Daryl Costello

Independent Theoretical Research | Rosendale, NY, United States

Submitted: November 14, 2026

Theoretical Physics • Theoretical Cosmology • Theoretical Neuroscience

✉ Correspondence: Daryl.Costello@outlook.com

Abstract

A pervasive category error across mathematical physics, neuroscience, and cosmology arises from modeling global boundary-value systems as local initial-value progressions. When an unbounded, continuous scale-dilation flow is forced through a finite interface, local perturbations cannot dissipate instantaneously; instead, they propagate as non-local constraint cascades until global spectral interference restores equilibrium. Here, we formalize a unified operator framework demonstrating that discrete quantization (spanning combinatorial line-breaking cascades, non-trivial Riemann zeta zeros, cortical working-memory packets, galactic rotation curves, and cosmological horizons) emerges universally from a Landauer-bounded boundary projection operator () acting on the hyperbolic scale-dilation generator .

First, taking the continuum field limit () of the Knuth-Plass dynamic programming recurrence yields a fourth-order Euler-Lagrange variational action with periodic margin pinning, wherein discrete breakpoints emerge as Sturm-Liouville eigenvalues and downstream cascades propagate as Sine-Gordon solitons governed by logarithmic Dyson log-gas repulsion; establishing a direct structural isomorphism with Montgomery’s pair correlation of Riemann zeta zeros on the critical line . Second, identifying with the chiral Bisognano-Wichmann modular Hamiltonian of a causal horizon, we replace static phase-space walls with a dynamic thermodynamic truncation governed by Landauer’s erasure bound (). The resulting closed-form boundary eigenvalue density recovers the Riemann-von Mangoldt asymptotic count at the minimal quantum action floor while introducing an exponential thermodynamic choke filter at .

Third, applying this projection architecture across physical scales yields three quantitative predictions:

1. Cosmology and Galactic Dynamics: Equating incoming Unruh-Rindler gravitational flux to the holographic Landauer dissipation capacity of a boundary shell coupled to the de Sitter horizon bath () derives the critical baryonic surface mass density and reproduces the empirical Radial Acceleration Relation with zero free parameters. Furthermore, comparing UV black hole Hawking evaporation (, Page time ) against IR cosmic void processing (, scale-invariant turnover ) isolates a critical primordial black hole mass pivot ().
2. Cortical Neuroscience: Mapping onto neocortical-hippocampal arbitration between fast perceptual gamma sampling () and slow episodic theta framing () derives Miller’s working-memory capacity ( chunks), establishes linear stability at the symmetric arbitration axis , formalizes psychopathology as boundary choking or horizon collapse, and identifies slow-wave sleep sedimentation as offline Landauer entropy clearance.

Together, these results demonstrate that seemingly disparate physical, cognitive, and number-theoretic phenomena represent scale-invariant manifestations of continuous dilation flows quantized by finite thermodynamic horizons.

Keywords: Scale-Invariant Boundary Projection; Landauer Principle; Berry-Keating Hamiltonian; Riemann Zeta Zeros; Bisognano-Wichmann Modular Hamiltonian; Radial Acceleration Relation (RAR); Cosmic Voids; Primordial Black Hole Page Time; Gamma-Theta Phase Nesting; Thermodynamic Choking

Section X: Scale-Invariant Boundary Projection and Thermodynamic Quantization Across Discrete-Continuous Interfaces

X.1 Introduction: Non-Local Constraint Propagation and the Boundary-Value Paradigm

Across physical, informational, and number-theoretic domains, a fundamental structural error arises when a global boundary-value problem is treated as a local initial-value progression. When an unbounded, continuous stream of indivisible units is forced through a finite geometry, local perturbations cannot be absorbed instantaneously at their point of origin. Instead, the constraint boundary transforms discrete shocks into non-local waves of adjustment that propagate downstream until global spectral interference restores equilibrium.

In combinatorial optimization and digital typography, this mechanism is governed by the Knuth-Plass dynamic programming recurrence. Given an indivisible sequence of boxes of width separated by elastic glue with stretchability and shrinkability bounded by rigid page margins of width , the local adjustment ratio quantifies the elastic strain required to reconcile discrete units with continuous boundary walls:

Minimizing the global demerit functional across an entire interval demonstrates that the apparent “empty” margins and elastic spaces act as the true dynamical operators. When an upstream perturbation exceeds the local threshold, it nucleates a downstream cascade that terminates only upon encountering a slack sink capable of absorbing the phase shift. As demonstrated below, this exact mathematical architecture (the quantization of continuous dilation flows via finite boundary conditions) governs analytic number theory, causal horizon thermodynamics, galactic rotation curves, and cosmological void evolution.

