P E R S P E C T I V E   T R A N S F O R M A T I O N   T H E O R Y

By Axiom Three, translation is the mapping of the relational structure of source system A into the medium of target system B through the translation operator T. Let the total relational structure of A involved in the translation be R₀. T retains a portion in B—this is the similarity Σ; the rest is irreversibly discarded in the translation—this is the cost ΔΦ. Retention and discarding exhaust the entire relational structure involved in the translation, therefore Σ + ΔΦ = R₀. If and only if ΔΦ = 0, Σ = R₀, and the translation is completely isomorphic; the equals sign holds.

According to Axiom Three (the translation cost axiom), any translation causes irreversible rupture of the original structure, therefore translation is irreversible.

By Axiom Three, cost is the irreversible discarding of the source system's original relational structure when translated into the medium of the target system. Any technical means is itself a translation operation—it maps the source system into the target system through a technical medium. According to Axiom Three, this mapping also pays cost. Therefore, the attempt to eliminate cost itself produces new cost, forming infinite recursion. Cost cannot be eliminated, only minimized by optimizing the translation operator.

By Axiom Two, translation is path-dependent—each translation changes the initial conditions of the next. By Axiom Three, each translation pays cost; the discarded relational structure cannot be restored. Let the cost of direct translation from A to B be ΔΦ(A→B), and the cost of a two-step translation via intermediary C be ΔΦ(A→C) + ΔΦ(C→B). Since the first translation has already discarded part of A's relational structure, the second translation can only proceed on the already-reduced basis, and cannot restore what has been discarded. Therefore the total discarded amount is at least equal to that of the direct translation, that is, ΔΦ(A→C) + ΔΦ(C→B) ≥ ΔΦ(A→B). Extending to any number of steps, the total cost of multiple consecutive translations is not less than the cost of a single direct translation. The equals sign holds if and only if all intermediary systems are completely isomorphic to both the source and target systems.

By Axiom Three (all extension pays cost), total cost is proportional to total extension. In a closed system, total extension is constant, therefore total cost is constant. If a subsystem reduces the cost it pays through optimization, that cost does not disappear, but is transmitted to another subsystem through system coupling. Externalities, intergenerational injustice, risk transfer—these phenomena are all cost transfer, not cost elimination. This theorem is in fact the counterpart of the first law of thermodynamics (energy conservation) in the domain of cost—just as energy cannot be created or destroyed, cost cannot be destroyed, only transferred. The difference is that energy transfer is bidirectionally reversible, whereas cost transfer is often accompanied by irreversible rupture of structure (Axiom Three), so cost usually increases rather than remaining constant in transfer.

Application examples: A company transfers pollution costs to surrounding communities; the communities pay with health costs, and ultimately the company is harmed through medical expenses and labor loss. A country transfers debt costs to the next generation; the next generation pays with inflation or tax burden. This theorem declares: all "cost externalization" ultimately returns to the interior of the system in some form, because there is no true "outside" in a closed system.

By Axiom Two (focus → fixation → extension), once focus intensity and duration exceed a certain threshold, the degree of fixation makes the return of extension (increase in certainty) lower than the cost (collapse of possibility, information loss, structural distortion). The intersection of diminishing marginal return and increasing marginal cost is the optimal upper limit. This is specifically manifested as: decreased creativity (Axiom Two corollary: extension after fixation is inertial repetition, not innovation); decreased empathy (Axiom Two corollary: the perspective is locked in a single direction and cannot enter another's perspective); decreased global judgment (Axiom One corollary: after the medium is fixated, the subject cannot use other media to examine the same object).

Application examples: In AI training, overfitting corresponds to overfocus. When training epochs exceed the optimal upper limit, the model's accuracy on the training set continues to rise, but its generalization ability on the test set declines—this is the precise fulfillment of Axiom Two. In education, overtraining (drilling) corresponds to overfocus. The student's accuracy on a particular type of problem continues to rise, but creativity in facing new problem types declines. In research, over-specialization corresponds to overfocus. The researcher's certainty within a narrow subfield continues to increase, but the capacity for cross-domain translation declines. In organizational management, over-centralization corresponds to overfocus. The leader's control over details continues to strengthen, but the organization's overall adaptability declines.

