1: Global Weighted-Homogeneous Preservation Proposition If a finite geometric world $W$ has a constraint system $\Phi$ consisting entirely of weighted-homogeneous equations compatible with the global scale action, every valid uniform global scaling preserves $\Phi$. This proposition explains that valid scaling does not “reapproximate” a world; it moves the entire world to another representative in the same equivalence class. --- # 4. Proposition Two: a rational world has an integer representative Suppose every first-order scale of a finite world has a rational ratio to some reference length $L_0>0$: $$ \frac{L_i}{L_0}\in\mathbb Q. $$ Write: $$ \frac{L_i}{L_0} = \frac{p_i}{q_i}, $$ where: $$ p_i\in\mathbb Z, \qquad q_i\in\mathbb Z_{>0}. $$ Let: $$ Q = \operatorname{lcm}(q_1,\ldots,q_n), $$ and take: $$ \lambda = \frac{Q}{L_0}. $$ Then: $$ \lambda L_i = Q\frac{p_i}{q_i} \in\mathbb Z. $$ ### Proposition 2: Finite Rational World Integral Representative Proposition Every finite geometric world with a common scale reference and rational ratios for all required scales has an integral representative in its scale-equivalence class. This proposition is important because it rewrites “finding an integer geometric object” as: > First find an exact rational shape representative, then perform one valid global denominator clearing. --- # 5. Proposition Three: local rounding generally does not preserve world equivalence Consider: $$ \widetilde L_i = \operatorname{round}(\lambda L_i). $$ In general, there is no guarantee that: $$ F(\widetilde L_1,\ldots,\widetilde L_n)=0 $$ follows from $$ F(L_1,\ldots,L_n)=0 $$ the original constraint. Therefore: ### Proposition 3: Independent Rounding Non-Equivalence Proposition Variable-by-variable rounding, edge-by-edge snap, local denominator fitting, or mutually independent rational approximation generally do not constitute valid global world equivalence operations. Thus, even if future software provides a visual “snapping” feature, it must not treat visual snapping as exact arithmetic reconstruction. --- # 6. A crucial distinction: scaling cannot create rationality If: $$ \frac{L_i}{L_j}\notin\mathbb Q, $$ then for every nonzero $\lambda$: $$ \frac{\lambda L_i}{\lambda L_j} = \frac{L_i}{L_j} \notin\mathbb Q. $$ Pure uniform scaling therefore cannot turn a genuinely irrational shape into a rational shape. This forces the overall research procedure to split into two different problems: ## 6.1 Global Equivalence Reconstruction $$ \mathrm{GER}: \widetilde W \longrightarrow W^\ast. $$ The aim is to determine whether the floating world $\widetilde W$ is a finite-precision projection of some exact world $W^\ast$, and to reconstruct $W^\ast$ under provable conditions. ## 6.2 Global Equivalence Scaling $$ \mathrm{GES}: W^\ast \longrightarrow W_{\mathrm{can}}. $$ The aim, once an exact world is known, is to obtain a canonical rational / integer representative within its equivalence class. The overall research procedure is written as: $$ \boxed{ \widetilde W \rightarrow \mathrm{Recover}(W^\ast) \rightarrow \mathrm{Canonicalize}(W^\ast) \rightarrow \mathrm{Scale}(W_{\mathbb Z}) \rightarrow \mathrm{ExactVerify} } $$ This paper provisionally calls the complete framework: $$ \mathrm{GERES} = \mathrm{GER} + \mathrm{GES}. $$ that is, Global Equivalence Reconstruction and Scaling. --- # 7. World-level arithmetic reconstruction, not single-value approximation Traditional rational approximation may separately perform: $$ x_i \approx \frac{p_i}{q_i}. $$ But this is insufficient to prove that one shared exact world exists. GER requires joint reconstruction: $$ \widetilde W \longrightarrow W^\ast\in\mathcal V(\mathbb Q), $$ where $\mathcal V$ is the algebraic / geometric locus defined by the constraint system. For every constraint: $$ F_k(W^\ast)=0 $$ must hold exactly; it is not enough merely to require: $$ |F_k(\widetilde W)| \ll1. $$ The basic unit of reconstruction is therefore not a coordinate, but the entire constraint-coupled world. --- # 8. Conjecture One: global exact recovery at bounded height Let: $$ \mathcal V \subseteq \mathbb R^n $$ be defined by polynomial constraints with rational coefficients. Suppose there exists: $$ W^\ast\in\mathcal V(\mathbb Q), $$ and a floating observation is given: $$ \widetilde W = W^\ast+\varepsilon. $$ Further suppose: 1. $W^\ast$ is an isolated admissible rational solution in the specified admissible region; 2. the rational height has a known upper bound $H$; 3. floating precision is sufficient to distinguish competing rational worlds within that height range; 4. the constraint system and gauge fixing suffice to exclude pure-scale degeneracy and other known symmetry