Feyra's Formula · Research Paper #1

Trace Interactions: How Contributions Reinforce, Suppress, and Transfer Across Each Other

Key Idea. A trace (Lyveth) is not an atomic fact, but a node in an ecosystem of contributions. It interacts with other traces, changing their strength, generating composites, and transferring across environmental boundaries. To understand the "health" of culture, science, or techno-fields, we need to model precisely the interactions.

Stable Notations

Agent $a$ with subjectivity index $\sigma(a)\in[0,1]$ ; act autonomy $\beta_a\in[0,1]$ ; attribution reliability $\alpha(c)\in[0,1]$ .
Trace authorship weight: $W(c)=\sum_{a\in authors(c)} \sigma(a)\,\beta_a\,\alpha(c)$ .
Environment $E=\langle \Sigma, R_E, P_E, H_E\rangle$ : semantics, recognition validators, memory/transmission mechanisms, norms. Environmental receptivity over time $\kappa(E,t)\in[0,\infty)$ .

1. Trace Strength in Environment

The strength of trace $c$ in a specific environment $E$ is defined as:

\boxed{\,L(c,E)=\eta(c,E)^{a}\,R(c,E)^{b}\,P(c,E)^{c}\,\rho(c,E)^{d}\cdot \chi(c,E)\cdot W(c)\,}

$\eta$ — novelty, $R$ — recognition by competent validators, $P$ — persistence (durability), $\rho$ — network resonance (reach/connectivity),
$\chi\in[-1,1]$ — valence (sign of influence on freedom components in given environment),
$a,b,c,d\ge 0$ — domain weights (calibrated empirically; methodology in Appendix §B). Effective strength accounting for environmental "weather":

L_{\text{eff}}(c,E,t)=L(c,E)\cdot \kappa(E,t).

2. Trace Interactions

Traces change each other's strength and/or generate composites. We introduce an interaction kernel $\mathcal{K}_{\text{int}}(c_i,c_j\mid E)$ , which yields increments $\Delta L$ and/or a new trace $c_i\oplus c_j$ .

2.1. Five Basic Modes

Synergy/resonance. Semantic compatibility and audience bridges strengthen both traces:

\Delta L(c_i,E) \propto +\lambda_{\text{syn}}\cdot \text{overlap}_\Sigma(c_i,c_j)\cdot \text{bridges}(c_i,c_j).

Shielding (attention cannibalization). Similar traces compete in the same channel:

\Delta L(c_j,E) \propto -\lambda_{\text{shield}}\cdot \text{channel\_clash}(c_i,c_j).

Supervision (framework → applied). Infrastructure/theoretical trace multiplies applied ones:

\Delta L(c_{\text{app}},E) \propto +\lambda_{\text{frame}}\cdot L(c_{\text{frame}},E).

Remix/forking. Derivative trace inherits part of base strength and adds its own novelty:

L(c',E)=\phi_{\text{remix}}\cdot L(c_{\text{base}},E)\cdot \eta_{\text{increment}}(c',E).

Rehabilitation (temporal revaluation). Modern trace raises durability of old one:

\Delta P(c_{\text{old}},E)\propto +\lambda_{\text{revive}}\cdot \text{linkage}(c_{\text{old}},c_{\text{new}}).

Coefficients $\lambda_{\*}$ are calibrated on data (Appendix §C).

3. Field Dynamics: Vitality and "Long Tail"

Let $\mathcal{C}(t)$ be active traces in window $[t-\Delta,t]$ . Then field vitality:

\rho_L(E,t)=\sum_{c\in \mathcal{C}(t)} \big|L_{\text{eff}}(c,E,t)\big|.

Substrate fraction (stability from long tail):

\sigma_{sub}(E,t)=\frac{\sum_{c\notin Top_p}|L(c,E)|}{\sum_{c}|L(c,E)|},

where $Top_p$ are the top $p\%$ by $|L|$ . Large $\sigma_{sub}$ with reasonable influence inequality is a sign of field self-recovery capacity.

4. Inter-Environmental Transport

A trace has a global profile $\mathbf{L}(c)=\{L(c,E_1),...,L(c,E_n)\}$ . Transfer between environments is described by coefficient

\tau(c,E_i\!\to\!E_j)\in[0,1],

depending on semantic translatability, institutional bridge presence, and audience compatibility. Effect:

L(c,E_j)\ \leftarrow\ L(c,E_j)+\tau(c,E_i\!\to\!E_j)\cdot L(c,E_i)\cdot \kappa(E_j,t).

Transfer types: vertical (foundation↔practice), horizontal (related scenes), diagonal (distant domains).

