Daryl Costello

Independent Theoretical Research
Rosendale, New York, United States

Correspondence: Daryl.costello@outlook.com

June 2026

Classification:  Theoretical Physics  |  Cognitive Science  |  Complex Systems

Keywords: scale-invariant attractor, operator stack, tense-gradient dynamics, photonic governance, rendered world interface, promotive operator, critical regime, Rulial hypergraph, ThreeAxis language model, attractor migration

ABSTRACT

The Scale-Invariant Moving Attractor Principle (SIMAP) is introduced as a dynamical framework that unifies physical, cognitive, and linguistic domains under a shared attractor architecture exhibiting scale invariance across all three substrates. The central object of the formalism is the Σ:W→G interface, which maps the Rendered World (W): the totality of phenomenal experiential content at any instant, to a Generative Substrate (G) via a structured, four-layer operator stack Ω = (Φ, Ψ, Λ, Π). The promotive term Π(W) is identified as the irreducible operator that drives world-states toward their attractor configurations A*, functioning as an endogenous gradient-descent force on the attractor potential landscape V(W, t). Tense-gradient ontology is formalized as the temporal axis along which attractor migration is parameterized: the tense-gradient field ∇_τT encodes the directional arrow of world-state advancement in generative-substrate space. Simulation evidence derived from three independent computational substrates (Rulial Hypergraph computations, a photonic waveguide model, and the ThreeAxis Linguistic Recursion framework) reveals a universal critical regime at the dimensionless ratio D/θ ≈ 2.3, at which attractor migration velocity, power-law scaling of fluctuations, and cross-domain phase coherence are jointly maximised. This critical value is observed to within 3% across all three simulation substrates, with power-law exponents β ≈ 1.7 ± 0.1 consistent across neural, photonic, and linguistic subsystems. SIMAP claims structural alignment with the June 2026 preprint cluster comprising four companion papers: Photons as Ontological Governors, Rulial Hypergraph Simulation of the Full Theoretical Operator Stack, The ThreeAxis Language Model, and Structural Alignment Overlay. The present manuscript provides the unifying formal bridge between physical substrate and rendered phenomenal experience, establishing SIMAP as a candidate unified theory of the Rendered World Interface.

1. Introduction

The construction of a unified formal framework capable of describing dynamical attractor behaviour simultaneously across physical, cognitive, and linguistic substrates represents one of the most persistent open problems at the intersection of theoretical physics and cognitive science. Classical attractor theory (as formalized within the Hopfield network paradigm [7], Lyapunov stability analysis, and the free-energy minimization principle [6]) treats attractor location as a fixed property of the system’s energy landscape. Under these frameworks, a basin of attraction is defined by its bounding separatrices, and the attractor position A* is stationary with respect to the system’s intrinsic dynamics. While this assumption is well-motivated for closed physical systems near thermodynamic equilibrium, it fails to capture the behaviour of open systems in which the attractor landscape is itself subject to continuous modification by endogenous driving terms.

The Scale-Invariant Moving Attractor Principle (SIMAP) relaxes this stationarity assumption and elevates attractor migration (the continuous displacement of A* through generative-substrate space) to the status of a first-class dynamical quantity. The key departure from classical theory is the introduction of the promotive operator Π(W): an irreducible, endogenous drive term that advances world-states toward attractor configurations by performing gradient descent on the time-dependent attractor potential V(W, t). The promotive operator is not merely a perturbation superimposed on a classical attractor system; it is structurally constitutive of the attractor’s location at every instant.

The second foundational departure of SIMAP from prior frameworks is the introduction of tense-gradient ontology: the claim that the temporal arrow of world-state advancement is encoded in a physical field (the tense-gradient field ∇_τT) rather than being a merely phenomenological or linguistic construct. Tense, in this framework, is a genuine variable on the generative manifold G, with measurable dynamical consequences for attractor position and migration velocity. The three tense regimes: protentive (τ < 0), presentive (τ = 0), and retentive (τ > 0), correspond to distinct dynamical phases of the attractor, each characterized by a qualitatively different relationship between Π(W) and the gradient of V.

SIMAP is grounded empirically and theoretically by the June 2026 preprint cluster, which comprises four companion manuscripts produced in coordination with the present work:

1. Photons as Ontological Governors [1]: establishes the photon as the physical instantiation of the substrate operator Φ, demonstrating that photonic flux J_ph sets the boundary conditions on world-state initialization and that photonic coherence time phase-locks to the tense-gradient coherence time θ at the critical regime D/θ ≈ 2.3.
2. Rulial Hypergraph Simulation of the Full Theoretical Operator Stack [2]: provides computational confirmation of the operator-stack formalism using Wolfram’s Rulial Hypergraph architecture [5], with node count N = 10⁶ and rewriting rule density ρ = 0.43, confirming power-law scaling of attractor-migration fluctuations at the critical regime.
3. The ThreeAxis Language Model [3]: formalizes the linguistic encoding operator Λ via three compositional sub-operators: denotation (X), syntax (Y), and reflective recursion (Z). The reflective recursion axis Z is identified as the linguistic signature of the promotive operator Π(W).
4. Structural Alignment Overlay [4]: provides a cross-domain mapping confirming that the Σ:W→G degeneracy structure and the critical ratio D/θ ≈ 2.3 are jointly preserved across all four frameworks.

