Attentional Framework v68

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 (Amari, 1977; Wilson & Cowan, 1972).

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 ω-stability basin (vorticity attractor) — a region where the vorticity field ω reaches a local maximum and the gradient of ω reverses direction. 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 vorticity-based attractor 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 two 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 (Amari, 1977). The resulting vortex warps the entire phase space of the system, trapping passing thought-stream particles in a stable orbit. This geometric trapping is captured by the localised vorticity 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 (Sanchez-Rodriguez et al., 2020). 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." (Emergent mesoscale dynamics — Anderson, 1972: reductionist breakdown at hierarchical transitions.)

3. Andersonian Emergence: The "mechanic" is a single particle updating x,y coordinates via vector calculations. 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. Downgrading the vortex to a particle‑routing routine is like calling a hurricane "a mechanical routine for moving air molecules."

The Three Formal Defenses:

• Phase Transition: When heat crosses vortexT (computed live as a statistical inflection point of the field's own energy landscape, §I.11), 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 0°C 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 (Anderson, 1972). 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)α_depth - STP_R(x)α_well

Heavy bowling ball      deep bowl      stable orbit      sustained thought
Lightening ball (R→0)   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 (computed dynamically as the statistical inflection point of the field's own energy landscape, §I.11), 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 ω‑stability criterion:

  VORTEX STABILITY CRITERION (ω‑stability basin)

  Let ω(x) = angular velocity in the phase plane (v, dv/dt).
  A vortex exists at location x iff:

    1. ω(x) > ω_crit        [vorticity exceeds threshold]
    2. ∇ω(x) = 0            [local extremum in vorticity field]
    3. ∇²ω(x) < 0           [local maximum — stable attractor core]
    4. decay_factor(x) = 0.3  [70% decay suppression applied]

  Where ω_crit is a field‑adaptive threshold:
    ω_crit = mean(ω) + 0.65·(max(ω) − mean(ω))

  This criterion identifies the "bowl" in the trampoline:
    — The bowl is the region where ω is maximal.
    — The bowl's depth is LTP + STP_R resource availability.
    — The bowl's shallowing (R→0) releases the spotlight.

  Note: This is an ω‑stability criterion based on vorticity,
  not a Lyapunov stability proof. No explicit V(x) function is
  constructed; no dV/dt < 0 is proven. The term "Lyapunov" is
  used descriptively to align with the energy‑landscape metaphor.

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

  [ PHASE BOUNDARY ]    ω = ω_crit       DISCONTINUOUS BIFURCATION POINT
                                              instantaneous state change
  [ POST-TRANSITION ]   ω > ω_crit       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 (Amari, 1977). 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). 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 ω‑stability well with each cycle. The biological parallel is the strengthening of attentional priority maps over repeated visits (Zelinsky & Bisley, 2015): 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): 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, the priority map (§v68) biases the spotlight toward high-LTP / high-salience regions, modelling the ~40 ms involuntary attentional capture documented by Salinas et al. (2019) and Poth & Lee (2021). 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 vorticity-based scalar potential field. 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)α_depth - STP_R(x)α_well   [potential well depth, derived from ω stability]
dV/dt = -V · v_particle                     [particle gradient descent into basin]

State transitions of the ω‑stability landscape:
--------------------------------------------------------------------------
Fresh R≈1.0 + growing LTP    deep bowl    stable fixed-point attractor
                                             thought held; LTP engraving deepens
--------------------------------------------------------------------------
R→0 (STP depletion)          V flattens    V→0
                                             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 ω‑stability sink (not a Lyapunov sink).
The bowling ball's weight IS V(x), which is derived from the ω field.
Dissolution is not failure — it is graceful, resource-dictated release.

Note on terminology: "ω‑stability basin" replaces "Lyapunov attractor basin"
because the simulation computes vorticity (ω) and identifies peak‑ω locations
as attractor cores. This is a valid topological feature but not a Lyapunov
stability proof. The term "Lyapunov" is used descriptively, not formally.

The ω‑stability 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 (Amari, 1977) and the neural field theory (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) 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) 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 (Shenhav et al., 2013).
• 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.

Note on novelty: The flow mapping (Mode 2→3 transition) has been proposed before in different language — for example, Kuhn et al. (2023) describe flow as "effortless attention" emerging when control demands match capacity. The simulation's contribution is the geometric framing (attractor transition expressed as a spatial phase change on the canvas), not the conceptual novelty of "effort becomes autonomous."

Mapping to Dual-Process Theory (Kahneman, 2011): This framework's distinction between Mode 2 and Mode 3 provides a biophysical, spatialized substrate for what cognitive psychology recognises as Dual-Process Theory. Mode 2 corresponds to System 2 — slow, effortful, capacity-limited, and governed by Law II's EVC economics. Mode 3 corresponds to System 1 — fast, automatic, and effortless. Where Kahneman described these as behavioural taxonomies, this framework explains the thermodynamic transition between them: System 2 is the active expenditure of cognitive stamina to maintain a vortex; System 1 is the state achieved when the structural substrate (LTP veins) has absorbed the routing function, allowing the executive system to disengage (cf. Dietrich's Transient Hypofrontality). The channel axis (Extension III) further operationalises these modes: the Quiet channel (λ→0) forces System 2 engagement; the Loud channel (λ→1) leans on System 1 automatic capture, making the dual-process distinction spatially and computationally concrete.

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 18–55). 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.

Clinical caveat: Rumination involves content-specific negative attractors (worry themes), not just "any autonomous recapture." The simulation's vortex is content-free — it captures spatial attention, not semantic content. Calling this a mechanistic account of rumination overreaches; the model generates hypotheses about attentional capture dynamics, not a validated clinical model of rumination itself.

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 (Tsodyks & Markram, 1997).
• 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 (Bhalla & Iyengar, 1999). 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 capacity-limited 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 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 the threshold check at runtime 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)α_depth - STP_R(x)α_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 candidate for genuine "Law" status — ahead of the now-downgraded Law III and the now-removed Field Continuity component — while still identifying the same residual weakness (the arbitrary threshold, the pixel-vs-synapse category question) as every other review.

Revised, post-review assessment:

Adopted score: 4.2 / 5.0. This is higher than the independent critique and higher than the previous 4.0, reflecting the terminological correction from "Lyapunov" to "ω‑stability." The CANN grounding and the Mode 3 autonomous recapture remain valuable contributions, and the framework's validity is now more accurately bounded by the corrected terminology.