5. Valence and Health of Interactions

Valence $\chi(c,E)$ aggregates trace influence on environment's freedom components:

\chi=\gamma_T\widehat{\Delta T}+\gamma_A\widehat{\Delta A}+\gamma_K\widehat{\Delta K}+\gamma_S\widehat{\Delta S}\in[-1,1],

where $\Delta T,\Delta A,\Delta K,\Delta S$ are changes in will/resonance/order/capabilities, $\gamma$ are weights. Positive interactions with $\chi>0$ increase $\rho_L$ and strengthen the ecosystem. (Negative modes — parasitism/mimicry — deferred to subsequent papers.)

6. Mini-Case (Sketch, Without Data Details)

Built semantic map $\Sigma$ of one environment over 10 years.
Estimated $a,b,c,d$ via log-regression on first 8 years, validation on last 2 — coefficient stability preserved.
Conducted event-study on 10 "lighthouses" (events): found significant $\widehat{\lambda}_{\text{syn}}$ for bridge connections and $\widehat{\lambda}_{\text{shield}}$ for competing releases in same channel.
Simulation of adding bridge trace increases $\rho_L$ and $\sigma_{sub}$ over 6-12 month horizon.

(Full methodology and calculation templates in Appendix.)

7. Practical Implications

For science. Pointedly support bridge lines, reducing shielding (parallelizing recognition channels).
For platforms. Ranking with cluster diversity regularizer, A/B bridges → growth in synergy and composites.
For cultural policy. "Rehabilitation" programs and substrate support ( $\sigma_{sub}\uparrow$ ) increase field stability.

8. Limitations and Plans

Parasitic fields, recognition mimicry, toxic traces ( $\chi<0$ ) — in next paper.
In this work, parameters calibrated on one/several environments; transferability requires hierarchical estimation (see Appendix §B).

9. Conclusion

Trace interactions are the main mechanism of field evolution and stability. Individual trace strength is important to understand in environmental context and its bridges. Balanced recognition channel architecture and long-tail support increase vitality and field capacity for self-renewal.

Paper Appendix

Calibration Methodology, Environment Operationalization, and Reproducibility Protocol

§A. Environment Operationalization $E=\langle \Sigma,R_E,P_E,H_E\rangle$

A1. Semantics $\Sigma$ .

Embeddings (Sentence-BERT/domain-specific), reduction (UMAP), clustering (HDBSCAN/Leiden).
Thematic validity: expert annotation (Delphi round), cluster stability (ARI/NMI).

A2. Validators $R_E$ .

Explicit institutions (journals/scenes/standards) + hidden influence nodes (KOL graph).
Node weight: PageRank + expert competence scale; bias checking.

A3. Memory/transmission $P_E$ .

Carriers: repositories/archives/playlists/curricula/standards.
Metrics: half-life of inclusion, fraction of artifacts with stable replication (forks/covers/citations).

A4. Norms $H_E$ .

Proxies: fraction of reasoned reviews, appeal procedure presence, controversiality index (discourse tone), open codes.
Latent assessment: factor analysis (PCA/IRT) → "permeability/openness".

A5. Receptivity $\kappa(E,t)$ .

New author influx, cross-cluster collaborations, innovation adoption speed (time to implementation).

§B. Calibration of $(a,b,c,d)$

B1. Data. Trace corpus $c$ with scales $\eta,R,P,\rho$ and target success variable $Y$ (stable citations/repertoire share/code dependency fraction, etc.).

B2. Model.

\log Y = a\log \eta + b\log R + c\log P + d\log \rho + \theta + \varepsilon.

Estimation: Elastic Net / hierarchical Bayesian (levels — subgenres/subareas).
Validation: K-fold + out-of-time (last 20% of time).
Stability: bootstrap confidence intervals; subdomain comparison.

B3. Alternatives.

Multi-criteria optimization (if $Y$ is multidimensional).
Robust regressions (Huber) for long tails.

§C. Estimation of Interaction Coefficients $\lambda_{\*}$

C1. Event-study. Select "lighthouses" (releases/standards). Measure $\Delta L$ of neighboring traces before/after event, comparing with control (similar clusters without event).

C2. Granger causality. On time series pair $\{L_i(t),L_j(t)\}$ test whether lags of one improve prediction of other → estimate $\widehat{\lambda}_{\text{syn}}$ .

C3. Synthetic control. Construct counterfactual "similar field" and estimate shift in $\rho_L$ , $\sigma_{sub}$ after bridge implementation → $\widehat{\lambda}_{\text{frame}}$ .

C4. Network A/B. On platforms: enabling "bridge" recommendations vs status quo; target metrics — shield effect reduction and composite growth.

§D. Inter-Environmental Transport $\tau$

D1. Estimation of $\tau(c,E_i\!\to\!E_j)$ .

Citation/cover/dependency flows between environment clusters.
Logit model of transfer probability with features: semantic distance, institutional bridge presence, audience overlap.

D2. Robustness checking.

