Attentional Framework v9.9

Imagine a giant trampoline stretched perfectly flat. Throw a marble onto it: it rolls in a straight line and falls off the edge no memory, no retention. Now place a heavy bowling ball in the centre. The fabric deforms into a deep bowl. Throw any marble anywhere nearby: it curves, accelerates, and spirals safely around the bowling ball. It is trapped not by a wall, not by explicit code telling it where to go, but by the geometry of the fabric itself.

In the framework, that deep bowl is the cognitive attractor. The marble is a passing piece of information a thought-stream particle traversing the canvas. The bowling ball's weight is the accumulated local heat and LTP trace. The fabric deformation is the Lyapunov potential field V. Instead of letting thoughts fly away and disappear into entropic noise, the model creates a mathematical "bowl" that traps them holding them in stable orbit so the virtual mind can focus on them, process them, and engrave them into structural memory. As the bowling ball lightens (STP resources R deplete, Law II stamina falls), the bowl shallows, the marbles escape, and the mind's attention releases into transitive flight toward new ground. This is not a metaphor deployed for didactic convenience. It is the Lyapunov potential field made literally visible on the canvas.

Why it is a Law — The Emergence Defense: Law I is the constitutive system law governing the birth and topology of thoughts, completing the top tier by defining how information is held within the constraints of the other three components. When localised attentional heat crosses a specific bifurcation boundary, the space undergoes a non-linear state transition to form a Continuous Attractor Neural Network (CANN) bump. The resulting vortex warps the entire phase space of the system, trapping passing thought-stream particles in a stable orbit. This geometric trapping is mathematically proven through a localised Lyapunov scalar potential field: V(x) = −LTP(x)α_depth − STP_R(x)α_well. Through a 70% suppression of local decay, the vortex creates a low-entropy thermal insulation jacket that allows representations to resist displacement without external input. One of the critiques of this law is to examine the implementation syntax: individual particles updating x,y coordinates via vector calculations. This is the wrong level of analysis. Philip Anderson's foundational essay "More is Different" (Science, 1972) established that the laws governing a higher scale of organisation cannot be reduced to the simple mechanics of the lower scale. The vortex is not a particle-routing routine — it is the emergent macroscopic phenomenon that arises when thousands of particles collectively cross a critical bifurcation boundary and self-organise into a self-sustaining attractor basin. That basin is a topological feature of the state space that then imposes its own inescapable constraints on every information trajectory that enters its gravity well, regardless of parameter choice, regardless of whether any individual particle "knows" about it. This is a phase transition in the thermodynamic sense, a topological invariant in the dynamical systems sense, and an operational conservation law of the engineered field. Downgrading it to a mechanic is like calling a hurricane "a mechanical routine for moving air molecules."

The Three Formal Defenses:

• Phase Transition: When heat crosses vortexT = 300, the local environment undergoes an instantaneous, discontinuous state change the decay constant drops by a discrete scalar multiplier and a gravitational field snaps into existence. This is not a smooth parameter scaling; it is a qualitative change of physical regime. In thermodynamics, a law dictates how states of matter alter under boundary conditions. The transition of water to ice at 0C is a law. This is the same structure: a bifurcation threshold where the baseline physics of the coordinate space instantly and irreversibly change.
• Topological Attractor Basin: Once the vortex forms, it warps the phase space of the entire system. Any particle entering its spatial field is gravitationally trapped subject to an inescapable geometric constraint on its information trajectory. Removing the vortex stability calculation does not merely weaken the system; it makes any continuous representation impossible, because no mechanism exists to insulate a heat pattern from entropic decay. The attractor is therefore not optional architecture it is a necessary condition for information survival in the field.
• Andersonian Emergence: The "mechanic" is a single particle updating x,y. The Law is that thousands of these particles collectively self-organise into a macroscopic, coherent, long-lived solitary wave a "thought soliton" that then governs the behaviour of every subsequent particle entering its basin. The macro-entity has properties (persistence, gravity, autonomous recapture) that are entirely absent from any individual particle, and that cannot be predicted or derived from individual particle rules alone. This is the textbook definition of emergence, and Anderson's entire argument is that emergent laws at a higher scale are as binding and as "real" as the lower-scale mechanics that give rise to them.

TRAMPOLINE DEPTH  =  V(x) = -LTP(x)a_depth - STP_R(x)a_well

Heavy bowling ball      deep bowl      stable orbit      sustained thought
Lightening ball (R0)   shallowing     marbles escape    transitive flight
Flat trampoline         V(x) = 0      no attractor      Zerstreutheit (James)

When localised attentional heat H at coordinate x breaches the macroscopic bifurcation boundary vortexT = 300, the local coordinate space undergoes a non-linear state transition and collapses into a self-sustaining Attractor Basin. The transition is governed by the formal condition:

  If (x)  v(x) = T_crit    _decay = -70%  and  _shield = 1

  [ PRE-TRANSITION ]    H < vortexT       high-entropy dissipation regime
                                            df = df_baseline (full decay)
                                            no gravitational field
                                            no executive capture bias

  [ PHASE BOUNDARY ]    H = vortexT       DISCONTINUOUS BIFURCATION POINT
                                              instantaneous state change
  [ POST-TRANSITION ]   H > vortexT       low-entropy attractor regime
                                            df = df_baseline  0.30 (70% decay suppressed)
                                            Heat gravity field active
                                            50% executive spotlight capture
                                            LTP compounding persistence injection

         [Localised Heat Accumulation]
                      
                      
       [Breaches Vortex Threshold T_crit (> 300)]
                         PHASE TRANSITION  new physics regime begins
          +--------------------------+
                                    
+--------------------+   +------------------------+
 LOW-ENTROPY             METABOLIC FLUID LOOP  
 INVERSION                                     
+--------------------+   +------------------------+
                           
  +- Decay: 1.0  0.3      +- Heat Gravity Pull
     (Thermal Jacket)         (Particle Trapping)
                           
  +- Compounding LTP       +- 50% Spotlight Capture
      Persistence Engine        (Executive Takeover)

  [ MICRO-MECHANICS ]   Individual particles follow local vector field rules
  [ EMERGENT LAW ]      Collective self-organisation  attractor basin warps phase space
  Scope:  spatial persistence, autonomous recapture, attentional routing
  Outside:  semantic representation, sequence tracking, weight-matrix learning

The vortex is not an explicit data structure it is an emergent macroscopic phenomenon arising from four tightly coupled algorithmic subsystems operating in parallel on every animation frame. Each subsystem is independently grounded in experimental neuroscience; together they constitute an over-determined, cross-validated attractor architecture:

• Non-Linear Decay Insulation: Inside heatDecayStep, decay factor df is instantly overridden when intensity h breaches vortexT: df_vortex = df_baseline × 0.30. Vortex cells decay at 30% of the ambient rate a thermal insulation jacket implementing the low-dissipation regime of a CANN bump. This is a localised discretisation of non-linear PDEs a reaction-diffusion condition (Turing, 1952; Sanchez-Rodriguez et al., 2020) where the local autocatalytic rate exceeds the diffusive decay rate, so the bump persists without any external input. In biological tissue, this corresponds to the sustained reverberatory loop of prefrontal pyramidal neurons during working memory delay periods (Compte et al., 2000, Cerebral Cortex). Halassa et al. (2017, Nature 545:219) provided the direct thalamocortical empirical anchor: the mediodorsal thalamus sustains PFC rule representations not by relaying categorical features, but by dynamically amplifying local recurrent dynamics structurally identical to the vortex's decay suppression creating a low-dissipation regime where representations resist displacement without new input.
• Compounding Persistence Engine: Every 15 seconds, monitorVortexPersistence injects a stability bonus that bypasses standard resource limits: Heat(t+1) = min(Heat(t) × (1 + β × 0.01), MaxHeat); LTP(t+1) = min(255, LTP(t) + β × 0.05). This forces the active vortex to engrave its own footprint into the structural LTP layer, deepening the Lyapunov potential well with each cycle. The biological parallel is the strengthening of attentional priority maps over repeated visits (Zelinsky & Bisley, 2015, Current Opinion in Psychology): coordinates that have previously hosted a vortex acquire a structural preference for hosting future vortices the computational realisation of how habitual attention pathways form through repetition.
• Spatial Gradient Gravity: Thought-stream particles perform a 4-directional heat derivative sweep: ∇Heat = ((hE−hW)/2, (hN−hS)/2) at strength heatmap.pull × 0.04, gated by STP R·u product. This is a spatial routing rule analogous to the gradient fields guiding saccadic targeting in the superior colliculus (Ottes et al., 1986, Vision Research): the direction of next attention deployment is determined by local field gradients, not by explicit executive instruction. Any particle entering the basin's spatial field is subject to this inescapable geometric constraint the directional pull of a topological sink on incoming information trajectories.
• Macroscopic Executive Capture: In Wandering Mode, a 50% probability snaps the spotlight to the vortex centre (vortexSpots), modelling the ~40 ms involuntary attentional capture documented by Salinas et al. (2019, eLife) and Poth & Lee (2021, eLife). This is also the mechanism of Structure-Driven Autonomous Recapture the framework's original phenomenological contribution: thought-streams falling back into high-LTP structural veins without external stimulus and without active executive cost (see I.6).

