Experiments

This page provides a comprehensive overview of the experiment suite validating the five axioms of Clock-Selected Compression Theory (CSCT). For exact definitions, loss functions, and equations, see the manuscript source in paper/main.tex.

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Experiment	Topic	Axiom Tested
EX1	Single-Channel Waveform Discretization	A1 (Streams)
EX2	Multi-Channel Relational Information	A3 (Multi-Clock)
EX3	Codebook Size (K) Dependency	A2 (Simplex)
EX4	Open vs Closed Regime	A4 (Irreversible Anchor)
EX5	Feature Binding via Common Clock	A3 (Multi-Clock)
EX6	Category Detection (Frozen Codebook)	A2 (Simplex)
EX7	Relational Internal Time	A3 (Multi-Clock)
EX8	Semantic Grounding	A2 (Simplex)
EX9	Syntax Inference	A5 (Barycentric Syntax)

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EX1 — Single-Channel Waveform Discretization

Purpose

EX1 examines the fundamental discretization capability of CSCT at the peripheral processing level. This experiment tests whether SingleGate architecture can achieve stable discrete quantization of single-channel waveforms without requiring complex relational processing.

Hypotheses

• H1 (SingleGate Sufficiency): For single-channel waveforms with self-referential anchoring, SingleGate architecture will achieve low reconstruction error.
• H2 (MultiGate Redundancy): MultiGate architecture will perform worse than SingleGate on this task, despite having greater representational capacity.
• H3 (Waveform Generalization): The performance advantage of SingleGate will hold across diverse waveform types.

Task Design

The task discretizes a single continuous waveform into a sequence of discrete codes while minimizing reconstruction error. Nine waveform types ensure generalization:

Waveform	Description
sine	Pure sinusoid — basic periodic discretization
chirp	Frequency sweep — time-varying frequency adaptation
am	Amplitude-modulated — envelope dynamics
fm	Frequency-modulated — instantaneous frequency changes
ecg	ECG-like with periodic sharp peaks — transient events
saw_bl	Band-limited sawtooth — asymmetric waveforms
composite	Sum of multiple frequencies — multi-scale representation
noisy	Sine + Gaussian noise — noise robustness
burst	Intermittent signal — onset/offset handling

Method

Parameter	Value
Anchor Configuration	Self-referential (y = x)
Codebook Size (K)	8
Transition Penalty (β)	50
Sequence Length (T)	200
Training Steps	500
Seeds	20 per condition

Metrics

• Reconstruction Loss (MSE): $\mathcal{L}_{\text{recon}} = |x - \hat{x}|^2$
• Transition Rate: Fraction of timesteps where discrete code changes
• Unique Codes: Number of distinct codes activated
• Code Entropy: Normalized entropy of code distribution

Results

Aggregate Statistics (180 paired wave×seed trials):

Architecture	Reconstruction Loss	95% CI
SingleGate	0.0381	[0.0364, 0.0399]
MultiGate	0.0902	[0.0858, 0.0946]

Paired difference: ΔRecon = 0.0521 (95% CI [0.0480, 0.0561]), d_z = 1.879

Per-Waveform Results (Click to expand)