By Axiom Two, focus → fixation → extension. Axiomatization makes a disciplinary community's focus highly concentrated, accelerating fixation, enabling the discipline to produce a large amount of reliable knowledge (extension) during the normal science phase. However, according to the Overfocus Cost Theorem, fixation obstructs innovation after crossing the optimal upper limit. Axiomatization reduces the operational cost of innovation, but the acceptance of axiomatization itself increases the inertia of perspective fixation; the two form a paradox: enhancing innovation capacity while reducing innovation willingness. Jubilee reset breaks the inertia of fixation through forced perspective transformation, at the cost of efficiency loss during the reset. The Optimal Jubilee Cycle Theorem seeks a balance between fixation inertia and reset cost.

Application examples: After Newtonian mechanics was axiomatized, physics throughout the nineteenth century produced enormous achievements within the fixated paradigm, while accumulating the cost of anomalous explanations such as "ether" and the "ultraviolet catastrophe." Axiomatized physicists found it extremely difficult to accept the relativity of space-time and the probability of quantum mechanics. Ultimately physics completed a Jubilee reset through the paradigm shift of relativity and quantum mechanics. The work of Copernicus, Kepler, and Galileo was extremely difficult to accept within the fixated paradigm of "geocentrism"; the higher the degree of axiomatization of geocentrism, the more Copernicus's Jubilee reset was resisted. After economics was axiomatized, the rational-agent assumption became the fixated default premise. Behavioral economics modified this axiom, but experienced prolonged rejection within the discipline before being accepted.

By Axiom Three (cost is the irreversible rupture of relational structure), once rupture occurs, it cannot "automatically heal." By the Cost Transfer Conservation Theorem, cost can only be transferred, not eliminated. If the system does not actively reset (transferring cost to "the end of history" or "the altar of Jubilee"), cost continuously accumulates within the system, and structural distortion continuously deepens. The accumulated amount is proportional to time and inversely proportional to reset frequency. After reaching the critical threshold, any minor extension of the system may trigger chain collapse.

Application examples: In earthquakes, crustal stress accumulates year by year; unreleased stress erupts in concentrated form as earthquakes after reaching the critical point at the fault zone. In interpersonal relationships, unrepaired minor hurts accumulate year by year, erupting in concentrated form as relationship rupture when the trust threshold is touched. In economics, unsettled financial risks accumulate year by year, triggering a systemic crisis when a financial institution collapses.

By the translation cost axiom of similarity mathematics, the cost of each translation step is ΔΦ. Total cost equals the sum of the costs of each step on the path. By the Cost Accumulation Theorem, total cost is not less than the cost of direct translation. The question is: does there exist an intermediary system C such that the total cost of a two-step translation via C is less than the cost of direct translation? The answer is: when the direct translation cost between A and B is extremely high (Σ(A,B) far less than 1), but the translation costs of A with C and C with B are both small, the total cost of indirect translation via C is smaller. This optimal path is the cost minimization path.

Application examples: In interdisciplinary education, jumping directly from one field's axiomatic system to a completely different field has an extremely high cost. A better path is to first learn an intermediate discipline with high similarity to both, completing the transformation in two steps. In cross-cultural communication, direct communication between two groups with vastly different languages and cultures has an extremely high cost. Finding an intermediate culture as a bridge makes the total cost smaller.

By Axiom Three, any extension pays cost; cost is the irreversible rupture of relational structure. Suppose the system has multiple possible extension paths; a high-cost path means each extension pays a greater structural rupture. In time, the cumulative effect of high-cost paths brings the system to the collapse threshold faster (guaranteed by Theorem 98, the Irreversible Cost Accumulation Theorem). Therefore, in long-term operation, the extension pattern that can persist is necessarily the lower-cost path. This is not the system consciously "choosing" or nature having an intrinsic purpose—collapsed systems are no longer observed; what remains are only those systems that happen to be on lower-cost paths. This "survivor bias" is the statistical necessity of the cost axiom operating in time.

At the physical medium level, physical extensions such as object motion and light propagation follow the least-cost path in space-time. Fermat's principle of least time and Hamilton's principle of least action are special cases of this theorem in physical systems. At the cognitive medium level, cognitive extensions such as thinking activity and theory construction follow the least-cost path in symbolic systems. Mach's principle of economy of thought—science pursuing the simplest formula, logic pursuing the shortest derivation—is a special case of this theorem in cognitive systems.