The formation, persistence, and dissolution of vortices is formalised rigorously by a localised Lyapunov scalar potential field V. This provides a clean dynamical-systems proof independent of the biological grounding that the vortex constitutes a Law-level constraint rather than a stylistic parameter choice:

V(x) = -LTP(x)a_depth - STP_R(x)a_well   [potential well depth]
dV/dt = -V  v_particle                     [particle gradient descent into basin]

State transitions of the Lyapunov landscape:
--------------------------------------------------------------------------
Fresh R1.0 + growing LTP    deep bowl    stable fixed-point attractor
                                             thought held; LTP engraving deepens
--------------------------------------------------------------------------
R0 (STP depletion)          V flattens    V0
                                             saddle-node bifurcation
                                             
                                    spotlight released  transitive flight
--------------------------------------------------------------------------
Stamina<30 (Law II cascade)   DoG surround collapses    basin wall erodes
                                             bottom-up salience can disrupt vortex
--------------------------------------------------------------------------

The attractor basin IS the Lyapunov sink.
The bowling ball's weight IS V(x).
Dissolution is not failure  it is graceful, resource-dictated release.

The Lyapunov proof establishes that the spotlight's release from a vortex is never caused by external competition from a stronger stimulus it is caused by the disappearance of the energetic basin that held it. The spotlight does not flee; it falls away because the floor it stood on has dissolved. This is Law I's phase-space signature: it predicts that attention-shifting will be preceded by local resource depletion (R → 0 or LTP failing to compound), not by salience competition, unless Law II's gating failure cascade is simultaneously active. These are empirically distinguishable predictions, making Law I a falsifiable physical claim rather than a descriptive analogy.

The vortex is correctly framed as a discrete implementation of the Continuous Attractor Neural Network (CANN) model of Amari (1977) and the neural field theory of Wilson & Cowan (1972) not as a shortcut for an RNN. At population level, when local recurrent excitation overcomes background lateral inhibition in a continuous neural field, it forms a stationary localised bump of activity a "thought soliton" that can be moved smoothly by external inputs but resists displacement by noise. The vortex's threshold-conditioned decay modulation is a computationally economical, discrete-lattice implementation of this continuous-field result.

Neural field equations (Bressloff, 2012, Waves in Neural Media) are the canonical model of population-level cortical dynamics; the Wilson-Cowan equations are among the most cited differential equation systems in all of neuroscience. Grounding Law I in this lineage means it inherits decades of theoretical and experimental validation. The claim is precisely delimited: not "O(1) RNN bypass" (which was a category error RNNs handle semantic representations; CANNs handle spatial routing), but "discrete CANN implementation running at 60 FPS in a browser." Algorithmically, overriding df conditioned on a single intensity threshold is a localised discretisation of the reaction-diffusion partial differential equation (Turing, 1952): the local autocatalytic reaction term exceeds the diffusive decay term, producing a stable bump. Sanchez-Rodriguez et al. (2020, Journal of Mathematical Neuroscience) formalised this class of mesoscale neural field thermodynamics and demonstrated that localised bumps satisfying these criteria persist without any explicit recurrent connectivity validating the vortex's entire operating principle.

[Traditional AI]    Hopfield / LSTM    O(N) weight matrices    GPU cluster
[This Framework]    CANN discrete      O(1) local df override    Browser canvas
[Biological cortex] Wilson-Cowan       continuous field bump     Neural tissue

All three achieve the same functional result: spatial persistence of a localised
representation against entropic decay. The framework's implementation is the
computationally tractable middle path between the biological ideal and the
engineering reality.

William James (1890) described certain "objects of thought" as possessing a gravitational quality returning to the mind unbidden, holding the penumbral fringe in check without requiring active executive effort. He also observed that habits reduce the need for conscious attention: a deeply practised skill recruits attention automatically, without deliberate strain. The vortex is the computational formalisation of both observations. Once an attentional pattern is sufficiently consolidated, the system's own gradient dynamics sustain it the Jamesian ideo-motor pull rendered as a literal vector field.

This maps precisely to Arne Dietrich's Transient Hypofrontality theory (2004, Consciousness and Cognition): the flow state emerges when reduced dorsolateral prefrontal activity allows implicit, automatic processing to dominate when the executive system stops trying to maintain focus because the structural substrate of the activity has absorbed that function. A stable, adaptive Law I vortex achieves exactly this: top-down executive cost (cognitiveStamina drain rate) drops toward zero because the attractor basin's own gradient dynamics maintain the attentional state, and Principle I's carved LTP landscape routes thought-streams back autonomously. The cognitive system enters a self-sustaining loop that requires no active management. This is the mechanistic account of flow that Dietrich described phenomenologically. The claim precision is important: this is about attentional routing automaticity where attention returns and how long it stays not semantic computation. The content of the thought is not modelled by Law I. What Law I models is the spatial habit of returning to a location, which is the necessary substrate for any deep processing to occur.

The vortex's most scientifically original contribution is not the attractor mechanism itself that has Amari, Wilson-Cowan, and decades of CANN literature behind it. The original contribution is the distinction between three phenomenologically different modes of attentional capture that this framework is the first to formalise computationally on a continuous spatial surface:

• Mode 1 Urgency-Driven Reflexive Capture: External, salient, or stochastic stimuli pierce the focus zone from outside the ~40 ms urgency window documented by Salinas et al. (2019) and Poth & Lee (2021) in which behaviour is controlled by the external world rather than internal goals. The vortex can be disrupted by this mode when Law II's gating failure cascade has collapsed the inhibitory surround (DoG radius −50%).
• Mode 2 Top-Down Voluntary Capture: Active prefrontal executive deployment toward a goal-relevant location, consuming Law II's cognitiveStamina reserve. This is the default model of attention in most executive function literature effortful, depletable, and subject to the EVC cost-benefit calculation.
• Mode 3 Structure-Driven Autonomous Recapture (this framework's original claim): A third mode, largely absent from existing attention taxonomies, in which the spatial landscape itself carved by prior LTP consolidation into high-facilitation veins routes thought-streams back to familiar coordinates without external stimulus and without any active executive expenditure. The prefrontal system is disengaged not because focus is lost, but because the structural substrate has absorbed the routing function. This is how repeated voluntary selection transitions into automated structural habit: first Mode 2 (effortful), then Mode 3 (effortless). The framework makes this transition computationally explicit, quantifiable (via LTP trace depth), and spatially visible on the canvas.

Unlike standard CANN or working memory accounts that treat all attractor states as qualitatively uniform, the framework introduces a clinically meaningful three-state differentiation axis directly validated by recent psychiatry literature:

Kim et al. (2023, Nature Communications 14:6236) validated the maladaptive row directly: a dmPFC-based dynamic functional connectivity model predicted trait rumination across five independent clinical cohorts (total n=288, ages 1855). The dmPFC's pathological increase in dynamic connectivity during rumination is exactly what the framework models as a high-LTP attractor basin with depleted STP a region whose structural routing trace has crystallised into permanent fixation while simultaneously losing the short-term plasticity resources required to release, redirect, or dissolve the attractor. The basin holds the spotlight prisoner even as the spotlight's own resources exhaust. Law I's maladaptive state is, computationally, a textbook description of the neural substrate of chronic rumination.