| Waveform | Gate | Recon. Loss | Transition Rate | Unique Codes | Code Entropy | |----------|------|-------------|-----------------|--------------|--------------| | am | MultiGate | 0.081 ± 0.015 | 0.205 ± 0.046 | 4.30 ± 1.00 | 0.570 ± 0.115 | | am | SingleGate | 0.045 ± 0.012 | 0.231 ± 0.030 | 3.85 ± 0.73 | 0.571 ± 0.041 | | burst | MultiGate | 0.083 ± 0.021 | 0.201 ± 0.035 | 3.50 ± 0.92 | 0.405 ± 0.098 | | burst | SingleGate | 0.025 ± 0.004 | 0.215 ± 0.022 | 3.85 ± 0.48 | 0.440 ± 0.038 | | chirp | MultiGate | 0.123 ± 0.011 | 0.253 ± 0.040 | 4.80 ± 0.87 | 0.585 ± 0.078 | | chirp | SingleGate | 0.042 ± 0.004 | 0.250 ± 0.010 | 4.80 ± 1.03 | 0.592 ± 0.026 | | composite | MultiGate | 0.083 ± 0.008 | 0.071 ± 0.018 | 4.55 ± 0.92 | 0.527 ± 0.095 | | composite | SingleGate | 0.053 ± 0.009 | 0.099 ± 0.014 | 4.20 ± 0.60 | 0.580 ± 0.044 | | ecg | MultiGate | 0.045 ± 0.030 | 0.109 ± 0.020 | 4.55 ± 0.80 | 0.543 ± 0.100 | | ecg | SingleGate | 0.022 ± 0.006 | 0.074 ± 0.016 | 4.05 ± 1.36 | 0.374 ± 0.056 | | fm | MultiGate | 0.123 ± 0.010 | 0.189 ± 0.040 | 4.85 ± 0.96 | 0.590 ± 0.099 | | fm | SingleGate | 0.043 ± 0.004 | 0.184 ± 0.018 | 5.00 ± 0.77 | 0.616 ± 0.021 | | noisy | MultiGate | 0.072 ± 0.012 | 0.209 ± 0.088 | 4.20 ± 1.08 | 0.511 ± 0.111 | | noisy | SingleGate | 0.033 ± 0.011 | 0.198 ± 0.042 | 3.90 ± 0.70 | 0.549 ± 0.037 | | saw_bl | MultiGate | 0.080 ± 0.012 | 0.107 ± 0.047 | 4.05 ± 1.07 | 0.491 ± 0.100 | | saw_bl | SingleGate | 0.036 ± 0.010 | 0.089 ± 0.012 | 3.55 ± 0.59 | 0.549 ± 0.037 | | sine | MultiGate | 0.122 ± 0.010 | 0.129 ± 0.035 | 4.50 ± 1.07 | 0.594 ± 0.094 | | sine | SingleGate | 0.043 ± 0.004 | 0.119 ± 0.015 | 4.75 ± 0.70 | 0.609 ± 0.026 |

Visual Analysis

Architecture Comparison (Representative Seed):

Left column (SingleGate): Robust phase-locking and stable discretization (red) accurately tracking input (gray). Right column (MultiGate): Fails to capture signal structure, resulting in collapsed representations (blue dashed).

Convergence Dynamics:

Key Findings

Both architectures produce comparable discrete-state statistics (transition rate, code usage, entropy), confirming that Axioms A2–A3 suffice for discretization. However, MultiGate substantially worsens reconstruction (large effect size), demonstrating that routing capacity is unnecessary and detrimental when no relational ambiguity exists.

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EX2 — Multi-Channel Relational Information

Purpose

EX2 validates the necessity of MultiGate architecture for processing multi-channel signals with time-varying relationships. This tests whether independent gating is required when channels exhibit asynchronous dynamics.

Hypotheses

• H1 (Failure of Shared Clock): SingleGate will fail to accurately reconstruct dual-channel signals with varying frequency ratios.
• H2 (Success of Independent Gating): MultiGate can assign distinct update timings to each channel.
• H3 (Relational Encoding): The combination of discrete codes will implicitly encode the frequency ratio k(t).

Task Design

Dual-channel dataset coupled by time-varying frequency ratio k(t):

• Channel 0 (Reference): $x_0(t) = \sin(2\pi f_0 t)$
• Channel 1 (Target): $x_1(t) = \sin(\int_0^t 2\pi f_0 k(\tau) d\tau)$
• Relational Parameter: k(t) ∈ [1.0, 2.0], piecewise constant (4 segments)

Method

Parameter	Value
Anchor Configuration	Reference anchor (y = x₀)
Codebook Size (K)	8
Sequence Length (T)	400
Training Steps	1500
Base Frequency (f₀)	5.0 Hz
Seeds	20 per condition

Results

Architecture	Recon. Loss	Unique Codes	Code Entropy
MultiGate	0.063 ± 0.007	7.85 ± 0.36	0.920 ± 0.042
SingleGate	0.171 ± 0.039	6.85 ± 0.57	0.846 ± 0.051

Paired difference: ΔRecon = −0.108 (95% CI [-0.127, -0.090]), d_z = −2.561

Visual Analysis

Key Findings

Double Dissociation: SingleGate excels at single-channel tasks (EX1) while MultiGate excels at relational tasks (EX2). This validates the functional division: peripheral processing (SingleGate) for stable integration, central processing (MultiGate) for relational abstraction.

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EX3 — K-Dependency Analysis

Purpose

EX3 investigates the geometric nature of discretization. Does CSCT perform “Lebesgue-like” integration (partitioning value/amplitude space) rather than “Riemann-like” integration (partitioning time domain)?

Hypotheses

• H1 (K=2, Sign Detection): Minimal capacity captures signal sign; transitions align with zero-crossings.
• H2 (Lebesgue-like Partitioning): For K>2, system partitions amplitude range into discrete bins.
• H3 (Velocity Constraint): Transitions are discouraged near extrema (where velocity ≈ 0).