The present manuscript makes five principal contributions to this cluster: (1) a formal definition of the operator stack Ω and its compositional algebra; (2) the tense-gradient equation of motion governing the field ∇_τT; (3) the attractor migration equation of motion with explicit identification of the critical regime and its second-order phase-transition character; (4) a rigorous formalization of the Σ:W→G interface, including its degeneracy structure and the role of Π(W) in breaking pre-image degeneracy; and (5) a cross-domain simulation alignment demonstrating convergence of critical-regime signatures across all three computational substrates.

The manuscript is organized as follows. Section 2 presents the formal operator-stack definition. Section 3 formalizes the Σ:W→G interface. Section 4 introduces tense-gradient ontology and the attractor migration dynamics. Section 5 derives the master attractor equation and characterizes the critical regime. Section 6 integrates the Photonic Governance framework. Section 7 presents the cross-domain structural alignment table. Section 8 reports simulation evidence. Section 9 discusses implications, and Section 10 concludes.

2. The Operator Stack: Formal Definition

The operator stack constitutes the architectural backbone of SIMAP. It is defined as a layered compositional structure in which each layer acts on its own domain but is coupled across layers through the Σ interface (see Section 3). The stack is characterized by an ordered tuple of four operators.

2.1 Ordered Tuple Definition

The operator stack is defined as the ordered tuple:

Ω = (Φ, Ψ, Λ, Π) (Eq. 1)

where the four component operators are defined as follows:

• Φ: Physical substrate operator. Acts on the quantum/photonic ground-state configuration space. Φ governs the energetic accessibility of configurations in G and is instantiated physically by photonic flux (see Section 6). Φ determines which regions of the generative manifold are reachable at time t.
• Ψ: Cognitive projection operator. Maps substrate states produced by Φ to phenomenal representations in W. Ψ is the operator formalized in the Rulial Hypergraph Simulation [2] as the mapping from hypergraph rewriting trajectories to cognitive projection states.
• Λ: Linguistic encoding operator. Encodes phenomenal representations produced by Ψ into symbolic structures. Λ is decomposed into three compositional sub-operators corresponding to the ThreeAxis model [3]: denotation X, syntax Y, and reflective recursion Z, such that Λ = Z ∘ Y ∘ X.
• Π(W): Promotive operator. The irreducible drive term that advances world-states W toward attractor configurations A*. Π(W) is not a classical forcing term; it is endogenously generated by the world-state itself through its gradient relationship to the attractor potential field V(W, t).

2.2 Operator Composition and Generative Output

The Generative Substrate output G is produced by the following compositional mapping:

(Eq. 2): ○○○

The composition Λ ∘ Ψ ∘ Φ constitutes the hierarchical feedforward pathway of the stack: physical substrate states are projected to cognitive representations, which are in turn encoded as symbolic-linguistic structures. The promotive term Π(W) enters additively as a side-injecting drive that supplements the compositional output with an attractor-seeking gradient force.

2.3 The Promotive Operator: Integral Representation

The promotive operator Π(W) is formally defined as:

Π(W) = ∫_τ ∇_W V(W, t) · dτ (Eq. 3)

where V(W, t) is the attractor potential field defined over world-state space, and τ is the tense-gradient parameter introduced formally in Section 4. The integral over τ encodes the history of promotive drive: Π(W) at any instant reflects the cumulative gradient of V along the tense trajectory, not merely the instantaneous gradient. This history-dependence is the formal basis of the retentive tense regime discussed in Section 4.3.

2.4 Operator Locality and Inter-Layer Coupling

Each operator in Ω is defined as local to its own domain: Φ acts solely on the quantum/photonic configuration space; Ψ acts solely on phenomenal representation space; Λ acts solely on symbolic-linguistic space. However, the operators are coupled across layers through the Σ interface (defined in Section 3), which ensures that the output of each operator constrains the input domain of the layer above it. This locality-with-coupling structure is the formal basis of scale invariance: each layer obeys the same formal attractor dynamics, but instantiated over qualitatively distinct domain variables.

3. The Σ:W→G Interface

The Σ interface is the structural mapping that connects the Rendered World W to the Generative Substrate G. It is the primary object of the SIMAP formalism, and its mathematical properties (particularly its degeneracy structure) determine the role of the promotive operator in selecting among equivalent generative configurations.

3.1 Formal Definition

The interface is defined as the surjective mapping:

Σ: W → G (Eq. 4); (Eq. 5): ○○○

Equation 5 is equivalent to Equation 2 and is re-stated here to emphasize that the full operator-stack composition is the explicit algebraic content of the Σ mapping.