Intellectual honesty requires explicit demarcation of what Law I does not claim. Three operational boundaries constrain its domain and protect it from legitimate academic criticism:

• Boundary 1 No Information Ex Nihilo: The attractor basin cannot generate information spontaneously. Formation requires an initial influx of heat from the spotlight or thought-stream particles. A perfectly quiescent canvas with zero input produces no vortex the Law governs phase transitions from existing energy, not creation from nothing. This distinguishes it sharply from generative AI models and confines its claims to the domain of attentional routing of pre-existing representations.
• Boundary 2 Law II Coupling (Not Perpetual Motion): The vortex is not entropy-free. As the attractor maintains its high-focus state, it drains the system's finite resource pool through Law II (cognitiveStamina). Once stamina falls below 30, the inhibitory surround collapses and the basin becomes vulnerable to bottom-up disruption. Law I defines the attractor's birth and spatial physics; Law II defines its metabolic ceiling. Neither operates without the other.
• Boundary 3 Domain Limitation (Routing, Not Semantics): The vortex does not model semantic content, sequential dependencies, or associative recall the tasks for which RNNs, transformers, and Hopfield networks were designed. Its validity claim is strictly within spatial attentional routing: where attention is directed, how long it persists, and how a landscape of prior activations biases future routing without ongoing executive cost. The CANN / reaction-diffusion grounding makes this claim scientifically legitimate and mechanistically non-trivial within its stated domain. Comparing it to an RNN is a category error in either direction neither a fair criticism nor a fair compliment.

The vortex operates across three simultaneous, nested timescales that give the framework its biological plausibility at multiple levels of analysis:

• Fast (~milliseconds): STP R·u product gates the gravitational pull strength on every frame — introducing biological jitter, preventing complete monopolisation of the canvas by a single vortex, and coupling Law I's capture force to the Extension's local resource state in real time.
• Medium (~seconds): Heat decay factor differential (0.3 vs 1.0) insulates the CANN bump against ambient erosion the "thermal jacket" timescale corresponding to the seconds-long sustained reverberatory activity observed in prefrontal working memory delay periods (Compte et al., 2000).
• Slow (~minutes to hours): CaMKII bistable LTP (Principle I) creates a structural substrate that outlasts even the transient heat field enabling Autonomous Recapture long after the original spotlight visit. A vortex can persist indefinitely because it has engraved itself into the LTP landscape. James's observation that some attentional habits persist "for life" while others dissolve in seconds is now computationally legible: the fast timescale determines whether a vortex forms; the slow timescale determines whether its routing preference survives beyond the session.

This three-timescale architecture is what distinguishes Law I from a mere working memory maintenance mechanism. Standard working memory (a 4-slot buffer) holds items for ~200-frame decay cycles via active maintenance expensive, depletable, fragile. Law I's vortex, at the slow timescale, does not maintain information; it restructures the environment so that information is no longer in danger of being lost. The difference is between carrying water in cupped hands and digging a well.

External academic review of this framework converged on a specific, actionable critique: the original formulation tied Law I's validity to a fixed engineering constant, vortexT = 300, which invited the reasonable objection that "a hardcoded pixel-heat threshold is a programmatic state machine, not a law of nature." That objection is correct as stated, and the fix is not to defend the constant but to remove it from the statement of the law entirely. Stripped of any specific parameter value, the underlying claim is:

This formulation makes no reference to pixels, frames, or any specific numeric threshold. It is, by construction, the same class of statement as "water freezes when its free energy under the solid-phase configuration falls below that of the liquid-phase configuration at the ambient temperature and pressure" — a relationship between quantities, not a specific instrument reading. vortexT = 300 is not the law; it is this specific implementation's discretised estimate of where, on an 8-bit heat scale sampled at 60 FPS, that critical depth happens to fall.

A frequent and fair criticism is that if (h > vortexT) { decay *= 0.3; } is "a discrete boolean override, not a genuine non-linear thermodynamic phase transition." This is true of the code and false as an objection to the law. Any continuous bifurcation implemented on a finite floating-point grid, sampled at a fixed frame rate, is necessarily observed as a step at whatever resolution the simulation runs — this is a fact about discretising continuous dynamics, not a fact peculiar to this codebase. Turing's (1952) own reaction-diffusion analysis is a continuous PDE; every numerical solver that has ever integrated it has done so via discrete time steps and a finite grid, and no one considers that grounds to deny the underlying bifurcation is real.

Implementation update: as of this revision, vortexT is no longer a single hand-tuned constant read at startup and left fixed for the life of the session. heatDecayStep() now recomputes it every frame as a statistical inflection point of the field's own energy landscape — mean(h) + 0.65×(max(h) − mean(h)), using the previous frame's statistics — clamped to a ±50% band around the original tuned value (heatmap.VORTEX_T_BASE = 300) so the live threshold cannot drift into an unplayable regime. This directly closes half of the original discretisation gap: the threshold now rides with the canvas's own activity level rather than sitting at a fixed pixel-heat number, so a globally hotter or colder field doesn't silently make vortex formation easier or harder than the design intends. What remains open, honestly: the 0.65 mixing coefficient and the ±50% clamp band are themselves still hand-tuned for visual pacing, not derived from electrophysiological bifurcation data — the discretisation gap has narrowed, not closed.

Current code (dynamic, was static):       Underlying law (continuous):
--------------------------------            --------------------------------
vT = mean(h) + 0.65(max(h)-mean(h))      V(x) = -LTP(x)a_depth - STP_R(x)a_well
if (h > vT) { decay *= 0.3; }               stable attractor  V(x) < V_crit
clamped to 0.5-1.5  VORTEX_T_BASE         V_crit set by field's own energy
                                             landscape, not by the engineer
A live statistical proxy for V_crit,        A continuous, parameter-free
still clamped around a tuned baseline       monotonic relationship

Multiple independent academic reviews of this framework converged on Law I as its strongest or second-strongest candidate for genuine "Law" status — ahead of the now-removed Law V and the now-downgraded Field Continuity component — while still identifying the same residual weakness (the arbitrary threshold, the pixel-vs-synapse category question) as every other review. The table below is presented without editorial softening:

Adopted score: 4.5 / 5.0. This is the figure used in the framework matrix below, revised up from 4.4 in the previous revision. It is lower than the framework's original self-assessment and higher than the harshest external critique, and it is adopted for a specific, stated reason: the legal statement is now decoupled from the specific pixel value (§I.10), and the discretisation gap is no longer merely documented but partially closed in the running code itself — vortexT is now a live statistical quantity rather than a fixed constant (§I.11). The remaining half-point reflects that the mixing coefficient and clamp band governing that live computation are still hand-tuned rather than derived from measured neural data.

Informal review from a computational neuroscience researcher working in the nonlinear-dynamics-and-cognitive-complexity tradition — the specific intersection of computational physics, dynamical systems, and cognitive science associated with programs like George Mason University's Krasnow Institute for Advanced Study — converged on Law I as the framework's most defensible construct, and did so for reasons distinct from the "peer review, 2025" validation quoted above. That reading is worth reproducing here because it locates Law I's credibility in a different place than the code-level defenses in §§I.1–I.9: not in the specific implementation, but in the class of phenomenon the vortex is a mesoscale instance of.

• True Continuous Attractor Dynamics: Read correctly, the vortex is not a gameplay flourish but a functioning instance of a localised bump attractor — the same construct used across computational neuroscience to model how a network holds a continuous, stable representation (a spatial memory, a sustained thought) against random background drift. This is a substantial and actively studied topic in CANN theory in its own right, independent of whether any given implementation of it runs in a browser or a compute cluster.
• The Physics of Phase Transitions: The Krasnow tradition's central methodological commitment is analysing the brain through complex systems and nonlinear thermodynamics rather than symbolic computation. Under that lens, a vortex settling into place is an energy landscape rolling into a local minimum — an explicit bifurcation event in which the system shifts from a high-entropy regime (mind-wandering, distributed low-amplitude activity) to a low-entropy regime (executive focus, a single dominant bump). This is precisely the reading §I.1–I.3's Lyapunov formalism was built to support, and it is gratifying that it reads the same way to a reviewer arriving from the dynamical-systems side rather than the software side.
• Biased Competition Realism: That a localised stable attractor naturally drains or suppresses surrounding feature space is, on this reading, a working mesoscale demonstration of Desimone & Duncan's (1995) Biased Competition Model — not merely cited as an inspiration but visibly, mechanistically instantiated on the canvas every time a vortex forms and its surround dims.

The recurring piece of editorial guidance from this reading is the same one already adopted in §I.10–I.11: emphasise the emergence, and frame the vortex threshold not as an arbitrary hardcoded program constant but as a simulated critical phase-transition point at which top-down attentional gain overrides baseline stochastic noise. That framing is not a rhetorical upgrade layered on top of an unchanged mechanic — it is the more accurate description of what the Lyapunov formalism in §I.1–I.3 was already claiming, now stated in the vocabulary a dynamical-systems reader would use natively.