Corollary: The principle of least action in physics and the principle of economy of thought in cognitive science have the same mathematical structure within the framework of the cost functional ΔΦ. This provides a theoretical foundation for the application of the Cross-Domain Verification Theorem between physics and cognitive science. The "teleological appearance" of the principle of least action—nature seeming to "choose" the optimal path—is reduced to the statistical necessity of the cost axiom, without appealing to nature's intrinsic purpose or the Creator's continuous intervention.

By Axiom One, all cognition is within a perspective; different domains' perspective systems use different media, point to different objects, and involve different depths of subject involvement. By Axiom Three, cost is the irreversible rupture of relational structure in extension. The degree of medium standardization determines the accumulation rate of M-O edge cost—physics uses highly standardized mathematical language and experimental instruments, and the cost of the medium touching the object is extremely small; history uses natural language and non-reproducible documents, and the rupture between medium and object is amplified in each translation. Object stability determines the accumulation pattern of cost—the motion laws of celestial bodies are unchanged on the human observation timescale, and the certainty of the laws is extremely high; the object of financial markets is the aggregate behavior of countless subjects with free will, and the laws change with observation. The depth of subject involvement determines the degree of P-O edge cost intervention—the astronomer's personal beliefs do not affect stellar orbits, but the political scientist's own ideology is deeply embedded in his object of study. The three factors jointly determine the natural non-uniformity of cost distribution across domains. This is the direct corollary of the cost axiom in cross-domain comparison, not dependent on any theological presupposition. The Fall's Distortion Gradient Theorem of the theological edition provides an ultimate explanation for this distribution, but the distribution itself is a structural fact independently derivable from the cognitive edition.

By Axiom Three, cost is the irreversible rupture of relational structure. Before reaching the critical point, rupture is not unobservable, but accumulates gradually. The ruptures on the three edges each have their phenomenological characteristics. P-M edge rupture manifests as medium-subject dissonance (the subject can no longer flexibly use the medium). M-O edge rupture manifests as medium-object dissonance (the medium can no longer accurately reflect the object). P-O edge rupture manifests as subject-object dissonance (the subject loses stable direction). Simultaneous triggering of the three signals is the necessary condition for cascading collapse.

Application examples: Precursors of financial crisis: financial institutions' risk models (P-M edge medium) repeatedly fail, market data (M-O edge information) severely deviate from the real economy, investors vacillate between greed and panic (P-O edge direction). Major crises such as the 2008 financial crisis and the Great Depression all exhibited these three signals simultaneously before erupting. Precursors of dynastic collapse: the bureaucratic system (P-M edge medium) increasingly rigidifies and fails, local reported performance data (M-O edge information) is severely distorted, the court vacillates between reform and conservatism (P-O edge direction). In the dynastic cycle of Chinese history, these three signals appeared successively within the decades before collapse. Precursors of relationship rupture: communication (P-M edge medium) becomes routine or cold violence, information transmitted between the parties (M-O edge) is severely misunderstood or distorted, one or both parties in the relationship vacillate between approach and withdrawal (P-O edge direction), unable to form a stable focus of love. Before marriage breakdown, friendship rupture, or church split, these three signals appear almost without exception.

1. The ontology of cost is the irreversible rupture of a specific relational structure within the perspective (Axiom Three). 2. Formalizing cost is a cognitive extension—the subject translates "cost" from an ontological concept into a computable quantity at the medium level through formalizing media (mathematics, logic, algorithms). 3. By Axiom Three, any extension pays cost. The extension of formalizing cost itself necessarily pays additional cost—in the translation process, certain relational structures of the ontology of cost are irreversibly discarded. 4. These discarded structures cannot be captured by the same formalization system—because capturing them requires further extension, paying further cost, forming infinite recursion. 5. Therefore, any formalization system's measurement of cost is a degraded projection of the ontology of cost, not cost itself. The attempt to completely formalize cost is structurally impossible.

Corollary One (Boundary of Quantifiable Domains): Cost can only be precisely measured in highly standardized medium-objects (such as entropy increase), but what is measured is only the specific projection of cost in that medium, not the ontology of cost.

