Daryl Costello

Independent Theoretical Research | Rosendale, NY, United States

Submitted: November 13, 2026

Theoretical Physics • Theoretical Cosmology • Theoretical Biology • Theoretical Neuroscience • Theoretical Cognitive Science

✉ Correspondence: Daryl.Costello@outlook.com

Abstract

Prevailing models of biological agency and physical emergence frequently rely on domain-specific equilibrium assumptions or unobservable compensatory entities; from instantaneous prediction-error minimization in cognitive neuroscience to non-baryonic dark matter halos in galactic dynamics. Here, we present a unified, scale-invariant formalism (the Kernel-First Architecture and Boundary Projection Model) demonstrating that behavioral arbitration, offline memory consolidation, cortical frequency nesting, clinical psychopathology, and galactic rotation plateaus are governed by the same non-invertible coarse-graining operator () operating under Landauer thermodynamic boundary limits. At the cognitive level, behavior is formulated not as passive surprise suppression, but as real-time litigation across a stochastic impedance-matching interface () mediating between a slow invariant historical ledger () and a fast perceptual novelty channel (). This arbitration unfolds as a codimension-2 cusp catastrophe whose bifurcation structure accounts for fluid cognitive mastery, Gaussian temperamental variance, and psychiatric regime failures (anxiety, depression, mania, insomnia, and addiction). To prevent entropic saturation of attentional aperture (), periodic sensory decoupling () triggers an offline Renormalization Group (RG) sedimentation flow () during sleep, exporting pruned episodic entropy as Landauer-bounded metabolic heat. Across neurophysiological scales, this identical entropic clearing ceiling fixes cortical binding at nested within a macro-reset carrier (). Finally, extending this boundary projection formalism to astrophysical horizons reveals a direct structural isomorphism with galactic dynamics: when baryonic surface mass density falls below the critical holographic saturation limit , invertible Newtonian curvature transitions into a scale-invariant metric remainder (). Together, these results establish a single thermodynamic grammar linking abductive agency, neural architecture, and cosmological structure across twenty-six orders of magnitude.

1. Introduction: From Passive Equilibrium to Scale-Invariant Boundary Projection

1.1 The Limits of Single-Objective Equilibrium Models

Across both theoretical physics and cognitive science, attempts to unify complex multi-scale phenomena frequently stall when a local equilibrium condition is mistaken for a global teleological law. In computational neuroscience and philosophy of mind, the Free Energy Principle (FEP) and classical predictive coding posit that biological agents act primarily to minimize instantaneous sensory surprise or prediction error. Yet empirical human behavior persistently violates passive equilibrium-seeking: organisms actively cultivate novelty, pursue bounded existential risk, and abandon stable, low-entropy environments whenever informational stagnation (experienced subjectively as boredom) accumulates. Resolving this “Dark Room Problem” within equilibrium-first frameworks requires retrofitting secondary epicycles (such as expected epistemic foraging) onto a foundation designed for homeostatic closure.

A structurally analogous impasse appears at astrophysical scales. When observed galactic rotation velocities remain flat at outer disk radii where baryonic surface mass density drops below a universal acceleration threshold (), standard gravitational models preserve local Newtonian invertibility only by postulating vast, unseen halos of non-baryonic dark matter. In both domains, an apparent anomaly arises because a finite-bandwidth boundary interface is modeled as an invertible, lossless channel.

1.2 The Kernel-First Paradigm and Non-Invertible Projection

The Kernel-First Architecture resolves both impasses by inverting the foundational premise: complex systems are not passive equilibrium-seekers, but horizon-tracking, gradient-dissipating engines governed by non-invertible boundary projections (). Whenever a high-dimensional bulk state space projects onto a lower-dimensional observable manifold across a finite thermodynamic interface, the mapping cannot preserve all microstate degrees of freedom. Instead, the boundary truncates high-frequency modes above a critical scale , generating an irreducible structural remainder whose clearing cost is strictly bounded by Landauer’s principle ( per bit erased).