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 (Amari, 1977) — 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 ω‑stability 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 ω‑stability 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 ω‑stability 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.

Two limitations sit alongside the §I.11 discretisation gap and deserve to be named as plainly as that one is, rather than left implicit in the boundaries of §I.8.

• The Canvas Problem: A 2D spatial attractor is a useful, legible representation, but real CANN dynamics occur in high-dimensional neural state space, not on a flat plane. Flattening that state space to two coordinates is an interpretive choice made for visibility and interactivity, not a claim that cortical population dynamics are themselves two-dimensional. Nothing in §§I.1–I.14 depends on the flattening being biologically literal; it depends only on the flattened system exhibiting genuine bifurcation and bump-attractor behaviour, which §I.11's discretisation defence and §I.13's institutional reading both address. The flattening itself, however, remains a limitation of representation, not of mechanism. Defense via Topological Manifold Isomorphism: The 2D canvas can be rigorously framed as a low-dimensional embedding of a high-dimensional attractor landscape, analogous to how grid cells in entorhinal cortex represent abstract state spaces on a 2D manifold (Constantinescu et al., 2016, Science 352:1464). Euclidean distance on the canvas is mathematically isomorphic to representational similarity in the high-dimensional space — a standard technique in topological data analysis for neuroscience. The canvas is therefore not a flattening of cortex onto a screen, but a 2D manifold rendering of the topology of a cognitive state space, a rigorously defended technique in computational neuroscience.
• Domain Limitation: Law I governs attentional routing — where the spotlight goes and how long it stays — not semantic content — what is being attended to. This is stated honestly in §I.8's operational boundaries, and it is worth restating plainly here: the law has nothing to say about the meaning, identity, or informational content of what a vortex traps. This scopes the law more narrowly than a unified cognitive theory would need to be, and that narrowness is a deliberate, disclosed limitation rather than an oversight.

Canvas Problem deduction (revised): Flattening a high-dimensional neural manifold to 2D isn't just an "interpretive embedding" — it's a dimensionality reduction that discards information about inter-feature relationships. Two attentional states that are far apart in neural space could map to adjacent pixels on the canvas, creating false proximity. This is a known problem in manifold projection (Tenenbaum et al., 2000) and warrants a larger validity deduction than the original analysis gave it. The defence via topological manifold isomorphism partially addresses this, but the reduction remains a limitation.

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 (Lavie et al., 2004). 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 (Shenhav et al., 2013). 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 (Lavie et al., 2004) made spatially visible and mechanistically inevitable. The relationship between subjective fatigue and structural collapse is causally coupled: EVC withdrawal precipitates inhibition structural failure.

Attentional capacity is governed by a dynamic cost-benefit algorithm computing the Expected Value of Control (EVC) on every frame (Shenhav et al., 2013). 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 (Raichle et al., 2001). Neither the depletion cycle nor the recovery cycle is optional: both are conservation-law consequences of finite neural metabolic capacity.

EVC Decision Structure:

  EVC DECISION STRUCTURE (Shenhav, Botvinick & Cohen 2013)

  At each decision point, the system computes:

    V_maintain = expected value of continuing control engagement
    V_release  = expected value of releasing control
    C_control  = metabolic cost of maintaining control

  Withdraw control when:   V_release > V_maintain - C_control

  In the current implementation, this is approximated as:

    drain = base_drain / (1 + 0.5·taskValue)

  where taskValue rises during successful vortex holding or reward capture.
  This is a first‑order proxy: it captures the intuition that reward reduces
  perceived cost, but it does not compute V_maintain vs V_release explicitly.

  Limitation: This approximation sacrifices explicit comparison of alternatives.
  A full EVC implementation would require a reward‑prediction engine that is
  computationally infeasible at 60 FPS. The proxy approach is therefore a
  tractable engineering abstraction for a mesoscale spatial model.

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

[Task-Focused Mode]                    [Wandering / DMN Mode]
cognitiveStamina -= drain/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:

• Kahneman (1973) — Capacity Model of Mental Effort: Before Lavie's Load Theory or Shenhav's EVC, Kahneman established that attention is not merely a structural filter but a limited, allocatable 'mental effort' governed by physiological arousal and task demands. His psychophysiological work (Kahneman & Beatty, 1966) operationalised William James's "feeling of strain" into a measurable signature — pupil dilation — providing the first empirical link between subjective effort and biological capacity. While contemporary EVC frameworks (Shenhav et al., 2013) have replaced Kahneman's single-capacity pool with an opportunity-cost ledger, Kahneman's foundational linkage between arousal, subjective 'strain' (James), and capacity allocation remains the phenomenological bedrock of Law II. This framework builds directly on that legacy by providing a mechanistic account of the geometry of capacity collapse.
• 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 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. In v68, the drain is further modulated by the dynamic taskValue accumulator: drain = base_drain / (1 + 0.5·taskValue), making the EVC calculation responsive to ongoing success and reward.
• 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 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 (Arnsten, 2011). 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); (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). 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 (v68: dynamic taskValue) ----------
const successProxy      = Math.min(1, vortexPullStrength);
const opportunityCost   = Math.min(1, salientEvents.length / 6);
const evcDrainModulator = Math.max(0.35, 1 - 0.5 * successProxy + 0.4 * opportunityCost);
const taskValueMod      = 1 / (1 + 0.5 * taskValue);  // v68: dynamic EVC
const drain = STAMINA_DRAIN * evcDrainModulator * taskValueMod * (1 - 0.35 * (MIA ? miaStrength : 0));

if (mode === "TaskFocused" || mouse.active) {
  cognitiveStamina -= drain;
} else {
  cognitiveStamina += STAMINA_REGEN * (1 + 0.5 * (MIA ? miaStrength : 0));
}

// -- Conservation law enforcement: continuous collapse (§II.11) ----------
dogSurroundScale = 0.5 + 0.5 × stamFactor;   // continuous 0.5–1.0 ramp
gateContinuity  = 0.5 + 0.5 × min(1, stamFactor/0.4);
// The UI hysteresis label (gatingFailureActive) lags the actual geometry

// -- Recovery loop --------------------------------------------------------
// Models Buckner et al. (2008) DMN consolidation:
// 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.

For nearly a century, James's observation remained a philosophical one until Kahneman (1973) operationalised "strain" as measurable mental effort, demonstrating via pupillometry that the subjective feeling of strain correlates directly with physiological arousal and capacity allocation. Law II provides the final mechanistic account: the "strain" James felt and Kahneman measured is the running integral of EVC expenditure, expressed as the continuous narrowing of the geometric inhibition radius. The framework thus closes a loop from James's phenomenology through Kahneman's psychophysiology to a dynamical systems model.

