Mathematical Foundations

The system's fairness and efficiency are guaranteed by formal mathematical properties.

Recognition Distribution

Priority Weights

Each entity allocates 100% of priority weight among recipients or categories:

∀ Entity E: Σ Priority(E → Others) = 100%

Properties:

• Non-transferable (cannot be bought, sold, or traded)

• Dynamically adjustable

• Self-priority permitted: Priority(E → E) ≥ 0

• Continuous values: Priority(E → X) ∈ [0%, 100%]

Priority Allocation

Example:

Organization A allocates recognition:
- Partner B: 30%
- Partner C: 25%
- Service D: 20%
- Ally E: 15%
- Self: 10%
Total: 100%

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Recognition and Allocation

Recognition Weights

Each entity's recognition determines how they allocate capacity:

Allocation_i ∝ Recognition_i / Σ Recognition

Key Property: Allocation is proportional to recognition share.

Example

Organization A recognizes:
- Partner B: 50%
- Partner C: 30%
- Partner D: 20%

With 100 units capacity:
- Allocation to B: 50 units
- Allocation to C: 30 units  
- Allocation to D: 20 units

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True vs. False Recognition

True Recognition: Recognition of contribution to the realization of priorities that enables the continued realization of priorities (self-sustaining).

False Recognition: Recognition of contribution to the realization of priorities that impairs the continued realization of priorities (self-terminating).

The Causality Chain

GIVEN:

Total Recognition = 100%
True ∩ False = ∅ (mutually exclusive)
Capacity Directed ∝ Recognition Share

IMPLICATIONS:

↑ False Recognition
⟹ ↓ True Recognition (budget constraint)
⟹ ↓ Alignment (α)
⟹ ↑ Capacity to non-beneficial partners
⟹ ↓ Capacity to beneficial partners
⟹ ↓ Goal Achievement
⟹ Immediate incentive to revoke false recognition
⟹ Free-rider loses allocation

Mathematical Formulation:

Let:
  R_true = True recognition allocation
  R_false = False recognition allocation
  R_total = 100%

Then:
  R_total = R_true + R_false
  
If R_false increases:
  R_true = R_total - R_false decreases
  
Since Allocation ∝ Recognition:
  Capacity_beneficial = C × (R_true / R_total) decreases
  Capacity_non-beneficial = C × (R_false / R_total) increases
  
Therefore:
  Goal_Achievement = f(Capacity_beneficial) decreases

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Alignment Mathematics

Alignment (α)

Alignment measures how closely your capacity allocation matches true recognition:

Alignment (α) = Σ_i min(Allocation_i / Capacity, TrueRecognition_i)

Where:

• Capacity = Your total available capacity

• Allocation_i = Capacity you give to partner i

• TrueRecognition_i = Actual proportion of contribution to your goal realization

Properties:

• α ∈ [0, 1]

• α = 1 when Allocation_i / Capacity = TrueRecognition_i for all i (perfect alignment)

• α = 0 when allocations completely mismatch true recognition

Perfect Alignment

Perfect Alignment: Allocation_i / Capacity = TrueRecognition_i for all i

When actual allocation proportions exactly match true recognition proportions,
alignment = 1.

Alignment Velocity (v)

Alignment Velocity measures how fast alignment improves:

Velocity (v) = ΔAlignment / ΔTime

Interpretation:

• v > 0: Getting more aligned (learning, correcting)

• v < 0: Getting less aligned (degrading)

• v = 0: Stable (either perfect or stuck)

Maximizing Alignment Velocity:

Entities are incentivized to maximize v through:

1. Transparency - Real-time visibility into allocations and outcomes

2. Sovereignty - Unilateral power to reallocate instantly

3. Revocability - Instant withdrawal of allocation

4. Discovery - Low-friction mechanisms to find better partners

5. Low Switching Costs - Minimal transaction costs for reallocation

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Allocation Formulas

Proportional Allocation

Allocation proportional to recognition:

Allocation_i = Capacity × (Recognition_i / Σ Recognition)

Constrained Allocation

Subject to capacity and need constraints:

Final_Allocation(Recipient, Provider) = Allocation_i

Subject to:
  Σ_j Allocation_j ≤ Capacity (provider constraint)
  Σ_i Allocation_i ≤ Need (recipient constraint)

Key Property: Allocation preserves recognition proportions while respecting constraints.

Need Update

For next calculation round:

Remaining_Need = max(0, Declared_Need - Total_Received)

where Total_Received = Σ Final_Allocation(All_Providers)

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Formal Properties

Property 1: Recognition Accuracy Incentive

Analysis: Recognition accuracy emerges from mathematical necessity through the true/false recognition framework.

Key observations:

• False recognition displaces true recognition (budget constraint)

• Displaced capacity flows to non-beneficial partners

• Goal achievement decreases

• Immediate incentive to correct recognition

This creates natural selection pressure for accurate recognition without imposed metrics.

Property 2: Proportional Preservation

Property: If you express that Recipient A should receive twice as much as Recipient B (through recognition), the system allocates approximately twice as much capacity to A when feasible given constraints.

Mathematical Statement:

X_ij / X_ik ≈ P_ij / P_ik

The proportional relationships you express are preserved in the final allocation.

Property 3: Least Biased Solution

Property: Among all possible allocations satisfying the constraints, the system selects the one that introduces the least additional bias beyond what entities express.

This is the entropy-maximizing (information-theoretically optimal) solution.

Property 4: Constraint Propagation

Property: When constraints bind (e.g., a recipient reaches capacity), the effects propagate through the network.

Capacity that cannot flow to a full recipient automatically redistributes to other compatible needs according to expressed preferences.

Property 5: Equilibrium Convergence

Property: The system converges to a stable equilibrium where no entity can improve their allocation quality (measured by preference satisfaction) without degrading someone else's.

This is a Pareto-efficient outcome.

Property 6: Determinism

Property: Same network state yields identical allocations.

Formal Statement:

∀ Network States S₁, S₂:
If Recognition(S₁) = Recognition(S₂) AND
   Capacity(S₁) = Capacity(S₂) AND
   Needs(S₁) = Needs(S₂)
Then Allocations(S₁) = Allocations(S₂)

Implication: Multiple independent calculations yield identical results. No randomness, no arbitrary choices.

Recognition Accuracy Incentive

Recognition accuracy emerges from mathematical necessity. Participants define their goals subjectively, but achieving them depends on objective access to resources and contributions.

GIVEN:

Total Recognition = 100%
True ∩ False = ∅
Capacity Directed ∝ Recognition-Share

IMPLICATIONS (The Causality Chain):

↑False Recognition
⟹↓True Recognition
⟹↓Alignment (α)
⟹↑Capacity Directed to non-beneficial
⟹↓Capacity Directed to beneficial
⟹↓Goal Achievement
⟹Immediate incentive to revoke recognition
⟹Free-rider loses allocation

The causality chain: False recognition → Displaced capacity → Worse outcomes → Immediate incentive to correct → Free-rider loses allocation.

Key Implication: The system creates natural incentives for true recognition. Inflation or misattribution of contribution to priority realization only decreases connection to actually beneficial partners. Participants that maintain True Recognition better-align their capacity allocation and achieve better outcomes.