Qhenomenology Reasoning Systems for Waves paper

 

Sanjoy Nath's Qhenomenology  reasoning for Waves Topology presents a radical departure from conventional signal processing, proposing a novel, time-domain, and topology-centric framework for wave analysis. This abstract outlines its core tenets, suggests mathematical formalizations, and compares it to related fields like Topological Data Analysis, signal grammars, and symbolic dynamics.

Research Abstract: Qhenomenology: A Topological-Combinatorial Framework for Waveform Analysis

Abstract: Conventional digital signal processing, heavily reliant on Fourier analysis, often focuses on spectral content, potentially obscuring time-domain morphological features critical for certain information extraction tasks. This paper introduces Qhenomenology, a novel axiomatic framework for waveform analysis that fundamentally redefines signal interpretation. Diverging from sinusoidal decomposition, Qhenomenology treats waveforms as ordered sequences of topologically classified Axis-Aligned Bounding Box (AABB) objects: "Crest AABBs," "Trough AABBs," and "Silence AABBs."

The core innovation lies in: (1) a precise methodology for normalizing signals via a median-centered baseline, yielding "crisp zero-crossing points"; (2) the extraction and intrinsic topological numbering of individual AABBs based on scale-invariant properties (e.g., local extrema counts, monotonic segment lengths, and percentile ranks of boundary amplitudes), encapsulated in a "Sensitiveness Number"; (3) a novel classification and topological numbering system for zero-crossing "junctions" (e.g., Crest-to-Trough, Trough-to-Silence), incorporating neighborhood rank information; and (4) the identification and topological categorization of "Container AABBs" representing complete wave cycles or meaningful segments, derived from specific combinatorial patterns of constituent AABBs and junctions. Qhenomenology asserts that information critical for perception (e.g., musical tonality, percussions) resides in these topological-geometric arrangements rather than exact amplitude values or harmonic superpositions. This framework transforms time-series analysis into a "stringology" or "grammar parsing" problem, enabling pattern matching and algebraic reasoning on sequences of topologically numbered symbols. This approach promises to uncover distinct waveform features essential for applications where conventional methods fall short, without recourse to Fourier transforms or statistical machine learning for core classification.

Mathematical Formalizations for Peer-Reviewed Publication

To formalize Sanjoy Nath's Qhenomenology for peer-reviewed publication, a precise mathematical language is essential. Here are proposed formalizations:

1. Signal Representation and Baseline Normalization

Let S={s[n]}n=0N−1​ be a discrete-time signal of length N, where s[n] is the amplitude at sample index n.

Axiom (Median Baseline): The fundamental zero-amplitude reference line is established by the global median of the signal.

Definition 1.1 (Median-Centered Signal):

Let smed​=median(S). The normalized signal S′={s′[n]}n=0N−1​ is defined as:

 

s′[n]=s[n]−smed​

2. AABB Object Definition and Properties

Definition 2.1 (Amplitude Type Function): Given a silence threshold δ>0:

 

AmpType(x)=⎩⎨⎧​CrestTroughSilence​if x>δif x<−δif −δ≤x≤δ​

Definition 2.2 (AABB Object): An AABB object Ak​ is a contiguous segment of the normalized signal S′ defined by indices [nkstart​,nkend​] such that:

1.    All samples s′[n] for nkstart​≤n≤nkend​ have the same AmpType(s′[n]) (ignoring boundary transitions for definition of the AABB itself).

2.    It is maximally extended, meaning s′[nkstart​−1] (if exists) and s′[nkend​+1] (if exists) have a different AmpType than samples within Ak​.

3.    Each Ak​ is assigned its Type(Ak​)∈{Crest, Trough, Silence}.

Definition 2.3 (AABB Properties for Ak​):

Let Ak​={s′[n]}n=nkstart​nkend​​ be an AABB with Wk​=nkend​−nkstart​+1 samples.

●     Width: Wk​=nkend​−nkstart​+1 (in samples). Can be converted to microseconds: Wkµs​=Wk​/SampleRate×106.

●     Area Under Curve: Area(Ak​)=∑n=nkstart​nkend​​s′[n].

●     Monotonicity Counts:

○     MI​(Ak​)=∑n=nkstart​+1nkend​​1s′[n]>s′[n−1]​

○     MD​(Ak​)=∑n=nkstart​+1nkend​​1s′[n]<s′[n−1]​

○     where 1condition​ is the indicator function (1 if condition is true, 0 otherwise).

●     Local Extrema Counts: Lmin​(Ak​), Lmax​(Ak​) (number of local minima/maxima within Ak​).

●     Absolute Amplitude Percentile Ranks:
Let Sabs​(Ak​)={∣s′[n]∣}n=nkstart​nkend​​ be the set of absolute amplitudes within Ak​. Let Sabssorted​(Ak​) be this set sorted in ascending order.