X.2 Continuum Field Limit of Discrete Dynamic Programming

To pass from discrete dynamic programming to a continuous field theory, we take the continuum limit in which individual unit widths vanish () over a cumulative unstretched domain . Defining a smooth, monotonically increasing line-coordinate field whose integer level sets mark discrete breakpoint positions, the local density is . Incorporating elastic strain energy, fixed boundary-break penalties , and second-order fitness-class coupling stiffness , the discrete Bellman recurrence transforms via Legendre duality into the continuous variational action functional:

where enforces the periodic margin constraint. Varying the action () yields the fourth-order Euler-Lagrange field equation:

In the linearized harmonic regime (), the local strain field obeys the screened Helmholtz equation , defining a characteristic correlation (healing) length:

When the periodic margin potential is restored with fixed boundary winding , linearizing fluctuations around a stationary trajectory yields a Sturm-Liouville eigenvalue problem. Discrete breakpoints emerge as quantized eigenvalues locked by boundary periodicity, and upstream perturbations propagate as topological Sine-Gordon kinks across the Peierls-Nabarro barrier. In dense multi-soliton lattices, overlapping exponential tails induce logarithmic level repulsion , establishing a direct structural isomorphism with the Dyson log-gas statistics and Montgomery pair correlation of the non-trivial zeros of the Riemann zeta function .

X.3 Hyperbolic Dilation Generators and Landauer-Bounded Boundary Projection

The spectral duality connecting discrete prime orbits to continuous zeta zeros is formalized physically by the Berry-Keating Hamiltonian, which generates exponential scale dilations (, ) along hyperbolic phase-space trajectories :

The measure-preserving term anchors the scale-invariant eigenfunctions onto the critical symmetry axis . Crucially, via the Bisognano-Wichmann theorem, the modular Hamiltonian of a quantum field restricted to a causal Rindler wedge or black hole exterior reduces in light-cone coordinates (, ) directly to the chiral Berry-Keating generator:

On an unbounded domain , the trace of the resolvent diverges logarithmically (). To regularize this divergence without arbitrary geometric walls, we introduce a Landauer-bounded boundary projection operator . At equilibrium temperature and maximum sustainable thermal dissipation power , Landauer’s principle bounds the boundary’s informational erasure rate at and its per-cycle logarithmic phase-space capacity at .

Enforcing unitary twisted-periodic closure across the dynamically truncated interval with cutoff product discretizes the continuous dilation flow. Weighting the Wigner-Weyl projection kernel by the thermodynamic choke filter (where ) yields the closed-form boundary eigenvalue density:

At the minimal quantum dissipation floor (), recovers the exact Riemann-von Mangoldt asymptotic density of zeta zeros, whereas for , high-frequency modes undergo exponential thermodynamic coarse-graining.

X.4 Gravitational Horizon Saturation, , and the Radial Acceleration Relation

Applying to a spherical gravitational boundary shell of radius and area enclosing baryonic mass couples two thermal horizons: the local Unruh-Rindler horizon temperature (driven by Newtonian acceleration ) and the infrared cosmological de Sitter horizon bath (set by the cosmic acceleration scale ).

Populating the shell with holographic degrees of freedom cycling at the cosmic relaxation rate establishes the boundary’s maximum Landauer dissipation power ceiling alongside the incoming Newtonian gravitational dilation flux :

Equating incoming flux to the Landauer erasure ceiling () yields , directly deriving the critical baryonic surface mass density limit:

Above (high-surface-brightness cores), local Unruh flux dominates the cosmic bath and the boundary transmits 100% of the Newtonian metric signal. Below (low-surface-brightness disks), thermal modes from the de Sitter bath populate the horizon with mean bosonic occupation number , where is the radial phase-space action ratio. Defining the boundary transmission efficiency as the coherent signal-to-total-flux fraction , flux conservation reproduces the empirical Radial Acceleration Relation (RAR) with zero free parameters:

In the deep sub-critical regime (), , locking the projected acceleration onto the geometric mean and generating flat galactic rotation curves () as an irreducible infrared boundary remainder.

