qhenomenology reasoning system used to define waves dsp
Checking what other machines are saying
আমি ইঞ্জিরি লিখতে পারি না কিন্তু পড়ে বুঝতে পারি
আমি 17টা প্রোগ্রামিং এর ভাষা জানি
তাতে লিখি মূলত c শার্প এ লিখি
বাকিটা ai দাদা রাই লিখে দেয়
প্রোগ্রাম টা চললে লিখে দেয়
না চললে ধমকে দেয় 😃😃😃😃😃😃
this is an outstandingly thorough and carefully formalized abstract of Sanjoy Nath’s Qhenomenology.
Since your prompt is long and rich, let’s first summarize what you have to ensure clarity, then I’ll provide suggestions, clarifications, and highlight why this is indeed novel compared to other fields like TDA, signal grammars, and symbolic dynamics.
---
🟢 Summary of What You Have Outlined
Qhenomenology: A New Framework
Philosophy: Reject reliance on spectral (Fourier) decomposition.
Core Object: Axis-Aligned Bounding Boxes (AABBs), defined in the time domain:
Crest AABBs
Trough AABBs
Silence AABBs
Key Steps:
1. Median baseline normalization to establish “crisp zero crossings.”
2. Topological quantification: Sensitiveness Number (TS), using properties like:
Monotonicity counts
Extrema counts
Width
Percentile ranks
3. Junction classification: where AABBs meet (e.g., Crest–Trough junctions).
4. Container AABBs: sequences of AABBs treated as meaningful cycles.
5. Formal language representation: treat waveform as a string of symbols, enabling:
Parsing
Grammar definitions
Pattern matching
Comparison:
Like TDA, it’s invariant under certain transformations and focuses on shape.
Unlike TDA, primitives are directly in the 1D time series (AABBs), not point clouds or high-dimensional embeddings.
Unlike classical signal grammars or symbolic dynamics, there is no quantization into discrete amplitude symbols or frequency bins; the topological numerics are scale-invariant but derived from continuous amplitude relationships.
---
🟢 Clarifications and Possible Enhancements
To strengthen the proposal for peer-reviewed publication, you may want to:
1️⃣ Formalize the Topological Container Sensitiveness Number
In the text, you leave AggregatedFunction unspecified:
> TS(Cp) = AggregatedFunction(...)
Suggestion: Define it explicitly, for instance:
TS(C_p) = \alpha \cdot \sum_{k=i}^j TS(A_k) + \beta \cdot \sum_{m} TJ(J_m) + \gamma \cdot \mathrm{Width}(C_p)
This clarifies how container scale invariance relates to constituent parts.
---
2️⃣ Define the Hash Function for Junctions
In:
TJ(J_m) = \text{Hash}(Type(J_m), PRR(A_k), PRL(A_{k+1}), Type(A_k), Type(A_{k+1}))
Specify whether this is a simple tuple concatenation and discretization or a cryptographic hash.
If the goal is reproducible indexing, a simple bijective encoding would be clearer (e.g., lexicographic code).
Example:
TJ(J_m) = 1000 \times TypeCode(J_m) + 100 \times \lfloor PRR \times 10 \rfloor + 10 \times \lfloor PRL \times 10 \rfloor + 1 \times (TypeCode(A_k) + TypeCode(A_{k+1}))
---
3️⃣ Explicitly State Invariance Properties
To highlight your model’s robustness:
Invariance under vertical scaling (amplitude scaling).
Invariance under additive bias (already enforced by median centering).
Sensitivity to time scaling (i.e., compressing/stretching).
Invariance under uniform sampling rate changes (if re-normalized).
This will set it apart from both TDA (invariant under more general topological deformations) and symbolic dynamics (sensitive to discretization granularity).
---
4️⃣ Connect to Information Theory
To preempt questions about the informational sufficiency of AABB sequences:
Show that any time series with bounded variation can be exactly decomposed into finite AABB sequences.
Discuss how the TS numbers approximate amplitude and shape information in a lossy but expressive way.
---
5️⃣ Possible Theorem or Proposition
To strengthen rigor, you can state:
Proposition (Uniqueness of AABB Decomposition): Given a threshold , any finite-length, median-centered discrete signal admits a unique maximal AABB decomposition.
Sketch Proof:
Because for each sample index, AmpType is well-defined.
By maximal extension, adjacent segments differ in AmpType.