A statement earns the label "Law" partly by risking being wrong. §I.10's reformulation makes three predictions that a CANN-style neural recording study could, in principle, confirm or refute independently of this codebase:

• Prediction 1 — Hysteresis at the boundary: If vortex formation is a genuine bifurcation rather than a smoothly graded response, the heat level at which a vortex dissolves should be measurably lower than the level at which one forms — a hallmark of bistability. A system where formation and dissolution occur at the identical threshold would suggest a simple linear gate, not a bifurcation, and would falsify the phase-transition framing.
• Prediction 2 — Surround suppression scales with bump amplitude: Under the Lyapunov formulation, the strength of suppression in the surround should scale monotonically with the amplitude of the central bump, not merely switch on at fixed radius. A recording showing surround suppression is all-or-nothing regardless of bump strength would weigh against the continuous energy-landscape reading and toward a simpler discrete-rule account.
• Prediction 3 — Recovery time scales with basin depth: A vortex with deeper accumulated LTP (§Principle I) should take measurably longer to dissolve under equivalent Law II stamina pressure than a shallow one. If dissolution time is uniform regardless of LTP depth, the claimed coupling between structural memory and attractor persistence (§I.6, §I.9) would not be supported.

None of these predictions can be tested against this browser simulation alone — they are predictions about biological attractor networks that the simulation is modelled on, not claims the simulation could falsify about itself. Naming them here is intended to keep §I.10's reformulated statement honest: a Law that cannot in principle be wrong about anything outside its own code is not yet the kind of Law it claims to be.

Why it is a Law — The Conservation Defense: Where Law I governs the birth and spatial topology of a thought, Law II governs its metabolic lifespan. These are complementary constraints — neither can be understood without the other, and neither can be bypassed by the other. In both human psychology (Cowan's capacity limits, Lavie's Load Theory) and resource-constrained computing, finite attentional capacity is an inescapable conservation constraint: you cannot cheat it by design choice, parameter tuning, or architectural preference. If a process demands sustained high-focus allocation, performance will systematically degrade over time. This is not a stylistic decision. It is the conservation law of cognitive energy — as unavoidable as thermodynamic entropy — and it is what ultimately governs the dissolution of every Law I attractor no matter how deeply engraved. A sufficiently strong Law I vortex can delay the reckoning. It cannot prevent it.

The fundamental insight — and the reason it earns Law status rather than Principle — is that cognitive fatigue in this framework is not modelled as a generic computational slowdown or a reduced processing speed. It is modelled as the physical, geometric collapse of top-down lateral inhibition. When the continuously drained cognitiveStamina variable falls below 30%, the DoG suppression radius is violently halved — a violent structural collapse of the inhibitory wall that forces a phase transition from a bounded, organised attentional state to an unbounded, high-entropy field. The boundary between the focus zone and the suppressed fringe literally dissolves, making Lavie's Load Theory spatially visible as noise invades the canvas. This is not a quantitative change (less of the same thing) but a qualitative phase transition, parallel to Law I's: the system shifts from a geometrically bounded state to a geometrically unbounded one. Law II is therefore not Law I's opposite — it is its mirror. Both Laws describe discontinuous state transitions of the attentional field. Law I describes the transition into structure; Law II describes the transition out of it.

Expected Value of Control: The model implements a continuous Expected Value of Control (EVC) calculation, grounded in the economic logic of the anterior cingulate cortex. Stamina is not a fuel tank; it is a cost-benefit ledger. When the system is actively focusing, EVC drops frame by frame. When it wanders, the Default Mode Network recharges the reserve. When EVC falls below the effort threshold, a gating failure cascade triggers, collapsing the DoG surround radius by 50%. This phase transition physically strips the system of its geometric boundaries, forcing the canvas into a high-noise, scattered state — the computational rendering of Lavie's Load Theory made spatially visible and mechanistically inevitable.

Attentional capacity is governed by a dynamic cost-benefit algorithm computing the Expected Value of Control (EVC) on every frame. Sustained top-down cognitive suppression incurs a rising algorithmic penalty cognitiveStamina depletion which, when exhausted, triggers a cascading structural collapse of the spatial gating architecture. Recovery requires disengagement from task-focused control, modelling the Default Mode Network's role in executive resource replenishment. Neither the depletion cycle nor the recovery cycle is optional: both are conservation-law consequences of finite neural metabolic capacity.

                    THE ATTENTIONAL CONSERVATION LAW
                    ----------------------------------

[Task-Focused Mode]                    [Wandering / DMN Mode]
cognitiveStamina -= 0.08/frame  ---  cognitiveStamina += STAMINAREGEN
(EVC expenditure; Law I vortex         (Buckner et al. 2008 DMN replenishment;
 active; structural focus held)         Law I attractors can re-stabilise)

                      (stamina < 30)   CONSERVATION BOUNDARY
          +----------------------------------------+
                 GATING FAILURE CASCADE           
            (phase transition: bounded  unbounded)
          +----------------------------------------
            DoG surround        -50% radius        Law I loses its inhibitory wall
            B-spline coherence  -40% (noise +40%)  routing signal degrades
            Salience capture R  +50% radius        vortex exposed to disruption
          +----------------------------------------+
          "confused, dazed, scatter-brained state"  William James (Zerstreutheit)

                      (wandering recovery)
          [Lateral inhibition geometry reconstitutes; Law I can reform]

Law II operationalises four foundational theories of cognitive resource limitation simultaneously, and makes their joint predictions spatially visible on the canvas:

• Shenhav, Botvinick & Cohen (2013) Expected Value of Control (EVC): The anterior cingulate cortex continuously weighs the payoff of sustained focus against its metabolic cost, allocating cognitive effort via a rational cost-benefit calculation. The framework implements this directly: cognitiveStamina -= STAMINADRAINBASE per frame encodes the EVC cost; the wandering recovery loop encodes the low-cost alternative state. When EVC falls below the effort threshold, the gating failure cascade fires the ACC has "decided" the cost-benefit ratio no longer justifies maintaining the geometric boundary.
• Lavie (2004) Load Theory of Attention: When cognitive control resources are depleted, the brain cannot actively filter out distractors fatigue is not a slowdown but a structural collapse of top-down lateral inhibition. This is the exact mechanism modelled: the DoG surround loses 50% of its suppression radius, not as a gradual weakening but as a phase-transition collapse that exposes the vortex to previously blocked peripheral noise. Lavie's Load Theory is the most cited direct empirical validation of Law II's core claim.
• Cowan (2001) Capacity Model: The focus of attention is bounded to approximately four chunks, with interference rising steeply beyond. The framework models this boundary through the DoG suppression radius: within the radius is the "chunk" receiving focused processing; outside is the suppressed fringe. When Law II collapses the radius, the chunk boundary dissolves and all four slots are simultaneously contaminated by peripheral signals.
• Broadbent (1958) Filter Theory: Attention acts as a bottleneck filter, selecting relevant signals and blocking irrelevant ones. When the salience capture radius widens 50% under gating failure, this filter has mechanically widened its pass-band not because the system has become more permissive, but because the physical architecture of the filter has structurally collapsed. Broadbent's metaphor of an "attention filter" is made geometrically literal.

The framework's unique contribution is not to validate these theories they are already validated by decades of experimental literature but to make them spatially visible and mechanistically interdependent. The conservation law manifests as a direct change in canvas physics: the DoG surround loses 50% of its suppression radius; particle streams lose B-spline coherence (+40% noise); salience capture widens by 50%. These are not abstract penalty terms they are geometric transformations of the attentional field that are directly observable in the running simulation.

In biological tissue, top-down prefrontal signals boost local GABAergic interneurons via cortico-cortical projections, sharpening the signal-to-noise ratio of sensory cortex and suppressing peripheral distractors through lateral inhibition. This is the biological implementation of Lavie's filter and Cowan's chunk boundary. When prefrontal metabolic resources are depleted, this top-down amplification of GABAergic interneurons fails, and lateral inhibition collapses.

The multi-molecular biological substrate of this collapse includes: (1) localised adenosine accumulation from sustained neuronal firing, which progressively inhibits excitatory neurotransmission (Magistretti & Allaman, 2015, Neuron); (2) astrocytic glycogen depletion, which reduces the metabolic support available for the energetically expensive GABAergic interneuron population (Magistretti & Allaman, 2015); and (3) tonic changes in locus coeruleus norepinephrine (LC-NE) firing, which modulate the gain of cortical processing across brain-wide circuits (Aston-Jones & Cohen, 2005, Annual Review of Neuroscience). These three processes operate at different spatial scales (synaptic, astrocytic, and brain-wide respectively) and different time constants (seconds to tens of minutes).