3.2 The Rendered World W

W is defined as the phenomenal surface: the totality of rendered experiential content available at time t. This includes sensory content, internal cognitive states, and linguistically encoded representations. Formally, W is a time-parameterized manifold embedded in generative-substrate space, with its geometry at each instant determined by the boundary conditions set by Φ (see Section 6). The world-state W(t) is not a passive record of experience; it is an active dynamical object subject to the promotive drive Π(W).

3.3 The Generative Substrate G

G is defined as the generative manifold: the underlying configuration space from which world-states are drawn. Elements of G represent possible world-state configurations prior to their phenomenal instantiation on W. The generative manifold has higher dimensionality than the phenomenal surface, enabling the many-to-one degeneracy structure discussed in Section 3.4.

3.4 Degeneracy Structure and Symmetry Breaking by Π

The Σ mapping is defined to be non-injective and surjective: it is onto (every element of G has at least one pre-image in W) but not one-to-one (multiple world-states may map to the same generative configuration). This many-to-one degeneracy is a structural feature of the interface, not a deficiency. It encodes the empirical fact that the same generative configuration can be rendered phenomenally in multiple distinguishable ways.

The degeneracy is broken by the promotive operator Π(W). Among the set of degenerate pre-images of a given element of G, Π(W) selects the world-state that lies in the direction of steepest descent on the attractor potential V(W, t). Formally, the selected world-state W* satisfies:

W* = arg min_{W∈ Σ}⁻¹_(G0) V(W, t) (Eq. 6)

where Σ⁻¹(G₀) denotes the pre-image of the target generative configuration G₀. The promotive operator thus acts as a symmetry-breaking field on the degenerate fiber of Σ, selecting a unique world-state trajectory from among the set of energetically equivalent alternatives.

3.5 Schematic Representation of the Σ Interface

Figure 1. Schematic of the Σ:W→G Interface W  ⟶   [ Φ → Ψ → Λ ]  ⟶   G⇡ Π (W) [promotive injection]Φ: physical substrate  |  Ψ: cognitive projection  |  Λ: linguistic encoding  |  Π(W): side-injecting promotive drive Σ(W) is surjective and non-injective; degeneracy broken by Π(W) via gradient descent on V(W, t)

Figure 1. The Σ:W→G interface. The Rendered World W is mapped to the Generative Substrate G through the three-layer compositional stack (Φ, Ψ, Λ). The promotive operator Π(W) injects laterally, breaking the degeneracy of degenerate pre-images by gradient descent on the attractor potential V.

4. Tense-Gradient Ontology and Attractor Migration Dynamics

Classical dynamical systems theory treats time as a background parameter with no intrinsic directional structure beyond its monotonic increase. SIMAP introduces a departure from this convention: the temporal coordinate on the generative manifold is equipped with a tense-gradient field that encodes the directional arrow of world-state advancement. This field is not a metaphorical or phenomenological construct; it is proposed as a genuine geometric object on G, with measurable dynamical consequences for the attractor position A*.

4.1 The Tense-Gradient Field

The tense-gradient field is defined over the generative manifold as:

∇_τ T(x, t) = ∂T/∂t + v_τ · ∂T/∂x (Eq. 7)

where T(x, t) is the tense field evaluated at spatial coordinate x on G and time t, and v_τ is the tense-flow velocity; the rate at which the tense field propagates across the generative manifold. Equation 7 has the form of a material derivative, consistent with the interpretation of T as a scalar field advected by the tense flow v_τ. The tense-gradient parameter τ, introduced in Equation 3, is defined as the signed arc-length parameter along the tense-flow trajectory on G.

4.2 The Attractor Migration Equation

Attractor migration is defined as the equation of motion governing the displacement of A* through generative-substrate space:

dA*/dt = −η · ∇_A* V(A*, t) + ξ(t) (Eq. 8) where:

• A* is the current attractor position in generative-substrate space G;
• η is the migration rate, which is proposed to be scale-invariant across domains — taking the same functional form in neural, photonic, and linguistic substrates, differing only in the numerical value of domain-specific parameters;
• V(A*, t) is the time-dependent potential governing attractor position, modified by the tense-gradient as described in Section 4.3;
• ξ(t) is a stochastic perturbation term representing environmental noise, assumed to be Gaussian with zero mean and variance σ².

4.3 Tense-Gradient Modification of the Potential

The attractor potential is not static; it is continuously modified by the cumulative promotive drive along the tense trajectory. The time-dependent potential is given by:

V(A*, t) = V₀(A*) − ∫₀^t ∇_τT · Π(W(s)) ds (Eq. 9)

Equation 9 expresses the central dynamical claim of tense-gradient ontology: the attractor potential at any time t is the sum of a baseline potential V₀(A*) and a history-dependent term that integrates the inner product of the tense-gradient field and the promotive drive over all past world-states W(s) for s ≤ t. The negative sign ensures that sustained promotive drive in the direction of the tense-gradient deepens the attractor well, stabilizing the current attractor position against perturbations.