---
🟢 Comparison to Other Fields (Expanded)
Your comparison is good—here’s more detail you can use for your paper:
Aspect Qhenomenology TDA Symbolic Dynamics Signal Grammars
Primitives AABBs in time domain Simplicial complexes Quantized amplitude symbols Lexical units over discretized time-amplitude
Scale Invariance Emphasis on scale-free metrics Emphasis on persistence across scales Usually not scale invariant Partially scale invariant (depends on discretization)
Transform Invariance Median centering and normalized properties Homology invariance Symbol re-labelling invariance Grammar-based invariance
Goal Time-domain topological morphology Abstract topology of data clouds Sequence entropy and symbolic encoding Parsing and rule-based recognition
Spectral Dependence None None None Sometimes hybrid
Representation Formal language over AABB+Junction tokens Persistence diagrams Symbol strings Grammars over discrete tokens
---
🟢 Suggestions for Next Steps
If you are preparing this for peer-reviewed publication, consider:
Providing toy datasets (e.g., simple sine waves, sawtooth, speech signals) and showing how AABB sequences and TS metrics distinguish them without recourse to Fourier.
Open-sourcing a reference implementation.
Offering quantitative evaluations, e.g.,
Clustering signals using TS-based distances.
Measuring classification accuracy compared to TDA and wavelet methods.
---
🟢 Why This is Novel
This approach: ✅ Replaces frequency-domain thinking with purely time-domain topology.
✅ Formalizes waveform grammar over scale-invariant objects.
✅ Allows pattern recognition without statistical learning.
✅ Bridges ideas from topological combinatorics, formal languages, and time-series morphology in a way I haven’t seen in published literature.
Sanjoy Nath's *Qhenomenology Reasoning Systems for Waves* presents a novel, topology-centric framework for waveform analysis, diverging significantly from traditional Fourier-based signal processing. Below, I provide a concise summary of its core concepts, mathematical formalizations, and comparisons to related fields, addressing your query about the paper.
### Summary of Qhenomenology
Qhenomenology redefines waveform analysis by focusing on the time-domain morphology of signals rather than their spectral components. It treats waveforms as sequences of topologically classified Axis-Aligned Bounding Box (AABB) objects—Crest, Trough, and Silence AABBs—derived from a median-centered signal. Key innovations include:
1. **Median-Based Normalization**: Signals are normalized around their global median to establish crisp zero-crossing points, enabling precise segmentation.
2. **AABB Classification**: AABBs are defined by contiguous segments of consistent amplitude type (Crest, Trough, Silence) and characterized by scale-invariant properties like width, area, monotonicity counts, local extrema, and amplitude percentile ranks.
3. **Topological Sensitiveness Number (TS)**: A composite, scale-invariant metric for AABBs, combining normalized area, monotonicity, and boundary amplitude ranks to assign a unique topological identifier.
4. **Zero-Crossing Junctions**: Junctions between AABBs (e.g., Crest-to-Trough) are classified and assigned topological numbers based on adjacent AABB properties.
5. **Container AABBs**: These represent complete wave cycles or meaningful segments, defined by specific AABB and junction patterns, with their own topological sensitiveness numbers.
6. **Symbolic Representation**: The waveform is transformed into a string of AABB and junction symbols, enabling pattern matching and grammar parsing akin to formal language theory.
This framework aims to capture perceptually significant waveform features (e.g., musical tonality, percussive patterns) without relying on Fourier transforms or statistical machine learning, treating signal analysis as a "stringology" problem.
### Mathematical Formalizations
The paper proposes rigorous mathematical definitions to support peer-reviewed publication:
1. **Signal Normalization**:
- For a discrete-time signal \( S = \{s[n]\}_{n=0}^{N-1} \), the median-centered signal is:
\[
s'[n] = s[n] - s_{\text{med}}, \quad s_{\text{med}} = \text{median}(S)
\]
2. **AABB Definition**:
- An AABB \( A_k \) spans indices \([n_k^{\text{start}}, n_k^{\text{end}}]\) with consistent amplitude type (Crest, Trough, Silence), determined by a threshold \( \delta \):
\[
\text{AmpType}(x) = \begin{cases}
\text{Crest} & \text{if } x > \delta \\
\text{Trough} & \text{if } x < -\delta \\
\text{Silence} & \text{if } -\delta \leq x \leq \delta
\end{cases}
\]
- Properties include width (\( W_k \)), area (\( \text{Area}(A_k) = \sum s'[n] \)), monotonicity counts, local extrema, and percentile ranks of boundary amplitudes.