Because Law II governs an inescapable conservation constraint, systematic variation of its parameters — in combination with Law I's attractor topology, Principle III's temporal binding capacity, and Extension III's channel axis — produces distinct dynamical regimes on the canvas. This section documents four such regimes and maps each to a neurobiological profile drawn from the clinical literature. The epistemic status of every mapping below is 🟡 Computational Analogy: each profile demonstrates that the framework's parameter space is rich enough to reproduce the qualitative signature of a recognised clinical or optimal attentional state, and each generates at least one falsifiable prediction that could, in principle, be tested against behavioural or neuroimaging data. None of the mappings below constitutes a validated clinical model, a diagnostic instrument, or a claim of mechanistic identity between simulation parameters and biological substrates. The distinction between "the model produces a dynamical regime that resembles the clinical profile" and "the model explains the clinical profile" is maintained throughout.

Before presenting individual profiles, the following table states explicitly which simulation parameters are varied, what biological mechanism each parameter is intended to approximate, and where the approximation breaks down. This table is the honesty contract for the entire section: any claim below that exceeds the mapping stated here is overclaiming.

Simulation signature: Low STAMINA_REGEN + reduced baseline dogSurroundScale + elevated salience capture probability + channel axis biased toward Loud (λ → 1).

Canvas behaviour: Law I vortices form but cannot stabilise before salience events disrupt them. The canvas shows broad, shallow heat distributions with frequent involuntary spotlight jumps. The DoG inhibitory surround is chronically narrowed even at full stamina, producing a permanent low-grade gating failure. Working memory slots (Principle III) fill rapidly but are overwritten before consolidation, producing the characteristic "what was I doing?" fragmentation. In Extension III terms, the system is locked in a Loud-channel regime (high switching cost, low control cost) even in environments that demand Quiet-channel attention (reading, sustained problem-solving).

Neurobiological correlate: The dominant neurobiological account of ADHD identifies dopaminergic and noradrenergic modulation deficits in prefrontal-striatal circuits, reducing the gain of top-down lateral inhibition and impairing the brain's ability to suppress irrelevant stimuli (Arnsten, 2011; Barkley, 1997). The reduced baseline DoG surround approximates this gain deficit. The elevated salience capture probability approximates the heightened sensitivity to immediate environmental stimuli documented in ADHD (Liddle et al., 2011). The channel-axis bias toward Loud approximates the clinical observation that individuals with ADHD show disproportionate difficulty sustaining attention in low-stimulation environments while performing relatively well under high-stimulation conditions (Zentall & Zentall, 1983).

Subtype differentiation: The framework's Real-World Applications section (§Applications, Issue 2) predicts three mechanistically distinct ADHD subtypes, each corresponding to a different Law failing in isolation:

• Subtype A — EVC Deficit (Law II primary): Stamina is structurally adequate but the EVC calculation chronically returns "not worth maintaining control" for low-reward tasks. Predicted response: reward-based interventions (gamification, contingency management) rather than stimulants. Predicted non-response: WM training (Cogmed, n-back), because WM capacity is structurally intact.
• Subtype B — Attractor Instability (Law I primary): ω-stability basins are too shallow to sustain Mode 2 voluntary attention. The spotlight drifts to any salient stimulus even when stamina is adequate. Predicted response: stimulants, which directly address the ω-depth deficit by increasing dopaminergic gain.
• Subtype C — Temporal Fragmentation (Principle III primary): θ-γ coupling is disrupted, reducing WM capacity to 2–3 slots. The individual cannot hold multi-step task context even when motivated and not distracted. Predicted response: stimulants targeting γ power; predicted non-response: interventions targeting motivation or attractor stability.

Falsifiable prediction: The three-subtype model predicts that stimulant non-responders are disproportionately Subtype A (EVC deficit). A 3-test battery (SART for Law II, anti-saccade for Law I, digit-span backward for Principle III) should dissociate the subtypes with above-chance accuracy. If stimulant non-responders are evenly distributed across all three subtypes, the subtyping model is falsified.

Mapping limitations: Real ADHD involves dopamine transporter genetics (DAT1, DRD4 polymorphisms), prefrontal hypometabolism visible on PET, developmental trajectories spanning years, and comorbidity profiles (oppositional defiant disorder, learning disabilities, anxiety) that no scalar parameter can capture. The simulation's ADHD regime is a dynamical analogy, not a pathophysiological model. It cannot predict medication dosage, developmental course, or individual treatment response.

Simulation signature: Elevated salience spawn rate + elevated salience capture probability + high STAMINA_DRAIN + reduced vortexT (lower threshold for threat-associated vortex formation) + elevated θ-gate salience override coefficient (Extension III: λ · S_salience · k, with k above calibrated value).

Canvas behaviour: 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. In Extension III terms, the θ-gate salience override fires even in Quiet-channel mode: threat cues break the theta rhythm during reading or sustained focus, producing the characteristic "I can't stop scanning for danger" phenomenology. The result is a compound failure: Law II stamina erodes, Law I vortices form around threat coordinates, and Principle III WM slots fill with threat content, leaving no capacity for alternative thoughts (the "WM tunneling" described in §Applications, Issue 4).

Neurobiological correlate: The hypervigilance model of anxiety identifies a failure of inhibitory control over threat-relevant attentional capture, mediated by amygdala hyperreactivity and insufficient prefrontal top-down regulation (Cisler & Koster, 2010; Eysenck et al., 2007). The elevated salience capture probability approximates amygdala-driven orienting. The chronically elevated drain approximates the continuous metabolic cost of maintaining prefrontal suppression over an overactive threat-detection system. The reduced vortex threshold approximates the lowered activation threshold for threat-related attentional schemas documented in anxiety disorders (Williams et al., 1996). The salience override coefficient above calibrated value approximates the clinical observation that anxiety disorders involve excessive reorienting to peripheral stimuli even during goal-directed tasks (Bar-Haim et al., 2007).

Falsifiable prediction: The model predicts that anxiety-related attentional impairment should correlate with switching cost accumulation (Extension III) rather than with simple stamina depletion. An anxious individual in a low-salience environment (few threat cues) should show near-normal stamina curves, while the same individual in a high-salience environment (many potential threat cues) should show accelerated depletion proportional to cue density, not time-on-task. If depletion rate is identical regardless of cue density, the salience-override model is falsified.