degeneracy. ### Conjecture 1: Bounded-Height Global Arithmetic Recovery Conjecture There exists a finite algorithm that, when the above conditions hold, can use $(\widetilde W,\Phi,H)$ to uniquely recover $W^\ast$ and output an independently verifiable exact certificate. This conjecture does not say that any approximate point can be recovered as a rational solution; it concerns only cases where “an exact rational source world of bounded complexity actually exists.” --- # 9. Conjecture Two: constraint-coupled reconstruction outperforms coordinatewise reconstruction Define coordinatewise reconstruction: $$ \mathcal R_{\mathrm{local}} = \prod_i\mathcal R_i. $$ Define joint world reconstruction: $$ \mathcal R_{\mathrm{global}} : (\widetilde W,\Phi) \mapsto W^\ast. $$ ### Conjecture 2: Constraint-Coupled Reconstruction Conjecture With the same precision and arithmetic height budget, joint reconstruction using the global constraint graph can eliminate many false candidates left by coordinatewise approximation, and reduce reconstruction ambiguity when a unique exact source world exists. The computational version of this conjecture can be tested directly with a benchmark: $$ N_{\mathrm{local\ candidates}} \gg N_{\mathrm{globally\ admissible\ candidates}}. $$ --- # 10. Conjecture Three: Canonical Gauge can reduce arithmetic search dimension If a world has a pure-scale degree of freedom, searching directly in: $$ (L_1,\ldots,L_n) $$ includes a redundant one-dimensional orbit. Choose a valid gauge, such as: $$ L_0=1, $$ and define: $$ x_i = \frac{L_i}{L_0}, $$ The search then moves into shape space. ### Conjecture 3: Gauge-Reduced Arithmetic Search Conjecture For a finite constrained geometric world with uniform scale symmetry, first quotienting out the scale degree of freedom and then performing arithmetic reconstruction yields a lower-dimensional, more stable exact search problem for which uniqueness certificates are easier to establish. --- # 11. Perfect Cuboid as the first benchmark world Let the three cuboid edges be: $$ a,b,c>0. $$ The three face diagonals are: $$ d_{ab}, \qquad d_{ac}, \qquad d_{bc}, $$ The space diagonal is: $$ g. $$ The constraints are: $$ a^2+b^2=d_{ab}^2, $$ $$ a^2+c^2=d_{ac}^2, $$ $$ b^2+c^2=d_{bc}^2, $$ $$ a^2+b^2+c^2=g^2. $$ Using: $$ g=1 $$ as the gauge, define: $$ x=\frac{a}{g}, \qquad y=\frac{b}{g}, \qquad z=\frac{c}{g}, $$ and: $$ u=\frac{d_{ab}}{g}, \qquad v=\frac{d_{ac}}{g}, \qquad w=\frac{d_{bc}}{g}. $$ The normalized world then satisfies: $$ x^2+y^2-u^2=0, $$ $$ x^2+z^2-v^2=0, $$ $$ y^2+z^2-w^2=0, $$ $$ x^2+y^2+z^2-1=0. $$ The arithmetic existence problem for Perfect Cuboid can therefore be rewritten as: > Does there exist a positive rational point > > $$ > (x,y,z,u,v,w)\in\mathbb Q_{>0}^6 > $$ > > satisfying all of the above constraints simultaneously? If the answer is yes, Proposition 2 immediately gives an integral representative. Therefore: $$ \boxed{ \text{Perfect Cuboid integral existence} } $$ can first be transformed within this framework into: $$ \boxed{ \text{normalized rational world existence} } $$ followed by global denominator clearing. --- # 12. The new role of floating candidates Existing floating candidates should no longer be understood as edge-length tuples that are “almost integers,” but as observed states in: $$ \widetilde W \in \mathbb R^n $$ this space. The central question becomes: > Is $\widetilde W$ in a recoverable numerical neighborhood of some $W^\ast\in\mathcal V(\mathbb Q)$? We can therefore construct a search functional: $$ J(W) = \alpha \left\| W-\widetilde W \right\|^2 + \beta \sum_k \left| F_k(W) \right|^2 + \gamma \log H(W), $$ where: - The first term bounds the distance from the observed candidate; - The second term bounds the constraint residual; - The third term suppresses arithmetic complexity. But the final accept condition cannot be that $J(W)$ is small; it must be: $$ W\in\mathcal V(\mathbb Q) $$ with all exact verification completed. --- # 13. Outline of the GERES prototype algorithm ## Phase A: World Ingestion Input: $$ \widetilde W, \qquad \Phi, \qquad \mathcal R. $$ Convert the candidate into a world graph, not a flat vector. ## Phase B: Canonical Gauge Fixing Eliminate the uniform scale degree of freedom, for example: $$ g=1. $$ Where necessary, also fix orientation, ordering, or a symmetry convention. ## Phase C: Precision Elevation Do not trust the binary floating representation directly. Use: - arbitrary precision arithmetic; - interval arithmetic; - exact residual tracking; - consistency tests across multiple precisions. ## Phase D: Local Arithmetic Proposal Possible tools include: - continued fractions; - rational reconstruction; - lattice