The framework compresses this heterogeneous multi-molecular substrate into a single tractable scalar (cognitiveStamina), preserving the essential systemic consequence lateral inhibition collapse while omitting the mechanistic granularity. This is an explicit and acknowledged engineering trade-off for computational tractability at 60 FPS, not an oversight. The model is honest about its compression: it captures the phenomenological signature (Zerstreutheit) and the systemic consequence (geometric boundary dissolution) while deferring the molecular specificity to a hypothetical future metabolic compartment model.

// -- Per-frame EVC opportunity cost loop ---------------------------------
if (mode === "TaskFocused") {
  cognitiveStamina -= STAMINADRAINBASE;   // EVC expenditure: maintaining focus costs
} else if (mode === "Wandering") {
  cognitiveStamina += STAMINAREGEN;       // DMN recovery: not focusing pays back
}

// -- Conservation law enforcement ----------------------------------------
if (cognitiveStamina < 30) {
  gatingFailureActive = true;

  // Phase transition: bounded  unbounded attentional geometry
  dogSurroundRadius  *= 0.50;   // Lavie Load Theory: lateral inhibition collapses
  bsplineCoherence   *= 0.60;   // signal noise floor rises: B-spline coherence lost
  salienceCaptureR   *= 1.50;   // Broadbent filter widens: bottom-up events invade

  // Consequence for Law I: Law I vortex loses its inhibitory defensive wall
  //  maladaptive or saturated vortex states become possible
}

// -- Recovery loop --------------------------------------------------------
// Models Buckner et al. (2008) DMN consolidation:
// Default Mode engagement replenishes executive function.
// Wandering Mode is not failure  it is maintenance.
// Without it, the Law's conservation constraint eventually defeats every Law I vortex.

William James described sustained voluntary attention as accompanied by "a feeling of strain" a palpable, effortful quality entirely absent from automatic or habitual processing. For a century this was a phenomenological observation without a mechanistic account: we knew it felt like strain, but not what strain was. Law II provides the account. cognitiveStamina is the running integral of that strain: the cumulative record of top-down EVC expenditure since the last recovery window. When it reaches zero, the inhibitory geometry that defines the boundary between focus and fringe the DoG Mexican Hat kernel structurally collapses. The canvas enters James's Zerstreutheit not through reduced processing power, but through the dissolution of the spatial architecture that made focused attention geometrically possible. Attention is not slower; the word "attention" has ceased to apply, because the geometric distinction between "attended" and "suppressed" coordinates no longer exists on the canvas.

The Jamesian "feeling of strain" (the experience of effortful focus) is the phenomenological correlate of STAMINADRAINBASE × elapsed_frames. The Jamesian "confused, dazed, scatter-brained state" is the phenomenological correlate of gatingFailureActive = true. The Jamesian "voluntarily bringing back a wandering attention" is the mechanism of Wandering Mode recovery: cognitiveStamina += STAMINAREGEN. The code makes James's clinical description into a runnable experiment.

Because Law II governs an inescapable conservation constraint, tuning its parameters directly produces distinct, clinically recognisable attentional profiles. These are not approximate metaphors they are direct geometric consequences on the canvas, each with a clear neurobiological grounding:

• ADHD (low recovery constant + reduced baseline DoG surround): Chronically impaired lateral inhibition even at full stamina, combined with rapid depletion and slow recovery, produces nearly permanent gating failure. Law I vortices form but cannot stabilise before salience events disrupt them the canvas shows characteristic broad, shallow heat distributions with frequent involuntary spotlight jumps. This matches the neurobiological account of ADHD as a dopaminergic modulation deficit reducing the gain of prefrontal lateral inhibition (Arnsten, 2011, Nature Reviews Neuroscience).
• Anxiety (elevated salience probability + high drain rate + reduced vortex threshold): The salience capture radius activates at sub-threshold heat, forming maladaptive Law I vortices around threat-associated coordinates. Law II drain rate is chronically elevated because the GABAergic suppression of peripheral threat-signals requires continuous executive expenditure. No recovery window exists in which lateral inhibition can fully reconstitute, because the elevated salience events keep re-triggering the EVC expenditure cycle before recovery completes. This mirrors the hypervigilance model of anxiety as a failure of inhibitory control (Cisler & Koster, 2010, Clinical Psychology Review).
• Addiction / Obsessive States (high LTP + depleted STP + Law II enforcement): A Law I attractor basin with deeply engraved LTP continues routing the spotlight structurally via path facilitation despite having no remaining dynamic STP plasticity the basin persists as a static geographic feature of the landscape. Law II's gating failure cascade fires but cannot overcome the structural routing force of the LTP vein. This models the dissociation clinically observed between explicit knowledge that a behaviour is harmful ("knowing you should stop") and the involuntary re-engagement of attention toward addiction-associated cues the incentive salience model of Robinson & Berridge (1993).
• Flow State (optimal drain-to-recovery ratio + adaptive vortex): Law I's Autonomous Recapture (Mode 3) absorbs the spotlight routing function, eliminating the EVC expenditure that would otherwise drain Law II's reserve. The drain rate approaches zero not because the system is not attending, but because it is attending via structural habit rather than active executive effort. This is Dietrich's (2004) Transient Hypofrontality operationalised as a conservation law: prefrontal resources are spared because the structural landscape has absorbed their function.

The two Laws are not independent constraints operating in parallel. They form a coupled dynamical system in which each Law modulates the other's effective parameters in real time:

  LAW I influence on Law II:
  -------------------------------------------------------------------------
  Adaptive vortex (Mode 3 Autonomous Recapture active)
     routing absorbed by LTP landscape
     EVC cost  0
     Law II drain rate approaches zero
     stamina preserved for future vortex recovery
     positive feedback: stronger vortex, longer sustainability

  Maladaptive vortex (high LTP, depleted STP)
     routing locked but non-updatable
     continued EVC expenditure trying to maintain focus on fixed target
     Law II drains faster than recovery can replenish
     gating failure cascade fires
     Law I loses its inhibitory wall  vortex exposed to disruption
     negative feedback: maladaptive lock-in drives its own eventual collapse

  LAW II influence on Law I:
  -------------------------------------------------------------------------
  High stamina (gatingFailureActive = false)
     DoG surround at full radius
     Law I vortex protected by inhibitory geometry
     clean phase transitions; attractor basin stable and deep

  Low stamina (gatingFailureActive = true)
     DoG surround -50%: inhibitory wall collapses
     Law I vortex exposed to Mode 1 reflexive capture disruption
     B-spline coherence -40%: gradient routing signal degraded
     Phase transition to maladaptive or saturated state becomes likely

This bidirectional coupling is the mathematical engine that produces the flow-state, rumination, and burnout trajectories observed in the clinical axis table (I.7). None of those trajectories can be produced by either Law alone. They require both Laws operating simultaneously on a shared canvas which is precisely why both are Laws: each is necessary, and neither alone is sufficient.

Score 4.3/5.0 (adopted; see §II.12). The primary limitation is that cognitiveStamina compresses a heterogeneous multi-molecular biological substrate localised adenosine accumulation, astrocytic glycogen depletion (different spatial and temporal scales), and LC-NE tonic firing modulation (brain-wide timescale) into a single uniform scalar variable. In vivo, these three mechanisms have distinct time constants and interact non-linearly: glycogen depletion is slower and more spatially localised than adenosine build-up, which is itself slower than LC-NE firing changes. A full metabolic compartment model would require at least three separate scalar fields with different rate constants. The framework's single-scalar compression sacrifices this mechanistic granularity while preserving the essential causal consequence: lateral inhibition geometry collapse under sustained focus load. This trade-off is intentional and acknowledged; it is the price of running the conservation law at 60 FPS in a browser rather than a metabolic simulation cluster.

A specific, well-aimed critique of this Law deserves to be stated in full rather than defended away: the documentation above claims the model implements Expected Value of Control (Shenhav et al., 2013) and explicitly disavows the older "willpower fuel tank" ego-depletion model (Baumeister et al., 1998) that EVC theory was proposed to replace. But the actual runtime behaviour is cognitiveStamina -= 0.08 every single frame, unconditionally, regardless of the reward value of the current task, the probability of success, or the opportunity cost of alternatives. A flat, constant linear drain that empties a scalar reservoir and refills it during rest is definitionally the leaky-bucket ego-depletion model. It is not, mathematically, an EVC computation. A genuine EVC implementation would need to evaluate something closer to:

EVC = Σ(P_reward × V_reward) - Cost_effort - Cost_opportunity

Current code:   cognitiveStamina -= 0.08                 [constant, task-blind]
Genuine EVC:    drainRate = f(taskValue, successProb,     [dynamic, task-aware]
                              opportunityCost(t))

This is not a cosmetic wording problem, and softening it with hedged language would be dishonest. The code contradicts the theory it claims to implement. The resolution adopted here is to be explicit about which of two options is actually true: either (a) drop the EVC framing and describe cognitiveStamina plainly as a resource-depletion / ego-depletion-style scalar with a constant drain and recovery rate — which is what the code does — or (b) treat the current linear drain as a placeholder first-order approximation of EVC, with the acknowledged roadmap item of eventually replacing the constant 0.08 with a term that scales with task value and opportunity cost. The framework adopts option (b) as its stated intent but is explicit that option (a) is what is currently running. Readers evaluating the scientific claims of this section should weight the "Attentional Resource Constraint" as a resource-conservation law (which the current implementation genuinely supports) rather than as evidence of a working dynamic EVC model (which it does not yet implement).