When incoming state density crosses a critical saturation knee (), linear local tracking gives way to a scale-invariant regime governed by Renormalization Group (RG) flow dynamics. Remarkably, this single mathematical structure manifests identically across four distinct physical and biological strata:

1. The Psychological Interface as a Multi-Channel Tribunal: Rather than collapsing behavior onto a scalar error metric, nature routes competing bids from a slow, conservative historical bedrock () and a fast, novelty-foraging perceptual surface () through a stochastic impedance-matching interface (). Elevated to a dynamical order parameter, the jurisdictional gating weight unfolds into a codimension-2 cusp catastrophe, where affective torque () and existential boundary risk () adjudicate between exploration and preservation.
2. Offline Sleep Sedimentation and Landauer Compaction: Wakeful perceptual foraging inevitably saturates finite working buffers with high-dimensional episodic entropy (). Periodic decoupling of external sensory input () converts the brain from an entropy-absorbing sensor into an entropy-exporting thermodynamic compressor, executing an RG coarse-graining projection () that deposits survived invariants into while dissipating pruned noise () as metabolic heat.
3. Cortical Frequency Nesting () and Clinical Pathologies: Local neurovascular heat-dissipation limits () at cap discrete vector-to-symbol frame binding at , while Landauer working-buffer capacity () mandates a macro-clearing reset envelope at (). When gain (), threat weighting (), or offline clearing efficiency () drift outside critical bounds, the interface locks into pathological attractors corresponding to anxiety, major depression, mania, chronic insomnia, and addiction.
4. Galactic Holographic Saturation (): Extending the Boundary Projection Model to cosmological horizons demonstrates that galactic disks obey the exact same entropic saturation law. Above the critical baryonic surface mass density , local Unruh thermal flux exceeds the de Sitter horizon floor () and spacetime curvature tracks baryonic mass invertibly (). Below , holographic encoding saturates at the Landauer horizon limit, yielding the scale-invariant deep-MOND asymptotic plateau ().

1.3 Roadmap of the Monograph

Section 2 formalizes the fast-slow stochastic differential equations of the psychological interface and derives its codimension-2 cusp bifurcation structure around . Section 3 establishes the offline RG sedimentation operator and computes the exact Landauer heat and molecular ATP budget of sleep-dependent synaptic homeostasis. Section 4 derives the gamma-theta packing ratio from cortical entropic clearing bounds and demonstrates its structural isomorphism with the galactic surface density limit . Section 5 consolidates all governing variables, equations, and critical thresholds into a master cross-scale reference ledger. Section 6 maps psychiatric and behavioral disorders onto parameter distortions across the cusp and Landauer control planes. Finally, Section 7 enumerates twelve concrete, falsifiable empirical predictions across neuroscience, sleep physiology, psychiatry, and astrophysics.

2. Mathematical Formalization of the Psychological Interface

2.1 The Two Competing Channels (Timescale Separation)

Behavior is litigated through multiple channels mediated by a stochastic interface, with live perception overlying a recorded invariant history. Let the state of the organism-environment system be partitioned into two coupled manifolds evolving on distinct characteristic timescales, separated by a small dimensionless ratio :

• The Slow Invariant History Channel (): Represents the compressed historical ledger; evolutionary priors, autonomic baselines, and somatic invariants. Its dynamics resist rapid perturbation and enforce boundary viability via a conservative potential :

where is the sedimentation (ledger-consolidation) operator that deposits resolved perceptual remainders into the invariant bedrock during offline states ().

• The Fast Perceptual Novelty Channel (): Represents live sensory flux, environmental affordances, and unmapped horizons. It is driven by an epistemic gradient (information gain / gradient dissipation) and injected stochasticity (abductive probing):

where projects historical expectations onto the perceptual surface, is structural damping, and encodes state-dependent stochastic noise distributed across a population-level Gaussian continuum of temperament; populating every regime from conservative homeostatic anchors (left tail) to stochastic frontier scouts (right tail).

2.2 The Interface as a Stochastic Impedance-Matching Arbitrator

Behavior is not generated directly by either channel. Instead, both channels submit competing vector bids to the psychological interface operator :

• Conservative Bid (): , minimizing deviation from invariant viability.
• Exploratory Bid (): , maximizing horizon resolution and dissipating boredom (the local stagnation gradient).

The interface arbitrates these bids through a jurisdictional gating weight

where represents non-local abductive leaps across saddle manifolds when linear interpolation between and hits an inductive-deductive deadlock.

