One Bit, Two Sides, Three Theorems: Records, Hinges, and Mirror-Paired Transitions
Abstract
One Bit, Two Sides, Three Theorems: Records, Hinges, and Mirror-Paired Transitions A Bit, a Hinge, and Two Hands: The Exact Anatomy of a Finite Pregeometric Transition Before geometry supplies distance or orientation, a finite theory must still answer three concrete questions: what can represent a Boolean event, where can a two-sided transition occur, and what exact map carries it into mirror-related outputs? We answer them with three linked theorem families.
Records. Equality of normalized first-order exposures gives eta_A(m_evt)=eta_A(q_rec) if and only if q_rec=m_evt ; hence a declared untouched/touched event fixes the compatible candidate record alphabet at two.
Hinges. In the unsaturated sector 0 <= T < M , under the declared two-sided gate, activity somewhere is equivalently expressed by a maximum over hinges of the minimum of their two pressure shares, with closed path and cycle counts, transfer-matrix probabilities, and a binary firing law for M >= 4 .
Mirror-paired transitions. At an active hinge, an explicit map V: C^8 -> C^32 satisfies V^dagger V=I_8 . The declared alphabet normalization gives g=(1/2)ln 2 ; within the declared product-form transition class, covariance and transport then fix g_L=g_R=g_o=g , and the declared mirror obeys M_out V_g_o=V_-g_o S_in .
Exact reach. The clique complex induced by the transition's direct factor scopes has dimension at most two, its hinge address remains controller input, and its orientation output is coherent but not a persistent zero-error record. In addition, no spectral function of P_V is a connected graph Laplacian on the 32 output labels. These results make the construction a precise finite input for later geometric models.
Files and reproducibility. This record contains exactly four direct uploads: the manuscript PDF, Supplemental Material PDF, Bulmash_Boolean_Hinge_Chirality_Source.zip , and RIGHTS_NOTICE.txt . The source archive contains the LaTeX sources, bibliography, dependency map, deterministic figures and build instructions, and a standard-library finite verifier with frozen expected output.
Rights. Scientific text and assets are All Rights Reserved. Identified build and verification software inside the source archive is MIT-licensed. The separately uploaded rights notice states the controlling file-level boundary.
Lineage. Quantum Genesis v6 is the broader public framework and derivation lineage ( DOI 10.5281/zenodo.21744228 ). This dedicated record reproduces every premise and proof it uses. Its project DOI is 10.5281/zenodo.21961046 .
Publication status. Author-formatted, non-peer-reviewed preprint. DOI 10.5281/zenodo.21961046 identifies this dedicated project record.
Complete record: ORCID 0009-0003-7030-144X · all Zenodo deposits