This document establishes the ontological status of mathematics within the Energy-Efficiency Theory (EET) framework. Mathematics is not an independent reality standing alongside the physical world, not a pure mental construction, not an arbitrary symbolic game, not an eternal Platonic entity. Mathematics is the structural space within the Rule-Causal domain --- a sub-model generated by the application of the modeling rules (causality, logic, the seven operations of the Generative Grammar, MEER maximization). It constructs stably invariant formal structures under the transformational grammar provided by logic, and is continuously revised and extended in interaction with physical cognitive models. Core Constitutional Position: Mathematics is a sub-model generated by the modeling rules. It does not define new irreducible modeling rules; it describes the structural space that the constitutional rules of causality and logic generate. Its constitutional status is Satellite Paper (SAT) --- dependent on and parasitic upon the constitutional mother texts (Causality v1. 7 and Logic v1. 2). The Ontology of Mathematics belongs to the Cognitive Constitution. Its employment of physical-constitutional concepts (cognitive heat engine, Ben-Shi dynamics) constitutes instrumental application by a cognitive model, not constitutional dependence. The Ultimate Engine --- Causal Arbitrage: The pursuit of mathematical necessity is the maximal-MEER form of causal arbitrage in the virtual domain. By stipulating axioms and derivation rules (Presupposition of Cause), the cognitive system converts the high-cost, ongoing verification of physical causality into the low-cost, one-time application of symbolic consistency. Mathematical structures, once constructed, can be invoked at near-zero marginal cost to generate predictions about physical systems---the ultimate arbitrage return: temporal buffer maximized, response energy minimized. Core Constitutional Contribution --- Completeness Feedback: Mathematics possesses an inherent drive toward systematization. This drive leads it to systematically probe the limits of the modeling rules themselves---in consistency, completeness, decidability, and categoricity---and to feed these limits back to the rule-makers. This Completeness Feedback is the endogenous engine of the modeling rules' self-refinement. Mathematics is thus not merely the passive product of the modeling rules; it is the ``constitutional mirror''---the only device through which the modeling rules can see their own limitations. Core Insight --- Rhetoric of Rupture, Operational Continuity: What the mathematical community has historically narrated as ``crises'' were, from the constitutional perspective of the mathematics mother model, Divides---internal symmetric divisions that produced new structural spaces while preserving the old ones. The rupture was rhetorical; the operation was continuous. The mathematical mother model never contracts, only expands (the Principle of Elastic Expansion). Mathematics as a Language/Symbol Protocol: Mathematics occupies the limit endpoint of the parameter-sharing / trigger-precision trade-off (Language v1. 0). It sacrifices parameter-sharing---accepting that only a tiny fraction of the population can decode its signals---to achieve near-perfect trigger precision and cross-generational transmission fidelity. Mathematical symbols are second-order externalizations: they refer not to perceptual objects but to operations on already-established distinctions. Mathematical texts are frozen operation sequences---encoded instructions for re-executing cognitive operations, not passive records of results. The decoding cost of mathematical protocol is not significantly lower than its encoding cost---mathematical knowledge cannot ``diffuse'' but must be rebuilt by each individual through the Conscious Self's active re-execution of the encoded operations. Mathematics as a Cognitive Heat Engine: The mathematical system operates as a cognitive heat engine (Xu-Shi v3. 1) governed by the four universal laws of Ben-Shi dynamics (Ben-Shi v3. 0). New axiom stipulation provides the high-Xu heat source; theorem Encapsulation is the work; encapsulated theorems form the low-Xu heat sink. The system's capacity for flexible oscillation between exploration and consolidation monotonically contracts over its lifetime (Ben-Shi Irreversibility), and its metacognitive precision decays with constraint accumulation (XQ Decay Law). Structure pathology---the accumulation of formally valid but low-MEER structures---complements the grammar pathology of logic. Mathematical Intuition, Aesthetics, and Genius: Platonic intuition is the phenomenological correlate of Algorithmic Submersion---Divide and Encapsulate operations hardened to invisibility through repeated successful application. Mathematical elegance is the Emotional Self's direct phenomenological readout of MEER: Elegance (P) = structurally necessary conclusions / (derivation steps axioms and lemmas invoked). Mathematical genius is the precise synchronization of the Automatic Self (submerged intuitions), Emotional Self (aesthetic navigation of MEER), and Conscious Self (explicit proof audit). Unified Resolution of Six Major Problems: enumeratelabel= (*) Wigner's problem: the effectiveness of mathematics comes from joint evolution---physical and mathematical cognition share the same underlying modeling grammar and co-evolve in interaction with real space; Benacerraf's problem: dissolved, not resolved---mathematical cognition requires no cross-domain channel because it shares the same operational grammar with physical cognition; The fragmentation of foundations: set theory, category theory, and homotopy type theory are different schemes of the Presupposition of Cause, each valid within its own Root Cut boundary; Undecidability: the ``charter of blindspots'' of the modeling rules---the constitutional boundary of what any finite formal system can determine; Stratified effectiveness: the applicability of mathematics is stratified because causal compression itself is stratified; Platonic intuition: the first-person experience of hardened Divides---operations submerged to invisibility perceived as eternal entities. enumerate The document establishes the complete interfaces between mathematics and the EET constitutional system, articulates its own constitutional boundaries and meltdown conditions, and provides the systematic framework for understanding mathematics as both the product of the modeling rules and the driver of their self-refinement. Keywords: Ontology of mathematics; Rule-Causal domain; structural space; Presupposition of Cause; joint evolution; completeness feedback; causal arbitrage; Wigner's problem; MEER; virtual domain; encapsulation network; rhetoric of rupture; operational continuity; structure pathology; mathematical intuition; mathematical aesthetics; mathematical protocol; parameter-sharing / trigger-precision trade-off; frozen operation sequences; decoding cost asymmetry; elegance as cognitive operational efficiency; cognitive heat engine; Energy-Efficiency Theory
Hongpu Yang (Sat,) studied this question.