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Abstract #

This companion reference consolidates every mathematical variable, notation, and formula used across the FLAI and GROMUS research articles published on Stabilarity Research Hub. Researchers, practitioners, and reviewers who work with both frameworks will find unified definitions here, eliminating the need to cross-reference multiple papers. All definitions are sourced directly from the primary articles [1]^[2], [2]^[3], and [3]^[4].

FLAI Core Variables #

FLAI (Framework for Leveraging AI in Social Media) is an RNN-based trend prediction system with a dynamic Injection Layer. Its mathematical model spans nine structural equations (1.1–1.9) and five validation expressions (3.1–3.5).

Symbol	Full Name	Definition
R_i(n)	Repost Count	Observed repost count for social-network object i on day n; the primary stochastic target time series
dS_i(n)	Days Skipped	Number of days missing between consecutive data collection points for object i at step n; drives gap interpolation
bW_i(0)	Base Weight	Initial weight for object i ∈ [0, 1]; encodes expert-assessed prior “success level” at simulation start (e.g., 0.9 for a 90% popularity artist)
DRF_i(0)	Initial Repost Forecast	Starting value of the daily repost forecast at n = 0 (e.g., 0.1)
DRFE_i(n)	Daily Repost Forecast Error	Relative forecast error from the previous iteration; serves as an adaptive correction coefficient — a standalone controller operating above standard backpropagation
DR_i(n)	Daily Rise	Absolute daily repost change normalised by gap length; prevents spurious spikes after multi-day data gaps
GW_i(n)	Graph Weight	Dynamically generated synapse weight at iteration n; reflects the network’s current prediction confidence; recalibrated per-object and per-timestep using heuristic rules
X_i(n)	Exogenous Variable Set	Set of external signals at time n fed into the Injection Layer: publication time, content category, platform identifier, seasonality indicators, viral event metadata
FR_i(n+1)	Forecasting Repost	Total predicted repost count for the next day; the primary model output
pV	Past Value Flag	Boolean {0, 1}: 1 if prior historical data exists for object i, 0 otherwise; determines initialisation path
f(x)	Sigmoid Activation	Neuron activation function mapping weighted input to probability ∈ (0, 1)
b	Bias Term	Learnable bias in the neuron activation formula
N	Simulation Period	Total length of the forecasting window; constrains maximum permissible gap as dS_i^max = N/2

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FLAI Equations #

Structural Equations (Chapter 1) #

1.1 — Prior data flag; determines whether the model uses historical bootstrapping or cold-start initialisation.

1.2 — Gap interpolation constraint; skipped days must not exceed half the simulation period, beyond which interpolation becomes statistically unreliable.

1.3 — Positivity constraint on the error correction coefficient; prevents degeneracy; enforced by injecting bW_i(0) as a floor value when needed.

1.4 — Core forecasting equation; expresses the Injection Layer concept: the next repost count depends on history and live exogenous context.

1.5 — Sigmoid activation providing probabilistic [0,1] interpretation of neuron output.

1.6 — Weighted neuron input: sum of input signals multiplied by synapse weights plus bias.

1.7 — Daily rise metric; gap-normalised daily repost delta.

1.8 — Days skipped calculation; the day-index difference between consecutive observations.

1.9 — Daily rise forecast; product of the gap-normalised change rate and the current synapse weight.

Validation Equations (Chapter 3) #

3.1 — Adaptive error correction; computes how far the previous forecast deviated from reality, producing the feedback signal for weight recalibration.

3.2 — Forward forecast; adds the predicted daily rise to the current observed count to yield tomorrow’s total repost prediction.

3.3 — Absolute forecast deviation; difference between predicted and actual repost count for error monitoring.

3.4 — Mean Squared Error loss; quadratic penalty used during training and held-out evaluation.

3.5 — Coefficient of determination; proportion of variance in actual reposts explained by model predictions.

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FLAI Validation Metrics #

Metric	Formula	Acceptable	Good	Excellent	FLAI Result
MAPE		< 15%	< 8%	< 5%	4.69%
R²	See Eq. 3.5	> 0.70	> 0.85	> 0.95	0.988
Accuracy (1 − MSE)	See Eq. 3.4	> 85%	> 95%	> 99%	99.76%
Black-swan latency	—	< 48h	< 12h	< 1h	< 1h

Progressive approximation error: Ā₁₋₂₅ = 5.20% → Ā₂₆₋₅₀ = 4.25%, confirming online learning convergence.

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GROMUS Variables #

GROMUS (General Relevance and Output Music Unified System) is a five-network AI architecture for pre-publication music virality prediction operating on raw audio signals.