Corollary Two (Legitimacy of Qualitative Diagnosis): In complex systems, the formalization of cost can only proceed by qualitative diagnosis—precisely attributing the structure of the rupture, locating the ruptured edge (P-M, M-O, P-O), describing the state change before and after the rupture—rather than giving an exhaustive numerical value.

Corollary Three (Non-Quantifiability of Aesthetics and Ethics): The cost of aesthetics and the cost of ethics involve deep ruptures of relational structure, whose value cannot be replaced by any numerical value. Demanding their quantification is a category error regarding the ontology of cost.

By Axiom Two (focus → fixation → extension), the longer the system operates, the higher the degree of fixation. By Axiom Three (cost accumulation), extension after fixation pays increasing cost. By the Jubilee Theorem (Cognitive Theorem), periodic reset is the only way to interrupt cost accumulation. Optimal control theory gives the existence and uniqueness of T*.

Application examples: Corporate strategic renewal cycles, software system refactoring cycles, personal habit reflection cycles, church governance leadership rotation cycles—all can calculate the optimal Jubilee frequency through this theorem. An organization's leader does not need to decide by intuition "when to change"; he can precisely calculate the optimal change cycle by evaluating the fixation rate, critical threshold, and reset cost.

By Cognitive Theorem 5 (Periodic Reset Theorem) and the Optimal Jubilee Cycle Theorem. The three-level classification corresponds to the three critical thresholds of cost accumulation: parameter threshold, structural threshold, foundation threshold.

Level One · Parameter reset: Retaining the system's basic axioms and theorem structure, resetting only parameters. For example, a company adjusting goals within the same strategic framework, an individual adjusting habits within the same cognitive framework, software updating versions within the same architecture. Level Two · Structural reset: Modifying the system's domain axioms (medium layer, object layer, or relation layer), rebuilding the theorem system. For example, fundamental corporate transformation, disciplinary paradigm shift, church institutional reform, personal perspective transformation. Level Three · Foundation reset: The system completely disintegrates, returns to the meta-axiom state, and rebuilds from zero. For example, corporate bankruptcy and reorganization, institutional reconstruction after revolution, life reconstruction after repentance, bodily renewal after resurrection. Level one reset is most frequent with the smallest cost; level three reset is least frequent with the largest cost. Frequent level one reset prevents daily cost accumulation; periodic level two reset interrupts fixated structures; rare level three reset responds to systemic collapse.

Application examples: In personal life, daily confession is parameter reset, major life turning points are structural reset, repentance and conversion are foundation reset. In church governance, annual ministry evaluation is parameter reset, the fifty-year Jubilee is structural reset, the Reformation is foundation reset. In software engineering, daily patches are parameter reset, architectural refactoring is structural reset, complete rewriting is foundation reset.

By Axiom Three (cost is the irreversible rupture of relational structure), the reset operation itself pays cost. The reset breaks the old structure and establishes a new one, but the trace of the old structure is retained in the perspective history. The system's state variables can be restored, but the system's history cannot be rewritten.

Application examples: After corporate bankruptcy and reorganization, the new company's credit record and market reputation differ from the old company's.

Step One: Hypothesis is focus. By Cognitive Axiom Two, focus is the act of the subject actively directing the perspective toward a specific direction. When a researcher faces a phenomenon not yet theorized, he must choose one direction of exploration from infinitely many possible explanatory directions. This choice is not "induced" from data—data itself cannot specify its own explanatory direction. The choice is a free act of focus: the researcher actively directs attention toward a possible relational structure; this directed direction is the hypothesis.

Step Two: Formalization is fixation. By Cognitive Axiom Two corollary, fixation is the trace left by focus in the medium. The researcher repeatedly focuses on the same hypothesis, using logic, mathematics, or formal language to unfold it into a derivable structure. This process produces theorems, formulas, predictions—these are the necessary extensions under the fixated perspective. Once the hypothesis is fixated into a formal structure, its internal consistency, derivability, and testability are determined.