Method

Parameter	Value
Architecture	SingleGate
Input	Pure sine wave (f₀ = 5.0 Hz)
Codebook Size (K)	{2, 4, 8, 16}
Sequence Length (T)	300
Training Steps	2000
Seeds	20 per K

Results

K	Recon. Loss	Transitions	Zero-Cross Ratio	Extrema Ratio
2	0.101 ± 0.004	10.2 ± 0.8	1.000	0.000
4	0.041 ± 0.004	21.8 ± 2.6	0.438	0.000
8	0.021 ± 0.005	36.6 ± 3.4	0.449	0.000
16	0.015 ± 0.002	48.0 ± 4.0	0.546	0.000

Visual Analysis

Key Findings

• K=2: 100% of transitions occur at zero-crossings (sign detection)
• All K: 0% of transitions occur at extrema (velocity constraint confirmed)
• Reconstruction improves monotonically with K (capacity-driven refinement)

The system performs value-space partitioning (Lebesgue-like), not time-domain sampling (Riemann-like).

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EX4 — Anchor Role (Noise Floor vs. Drift)

Purpose

EX4 examines the trade-off between short-term accuracy (noise filtering) and long-term stability (drift prevention) in anchored vs. autonomous systems.

Hypotheses

• H1 (Short-term Noise Cost): Closed system shows lower error initially due to noise filtering.
• H2 (Long-term Drift Prevention): Open system shows lower error long-term due to drift prevention.
• H3 (Crossover Point): A crossover T_cross exists, increasing with noise σ.

Task Design

• Target: Sine wave x(t) = sin(t)
• Internal Model Imperfection: 3% frequency error (accumulates drift)
• Anchor: Noisy sawtooth y(t) = saw(t) + N(0, σ²)
• Conditions: σ ∈ {0.05, 0.10, 0.20, 0.30}

Results

Noise (σ)	Short MSE (Closed)	Short MSE (Open)	Long MSE (Closed)	Long MSE (Open)	T_cross
0.05	0.017	0.013	0.460	0.012	87
0.10	0.017	0.046	0.460	0.043	167
0.20	0.017	0.149	0.460	0.147	309
0.30	0.017	0.272	0.460	0.273	435

Visual Analysis

Key Findings

• Short-term: Closed system wins when σ ≥ 0.10 (noise filtering advantage)
• Long-term: Open system wins across all noise levels (drift prevention)
• Crossover: T_cross scales with σ (lower noise → earlier crossover → anchor more valuable)

Irreversible anchors provide long-term stability, not immediate accuracy.

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EX5 — Binding Problem

Purpose

EX5 tests whether a shared anchor clock can preserve relative phase relationships between modules despite independent frequency drifts, solving the classic “binding problem.”

Hypotheses

• H1 (Unbounded Drift without Anchor): Relative phase error grows unbounded without shared reference.
• H2 (Bounded Error with Anchor): Relative phase error remains bounded with shared anchor.

Task Design

• Target Signals: Two sine waves with fixed phase offset (φ = 1.5 rad)
• Drift Condition: ±1.5% frequency error in opposite directions
• Anchor Noise: σ = 0.1 rad

Results

Condition	MSE (Early)	MSE (Late)	Stability Duration	PLV (Late)
Open	0.008	0.009	1.000	0.998
Closed	0.024	3.697	0.054	0.935

Visual Analysis

Key Findings

Binding can emerge from shared temporal reference without direct inter-module communication. The shared anchor acts as “temporal glue,” synchronizing internal clocks to preserve relational structure.

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EX6 — Frozen-Code Category Recognition

Purpose

EX6 tests whether new inference can occur after learning is stopped. A MultiGate model is trained on a subset of Lissajous categories, then the codebook is frozen. Can it detect a withheld category?

Task Design

Shape	Frequency Ratio	Status
ShapeA (Circle)	1:1	Trained
ShapeB (Figure-8)	2:1	Trained
ShapeC (Trefoil)	3:1	Withheld

Method

Parameter	Value
Architecture	MultiGate
Codebook Size (K)	16
Training Steps	2000
Seeds	50

Results

Metric	Trained Shapes	Withheld Shape
MSE	0.050 ± 0.008	0.184 ± 0.035
MSE Ratio	—	3.79× ± 0.91
Detection Success	—	100% (50/50)

JSD (within-trained vs. to-withheld): ΔJSd = −0.024 ± 0.048 (not discriminative)

Visual Analysis

Key Findings

• MSE ratio provides robust OOD detection (100% success, all seeds > 2.0×)
• JSD fails to distinguish withheld from trained categories (codebook reuse phenomenon)
• Ungrounded Symbol Acquisition: Codes are assigned but lack reconstructable meaning outside the convex hull

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EX7 — Relational Internal Time

Purpose

EX7 examines whether internal time (measured as transition rate) scales with external anchor tempo—testing that subjective time is flux-driven, not absolute.