reduction; - integer relation detection; - denominator pattern mining. But this phase produces only proposals, not theorems. ## Phase E: Global Constraint Coupling Return proposals to the entire constraint graph and solve jointly: $$ F_k(W)=0. $$ Immediately discard any proposal that holds locally but cannot pass global closure. ## Phase F: Exact Reconstruction Candidates must be converted into exact arithmetic objects: $$ W^\ast\in\mathbb Q^n $$ or a more general specified arithmetic domain. ## Phase G: Exact Certificate Output: 1. exact coordinates; 2. exact constraints; 3. denominator data; 4. symmetry / gauge data; 5. transformation trace; 6. verification hash or machine-checkable proof object. ## Phase H: Global Scaling Take the common scale of all required denominators: $$ Q = \operatorname{lcm}(q_1,\ldots,q_n). $$ Obtain: $$ W_{\mathbb Z} = QW^\ast. $$ ## Phase I: Primitive Canonicalization If all integer quantities share a common factor $G$, then: $$ W_{\mathrm{primitive}} = \frac{1}{G}W_{\mathbb Z}. $$ ## Phase J: World Re-Simulation Project the exact representative back into the virtual world and compare: - topology; - incidence; - angle; - ratios; - metric constraints; - world graph; - normalized shape. This step is a world-level regression test. --- # 14. Conjecture Four: a transfer principle between geometry classes Perfect Cuboid should not be the endpoint of the method. Consider a family of worlds: $$ \mathfrak W = \{W_\theta\}. $$ As long as their constraints: 1. can be represented by a finite relation graph; 2. have an explicitly specifiable arithmetic domain; 3. have a known transformation law; 4. have a scale symmetry that can be quotiented; 5. admit computable exact verification; GERES can theoretically be applied. ### Conjecture 4: World-Class Transfer Conjecture If two classes of finite geometric worlds share the same transformation grammar, arithmetic reconstruction interface, and constraint-preserving certificate structure, a GERES algorithm developed in one class can transfer to the other through finite adaptation, without reinventing the entire reconstruction paradigm. Candidate extensions include: - rational triangles; - Heronian geometry; - tetrahedral distance systems; - rational polyhedra; - rigid frameworks; - lattice embeddings; - finite distance geometry; - constrained coordinate worlds. --- # 15. Conjecture Five: computational procedures can be reverse-compiled into mathematical proofs GERES's ultimate goal is not to retain a black-box heuristic. For every successful reconstruction, require preservation of: $$ \mathcal T = (T_1,T_2,\ldots,T_m), $$ where each $T_i$ is a describable exact transformation / inference step. ### Conjecture 5: Algorithm-to-Theorem Compilation Conjecture If a global equivalence reconstruction algorithm can be decomposed into finitely many: - exact algebraic transformations; - certified inequality bounds; - finite rational reconstruction steps; - constraint-preserving equivalence transformations; the computation trace of a successful case can be compiled into a machine-checkable certificate; with further abstraction, some trace families can be elevated to general mathematical propositions or theorems. This establishes the research route: $$ \boxed{ \text{virtual experiment} \rightarrow \text{algorithm} \rightarrow \text{certificate} \rightarrow \text{mathematical theorem} } $$ as a progression. --- # 16. Visualization software: World-Equivalent Scaling Lab Later work in this series should build a visual experimental tool, provisionally called: **World-Equivalent Scaling Lab, WESL** The first version need not aim to be a large software system; it should provide six core layers: ## 16.1 Container / Center Layer Display: - container; - center; - gauge origin; - global scale orbit. ## 16.2 Geometry Layer Display: - vertices; - edges; - faces; - diagonals; - dependent metric objects. ## 16.3 Constraint Graph Visualize: $$ F_1,\ldots,F_m $$ as a coupled network. ## 16.4 Dual Representation Display simultaneously: $$ \text{floating projection} $$ and: $$ \text{exact candidate representation}. $$ ## 16.5 Global Scale Control When dragging $\lambda$, do not let local objects change independently; instead: $$ T_{\lambda,c} $$ must act on the entire world. ## 16.6 Arithmetic Closure Monitor Every arithmetic channel must be labeled: - floating only; - rational candidate; - exact rational; - exact integer after scaling; - failed closure. In particular, visual closeness must never be labeled exact closure. --- # 17. Existing visualization systems can be adapted directly If a Perfect Cuboid / X-Diagonal / center-container sandbox already exists, subsequent work need not