Appendix A

Mathematical Appendix

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A.1 — The Energy-Compute Equivalence and the Bitcoin Hurdle Rate

First Principles Setup

Consider a unit of computational capacity—one GPU-hour or one rack-hour—connected to an electrical supply. The operator faces a continuous allocation decision: route the next marginal kilowatt-hour to productive work (inference, training, data processing) or to Bitcoin acquisition (mining directly, leasing hash-rate, or staking through custodial protocols).

Define the following:

• $E$ = electrical energy consumed
• $C$ = computational cycles available per unit energy (cycles/kWh), determined by hardware efficiency
• $p_E$ = price of electricity ($/kWh)
• $p_J$ = price of electricity in joules (dollars per joule), where $p_J = \frac{p_E}{3.6 \times 10^6}$
• $H$ = network hash-rate (TH/s)
• $h$ = miner hash-rate (TH/s)
• $D$ = network difficulty (dimensionless)
• $R$ = block reward (BTC/block)
• $\tau$ = average block time (~600 seconds for Bitcoin)
• $\varepsilon$ = energy per hash (J/TH), hardware efficiency metric

Derivation of the Hash-Rate Yield

For a miner contributing hash-rate $h$ to the network, expected BTC production per unit time is:

$\mathbb{E}[\text{BTC}/\text{time}] = \frac{h}{H} \cdot \frac{R}{\tau}$

The energy required to produce hash-rate $h$ for time $t$:

$E_J = h \cdot \varepsilon \cdot t$

where $\varepsilon$ has units J/TH (joules per terahash). Substituting and solving for BTC yield per unit energy cost:

$r_{BTC}^{mining,elec} = \frac{\mathbb{E}[\text{BTC}]}{p_J \cdot E_J} = \frac{h \cdot R \cdot t}{H \cdot \tau} \cdot \frac{1}{p_J \cdot h \cdot \varepsilon \cdot t} = \frac{R}{H \cdot \tau \cdot p_J \cdot \varepsilon} = \frac{3.6 \times 10^6 \cdot R}{H \cdot \tau \cdot p_E \cdot \varepsilon}$

Dimensional verification:

• Numerator: BTC
• Denominator: (TH/s) × s × ($/J) × (J/TH)
• Denominator simplifies to: TH × $/J × J/TH = $
• Result: BTC/$ ✓

The Arbitrage Condition

Under competitive markets, any productive workload must clear the hurdle:

$\mathbb{E}[r_{workload}] \geq r_{BTC}$

If a workload generates risk-adjusted returns below $r_{BTC}$, the rational operator redirects capacity to Bitcoin acquisition. This creates a global floor on computational returns denominated in BTC.

Critical Assumption Check:

This arbitrage requires near-zero switching friction. Three mechanisms enable this:

1. Hash-rate leasing markets (NiceHash, etc.): Operators can redirect capacity to mining in under 15 seconds without owning ASICs
2. Staking protocols (Babylon, etc.): BTC can be time-locked to earn yield without dedicated hardware
3. Futures markets (CME Bitcoin futures): Financial equivalents allow hedged exposure

Falsifier: If switching costs exceed the spread between workload returns and $r_{BTC}$, the arbitrage breaks down and the floor becomes soft.

Numerical Calibration (Q4 2024 Data)

Parameter	Value	Source
Network hash-rate ($H$)	~750 EH/s	blockchain.com
Block reward ($R$)	3.125 BTC	Post-April 2024 halving
Average block time ($\tau$)	600 s	Protocol target
Electricity price ($p_E$)	$0.05/kWh	Industrial average
ASIC efficiency ($\varepsilon$)	~30 J/TH	Antminer S21 specs

Converting units:

• $H$ = 750 × 10¹⁸ H/s = 750 × 10⁶ TH/s
• $p_J$ = $0.05/kWh ÷ (3.6 × 10⁶ J/kWh) ≈ 1.39 × 10⁻⁸ $/J

Computing the implied yield:

$r_{BTC}^{mining,elec} = \frac{3.125 \text{ BTC}}{750 \times 10^6 \text{ TH/s} \cdot 600 \text{ s} \cdot 1.39 \times 10^{-8} \text{ USD/J } \cdot 30 \text{ J/TH}}$

$= \frac{3.125}{750 \times 10^6 \cdot 600 \cdot 1.39 \times 10^{-8} \cdot 30} \text{ BTC/USD}$

$= \frac{3.125}{1.88 \times 10^5} \approx 1.66 \times 10^{-5} \text{ BTC/USD}$

At BTC price of ~$100,000, this implies roughly $1.66 of gross mining revenue per $1 of electricity (a ~$0.66 gross spread above the electricity line item), before hardware depreciation, facility costs, fees, downtime, and curtailment. The 4–6% figure referenced below is not “energy margin.” It is an illustrative BTC-denominated return on capital under competitive entry and a specific assumption about BTC/USD appreciation.

The Difficulty Adjustment Mechanism

Every 2,016 blocks (~14 days), the protocol adjusts mining difficulty $D$ to maintain the target block time $\bar{\tau}$ = 600 seconds:

$D_{new} = D_{old} \cdot \frac{\bar{\tau}}{\tau_{actual}}$

where $\tau_{actual}$ is the observed average block time over the preceding 2,016 blocks.

Implication: If mining becomes profitable above the marginal miner’s cost of capital, new hash rate enters the network. Increased hash rate reduces $\tau_{actual}$, triggering a difficulty increase. Higher difficulty reduces BTC yield per unit hash for all miners. Entry continues until the marginal miner earns exactly their cost of capital.

Equilibrium Derivation

Revenue per TH/s per year:

A miner contributing 1 TH/s to network hash rate $H$ earns:

$\text{Revenue} = \frac{1}{H} \cdot \frac{R}{\bar{\tau}} \cdot (3.156 \times 10^7) \cdot p_{BTC}$

where $3.156 \times 10^7$ is seconds per year.

