Skip to main content
FRC.v2
|
|

series

FRC 800 series

12 items

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

RelatedkuramotoReciprocityCoherencelift-obstructionno-goresonanceAI and computationborn-reciprocity

Frontier research (8)

FrontierPaperv0.12026-07-15

Dimension Threshold for FRC Reciprocity v0.1: One-Parameter and Qubit Lift Obstructions

The FRC ledger involution D(s,y)=(-y,-s), with normalized entropy s=S_mu/k*_mu and y=ln C, cannot act nontrivially on any one-parameter family whose coherence is strictly monotone and whose normalized entropy decreases with log coherence. The preserved coordinate v=s-y is then injective, so every lift is the identity restricted to u=s+y=0. Exact normalized reciprocity ds+dy=0 sharpens the result: a connected curve lies on u=c; if c is nonzero there is no same-family lift, while c=0 permits only the trivial identity on a ledger-injective family. The wrapped-Cauchy family supplies an exact Poisson-kernel example with stationary coherence 1/sqrt(3) and algebraic fixed ledgers. The full qubit state space crosses the dimension threshold: its two-dimensional ledger image contains a nonempty D-invariant region K with a one-dimensional fixed leaf and admits explicit inequivalent set-theoretic fiber lifts. Those lifts are not physical promotions. No incoherent operation, no unital qubit CPTP channel, and no unitary or antiunitary Wigner symmetry realizes D on all of K. A general nonunital coherence-generating CPTP or resource-assisted lift remains open.

FrontierPaperv0.22026-07-15

FRC and Quantum Born Reciprocity v0.2: A Present-Structure Obstruction to Majid-Style Self-Duality

FRC has an exact normalized entropy-coherence ledger involution D(s,y)=(-y,-s), a typed category of operational registers, and now both one-parameter and full-qubit lift obstructions. This paper asks whether those results constitute Born reciprocity or Majid-style representation-theoretic self-duality. They do not. The canonical Born exchange B_a(q,p)=(ap,-q/a) is order four and symplectic, whereas D is order two and anti-symplectic on the ambient ledger plane; indeed their different orders prevent conjugacy even by a bijection. The ledger datum also does not reconstruct a state-space action, and FRC 830.004 shows that this non-reconstruction occurs in natural qubit fibers. A primitive Hopf algebraization makes D only a Hopf automorphism of one chosen ledger algebra, without an independently defined dual object, pairing, semidualisation, representation exchange, or physical state-observable map. The general one-parameter identity-lift obstruction excludes every nonidentity lift under its monotonicity hypotheses, while the full qubit ledger excludes incoherent operations, unital qubit CPTP channels, and unitary or antiunitary Wigner symmetries as global lifts. A general nonunital, coherence-generating CPTP lift remains open. Present FRC therefore has exact coordinate, typed-morphism, dimension-threshold, and scoped no-go mathematics, but not a Born- or Majid-style self-duality.

FrontierPaperv0.12026-07-14

Exact Phase-Family Test of FRC Duality v0.1: A Lift Obstruction on the von Mises Manifold

The fixed-mean von Mises family provides an exact test of whether the FRC ledger involutions classified in FRC 830.001 lift from derived coordinates to probability distributions. The family has genuine information-geometric structure: concentration kappa and mean resultant r are natural and expectation coordinates, the log-partition function is strictly convex, and minus the differential entropy is its Legendre dual potential. That structure is not the FRC ledger exchange. For the ledger ell(kappa)=(S(kappa),ln r(kappa)), introduce u=S+ln r and v=S-ln r. The function v is strictly decreasing, while u is strictly unimodal with its unique maximum at kappa r=1. Because D(x,y)=(-y,-x) preserves v, every proposed D lift must fix kappa and can exist only at the two isolated roots of u=0. The broader half-plane involutions R_p(x,y)=(-x-2y+p,y) likewise force kappa to remain fixed and survive at no more than two isolated fixed ledgers. Neither class lifts on any open concentration domain. The point kappa r=1 is a maximum of the system-only ledger total, not a D-fixed point or structural self-duality. FRC therefore retains a P2 coordinate involution and gains a precise P3 lift obstruction in this family; it does not gain a phase-family physical or Majid-style self-duality.

FrontierPaperv0.12026-07-14

Operational Registers and Reciprocity-Preserving Morphisms v0.1

FRC uses several legitimate coherence routes, including phase order, the von Mises mean resultant, quantum purity, basis-dependent off-diagonal interference, and platform-specific composites. This paper formalizes how those routes may share a framework without being declared the same observable. A weak reciprocity map preserves the pulled-back ledger one-form up to a nonzero multiplier. A strong affine register morphism additionally supplies a state or model map and a commuting ledger square. Both classes compose; the strong class forms a category with explicit identities, associativity, exact-preserving and signed subcategories, and an invertible groupoid. One-form preservation alone is shown insufficient by an exact counterexample. The von Mises family supplies a nontrivial strong morphism into the phase-distribution register: the state-space inclusion commutes exactly with the entropy and mean-resultant ledger, but is not an isomorphism. A qubit no-go theorem proves that purity and fixed-basis off-diagonal coherence admit no single-valued ledger adapter under the identity state map, even though both are valid operational routes. Present FRC registers therefore form a typed formal category populated by a sparse graph of verified arrows, not one proven physical equivalence class.

FrontierPaperv0.12026-07-14

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.

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.

Philosophical work (1)

PhilosophicalPaperv0.12026-07-14

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.

Archive / development history (3)