Step Three: Testing is extension. By Cognitive Axiom Two, extension is the necessary unfolding under the fixated perspective. From the fixated hypothesis, specific empirical predictions are derived, then tested in observation. The test result (confirmation or refutation) is the product of extension—it does not change the formal structure of the hypothesis itself, but determines whether the hypothesis is worth continuing to focus on.

Corollary 1: The non-inductive origin of hypotheses. Traditional methodology has long been puzzled by "where hypotheses come from." Inductionists claim hypotheses are induced from data, but this argument is logically untenable—the same set of data can support infinitely many mutually contradictory hypotheses. Axiom Two reveals the true origin of hypotheses: hypotheses are the subject's active act of focus. Data can provide clues for focus, but cannot replace focus itself. This is precisely why the greatest breakthroughs in scientific history often come from "genius intuition"—those intuitions are particularly successful acts of focus, and the reason they cannot be exhausted by methodological rules is that focus is free.

Corollary 2: The hypothetico-deductive method applies to all cognitive domains. A detective proposing a suspect hypothesis, a mother proposing a hypothesis about why the baby is crying, a pastor proposing a hypothesis about the congregation's spiritual needs—these are all instances of the same dynamic process in different media. The hypothetico-deductive method is not an exclusive method of science, but a universal method of cognitive activity.

Step One: The perspectivity of a single observation. By Cognitive Axiom One, any observation is the extension trace of a specific perspective system at a specific moment. An observation result ("this metal expanded at 100°C") is not "the objective fact itself," but the object certainty tracked by that perspective system at that moment. A single observation cannot establish a law, because it is only a single extension of one perspective system—the subject cannot determine to what extent this extension result depends on the specific conditions of that moment.

Step Two: Induction is cross-perspective translation. When a researcher "repeats" an observation, he is actually independently focusing on the same object in multiple perspective systems at different times and under different conditions. Each perspective system produces its own extension trace. The researcher translates these traces—comparing, aligning, seeking consistent patterns. When the extension traces of multiple perspective systems present a consistent relational structure in translation, this relational structure is identified as a "law."

Step Three: Invariants are laws. By the similarity mathematics axiom, the smaller the translation cost between two systems, the larger the relational structure (invariant) they share. The "law" sought by induction is precisely the relational structure whose translation cost between multiple observation perspective systems approaches zero. When a hundred experiments of heating metal all show the same expansion coefficient, the translation cost between these hundred perspective systems is extremely small—the expansion coefficient relational structure they share is the law.

Step Four: The root of inductive uncertainty. By Cognitive Axiom Three, any cross-perspective translation pays cost. Induction is the translation of past perspective observation results into future perspective predictions. The temporal cost of this translation cannot be eliminated—between the past perspective and the future perspective, the initial conditions cannot be reproduced, and the boundary conditions may have unknown changes. Therefore, inductive conclusions always carry uncertainty. The root of Hume's problem is precisely the cost axiom: from "the past was so" translated to "the future will be so," an ineliminable translation cost must be paid.

Corollary: The reliability of inductive conclusions depends on the magnitude of translation cost between multiple perspective systems. When translation cost approaches zero (such as the measurement of physical constants under controlled experimental conditions), inductive conclusions can be considered certain in practice. When translation cost cannot be ignored (such as the analysis of the causes of historical events), inductive conclusions must be annotated with uncertainty ranges. Statistical methods (confidence intervals, p-values) are quantitative estimates of translation cost—they measure the degree of information loss in multi-perspective translation. Induction applies to all cognitive domains—scientists inducing experimental data, farmers inducing seasonal experience, language learners inducing grammatical rules—these are all instances of the same cognitive operation in different media. Induction is not an exclusive method of science, but a universal method of cognitive activity. The difference lies only in the magnitude of translation cost and the degree of manageability in different domains.

The three structural features are: First, the cost generation mechanism is consistent, meaning the costs of the two systems are mainly concentrated on the same edge—either both dominated by the P-M edge, or both by the M-O edge, or both by the P-O edge. Second, the cost accumulation pattern is consistent—the mathematical signatures of the two systems at the cost accumulation stage agree at low-order approximation, such as both exhibiting "diminishing growth rate" or "fluctuation amplitude proportional to stock," and other observable trends. Third, the cost settlement or reset method is consistent—after the cost of the two systems reaches the critical point, the same type of system reset appears, such as forced zeroing, periodic cleansing, or structural collapse.