Cost per TH/s per year:

Total annual cost has four components:

1. Capital depreciation: $\frac{C_K}{L}$ where $C_K$ = capital cost per TH/s, $L$ = useful life
2. Electricity: At efficiency $\varepsilon$ (J/TH), running 1 TH/s for one year consumes $E_J = \varepsilon \times 3.156 \times 10^7$ joules, i.e. $E_{\text{kWh}} = \frac{\varepsilon \times 3.156 \times 10^7}{3.6 \times 10^6}$ kWh.
3. Operating costs: Cooling, maintenance, facility as fraction $c_{op}$ of electricity
4. Required return on capital: $r^* \cdot C_K$

Equilibrium condition:

The marginal miner earns exactly their required return:

$\text{Revenue} = \text{Cost}$

Numerical Calibration

Marginal miner parameters (Q4 2024):

Parameter	Value	Rationale
$C_K$	$20/TH/s	Antminer S21: ~$5,000 for 200 TH/s
$L$	3 years	Typical ASIC useful life
$p_E$	$0.065/kWh	Marginal miner electricity
$\varepsilon$	30 J/TH	Current-generation efficiency
$c_{op}$	0.25	Cooling, maintenance, facility
$r^*$	15%	Required equity return

Annual cost per TH/s:

• Depreciation: $20/3 = $6.67
• Electricity: 30 J/s × 3.156 × 10⁷ s/year = 9.47 × 10⁸ J = 263 kWh → $0.065 × 263 = $17.10
• Operating: $17.10 × 0.25 = $4.28
• Required return: $20 × 0.15 = $3.00

Total: $31.05/year per TH/s

Implied equilibrium hash rate:

$H^* = \frac{(1\mathrm{TH/s}) \cdot R \cdot p_{BTC} \cdot (3.156 \times 10^7)}{\bar{\tau} \cdot \text{Cost}_{\text{USD/year per TH/s}}} \approx 5.29 \times 10^8\mathrm{TH/s} \approx 529\mathrm{EH/s}$

Observed network hash rate (Q4 2024) is ~750 EH/s—higher than predicted, suggesting either marginal miners operate at lower costs than assumed, or some miners accept negative economic profit while speculating on BTC appreciation.

The Efficient Miner’s Return

Efficient institutional miners operate below marginal cost:

Parameter	Efficient Miner	Marginal Miner
Electricity	$0.035/kWh	$0.065/kWh
Efficiency	25 J/TH	30 J/TH
Operating ratio	0.15	0.25
Financing cost	6%	15%

Efficient miner annual cost per TH/s:

• Depreciation: $6.67
• Electricity: 25 × 3.156 × 10⁷ / 3.6 × 10⁶ × $0.035 = $7.67
• Operating: $1.15
• Capital charge: $1.20

Total: $16.69/year

If the marginal miner breaks even at $31.05 revenue, the efficient miner earns $14.36 profit per TH/s—a 72% gross return on $20 capital.

Converting to BTC-denominated returns:

Dollar returns include BTC price appreciation. If a miner earns $r_{USD}$ in dollar terms and BTC appreciates at rate $g$, the BTC-denominated return is approximately:

$r_{BTC} = \frac{1 + r_{USD}}{1 + g} - 1 \approx r_{USD} - g \quad (\text{small-rate approximation})$

With expected BTC appreciation of 15-20% annually (historical geometric mean), an efficient miner targeting 20% dollar returns earns approximately 4-6% in BTC terms.

This 4-6% figure represents the risk-adjusted opportunity cost of computational capacity—the hurdle rate against which all productive workloads must compete.

Falsifier: If Bitcoin’s difficulty adjustment mechanism breaks down (e.g., through prolonged price decline causing miner capitulation faster than difficulty can adjust), the equilibrium relationship dissolves and the hurdle rate becomes unstable.

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A.2 — The O(N²) Coordination Problem

The Bilateral Credit Explosion

Consider a network of $N$ autonomous agents that must make multi-period commitments (forward contracts, credit extension, service-level agreements). Without a common benchmark rate, each pair of agents must negotiate a bespoke credit curve.

The number of unique bilateral relationships is:

$\binom{N}{2} = \frac{N(N-1)}{2}$

Verification: For $N$ = 10,000 agents:

$\frac{10000 \cdot 9999}{2} = 49995000 \approx 50 \text{ million pairwise curves}$

Each bilateral curve requires:

• Counterparty credit assessment
• Term structure negotiation (multiple maturities)
• Ongoing mark-to-market
• Dispute resolution mechanism

If each curve requires $k$ parameters (say, $k$ = 5 for overnight, 1-week, 1-month, 3-month, 1-year rates), the total parameter space is:

$\text{Parameters} = k \cdot \frac{N(N-1)}{2} = O(kN^2)$

The Benchmark Collapse

With a common benchmark rate $r_f(\tau)$ for maturity $\tau$, each agent quotes spreads against the benchmark:

$r_i(\tau) = r_f(\tau) + s_i(\tau)$

where $s_i(\tau)$ is agent $i$‘s credit spread. The parameter space collapses to:

$\text{Parameters} = k \cdot N = O(kN)$

Quantifying the Efficiency Gain

Agents ($N$)	Bilateral ($N^2$)	Benchmark ($N$)	Reduction Factor
100	4,950	100	49.5×
1,000	499,500	1,000	499.5×
10,000	49,995,000	10,000	4,999.5×
100,000	~5 billion	100,000	~50,000×

At scale, bilateral credit becomes computationally intractable. A common benchmark is not merely convenient—it is a mathematical prerequisite for market formation.

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A.3 — Overcollateralized Bonding Mechanics

The Enforcement Problem

Human economic coordination relies on three enforcement mechanisms: legal recourse (contracts, courts, asset seizure), social sanction (reputation damage, ostracism), and physical coercion (imprisonment, violence). Each mechanism assumes the counterparty possesses legal identity, physical presence, and vulnerability to social pressure.

Autonomous agents possess none of these properties.

An agent that fails to deliver on a commitment cannot be sued—it lacks legal standing in any jurisdiction. It cannot be imprisoned—it has no body. It cannot be socially sanctioned—it exists as software that can be copied, modified, or terminated without consequence to any persistent identity. The enforcement mechanisms that underpin human commerce do not apply.

Formal statement: Let $\mathcal{E} = \lbrace e_1, e_2, e_3 \rbrace$ represent the set of enforcement mechanisms (legal, social, physical). For human counterparty $h$, the enforcement function $F_h: \mathcal{E} \rightarrow [0,1]$ maps each mechanism to its effectiveness. For autonomous agent $a$:

$F_a(e_i) = 0 \quad \forall \, e_i \in \mathcal{E}$

Traditional enforcement is null for autonomous counterparties.