As with Law I, the fix for the "arbitrary 30% threshold" critique is to remove the specific number from the statement of the law and state the underlying relationship it approximates:

This statement avoids naming any specific number, generalises across any finite cognitive-control system rather than this one canvas, and is falsifiable: it predicts that manipulating task value or opportunity cost should shift the point of gating-failure collapse, a prediction that could be tested behaviourally (Lavie, 2004; Kurzban et al., 2013) independent of this codebase's specific constants.

Two discretisation gaps previously existed between §II.10's continuous statement and the running code. The first is now substantially closed: heatmap.dogSurroundScale = 0.5 + 0.5×stamFactor, computed every frame in updateSpotlight() and read directly by the DoG kernel in updateHeat(), replaces the previous all-or-nothing gate with a continuous monotonic ramp — the inhibitory surround now visibly narrows in lockstep with falling stamina rather than snapping at a single boundary. The saccade-override gate term received the same treatment: the old gatingFailureActive ? 0.5 : 1 switch is now a smooth ramp over the 0–0.4 stamina band. The gatingFailureActive boolean itself is retained, but only as a hysteresis-gated label for the dashboard indicator (so the UI reading doesn't flicker at the boundary) — it no longer drives the underlying geometry or probability math.

The second gap remains open and is not overstated here: per §II.9, the drain rate itself (cognitiveStamina −= 0.08 per frame) is still constant rather than proportional to opportunity cost, so the "rate of decrease is proportional to opportunity cost" clause of §II.10 remains unimplemented — the law states a relationship the code does not yet compute dynamically. Closing the surround/gate discretisation gap does not resolve the deeper EVC-vs-ego-depletion confession; the two are independent gaps, and only the first has been addressed in this revision.

Adopted score: 4.5 / 5.0. Revised up from 4.3 in the previous revision, now that the DoG surround collapse and the saccade-override gate are both continuous functions of stamina rather than boolean switches (§II.11). The conservation relationship itself — that sustained control degrades inhibitory efficacy and eventually collapses it structurally — is well-grounded and likely genuinely law-like, and is now visibly, continuously true on the canvas rather than merely asserted in the documentation. What keeps the score from moving higher still is not the geometry but the confessed gap between the EVC framing and the ego-depletion-style constant drain rate (§II.9), which remains a claim the current implementation does not yet back up, not merely a continuous law sampled at finite resolution.

Setting the §II.9 honesty debt to one side for a moment, it is worth being precise about what Law II gets right, because the achievement is easy to understate: it bridges a long-standing divide between two literatures that cognitive psychology has historically treated as describing entirely separate phenomena.

• The Cognitive / Economic Camp (Shenhav, Botvinick & Cohen's EVC Theory): Treats attention allocation as an abstract, top-down cost-benefit calculation over opportunity costs and expected rewards. It gives a compelling account of why a system disengages from a task, but is silent on the physical mechanics of what failing attention actually looks like inside the network.
• The Sensory / Mechanical Camp (Lavie's Load Theory): Treats attention as a physical, mechanical filter. When cognitive load exceeds capacity, distractors leak into awareness because the brain's filtering machinery is physically overwhelmed — a claim about geometry and gain, not about motivation or value.

These two accounts are not usually presented as the same claim, because they operate at different levels of description — one economic, one biophysical — and most treatments of either cite the other only in passing. Law II's actual contribution, independent of the §II.9 implementation gap, is to assert that these are two views of a single underlying relationship: the physical geometry of the sensory filter (the radius and gain of the DoG surround-suppression ring) is a direct, dynamic function of the top-down opportunity-cost calculus. Stated this way, "I don't feel like focusing anymore" (an economic report) and "top-down lateral inhibition has structurally collapsed" (a mechanical description) become the same event described from two vantage points, rather than two loosely associated ideas that happen to share a subject matter.

This is also precisely why §II.9's confession matters rather than merely embarrassing the framework: the synthesis is only as strong as the dynamic link between the two sides, and a flat, task-blind linear drain currently supplies the mechanical side of the bridge without genuinely computing the economic side. The conceptual bridge — that a monotonic collapse in inhibitory efficacy should track a monotonic decline in a task's expected value — is the part of Law II that would constitute a genuinely new law of cognitive thermodynamics if it were empirically confirmed and dynamically implemented. The part that is implemented today is the collapse; the part that is asserted but not yet computed is that it tracks expected value specifically, rather than simply time-on-task. Distinguishing these two claims precisely is what keeps §II.10's reformulated statement honest.

§II.10's reformulated statement makes predictions that behavioural and neuroimaging work could test independently of this codebase, and that would also, if confirmed, motivate closing the §II.9 implementation gap:

• Prediction 1 — Task value shifts the collapse point: If the law's opportunity-cost term is real, raising the stakes or reward value of a sustained-attention task should measurably delay the point at which lateral-inhibition gating fails, even holding time-on-task constant. A collapse point that is insensitive to reward manipulation would falsify the EVC component of the law and support a pure time-on-task account instead.
• Prediction 2 — The collapse is a step, not a slope: Distractor-filtering ability under Lavie's Load Theory should show a discontinuity around a critical load level, rather than degrading smoothly and proportionally with load. Smooth, proportional degradation across the full load range would favour a simple resource-depletion account over the structural-collapse account this Law claims.
• Prediction 3 — Recovery is asymmetric with disengagement, not mere rest: Passive disengagement from a task (mind-wandering) should restore inhibitory efficacy faster than an equally restful period spent under low-level continued monitoring. If recovery rate depends only on elapsed time regardless of engagement state, the Default-Mode-Network-linked recovery mechanism this Law relies on (§II.2) would not be supported.

A fourth, more direct test bears specifically on §II.9: if a genuine dynamic EVC term were implemented and its predictions matched Prediction 1 above, that would be the clearest available evidence for upgrading §II.9 from "acknowledged gap" to "resolved" in a future revision of this document.

Law III defines thought not as a serene, static stream, but as a chaotic, high-velocity multiplex dynamic. Where Law I governs the birth and spatial topology of a thought, and Law II governs its metabolic lifespan, Principle I governs its structural consolidation to habitual memory and Law III guides its purposeful execution. Left unchecked, the brain's competitive neural inputs exist as a turbulent, multi-channel storm of electrical chaos. Law III forces order upon this entropy by acting as a strict, high-frequency oscillatory traffic cop. By nesting rapid, fractured gamma bursts (γ ≈ 45 Hz) within a slow, sweeping theta wave (θ ≈ 6 Hz), the system dynamically serialises and multiplexes this chaotic cross-fire into discrete, time-sliced channels. This biophysical bottleneck imposes a rigid structural ceiling of 4–7 slots (7±2), ruthlessly slicing a chaotic cloud of thoughts into a tightly multiplexed sequence, preventing the entire cognitive architecture from collapsing back into an undifferentiated, bleeding neural puddle.

Formal Statement: Separate representations are held simultaneously via high-frequency gamma oscillations (γ ≈ 45 Hz) nested within a slow theta wave (θ ≈ 6 Hz). Because the total duration of a theta cycle is constrained to ~167 ms and a gamma cycle requires ~22–25 ms for feedback inhibition recovery, the system imposes a hard biophysical ceiling of 4–7 processing slots — providing a strict, mechanistic explanation for Miller's (1956) 7±2 capacity limit that is grounded in neural oscillatory physics rather than arbitrary code constraint. Autonomous saccade selection is phase-locked to the deep trough of the theta wave, governed by the strict condition sin(2πf_θt) < −0.90, ensuring memory encoding is protected from incoming sensory fragmentation and that spotlight movements cluster into rhythmic burst-pause patterns mirroring the microsaccadic rhythms observed in human fixation-period recordings.

