Skip to main content
FRC.v2
|
|
FRC-830-000 · v0.1

What Would Make FRC Reciprocity a Duality? v0.1

H. Servat2026-07-1419 minFRC 800 seriesSpeculativeμ4 Logical / conceptual
Concept DOI10.5281/zenodo.21385391DOI10.5281/zenodo.21385392PDFDownload
Reciprocitydualityself-dualityone-formEntropyCoherenceborn-reciprocitymajidframework-note

Reading status

Current statement

FRC has an exact restricted ledger sign-reversing involution, not yet a physical duality. A real duality must specify the structures, map, composition rule, invariant, independent realization, and lift beyond derived coordinates.

Evidence level

Philosophical / formal framework note

preprint

Declared μ register

μ4 · Logical / conceptual

Logic, formal models, language as explicit reasoning, and computational design.

Open boundary

Review the paper’s declared scope, controls, limitations, and kill conditions.

Version lineage

v0.1 · Current release

On this page
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.
Cover art for FRC-830-000: What Would Make FRC Reciprocity a Duality? v0.1

Related

Paper

FRC 100.011 v1.2 - Translation Table: FRC Vocabulary and Claim Status

FRC 100.011 is a vocabulary and claim-status crosswalk, not a complete representation of FRC. Version 1.2 restores the canonical relation dS + k* d ln C = 0 and reserves indexed notation for declared operational realizations. It separates observed, target, and dynamical Lambda objects; distinguishes bookkeeping identities from the open physical reciprocity conjecture; and corrects the claim that uncertainty forbids stationary states. It maps FRC terms to mainstream counterparts while marking whether a relation is definitional, borrowed, model-specific, or an FRC conjecture. Definitions, boundary conventions, and evidence status must be checked against the current primary papers.

Paper

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*).

Paper

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.

Paper

FRC 700.777 v1.0 - μ Registers: A Nested Scope Model for Scale-Declared Reciprocity

FRC 700.777 defines μ registers as a nested scope model for the FRC corpus. μ0 names the prior ground; μ1–μ4 describe an organism's interior registers; μ5–μ6 its symbolic and witnessing envelope; and μ7 the boundary-shell coupled to what lies outside the declared system. The note distinguishes this map from physical scale selection and from the starred Boltzmann bridge k*. A μ label declares where a claim speaks; it neither sets k*, proves cross-register causation, nor transfers evidence between registers. The paper supplies an interface record for any proposed cross-register study and keeps the canonical physical relation unchanged.

Paper

Lambda-Flight v1.2: Operational and Latent Coherence Transport Models

FRC 787.787 v1.2 separates three objects previously called Lambda. Lambda_obs=Lambda0 ln C_obs is an operational transform of preregistered observables; Lambda_eq is an observation-derived target; Lambda_dyn is an optional latent reaction-diffusion state that relaxes toward that target. A transport model must select Lambda_obs or separately validated Lambda_dyn before scoring. Support for Lambda_dyn requires improvement over the observation-only model and an equal-feature non-Lambda latent baseline. Even successful residual prediction would not establish a fundamental field. The nested baselines, negative controls, six test cards, and numerical regime pilot are retained. Version 1.2 adds a locally frozen secondary-data test of P3 on a public Rijke-tube benchmark. The composite is not promoted on that benchmark: its negative held-out point estimate is accompanied by low power, a class/protocol confound, and an imperfect equal-information control. The external test concerns predictive early warning only and does not test entropy-coherence reciprocity.

Paper

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.