FodaOS Unified Field Theory Series VI: The Brain-Inspired Cognitive Architecture — Dual-Path Reasoning, Reservoir Computing on the Residueon Field, and the PCD/RRC Dual Criteria

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-18
DOI
https://doi.org/10.5281/zenodo.22822343
Primary Topic
Neural Networks and Reservoir Computing
Type
preprint
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preprint

FodaOS Unified Field Theory Series VI: The Brain-Inspired Cognitive Architecture — Dual-Path Reasoning, Reservoir Computing on the Residueon Field, and the PCD/RRC Dual Criteria

Pan JinWen
Zenodo (CERN European Organization for Nuclear Research)
Neural Networks and Reservoir Computing
preprint

FodaOS Unified Field Theory Series VI: The Brain-Inspired Cognitive Architecture — Dual-Path Reasoning, Reservoir Computing on the Residueon Field, and the PCD/RRC Dual Criteria

Pan JinWen
preprint en

Abstract

This paper is the sixth installment of the FodaOS Unified Field Theory series. The previous five installments established, respectively, the physical unified field ([Paper1] the complex Ginzburg–Landau master equation), the computational unified field ([Paper2] the five-operator arithmetic resonance computer ARC), the mathematical unified field ([Paper3] the exactification of algebraic quantum mechanics on GF(65537)), the unified primitive ([Paper4] the residueon trichotomy), and the bootstrap compiler ([Paper5] the residueon compiler). Taking the Brain-Inspired Cognitive Architecture as its subject, this paper adopts dual-path reasoning as the architectural backbone (right path: interference field = intuition; left path: collision algebra = logic), residueon reservoir computing as the substrate of memory and learning (frozen field + linear readout), and the PCD/RRC dual criteria as the verdict-and-immunity instrumentation running through the entire text, converging all the assets of the previous five papers into one cognitive architecture — the Dual-Path Cognitive Architecture. The core results fall into five layers. (i) The ontological layer — the residueon (zk,rk,nk)(zk,rk,nk) simultaneously carries particle nature (pole position), wave nature (residue amplitude), and topological nature (pole order), and strictly contains the qubit and the classical bit as degenerate special cases (Theorem VI1); the neuron↔residueon mapping and the population field theorem (Theorem VI3): under the kernel K(x,xj)=1/(x−xj)K(x,xj)=1/(x−xj) the residue theorem yields "global activity = the sum of local residues"; all four core features of the qubit fail at the neuronal level (Proposition VI2) — the qubit is not the hardware of the human brain, but the grammar of cognition. (ii) The dual-path reasoning layer (Pillar One) — the right path, an all-pole interference field, carries intuition: the continuum spectrum is rehabilitated from "noise" into a carrier of associative memory, and the interference findings of quantum cognition acquire a deterministic counterpart in the GF domain; the left path, a soliton collision algebra, carries logic (20/20 exact selective superposition at τ=10−5τ=10−5); arbitration is carried by the phase-coherence divergence of the PCD, D(Q)=1−∣∑ui∣/∑∣ui∣D(Q)=1−∣∑ui∣/∑∣ui∣, constituting the compute-allocation economics of System 1/System 2 (measured: 79.7% of contributions can be discarded by gating with deviation <0.0012%); the chirality duality correction: separation uses the grade involution A^A^ (idempotent, orthogonal, zero-cost in GF implementation) while translation uses the Hodge star (Theorem VI4: ∗2=−id∗2=−id, a complex structure with eigenvalues ±i=±256±i=±256), and the algebraic mechanism of insight is "odd × odd = even" (Proposition VI5); the statistical identity of the PCD = circular variance = 1 − Kuramoto order parameter (Theorem VI15), and it is gauge invariant — contradictions cannot be masked by recoloring. (iii) The residueon reservoir layer (Pillar Two) — reservoir = fixed wave-field medium + linear observation (pool = pole spectrum, readout = observation operator); the entanglement criterion for field multiplication (Theorem VI7) and the echo–entanglement duality (Theorem VI8: λmax⁡↔e−Γλmax↔e−Γ), with entanglement lifetime proportional to the pole order; honest capacity bounding (Proposition VI9: all O(2N)O(2N) claims revoked, storage is O(∑kSk2)O(∑kSk2)); group-theoretic stability (Theorem VI10: gate residue evolution is naturally bounded, and the empirical prerequisite of the reservoir spectral radius is eliminated by group-theoretic identities); the bootstrap internalization theorem (Theorem VI12: measured Lipschitz L∈[0.821,0.878]<1L∈[0.821,0.878]<1, convergence exponent α=0.234α=0.234 bit/iteration — the six-beat closed loop of "logic internalized into intuition", with the robot persistent-memory bridge as the engineering existence proof); the gate-residual adapter = a residueon-table version of LoRA (Proposition VI11: 40 integers per person recover 8/8 personas); the distilled generative head overtakes the engine head for the first time (55.94% vs 52.72%, +3.2 pp); row-append cognition (Propositions VI26–VI29: field frozen + sealed interface rows appended, learning as certification). (iv) The criteria-and-immunity layer (Pillar Three) — the PCD (the phase ruler, phase-coherence divergence) and the RRC (the curvature ruler f3−2f2+f1f3−2f2+f1, residue-rank condition) are orthogonally complementary and jointly constitute algebraic diagnostics, yielding the dual disease signature table shared by brain diseases and AI model diseases (Proposition VI13); causal impedance T(e)=σ(−10 PCD+2)T(e)=σ(−10PCD+2) with four-point calibration 0.88/0.50/0.27/0.04 (Theorem VI16); the Kuramoto global-synchronization pathology criterion (Theorem VI19: 0.3<R<0.70.3<R<0.7 is the healthy critical zone, R→1R→1 is epilepsy = 99.6% phase locking); the Landau order parameter and the amended three laws of cognition (Theorem VI17 / Proposition VI18); the Bragg firewall against hallucination; the three-level verdict (Proposition VI14): at the physical level "qubits in the brain" has no evidence and is falsifiable (decoherence 10⁻¹³ s vs neurons 10⁻³ s), at the structural level "cognition uses quantum algebraic structure" holds and the present framework is its existence proof, and at the ontological level the representation claim holds while the hard problem of consciousness remains open. (v) The measurements-and-boundaries layer — pure-ψ inference at 39.2 t/s with trajectories bit-identical to the CPU token by token; W16M weights resident on GPU at 6.60 ms/token (4.1×) with the three adapters cross-checking bit-identical; the q15cuda inverted table at 395–479 tok/s, bit-exact; ΨDB PoolCore at 8.35 million ops/s; the complete verdict on the reservoir readout (the §17.3 verdict quick-reference table: engine head = the generic-readout upper bound; learned readout = narrow-domain specialization plus distilled generation; scaling saturates; CE calibration 6.56%→47.44%); the pole page table gives the precise address of selective forgetting (§17.6); black-box perfect factorization and the sphere ledger dashboard (GF perfect factorization under 134× cancellation, health criterion A=1/(1−PCDledger)A=1/(1−PCDledger)); environment-interactive learning places the cognitive agent into a many-body training field (§18: external-feedback poles and a backpropagation-free Hebbian closed loop). Honest boundaries: capacity claims are revoked at their information-theoretic upper bounds (VI9); the failure domain of the gf_sqrt formula is registered as a P-level fix item; all accuracies cited are clean in-domain accounts after the early readout leakage audit; throughout, GF(65537) pure integers with zero floating point, zero randomness, and zero dissipation (the Three-Zero Principles), and every formula is machine-verified item by item by the Appendix B programs.

Zenodo (CERN European Organization for Nuclear Research)
Neural Networks and Reservoir Computing
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