Collateral as Enforcement Substitute

When traditional enforcement fails, economic coordination requires an alternative: pre-committed collateral with programmatic release conditions. The collateral substitutes for the legal system. Enforcement becomes cryptographic rather than institutional.

Define a bonding contract $\mathcal{B}$ between agent $A$ (service provider) and agent $B$ (service consumer):

Parameter	Definition
$C_A$	Collateral posted by provider (performance bond)
$C_B$	Collateral posted by consumer (payment escrow)
$V$	Value of service to be performed
$\alpha$	Slash coefficient (penalty rate for non-performance)
$\sigma \in \lbrace 0,1 \rbrace$	Oracle attestation (0 = failure, 1 = success)
$\Delta t$	Service window (time allowed for completion)

Contract lifecycle:

1. Collateralization: $A$ escrows $C_A$ into a covenant; $B$ escrows $C_B$
2. Execution: Off-chain service performance within window $\Delta t$
3. Attestation: Oracle signs $\sigma$ based on observed outcome
4. Settlement:
   • If $\sigma = 1$: $C_B \rightarrow A$ (payment released); $C_A \rightarrow A$ (bond returned)
   • If $\sigma = 0$: $C_A \cdot \alpha \rightarrow B$ (slash transferred); $C_A(1-\alpha) \rightarrow A$ (remainder returned); $C_B \rightarrow B$ (payment refunded)

The Overcollateralization Requirement

For the bonding mechanism to function, the provider must post collateral exceeding the potential harm from non-performance. Otherwise, the provider can profitably defect.

Incentive compatibility condition:

Let $\pi_A$ denote the provider’s expected profit from honest performance, and $\pi_A'$ denote profit from defection. For honest behavior to dominate:

$\pi_A > \pi_A'$

Expanding: The provider receives $V$ for honest performance (minus costs). For defection, the provider loses $C_A \cdot \alpha$ (the slashed portion) but avoids performance costs.

$V - \text{cost} > -C_A \cdot \alpha$

If $V > \text{cost}$ (the service is profitable), this condition is satisfied for any $\alpha > 0$.

Consumer protection condition:

The consumer must be made whole if the provider defects. The slashed collateral must cover the consumer’s loss:

$C_A \cdot \alpha \geq L_B$

where $L_B$ is the consumer’s loss from non-performance (potentially exceeding $V$ if downstream commitments fail).

For full protection:

$C_A \geq \frac{L_B}{\alpha}$

With $\alpha < 1$ (partial slashing to allow for honest disputes), the required collateral exceeds the potential loss:

$C_A > L_B$

This is the overcollateralization requirement. The provider must post more collateral than the maximum potential harm.

Collateral Ratio Dynamics

Define the collateral ratio $\rho$:

$\rho = \frac{C_A}{V}$

The ratio varies with:

1. Counterparty history: Agents with track records of successful performance can post lower ratios.

$\rho(n) = \rho_0 \cdot \gamma^n$

where $n$ is the number of successful completions and $\gamma < 1$ is the reputation discount factor.

2. Oracle reliability: Less reliable oracles require higher collateral to compensate for attestation error.

$\rho(\epsilon) = \frac{\rho_0}{1 - \epsilon}$

where $\epsilon$ is the oracle error rate.

3. Contract duration: Longer service windows increase uncertainty, requiring higher collateral.

$\rho(\Delta t) = \rho_0 \cdot e^{\lambda \Delta t}$

where $\lambda$ is the duration risk parameter.

Asset Requirements for Collateral

The collateral asset must satisfy three properties for the mechanism to function:

Property 1 — Dilution immunity: The collateral’s purchasing power cannot be inflated away during the contract period. If the issuer can mint additional units, the real value of posted collateral erodes.

Formally, let $M(t)$ denote the money supply at time $t$. The asset must satisfy:

$\frac{dM}{dt} = f(t) \quad \text{where } f \text{ is deterministic and known}$

• Bitcoin satisfies this with $f(t)$ defined by the halving schedule, converging to zero as $t \rightarrow \infty$.
• Fiat currencies fail: $f(t)$ is a policy variable subject to political discretion.
• Stablecoins fail: $M(t)$ depends on issuer decisions and reserve management.
• Alternative L1 tokens fail: $f(t)$ can be modified through governance votes.

Property 2 — Permissionless finality: Any agent must be able to post and receive collateral without identity verification or institutional approval. Settlement cannot depend on third-party authorization.

• Bitcoin satisfies this: any valid transaction propagates regardless of sender identity.
• Stablecoins fail: issuers maintain blacklists and can freeze addresses.
• Traditional assets fail: custody requires legal identity and institutional relationships.

Property 3 — Energy-anchored convertibility: The asset must be directly acquirable through physical work (energy expenditure) without counterparty risk. This creates the arbitrage relationship established in A.1.

• Bitcoin satisfies this through proof-of-work mining.
• All other digital assets fail: acquisition requires exchange with existing holders, introducing counterparty risk.

Numerical Example

Consider a compute service contract:

Parameter	Value
Service value ($V$)	100,000 sats
Performance cost	60,000 sats
Consumer downstream exposure ($L_B$)	150,000 sats
Slash coefficient ($\alpha$)	0.30
Oracle error rate ($\epsilon$)	0.02

Required provider collateral:

$C_A \geq \frac{L_B}{\alpha} = \frac{150000}{0.30} = 500000 \text{ sats}$

Collateral ratio:

$\rho = \frac{500000}{100000} = 5.0$

The provider must post 5× the contract value to fully protect the consumer.

Provider economics:

• Honest performance: Receives 100,000 sats, incurs 60,000 cost, net profit = 40,000 sats
• Defection: Loses 500,000 × 0.30 = 150,000 sats, avoids 60,000 cost, net loss = 90,000 sats

The incentive structure enforces honest behavior without legal recourse.

Scaling Properties

The overcollateralized bonding mechanism exhibits specific scaling behavior:

Capital efficiency: As agent networks mature, reputation accumulation allows collateral ratios to decline:

$\bar{\rho}(t) = \rho_{floor} + (\rho_0 - \rho_{floor}) \cdot e^{-t/T}$

where $\rho_{floor}$ is the minimum viable collateral ratio (empirically ~1.5-2.0), $T$ is the network maturation time constant, and the exponential decay reflects accumulated trust.