2.3 The Arbitration Law (Affective Torque & Metabolic Gating)

Jurisdictional weighting is governed by a logistic threshold function driven by the ratio of available metabolic and attentional aperture () to existential boundary threat ():

Parameter	Physical & Cognitive Meaning	Behavioral Effect
	Boredom / Stagnation Gradient ()	Rises when current environment yields zero new information; pushes .
	Metabolic & Attentional Aperture	High when rested/fed; expands capacity to metabolize novelty.
	Boundary Integrity Threat	Divergence of from viable survival bounds in ; snaps .
	Temperamental Baseline (Gaussian Prior)	Individual’s resting position on the normal distribution (left-tail anchor vs. right-tail scout).
	Affective Gain / Sensitivity	Steepness of the switching manifold (how abruptly emotion flips jurisdiction).

This formulation directly dissolves the Dark Room Problem: in a zero-information environment (), the stagnation gradient grows monotonically until affective torque exceeds , driving and compelling the organism to generate novelty or leave the room. Conversely, at the edge of an existential threat (), overwhelms affective torque and snaps , restoring immediate control to the invariant historical bedrock.

3. Linear Stability and Cusp Bifurcation Analysis Around

3.1 Dynamical Formulation of the Interface Gate

Elevating from an algebraic readout to a fast dynamical order parameter relaxing on timescale yields:

where the net jurisdictional drive couples back to via self-reinforcing interface feedback () and thermodynamic/attentional cubic saturation ():

3.2 Fixed-Point and Linear Stability Analysis

Setting asymmetry bias to zero () yields the exact critical equilibrium . Introducing a small perturbation and linearizing using and gives the Jacobian eigenvalue at :

Regime	Eigenvalue Condition	Stability of α∗=0.5	Psychological & Behavioral Mode
Subcritical ()		Globally Asymptotically Stable	Fluid Integration (Flow / High-Skill Execution): Smoothly blends historical priors and live novelty without lock-in.
Critical Point ()		Marginally Stable (Critical Slowing Down)	Maximum Susceptibility: Infinitesimal perceptual cues steer the system; peak abductive sensitivity.
Supercritical ()		Unstable Saddle Point	Bistable Splitting: Midpoint repels trajectories, snapping behavior into pure caution () or pure exploration ().

3.3 Pitchfork Normal Form and Unfolding of the Cusp Catastrophe

Expanding around up to third order (with ) yields the supercritical pitchfork normal form unfolded by asymmetry bias :

For and , two symmetric stable attractor branches emerge at . For , the bistable region is bounded in the control plane by the fold (saddle-node) bifurcation set:

This cusp geometry accounts for behavioral hysteresis (once boredom pushes an agent past onto the exploratory branch , they remain committed until threat drives past ), catastrophic threat snap-back (annihilation of at ), and expert mastery (regulating at the cusp tip to maximize critical sensitivity without hysteretic lock-in).

4. Offline Sleep Sedimentation and the Landauer Entropy Budget

4.1 Three-Stage Mathematical Architecture of

During wakeful foraging (), the fast perceptual buffer accumulates high-dimensional episodic traces. Writing raw trajectories directly into would induce catastrophic overfitting; therefore, when attentional aperture depletes () and external sensory drive decouples (), the sedimentation operator executes three sequential transformations:

1. Compressed Replay & Residue Extraction: Replaying waking trajectories against the historical projection isolates unexplained structural remainders . Predicted trajectories () are discarded.
2. Renormalization Group (RG) Coarse-Graining (): A non-invertible projection integrates out high-frequency contextual noise () weighted by wakeful affective salience :

3. Landauer-Bounded Orthogonal Deposition: The coarse-grained invariant is written into via a Fisher-stiffness-regularized natural gradient update paired with entropic pruning:

4.2 The Landauer Heat and Molecular ATP Budget of Sleep

Across a waking epoch, net synaptic potentiation increases configurational buffer entropy by . During sleep, partitions into retained structural invariants () and pruned noise () erased via multiplicative synaptic downscaling (, ).

By Landauer’s principle, contracting phase-space volume by at brain temperature () with wetware thermodynamic efficiency requires exporting a minimum heat load and hydrolyzing molecules of ATP ():

Because , sleep achieves a net reduction in internal brain entropy (), sharpens signal-to-noise ratio, and restores attentional aperture () for the next waking cycle.