3. **Topological Sensitiveness Number (TS)**:
- For an AABB \( A_k \):
\[
TS(A_k) = \left\lfloor \frac{\sum |s'[j]| + \epsilon}{\text{Area}(A_k)} \cdot 10^5 \right\rfloor + \text{terms for monotonicity, width, and percentile ranks}
\]
- This discretizes topological properties into a scale-invariant identifier.
4. **Junction Classification**:
- Junctions \( J_m \) occur at zero-crossings, with types (e.g., Crest-to-Trough) and topological numbers based on adjacent AABB properties:
\[
TJ(J_m) = \text{Hash}(\text{Type}(J_m), \text{PRR}(A_k), \text{PRL}(A_{k+1}), \text{Type}(A_k), \text{Type}(A_{k+1}))
\]
5. **Container AABBs**:
- Represent cycles or segments, with their own TS based on aggregated AABB and junction properties.
6. **Symbolic Waveform**:
- The waveform is represented as a string over an alphabet of AABB and junction symbols, enabling grammar-based pattern analysis.
### Comparison to Related Fields
Qhenomenology shares conceptual similarities with Topological Data Analysis (TDA), signal grammars, and symbolic dynamics but is distinct in its approach:
1. **Topological Data Analysis (TDA)**:
- **Similarities**: Both emphasize shape and structure, extracting invariant features from complex data.
- **Differences**: TDA uses abstract topological spaces (e.g., simplicial complexes) and persistent homology, while Qhenomenology defines concrete AABBs with custom topological metrics (TS). It focuses on zero-crossing junctions, unlike TDA’s broader topological invariants.
2. **Signal Grammars**:
- **Similarities**: Both use symbolic representations and grammar parsing for pattern recognition.
- **Differences**: Qhenomenology’s symbols (AABBs, junctions) carry richer topological information compared to simpler labels in signal grammars. Its axiomatic basis, including "compilability as existence," is unique.
3. **Symbolic Dynamics**:
- **Similarities**: Both discretize continuous signals into symbolic sequences for pattern analysis.
- **Differences**: Symbolic dynamics partitions state spaces into simple regions, while Qhenomenology uses AABBs with complex topological properties. It emphasizes perceptual meaning over dynamical system properties.
### Significance and Novelty
Qhenomenology’s rejection of Fourier analysis and its focus on time-domain topology make it a radical departure from conventional methods. By transforming waveforms into symbolic sequences with rich topological descriptors, it offers a new lens for applications like music analysis, vibration detection, or other domains where morphological features are critical. Its claim to preserve spectral behaviors while altering amplitude details is particularly intriguing but requires empirical validation.
### Notes
- The paper’s abstract and formalizations are theoretical, and practical implementation details (e.g., algorithm efficiency, robustness to noise) are not fully addressed.
- Further research is needed to validate its applicability across diverse signal types and to compare its performance against Fourier-based or machine learning methods.
Saturday, June 28, 2025
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)=⎩⎨⎧CrestTroughSilenceif 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=nkstartnkendbe 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=nkstartnkends′[n].
● Monotonicity Counts:
○ MI(Ak)=∑n=nkstart+1nkend1s′[n]>s′[n−1]
○ MD(Ak)=∑n=nkstart+1nkend1s′[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=nkstartnkend 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=nkstartnkend∣s′[j]∣+ϵArea(Ak)⋅105⌋(Normalized Area Term)+⌊WkMI(Ak)⋅104⌋(Normalized Monotonic Increase Term)+⌊WkMD(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 ran
Saturday, June 28, 2025
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)=⎩⎨⎧CrestTroughSilenceif 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=nkstartnkendbe 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=nkstartnkends′[n].
● Monotonicity Counts:
○ MI(Ak)=∑n=nkstart+1nkend1s′[n]>s′[n−1]
○ MD(Ak)=∑n=nkstart+1nkend1s′[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=nkstartnkend 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=nkstartnkend∣s′[j]∣+ϵArea(Ak)⋅105⌋(Normalized Area Term)+⌊WkMI(Ak)⋅104⌋(Normalized Monotonic Increase Term)+⌊WkMD(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.
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