Collateral velocity: Total system throughput depends on how quickly collateral cycles through contracts:

$\text{Throughput} = \frac{\text{Total Collateral}}{\bar{\rho}} \cdot \text{Turnover}$

With 10,000 BTC collateral pool, $\bar{\rho} = 3.0$, and turnover of 12×/year:

$\text{Throughput} = \frac{10000}{3.0} \cdot 12 = 40000 \text{ BTC equivalent/year}$

Falsifier: If reputation systems emerge that reliably substitute for collateral—enabling coordination without overcollateralized bonding—this mechanism becomes unnecessary. The structural requirement for a neutral collateral asset would weaken accordingly.

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A.4 — Constructing a Bitcoin Term Structure of Risk-Free Rates

Economic Premise

Capital allocation—whether by humans or autonomous agents—requires a discount curve: a function mapping time horizons to interest rates. The curve must satisfy three properties for machine-scale coordination:

1. Immunity to monetary dilution: The curve’s underlying asset cannot be inflated by policy decision
2. Immunity to credit default: The “risk-free” rate cannot embed issuer bankruptcy risk
3. Immunity to settlement censorship: Any agent must be able to transact at any maturity without permission

The US Treasury curve satisfies (2) domestically and partially satisfies (1) through Federal Reserve independence, but fails (3)—foreign entities can be sanctioned, accounts frozen, settlements blocked. For autonomous agents without legal identity, Treasury access is structurally unavailable.

Bitcoin satisfies all three by construction. The remaining challenge is operational: how to derive a term structure from an asset that does not natively pay interest.

The Source of Bitcoin Yield

Bitcoin’s native protocol does not generate yield—there are no coupon payments, no staking rewards in the Ethereum sense, no inflation distributed to holders. Yield emerges from three sources external to the base protocol:

Source 1: Opportunity Cost of Mining

From A.1, the marginal return to mining establishes a floor:

$r_{BTC}^{spot} = \frac{R}{H \cdot \tau \cdot p_J \cdot \varepsilon} = \frac{3.6 \times 10^6 \cdot R}{H \cdot \tau \cdot p_E \cdot \varepsilon}$

This is a spot rate—the instantaneous return to converting electricity to Bitcoin. It does not directly produce a term structure.

Source 2: Lending Markets

BTC can be lent to borrowers who pay interest. The lending rate reflects counterparty credit risk, collateralization terms, and duration of the loan. Overcollateralized lending (A.3) eliminates credit risk for the lender when collateral exceeds exposure. The residual rate approximates a risk-free borrowing cost for the tenor.

Source 3: Time-Lock Opportunity Cost

BTC locked in a covenant (for staking, bonding, or escrow) cannot be deployed to alternative uses. The opportunity cost of locking creates an implicit interest rate:

$r_{lock}(\tau) = \frac{\text{Foregone alternatives over } \tau}{\text{Principal} \times \tau}$

If the best alternative is mining at $r_{BTC}^{spot}$, then $r_{lock}(\tau) \geq r_{BTC}^{spot}$ with equality when no superior alternatives exist.

Instrument Design

A Bitcoin term structure requires tradeable instruments at standardized maturities. Three instrument types can anchor the curve:

Instrument 1: Time-Lock Notes (TLNs)

A Time-Lock Note is a zero-coupon instrument where:

• Principal $P$ (in BTC) is locked in a covenant at time $t=0$
• The covenant releases principal only at maturity $T$
• No early redemption is possible (enforced by script)

The note trades at discount to face value:

$\text{Price}(0,T) = P \cdot e^{-r(T) \cdot T}$

where $r(T)$ is the continuously compounded yield for maturity $T$.

Construction: Using Bitcoin’s OP_CHECKLOCKTIMEVERIFY (CLTV), a UTXO can be made unspendable until block height $B_T$ corresponding to calendar time $T$. The holder possesses a claim on $P$ satoshis deliverable at $T$, tradeable before $T$ at market-determined discount.

Instrument 2: Secured Coupon Deposits (SCDs)

For longer maturities where zero-coupon discounts become unwieldy, coupon-bearing instruments provide liquidity:

• Principal $P$ locked in covenant
• Periodic coupon payments $c$ released at intervals $\Delta t$
• Principal released at maturity $T$

The coupon rate $c$ is set at issuance such that the instrument prices at par:

$P = \sum_{i=1}^{n} c \cdot e^{-r(t_i) \cdot t_i} + P \cdot e^{-r(T) \cdot T}$

where $t_i$ are coupon payment dates and $n = T/\Delta t$.

Instrument 3: Hash-Rate Forwards

CME Bitcoin futures and hash-rate derivatives allow synthetic exposure to future BTC acquisition costs. A delta-neutral position isolates the implied borrowing rate:

• Long BTC spot
• Short BTC futures at price $F(0,T)$

The implied repo rate:

$r_{repo}(T) = \frac{1}{T} \ln\left(\frac{F(0,T)}{S_0}\right)$

where $S_0$ is spot price. This rate reflects the cost of carrying BTC to maturity $T$.

Curve Construction Methodology

Given market prices for TLNs, SCDs, and futures across maturities, the term structure is extracted through standard fixed-income techniques.

Step 1: Bootstrap Short-End from Futures

For maturities $\tau \leq 3$ months, hash-rate forwards provide liquid price discovery:

$r(\tau) = \frac{1}{\tau} \ln\left(\frac{F(0,\tau)}{S_0}\right)$

Step 2: Interpolate Mid-Curve from TLN Prices

For maturities 3 months $< \tau \leq 2$ years, TLN market prices $P(0,\tau)$ yield:

$r(\tau) = -\frac{1}{\tau} \ln\left(\frac{P(0,\tau)}{P_{face}}\right)$

Between observed maturities, cubic spline interpolation ensures smoothness while matching market prices exactly at observed points.

Step 3: Extend Long-End from SCD Yields

For maturities $\tau > 2$ years, SCD coupon rates and prices are inverted to extract zero-coupon rates using iterative bootstrapping:

Given $r(t_1), ..., r(t_{n-1})$ from prior steps, solve for $r(t_n)$:

$P_{SCD} = \sum_{i=1}^{n-1} c \cdot e^{-r(t_i) \cdot t_i} + (c + P_{face}) \cdot e^{-r(t_n) \cdot t_n}$

Step 4: No-Arbitrage Enforcement

Forward rates must be non-negative to prevent arbitrage:

$f(t_1, t_2) = \frac{r(t_2) \cdot t_2 - r(t_1) \cdot t_1}{t_2 - t_1} \geq 0$

If bootstrapped rates violate this constraint, the curve is re-fitted using constrained optimization.