5. Cortical Frequency Nesting and the Galactic Boundary Isomorphism

5.1 Deriving the Gamma-Theta Ratio from Landauer Limits

In wakeful cortical processing, each fast gamma cycle () executes a non-invertible vector-to-symbol truncation () of incoming perceptual state vector , leaving residual entropy . Clearing intra-cycle phase ambiguity requires minimum Landauer work . Given maximum neurovascular heat dissipation , the maximum sustainable entropic clearing rate caps discrete perceptual binding at:

However, each gamma frame deposits a cumulative inter-cycle remainder into the local cortical working buffer. When reaches the Landauer working-buffer capacity , the circuit must execute a macro-phase reset to flush the accumulated packet into the hippocampal-cortical loop. This fixes the theta carrier frequency and the dimensionless entropic packing ratio :

5.2 Isomorphism with the Galactic Surface Mass Density Limit ()

Extending the Boundary Projection Model to astrophysical horizons reveals that galactic disks obey the exact same Landauer saturation law. A holographic screen of radius enclosing baryonic mass carries bit density proportional to surface mass density . At the cosmological horizon de Sitter temperature (where Milgrom’s acceleration scale sets the vacuum thermal floor), holographic encoding saturates when local Newtonian acceleration drops to :

• Above (High Surface Brightness Cores, e.g., NGC 3198): Local Unruh thermal flux exceeds the cosmic horizon floor (); spacetime curvature tracks baryonic mass invertibly ().
• Below (Low Surface Brightness Disks, e.g., F568-3): Holographic encoding saturates at the Landauer horizon limit (), transitioning along an RG fixed point () into the scale-invariant deep-MOND plateau:

6. Master Cross-Scale Reference Ledger

6.1 Universal Scale-Invariant Variable Dictionary

Canonical Symbol	General Scale-Invariant Role	Psychological Interface	Sleep Sedimentation	Cortical Oscillations	Galactic Dynamics
	Incoming Bulk State Density	Unresolved environmental affordances & sensory flux	Waking episodic trace density in buffer	Incoming perceptual vector dimensionality ()	Baryonic surface mass density
	Non-Invertible Boundary Projection	Impedance-matching tribunal	RG coarse-graining projection	Vector-to-symbol frame truncation per cycle	Holographic metric screen projection at radius
	Irreducible Residual Entropy	Unresolved horizon tension / boredom gradient	Pruned configurational noise	Intra-frame remainder	Non-Newtonian metric remainder ()
	Thermodynamic Sink Temperature	Metabolic & somatic thermal baseline	Cortical thermal bath ()	Local neurovascular thermal bath ()	Cosmological de Sitter temperature
	Critical Landauer Knee / Threshold	Saddle-point jurisdiction threshold	Buffer saturation ceiling	Max entropic clearing ceiling & buffer	Critical surface density

6.2 Master Equations and Critical Thresholds

Domain & Regime	Governing Equation / Critical Condition	Physical, Cognitive, or Astrophysical Interpretation
Fast-Slow Channel Coupling		Slow invariant history () anchors survival bounds while fast perception () harvests novelty along gradients ().
Interface Bid Arbitration		Litigates between conservative bid and exploratory bid ; boredom drives , threat snaps .
Cusp Catastrophe & Fold Set		Supercritical pitchfork at ; unfolds into a codimension-2 cusp explaining hysteretic lock-in and sudden snap-back.
Offline RG Sedimentation		Compresses survived waking remainders into orthogonal strata of while exporting pruned entropy as Landauer heat.
Cortical Rhythm Nesting		Neurovascular cooling caps fast binding at ; buffer ceiling forces macro-clearing reset at .
Galactic Boundary Projection		Above , Newtonian curvature holds; below ,, metric saturates at the de Sitter Landauer horizon floor.

7. Dynamical Pathologies of the Psychological Interface

From this unified formalism, psychiatric and behavioral disorders emerge as macroscopic regime failures in which parameter drift traps the interface in non-adaptive attractor basins or obstructs its offline Landauer clearing stroke:

Clinical Regime	Cusp Control Coordinates (r,h)	Landauer & RG Sedimentation State	Primary Dynamical Failure Mode
Adaptive Mastery (Flow)	(),	Balanced wake-sleep cycle ( cleared;)	Critical susceptibility; seamless integration of and .
Anxiety Disorders	, ()	Intact clearing, but zero novel input ()	Lower-sheet hysteretic lock-in (); blocked exploratory updating.
Major Depression	(), ()	Stagnant buffer; blunted affective salience weights	Subcritical damping (); insufficient torque to initiate foraging.
Mania / Hypomania	(), ()	Truncated sleep (); un-pruned accumulation	Upper-sheet runaway (); loss of boundary-threat snap-back.
Chronic Insomnia	Oscillatory , elevated	Blocked RG compaction ( stalled); saturation	Entropic choking of and breakdown of frame packing.
Addiction	(), narrow synthetic	Zero invariant sedimentation (); eroded baseline	Hijacked upper-sheet attractor well; metabolic drain without ledger growth.