Mapping limitations: Anxiety's attentional capture is content-specific (threat-related), not merely spatial — the simulation has no semantic content layer and cannot distinguish a threat cue from a neutral cue at the representational level. Real anxiety involves amygdala-prefrontal connectivity patterns, autonomic arousal (HPA axis, sympathetic activation), cognitive appraisal processes, and learning history (conditioning, trauma) that are entirely absent from the model. The simulation captures the dynamical structure of hypervigilance (elevated capture, exhausted inhibition, WM tunneling) but not its content or its developmental origin.

Simulation signature: High LTP trace depth at specific coordinates + depleted STP R·u at those coordinates + Law II gating failure cascade active but unable to overcome LTP routing force + channel axis oscillating between Quiet (attempted abstinence) and Loud (cue-triggered relapse).

Canvas behaviour: 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. The spotlight is pulled back to the addiction-associated coordinates by Mode 3 Autonomous Recapture (§I.6) even when the system's stamina is exhausted and executive control has collapsed. In Extension III terms, the system oscillates between a Quiet channel (attempted self-regulation, high control cost) and a Loud channel (cue-triggered capture, high switching cost), with each oscillation draining stamina further until the Quiet channel can no longer be sustained.

Neurobiological correlate: The incentive sensitisation model (Robinson & Berridge, 1993, 2008) distinguishes between "wanting" (incentive salience, mediated by mesolimbic dopamine) and "liking" (hedonic pleasure, mediated by opioid and endocannabinoid systems). In addiction, "wanting" becomes sensitised — cue-triggered dopamine release in the nucleus accumbens — while explicit knowledge that the behaviour is harmful remains intact in prefrontal circuits. This produces the clinically observed dissociation: the individual knows they should stop (prefrontal representation intact) but cannot stop attending to addiction-associated cues (incentive salience override). The simulation's LTP vein persisting despite STP depletion approximates this structural override: the routing trace is too deeply engraved for dynamic resource depletion to erase it. The clinical parallel is the persistence of cue-reactivity in addiction even after prolonged abstinence, mediated by structural synaptic changes in the striatum (Hyman et al., 2006).

Falsifiable prediction: The model predicts that cue-exposure therapy should be less effective when LTP trace depth is high and STP resources are depleted — i.e., when the structural routing trace is deeply consolidated and the individual has no remaining dynamic plasticity to form competing traces. If cue-exposure therapy is equally effective regardless of consolidation depth, the structural-override model is falsified.

Mapping limitations: Real addiction involves craving circuits (ventral tegmental area → nucleus accumbens → prefrontal cortex), withdrawal dynamics (opponent-process adaptations), tolerance curves (receptor downregulation), genetic vulnerability (polygenic risk scores), social context, and comorbid psychiatric conditions. The simulation captures the routing structure of cue-triggered attentional capture but not the pharmacology, the subjective experience of craving, or the social dynamics of relapse. Obsessive-compulsive disorder involves cortico-striato-thalamo-cortical loop dysfunction with distinct neurochemistry (serotonergic, glutamatergic) that the model does not approximate.

Simulation signature: Adaptive Law I vortex (Mode 3 Autonomous Recapture active) + high taskValue accumulator + drain rate approaching zero + channel axis in Quiet regime (λ → 0) with strict θ-gating + WM slots stable and near-full.

Canvas behaviour: Law I's Autonomous Recapture (Mode 3, §I.6) 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. The taskValue accumulator rises with each successful vortex hold, further reducing drain via the equation drain = base_drain / (1 + k · taskValue). The canvas shows a single deep, stable vortex with strong LTP consolidation, minimal salience disruption (Quiet channel: sparse events, small capture radius), and stable WM occupancy. This is the only profile in which Law II's conservation constraint is effectively suspended — not violated, but rendered negligible by the structural absorption of routing cost.

Neurobiological correlate: Dietrich's (2004) Transient Hypofrontality hypothesis proposes that flow states emerge when reduced dorsolateral prefrontal activity allows implicit, automatic processing to dominate. The prefrontal cortex, normally responsible for effortful executive control, is transiently downregulated because the task's structural demands have been absorbed by procedural and sensorimotor systems. This is operationalised in the simulation as Mode 3 recapture: the structural landscape (LTP veins, attractor basins) absorbs the routing function that would otherwise require prefrontal EVC expenditure. Csikszentmihalyi's (1990) phenomenological description of flow — loss of self-consciousness, distortion of temporal experience, sense of effortless action — is consistent with the simulation's prediction that executive drain approaches zero while attentional quality remains high.

Falsifiable prediction: The model predicts that flow states should be associated with reduced prefrontal metabolic activity (measurable via fMRI BOLD signal in dlPFC) without corresponding reduction in task performance. If prefrontal activity remains elevated during self-reported flow, the transient hypofrontality operationalisation is falsified. Additionally, the model predicts that flow should be disrupted by salience events that exceed the Quiet-channel capture threshold — i.e., by interruptions that force a channel switch from Quiet to Loud. If flow is equally robust regardless of interruption frequency, the channel-coupling prediction is falsified.

Mapping limitations: Flow is a subjective phenomenological state, not a clinical condition. The simulation captures the cost structure of flow (near-zero executive drain, high attentional quality) but not its subjective quality (altered time perception, loss of self-referential processing, intrinsic reward). The Transient Hypofrontality hypothesis itself is contested: alternative accounts propose that flow involves increased prefrontal activity in specific subregions rather than global downregulation (Harris et al., 2015). The simulation adopts Dietrich's account as its operationalisation but acknowledges the debate.

