
This paper constitutes Stage X of an independent Phase–Scalar research program developed between 2025–2026. It introduces the Phase–Scalar Spiral as a universal diagnostic for identifying why complex systems fracture as they scale. Across mathematics, physics, distributed computing, artificial intelligence, biological systems, and organizational systems, recurring failure modes appear when systems scale: paradoxes in infinite constructions, divergences and "ghosts" in certain quantization programs, coordination breakdowns in distributed systems, hallucinations in large language models, dysregulated growth in biological systems, and metric-induced breakdowns in human institutions. These phenomena are typically treated as domain-specific anomalies. This paper proposes a unifying diagnostic explanation: many scale-failures are representational mismatches generated when global measurement and abstraction (Scalar) outrun local relational coordination (Phase). Building on Phase–Scalar Reconstruction (PSR) and the Spiral Coordinate System (SCS), we formalize a general principle: coherence is preserved when Phase precedes Scalar at each recursive level of growth. We present the Phase–Scalar Spiral as an applied synthesis: a cross-domain lens that explains why locality-preserving methods (e.g., renormalization, neighborhood-based computation, context-grounded inference) stabilize systems, while scalar-first expansions often produce artifacts (divergence, paradox, hallucination, institutional dysfunction). The contribution is methodological and diagnostic: it proposes no new physical laws and does not modify established domain formalisms. Instead, it offers a disciplined way to classify when a problem is ill-posed due to category collapse, and how to restore coherence by re-establishing local coordination before re-scaling.
coordination, quadratic gravity, ghosts, PSR-B, representational mismatch, Phase and Scalar Time, infinite sets, renormalization, Spiral Coordinate System, boundary enforcement, cross-domain synthesis, temporal representation, systems theory, cancer biology, distributed systems, locality preservation, PSR, CAP theorem, diagnostic methods, organizational coherence, Phase–Scalar Reconstruction, scaling failure, AI hallucination, Phase–Scalar Spiral, coherence, biological coherence, measurable graphs, system coordination, SCS
coordination, quadratic gravity, ghosts, PSR-B, representational mismatch, Phase and Scalar Time, infinite sets, renormalization, Spiral Coordinate System, boundary enforcement, cross-domain synthesis, temporal representation, systems theory, cancer biology, distributed systems, locality preservation, PSR, CAP theorem, diagnostic methods, organizational coherence, Phase–Scalar Reconstruction, scaling failure, AI hallucination, Phase–Scalar Spiral, coherence, biological coherence, measurable graphs, system coordination, SCS
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