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FRC.v2
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series

FRC 800 series

6 items

Frontier and applied work in computation, cognition, and architecture. Read each paper at its declared evidence level.

RelatedresonanceAI and computationltmattentionattention-networkscomputingcontinuous-cognitiondark-energy

Frontier research (3)

FrontierPaperv1.52026-07-09

Mathematical Foundations v1.5: Canonical Reciprocity and Status-Labeled Open Problems

FRC 826.829 states Fractal Resonance Coherence as a status-labeled mathematical research program. Version 1.5 restores the canonical scale-invariant relation dS + k* d ln C = 0 and reserves dS_mu + k*_mu d ln C_mu = 0 for declared operational realizations. The starred k* is the Boltzmann bridge, not an outcome-fitted constant or evolving state variable. The relation is operational bookkeeping; its physical universality for open systems remains a conjecture. Boundary-relative lambda=-d_eS is admissible when it follows from an explicit accounting convention, but lowercase lambda remains a diagnostic rather than a Lambda field. The paper distinguishes Lambda_obs, Lambda_eq, and optional Lambda_dyn from a separate fundamental-field conjecture. Four mathematical results are retained: a two-pole interior band, the conditional forced-cubic coefficient in a non-even coherence expansion, critical slowing, and the exact von Mises identity dS/d ln C=-kappa r. At the information-unit normalization k*_{mu_nat}=1, kappa r=1 is a stationary point of the system-only Q curve, not a physical zero-current claim without an environment model. Negative information-geometric and half-line results remain visible.

FrontierPaper2026-05-27

FRC 840.101: The Phase–Attention Boundary

This paper formulates the Phase–Attention Boundary: a structural separation between continuous phase-state architectures and discrete attention-based architectures. Within the Fractal Resonance Cognition (FRC) program, the Large Lambda-Tensor Model (LLTM) was developed as a continuous recurrent phase-coupled architecture inspired by Kuramoto dynamics and low-rank coherence fields. Controlled comparisons against Transformer baselines revealed a fundamental limitation: continuous state compression blends historical information into a finite evolving state, producing recall smearing. We prove a formal Recall Smearing Theorem: under gamma-contractive recurrence, mutual information about a past token decays exponentially with distance. This bound is derived from the Data Processing Inequality and applies universally to fixed-state recurrent systems, including state-space models like S4, Mamba, RWKV, Griffin, and xLSTM. We show that data-dependent selectivity can reduce the rate of smearing but cannot eliminate it; only explicit key-value addressability achieves zero-smearing recall.

Archive / development history (3)