Mathematical Foundations of the Information Isomorphism Law: A Complete Formalization of the Three Ontological Levels, Detection Sufficiency, and Exact Permutation Control

The Information Isomorphism Law (Tang, 2026ac) claims that in complex systems grounded in objective data and exhibiting regime switching, statistical structures carry information about regime structures. The law contains three progressive ontological levels—existence (information traces exist), correspondence (systematic correspondence exists between traces and regimes), and precision (encoding is precise on truth-known systems). This paper surveys the three papers of the mathematization series of the Information Isomorphism Law (Tang, 2026aq–2026as), and formally declares that the three levels have acquired complete mathematical foundations in the dimension of structural break detection. The sufficiency direction is established in two steps: Tang (2026af) proves the Global Maximum Point Invariance Theorem in the linear model—the unified shape function attains a global unique maximum at the true boundary, and the standardized limits of Chow F/n, Wald*/n, and LR/n share this maximum point; Tang (2026aq) generalizes it to the L² function space, proving that the L² unified shape function attains a global unique maximum at the true boundary, that the multi-regime sufficiency theorem gives a global minimum of the functional-space BIC at the true boundary set, and that the spectral weighted sufficiency theorem establishes sufficiency uniformly over the entire spectral weighting family. The X-dependent convex combination lemma provides the geometric foundation for the multi-regime generalization; the detection information loss corollary and the detection resolution–information lower bound duality proposition quantitatively connect detection geometry with the estimation information limit. The necessity direction is established by Tang (2026ar): under the null hypothesis, the unified shape function tends to zero in probability everywhere (at rate O_p(n^{-1} log n)); the finite-sample false-peak control theorem proves that the permutation test exactly controls the false positive rate; the extreme value limit theorem proves that the standardized statistic converges to the Brownian-bridge supremum distribution (aligned with Andrews, 1993); the permutation test power theorem and the local power theorem complete the full characterization under the alternative. The bijection is completed by Tang (2026as) through the conjunction: a regime switch exists if and only if the population unified shape function attains a global unique maximum; the Local Bijection Theorem, based on the one-sided kernel locally weighted unified shape function, generalizes this to a pointwise correspondence in the multi-regime model; and the pseudo-peak phenomenon proposition of global splitting establishes the necessity of localization through an explicit counterexample. This paper gives a Preliminary Establishment Statement: in the statistical dimension of structural break detection, the three ontological levels of the Information Isomorphism Law have all been mathematically formalized, and the conjunction of the sufficiency and necessity directions constitutes the bijection theorem. This paper provides a cross-disciplinary verification matrix that maps the sufficiency and necessity directions to empirical findings in six disciplines—finance, astronomy, statistical physics, particle physics, chemistry, and nuclear physics; consolidates open problems IIL-T1 to IIL-T22; and provides a concrete tool list for subsequent research. Together with the Switch Measure Theory survey (Tang, 2026al) and the Factor Hierarchy Law mathematization survey (Tang, 2026ap), this paper constitutes the complete survey of the three mathematical series of the Factor Hierarchy framework. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently directed by the author. DeepSeek assisted with code implementation, data presentation, and text drafting. The author takes full academic responsibility for the final content.