Task Design

Train once on “Standard World” (f₀ = 0.5 Hz), then deploy (frozen weights) to:

• Standard: 0.5 Hz (baseline)
• Fast: 0.5 → 1.5 Hz (3× acceleration)
• Slow: 0.5 → 0.15 Hz (0.3× dilation)
• Null: No input (zero signal)

Results

World	Unique Codes	Transitions	Trans./sec	Dilation Factor
Fast	4.00	90.2 ± 9.9	4.51 ± 0.49	1.60×
Standard	4.00	56.4 ± 11.5	2.82 ± 0.58	1.00×
Slow	4.00	29.3 ± 3.3	1.47 ± 0.17	0.52×
NULL	1.00	0.0	0.00	—

Visual Analysis

Key Findings

• Content invariance: Same 4 codes used across all non-null worlds
• Duration adaptation: Transition rate scales with environmental tempo
• Null halt: Internal time stops when input flux = 0

The system processes “change,” not “time”—a form of cognitive relativity.

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EX8 — Semantic Grounding via Convex Hull

Purpose

EX8 investigates whether semantic grounding requires geometric compatibility within the learned simplex (Axiom A2). Can a withheld primitive be inferred after codebook freezing?

Task Design

Training: {A, B, A+B, A+C, B+C} — Primitive C never shown alone

Test (Frozen): Can the model reconstruct C using only the anchor?

Geometric Conditions:

• IN_HULL: C = αA + (1−α)B (inside convex hull)
• RANDOM: C randomly initialized
• OUT_HULL: C ⊥ A, C ⊥ B (orthogonal = outside hull)

Results

Condition	Withheld Similarity	Success Rate	Unique Code Acquisition
IN_HULL	0.979 ± 0.025	96.7% (29/30)	96.7%
RANDOM	0.682 ± 0.370	53.3% (16/30)	66.7%
OUT_HULL	0.701 ± 0.242	16.7% (5/30)	80.0%

Statistical significance: Kruskal–Wallis H = 42.52, p = 5.85 × 10⁻¹⁰

Key Findings

• Convex hull membership determines grounding: IN_HULL achieves near-perfect recovery
• OUT_HULL demonstrates Ungrounded Symbol Acquisition: 80% acquire unique codes, but only 16.7% succeed in reconstruction
• The code is manipulated, but lacks reconstructable meaning—a mathematical instantiation of the Chinese Room argument

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EX9 — Syntax Inference via Barycentric Interpolation

Purpose

EX9 tests whether syntactic inference can arise once meaning has been discretized. Can a system infer an unseen composition rule under reconstruction-only learning?

Task Design

Training: {A, B, C, A+B, A+C} — Composite B+C withheld

Test (Frozen): Can the model reconstruct B+C?

Results

Condition	Withheld Similarity (B+C)	Success Rate
IN_HULL	0.890 ± 0.138	66.7% (20/30)
RANDOM	0.767 ± 0.253	33.3% (10/30)
OUT_HULL	0.742 ± 0.168	13.3% (4/30)

Statistical significance: Kruskal–Wallis H = 19.13, p = 7.00 × 10⁻⁵

Key Findings

• Syntax emerges as barycentric interpolation: The system learns $f: y_{B+C} \mapsto \frac{1}{2}v_B + \frac{1}{2}v_C$
• Unlike meaning (EX8), syntax benefits from geometric scaffold even when withheld primitive is outside hull
• But performance still degrades in OUT_HULL, confirming syntax remains geometrically constrained

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Summary Table

Exp.	Task	Key Finding	Axiom
EX1	Single-channel discretization	SingleGate suffices for self-anchored simple signals	A1
EX2	Two-channel relation	MultiGate superior for complex relational dynamics	A3
EX3	Codebook size dependency	K-dependency follows Lebesgue-like geometry	A2
EX4	Noise vs. drift	Irreversible anchors provide long-term stability	A4
EX5	Feature binding	Common clock solves binding without concatenation	A3
EX6	Category recognition	MSE ratio detects novel categories (100% success)	A2
EX7	Relational time	Internal tempo emerges from flux dynamics	A3
EX8	Semantic grounding	Meaning requires convex-hull membership (96.7% vs 16.7%)	A2
EX9	Syntax inference	Syntax emerges via barycentric interpolation (66.7% vs 13.3%)	A5

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For full details, see the manuscript: 10.5281/zenodo.18408862