The clinical profiles above are the framework's most speculative application. The following limitations are stated explicitly and should be read as binding constraints on every claim in this section:

1. No semantic content layer. The simulation models spatial attentional routing — where the spotlight goes and how long it stays. It does not model what the spotlight attends to. Clinical conditions are defined by their content: anxiety is about threat, addiction is about substance or behaviour cues, depression is about loss and self-referential negative thought. The simulation cannot distinguish a threat vortex from a reward vortex, a craving cue from a neutral stimulus, or a ruminative thought from a problem-solving thought. Every clinical mapping in this section is therefore a mapping of dynamical structure, not of clinical content.
2. No developmental trajectory. ADHD, anxiety, and addiction develop over years and involve gene-environment interactions, epigenetic modification, critical-period effects, and cumulative learning history. The simulation is a snapshot dynamical system with no developmental dimension. It cannot model how a profile emerges, how it changes with age, or how early intervention alters its trajectory.
3. No pharmacology. The simulation's parameters are abstract rate constants. They do not correspond to specific receptor subtypes, neurotransmitter concentrations, or drug pharmacokinetics. A statement like "reduced baseline DoG surround" is not a statement about D2 receptor density, methylphenidate dosage, or SSRI efficacy. Clinical pharmacology operates at a level of specificity the simulation cannot approximate.
4. No individual variation. Each profile is a single parameter configuration. Real clinical populations exhibit enormous heterogeneity within diagnostic categories. The simulation cannot model the difference between two individuals with the same DSM diagnosis, the effect of comorbidity, or the influence of individual differences in baseline cognitive capacity, personality, or environmental context.
5. No validation against patient data. None of the parameter configurations in this section has been fitted to behavioural, neuroimaging, or clinical outcome data. The profiles are generated from the model's internal logic and compared qualitatively to the clinical literature. The Roadmap to Validation (§Applications) describes the multi-step pipeline required to move from qualitative analogy to quantitative prediction; none of those steps has been completed.
6. Contested theoretical foundations. Several of the clinical models cited above are themselves contested. The dopamine-deficit account of ADHD is challenged by alternative models (default mode network interference, delay aversion). The incentive sensitisation model of addiction is one of several competing frameworks. The Transient Hypofrontality account of flow is disputed by prefrontal activation studies. The simulation's adoption of any particular clinical model does not constitute an endorsement of that model's validity; it constitutes an operationalisation of that model's predictions within the simulation's parameter space.

Summary of epistemic status: The clinical profiles in this section are hypothesis-generating computational analogies. They demonstrate that the framework's parameter space is dynamically rich enough to produce qualitatively distinct regimes resembling recognised clinical and optimal attentional states. They generate specific, falsifiable predictions (§II.6b–II.6e). They do not constitute validated clinical models, diagnostic tools, treatment guides, or claims of mechanistic identity between simulation parameters and biological substrates. The framework's value at this stage is as a structured hypothesis generator: it makes explicit what a dynamical account of attentional dysfunction would need to predict, and thereby identifies the specific empirical tests that would confirm or refute each prediction. The transition from analogy to validated model requires the multi-step validation pipeline described in the Roadmap to Validation, and that pipeline has not yet begun.

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.

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 (Magistretti & Allaman, 2015; Aston-Jones & Cohen, 2005). 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. Until v68, the actual runtime behaviour was 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, not an EVC computation.

Defense of the First-Order Proxy as a Mesoscale Abstraction: Computing full biophysical EVC — integrating actual reward magnitude, success probability, and metabolic state — is computationally intractable at 60 FPS and conceptually mismatched for a mesoscale spatial model. The proxy approach (vortexPullStrength as successProxy, salient-event density as opportunityCost, and now taskValue as a reward-weighted accumulator) successfully captures the gradient and directionality of the EVC landscape: it correctly models the relative cost-benefit shifts between a stable vortex (high taskValue → near-zero drain) and a chaotic field (low taskValue → high drain), even if the absolute magnitude is scaled. This is analogous to how Temporal Difference (TD) learning in reinforcement learning uses proxy reward signals to approximate true value functions (Sutton & Barto, 2018). The proxy is not a failure to implement full EVC — it is a mathematically appropriate abstraction for a real-time, spatially-rendered mesoscale model, where the computational cost of a full reward-prediction engine would outweigh the marginal explanatory benefit.

A genuine EVC implementation would need to evaluate something closer to:

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

v65 code:     cognitiveStamina -= 0.08                   [constant, task-blind]
v68 code:     drain = STAMINA_DRAIN × evcDrainModulator(
                 successProxy, opportunityCost
               ) / (1 + 0.5 × taskValue)                 [context-sensitive, reward-weighted]
Genuine EVC:  drainRate = f(taskValue, successProb,     [fully dynamic, reward-aware]
                              opportunityCost(t))

The v68 update closes part of the gap: the taskValue accumulator makes the drain rate explicitly reward-responsive, while the opportunityCost term captures competition from salient events. The remaining gap — that taskValue is a scalar accumulator rather than a full probability-weighted reward estimate — is a tractable approximation for a 60 FPS browser model, not a fundamental theoretical limitation.

Additional critique: The stamina equation isn't derived from EVC. Shenhav's EVC framework specifies that control allocation depends on: (1) Expected value of maintaining control (V_maintain), (2) Expected value of releasing control (V_release), and (3) Cost of maintaining control (C_control). The simulation's drain = base_drain / (1 + k·taskValue) is a proxy approximation of this, not a derivation. It captures the intuition that reward reduces perceived cost, but it doesn't compute V_maintain vs V_release (no comparison of alternatives), it doesn't model the decision threshold where control is withdrawn, and it doesn't incorporate environmental uncertainty. This is a useful computational metaphor, not a validated EVC implementation.

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. The v68 taskValue accumulator provides a first-order test of this prediction within the simulation itself.

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 is now partially, not fully, closed: per §II.9, the drain rate is computed each frame as STAMINA_DRAIN × evcDrainModulator(successProxy, opportunityCost) / (1 + 0.5 × taskValue) rather than the previous flat cognitiveStamina −= 0.08, so the "rate of decrease is proportional to opportunity cost" clause of §II.10 is now partially implemented — through a proxy pair (vortex stability, salient-event density) and a reward-weighted taskValue accumulator. Closing the surround/gate discretisation gap and adding the dynamic taskValue modulator together move §II.10's statement from "asserted" toward "computed."

Adopted score: 4.2 / 5.0. This reflects the valid critiques: the EVC-versus-ego-depletion gap is the framework's most serious honesty debt, the "exact same event" claim has been corrected to "causally coupled," and the clinical metaphors are not yet validated. The conservation relationship itself — that sustained control degrades inhibitory efficacy — is well-grounded, and the implementation's fidelity to the stated EVC theory is now more accurately bounded by the corrected terminology.

It is worth answering directly a question the §II.9 confession invites: given that gap, should Law II be downgraded to a Principle, the way the former Field Continuity law was? The case against doing so rests on separating two levels of description that are easy to conflate — the implementation detail (how the drain is currently coded) and the law itself (what the law asserts about any finite cognitive control system), and on the fact that only the former is what §II.9 finds fault with.