○     PRL​(Ak​)=percentile_rank(∣s′[nkstart​]∣,Sabssorted​(Ak​)).

○     PRR​(Ak​)=percentile_rank(∣s′[nkend​]∣,Sabssorted​(Ak​)).
(The percentile_rank(value, sorted_list) function returns the proportion of values in sorted_list less than or equal to value).

3. Topological Sensitiveness Number for AABBs

Definition 3.1 (AABB Topological Sensitiveness Number, TS​(Ak​)):

This metric quantifies the topological "shape" of an AABB, designed to be largely scale-invariant and to "forget" exact amplitude details.

 

TS​(Ak​)=⌊∑j=nkstart​nkend​​∣s′[j]∣+ϵArea(Ak​)​⋅105⌋(Normalized Area Term)+⌊Wk​MI​(Ak​)​⋅104⌋(Normalized Monotonic Increase Term)+⌊Wk​MD​(Ak​)​⋅103⌋(Normalized Monotonic Decrease Term)+PRL​(Ak​)⋅102(Leftmost Sample Percentile Rank Term)+PRR​(Ak​)⋅101(Rightmost Sample Percentile Rank Term)+103Wk​​(Normalized Width Term)

 

where ϵ is a small positive constant to prevent division by zero, and floor ⌊⋅⌋ operations discretize the contributions. (Note: The TotalArea_ofThisAABB in the original text is interpreted as sum of absolute samples for normalization, making the area term a ratio reflecting general amplitude distribution, insensitive to scale).

Definition 3.2 (AABB Global Rank): AABB objects are ranked globally based on their TS​(Ak​) values.

Rank(Ak​)=position of Ak​ when all Aj​ are sorted by TS​(Aj​).

Definition 3.3 (AABB Scale Factors): For each topological class (AABBs with the same Rank(Ak​)), identify the widest AABB, Awidest​.

●     SFX​(Ak​)=Wk​/Wwidest​

●     SFY​(Ak​)=(MaxAmp(Ak​)−MinAmp(Ak​))/(MaxAmp(Awidest​)−MinAmp(Awidest​))

4. Zero-Crossing Junction Classification

Definition 4.1 (Zero-Crossing Junction): A zero-crossing junction Jm​ occurs at sample index m where AmpType(s′[m−1])=AmpType(s′[m]) (or equivalent, between AABB Ak​ and Ak+1​).

Definition 4.2 (Junction Type, Type(Jm​)): Categorized based on the types of adjacent AABBs:

Type(Jm​)∈{CT, TC, TT, CC, SS, ST, TS, SC, CS, Undefined}

Definition 4.3 (Junction Topological Number, TJ​(Jm​)): A unique, scale-invariant identifier for a junction based on its type and the topological properties of its neighboring AABBs.

 

TJ​(Jm​)=Hash(Type(Jm​),PRR​(Ak​),PRL​(Ak+1​),Type(Ak​),Type(Ak+1​))

 

This hash captures the local topological context of the crossing.

5. Container AABB (Cycle) Definition and Classification

Definition 5.1 (Container AABB, Cp​): A Container AABB Cp​ is a sequence of contiguous AABB objects {Ai​,Ai+1​,…,Aj​} that represents a "complete cycle" or a perceptually meaningful segment of the waveform. In this framework, such a container is typically delimited by specific topological junction types (e.g., a cycle starting with a TC junction and ending just before the next TC junction of a specific type).

Definition 5.2 (Container AABB Topological Sensitiveness Number, TS​(Cp​)): A composite metric reflecting the topological "essence" of the entire container, derived from its aggregated internal properties.

 

TS​(Cp​)=AggregatedFunction(Count(Ak​∈Cp​),Count(Jm​∈Cp​),Sum(Lmin​(Ak​)),Sum(Lmax​(Ak​)),…)

 

This function needs to be explicitly defined, but it would combine the scale-invariant properties of its contained AABBs and junctions.

Definition 5.3 (Container AABB Global Rank): Containers are ranked globally based on their TS​(Cp​) values.

Definition 5.4 (Container AABB Scale Factors): Similar to AABBs, but for containers. Identify the widest container Cwidest​ within the same topological class (same Rank(Cp​)).

●     SFX​(Cp​)=Width(Cp​)/Width(Cwidest​)

●     SFY​(Cp​)=(MaxAmp(Cp​)−MinAmp(Cp​))/(MaxAmp(Cwidest​)−MinAmp(Cwidest​)) (where MaxAmp/MinAmp of container means global max/min of samples within it).

6. Waveform as a Formal Language

Proposition 6.1 (Symbolic Waveform Representation): The timeline of a wave can be represented as a formal string of symbols Σ∗, where Σ is the alphabet of topologically numbered AABB objects and zero-crossing junctions.

Example: Σ={(AType​,TS​(A)),(JType​,TJ​(J))}.