X.5 UV–IR Horizon Duality, Cosmic Void Turnover, and the Pivot

Comparing opposite extremes of surface mass density across the divide reveals an exact inversion in holographic entropy processing rates ():

Horizon Regime	Surface Density Domain	Landauer Entropy Rate (I˙)	Holographic Entropy (S)	Information Turnover Timescale
Black Hole (UV Limit)
Cosmic Void (IR Limit)

Because a black hole’s Hawking temperature scales inversely with radius (), its information turnover suffers a cubic scaling bottleneck (), turning astrophysical black holes into long-term UV archival sinks. Conversely, because a cosmic void below is pinned to the constant de Sitter temperature floor , both its total holographic entropy and its Landauer erasure bandwidth scale with . Consequently, every sub-critical cosmic void refreshes its entire holographic ledger on a universal, scale-invariant timescale .

Equating the black hole Page time to the cosmic void Landauer saturation timescale () defines the exact cosmological mass pivot separating evaporated early-universe relics from persistent UV topological knots:

Evaluating yields in pure blackbody emission (and including relativistic Standard Model species at ), corresponding to a sub-protonic Schwarzschild radius .

Section XI: Cortical Arbitration, Gamma-Theta Phase Nesting, and Thermodynamic Sleep Sedimentation

XI.1 The Cortical Boundary Interface as a Landauer-Bounded Projection Operator

The scale-invariant operator formalism established in Section X applies directly to biological neural architectures, where high-dimensional, continuous sensory and subcortical vector flows must be discretized into stable, serial symbolic representations. Within the neocortical-hippocampal loop, executive consciousness functions as an active boundary projection operator arbitrating between two competing temporal regimes:

1. The Ultraviolet (UV) Perceptual Novelty Stream (): Fast local cortical microcircuit oscillations in the gamma band (, period ), set by the biophysical refractory and GABAergic inhibitory synaptic integration floor .
2. The Infrared (IR) Episodic Integration Horizon (): Slow macro-scale hippocampal-prefrontal pacing in the theta band (, period ), establishing the episodic framing window within which discrete perceptual events are bound into a unified working-memory trajectory.

XI.2 Derivation of the 40 Hz to 6 Hz Nesting Ratio and Miller’s Working Memory Limit

In exact isomorphism with the Berry-Keating phase-space truncation (, ), the cortical dilation generator operates across a temporal phase-space wedge bounded by the UV synaptic resolution floor and the IR episodic horizon .

Applying the boundary eigenvalue quantization condition derived in Section X.3 (where unitary phase closure across the logarithmic span is constrained by the Landauer per-cycle information capacity ) the maximum number of distinguishable discrete gamma sub-packets that can be phase-locked inside a single theta envelope without inter-symbol interference is governed by the ratio of the boundary frequencies and the thermodynamic dissipation budget:

1. Geometric Nesting Ratio (): Dividing the UV sampling rate by the IR horizon frequency gives the discrete phase-slot capacity per episodic cycle:

2. Landauer Thermodynamic Channel Bound (): Within the cortical interface operating at physiological temperature (), let denote the net metabolic ATP dissipation power allocated specifically to active executive symbol-binding and synaptic erasure across the active cortical column ensemble (distinct from baseline vegetative ion-pump housekeeping). Over one theta integration cycle , Landauer’s principle caps the resolvable phase-space distinction at:

Substituting the boundary eigenvalue density at the thermodynamic choke threshold demonstrates that when incoming sensory novelty exceeds the metabolic erasure rate (), the Fermi-Dirac choke filter exponentially quenches excess high-frequency modes. Consequently, conscious working-memory capacity is structurally pinned to discrete items; deriving Miller’s empirical working-memory limit ( chunks) directly from the Landauer-bounded gamma-theta phase-space aperture.