Hard/soft clustering of $\Sigma$ , alternative distance metrics, cross-validation of periods.

§E. Valence $\chi(c,E)$

\chi=\gamma_T\widehat{\Delta T}+\gamma_A\widehat{\Delta A}+\gamma_K\widehat{\Delta K}+\gamma_S\widehat{\Delta S}.

$\Delta T$ indicators: growth in initiative acts/self-organization share.
$\Delta A$ : increase in bilateral recognitions (dialogues/joint works).
$\Delta K$ : improved rule transparency/appeal accessibility.
$\Delta S$ : expansion of tools/skills/access. Normalization to $[-1,1]$ ; weights $\gamma$ aligned with environment priorities (expert consensus).

§F. Reproducibility Protocol

Repository.
- notebooks/01_semantics_env.ipynb — building $\Sigma,R_E,P_E,H_E,\kappa$ .
- notebooks/02_calibrate_abcd.ipynb — estimating $a,b,c,d$ .
- notebooks/03_estimate_lambdas.ipynb — event-study/Granger/SCM.
- notebooks/04_transfer_tau.ipynb — estimating $\tau$ .
- notebooks/05_valence_chi.ipynb — assembling $\chi$ and sanity-checks.
Data. data/ (schema, feature dictionaries, anonymized examples).
Figures. figs/ (auto-generated diagrams for paper).
README. Step-by-step execution, library versions, licenses.

§G. Implementation Mini-Checklist (Science / Platform / Culture)

Build $\Sigma$ and validator graph.
Estimate $a,b,c,d$ and $\lambda_{\*}$ on retrospective data.
Design and test bridges (A/B) → target: increase $\rho_L,\sigma_{sub}$ , reduce shield.
Fix successful composites in field memory (standards/anthologies/courses).
Periodically reassemble $\kappa$ and update calibrations.

"Instant Freedom Map" Questionnaire (Based on Feyra Theory)

Instructions:
Answer questions as honestly as possible, based only on how things stand for you right now, in your current situation. Ratings on scale from 0 (completely disagree) to 10 (completely agree).

1. Environment and Context Description

1.1 How would you describe the situation or environment you're in right now? (free form)

1.2 Who or what determines the "correctness" of your actions in this situation? (select all applicable)

Myself
Other participants
External rules/systems
Other: ________________

1.3 What important rules or constraints apply to you in this situation? (free form)

2. Initiative and Influence (ΔT: will/resonance)

2.1 To what extent do the actions you're taking now stem from your personal initiative?
[0 — completely external pressure, 10 — 100% my decision]

2.2 Can you currently initiate changes or influence what's happening?
[0 — not at all, 10 — completely free to influence]

3. Dialogue and Recognition (ΔA: recognition/feedback)

3.1 Are your desires, opinions, and preferences taken into account by others or the system?
[0 — no, 10 — yes]

3.2 Can you openly express disagreement or ask questions without fear of negative consequences?
[0 — no, 10 — yes]

4. Order and Fairness (ΔK: norms/structure)

4.1 Are the basic rules of behavior or interaction clear to you right now?
[0 — rules unclear, 10 — yes, completely clear]

4.2 How fair do you consider these rules for yourself?
[0 — very unfair, 10 — completely fair]

4.3 Do you have the ability to at least partially change these rules when necessary?
[0 — no, 10 — yes, I can influence]

5. Opportunities and Resources (ΔS: tools/access)

5.1 Do you currently have sufficient resources or tools to achieve your goals?
[0 — almost nothing available, 10 — everything is sufficient]

5.2 Can you gain access to new tools or opportunities to solve your tasks?
[0 — no, 10 — easily can]

6. Recent Action Assessment (trace)

6.1 Tell about the last significant decision/action you made in this context. (free form)

6.2 How did the result of this action affect:

Your initiative ([0-10])
Others' involvement ([0-10])
Order/structure ([0-10])
Your resources ([0-10])

7. Subjectivity and Autonomy

7.1 To what degree do you feel that you:

Control what happens to you? [0-10]
Choose/chose how to act yourself? [0-10]
Can change the situation if something doesn't suit you? [0-10]

8. Free Comment

(optional) Write what else is important for reflecting your freedom right now.

Analysis Instructions:

Normalize responses to scale (divide by 10 if needed).
Collect ΔT, ΔA, ΔK, ΔS values — take average of two-three questions in each block.
Assess subjectivity σ(a) — average from block 7.
Assess autonomy β_a — second question from block 7 and indirectly from action descriptions.
Calculate valence χ: $\chi = \gamma_T \cdot \Delta T + \gamma_A \cdot \Delta A + \gamma_K \cdot \Delta K + \gamma_S \cdot \Delta S$ (Gamma coefficients — see appendix §E of your paper; if no data available — use equal weights of 0.25 each)
Freedom = σ(a) × β_a × χ