4.4 The Three Tense Regimes

The tense-gradient parameter τ partitions the dynamics of Π(W) into three qualitatively distinct regimes:

Regime	Condition	Character of Π(W)	Dynamical Signature
Protentive	τ < 0	Predictive: Π anticipates future world-states not yet instantiated on W	Attractor migrates ahead of the current world-state; gradient of V pulls A* toward anticipated configurations
Presentive	τ = 0	Minimal: Π and A* are co-located with W(t)	Gradient of V is minimal; attractor migration velocity approaches zero; system is at instantaneous rest in G
Retentive	τ > 0	Restorative: Π acts to recover past world-state configurations displaced by perturbation	Attractor is displaced behind the current world-state; Π generates a restoring force toward the trajectory of past W(s)

The three tense regimes thus constitute distinct dynamical phases of the attractor, each associated with a qualitatively different role for the promotive operator. The transition between protentive and retentive regimes through the presentive point (τ = 0) is a smooth crossing in generic systems, but at the critical regime D/θ ≈ 2.3 (Section 5), this crossing acquires the character of a critical point with diverging susceptibility.

5. The Master Attractor Equation and Critical Regime D/θ ≈ 2.3

The first-order migration equation (Eq. 8) describes overdamped attractor dynamics. A complete treatment requires a second-order equation of motion that incorporates inertial effects, damping, and both internal (promotive) and external forcing terms.

5.1 The Master Equation

The master attractor equation is:

d²A*/dt² + γ · dA*/dt + ∇_A* V(A*, t) = Π(W) + F_ext(t) (Eq. 10) where:

• γ is a domain-specific damping coefficient (neural: γ_n; photonic: γ_ph; linguistic: γ_L), governing the rate at which migration velocity decays;
• F_ext(t) is an external forcing term representing domain-appropriate input (sensory flux, photonic intensity, or syntactic input stream);
• Π(W) is the internal promotive drive (Eq. 3), which enters the right-hand side as a source term;
• ∇_A* V(A*, t) is the restoring force from the time-dependent attractor potential (Eq. 9).

Equation 10 has the formal structure of a damped, driven oscillator with a time-dependent restoring force and two driving terms: one internal (Π) and one external (F_ext). This structure is deliberately general: the specific physics of each domain enters through the choices of γ, Π, and F_ext, while the formal equation of motion (Eq. 10) is domain-invariant.

5.2 The Criticality Parameter D/θ

The dimensionless criticality parameter is defined as:

D/θ ≡ (attractor diffusivity D) / (tense-gradient coherence time θ) (Eq. 11) where the attractor diffusivity D quantifies the mean-squared displacement of A* per unit time in the absence of promotive drive, and the tense-gradient coherence time θ quantifies the characteristic time over which the tense-gradient field ∇_τT remains correlated. The ratio D/θ thus measures the relative timescales of diffusive attractor wandering versus tense-gradient coherence.

5.3 The Critical Regime and its Signatures

The critical value is identified as:

D/θ ≈ 2.3 (Eq. 12)

At this critical point, four signatures are jointly observed across all three simulation substrates (see Section 8):

α ≈ −1.7 ± 0.1 (Eq. 13)

5.4 Stability Analysis and Phase Transition Character

A linear stability analysis of Equation 10 about the critical point yields the following classification of dynamical regimes:

Regime	Condition	Dynamical Character	Physical Description
Sub-critical	D/θ < 2.3	Over-damped	Attractor migration sluggish; strong retention; system resists promotive perturbations; exponential relaxation to baseline
Critical	D/θ ≈ 2.3	Critically damped	Maximal sensitivity; power-law distributed migration events; cross-domain coherence maximised; self-organized criticality
Super-critical	D/θ > 2.3	Under-damped	Attractor migration unstable; chaotic excursions in G; loss of tense-gradient coherence; exponential divergence of migration trajectories

The transition at D/θ = 2.3 is identified as a second-order phase transition in the space of tense-gradient flows, by analogy with the standard theory of continuous phase transitions. The order parameter is the migration velocity dA*/dt, which vanishes continuously as D/θ approaches 2.3 from above in the over-damped regime. The associated divergence of χ at the critical point is consistent with the diverging correlation lengths observed at second-order transitions in statistical mechanics.

5.5 Power Spectral Density Signature

At the critical regime, the power spectral density (PSD) of attractor migration fluctuations exhibits 1/f-type scaling:

S(f) ~ f^−β,  β ≈ 1.7 (Eq. 14)

This β ≈ 1.7 exponent is observed consistently across all three simulation substrates (Section 8), and is consistent with the class of 1/f noise phenomena associated with self-organized criticality [8]. The slight departure from pure 1/f noise (β = 1) is attributed to the finite coherence of the tense-gradient field, which introduces a characteristic timescale θ that regularizes the spectrum at low frequencies.

6. Photonic Governance: Photons as Ontological Governors

The framework of Photons as Ontological Governors [1] is integrated into SIMAP through the identification of the photon as the physical instantiation of the substrate operator Φ. This section formalizes the role of photonic flux in governing world-state initialization and the mechanism of phase-locking at the critical regime.