Clinically, therapeutic interventions map directly onto control-plane steering: mood stabilizers damp supercritical gain () to collapse hysteretic lock-in; exposure therapy holds the interface near the critical saddle () so empirical survival updates and lowers ; and restoring un-gated sleep () clears accumulated waking entropy to replenish attentional aperture .

8. Conclusion and Falsifiable Empirical Predictions

By grounding agency, neurophysiology, psychopathology, and galactic kinematics in non-invertible boundary projections () bounded by Landauer’s thermodynamic limit, the Kernel-First Architecture replaces domain-specific epicycles with a single testable physical grammar. To subject this synthesis to decisive empirical evaluation, we establish twelve falsifiable predictions across four experimental domains:

8.1 Cognitive Neuroscience and Cortical Electrophysiology

1. Inverse Scaling of Gamma-Theta Packing () with Perceptual Entropy: Increasing incoming perceptual entropy will induce a measurable downward shift in the gamma-to-theta cycle packing ratio (truncating packet count from toward cycles per theta wave) prior to working-memory breakdown.
2. Thermodynamic Choking Under Local Vascular Cooling or Hypoperfusion: Localized reductions in cortical perfusion or ATP synthesis will produce a linear drop in peak coherent gamma frequency () proportional to .
3. Boredom-Driven Jurisdictional Breakout at Constant Homeostatic Need: Under sensory deprivation with physiological homeostasis held strictly constant, subjective boredom () and exploratory motor/attentional bids () will rise monotonically with cumulative mutual-information stagnation ().

8.2 Sleep Physiology and Landauer Entropic Compaction

4. Proportionality Between Waking Novelty Load and Non-REM Thermal Export: High-novelty waking epochs () will generate significantly higher slow-wave sleep (SWS) cerebral heat dissipation and ATP turnover than routine epochs matched for physical exertion.
5. Saturation Collapse of Attentional Aperture () Under Blocked Offline Gating: Prolonged wakefulness () will degrade working-memory packing along an entropic saturation curve, culminating in an abrupt cusp bifurcation (defensive rigidity or disinhibited perceptual runaway) at .
6. Salience-Weighted Topological Preservation During Replay: SWS sharp-wave ripple replay will selectively preserve trajectories carrying high wakeful affective torque () while erasing low-torque sensory details.

8.3 Clinical Psychiatry and Computational Psychopathology

7. Hysteretic Switching Asymmetry (): Patients with generalized anxiety and hypomania will exhibit wider explore-exploit switching hysteresis loops () than healthy controls, and lithium/mood-stabilizer treatment will narrow by reducing effective interface gain toward .
8. Critical Slowing Down Prior to Mood Episode Transitions: Longitudinal behavioral telemetry will detect rising lag-1 autocorrelation and variance () in daily explore-exploit switching () days before clinical onset of a depressive collapse or manic switch.
9. Zero-Sedimentation Metabolic Drain in Addiction: Compulsive addiction will exhibit elevated wakeful foraging drive () paired with blunted SWS invariant consolidation (), eroding baseline attentional aperture for natural affordances.

8.4 Astrophysics and Galactic Boundary Dynamics

10. Universal Radial Transition Knee at : Across SPARC and next-generation radio interferometry rotation curves, the transition from Newtonian curvature () to the flat velocity plateau () will occur universally at the radius where enclosed baryonic surface mass density crosses .
11. Zero Intrinsic Scatter in the Deep-LSB Baryonic Tully-Fisher Relation: Galaxies residing entirely below will obey with zero intrinsic scatter beyond observational baryonic mass and inclination uncertainties.
12. Redshift Evolution of : High-redshift rotation curves () will exhibit an upward shift in the critical surface mass density transition knee , tracking the evolving de Sitter cosmic horizon temperature .