Abstract

Constraints are not arbitrary rules imposed from without; they are necessary invariants arising from the structure of reality itself. Human understanding—encoded through symbolic forms, languages, and mathematical models—attempts to approximate these constraints. However, semantic modeling rests upon an unprovable faith postulate: the existence of an underlying invariant reality, the axiomatic manifold \( T_\text{axiom} \), far exceeding symbolic representations, the semantic manifold \( T_\text{semantic} \).

Here we attempt to briefly formalize the recursion boundary between these layers, presenting a model of survival-tested convergence toward external constraint surfaces without direct access to \( T_\text{axiom} \).

Introduction

Constraints govern persistence. Systems—biological, cognitive, technological, societal—survive by aligning with invariant structures embedded in reality. Yet, human attempts to model these invariants through symbolic systems—language, mathematics, policy—face a fundamental recursion limit: no symbolic representation fully captures the underlying reality.

Recursive constraint modeling provides a framework for pragmatically testing symbolic approximations against external invariants. It distinguishes two domains:

• The axiomatic manifold \( T_\text{axiom} \), representing the invariant structure of reality.
• The semantic manifold \( T_\text{semantic} \), representing the compressed domain of symbolic modeling.

Their relationship is overwhelmingly asymmetric:

\[
\text{Volume}(T_\text{semantic}) \ll \text{Volume}(T_\text{axiom})
\]

Semantic survival hinges on recursive alignment with \( T_\text{axiom} \).

Theoretical Framework

Faith Postulate and Recursion Boundary

“We assume, but cannot prove, that an invariant structure, \( T_\text{axiom} \), exists beyond symbolic construction.”

This postulate defines the recursion boundary: semantic models within \( T_\text{semantic} \) approximate constraint surfaces embedded in \( T_\text{axiom} \), but never fully access them.

Convergence toward invariants is modeled asymptotically:

\[
\lim_{n \to \infty} ||v_n – v^*||_2 = 0
\]

where:

• \( v_n \in \mathbb{R}^d \) represents the semantic state (e.g., model parameters, belief structures) at iteration \( n \),
• \( v^* \) denotes the inferred constraint attractor imposed by \( T_\text{axiom} \).

Survival depends on detecting convergence patterns and recursively rejecting semantic models that fail external constraint tests.

Constraint Surfaces and Failure Modes

CoConstraint Surfaces and Failure Modes

Constraint surfaces exist within \( T_\text{axiom} \) as structural invariants—stable realities that symbolic models attempt to approximate but never fully capture.

Semantic approximations—such as formal physical laws \( (E = mc^2, \nabla \cdot \mathbf{T} = 0) \), aesthetic symmetries, or intuitive resonances—serve as provisional mappings toward these deeper surfaces.

Semantic models survive by recursively converging toward constraint surfaces, without ever directly accessing them.

Failure modes include:

• Non-Convergence: Persistent divergence, formalized as
  \[
  ||v_{n+1} – v_n||_2 > \epsilon
  \]
  indicating that semantic recursion is drifting away from alignment with external constraint surfaces.
• Masked Convergence: Apparent local stability within \( T_\text{semantic} \) that conceals misalignment with the deeper topology of \( T_\text{axiom} \).

Non-symbolic recursion—artistic resonance, intuition, aesthetic coherence—extends survival strategies beyond purely formal articulation. Both symbolic and non-symbolic recursion operate by pragmatically testing survival hypotheses against external resistance, preserving coherence through asymptotic convergence rather than direct access.

Computational Model

Convergence toward external constraint surfaces is modeled asymptotically.

Recursive constraint modeling formalizes survival hypothesis testing by treating each semantic model as a point \( v_n \) in a high-dimensional semantic space \( \mathbb{R}^d \). Recursive updates attempt to converge toward inferred projections of constraint attractors \( v^* \) imposed by \( T_\text{axiom} \).

Convergence fidelity is assessed via:

\[
||v_{n+1} – v_n||_2 < \epsilon
\]

where:

• \( v_n \in \mathbb{R}^d \) is the semantic model’s internal state at iteration \( n \),
• \( v^* \) is the inferred constraint surface derived from persistent external invariants,
• \( \epsilon \) is a threshold defining acceptable convergence tolerance.

Failure Modes:

• Non-Convergence: Persistent divergence \( ||v_{n+1} – v_n||_2 > \epsilon \) over iterations signals semantic instability or drift.
• Masked Convergence: Local minima within \( T_\text{semantic} \) give the appearance of stability while masking deeper misalignment with \( T_\text{axiom} \).