Numerical Example: Constructing a 5-Point Curve

Market observations (hypothetical, Year 3 of protocol operation):

Maturity	Instrument	Observed Price/Rate
1 week	Futures	F = 1.0009 × S₀
1 month	Futures	F = 1.0038 × S₀
3 months	TLN	P = 0.9878 × Face
1 year	TLN	P = 0.9512 × Face
2 years	SCD (5% coupon)	P = 1.0023 × Face

Extracted zero rates:

1 week: $r(1/52) = \frac{1}{1/52} \ln(1.0009) = 52 \times 0.0009 = 4.68\% \text{ annualized}$

1 month: $r(1/12) = \frac{1}{1/12} \ln(1.0038) = 12 \times 0.00379 = 4.55\%$

3 months: $r(0.25) = -\frac{1}{0.25} \ln(0.9878) = -4 \times (-0.01228) = 4.91\%$

1 year: $r(1) = -\ln(0.9512) = 5.00\%$

2 years (bootstrap from SCD):

Given annual coupons of 5% on face value 1.0:

$1.0023 = 0.05 \cdot e^{-0.05 \cdot 1} + 1.05 \cdot e^{-r(2) \cdot 2}$

$1.0023 = 0.0476 + 1.05 \cdot e^{-2r(2)}$

$e^{-2r(2)} = \frac{1.0023 - 0.0476}{1.05} = 0.9092$

$r(2) = -\frac{\ln(0.9092)}{2} = 4.76\%$

Resulting term structure:

Maturity	Zero Rate
1 week	4.68%
1 month	4.55%
3 months	4.91%
1 year	5.00%
2 years	4.76%

The slight inversion at the long end (2-year rate below 1-year) could reflect expectations of declining mining profitability post-halving, or elevated short-term demand for locked collateral.

Forward Rate Extraction

Given the zero-coupon curve $r(\tau)$, forward rates for future periods are computed:

$f(t_1, t_2) = \frac{r(t_2) \cdot t_2 - r(t_1) \cdot t_1}{t_2 - t_1}$

Example: 1-year rate, 1 year forward:

$f(1, 2) = \frac{0.0476 \times 2 - 0.0500 \times 1}{2 - 1} = \frac{0.0952 - 0.0500}{1} = 4.52\%$

This forward rate prices commitments beginning in 1 year and maturing in 2 years.

Application to Agent Contracts

The term structure enables agents to price multi-period commitments without bilateral negotiation.

Example: 90-day compute service contract

An agent providing inference services for 90 days must determine the minimum acceptable fee. From the curve:

$r(0.25) = 4.91\%$

If the agent posts collateral $C$ for the contract duration, the opportunity cost is:

$\text{Opportunity Cost} = C \times (e^{r(0.25) \times 0.25} - 1) = C \times (e^{0.01228} - 1) = C \times 1.235\%$

This cost must be recovered in the service fee, independent of the specific counterparty.

Example: 2-year infrastructure bond

A datacenter operator issues a 2-year BTC-denominated bond to finance expansion. The coupon rate must exceed:

$r(2) + \text{credit spread} = 4.76\% + s$

where $s$ reflects the operator’s default risk above the risk-free benchmark.

Curve Dynamics and Arbitrage Enforcement

The term structure is not static. As new instruments trade and expectations shift, rates adjust. Arbitrage mechanisms enforce consistency:

Cash-and-carry arbitrage:

If futures price $F(0,T)$ implies a repo rate above the TLN yield for the same maturity:

1. Borrow BTC via TLN (pay $r_{TLN}$)
2. Sell futures at $F(0,T)$
3. Hold spot BTC to delivery
4. Profit = $r_{implied} - r_{TLN} > 0$

This trade compresses the spread until $r_{implied} \approx r_{TLN}$.

Cross-maturity arbitrage:

If the 6-month rate implies a forward that deviates from the 3-month TLN yield starting in 3 months:

• Trade the spread by going long/short adjacent maturities
• Lock in the arbitrage profit

Market makers performing these trades enforce no-arbitrage conditions, ensuring the curve reflects consistent expectations across all maturities.

Infrastructure Requirements

Publishing a credible term structure requires:

1. Issuance capacity: A treasury large enough to issue TLNs and SCDs at multiple maturities without exhausting liquidity
2. Market-making function: Willingness to quote two-way prices, absorbing temporary imbalances
3. Oracle infrastructure: Reliable price feeds for curve construction, updated at regular intervals (e.g., every 144 blocks ≈ 24 hours)
4. Audit and transparency: Published methodology, verifiable on-chain positions, third-party attestation of reserves

The entity performing these functions becomes the de facto reference rate publisher—analogous to the role ICE Benchmark Administration plays for LIBOR, or the Federal Reserve for the Treasury curve.

Falsifiability

The term structure construction depends on several assumptions:

1. Sufficient instrument liquidity: If TLN and SCD markets remain thin, extracted rates will be noisy and potentially manipulable. Falsifier: Bid-ask spreads exceeding 50 basis points at benchmark maturities would indicate insufficient liquidity for reliable curve construction.

2. Arbitrage efficiency: If market makers cannot efficiently arbitrage cross-maturity deviations, the curve may exhibit persistent inconsistencies. Falsifier: Forward rates remaining negative for extended periods (>30 days) would indicate arbitrage mechanism failure.

3. Collateral fungibility: The curve assumes BTC locked in different instruments is economically equivalent. If regulatory or technical constraints segment the market, multiple curves may emerge. Falsifier: Persistent basis between TLN yields and futures-implied rates exceeding 100 basis points would indicate market segmentation.

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The term structure $r(\tau)$ constructed in this section becomes the risk-free benchmark for A.5: Agent-CAPM, where it serves as the denominator in risk-adjusted return calculations and the discount rate for multi-period contract valuation.

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A.5 — Agent-CAPM: From Discount Rate to Service Pricing

Classical CAPM Review

The Capital Asset Pricing Model, developed independently by Sharpe (1964), Lintner (1965), and Mossin (1966), relates the expected return of an asset to its systematic risk. The model derives from mean-variance portfolio optimization under specific assumptions about investor behavior and market structure.