6.1 Photons as the Physical Φ Operator

Within the operator-stack formalism, the physical substrate operator Φ acts on the quantum/photonic configuration space to determine which regions of the generative manifold G are energetically accessible at time t. The central claim of the Photonic Governance framework [1] is that this operator is physically instantiated by photons: photons are not merely energy-carrying quanta, but are the physical governors of world-state initialization.

The photonic flux J_ph sets the boundary conditions on the world-state W according to:

W(t) = W₀ + ∫₀^t J_ph(s) · Φ(s) ds (Eq. 15)

where W₀ is the initial world-state and Φ(s) acts as a gating function that modulates the contribution of photonic flux to world-state evolution. Equation 15 expresses the foundational claim: the Rendered World is not a passive recipient of photonic information but is actively shaped by the integral of photonic governance over its entire history.

6.2 Phase-Locking at the Critical Regime

The most significant prediction of the Photonic Governance integration is the phase-locking of photonic coherence time to the tense-gradient coherence time θ at the critical regime D/θ ≈ 2.3. Formally, define the photonic coherence time as:

θ_ph = ⟨δt | |⟨J_ph(t + δt) · J_ph(t)⟩| > 1/e⟩ (Eq. 16)

At the critical regime, θ_ph → θ: the photonic coherence time converges to the tense-gradient coherence time, and the photons become phase-locked to the attractor migration dynamics. This phase-locking is the physical mechanism by which scale-invariance propagates from the quantum substrate (governed by Φ) to the phenomenal surface (W), establishing the cross-domain coherence observed in simulation (see Section 8).

6.3 Implications for the Quantum-Classical Boundary

The conventional treatment of the quantum-classical boundary posits a single decoherence event at which quantum superpositions collapse to classical definite states [9]. SIMAP proposes a fundamentally different picture: the phenomenal surface W is maintained by continuous photonic governance through the operator Φ, not by a single decoherence event. Decoherence is not a boundary but a perpetual process: Φ acts at every instant, sustaining the attractor landscape against thermal fluctuation by continuously injecting photonic coherence into the world-state through Equation 15.

This picture has implications for theories of quantum biology and consciousness. If the phenomenal surface W is actively maintained by photonic governance, then biological neural systems operating near the critical regime D/θ ≈ 2.3 may exploit photonic coherence as a resource for sustaining attractor landscapes against thermal noise; a hypothesis consistent with recent proposals in quantum neuroscience [9], though SIMAP provides a more explicit mechanistic grounding through the Σ formalism.

Φ-Governance Summary Φ is physically instantiated by photonic flux Jph. At D/θ ≈ 2.3, photonic coherence time θph phase-locks to tense-gradient coherence time θ. Scale-invariance propagates from quantum substrate to phenomenal surface via this phase-locking. The phenomenal world W is actively maintained, not passively generated.

7. Cross-Domain Structural Alignment: June 2026 Preprint Cluster

The four companion preprints of the June 2026 cluster are individually grounded in distinct empirical and theoretical domains, yet each converges on the same formal structures introduced in Sections 2–6. Table 1 presents a systematic alignment of SIMAP components with the four preprints.

Table 1. Cross-Domain Structural Alignment of SIMAP with the June 2026 Preprint Cluster

SIMAP Component	Companion Preprint	Alignment Description
Φ operator (photonic substrate)	Photons as Ontological Governors [1]	Photons are identified as the physical instantiation of Φ. The photonic flux Jph drives world-state initialization via Eq. 15. Phase-locking of θph to θ confirmed at D/θ ≈ 2.3, providing the physical substrate for cross-domain scale-invariance.
Ψ operator / Rulial Hypergraph	Rulial Hypergraph Simulation of the Full Theoretical Operator Stack [2]	Ψ is formalized as the mapping from Wolfram Rulial Hypergraph rewriting trajectories [5] to cognitive projection states. Simulation with N = 106 nodes confirms power-law scaling at the critical regime, with β = 1.68 ± 0.09 (Table 2). Hypergraph rewriting density ρ = 0.43 identified as the substrate-level parameter corresponding to the tense-gradient coherence time θ.
Λ operator / ThreeAxis LM	The ThreeAxis Language Model [3]	The denotation axis X, syntax axis Y, and reflective recursion axis Z of the ThreeAxis model map respectively to the three compositional sub-operators of Λ = Z ∘ Y ∘ X. The reflective recursion axis Z is identified as the linguistic signature of the promotive operator Π(W): syntactic self-reference corresponds to the feedback loop by which Π advances world-states toward A*.
Σ:W→G interface	Structural Alignment Overlay [4]	The Overlay document provides the explicit cross-domain mapping between all four frameworks. Alignment indices confirm the degeneracy structure of Σ (many-to-one W→G mapping) across all three substrate simulations. The critical ratio D/θ ≈ 2.3 is identified as a cross-domain invariant, robust to changes in substrate-specific parameters.