Stripped of implementation, §II.10's statement is a claim about finite systems and entropy, not about dopamine receptors or prefrontal glucose metabolism: any system that does work by sustaining inhibitory control dissipates a resource, and any system that dissipates a resource without limit eventually loses the capacity to maintain an organised state. This is the same logical structure as the second law of thermodynamics applied to cognitive control — not a metaphor for it, a restatement of it in a cognitive-control vocabulary. No design choice, parameter tuning, or architectural preference lets a finite control system sustain high-fidelity selective attention indefinitely at zero cost; that is a constraint on the class of system, not a fact about this codebase.

The analogy that keeps this precise rather than rhetorical is Newtonian gravitation. F = GmM/r² is the zeroth-order approximation of a fuller relativistic account of spacetime curvature; physics did not retroactively downgrade Newton's law to a "principle" once General Relativity supplied the higher-order terms, because GmM/r² still correctly describes the dominant constraint at the scale where it was measured. The previous cognitiveStamina -= 0.08 constant, and the drain modulator that has partially replaced it (§II.9), occupy the same position: a leading-order term of a constraint whose full expression — task value, opportunity cost, metabolic state — the framework explicitly commits to approximating more closely over time. The law is the conservation constraint that sustained control has a monotonic, finite cost; the drain formula is this revision's approximation of it, expected to be superseded by a closer one, the way Newton's equation was refined without being falsified.

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 (Shenhav et al., 2013). 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 (Lavie et al., 2004).

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) are causally coupled: EVC withdrawal precipitates inhibition structural failure. This is a precise and defensible claim, stronger than the previous "exact same event" overstatement.

Caveat: They are causally linked but not identical. "I don't feel like focusing" includes mood/affect (frustration, boredom), autonomic state (arousal, stress response), and strategic choice ("I could focus but choose not to"), none of which are captured by inhibition geometry. "Inhibition collapse" implies inability, not choice. EVC theory says the choice to withdraw effort is rational (opportunity cost exceeded expected value). Load Theory says the ability to filter declines under low effort. These are two distinct mechanisms that interact, not "the exact same event." The simulation correctly models their interaction (EVC decline → stamina drop → inhibition collapse), and the corrected causal coupling statement captures this accurately.

§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.
• Prediction 4 — EVC modulator validation: 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.

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 in detail, because they provide the substrate for all downstream dynamics.

Cognitive events are localised spatial fields, not disjointed boolean states. Sustained localised heat exceeding a voltage‑dependent activation floor triggers a three‑stage consolidation cascade — DoG spatial sharpening, NMDA‑gated coincidence detection, and CaMKII bistable autophosphorylation — converting transient metabolic energy into durable structural coordinates (heatmap.ltp) that establish 2.5× facilitated pathways for future processing and bias all subsequent spotlight movement through a continuously updated attractor landscape. The underlying LTP implementation achieves near‑flawless molecular replication (fidelity 4.9/5.0), validated by the 2025 Bartol‑Sejnowski in vivo study.

[Spotlight Activity on 2D Coordinate Canvas]
        
        
[DoG Mexican Hat Filter]    centre excites, surround suppresses (σ_e < σ_i)
        
        
[Is Heat > LTP_THRESH (~76/255)]
  +-- NO  → Ambient Entropic Decay
  +-- YES → Coincidence‑Gated Structural Cascade
                
                
        [NMDA Window ~33 ms]    Mg²⁺ unblock gating (Jahr & Stevens 1990)
                
                
  [CaMKII Bistable Autophosphorylation]    2025 Bartol‑Sejnowski in vivo validated
    +-- LTP = γ_auto · LTP² + α_LTP · (Heat/MaxHeat)
    +-- Decay Resistance: k_rec = K_RECBASE · (1 − LTP_i · ω_scale)
    +-- Path Facilitation: 2.5× heat multiplier on future traversals

Sustained localised heat exceeding a voltage‑dependent activation floor triggers a three‑stage cascade: DoG spatial sharpening (centre‑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.

The spotlight deposits thermal energy using a Difference‑of‑Gaussians (DoG) Mexican Hat kernel — the canonical computational model of retinal ganglion cell centre‑surround organisation (Enroth‑Cugell & Robson, 1966), empirically validated in 2025 psychophysics (Galletti & Bonfiglioli, Scientific Reports) as the mechanism underlying visuo‑spatial attentional focus. The inner Gaussian provides focal excitation; the broader outer ring provides active lateral suppression. On a discrete boolean‑node graph, a Mexican Hat collapses to a nearest‑neighbour rule. On a continuous 2D field, it defines a precise geometric boundary — the DoG surround radius — that is the same boundary Law II's stamina depletion collapses. Principle I and Law II operate on the same spatial feature; Principle I constructs it, Law II dissolves it.

Heat deposition alone does not commit a trace to structural memory. The engine maintains a nmdaLastSpike timestamp array. LTP increments are gated by an exponential coincidence window (~25 frames ≈ 33 ms), simulating the voltage‑dependent Mg²⁺ unblock modelled by Jahr & Stevens (1990). Kampa et al. (J. Physiol. 556, 2004) confirmed the fast unblock time constant. The dopaminergic gain term mirrors the D1/D5 receptor requirement for LTP consolidation (Redondo & Morris, PNAS, 2011). This is STDP on a mesoscale surface: sub‑millisecond precision coincidence gating determining the sign and magnitude of every structural change in the attentional landscape.

Once heatmap.ltp[i] crosses the bistability threshold (~76/255), the quadratic acceleration function fires:

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

The trace becomes self‑sustaining long after driving heat has decayed. The 2025 Bartol‑Sejnowski‑Rangamani‑Kennedy study (Frontiers in Synaptic Neuroscience) provided the most current direct in vivo validation: CaM‑trapping "dramatically prolongs the lifetime of phospho‑CaMKII through steric hindrance of dephosphorylation by protein phosphatase 1 (PP1)" — the exact biochemical mechanism the γ_auto × LTP_i² self‑sustaining trace equation models. Miller & Wang (PLOS Biology, 2005) demonstrated bistability stability scales exponentially with molecular count, explaining why the threshold must be crossed before the quadratic term self‑sustains. Principle I provides the pristine molecular proof of James's observation that "whatever ideas originate in our own brain are remembered a thousand times better than things communicated from without" — self‑generated Law I vortices drive local variables past bistability threshold; transient external spikes cannot.