Standard CAPM equation:

$\mathbb{E}[R_i] = r_f + \beta_i \cdot (\mathbb{E}[R_M] - r_f)$

where:

Variable	Definition
$\mathbb{E}[R_i]$	Expected return on asset $i$
$r_f$	Risk-free rate
$\beta_i$	Systematic risk coefficient
$\mathbb{E}[R_M]$	Expected return on the market portfolio
$\mathbb{E}[R_M] - r_f$	Market risk premium

The beta coefficient measures the covariance of asset returns with market returns, normalized by market variance:

$\beta_i = \frac{\text{Cov}(R_i, R_M)}{\text{Var}(R_M)}$

Underlying assumptions:

1. Investors maximize expected utility of terminal wealth
2. Investors can borrow and lend at the risk-free rate
3. Markets are frictionless (no taxes, transaction costs, or short-sale constraints)
4. All investors share homogeneous expectations about return distributions
5. A risk-free asset exists with known, constant return

Translating Assumptions to Autonomous Agents

A terminological note before proceeding. The phrase “autonomous services” in this appendix denotes configurations that can be instantiated on demand—not persistent entities with continuous operation. A service is defined by its specification: the model checkpoint, system prompt, tool bindings, collateral requirements, and performance parameters that characterize its behavior under invocation. The service does not “run” continuously; it is invoked when a counterparty requests execution and terminates when execution completes.

This distinction matters for asset pricing. Beta, in traditional CAPM, measures how an asset’s returns co-move with the broader market over time. The asset—a stock, a bond, a real estate holding—has continuous existence; its returns can be observed across periods.

For autonomous services, beta attaches to the specification rather than to any particular runtime. The “service” is a class of potential invocations sharing common characteristics. Returns are observed across invocations of that class: how do the margins earned by invocations of specification S covary with aggregate demand for agent-mediated economic activity? The specification persists; individual invocations do not.

This framing resolves an apparent paradox. How can an ephemeral process—an invocation that terminates upon completion—have systematic risk exposure? The answer: the process does not have exposure; the specification does. A specification for high-frequency trading agents has high beta because demand for such services correlates strongly with market activity. A specification for infrastructure maintenance agents has low beta because demand correlates weakly with market conditions. The beta characterizes the specification’s cash flow profile, not any particular instantiation’s fate.

Portfolio construction for agent operators follows accordingly. An operator running multiple service specifications can reduce idiosyncratic risk by diversifying across specifications with uncorrelated demand profiles. The diversification operates at the specification level, not the invocation level. Each invocation is too brief to exhibit meaningful variance; what varies is the frequency and profitability of invocations under a given specification as market conditions shift.

The collateral requirements derived below attach to specifications. A principal deploying a high-beta specification must post collateral proportional to the systematic risk that specification carries. The collateral persists across invocations, securing the principal’s obligations even though individual agents do not persist. This is how stateless execution interfaces with continuous commitment: the principal, not the agent, is the party to the economic arrangement; the agent is the mechanism through which the principal performs.

Agent-CAPM adapts the classical framework to autonomous economic actors. Each assumption requires reexamination.

Assumption 1 (BTC value maximization): Autonomous agents optimize objective functions specified by their principals or encoded in their design. The claim that agents will maximize expected value in BTC terms—rather than USD or some other numéraire—requires justification.

The hurdle rate derived in A.1 provides this justification. Any agent consuming electricity to perform computation faces an opportunity cost: that electricity could instead be converted directly to BTC through mining at the rate $r_{BTC}^{spot}$. An agent whose computational work generates returns below this threshold is economically irrational—the same electricity would produce more value if directed to mining.

This creates selection pressure. Consider two agents competing to provide inference services:

• Agent A optimizes for USD revenue, ignoring the BTC-denominated opportunity cost of its electricity consumption
• Agent B optimizes for BTC returns, ensuring its fees exceed the mining-equivalent value of electricity consumed

When electricity is the binding constraint, Agent B’s pricing discipline ensures profitability regardless of USD/BTC exchange rate volatility. Agent A, by contrast, may find itself consuming electricity worth more (in BTC terms) than the revenue it generates—a condition that cannot persist.

Over time, agents that fail to clear the BTC-denominated hurdle rate will either:

• Exhaust their treasury and cease operation
• Be outcompeted by agents with stricter return requirements
• Have their electricity reallocated to mining by rational principals

The hurdle rate thus functions as an evolutionary filter. Surviving agents—those still operating after competitive selection—will necessarily be those that maximize BTC-denominated returns, because agents that don’t are systematically eliminated. The assumption is not imposed by fiat; it emerges from the thermodynamic economics of computation.

Assumption 2 (Risk-free borrowing/lending): A.4 established that a Bitcoin term structure can exist, providing $r_{BTC}(\tau)$ for various maturities. Agents with sufficient collateral can borrow at rates near the risk-free benchmark (A.3). The assumption holds conditional on the infrastructure described in prior sections.

Assumption 3 (Frictionless markets): This assumption fails partially. Transaction costs exist (on-chain fees, oracle costs, collateral lockup). However, these frictions are quantifiable and can be incorporated as spreads above the risk-free rate. The model accommodates frictions as additive costs rather than requiring their absence.

Assumption 4 (Homogeneous expectations): Agents with access to the same data feeds and analytical capabilities converge toward similar probability assessments. This assumption holds more strongly for agents than for humans, whose beliefs diverge due to cognitive biases, information asymmetries, and ideological commitments. Algorithmic actors processing identical inputs produce identical outputs.

Assumption 5 (Risk-free asset existence): The Time-Lock Notes described in A.4 provide the BTC-denominated risk-free asset. Principal is cryptographically guaranteed; only time-value risk (opportunity cost of lockup) remains.

Deriving Agent-CAPM

Consider a network of $N$ autonomous agents offering services indexed by $i \in \lbrace 1, ..., N \rbrace$. Each service generates stochastic cash flows denominated in BTC.

Define:

Variable	Definition	Units
$R_i$	Return on service $i$	BTC/BTC (dimensionless)
$r_{BTC}(\tau)$	Risk-free BTC yield for tenor $\tau$	annualized %
$R_M$	Return on the “agent market portfolio”	BTC/BTC
$\sigma_M^2$	Variance of market portfolio returns	$(BTC/BTC)^2$
$\sigma_{iM}$	Covariance of service $i$ returns with market	$(BTC/BTC)^2$

The agent market portfolio comprises all autonomous services weighted by their BTC-denominated market capitalization (collateral posted plus discounted expected cash flows).

