Mathematical Proofs

Formal proofs of the economic and probability models underlying PlayOrbs.

Proof 1: Kill Reward Exact Sum

Claim: The sum of all kill rewards equals the bounty pool.

Given:

• bounty_pool: Total bounty pool
• N: Number of players
• w(A) = 1 + log₂(A): Weight function for A alive players
• W = Σ w(A) for A from 2 to N: Total weight

Kill reward formula:

per_kill(A) = floor(bounty_pool × w(A) / W)

Proof:

1. Without flooring:

   Σ per_kill(A) = Σ (bounty_pool × w(A) / W)
                 = bounty_pool × Σ w(A) / W
                 = bounty_pool × W / W
                 = bounty_pool

2. With flooring, let r(A) = bounty_pool × w(A) / W - floor(bounty_pool × w(A) / W) be the fractional remainder.

3. Total remainder:

   R = Σ r(A) where 0 ≤ r(A) < 1
   0 ≤ R < N-1

4. Since R is an integer (difference between integers), R ∈ {0, 1, ..., N-2}.

5. Implementation distributes remainder to early kills:

   for i in 0..R:
       schedule[i] += 1

6. After adjustment:

   Σ per_kill(A) = Σ floor(bounty_pool × w(A) / W) + R
                 = bounty_pool - R + R
                 = bounty_pool ∎

Proof 2: Emission Decay Convergence

Claim: Total emissions converge to a finite sum regardless of rounds played.

Given:

• E₀ = 10: Initial emission per round
• d = 0.85: Decay rate per epoch
• e = 10: Rounds per epoch

Per-round emission at round r:

E(r) = E₀ × d^(floor(r/e))

Total emissions after R rounds:

T(R) = Σ E(r) for r from 0 to R-1

Proof:

1. Group by epoch n:

   T(R) = Σ (e × E₀ × d^n) for complete epochs
        + (R mod e) × E₀ × d^(floor(R/e))

2. For infinite rounds (upper bound):

   T(∞) = Σ (e × E₀ × d^n) for n from 0 to ∞
        = e × E₀ × Σ d^n
        = e × E₀ × (1 / (1 - d))
        = 10 × 10 × (1 / 0.15)
        = 666.67 PORB per epoch series

3. Therefore:

   T(∞) ≤ 666.67 PORB

4. With per-round probability < 1, actual emissions are lower:

   Expected(∞) < 666.67 PORB ∎

Corollary: The 100M supply cap will never be reached by emissions alone.

Proof 3: Fee Distribution Conservation

Claim: All entry fees are distributed without loss or gain.

Given:

• total_payment: Entry amount
• vault_rate = 0.80
• protocol_rate = 0.20

Proof:

1. Primary split:

   to_vault = floor(total × vault_rate)
   to_fees = total - to_vault

2. Since floor(total × 0.80) + (total - floor(total × 0.80)) = total:

   to_vault + to_fees = total ✓

3. Protocol fee split:

   lp_vault = floor(dev_fee / 2)
   actual_dev = dev_fee - lp_vault
   lp_vault + actual_dev = dev_fee ✓

4. Referral split:

   referral = floor(actual_dev × 0.10)
   final_dev = actual_dev - referral
   referral + final_dev = actual_dev ✓

5. Total conservation:

   to_vault + lp_vault + final_dev + referral
   = to_vault + dev_fee
   = to_vault + to_fees
   = total ∎

Proof 4: Bounty Inheritance Invariant

Claim: Bounty inheritance never creates new value.

Invariant: Σ bounty_earned ≤ bounty_pool for all players at all times.

Proof by induction:

1. Base case (round start):

   Σ bounty_earned = 0 ≤ bounty_pool ✓

2. Inductive step (kill at frame f):

   a. Kill payment from schedule:

   killer.bounty_earned += per_kill(A)
   Σ bounty_earned increases by per_kill(A)

   b. By Proof 1, Σ per_kill ≤ bounty_pool, so:

   Σ bounty_earned ≤ bounty_pool ✓

   c. Inheritance transfer:

   uncashed = victim.bounty_earned - victim.cashed
   killer.bounty_earned += uncashed
   victim.bounty_earned -= uncashed

   d. Net change in sum:

   Δ = +uncashed - uncashed = 0

   e. Therefore:

   Σ bounty_earned (after) = Σ bounty_earned (before)

3. Inheritance only moves value, never creates it. ∎

Proof 5: Hazard Rate Probability

Claim: The emission probability function produces values in [0, 1].

Emission probability formula:

P(emit) = 1 - e^(-λ × activity)

Where:

• λ = 1.0 (hazard rate)
• activity ≥ 0 (cumulative orb-seconds)

Proof:

1. For activity = 0:

   P = 1 - e^(-1 × 0) = 1 - e^0 = 1 - 1 = 0 ✓

2. For activity → ∞:

   P = 1 - e^(-∞) = 1 - 0 = 1 ✓

3. For all activity ≥ 0:

   -λ × activity ≤ 0
   e^(-λ × activity) ∈ (0, 1]
   1 - e^(-λ × activity) ∈ [0, 1) ✓

4. P is monotonically increasing in activity:

   dP/d(activity) = λ × e^(-λ × activity) > 0 ∎

Proof 6: Tie-Break Uniqueness

Claim: The tie-break rule chain always produces exactly one winner.

Given:

• framesAlive: Non-negative integer
• kills: Non-negative integer
• rosterIndex: Unique integer per player

Proof:

1. Compare by framesAlive:

   • If different → unique maximum exists
   • If tied → proceed to next criterion

2. Compare by kills:

   • If different → unique maximum exists
   • If tied → proceed to next criterion

3. Compare by rosterIndex:

   • By construction, each player has unique index
   • Therefore, minimum rosterIndex is unique

4. The comparison chain always terminates at step 3 or earlier with exactly one winner. ∎

Proof 7: Log-Weight Monotonicity

Claim: Kill rewards decrease as fewer players remain.

Weight function:

w(A) = 1 + log₂(A)

Proof:

1. For A₁ > A₂:

   log₂(A₁) > log₂(A₂)  (log is monotonic)
   1 + log₂(A₁) > 1 + log₂(A₂)
   w(A₁) > w(A₂)

2. Since W is constant and bounty_pool is constant:

   per_kill(A₁) = bounty_pool × w(A₁) / W
   per_kill(A₂) = bounty_pool × w(A₂) / W

3. Therefore:

   per_kill(A₁) > per_kill(A₂) when A₁ > A₂ ∎

Summary

Proof	Ensures
Kill Reward Exact Sum	No bounty lost or created
Emission Decay Convergence	Finite token supply
Fee Distribution Conservation	All fees accounted for
Bounty Inheritance Invariant	No value minting
Hazard Rate Probability	Valid probability function
Tie-Break Uniqueness	Deterministic winner
Log-Weight Monotonicity	Early kills pay more

Next Steps

• Constants Reference - System parameters
• Epoch Decay - Decay algorithm
• Kill Rewards - Bounty schedule