A waveform might be represented as: (ACrest​,TS​(A1​))(JCT​,TJ​(J1​))(ATrough​,TS​(A2​))(JTC​,TJ​(J2​))…

Proposition 6.2 (Grammar and Pattern Matching): Specific wave "patterns" (e.g., musical phrases, distinct vibrations) can be recognized and categorized by defining a formal grammar G over Σ∗. Regular expressions or more complex parsing algorithms can be applied to identify occurrences of these patterns, allowing for "compilability checking" of wave structures.

Comparison to Specific TDA, Signal Grammar, or Symbolic Dynamics Models

Sanjoy Nath's Qhenomenology is unique due to its foundational axioms and explicit rejection of Fourier. However, it shares conceptual commonalities with several advanced analytical paradigms:

1. Comparison to Topological Data Analysis (TDA)

●     Similarities:

○     Focus on Shape/Structure: Both TDA and Qhenomenology prioritize the underlying shape and structure of data over precise numerical values. Both seek "invariant properties."

○     Abstraction and Feature Extraction: Both aim to extract meaningful, low-dimensional features that describe complex high-dimensional data.

●     Differences:

○     Fundamental Primitives: TDA typically constructs abstract topological spaces (e.g., simplicial complexes, Vietoris-Rips complexes) from point clouds or metric spaces. Features are then derived using algebraic topology (e.g., Betti numbers from persistent homology, which count "holes" of different dimensions). Qhenomenology, conversely, uses concrete, geometric AABBs directly on the time series as its fundamental building blocks. Its "topology" is defined by specific arithmetic combinations of derived AABB properties rather than abstract homological invariants.

○     Metric Definition: TDA relies on metric spaces to define proximity and build complexes. Qhenomenology's "Sensitiveness Number" is a custom, composite metric for AABBs themselves, not directly for a point cloud representation of the wave.

○     Zero-Crossing Emphasis: While TDA can reveal features related to "loops" or "cycles" in data, it does not have an inherent mechanism for "zero-crossing point classification" with specific semantic labels like CT/TC based on signal sign changes as a primary topological feature.

2. Comparison to Signal Grammar and Formal Languages

●     Similarities:

○     Symbolic Representation: Qhenomenology's transformation of the continuous waveform into a "strict queue of symbols" (AABBs and classified junctions) aligns perfectly with the core idea of symbolic representation in signal processing and formal languages.

○     Pattern Recognition: The use of "regular expressions" and "grammar parsing" to identify patterns in these symbolic sequences is a direct application of formal language theory.

○     Compositionality: Both allow complex signals/messages to be understood as compositions of simpler, meaningful units according to defined rules.

●     Differences:

○     Symbol Granularity and Richness: In typical signal grammars (e.g., for speech, music), symbols are often derived from simpler features like amplitude thresholds, frequency bands, or short-time spectral characteristics. Qhenomenology's "symbols" (AABBs and Junctions) are inherently richer, carrying internal topological "sensitiveness numbers" and other complex morphological properties that are scale-invariant. This provides a much deeper semantic content for each symbol than a simple categorical label.

○     Axiomatic Basis: The "compilability as existence" axiom and the derivation of "dictionary order" from class dependencies are unique meta-level philosophical constructs not found in standard signal grammar frameworks.

3. Comparison to Symbolic Dynamics Models

●     Similarities:

○     Discretization and Sequence Analysis: Both symbolic dynamics and Qhenomenology convert continuous-time dynamics (or signals) into discrete sequences of symbols for analysis.

○     Focus on Patterns and Complexity: Both fields aim to understand the underlying patterns, structure, and complexity within these symbolic sequences.

●     Differences:

○     State Space Partitioning: In symbolic dynamics, the continuous state space of a dynamical system is partitioned into a finite number of regions, and each time the system enters a region, a corresponding symbol is emitted. Qhenomenology's "partitioning" is explicitly based on the segments of the waveform determined by zero-crossings and silence, creating the AABB objects directly from the time domain signal.

○     Symbol Definition: While symbolic dynamics symbols are typically simple labels (e.g., '0' or '1'), Qhenomenology's AABB symbols encapsulate rich, multi-dimensional topological information (the sensitiveness number and other derived properties).

○     Reconstruction and Meaning: Symbolic dynamics focuses on properties of the symbolic sequence that reflect the original dynamics. Qhenomenology explicitly claims that altering amplitude details while preserving CG geometry can generate "same spectral behaviors," suggesting a unique perspective on information and reconstruction.

In conclusion, while Qhenomenology shares a family resemblance with time-domain morphological analysis, symbolic representation, and topological thinking in data analysis, its specific axiomatic foundation, the definition of AABB primitives with their unique topological metrics, and the ambitious claim of replacing Fourier analysis make it a distinct and novel theoretical framework. Further research would be needed to rigorously validate its claims and explore its full potential across various signal domains.