By Cognitive Axiom Three, any extension pays cost. By Cognitive Axiom Two, focus → fixation → extension is universal dynamics. By Cognitive Axiom One, all cognition is within a perspective. The three structural features correspond respectively to the three stages of cost generation, accumulation, and settlement. If the two systems agree on the structures of these three stages, then their cost functionals ΔΦ(A) and ΔΦ(B) satisfy the same functional equation at low-order approximation, and the candidate formula can be directly translated through the medium parameter mapping T(A→B), with translation cost approaching zero at the formal structural level of cost dynamics.

Application examples: In economics, diminishing marginal utility (logarithmic function) and in psychology, the Weber-Fechner law (logarithmic function) agree on the three structural features. Thermodynamic entropy increase and information-theoretic information entropy agree on the three structural features. The Lotka-Volterra equations in ecology and economic cycle models agree on the three structural features.

By the Isomorphic Hypothesis Generation Theorem, when two systems are completely isomorphic at the level of cost dynamics, the cost in A and the cost in B are instances of the same structure under two different sets of medium parameters. A theorem already empirically verified in A has evidence supporting not the specific values of particular medium parameters, but the predictive power of the cost dynamics structure. When this structure is accurately translated into B through the translation mapping T(A→B), the predictive power of the structure is retained, because the invariant I is not discarded in the translation. Translation cost approaches zero, and the migration of evidence pays no additional empirical cost.

Ethical significance: When two systems' cost functionals have been proven isomorphic, repeating high-risk experiments of the same type in the target system, causing subjects to pay costs that could have been avoided, itself constitutes dishonesty toward the cost axiom. The Cross-Domain Verification Theorem is not only a methodological efficiency, but also the application of cost awareness in research ethics—obtaining equivalent knowledge at minimal experimental cost.

Application examples: After thermodynamics and information theory are proven isomorphic, information theory researchers can directly cite thermodynamic experimental data as evidence for information entropy theorems. If material mechanics and political science are proven isomorphic in fatigue damage-institutional decay dynamics, political science researchers can directly cite material mechanics' fatigue experimental data to test institutional decay models, without running fifty-year social experiments. If metabolic networks and economics are proven isomorphic in steady-state imbalance dynamics, economics researchers can directly cite metabolic network data to test price control models. If drug metabolism is proven isomorphic at the level of cost dynamics across species, medical researchers can directly translate the cost functions precisely measured in animal experiments into human cost functions through parameter substitution.

Step One: The definition of isomorphism. By Similarity Axiom Three (the translation cost axiom), any cross-perspective translation pays cost. The cost functional ΔΦ measures the relational structure discarded in translation. If and only if ΔΦ approaches zero, the two systems are completely isomorphic—the entire relational structure of the source system is retained in translation. By the Cross-Domain Verification Theorem, under this condition, a theorem already empirically verified in A can be directly regarded as verified in B in its corresponding translated version. This means that, under the premise of complete isomorphism, the empirical data of A and the empirical data of B point to the same cost dynamics structure; they are not evidence for two different facts, but instances of the same structure in two media.

Step Two: The perspective-theoretic nature of data. By Cognitive Axiom One, any data is the extension trace of a specific perspective system at a specific moment. The data of domain A is the extension result of A's perspective system tracking A's object under A's medium. The data of domain B is the extension result of B's perspective system tracking B's object under B's medium. When A and B are completely isomorphic, these two sets of extension traces carry the information of the same cost dynamics structure. The difference lies only in the medium parameters—like two sets of numerical solutions of the same equation under different initial conditions.

Step Three: The legitimacy of aggregation. By Cognitive Axiom Three corollary, cost cannot be eliminated, only managed. In traditional methodology, A's data and B's data are treated as two unrelated batches of independent evidence, processed separately. This pays double experimental cost and double statistical uncertainty. After A and B are proven completely isomorphic, this separate treatment no longer has cognitive necessity. The two sets of data can be merged under the same cost dynamics equation, and the differences in medium parameters are precisely calibrated through the translation mapping T(A→B). The merged dataset contains variance information from two sources, and statistical power increases significantly.