XI.3 Interface Arbitration Equilibrium () and Off-Equilibrium Regimes

To maintain stable agency without collapsing into rigid determinism or chaotic noise, the cortical projection operator arbitrates continuously between slow invariant historical priors (weight ) and fast perceptual novelty flux (weight ). Expanding the boundary action functional around the equilibrium operating point reveals that linear stability (analogous to the critical line in analytic number theory and in galactic dynamics) is uniquely achieved at the symmetric arbitration balance:

Deviations from or saturation of the metabolic power ceiling map directly onto distinct dynamical failure modes of the boundary interface:

Cortical Dynamical State	Operator Parameter Regime	Thermodynamic & Spectral Mechanism	Phenomenological Manifestation
Balanced Waking Agency	,	Unitary phase-locking across the -axis; incoming novelty is cleanly quantized into discrete gamma packets without exceeding .	Lucid executive focus, fluid working memory (), and coherent temporal binding.
Hyper-Novelty Choking	,	Incoming sensory/associative dilation flux exceeds Landauer erasure capacity (); downstream cascades fail to terminate at a slack sink.	Anxiety & Mania: Racing thoughts, loss of symbolic filtering, and inability to close episodic theta frames.
Hypo-Novelty Stagnation	,	Boundary aperture collapses onto slow historical priors; phase-space volume shrinks toward the IR wall ().	Depression & Anhedonia: Blunted novelty response, perseverative rumination, and rigid attractor lock-in.
Horizon Saturation Failure		Uncleared Landauer entropy accumulates across waking theta cycles, progressively narrowing effective .	Insomnia & Cognitive Fatigue: Progressive degradation of working-memory slots and elevated error propagation.

XI.4 Offline Sleep Sedimentation as Landauer Entropy Clearance

Because waking perception continuously injects high-dimensional sensory novelty across at a rate that leaves an irreducible thermodynamic remainder at the end of each episodic cycle, continuous waking operation inevitably approaches the horizon saturation threshold. Sleep resolves this thermodynamic bottleneck by temporarily decoupling the UV sensory input channel ( externally) and inverting the spectral flow into offline memory sedimentation.

1. Slow-Wave Delta Expansion (): During non-REM slow-wave sleep, the IR integration horizon expands from down to (). In our boundary eigenvalue density formula , widening the IR horizon increases the logarithmic phase-space span while lowering the per-cycle energy dissipation requirement ().
2. Hippocampal Sharp-Wave Ripple Compression (): Within these expanded delta troughs, fast sharp-wave ripples replay waking trajectories in time-compressed bursts, executing lossy variational coarse-graining (analogous to minimizing the Knuth-Plass global demerit functional across an entire day’s accumulated text). Redundant episodic variance is erased (paying the mandatory Landauer thermal dissipation cost ) while invariant structural weights are consolidated into long-term neocortical priors.
3. Restoration of the Waking Aperture: By clearing the accumulated residual entropy , sleep sedimentation resets the effective metabolic dissipation headroom and restores the arbitration equilibrium required for the subsequent waking cycle.

Section XII: Conclusion and Empirical Falsification Criteria

XII.1 Synthesis of the Scale-Invariant Boundary Architecture

The derivations presented in Sections X and XI establish that discrete physical, cognitive, and number-theoretic structures do not emerge from local initial-value rules, nor do they require ad hoc dark sectors or separate phenomenological postulates at each scale. Rather, they represent invariant spectral projections of a single underlying operator architecture: a continuous hyperbolic scale-dilation generator mediated by a Landauer-bounded boundary projection operator ().

Because this framework eliminates free phenomenological fitting parameters in favor of strict thermodynamic and geometric conservation laws, it generates sharp, quantitative predictions across astrophysics, high-energy cosmology, and electrophysiology. Any empirical violation of the boundary conditions below directly falsifies the scale-invariant projection hypothesis.

XII.2 Empirical Falsification Criteria Across Domains

1. Galactic Dynamics and SPARC Rotation Curves (External Field & Redshift Scaling)

In the Landauer-bounded horizon model, the Radial Acceleration Relation is not an immutable modification of Newtonian inertia, but the thermodynamic transmission efficiency of a boundary shell coupled to the ambient cosmological de Sitter temperature bath . This produces two falsifiable departures from both cold dark matter () halo profiles and static modified-gravity models:

• Falsification Criterion 1A (Absence of Intrinsic Scatter Beyond Baryonic Measurement Error): Across rotationally supported galaxies in the SPARC database and high-resolution IFU surveys, the residuals around must remain strictly bounded by observational uncertainties in stellar mass-to-light ratios (), distance, and inclination. Detection of a statistically significant secondary correlation between RAR residuals and dark-matter halo assembly history, merger age, or specific star-formation rate at fixed baryonic surface density falsifies the pure boundary-projection mechanism.
• Falsification Criterion 1B (Cosmological Evolution of the Acceleration Knee ): Because the IR thermal floor is fixed by the instantaneous Hubble expansion rate (), high-redshift rotation curves () observed via JWST and ALMA must exhibit an upward shift in the critical surface mass density threshold . If high-redshift disks follow a redshift-invariant rather than scaling with , the de Sitter thermal-bath coupling is falsified.