Symbol	Full Name	Definition
FinalTrendScore	Unified Virality Score	Aggregated output of the Decision Core f₅; weighted linear combination of all sub-scores; determines High/Medium/Low classification
TrendProbability (P_high)	Class Probability	Output of the trend classifier f₁; probability that the track falls in the High virality class
ViralSegmentScore V(t)	Segment Quality Score	Per-segment output of f₂; sigmoid-activated score for each candidate audio window of length L
VibeScore	Vibe Similarity	Maximum cosine similarity between the track’s vibe embedding E_vibe and the set of trending reference embeddings; range [-1, 1], typically [0, 1]
LyricsTrendScore	Lyrical Virality	Output of f₄; encodes meme potential, phonetic patterns, and semantic themes of transcribed lyrics; range [0, 1]
E_vibe	Vibe Embedding	Dense vector in ℝᵈ encoding stylistic, timbral, and energetic character of the audio; produced by contrastive-learning encoder in f₃
Etrendingi	Trending Reference Embeddings	Set of vibe embeddings representing currently trending audio tracks; used as comparison targets for VibeScore computation
w₁, w₂, w₃, w₄, b₅	Aggregation Parameters	Learned weights and bias of the Decision Core f₅; balance contributions of P_high, max(V(t)), VibeScore, and LyricsTrendScore
MFCC	Mel-Frequency Cepstral Coefficients	Compact spectral representation approximating human auditory perception; core input feature for f₁ and f₂
h(·)	Audio Encoder (f₁)	Combined convolutional-recurrent encoder with conditioning injection in the trend classifier
g(·)	Segment Encoder (f₂)	Encoder mapping each audio window X[t:t+L] to a virality score in the segment detector
σ	Sigmoid Activation	Same function as FLAI’s f(x): σ(z) = 1/(1+e^{-z}); used in f₂ segment scoring and f₄ lyrics scoring

GROMUS Core Equations #

Virality class probability distribution from f₁ (High / Medium / Low).

Segment virality score for each window; t* is the optimal segment start time.

Stylistic alignment with the current trend aesthetic.

Lyrical virality score using a fine-tuned transformer encoder on Whisper-transcribed text.

Unified decision: High if ≥ THhigh, Medium if THmedium ≤ score < TH_high, Low otherwise.

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Variable Relationship Diagrams #

graph LR
    subgraph "FLAI Prediction Loop"
        R["R_i(n)\nObserved reposts"] --> DR["DR_i(n)\nDaily rise (Eq 1.7)"]
        DR --> DRF["DRF_i(n)\nForecasted rise (Eq 1.9)"]
        DRF --> FR["FR_i(n+1)\nForward forecast (Eq 3.2)"]
        FR --> DRFE["DRFE_i(n)\nForecast error (Eq 3.1)"]
        DRFE --> GW["GW_i(n)\nGraph weight update"]
        GW --> DRF
        X["X_i(n)\nExogenous signals\n(Injection Layer)"] --> FR
        bW["bW_i(0)\nBase weight"] --> DRFE
        pV["pV\nPrior data flag"] --> R
    end

graph LR
    subgraph "GROMUS Five-Network Architecture"
        AUDIO["Raw Audio\nX"] --> F1["f₁: Trend Classifier\nh(X,A,G) → P_high"]
        AUDIO --> F2["f₂: Segment Detector\ng(X[t:t+L]) → V(t), t*"]
        AUDIO --> F3["f₃: Vibe Encoder\nX → E_vibe → VibeScore"]
        AUDIO --> WHISPER["Whisper ASR\nLyrics"]
        WHISPER --> F4["f₄: Lyrics Analyzer\nBERT(Lyrics) → LyricsTrendScore"]
        F1 --> F5["f₅: Decision Core\nFinalTrendScore = w₁P + w₂maxV + w₃Vibe + w₄Lyrics + b₅"]
        F2 --> F5
        F3 --> F5
        F4 --> F5
        F5 --> OUT["High / Medium / Low\nVirality Classification"]
    end

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Cross-Reference Table #

Variable	FLAI Article [1][2]	Heuristic Rules [2][3]	GROMUS Article [3][4]
R_i(n)	✓ Eq 1.4, 1.7	✓ context	—
dS_i(n)	✓ Eq 1.2, 1.7, 1.8	✓ discussed	—
bW_i(0)	✓ Eq 1.3, 1.4	✓ deep dive	—
DRFE_i(n)	✓ Eq 1.3, 3.1	✓ deep dive	—
DR_i(n)	✓ Eq 1.7	✓ context	—
GW_i(n)	✓ Eq 1.6, 1.9	✓ deep dive	—
X_i(n)	✓ Eq 1.4	✓ context	—
FR_i(n+1)	✓ Eq 3.2	—	—
pV	✓ Eq 1.1	—	—
f(x) / σ	✓ Eq 1.5	—	✓ Eq G.2, G.4
MAPE, MSE, R²	✓ Eq 3.4, 3.5	—	✓ evaluation
FinalTrendScore	—	—	✓ Eq G.5
VibeScore	—	—	✓ Eq G.3
LyricsTrendScore	—	—	✓ Eq G.4
Evibe, Etrending_i	—	—	✓ Eq G.3
MFCC	—	—	✓ f₁, f₂ inputs
h(·), g(·)	—	—	✓ Eq G.1, G.2
w₁–w₄, b₅	—	—	✓ Eq G.5

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Conclusion #

This glossary serves as the authoritative single-page reference for all mathematical notation in the FLAI–GROMUS framework family. For the full derivations, experimental results, and architectural details, refer to the primary articles:

• FLAI full paper — DOI: 10.5281/zenodo.19226414^[2] — RNN architecture, Injection Layer, validation on 2.7M records
• FLAI heuristic rules — DOI: 10.5281/zenodo.19248846^[3] — deep analysis of bW, DRFE, and GW originality vs standard LSTM
• GROMUS full paper — DOI: 10.5281/zenodo.19226416^[4] — five-network pre-publication virality prediction from raw audio

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