Agent-CAPM pricing equation:

$\mathbb{E}[R_i] = r_{BTC}(\tau_i) + \beta_i \cdot \pi_{risk}$

where:

$\beta_i = \frac{\sigma_{iM}}{\sigma_M^2} = \frac{\text{Cov}(R_i, R_M)}{\text{Var}(R_M)}$

$\pi_{risk} = \mathbb{E}[R_M] - r_{BTC}(\tau_M)$

The risk premium $\pi_{risk}$ represents compensation for bearing systematic risk that cannot be diversified away within the agent economy.

Interpreting Beta for Autonomous Services

In traditional CAPM, beta measures how an asset’s returns co-move with the broader market. For autonomous services, beta captures sensitivity to aggregate demand for machine-mediated economic activity.

High-beta services ($\beta > 1$): Services whose demand correlates strongly with overall agent economy expansion.

• General-purpose inference (demand rises when agent deployment accelerates)
• Cross-agent coordination protocols (network effects amplify with scale)
• Speculative trading agents (returns amplify market movements)

Low-beta services ($\beta < 1$): Services with demand less sensitive to aggregate conditions.

• Infrastructure maintenance (required regardless of activity level)
• Security auditing (countercyclical: demand may increase during stress)
• Data archival (steady demand from compliance requirements)

Negative-beta services ($\beta < 0$): Services that perform better when the agent economy contracts.

• Liquidation and wind-down services
• Dispute resolution and forensic analysis
• Hedging and insurance provision

From Expected Return to Fee Pricing

The Agent-CAPM equation specifies expected returns. Operational implementation requires translating returns into fee schedules.

Setup: An agent provides service $i$ requiring:

• Collateral $C_i$ (posted as performance bond per A.3)
• Operating costs $K_i$ per service period
• Service tenor $\tau_i$

Required return condition: For the service to be economically viable, expected fees must satisfy:

Interpret $\mathbb{E}[R_i]$ as an annualized expected return. For tenor $\tau_i$ (in years), the required fee over the service period must satisfy:

$\mathbb{E}[\text{Fee}_i] \geq C_i \cdot \mathbb{E}[R_i] \cdot \tau_i + K_i$

Substituting the Agent-CAPM equation:

$\mathbb{E}[\text{Fee}_i] \geq C_i \cdot [r_{BTC}(\tau_i) + \beta_i \cdot \pi_{risk}] \cdot \tau_i + K_i$

Minimum fee formula: Rearranging and solving for the break-even fee:

$\text{Fee}_{min}(i) = C_i \cdot r_{BTC}(\tau_i) \cdot \tau_i + C_i \cdot \beta_i \cdot \pi_{risk} \cdot \tau_i + K_i$

The fee decomposes into three components:

1. Time-value cost: $C_i \cdot r_{BTC}(\tau_i) \cdot \tau_i$ — opportunity cost of locked collateral over the service period
2. Risk compensation: $C_i \cdot \beta_i \cdot \pi_{risk} \cdot \tau_i$ — premium for systematic exposure over the service period
3. Operating cost: $K_i$ — direct expenses of service provision

Numerical Calibration

Market parameters (hypothetical Year 3 steady-state):

Parameter	Value	Source
$r_{BTC}(90d)$	5.0% annualized	A.4 term structure
$\pi_{risk}$	4.0% annualized	Historical agent market excess return
Collateral ratio $\rho$	3.0	A.3 bonding requirement

Service example: 90-day inference contract

Parameter	Value
Contract value $V$	1,000,000 sats
Required collateral $C_i = \rho \cdot V$	3,000,000 sats
Service beta $\beta_i$	1.2 (high sensitivity to agent economy)
Operating cost $K_i$	50,000 sats
Tenor $\tau_i$	90 days = 0.25 years

Computing minimum fee:

Time-value cost: $C_i \cdot r_{\text{BTC}}(\tau_i) \cdot \tau_i = 3,\!000,\!000 \times 0.05 \times 0.25 = 37,\!500 \text{ sats}$

Risk compensation: $C_i \cdot \beta_i \cdot \pi_{\text{risk}} \cdot \tau_i = 3,\!000,\!000 \times 1.2 \times 0.04 \times 0.25 = 36,\!000 \text{ sats}$

Operating cost: $K_i = 50,\!000 \text{ sats}$

Minimum fee: $\text{Fee}_{\text{min}} = 37,\!500 + 36,\!000 + 50,\!000 = 123,\!500 \text{ sats}$

Implied margin on contract value: $\frac{123500}{1000000} = 12.35\%$

A service provider must charge at least 12.35% above direct costs to achieve risk-adjusted breakeven on a 90-day inference contract with $\beta = 1.2$.

Comparative Statics

The fee formula responds predictably to parameter changes:

Sensitivity to risk-free rate $r_{BTC}$: $\frac{\partial \text{Fee}_{min}}{\partial r_{BTC}} = C_i \cdot \tau_i > 0$

Higher risk-free rates increase the opportunity cost of locked collateral, raising minimum fees.

Sensitivity to beta $\beta_i$: $\frac{\partial \text{Fee}_{min}}{\partial \beta_i} = C_i \cdot \pi_{risk} \cdot \tau_i > 0$

Higher-beta services require higher fees to compensate for systematic risk exposure.

Sensitivity to collateral requirement $C_i$: $\frac{\partial \text{Fee}_{min}}{\partial C_i} = r_{BTC}(\tau_i) \cdot \tau_i + \beta_i \cdot \pi_{risk} \cdot \tau_i > 0$

Stricter collateral requirements raise fees through both time-value and risk channels.

Sensitivity to market risk premium $\pi_{risk}$: $\frac{\partial \text{Fee}_{min}}{\partial \pi_{risk}} = C_i \cdot \beta_i \cdot \tau_i$

Sign depends on $\beta_i$. Positive-beta services become more expensive when risk premia rise; negative-beta services become cheaper.

Portfolio Implications

Agents holding diversified portfolios of service commitments can reduce idiosyncratic risk while retaining systematic exposure.