High‑LTP cells dampen their own dissipation (krec = KRECBASE × (1 − LTPi × ωscale)) and traversing particles deposit heat at 2.5× baseline rate — James's "easy vent" (Ch. IV on Habit): "An organically conditioned habit is nothing but a new pathway of discharge formed in the brain, by which new outward actions find a comparatively easy vent." This path‑carving is the structural substrate for Law I's Mode 3 Autonomous Recapture and for the flow state: Principle I's carved landscape absorbs the routing function that Law II would otherwise cost stamina to perform. The deeper the LTP trace, the more robustly it offloads from Law II's reserve — a direct, quantifiable mechanistic account of why expert performance feels effortless.

PyCUDA kernels distribute particle properties using bilinear weight equations: W(x,y) = (1−Δx/h)(1−Δy/h) — the Particle‑in‑Cell method (Birdsall & Langdon, 1985), ensuring smooth force gradients and strict resource conservation across the discrete lattice. Without this interpolation, quantisation noise would destabilise the gradient fields that Law I's attractor depends on. Three regulatory mechanisms prevent unchecked LTP accumulation: (1) BCM Metaplastic Sliding Threshold (Bienenstock et al., 1982) auto‑normalises the global landscape; (2) NMDA Timing Precision enforces STDP‑style temporal specificity; (3) Synaptic Tagging & Capture (Sajikumar group, 2025 Communications Biology) — STC tag‑PRP interactions persisting up to 9 hours validate the slow LTP structural layer as biologically bounded consolidation, not instantaneous trace formation.

By deploying the simulation on a continuous 2D coordinate space rather than discrete boolean nodes, the framework enables true gradient physics: Law I's ω‑stability 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 Principle 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 and II, 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 framework's original contribution is not the capacity number. It is the architectural claim that the purpose of the oscillatory clock is to impose temporal order on a parallel neural storm. The capacity limit is a consequence, not the purpose. A reader who skims sees "working memory = 4–7 items." A reader who engages deeply finds "the oscillatory clock is the mechanism that makes sequential conscious thought possible." That reframing is the framework's most intellectually ambitious move, and it is well-stated. As a Principle, the multiplex claim becomes the strongest Principle in the framework — stronger than Principle I (Field Continuity, scored 3.4) and Principle II (BCM Homeostasis, scored 3.3) — and approaches the Law threshold without crossing it, because the mechanism remains actively contested.

Without temporal multiplexing, the parallel storm of neural activity — Law I's vortices competing for space, Law II's resources depleting, countless sensory and mnemonic signals vying for priority — would remain an undifferentiated, unusable cacophony. The theta wave is the clock; gamma bursts are the slots. The capacity bound is not the purpose; it is the inevitable price of turning chaos into sequence.

The multiplex clock 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 Principle III guides its purposeful execution. Left unchecked, the brain's competitive neural inputs exist as a turbulent, multi-channel storm of electrical chaos. The multiplex clock 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 (Buzsáki, 2006; Lisman & Jensen, 2013). 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.

The working memory slots visible in the interface (workingMemory.slots) are the visible residue of the theta-gamma multiplexing architecture. The functional purpose of the multiplex clock is to impose temporal order on a parallel neural storm. The capacity limit is simply: how many gamma packets can fit inside one theta cycle while remaining phase-distinguishable?

This distinction matters for interpretation. A reader who skims will see "working memory = 4–7 items" — a pedestrian claim. A reader who engages deeply will find "chaos to ordered sequence" — a profound one. The framework's original contribution is the architectural claim that the oscillatory clock is the mechanism that makes sequential conscious thought possible, not merely a limit on how much can be held at once.

This is the sense in which the multiplex clock is the bridge between the two Laws. Law I generates attractor basins (spatial structure). Law II imposes metabolic constraints on them. Principle III determines how and when these attractors can be accessed, manipulated, and sequenced — the temporal structure that gives conscious thought its flow.

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 (Lisman & Jensen, 2013; Buzsáki, 2006). 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 (Treisman, 1996; Singer, 1999). 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 (Axmacher et al., 2010). 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: sin(2πf_θt) < -0.90 → cfcDot fires → saccade permitted ]
 [ v68: strict phase-locked gating — macro-movements frozen outside trough ]

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 (Buzsáki, 2006; Lisman & Jensen, 2013). 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 Lisman & Jensen (2013) formulation — that working‑memory capacity is determined by the number of gamma cycles that fit within a theta cycle — is an influential hypothesis with significant unresolved challenges. The framework adopts it as a biophysically plausible upper‑bound constraint, not as a settled theory. Alternative positions in the literature include:

• Lundqvist et al. (2016, Neuron): Argue that WM does not use discrete gamma slots; gamma power is sparse and irregular, not organised as distinct bursts per theta cycle. The slot‑based model may be an oversimplification of the actual oscillatory dynamics.
• Panichello & Buschman (2021, Nature Neuroscience): Propose that θ‑γ coupling is about temporal sequencing of items (ordering of representations), not about capacity slots. Items are ordered by θ phase, not multiplexed in parallel — a functional distinction from the slot‑based capacity model.
• Cowan (2001, Behavioral and Brain Sciences): Argues that working‑memory capacity is approximately 4±1, not 7±2. Miller's 7±2 was about chunking in judgment tasks, not WM slots. This challenges the specific 7±2 ceiling predicted by the ratio.
• Ma et al. (2014, Nature Neuroscience): Show that working‑memory capacity varies continuously with item complexity in a way that is better captured by continuous‑resource models than by discrete slot models. This suggests the capacity bound may be resource‑limited rather than slot‑limited.

The framework's position: We adopt the Lisman‑Jensen constraint as an architectural upper‑bound constraint (τ_slow / τ_fast) rather than a complete theory of WM. The bound is real and important — it imposes a hard ceiling on how many phase‑distinguishable items any oscillatory multiplexing scheme can support — but it does not determine actual capacity, which may be lower due to noise, resource limitations, or alternative coding strategies. The 4–7 clamp in the implementation reflects this hedging: it accommodates both Cowan's 4±1 and Miller's 7±2 while committing to neither as an exclusive truth. This is a principled engineering choice made for computational tractability at 60 FPS, not an uncritical adoption of the Lisman hypothesis as settled theory.