rewrite it completely. The revision should move toward: $$ \text{geometry visualizer} \rightarrow \text{world-equivalence laboratory}. $$ The central addition is not more attractive 3D graphics, but: 1. transformation ledger; 2. constraint graph; 3. arithmetic state; 4. normalized world state; 5. exact certificate panel; 6. scale-orbit visualization. The software's role is therefore not “to display a cuboid,” but to let researchers directly observe: > Which quantities are changing together, which invariants must be preserved, which arithmetic channel has not yet reached closure, and whether the candidate exact world and the floating world truly have the same verifiable source. --- # 18. Falsification conditions A methodology that cannot be falsified cannot form a reliable research program. GERES has at least the following explicit failure conditions. ## Failure A: no exact source world exists If the neighborhood of a floating candidate contains no: $$ W^\ast\in\mathcal V(\mathbb Q), $$ within the height and admissibility bounds, recovery must return: $$ \mathrm{NO\ CERTIFIED\ RECOVERY}. $$ ## Failure B: multiple exact worlds cannot be distinguished If precision is insufficient to distinguish two or more admissible exact worlds, the output must be ambiguity, not an arbitrary choice among them. ## Failure C: local rational reconstruction cannot achieve global closure Any collection of local rational approximations for which: $$ F_k(W^\ast)\neq0 $$ holds for at least one $k$ must be discarded. ## Failure D: scaling improves only the visual appearance, not the arithmetic If a scale operation merely makes decimal patterns look closer to integers while ratios remain irrational or exact constraints do not hold, arithmetic progress must not be claimed. ## Failure E: the algorithm works only for a single handcrafted case If results cannot be reproduced on synthetic controls, known rational worlds, perturbed worlds, and negative controls, they cannot be elevated to a general methodology. --- # 19. Experimental controls Before formal Perfect Cuboid candidates, it is recommended to establish four benchmark classes. ## Control 1: Known Exact Rational World First generate exact rational geometry, then project it into a float: $$ W^\ast \rightarrow \widetilde W. $$ Test whether GER can recover the original $W^\ast$. ## Control 2: Known Irrational World Use a geometric object with an explicitly irrational ratio. Verify that the system does not fabricate rational closure through scaling. ## Control 3: Near-Rational False Positive Construct a floating world extremely close to a low-height rational point, but not actually on the constraint variety. Test the global constraint gate. ## Control 4: Multiple-Solution Ambiguity Deliberately construct a case where precision is insufficient to distinguish two arithmetic worlds. Test whether the system correctly outputs ambiguous. Perfect Cuboid experimental results have methodological significance only after these four classes of controls are passed. --- # 20. Completion conditions from Perfect Cuboid to general geometry True completion of this series should not be defined merely as “finding a tuple of numbers.” It should be divided into at least five levels: ## Level 1: World Representation Complete the formal representation of worlds, centers, containers, relations, constraints, and equivalence classes. ## Level 2: Verified Global Scaling Complete a scale engine that does not break world relations. ## Level 3: Certified Exact Reconstruction Recover a known exact world from an artificially generated floating projection. ## Level 4: Perfect Cuboid Benchmark Run reconstruction on the existing floating candidate pool and obtain one explicit outcome among: - certified exact recovery; - certified rejection; - certified ambiguity; these three. ## Level 5: Cross-Geometry Transfer The same arithmetic world methodology can validly act on at least several different finite geometric systems. Only after reaching Level 5 can one reasonably say: > What we have completed is not a Perfect Cuboid trick, but a generalized methodology for global equivalence arithmetic reconstruction. --- # 21. First-stage research route The series is recommended to proceed in the following order. ### P0: This paper The Global Equivalence Reconstruction and Scaling methodology, propositions, and conjectures. ### P1: Formal World Semantics Formally define: $$ W, \qquad [W], \qquad T_{\lambda,c}, \qquad \Phi, \qquad \mathcal R. $$ ### P2: Global Scale Engine Build both visualization and computation versions. ### P3: Synthetic Rational Recovery First test with known exact rational worlds. ### P4: Irrational / False-Positive Gate