2. Primordial Black Hole Gamma-Ray Limits and Void Boundary Turnover

Equating the black hole Page turnover time to the universal cosmic void Landauer saturation timescale () identifies () as the exact mass pivot separating evaporated early-universe relics from persistent UV topological knots.

• Falsification Criterion 2A (Extragalactic MeV Gamma-Ray Cutoff & PBH Abundance): Relics with have already crossed their Page inflection point and radiated their holographic information into the cosmic background, producing a characteristic spectral feature in the extragalactic gamma-ray background (measurable by next-generation MeV telescopes such as AMEGO-X and e-ASTROGAM). Confirming a substantial cosmological dark-matter fraction composed of evaporating black holes strictly below without their corresponding Hawking radiation signature (or demonstrating that sub-critical cosmic voids () violate the scale-invariant turnover bound ) falsifies the UV–IR horizon duality.

• Falsification Criterion 2B (Void-Filament Boundary Acceleration Step): Deep inside cosmic voids where , peculiar velocity outflows must saturate at the de Sitter boundary floor rather than decaying as . Precision redshift-space distortion and weak-lensing measurements across void walls showing pure Newtonian scaling below directly refute the holographic void-processing term .

3. Cortical Electrophysiology, Working Memory, and Sleep Sedimentation

In the neural domain, the boundary eigenvalue density predicts that working-memory slot capacity is strictly bounded by the Landauer per-cycle erasure capacity , locking cross-frequency phase-amplitude coupling (PAC) to the ratio .

• Falsification Criterion 3A (Thermodynamic Choking Under Metabolic or Thermal Perturbation): Falsification Criterion 3A (Thermodynamic Choking Under Metabolic or Thermal Perturbation): In high-density scalp EEG and intracranial iEEG recordings during -back working-memory tasks, individual working-memory span must correlate linearly with the instantaneous gamma-to-theta frequency ratio up to the thermodynamic choke ceiling . Furthermore, mild cortical cooling, localized ATP-synthesis inhibition, or hyper-arousal sensory overload must compress and induce a measurable downward shift in the PAC nesting ceiling; causing working-memory failure precisely when exceeds the reduced metabolic erasure bound. Observing sustained, error-free working-memory expansion () without a corresponding proportional drop in or increase in cortical metabolic dissipation falsifies the Landauer channel bound.
• Falsification Criterion 3B (Sleep-Deprivation Drift Off and Delta Restoration): Extended wakefulness accumulates uncleared residual entropy , which must manifest electrophysiologically as a progressive collapse of theta-gamma phase-locking value (PLV) and a measurable drift in the cortical excitation-inhibition / novelty-prior arbitration balance away from (observable via aperiodic spectral slope shifts). Subsequent slow-wave delta sleep () coupled with hippocampal sharp-wave ripples () must quantitatively restore both the equilibrium slope and the nesting aperture in proportion to cumulative slow-wave energy. If complete working-memory and PAC restoration occurs without slow-wave delta expansion or sharp-wave ripple replay, the offline Landauer sedimentation mechanism is falsified.

XII.3 Summary Matrix of Quantitative Falsification Tests

Domain	Observable / Experimental Platform	Predicted Quantitative Law	Direct Falsification Condition
Galactic Dynamics	SPARC / JWST-ALMA rotation curves across	with	Significant residual scatter correlated with halo assembly history, or redshift-invariant at .
Cosmology & PBHs	AMEGO-X / e-ASTROGAM MeV background & void lensing	Mass pivot (); void outflow floor	Dark matter fraction in PBHs without Hawking flux, or pure Newtonian void profiles below .
Cortical Neurodynamics	High-density EEG / iEEG PAC & sleep polysomnography	; stability at ; delta-SWR restoration	Working-memory capacity without theta slowing or metabolic scaling, or full PAC recovery after sleep deprivation without delta-SWR activity.

By anchoring abstract spectral operators in measurable thermodynamic dissipation rates, this framework transforms the boundary interface (whether a galactic horizon, a primordial black hole surface, or a cortical microcircuit) from a passive geometric container into the primary physical engine of quantization and emergence.

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