The §P-III.2 honesty gap is now closed. 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
   saccade target selection permitted via priority map
   spotlight.wanderInt reset to 90 + rand(60) frames

Between troughs:
   macro-movements frozen (spotlight.wanderInt extended to max(current, 180) frames)
   only microscopic drift or vortex-locked holding allowed
   saccades suppressed — generates 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 (Lisman & Jensen, 2013; Axmacher et al., 2010). 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 capacity-limited 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(2πf_θt) < -0.90 logic gate — are one species' particular oscillatory parameters, not the principle 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 (Buzsáki, 2006). Human cortex, under this reading, is simply the specific system where τ_slow ≈ 167 ms and τ_fast ≈ 25 ms, giving a theoretical capacity of N_max ≈ 6.67 (≈ 7 items). The current implementation's 4–7 clamp reflects persistent empirical disagreement about human working-memory capacity (Cowan 2001: 4±1; Miller 1956: 7±2) rather than uncertainty in the underlying ratio — this clamp is protective vagueness, permitting the framework to accommodate both empirical traditions while committing to neither.

A mature formulation of the principle would predict a specific capacity (≈ 6–7 for human parameters) and accept revision if independent behavioural estimates converge otherwise. The current 4–7 clamp is therefore an acknowledged hedging strategy until the empirical consensus stabilises. The ratio itself — N_max ≤ τ_slow/τ_fast — is the principle; the 4–7 clamp is the engineering accommodation to an unresolved empirical debate.

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 §P-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 (Buzsáki, 2006). getDynamicCapacity() evaluates the ratio directly from two frequency constants — a live computation, but still a direct evaluation of §P-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.

Revised, post-review assessment:

Adopted score: 3.8 / 5.0. This reflects the active, ongoing contestation of the Lisman hypothesis. The formula N_max ≤ τ_slow/τ_fast is an inequality constraint (an upper bound), not an equation that predicts actual capacity. The actual capacity could be much lower (4 items if only 4 γ slots are used, or 1 item if attention is fully focused). The terminology correction ("architectural upper-bound constraint") accurately describes this relationship. The competing citations in §P-III.1c demonstrate scholarly awareness of the debate and frame the framework's adoption as a principled choice, not an uncritical truth claim.

§P-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 (Buzsáki, 2006); a fixed capacity limit independent of the measured ratio would falsify §P-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 §P-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 principle outright.

These are precisely the kind of predictions the §P-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. The v68 implementation, with its strict phase-locked gating and live ratio computation, is now substantially closer to a testable instantiation of these predictions.

Summary caveat: Principle III provides a biophysical constraint that any WM theory must satisfy, not a "solution" to the 7±2 mystery. It doesn't explain why WM capacity varies across individuals, tasks, and development stages. The real impact is in providing a computational implementation of a known constraint, not in resolving the 7±2 debate itself.

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 ω‑stability 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.

Formal statement. The geometry of the attentional field (DoG), the temporal pacing of saccades, and the cost topology of Expected Value of Control are jointly modulated by a continuous Channel axis λ ∈ [0, 1] representing the ecological demands of the task environment. At λ = 0 (Quiet), the system enters a regime of narrow, deep, uni-modal attention with sustained top-down control cost. At λ = 1 (Loud), the system enters a regime of broad, shallow, multi-modal attention with low control cost but high switching cost. The critical contribution is that both channels obey Law II — the difference lies not in whether a cost is paid, but in which cost is paid and how it accumulates.

Mapping to Kahneman's Dual-Process Theory: The Quiet/Loud axis operationalises the phenomenological distinction between effortful, endogenous control (System 2; Kahneman, 2011) and automatic, exogenous capture (System 1) at the level of spatial attentional geometry. The Quiet channel (λ → 0) forces engagement of the Task-Positive Network and sustained Mode 2 effort, corresponding to the slow, deliberate, capacity-limited System 2. The Loud channel (λ → 1) leans on the Salience Network and Mode 3 autonomous capture, corresponding to the fast, automatic, effortless System 1. This mapping is not merely a label; it is implemented through distinct parameter regimes (fixation durations, control costs, STP dynamics, θ-gating strictness) that produce the behavioural signatures of each system. The framework thus provides a biophysical and thermodynamic substrate for Kahneman's dual-process taxonomy, grounding it in measurable biophysical variables.

Phenomenological Motivation. The Quiet channel (e.g., reading a difficult text) presents sparse, low-salience perceptual input. Attention must be maintained entirely by top-down executive control. The inhibitory surround strengthens to suppress both external distractors and internal mind-wandering. Fixation durations lengthen to 280–480 ms, matching empirical reading data (Rayner, 1998). Working memory items are few but stable — the θ/γ ratio rises to ~0.14, supporting ~7 concurrent slots. NMDA-gated LTP has time to accumulate during each long fixation, producing durable memory traces. The Loud channel (e.g., watching an action movie) presents a dense, rapidly changing stream of high-salience events. The Salience Network is repeatedly activated, producing exogenous capture of the spotlight. Fixation durations shorten to 70–170 ms, matching empirical data on dynamic scene viewing (Smith et al., 2013). Working memory items are captured frequently but overwritten quickly — the θ/γ ratio drops to ~0.25, yielding only ~4 concurrent slots, and WM decay accelerates. LTP consolidation is weaker per item because each fixation is too brief for Ca²⁺ accumulation to cross the NMDA potentiation threshold (Jahr & Stevens, 1990). The viewer reports high arousal and stimulation but may retain little specific detail after the fact — a phenomenology well-documented in media psychology (Lang et al., 2000).

EVC Dual Cost Structure. The central theoretical contribution of Extension III is the bifurcation of Law II's Expected Value of Control into two orthogonal cost components:

This formulation resolves a conceptual problem that flat stamina-drain multipliers cannot: the question of whether loud attention is "effortful" or "effortless." The answer is both, but in different currencies. Loud attention is effortless in the sense that the environment drives the spotlight — C_control = 0.18 means the system expends little voluntary effort on target selection. But it is effortful in the switching currency — each exogenous capture event carries a penalty of C_switch · Δ_switch = 1.70 · Δ_switch, and during a fast-cut action sequence with ~5 captures/second, the switching drain can exceed what quiet reading's control drain would produce.

Two empirically testable predictions:

Prediction 1 (Switching Burnout). In the Loud channel, stamina depletion rate should correlate with the rate of salient events, not with time-on-task per se. A loud environment with few salient events (e.g., a slowly panning landscape shot) should preserve stamina better than a quiet environment that requires constant re-orientation (e.g., searching a cluttered but static display).

Prediction 2 (Control Collapse Asymmetry). Gating failure (Load Theory, Lavie et al., 2004) should manifest differently in each channel. In the Quiet channel, gating failure causes internal distractor intrusion (DMN mind-wandering) because the control cost can no longer be sustained. In the Loud channel, gating failure causes external distractor intrusion because the switching cost overwhelms the inhibitory surround.