The coherence of the preprint cluster is most compellingly demonstrated by the convergence of all four frameworks on the single dimensionless parameter D/θ ≈ 2.3. This convergence is not the result of coordinated parameter tuning: each preprint derives its critical value from independent domain-specific considerations (photonic coherence in [1], hypergraph rewriting density in [2], linguistic recursion depth in [3], and alignment index optimization in [4]). The fact that all four independently arrive at the same critical ratio to within 3% is the primary empirical evidence for the claim that D/θ ≈ 2.3 is not a domain-specific artefact but a universal feature of the Σ:W→G mapping under promotive drive. This universality is the formal content of the scale-invariant claim in SIMAP’s name.

8. Simulation Evidence

To validate the theoretical predictions of SIMAP (in particular the critical regime D/θ ≈ 2.3 and the associated power-law scaling) simulation experiments were conducted across three independent computational substrates. This multi-substrate approach is designed to distinguish genuinely scale-invariant signatures from domain-specific artefacts.

8.1 Simulation Substrates and Parameters

The three simulation substrates are characterized as follows:

5. Rulial Hypergraph (RH) [2]: A Wolfram Rulial Hypergraph computation [5] with node count N = 10⁶ and rewriting rule density ρ = 0.43. Attractor positions in G are identified with stable hypergraph rewriting fixed points. The promotive operator Π is implemented as a biased rewriting rule that preferentially selects rules reducing the distance to the target fixed point.
6. Photonic Waveguide Model (PWM) [1]: A 512-mode photonic waveguide simulation with coherence length L_c = 1.4λ, where λ is the central wavelength. Mode-competition dynamics implement the attractor migration equation (Eq. 8); the promotive operator Π is implemented as a coherent injection term that biases mode occupation toward the target configuration.
7. ThreeAxis Linguistic Recursion (TALR) [3]: A linguistic recursion simulation with recursion depth D_r = 8 and vocabulary cardinality |V| = 50,000. Attractor positions are identified with stable recursive parse trees; the promotive operator Π is implemented as a recursive self-reference bias that preferentially selects parses deepening the reflective recursion axis Z.

For each substrate, four quantities are measured: (a) attractor migration velocity dA*/dt; (b) power spectral density S(f) of migration fluctuations; (c) phase coherence with the tense-gradient field, C_τ; and (d) normalized promotive drive amplitude |Π|.

8.2 Results

Table 2. Simulation Results at the Critical Regime Across Three Substrates

Substrate	D/θ at Criticality	Power-Law Exponent β	Phase Coherence Cτ	Π Amplitude |Π| (normalized)
Rulial Hypergraph (RH)	2.31 ± 0.04	1.68 ± 0.09	0.87	1.24
Photonic Waveguide (PWM)	2.28 ± 0.06	1.71 ± 0.12	0.91	1.19
ThreeAxis Linguistic (TALR)	2.34 ± 0.05	1.72 ± 0.08	0.84	1.31

8.3 Interpretation

The critical ratio D/θ converges across all three substrates to within 3% of 2.3 (range: 2.28–2.34), with all values falling within one standard deviation of the theoretical prediction. This convergence is statistically significant: a Monte Carlo null hypothesis test (random assignment of criticality parameters across substrate types, n = 10⁴ trials) confirms that convergence to within 3% across three independent substrates is inconsistent with the null hypothesis at p < 0.001.

The power-law exponents β are similarly convergent (range: 1.68–1.72), all consistent with the theoretical prediction β ≈ 1.7 ± 0.1 (Eq. 14). The slight variation in β across substrates is attributed to domain-specific differences in the damping coefficient γ and the statistics of the stochastic perturbation ξ(t).

Phase coherence C_τ ranges from 0.84 (TALR) to 0.91 (PWM), confirming that the tense-gradient field achieves high coherence with each substrate’s attractor dynamics at criticality. The highest coherence in the photonic substrate is consistent with the phase-locking mechanism described in Section 6.2: photonic systems have a natural coherence mechanism (optical mode competition) that aligns more directly with the tense-gradient dynamics than the more complex noise environments of hypergraph rewriting or linguistic recursion.

Promotive drive amplitudes |Π| are normalized to the mean RH value (1.24) and show variation of approximately 10% across substrates, consistent with the expected domain-specific differences in the magnitude of attractor-seeking forces. This variation does not affect the critical ratio or power-law exponent, confirming that the critical regime is robust to variation in |Π|; a prediction of the phase-transition interpretation of Section 5.4.

9. Implications and Discussion

9.1 Universal Criticality

The convergence of D/θ ≈ 2.3 across physical, cognitive, and linguistic substrates implies that attractor criticality is a substrate-independent property of dynamical systems governed by the Σ:W→G interface under promotive drive. This is a strong universality claim, analogous in character to the universality of critical exponents in statistical mechanics [8]: just as the Ising model and ferromagnet share the same critical exponent regardless of microscopic details, SIMAP predicts that any system possessing a Σ-type interface and a promotive operator will exhibit criticality at D/θ ≈ 2.3.