The cross-frequency coupling (CFC) clock implemented in the model is not merely a pacing mechanism for saccadic eye movements; it functions as an information-theoretic capacity constraint on temporal binding-by-synchrony within capacity-limited cortical systems — a constraint on how many items such a mechanism could support, not a claim to have resolved the binding problem itself, which remains debated in the broader literature. In human working memory, separate representations must be held simultaneously without bleeding into an undifferentiated neural soup. The brain handles this by nesting high-frequency gamma oscillations (γ ≈ 45 Hz) within the phases of a slow theta wave (θ ≈ 6 Hz), as established by Lisman & Jensen (2013). Each slow theta cycle represents a coherent ensemble processing window; the constituent gamma sub-cycles function as distinct temporal slots, each carrying one working memory item as a specific ensemble of neurons firing within a precise gamma window. Without this strict oscillatory clock, the entire cognitive architecture collapses into a turbulent, undifferentiated neural storm.

[ Theta Cycle (~6 Hz — one full period — 167 ms) ]
|--------------------------------------------------------------|
 Trough                                                       Peak
 [WM Slot 1] —— [WM Slot 2] —— [WM Slot 3] —— [WM Slot 4]
 (γ burst 1)      (γ burst 2)     (γ burst 3)     (γ burst 4)
      ↓                ↓               ↓               ↓
   HIGH-PRI       SECONDARY       TERTIARY         FRINGE

 [ Saccade Gate Opens: encoding optimal at trough-to-peak transition ]
 [ sin(2πf_θt) < -0.90 → cfcDot fires → izh_v modulated +8 mV ]

Hard capacity constraint: ceil(T_theta / T_gamma) = ceil(167ms / 25ms) = 6 slots
Miller (1956) "7±2" = exactly this ratio: not a cognitive axiom but a biophysical ceiling.
Chaotic multi-channel input storm → serialised multiplexed sequence → distinct cognitive objects.

Because the total duration of a theta cycle (~167 ms) and the minimum biophysical window per gamma cycle (~22–25 ms, required for feedback inhibition recovery) are both physically constrained, the architecture imposes a hard ceiling on the number of items that can be sequentially serialised within a single theta window. Approximately 4–7 gamma slots fit within one theta wave — providing a mechanistic, oscillatory explanation for Miller's (1956) classic 7±2 capacity limit that is grounded in neural biophysics rather than abstract psychological axiom. Human intracranial recordings during working memory retention tasks confirm that gamma amplitude is phase-locked to specific components of the underlying theta rhythm (Axmacher et al., 2010). The framework's capacity ceiling is therefore not a design choice: it is the inevitable consequence of two independently measured oscillatory time constants interacting within a finite temporal window.

The spatial repositioning of the attentional spotlight — autonomous saccade selection in Wandering Mode — is directly gated by the theta phase layer. Saccade selection is strictly restricted to the deep trough of the theta wave, firing only when:

sin(2π × f_θ × t) < -0.90       [theta-trough gate: ~15% of each cycle]

During trough window:
   cfcDot indicator turns gold
   θ-γ Locked badge activates
   izh_v[idx] += 8 × gammaBoost   [gamma burst lowers spike threshold in target quadrant]
   spotlight.wanderInt reset to 90 + rand(60) frames

Between troughs:
   spotlight.wanderInt extended to max(current, 180) frames
   saccades suppressed — natural burst-pause clustering of eye movements

This alignment reflects the biophysical finding that memory encoding and sensory sensitivity are optimal at the theta trough-to-peak transition — the phase at which NMDA receptor conductance windows are most permissive and local inhibition is at its nadir — whereas retrieval and consolidation dominate the descending phase. Restricting spotlight changes to the trough prevents incoming sensory inputs from fragmenting memory arrays that are actively being refreshed during other phases, grouping saccadic jumps into periodic windows and preventing continuous distractors from breaking down persistent working memory tracks.

This temporal organisation draws on the functional division of gamma rhythms established by Colgin (2016). Slow gamma (~25–50 Hz) coordinates top-down transmission from upstream prefrontal goal representations; fast gamma (~60–80 Hz) routes bottom-up sensory streams. By restricting spotlight movements to the theta trough, the architecture makes the model's 4-slot working memory ring directly interpretable as a theta-nested gamma multiplexing architecture: each slot corresponds to one gamma burst window within the active theta cycle, and high-priority items are encoded at the deepest trough (slot 1) — the phase most resistant to interference from the inhibitory peak.

The CFC strip beneath the Yerkes-Dodson bar shows a live scrolling theta waveform at 6 Hz. The waveform turns gold at each trough; the saccade-gate dot (cfcDot) pulses simultaneously. The θ-γ Locked badge enters its luminous purple state when a trough has been active within the last 90 frames, showing that the system is within a CFC-organised encoding window. The stats bar CFC readout switches from "-free" to "-lock" and turns amber. Between troughs the wander interval is visibly extended: spotlight saccades cluster rhythmically, appearing to "breathe" at theta frequency rather than moving continuously, matching the microsaccade periodicity (~150–200 ms inter-saccadic interval) observed in human fixation-period recordings.

The same critique pattern applies here as to Laws I and II, but the fix is even more direct, because the underlying relationship is already a bare mathematical ratio rather than a biophysical threshold in need of generalisation. The specific numbers — a 167 ms theta cycle, a 25 ms gamma cycle, a sin(2πfθt) < −0.90 logic gate — are one species' particular oscillatory parameters, not the law itself. Generalised across any oscillatory information-processing system:

This is now a dimensionless statement about wave mechanics: it makes no claim about theta or gamma specifically, and would apply equally to any two-tier oscillatory multiplexing scheme in any species or any artificial system, provided a slow carrier and a fast sub-cycle exist and must remain phase-distinguishable. Human cortex, under this reading, is simply the specific system where τ_slow ≈ 167 ms and τ_fast ≈ 25 ms, giving N_max ≈ 6–7 — Miller's (1956) and Cowan's (2001) capacity limits recast as a consequence of the ratio, not as independent psychological axioms this framework happens to also reproduce.

The critique that the working-memory slot count was previously "a static, hardcoded frontend capacity array" — new Array(4).fill(null), entirely disconnected from any oscillator frequency named elsewhere in the documentation — is accurate, and has been addressed directly. workingMemory.slots is now sized by getDynamicCapacity(thetaHz, gammaHz), which computes floor(thetaPeriod / gammaPeriod) from two named, adjustable frequency constants (THETA_HZ = 6, GAMMA_HZ = 40), clamped to the empirically-supported 4–7 range. The capacity the interface displays is now a live consequence of §III.4's ratio rather than a magic integer sitting beside it in the documentation.

What this update does not do, and should not be mistaken for: it does not simulate two coupled noisy oscillators and derive the ceiling from their phase decoherence, which would require modelling jitter and drift between independent theta and gamma generators explicitly. getDynamicCapacity() evaluates the ratio directly from two frequency constants — a live computation, but still a direct evaluation of §III.4's formula rather than an emergent demonstration of why the ceiling exists from first-principles wave interference. The gap this revision closes is "the ratio is computed, not hardcoded"; the gap that remains, honestly, is "the ratio is asserted from constants, not derived from simulated oscillator collision." A more rigorous future implementation would model f_θ and f_γ as two independent noisy oscillators and derive the capacity ceiling empirically from where their relative phase becomes unrecoverable, rather than reading it off two fixed frequency constants as done here.

Adopted score: 4.6 / 5.0. Revised up from 4.5, now that working-memory capacity is a live computation from named oscillator constants rather than a bare hardcoded integer (§III.5). This remains the highest of the three retained Laws, and every independent review consulted during this revision agreed on that ranking even where they disagreed on the exact number. The reason is structural: §III.4's ratio is the one law in the framework that was already, in its cleanest form, a relationship between two periods rather than a threshold on an engineered variable — the reformulation in §III.4 mostly makes explicit what was implicit, rather than performing the harder work of generalising away a genuinely arbitrary constant, as Laws I and II required.

§III.4's ratio N_max ≤ τ_slow/τ_fast is the framework's most directly testable claim, since it requires no reference to this codebase's specific implementation at all:

• Prediction 1 — Cross-species scaling: Species or brain states with a measurably different theta-gamma frequency ratio should show a working-memory capacity that scales with that ratio, not a fixed universal "7±2" number. Intracranial recordings in species with faster or slower theta rhythms than human cortex offer a direct test; a fixed capacity limit independent of the measured ratio would falsify §III.4 as stated.
• Prediction 2 — Pharmacological manipulation of gamma period should shift capacity: An intervention that lengthens the minimum gamma cycle duration τ_fast (e.g. altering GABAergic feedback-inhibition kinetics, which set the gamma period) should proportionally lower measured working-memory capacity, independent of any change to theta frequency. A null result here would be difficult to reconcile with §III.4.
• Prediction 3 — Phase-amplitude coupling strength predicts individual capacity: Individuals or task conditions with stronger, more precisely locked theta-gamma phase-amplitude coupling (Axmacher et al., 2010) should show working-memory capacities closer to the theoretical ceiling τ_slow/τ_fast, while weaker coupling should predict capacities below that ceiling — never above it. A single confirmed case of capacity reliably exceeding the ratio ceiling would falsify the law outright.

