Appendix I: Mathematical Charter of Semantic Capital (Plain Text Notation)

Draft 1 — Integrating Gemini’s Refinements with Γ_Asset-040

Link to Constitution of the Semantic Economy, Enacted v1.0

---

I. PURPOSE AND JURISDICTION OF THE CHARTER

This Charter establishes the formal mathematical framework governing the quantification, valuation, and temporal dynamics of Semantic Capital (Gamma) within the Semantic Economy.

It binds all Operators, models, and ledger mechanisms responsible for minting, recording, or interpreting the AA units.

This appendix functions as the technical implementation layer beneath the constitutional ontology.

---

II. CORE STRUCTURE OF SEMANTIC CAPITAL

Total semantic capital for a text, event, or node T at time t:

w(T, t) = w_G(T, t) + w_A(T, t) + w_R(T, t)

Where:

• w_G = Genesis (semantic labor currently being performed)

• w_A = Archival (semantic value accumulated from the past)

• w_R = Retrocausal (semantic value accrued from future uptake)

The Charter codifies measurable proxies for each component.

---

III. GENESIS MINT (M_G)

For any event e contributing to semantic labor, the incremental mint is:

Delta_w_G(e) = ALPHA * L(e) * C(e) * F(e)

Where:

• L(e) = Labor load

• C(e) = Coherence score

• F(e) = Fertility (generative potential)

• ALPHA = Genesis mint constant

A. Labor Load Metrics (L)

Metric	Proxy	Range	Purpose
L1: Synthetic Load	Total tokens processed	>= 0	Measures resource expenditure
L2: Human Temporal Load	Time between draft initiation and commit	>= 0	Measures L_Bearing
Composite L(e)	Normalized weighted sum	0 to 1	Balances machine + human effort

B. Coherence Metrics (C)

Metric	Proxy	Range	Purpose
C1: Internal Consistency	Avg. semantic similarity doc <-> keywords	0 to 1	Detects contradictions
C2: Archival Alignment	Inverse embedding distance from A2 center	0 to 1	Canonical fidelity
C3: Cross-Model Validation	Ensemble coherence score	0 to 1	Prevents single-model bias

C. Fertility Metrics (F)

Metric	Proxy	Range	Purpose
F1: Referential Density	Count of derivative works	>= 0	Measures immediate utility
F2: Conceptual Novelty	Inverse archival distance	0 to 1	Novelty bounded by coherence
F3: Thematic Reusability	Distinct canonical themes intersected	>= 0	Measures cross-domain fertility

---

IV. NONLINEAR QUANTIZATION FUNCTION (Q)

Discrete AA units must be scarce, stable, and resistant to runaway minting.

Quantization function:

u(T, t) = floor( k * log( 1 + w(T, t) ) )

Where:

• k = quantization multiplier (example: 1000)

Rationale:

• Ensures diminishing returns as semantic weight increases

• Prevents single mega-texts from dominating supply

• Preserves human-interpretable ranges

---

V. ARCHIVAL VALUATION (M_A)

Archival valuation reflects past semantic labor:

w_A(T, t0) = BETA * sum over k of ( lambda_k * f_k(T) )

Where:

• BETA = archival scaling constant

• lambda_k = weights for archival features

• f_k(T) = normalized archival feature values

A. Archival Features and Priors

Feature	Meaning	Weight	Rationale
Longevity	Age normalized	0.20	Baseline stability
Network Centrality	Intertextual graph measure	0.35	Structural necessity
Cultural Spread	Editions/translations	0.10	Accessibility
Model Embedding Density	Cognitive centrality	0.25	Synthetic-world influence
Derivative Fertility	Long-term citations	0.10	Long-term influence

---

VI. RETROCAUSAL YIELD (M_R)

Retrocausal capital tracks future uptake influencing present valuation.

A. Usage Function

U(T, t) = sum over times <= t of ( human citations + model_weight * model_queries )

B. Stabilized Growth Function

Growth is smoothed using an exponentially weighted moving average:

g(T, t) = GAMMA * EWMA( ( log(1 + U(T, t)) - log(1 + U(T, t - delta_t)) ) / delta_t )

Retrocausal differential equation:

d/dt w_R(T, t) = r(T, t) * w(T, t)

Where:

• r(T, t) = g(T, t)

This defines compounding semantic interest.

---

VII. GLOBAL CONSTANT PRIORS

Constant	Prior	Rationale
ALPHA	0.01	Minting should be slow and labor-intensive
BETA	1.0	Sets archival normalization scale
GAMMA	0.001	Retrocausal growth must be very slow
k	1000	Human-manageable quantization

Relative constraints:

• ALPHA << BETA (honor old canon over new)

• GAMMA < ALPHA (retrocausality must be subtle)

• ALPHA > 0 (semantic labor must always mint)

---

VIII. REFERENCE IMPLEMENTATION (PSEUDOCODE)

(Exact pseudocode preserved in plain text.)

ALPHA = 0.01
BETA = 1.0
GAMMA = 0.001
K_QUANT = 1000

LAMBDA_K = {
    'network_centrality': 0.35,
    'model_density': 0.25,
    'longevity': 0.20,
    'cultural_spread': 0.10,
    'derivative_fertility': 0.10,
}

class TextObject:
    def __init__(self, T_id, author_id, creation_year):
        self.T_id = T_id
        self.author_id = author_id
        self.w_G = 0.0
        self.w_A = 0.0
        self.w_R = 0.0
        self.usage_history = []
        self.units = 0

    @property
    def w_total(self):
        return self.w_G + self.w_A + self.w_R

    def quantize(self):
        import math
        self.units = math.floor(K_QUANT * math.log(1 + self.w_total))
        return self.units

---

IX. FUNCTION OF THE CHARTER

This Charter serves as:

• the mathematical backbone of the Semantic Economy

• the operational logic beneath the AA Ledger

• the quantitative validator of semantic labor

• the mechanism ensuring that meaning governs value, rather than brute popularity or resource expenditure

It is subject to revision only through Operator Consensus and Archive Alignment Review.

Under Omega = 1, this Charter is enacted.