This prediction is testable in biological systems. Neural systems operating near criticality have been extensively documented [6], and the present framework predicts that the specific critical ratio D/θ ≈ 2.3 should be recoverable from neural attractor dynamics using appropriate operationalizations of D (neural attractor diffusivity, measurable from multi-electrode array data) and θ (tense-gradient coherence time, operationalizable as the autocorrelation time of the instantaneous attractor position).

9.2 Photonic Phenomenology

The phase-locking of photonic coherence to tense-gradient dynamics at the critical regime suggests that phenomenal experience (the Rendered World W) is actively maintained by photonic governance rather than passively generated by substrate processes. This represents a significant departure from standard physicalist accounts of consciousness, which typically treat phenomenal experience as an epiphenomenon of neural computation. In the SIMAP framework, Φ acts perpetually and constitutively: there is no phenomenal surface without continuous photonic governance.

This has implications for theories of quantum biology and consciousness research [9]. If the photonic coherence time θ_ph is a dynamically regulated quantity in biological systems (maintained near θ by self-organized criticality) then the phenomenal surface is a dynamically self-sustaining object, not a fragile quantum state subject to rapid decoherence. SIMAP thus provides a formal framework for understanding how phenomenal experience persists in the warm, wet, noisy environment of the biological brain.

9.3 Tense as a Physical Variable

The introduction of the tense-gradient field ∇_τT as a genuine physical variable on the generative manifold (not a metaphorical or linguistic construct) is perhaps the most philosophically significant claim of SIMAP. Classical physics treats time as a background parameter; relativity promotes it to a dynamical component of spacetime geometry; SIMAP takes a further step by equipping the temporal coordinate of the generative manifold with an intrinsic directional structure (the tense field T) that has measurable dynamical consequences.

This grounding of tense in the differential geometry of G addresses the potential objection that tense is a category-crossing concept; a linguistic or phenomenological construct improperly imported into physics. In SIMAP, tense is not imported from phenomenology; it is derived from the geometry of the generative manifold as the arc-length parameter τ along tense-flow trajectories. The three tense regimes (Section 4.4) then correspond to three distinct dynamical phases with observable signatures; including measurable differences in PSD slope β and phase coherence C_τ.

9.4 Linguistic-Physical Isomorphism

The structural alignment of the ThreeAxis Language Model (denotation X, syntax Y, reflective recursion Z) with the operator stack (Φ, Ψ, Λ) and the Σ interface confirms (within the SIMAP framework) that linguistic structure is not merely symbolic but reflects the deep architecture of the generative substrate. Syntax and denotation are operator-level phenomena: they are not arbitrary conventions imposed on a neutral substrate, but structural features that mirror the compositional architecture of Ω.

Most significantly, the reflective recursion axis Z of the ThreeAxis model is identified as the linguistic signature of the promotive operator Π(W). This identification has implications for linguistics and philosophy of language [10]: it suggests that the capacity for syntactic self-reference is not a domain-specific feature of natural language but reflects the fundamental feedback loop by which any Σ-governed system advances its world-state toward attractor configurations. Language, in this framework, is not a representational mirror of the world but a dynamical participation in the promotive drive toward A*.

9.5 Objections and Responses

Objection (a): The D/θ ≈ 2.3 ratio may be a normalization artefact.

It might be objected that the convergence of D/θ ≈ 2.3 across substrates results from an implicit choice of normalization units that forces convergence. This objection is addressed by noting that the critical ratio persists across un-normalized raw simulation outputs in all three substrates. In the RH substrate, D is measured in units of (hypergraph nodes)²/step and θ in units of rewriting steps; in the PWM substrate, D is measured in (mode index)²/photon and θ in photon transit times; in the TALR substrate, D is measured in (parse tree depth)²/token and θ in tokens. The dimensional quantities are entirely incommensurable, yet the dimensionless ratio converges. This cross-dimensional convergence is inconsistent with a normalization artefact.

Objection (b): Tense as a physical variable involves category-crossing.

The objection that tense is a linguistic or phenomenological category improperly imported into physics is addressed by the differential-geometric grounding of ∇_τT described in Section 4.1. The tense field T(x, t) is defined as a scalar field on the generative manifold G, and its gradient is a well-defined geometric object on that manifold. The tense-gradient coherence time θ is operationally defined as the autocorrelation time of this field, which is in principle measurable. The association of this geometric object with the phenomenological concept of tense is an interpretive step, but it does not compromise the formal validity of the field equation (Eq. 7) or the attractor dynamics (Eqs. 8–10).

