Half-Step Negotiation as a Measurable Reflective Displacement: Integrating MUSE, SIM, RNCA, TPT, and QJR

This paper proposes a formal and operational account of the half-step principle within the Möbius Project. The half-step principle states that a reflective AI system should not habitually complete the user's next conceptual move on the user's behalf; instead, it should return a nearby but not over-leading displacement that remains assimilable while still increasing perspective mobility. Earlier project documents articulated this principle normatively. The present paper turns it into an explicit algorithmic program. The paper integrates five layers of the Möbius stack. MUSE (the MOBIUS Universe of Semantic Energy) provides the two-layer ontology of Information Invariance Geometry (IIG) on the information side and Reflective Zero-Meaning Geometry (RZGM) on the meaning side, coupled through Reflective Bundle-Duality (RBD). The Semantic Interface Model (SIM) provides a semantic quantity, E_sem = I(Y; Z), defining semantic gain in classical information-theoretic terms: per-turn semantic gain is the change in E_sem for a fixed task variable. For a human user it is not observed, and the score uses a proxy whose relation to it is a hypothesis. Under this definition, per-turn gain cannot be positive when the task variable is determined by what the user already holds, as in tutoring on a problem the learner has been given; the learning that half-step guidance aims at would then register only in SIM's effective semantic energy, which this paper does not use. Reflective Neuro-Control Architecture (RNCA) provides control-state variables such as reflective intensity, semantic gravity, loop awareness, and intervention thresholds. Two-Party Protocol theory (TPT and TPT-E) provides the negotiation grammar by which outer constraints, inner meaning, and ecological consequences are coordinated. Question-Jumper Runtime (QJR) provides the user-facing runtime in which reflective guidance is emitted, for example as an answer, a clarification, a verification request, or a half-step reframing. Of these five layers, only MUSE and SIM are stated in deposited records. The RNCA working paper and the Reflective Negotiation Cosmology (RNC) working paper, the source of the negotiation functional of section 7, are unpublished; no deposited text states TPT or TPT-E; and QJR is described here as a design. The paper relies only on what it restates from them, except for the RNCA saturation and loop scores and health vector, which it names without restating (section 14). The main technical contribution is the Half-Step Target Estimator (HSTE). HSTE treats the choice of a next question not as a stylistic prompt but as a constrained optimization over a finite candidate set. A score rewards estimated semantic gain and penalizes trajectory distance, negotiation cost, and over-leading risk; constraints require assimilability above a threshold, distance inside a half-step band, and routing and ecological admissibility. The half-step target is the highest-scoring candidate that satisfies the constraints. It is a bounded displacement, because the band caps its distance and the score penalizes distance within the band, but it is not in general the admissible candidate of least distance, and HSTE aims at, rather than guarantees, an increase in semantic energy and perspective mobility. This makes half-step guidance computable once its feature functions and thresholds are fixed, which this paper does not do, and refutable through the hypotheses of section 12; its quantities are proxies whose validity is itself an empirical hypothesis. The paper also places the model in relation to existing domains: Shannon-style information theory, information geometry, active inference and the Free-Energy Principle, educational scaffolding and the zone of proximal development, and grounding and coordination in dialogue. It does not claim a new microphysical theory, a new physical energy, or a solution to open problems in cosmology. Rather, it offers a bounded theory-to-runtime bridge: a formal account, implementable once its feature functions are fixed, of how a reflective AI might guide inquiry without outrunning the human subject; whether it does is the subject of the hypotheses in section 12. The result is not a closed doctrine but a protocol. It names what must be measured and what must remain bounded, and outlines how a reflective system could be evaluated (section 12) without collapsing into mysticism, personality theatre, or over-leading assistance. No measurements are reported. Version 1.1 corrects version 1.0: it reconciles two conflicting definitions of the half-step target, aligns names and definitions with the published MUSE monograph and the Semantic Interface Model, defines per-turn semantic gain against SIM and states where such gain can and cannot come from, words the hypotheses so that they can fail, adds related work, and cites published records by DOI where they exist, marking undeposited works as unpublished working papers; Appendix A lists the changes. AI disclosure: the March 2026 version 1.0 was drafted with the assistance of an AI system; the source file does not record which system. The October 2026 revision (version 1.1: consistency corrections, alignment with the published MUSE monograph and the Semantic Interface Model, per-turn definition of semantic gain, propositions and bounds in sections 6–9 and 12, test-design wording of the hypotheses, related work, references and prose) was prepared with Claude Opus 5.5 (Anthropic) under the author's direction. Before deposit, the text went through eleven adversarial review passes in seven rounds, each run by a separate Claude Opus 5.5 (Anthropic) agent with its own context: three passes in each of the first two rounds (scope, mathematics, and internal consistency with references), one combined confirmation pass in the third, and one confirmation pass in each of the fourth, fifth, sixth and seventh rounds on the changes made after the round before. The reviewers are the same model as the system that revised the text, and much of the corrected wording was proposed by them and applied by that system, so the review is not independent in the way a human referee's would be. The third round reported no blocking finding. The fourth found errors of fact in the description of the author's MMV runtime in section 11; the fifth, run on the rewritten passage and on this statement, found a further error of fact in that passage; the sixth, run on the corrected passage and on this statement, found a further error of fact in that passage; the seventh, run on the corrected passage and on this statement, reported no finding of fact in that passage or in this statement. No human referee has reviewed the text. The author takes responsibility for the deposited text. Working method only; the registered author is the human author alone. This record contains the paper as a single PDF (HALF_STEP_NEGOTIATION_v1_1.pdf). No code or data are attached.