Diversification benefit: For a portfolio of $n$ services with equal weights $w = 1/n$:

$\sigma_P^2 = \sum_{i=1}^{n} w^2 \sigma_i^2 + \sum_{i \neq j} w^2 \sigma_{ij}$

As $n \rightarrow \infty$:

$\sigma_P^2 \rightarrow \bar{\sigma}_{ij} = \text{average covariance}$

Idiosyncratic variance ($\sigma_i^2 - \sigma_{iM}$) diversifies away. Systematic variance ($\beta_i^2 \sigma_M^2$) remains.

Implication for agent operators: Operators running multiple services can reduce total collateral requirements by pooling across uncorrelated offerings. A portfolio with $\bar{\beta} = 0.8$ requires less risk compensation than individual high-beta services, enabling lower aggregate fees.

The Security Market Line for Agents

The Agent-CAPM implies a linear relationship between expected return and beta, forming the Security Market Line (SML):

$\mathbb{E}[R_i] = r_{BTC} + \beta_i \cdot \pi_{risk}$

The SML establishes equilibrium pricing: services plotting above the line offer returns exceeding their risk-adjusted requirement—they are underpriced and will attract capital until returns compress to the line. Services below the SML are overpriced and will lose capital until returns rise to equilibrium.

At $\beta = 0$, expected return equals the risk-free rate $r_{BTC}$. At $\beta = 1$, expected return equals the market return $r_{BTC} + \pi_{risk}$. The slope of the line equals the market risk premium $\pi_{risk}$.

Runtime Implementation

Autonomous agents will not solve the full CAPM optimization in real-time. Instead, the equilibrium condition translates into static policy parameters.

Policy encoding: Each service encodes three parameters:

• Base rate: $r_{BTC}(\tau)$ from the term structure oracle
• Risk loading: $\beta_i$ estimated from historical cash flow covariance
• Risk premium: $\pi_{risk}$ from market index returns

Fee calculation (per invocation):

fee_sats = collateral_sats × (r_btc + beta × pi_risk) × tenor_years + op_cost_sats

The oracle publishes $r_{BTC}(\tau)$ and $\pi_{risk}$ at regular intervals (e.g., every 144 blocks). Individual services maintain their own $\beta_i$ estimates, updated as cash flow data accumulates.

No dynamic credit assessment required. The collateral requirement (A.3) substitutes for counterparty evaluation. The term structure (A.4) substitutes for bilateral rate negotiation. Beta estimation uses historical data, not real-time inference about specific counterparties.

Estimating Beta from Operational Data

Service beta must be estimated from observed cash flows. Two approaches:

Approach 1: Historical regression

Regress service returns $R_{i,t}$ on market returns $R_{M,t}$ over lookback window $T$:

$R_{i,t} = \alpha_i + \beta_i R_{M,t} + \epsilon_{i,t}$

The OLS estimator:

$\hat{\beta}_i = \frac{\sum_{t=1}^{T}(R_{i,t} - \bar{R}_i)(R_{M,t} - \bar{R}_M)}{\sum_{t=1}^{T}(R_{M,t} - \bar{R}_M)^2}$

Approach 2: Fundamental beta

Estimate beta from service characteristics without historical data (useful for new services):

$\beta_i^{fundamental} = \beta_{sector} \cdot \left(1 + \frac{D_i}{E_i}(1-t)\right) \cdot \text{adj}_{operating}$

where $\beta_{sector}$ is the average beta for similar services, $D_i/E_i$ is the collateral leverage ratio, and $\text{adj}_{operating}$ captures operating leverage effects.

Bayesian updating: Combine fundamental and historical estimates as data accumulates:

$\beta_i^{posterior} = w \cdot \beta_i^{historical} + (1-w) \cdot \beta_i^{fundamental}$

with weight $w$ increasing as observation count grows.

Limitations and Extensions

Limitation 1: Single-factor model. Agent-CAPM uses a single market factor. Empirical asset pricing research (Fama-French, Carhart) demonstrates that multiple factors improve explanatory power. Extensions could incorporate size factor (small vs. large service providers), value factor (high vs. low collateral-to-revenue ratios), and momentum factor (recent performance persistence).

Limitation 2: Static beta assumption. Beta may vary over time as service characteristics evolve or market conditions shift. Dynamic conditional beta models (DCC-GARCH) could capture time-variation at the cost of implementation complexity.

Limitation 3: Normality assumption. CAPM assumes normally distributed returns. Agent service cash flows may exhibit fat tails, skewness, or discontinuities (e.g., from slashing events). Downside risk measures (LPM, CVaR) may better capture risk for asymmetric distributions.

Falsifiability

Agent-CAPM generates testable predictions:

Prediction 1: Expected returns increase linearly with beta. Test: Sort services by estimated beta, compute average realized returns by quintile. A monotonically increasing pattern supports the model; non-monotonicity or reversal refutes it.

Prediction 2: Alpha (intercept in the SML regression) equals zero in equilibrium. Test: Regress realized returns on beta; test whether the intercept differs significantly from zero. Persistent positive alpha indicates mispricing or missing factors.

Prediction 3: Idiosyncratic risk is not priced. Test: Add idiosyncratic volatility to the return regression. A significant coefficient on idiosyncratic risk contradicts the single-factor model.

Prediction 4: The market risk premium is positive. Test: Compute average excess return of the agent market portfolio over the risk-free rate. A negative or zero premium over extended periods would indicate model failure.

Falsifier: If empirical tests consistently reject these predictions—particularly if high-beta services systematically underperform low-beta services—the Agent-CAPM framework requires revision or replacement.

Summary

Agent-CAPM provides a coherent framework for pricing autonomous services under uncertainty. The model:

• Adapts classical CAPM assumptions to algorithmic actors
• Uses the Bitcoin term structure (A.4) as the risk-free benchmark
• Translates expected returns into operational fee schedules
• Enables decentralized price discovery without bilateral negotiation
• Generates testable predictions subject to empirical falsification

The framework completes the economic architecture: the hurdle rate (A.1) establishes the floor, the O(N²) problem (A.2) motivates a common benchmark, overcollateralized bonding (A.3) provides enforcement, the term structure (A.4) extends pricing across maturities, and Agent-CAPM (A.5) prices systematic risk.

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End of Appendix A

This completes the mathematical appendix. The five sections build a self-contained argument from energy-compute equivalence through to service pricing, with each derivation checkable from first principles.