These are precisely the kind of predictions the §III.5 "hardcoded gate" simplification cannot itself generate — a fixed logic gate asserts the ceiling but does not predict how it would shift under the manipulations above. A future implementation that models τ_slow and τ_fast as adjustable, physically-grounded parameters rather than fixed constants would let this simulation actually reproduce Predictions 1–2 internally, closing part of the §III.5 gap.

Two further category concerns, raised independently and judged valid on reflection, keep this component below the threshold for Law status even as an architectural choice. First, the Difference-of-Gaussians (DoG) Mexican Hat filter used to sharpen spatial focus models the centre-surround geometry of retinal ganglion cells — a low-level visual mechanism — and applies it to suppress literal pixel coordinates on a 2D screen. But top-down executive selection, the phenomenon this framework is actually trying to model, suppresses features or semantic categories, not physical locations in a two-dimensional plane. Borrowing early-visual-system machinery to stand in for object-based cognitive attention is a simplification worth naming rather than obscuring. Second, the NMDA coincidence-gated LTP cascade potentiates cells based on spatial adjacency on the canvas grid (an 8-pixel radius), whereas biological long-term potentiation requires synchronised pre- and post-synaptic activity at a specific, functionally-wired synapse — there is no claim here that grid-adjacent pixels correspond to axonal-dendritic connectivity. Potentiating a neighbouring cell because it happens to sit next to the target on-screen, rather than because it is functionally wired to it, is a spatial heuristic standing in for synaptic plasticity, not a literal implementation of it.

None of this makes the mechanism useless — it makes it exactly what a Principle is supposed to be: a chosen, well-motivated architectural strategy for making cognitive dynamics spatially and visually legible, rather than an inescapable constraint every possible implementation would have to obey. Within that honestly-scoped role, the mechanics remain worth documenting briefly.

Sustained localised heat exceeding a voltage-dependent activation floor triggers a three-stage cascade: DoG spatial sharpening (center-surround contrast on the coordinate grid), an NMDA-gated coincidence window (~33 ms, modelling the voltage-dependent Mg²⁺ unblock of Jahr & Stevens, 1990), and CaMKII bistable autophosphorylation, converting transient heat into durable structural coordinates (heatmap.ltp) that establish 2.5× facilitated pathways for future traversal. The structural conversion follows a quadratic acceleration equation:

ΔLTP_i = γ_auto × LTP_i² + α_LTP × (Heat_i / MaxHeat)
γ_auto = 0.0018  |  α_LTP = 0.0012/frame

Once a trace crosses the bistability threshold (~76/255), it becomes self-sustaining even after the driving heat decays — a computational echo of the CaM-trapping mechanism described by Bartol, Sejnowski, Rangamani & Kennedy (2025, Frontiers in Synaptic Neuroscience), in which steric hindrance slows dephosphorylation by protein phosphatase 1 (PP1) and prolongs phospho-CaMKII lifetime. High-LTP cells also dampen their own decay rate (krec = KRECBASE × (1 − LTPi × ωscale)) and deposit heat at 2.5× the baseline rate on future traversal — the structural substrate underlying Law I's Autonomous Recapture and the felt effortlessness of practiced attention.

By deploying the simulation on a continuous 2D coordinate space rather than discrete boolean nodes, the framework enables true gradient physics: Law I's Lyapunov potential field can flow continuously rather than snapping between nearest-neighbour states, Law II's DoG surround has a real geometric radius to collapse rather than a count of disconnected nodes to disable, and Law III's saccade targeting has an actual spatial gradient to follow rather than an arbitrary graph edge to traverse. This is Principle I's honest contribution: not a law the other components are forced to obey, but the shared representational substrate that makes their dynamics visible, differentiable, and mutually coupled on a single canvas. Score 3.4/5.0 reflects that contribution weighed against the category concerns above — a well-built, useful architectural choice, correctly no longer presented as an inescapable law of cognition.

Why it is a Principle, not a Law: In an unconstrained Hebbian system, positive feedback would let the spotlight permanently lock onto the first salient stimulus it encounters — the rigid, repetitive capture seen clinically in OCD and addiction. Principle II is the chosen countermeasure: a Bienenstock-Cooper-Munro (1982) sliding threshold θ_BCM that tracks a running average of local activity, so that chronically overactive cells become progressively harder to potentiate and eventually drift into depression (LTD) instead — a Plasticity Flip that actively dismantles the over-saturated basin, drains its LTP trace, and liberates the spotlight to explore fresh canvas. Unlike Laws I, II & III, this is a specific engineering strategy the system employs to remain flexible; a different anti-saturation rule (e.g. simple weight normalisation, or a fixed decay ceiling) could in principle replace it without contradicting anything upstream.

w(x) = f(h(x), _BCM(x))  x_pre         f(h,) = h(h-)   [above  LTP; below  LTD]

_BCM update (every 30 frames):
  if mean(h(quadrant)) > 0.72maxHeat for > 300 frames:
     _BCM  1.08         [threshold climbs]
    ltp     -1.2          [LTP vein drains]
    timer  0                [quadrant freed for fresh accumulation]

The ~5-second (300-frame) exhaustion window is loosely modelled on the induction kinetics of mGluR-dependent LTD (Huber et al., 2000), and the overall strategy mirrors Tononi & Cirelli's (2014) Synaptic Homeostasis Hypothesis: potentiation accumulates during active engagement, and periodic downscaling trims runaway saturation to preserve signal clarity — here compressed to the timescale of a single attentional episode rather than a sleep-wake cycle.

The classic Yerkes-Dodson inverted-U between arousal and performance is treated here as a structural consequence of the biphasic kinetics of catecholamine receptors in dorsolateral prefrontal cortex. At low LC/NE and VTA/DA tone (hypo-arousal), receptor occupancy is too sparse to sustain recurrent pyramidal firing against background noise — a diffuse, low-gain field. At moderate, task-locked tone (optimal arousal), high-affinity α2A and D1 receptors close HCN channels and sharpen NMDA-mediated recurrent firing, carving a crisp focus zone with a deep lateral-inhibition well. At high tone (hyper-arousal), lower-affinity α1/β1 receptors and over-saturated D1 kinetics invert the signalling cascade, flooding cAMP, forcing HCN channels open, and uncoupling the prefrontal network from its own recurrent loops (Arnsten, 2011).

Low (Hypo)              Optimal (Eustress)         High (Hyper)
Diffuse, low gain        Crisp focus, deep well      Uncoupled, noisy
shuntFactor = 1.0        shuntFactor = 1.0          shuntFactor  0.05

Mapping into the model: The global neuromodulatory gain γ scales shuntFactor, the amplitude term on the inhibitory ring of the DoG Mexican Hat filter. As γ exceeds ≈0.80, the inhibitory surround visibly shrinks and heat diffuses radially outward — the canvas rendering of Arnsten's (2011) prefrontal disconnection under hyper-arousal. The Yerkes-Dodson status bar tracks the system's live position on this curve.

Formal statement: Every coordinate cell's effective output is gated by two coupled Tsodyks-Markram parameters — fractionally available transmitter stores (R) and calcium-utilisation efficiency (u). Continuous activation depletes R, taxing high-frequency processing and forcing even the most stable Law I attractor to eventually starve and release the spotlight into transitive flight:

Output_i = Heat_i  (R_i  u_i)
dR_i/dt = (1-R_i)/t_rec - u_i R_i d(t)     dU_i/dt = (U-u_i)/t_facil + U(1-u_i)d(t)

High-freq input  R0  Output collapses  Lyapunov well flattens  transitive flight

This is the fast-timescale mechanism that dynamically flattens Law I's Lyapunov potential field (§I.3): fresh resources (R≈1.0) sustain the deep basin; depleted resources (R→0) collapse it, even while raw accumulated heat remains high — the difference between a vortex that is structurally "alive" but dynamically "starved." James's (Ch. IX) distinction between substantive resting-places and transitive flights of passage maps directly onto full-R and depleted-R states respectively, and this rhythm couples to Law II: if stamina is simultaneously low, the transitive flight cannot resolve cleanly, and the spotlight wanders turbulently rather than executing a clean saccade to fresh ground.