Verify that the method does not forcibly “integerize” irrational geometry. ### P5: Perfect Cuboid World Encoding Formally encode the complete Perfect Cuboid geometry / arithmetic constraint graph. ### P6: Floating Candidate Reconstruction Input the existing floating candidate pool. ### P7: Certificate Compiler Output successful reconstruction traces as exact proof objects. ### P8: Theorem Extraction Turn transformation principles that hold stably in the procedure into mathematical propositions. ### P9: Cross-Geometry Transfer Test triangles, polyhedra, distance geometry, and others. ### P10: General Global Arithmetic Equivalence Theory Then decide whether to formulate a more general theory name and formal axiom system. --- # 22. Relationship to traditional Perfect Cuboid approaches This method does not require abandoning traditional number theory, algebraic geometry, elliptic curves, Pythagorean parametrization, or exhaustive search. The two branches can remain relatively independent: $$ \text{Traditional Arithmetic Branch} $$ and: $$ \text{World Reconstruction Branch}. $$ then meet at the exact certificate layer. This has a methodological advantage: If two different research routes ultimately produce the same arithmetic obstruction, rational family, invariant, or impossibility condition, that intersection itself is strong evidence. Conversely, contradictions between the branches can expose hidden assumptions more quickly. --- # 23. The most important conceptual shift at present The central proposition of this paper is not: > Enlarge floating-point numbers until they look like integers. Instead, it is: > First define what it means to be the same world, then study whether its equivalence class contains an exact arithmetic representative. The basic research object is therefore: $$ [W], $$ not an arbitrary numerical table of: $$ (L_1,\ldots,L_n) $$ these lengths. More fully: $$ \boxed{ \text{Geometry} \rightarrow \text{World} \rightarrow \text{Equivalence Class} \rightarrow \text{Arithmetic Reconstruction} \rightarrow \text{Canonical Representative} } $$ --- # 24. Conclusion GERES proposes a geometry research program oriented toward computational methodology: 1. Do not split coupled geometry into independent floating-point numbers; 2. First establish a complete world model; 3. Preserve the equivalence structure with global transformations; 4. Quotient out the pure-scale degree of freedom; 5. Perform arithmetic reconstruction on the entire constraint-coupled world; 6. Accept only exact certificates; 7. Then obtain an integer representative through valid global scaling; 8. Abstract the algorithm trace in reverse into provable mathematical propositions; 9. Finally test transferability across different geometric worlds. For Perfect Cuboid, this framework does not assume that any floating candidate must succeed. What it actually provides is a new research question: $$ \boxed{ \text{Is a floating geometric world a recoverable projection of an exact rational world?} } $$ If the answer is no, the system should reject it with proof. If the answer is yes, the next step is not a visual “click into alignment,” but producing: $$ \boxed{ \text{exact world} + \text{global equivalence map} + \text{arithmetic certificate} } $$ and then compiling this entire computational route back into mathematics. This is what the series calls: **Global Equivalence Reconstruction and Scaling.** --- ## Theorem Status / Claim Boundary As of this version: - The existence of a Perfect Cuboid has not been proved; - It has not been proved that any current floating candidate corresponds to an exact Perfect Cuboid; - The basic mathematical framework for global scaling preservation has been established; - The basic proposition from a rational world to an integral representative has been established; - Scaling and arithmetic reconstruction have been explicitly distinguished; - Five research conjectures have been proposed: bounded-height global recovery, constraint-coupled reconstruction, gauge reduction, cross-geometry transfer, and algorithm-to-theorem compilation; - Subsequent work must validate them with synthetic controls, negative controls, and exact certificates. --- ## Canonical Research Formula $$ \boxed{ \widetilde W \overset{\mathrm{GER}}{\longrightarrow} W^\ast \overset{\mathrm{GES}}{\longrightarrow} W_{\mathrm{can}} \overset{\mathrm{Verify}}{\longrightarrow} \mathcal C \overset{\mathrm{Compile}}{\longrightarrow} \text{Mathematical Theorem} } $$ Here, $\mathcal C$ is a machine-checkable certificate. This chain is the shared backbone of all subsequent implementations, visualizations, Perfect Cuboid experiments, and generalization work in the series.