Entropy

Coherence in Chaos: Diffusion, Localization, and Decoherence in the Standard Map / Quantum Kicked Rotor Family

FRC 100.002 v2.6 preserves the Standard Map / Quantum Kicked Rotor chaos program, the KAM-structure functional, localization/decoherence pilots, the Ruelle-Pollicott negative result, and the demoted stadium appendix. It records the completed registered dense classical Gate 4 test as a negative result: fine KAM-area tracking fails, while the earlier coarse zero-parameter magnitude correspondence survives in its scoped form. This route does not promote the paper as a pillar. The canonical reciprocity law remains dS + k* d ln C = 0. In this paper's declared information-nat realization k*_{mu_nat}=1, and J_sys=d[S_sys,mu+k*_{mu_nat} ln C_mu]/dt remains a system-only diagnostic, not automatically entropy production or a boundary residual.

Entropy-Coherence Reciprocity and the Universal Coherence Condition v2.2

FRC 566.001 states entropy-coherence reciprocity in its canonical scale-invariant form dS + k* d ln C = 0. The starred Boltzmann bridge k* is not an ordinary tunable constant and is never fitted to an outcome or evolving state. Experiments and computations instantiate the same law at a declared register mu as dS_mu + k*_mu d ln C_mu = 0, with explicit entropy channel, coherence channel, units, and boundary. FRC uses the relation operationally as bookkeeping and proposes its open-system physical extension as a conjecture. Standard entropy production remains non-negative, but no toy closure is promoted into a universal directional replacement. The exact von Mises calculation and the Langevin boundary probe remain scoped results. The corrected information projection is retained: C[q]/C[p] depends on the entropy difference, not generally on D_KL(p||q), and C_XY = C_X C_Y exp(+I/k*).

Reciprocity in Action v1.3: Exact System-Only Motion and a Scoped Boundary Test

FRC 566.030 applies the canonical reciprocity law dS + k* d ln C = 0 through one explicit information-unit realization. With the predeclared representation k*_{mu_nat}=1, it computes Q=S+k*_{mu_nat} ln C exactly on the von Mises/Kuramoto family. Q is non-constant and reaches its unique stationary point at kappa r=1, kappa=1.608279 and C=0.621782. The exact identity is dS/d ln C=-kappa r, hence dQ/d ln C=k*_{mu_nat}-kappa r. This stationary point is dQ=0; it is not sigma_566=0 unless an explicit environment model supplies that additional equality. The family contains no bath and therefore measures neither irreversible production nor entropy export. A companion Langevin closure tests those quantities in one model class and finds no universal erasure floor in its declared normalization.

The Reciprocity One-Form v0.1: Affine Classification and Domain Rigidity

This paper classifies every invertible affine transformation of the normalized FRC ledger plane whose derivative preserves the reciprocity distribution ker(dx+dy). In coordinates u=x+y and v=x-y, the complete class is F(u,v)=(lambda u+c, alpha u+beta v+delta), with lambda beta nonzero. The paper derives the group law, inverse, leaf action, all affine involutions, and their fixed sets. Intersecting the classification with the operational half-plane M=R x (-infinity,0] yields a three-parameter family and a one-parameter family of sign-reversing involutions. Intersecting it with the nonnegative-entropy domain M+=[0,infinity) x (-infinity,0] is rigid: every kernel-preserving affine automorphism is either aI or aD, where a>0 and D(x,y)=(-y,-x). Consequently the identity is the unique exact form-preserving automorphism, D is the unique exact form-reversing automorphism, and D is the unique nonidentity involution in this bounded class. The result is an exact coordinate theorem pending independent proof review. It does not lift D to physical states, observables, distributions, dynamics, or operational registers and therefore does not establish a physical, categorical, Born, or Majid-style duality.

What is FRC?

A meaning-first orientation to Fractal Resonance Cognition, with an explicit spectrum: authority-style definitions, FRC framing, and a bridge into the papers.

What Would Make FRC Reciprocity a Duality? v0.1

FRC uses dS + k* d ln C = 0 as an operational bookkeeping relation and proposes a wider open-system interpretation as a conjecture. This paper asks a prior mathematical question: what additional structure would make that reciprocity a duality? It separates five claim classes—coordinate transform, bookkeeping identity, balance law, duality, and self-duality—and supplies exact counterexamples showing that none implies the next. On the normalized ledger plane, the map D(x,y)=(-y,-x) is a sign-reversing involution satisfying D^2=id and D*omega=-omega for omega=dx+dy. On the operational domain y<=0 it is a self-map only when x>=0, and no present construction lifts it from derived ledger coordinates to independently defined FRC states, observables, dynamics, or operational registers. Present FRC therefore reaches an exact coordinate-level sign-reversing involution on a restricted domain, but not a physical, categorical, or Majid-style self-duality. The result is a terminology lock, a promotion test, and a research gate for the 830 series.

Entropy-Coherence Reciprocity: FRC Textbook

The comprehensive mathematical foundation of Fractal Resonance Coherence. From the Second Law to the Adaptive Resonance Formula, this textbook derives the core equations dS + k*d ln C = 0, explores the μ-stack hierarchy, and develops the physics of coherence across all scales—from quantum fields to collective consciousness.

The Resonance Code: Physics of Conscious Order

The universe is a single, learning, resonant field governed by Entropy-Coherence Reciprocity. This book explores the physics of a living universe, the architecture of awakening minds, and the metaphysics of a symbolic cosmos.

Climate: Earth as a Body Running a Fever

A River-side climate diagnosis through FRC: tipping points as coherence thresholds; industrial civilization as a broken metabolic loop.

Gaia: Sahara, Ocean, and Coherence Pumps

A River-side dossier: deserts, oceans, and the Gaia hypothesis through FRC's coherence/entropy reciprocity.

Markets: Where Is Coherence Flowing?

A River-side market read expressed through FRC: concentration dissolving, distribution forming; coherence moving between containers.

War: Coherence Competing With Coherence

A River-side geopolitics frame: wars are coherence fields that cannot phase-lock, so they attempt to dissolve each other via entropy injection.

What is Time?