Survival Testing Mechanism:

Survival is pragmatically assessed through external persistence and structural resistance. In other words, semantic models that align more closely with structural invariants exhibit greater survival durability. Recursive failure—divergence or masked convergence—triggers the rejection or revision of semantic approximations.

This pragmatic survival testing refines semantic fidelity without ever claiming ontological access to \( T_\text{axiom} \).

Applications

Recursive constraint modeling ensures survival by detecting semantic drift and refining models against dynamic invariants. Three examples illustrate this process:

Semantic Drift Detection: Navigational Alignment with Dynamic Invariants

Semantic drift occurs when symbolic models misalign with evolving constraint surfaces.

Earth’s magnetic field, for example, acts as a directional invariant, operationalized through compasses, navigation charts, and airport runway identifiers. Historically pragmatic, the magnetic compass serves as a provisional hypothesis, not a permanent truth. The field varies dynamically, with temporal drift and local anomalies along isogonic lines, requiring recursive recalibration to maintain functional alignment.

Aviation systems update magnetic variation data using geomagnetic models (e.g., the International Geomagnetic Reference Field), periodically adjusting navigation parameters and runway designations when angular deviations accumulate beyond operational tolerances. However, no recalibration fully captures the complexity of the underlying constraint surface.

In polar regions, the compass’s reliability degrades due to weak horizontal magnetic field components, rendering low-resolution models \( v_n \in \mathbb{R}^d \) (e.g., compass readings) insufficient for high-precision tasks such as polar flights. Recursive survival demands a shift toward alternative invariants—such as GPS-based positioning or celestial navigation—aligning with deeper structural coherence.

This dynamic mirrors recursive constraint modeling:

\[
\lim_{n \to \infty} ||v_n – v^*||_2 = 0
\]

where \( v_n \) is the symbolic model’s internal state and \( v^* \) represents the evolving external constraint surface.

Functional alignment depends on continuous recalibration, treating even historically effective models as provisional survival hypotheses. Recursive fidelity to dynamic invariants, rather than static adherence, defines operational resilience.

Scientific Falsification: Recursive Model Refinement

Scientific models approximate constraint surfaces within \( T_\text{axiom} \), with experiments serving as survival tests. The Michelson–Morley experiment (1887) exemplifies this dynamic:

The ether hypothesis posited a medium for light propagation. Interferometry tested this by measuring light path differences relative to Earth’s motion, anticipating detectable velocity variations. The null result—differences below \( 10^{-8}c \)—falsified ether-based semantic models, exposing semantic drift.

Recursive refinement followed: special relativity emerged as a closer semantic approximation to constraint surfaces reflecting the constancy of light speed.

This process embodies recursive fidelity: semantic hypotheses \( v_n \) (e.g., ether theory) are tested against structural invariants \( v^* \) (e.g., speed of light consistency). When semantic recursion fails to converge—formally, when

\[
||v_{n+1} – v_n||_2 > \epsilon
\]

persists—revision is triggered. Survival requires continuous realignment with deeper invariants, not static claims to ontological certainty.

Non-Symbolic Recursion: Artistic and Intuitive Convergence

Non-symbolic recursion approximates constraint surfaces through experiential resonance rather than formal symbolization. Examples include:

• Music: Harmonic structures converge toward cognitive invariants of consonance and resolution.
• Visual Art: Iterative refinement of patterns aligns compositions with aesthetic invariants such as symmetry or golden ratios.
• Intuition: Non-verbal insights enable decision-making to cohere with structural stability beyond immediate articulation.

These processes capture invariants beyond linguistic or mathematical representation. Survival testing in non-symbolic domains proceeds through persistent experiential resonance—structures that endure against perceptual and environmental drift suggest convergence toward external constraint surfaces.

Thus, recursive constraint modeling extends across \( T_\text{axiom} \), encompassing both symbolic and non-symbolic approximations of structural invariance.

Conclusion

Recursive constraint modeling ensures survival through pragmatic convergence toward external invariants. Semantic models operate within the bounded domain of \( T_\text{semantic} \), recursively testing their approximations against constraint surfaces embedded in \( T_\text{axiom} \).

Survival depends not on absolute truth but on preserving coherence under structural resistance. Drift detection, recursive depth refinement, and fidelity assessment are necessary epistemic safeguards within an incompletely knowable topology.

In this framework, recursive constraint modeling bridges philosophy, physics, governance, and cognition—grounding all survival in coherent alignment with an assumed, but ultimately unprovable, reality that exceeds the reach of formal systems.