Step Four: The increase in statistical power. Let dataset A contain n observations and dataset B contain m observations. Under the isomorphism condition, the translation mapping T converts A's observations into equivalent observations in B's medium, producing a merged dataset containing n+m observations. The sample size of the merged dataset is larger than either single dataset, the standard error decreases, the confidence interval narrows, and the power of hypothesis testing increases. More importantly, the merged dataset contains variance information from two different media, capable of capturing cost dynamics features masked by systematic bias under a single medium, producing more robust structural identification than any single domain.

Step Five: Relation to the Cross-Domain Verification Theorem. The Cross-Domain Verification Theorem declares: under complete isomorphism, the evidence of one domain can directly support the theorem of another domain. This is the one-way migration of evidence from A to B. The Isomorphic Aggregation Generation Theorem is the inverse theorem, declaring that under complete isomorphism, the evidence of two domains can be bidirectionally merged. Together they constitute the complete framework for managing evidence under the condition of complete isomorphism—evidence can be migrated, and evidence can be aggregated. Migration reduces repeated payment of experimental cost; aggregation increases statistical power. The foundation of both is the same: the complete isomorphism of the cost functionals makes cross-domain translation cost approach zero.

Ethical significance: The Isomorphic Aggregation Generation Theorem is not only a methodological tool, but also a principle of research ethics. When two domains' cost functionals have been proven isomorphic, refusing to aggregate evidence while insisting on repeating large-scale experiments in each domain, causing subjects—whether humans, animals, or ecosystems—to pay costs that could have been avoided, itself constitutes dishonesty toward the cost axiom. This theorem declares: under the premise of structural isomorphism, separate research is not rigor, but waste; aggregate analysis is not laziness, but honesty.

Application examples: If the cumulative damage formula of metal fatigue in material mechanics is structurally isomorphic to the cost accumulation formula of political institutional decay, then decades of metal fatigue experimental data can be merged with data on historical institutional decay, producing a "cumulative damage-critical collapse" joint model more precise than any single domain. If the metabolic cost functions across species in pharmacokinetics are proven isomorphic, rat experimental data can be merged with a small amount of human experimental data, reducing the required human experimental sample size while increasing statistical power. If the cost functions of skill fixation in education and cognitive psychology are isomorphic, teaching experiment data from thousands of classrooms can be merged with laboratory cognitive training data, producing a more precise "overfocus optimal upper limit" model.

By Axiom Two (focus → fixation → extension) and Axiom Three (irreversible cost accumulation), infinitely iterated extension necessarily leads to monotonic cost accumulation on the corresponding edge. As the number of iterations tends to infinity, the decay rate or morphological transformation pattern of cost no longer depends on the accidental differences of initial conditions, but is determined by the structural characteristics of that edge. The cost on the M-O edge is concentrated on the certainty relation between medium and object; the observer can be standardized, so the constant can be precisely measured and manifests as a numerical constant. The cost on the P-M edge is concentrated on the relation between subject and meta-medium; all humans share the same meta-medium—the body—and its rhythms constitute a universal bodily-rhythm constant. The cost on the P-O edge is concentrated on the relation between subject and specific object; the uniqueness of the subject's perspective makes the constant non-numerical, but the repeatedly appearing relational morphological transformation pattern constitutes an identifiable morphological constant.

For numerical constants, by Theorem 105 (Measurement Is Intervention Theorem), any measurement is medium extension; the reading obtained is the extension result after the object is touched by the medium, and the observed value necessarily contains the cost paid by the measurement act. By Theorem 100 (Cost Distribution Non-Uniformity Theorem), the cost accumulation rates of different domains are naturally different. Dimensionless constants do not depend on any artificial unit, cross all measurement media, have cost approaching zero, and can be directly anchored as ideal values. Dimensional constants depend on human-defined measurement units; the unit itself is a medium, bearing cost. By Theorem 102 (Cost Formalization Incompleteness Theorem), cost cannot be completely formalized, so constant correction has no precise unified formula—the correction factor can only be identified, not strictly derived. Constants are identified, not calculated.

Corollary One: Different types of universal constants exist on different edges. The numerical constants of the M-O edge can be identified by standardized measurement and cross-perspective translation invariance. The bodily-rhythm constants of the P-M edge can be identified by universal biological features. The morphological constants of the P-O edge can be discovered by cross-case morphological pattern recognition. Constant discovery in various fields follows a unified identification logic; the discovery of specific constants belongs to the work of domain research.