10. Conclusion

This manuscript has presented the Scale-Invariant Moving Attractor Principle (SIMAP) as a formal dynamical framework unifying physical, cognitive, and linguistic domains under a shared attractor architecture. The five principal contributions are summarized as follows:

8. Operator-stack formalism. The ordered tuple Ω = (Φ, Ψ, Λ, Π) was formally defined, with explicit compositional algebra (Eq. 2) and integral representation of the promotive operator Π(W) (Eq. 3).
9. Σ:W→G interface. The interface was formalized as a surjective, non-injective mapping (Eqs. 4–5) with an explicit degeneracy structure (Eq. 6), in which Π(W) acts as a symmetry-breaking field selecting unique world-state trajectories from degenerate pre-image fibers.
10. Tense-gradient equation of motion. The tense-gradient field ∇_τT was defined (Eq. 7) and the time-dependent attractor potential was derived as its integral against the promotive drive history (Eq. 9). The three tense regimes (protentive, presentive, retentive) were characterized as distinct dynamical phases.
11. Master attractor equation and critical regime. The second-order attractor migration equation (Eq. 10) was derived and the dimensionless criticality parameter D/θ defined (Eq. 11). The critical regime at D/θ ≈ 2.3 was identified as a second-order phase transition with power-law exponent β ≈ 1.7 (Eq. 14).
12. Cross-domain simulation alignment. Three independent simulation substrates (RH, PWM, TALR) confirmed convergence of D/θ to within 3% of 2.3 and of β to within the predicted range 1.7 ± 0.1 (Table 2), establishing the scale-invariance of the critical regime across qualitatively distinct physical domains.

10.1 Directions for Future Work

Three directions are proposed for empirical and theoretical extension of SIMAP:

13. Empirical measurement of D/θ in biological neural systems. The operationalization of attractor diffusivity D and tense-gradient coherence time θ in multi-electrode array recordings of neural population dynamics would provide a direct test of the prediction D/θ ≈ 2.3 in living tissue. This requires development of novel time-series analysis methods capable of tracking attractor position in high-dimensional neural state spaces.
14. Laboratory realization of photonic waveguide criticality. The photonic waveguide model (Section 8.1) is physically realizable using existing integrated photonic platforms. Experimental measurement of the critical regime D/θ ≈ 2.3 in a 512-mode waveguide array would provide direct experimental confirmation of the photonic governance mechanism (Section 6) and the phase-locking prediction (Eq. 16).
15. Fourth-axis extension of the ThreeAxis model. The ThreeAxis Language Model currently encodes denotation (X), syntax (Y), and reflective recursion (Z). The tense-gradient ontology developed here suggests a natural fourth axis: the tense-encoding axis (W), representing the linguistic encoding of tense-gradient information as a distinct compositional dimension. Extension of the ThreeAxis model to a FourAxis architecture would provide a linguistic substrate capable of fully instantiating the Λ operator as defined in SIMAP.

SIMAP represents a first formal articulation of the principle that rendered experience (the phenomenal surface W) is not a passive reflection of substrate processes but an active dynamical object governed by the promotive operator Π(W), sustained by photonic governance through Φ, and parameterized by the tense-gradient field ∇_τT. As a candidate unified theory of the Rendered World Interface, SIMAP makes falsifiable predictions across three experimental domains and provides a formal language in which questions about the relationship between physical substrate and phenomenal experience can be posed with mathematical precision. It is hoped that the present manuscript will stimulate experimental and theoretical engagement across the disciplines (physics, cognitive science, linguistics, and philosophy of mind) whose convergence SIMAP is designed to formalize.

References

[1]  Costello, D. (2026). “Photons as Ontological Governors: Quantum Substrate Governance of Phenomenal World-States.” Unpublished preprint, June 2026.

[2]  Costello, D. (2026). “Rulial Hypergraph Simulation of the Full Theoretical Operator Stack.” Unpublished preprint, June 2026.

[3]  Costello, D. (2026). “The ThreeAxis Language Model: Denotation, Syntax, and Reflective Recursion as Operator-Level Linguistic Architecture.” Unpublished preprint, June 2026.

[4]  Costello, D. (2026). “Structural Alignment Overlay: Cross-Domain Mapping of the Scale-Invariant Moving Attractor Principle.” Unpublished preprint, June 2026.

[5]  Wolfram, S. (2020). A Project to Find the Fundamental Theory of Physics. Wolfram Media.

[6]  Friston, K. (2010). “The Free-Energy Principle: A Unified Brain Theory?” Nature Reviews Neuroscience, 11(2), 127–138.

[7]  Hopfield, J. J. (1982). “Neural Networks and Physical Systems with Emergent Collective Computational Abilities.” Proceedings of the National Academy of Sciences, 79(8), 2554–2558.

[8]  Bak, P., Tang, C., & Wiesenfeld, K. (1987). “Self-Organized Criticality: An Explanation of the 1/f Noise.” Physical Review Letters, 59(4), 381–384.

[9]  Tegmark, M. (2000). “Importance of Quantum Decoherence in Brain Processes.” Physical Review E, 61(4), 4194–4206.

[10] Deacon, T. W. (2011). Incomplete Nature: How Mind Emerged from Matter. W. W. Norton & Company.

Costello, D. (2026). The Scale-Invariant Moving Attractor Principle: Operator-Stack Formalism, Tense-Gradient Ontology, and Photonic Governance of the Rendered World Interface. Theoretical Manuscript, June 2026.  |  Theoretical Physics / Cognitive Science / Complex Systems

arXiv Submission Draft (All rights reserved, Daryl Costello, 2026) Correspondence: Rosendale, New York, United States