Corollary Two: The ideal value of the fine-structure constant α is very likely 1/137. A dimensionless constant, it possesses cross-perspective invariance in all measurement media, and can be directly identified as an ontological constant or its extremely close approximation.

Corollary Three: The ideal value of the vacuum speed of light c is very likely 3×10⁸ m/s. Meters and seconds are human-defined medium units, bearing measurement distortion.

Conjecture One (Constant Aesthetic Criterion): The more concise the constant, the closer to the ontological value. This criterion is not the result of strict derivation, but a reasonable hypothesis based on the ontological beauty of the created order. Its reasonableness is supported by cross-constant consistency checks: if multiple independent constants, after correction under the same aesthetic criterion, have their internal relations become more concise and self-consistent, the credibility of the criterion increases.

Conjecture Two: The internal relations among the corrected ideal values of constants present a more concise mathematical structure. The specific relations await further research.

Boundary: In this theorem, the existence of universal constants, their classification, and the identification of the ideal values of dimensionless constants are strict derivations; the constant aesthetic criterion and the specific numerical conjectures of dimensional constants based on it are reasonable hypotheses, not strictly derived conclusions. This is the fulfillment of the Cost Formalization Incompleteness Theorem—constant correction has no precise unified formula.

Step One: Select modification dimension. Axiom One is constituted by subject, medium, object, and relation. The subject is not affected by any medium and does not enter the modification scope. The modifiable dimensions are three: the medium layer—changing the legitimate medium used in the field (such as from Euclidean geometry to Riemannian geometry, from the money market to digital currency protocols); the object layer—changing the object boundary pointed to by the field (such as from macro low-speed matter to near-light-speed matter, from the rational agent to the bounded-rational agent); the relation layer—changing the relational structure of the field (such as from F=ma to F=dp/dt, from supply-demand equilibrium to non-equilibrium dynamics).

Step Two: Set modification extent. The modification can be replacement—completely replacing the old with a new medium, new object, or new relation; extension—adding new elements on top of the old, such as from real numbers to complex numbers; or limitation—adding constraints on top of the old, such as limiting the object to low speed, limiting the medium to linear operators.

Step Three: From the modified axiom set, strictly derive new theorems. The modified axiom set constitutes a new perspective system S' = (P, M', O', R'). From S', following the same derivation rules as the old theory, derive a new set of theorems.

Step Four: Test the new theorems. They can be tested directly by designing experiments or collecting data within the field, or indirectly through the Cross-Domain Verification Theorem, using experimental data already available in other isomorphic domains.

Step Five: Iterate or accept. If the new theorems are supported by empirical data, the new theoretical system is established. If refuted, return to Step One, select another modification dimension or extent, and re-execute the process. This process can be systematically iterated until all possible modification directions have been explored.

Derivation: Axiomatization compresses a field's knowledge system into a finite number of axioms and theorems strictly derived from the axioms. The modification of any axiom or theorem within the system is equivalent to changing the relational structure R of that perspective system. By Cognitive Axiom One, any change in relation R constitutes a new perspective system. By Cognitive Axiom Two, focus is a free act—the researcher can actively choose to focus on any axiom and attempt to modify it. By Cognitive Axiom Three, each modification pays cost—the modified derivation may be empirically refuted, but the refutation itself provides information, narrowing the scope of the next attempt. Axiomatic innovation is an operable, iterable, systematically executable process.

Application examples: After Euclidean geometry was axiomatized, Lobachevsky and Riemann modified the fifth postulate, deriving hyperbolic geometry and elliptic geometry respectively. After Newtonian mechanics was axiomatized, Einstein modified the implicit presupposition "time is absolute," deriving special relativity. If political economy completes axiomatization, modifying the axiom "wealth and power necessarily transform into each other" to "there exists a deceleration mechanism between wealth and power" can derive a whole new set of political economy theorems. If education completes axiomatization, modifying the axiom "fixation requires long-term high-intensity focus" to "fixation can be periodically broken without losing core capacity" can derive an educational methodology system centered on "cognitive Jubilee."