Appendix A: Mathematical Appendix

Precious metals and collectibles have an unforgeable scarcity due to the costliness of their creation. This once provided money the value of which was largely independent of any trusted third party. Precious metals have problems, however. Thus, it would be very nice if there were a protocol whereby unforgeably costly bits could be created online with minimal dependence on trusted third parties, and then securely stored, transferred, and assayed with similar minimal trust.

Mathematical Appendix

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Notation Reference

Symbol	Definition	Units
$E$	Electrical energy consumed	J or kWh
$p_E$	Electricity price	$/kWh
$H$	Network hash-rate	TH/s
$h$	Individual miner hash-rate	TH/s
$D$	Network difficulty	dimensionless
$R$	Block reward	BTC/block
$\tau$	Average block time	seconds
$\bar{\tau}$	Target block time (600s)	seconds
$\varepsilon$	Energy per hash (hardware efficiency)	J/TH
$r_{BTC}$	BTC-denominated return	annualized %
$p_{BTC}$	Bitcoin spot price	$/BTC
$C_K$	Capital cost per unit hash-rate	$/TH/s
$N$	Number of agents	count
$r_f(\tau)$	Benchmark rate for maturity $\tau$	annualized %
$s_i(\tau)$	Agent $i$'s credit spread	annualized %
$\mathcal{B}$	Bonding contract	—
$C_A$, $C_B$	Collateral (provider, consumer)	sats or BTC
$V$	Service value	sats or BTC
$\alpha$	Slash coefficient	dimensionless (0–1)
$\sigma$	Oracle attestation	binary (0 or 1)
$\rho$	Collateral ratio ($C_A / V$)	dimensionless
$r(\tau)$	Zero-coupon rate for maturity $\tau$	annualized %
$F(0,T)$	Futures price for delivery at $T$	$/BTC
$S_0$	Spot price	$/BTC
$R_i$	Return on service $i$	dimensionless
$R_M$	Return on agent market portfolio	dimensionless
$\beta_i$	Systematic risk coefficient	dimensionless
$\pi_{risk}$	Market risk premium	annualized %

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

Any compute capacity connected to electricity has a minimum value floor: the satoshis it could acquire by routing that electricity to Bitcoin mining. This floor disciplines all other uses of compute. The mathematics that follow derive this floor and establish its equilibrium properties.

Derivation of the Hash-Rate Yield

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

The energy required to produce hash-rate $h$ for time $t$: $E_J = h \cdot \varepsilon \cdot t$. Substituting and solving for BTC yield per unit energy cost:

Dimensional verification: Numerator: BTC. Denominator: (TH/s) × s × ($/J) × (J/TH) = $. Result: BTC/$. ✓

The Arbitrage Condition

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

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.

The arbitrage requires near-zero switching friction. Three mechanisms enable this: hash-rate leasing markets (NiceHash, etc.) allowing redirection in under 15 seconds, staking protocols (Babylon, etc.) enabling yield without dedicated hardware, and futures markets (CME Bitcoin futures) allowing hedged exposure. The arbitrage breaks down if switching costs exceed the spread between workload returns and $r_{BTC}$.

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:

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

Equilibrium and Calibration

Revenue per TH/s per year for a miner contributing 1 TH/s:

Total annual cost has four components: capital depreciation ($C_K/L$), electricity ($\varepsilon \times 3.156 \times 10^7 / 3.6 \times 10^6 \times p_E$ kWh), operating costs (fraction $c_{op}$ of electricity), and required return on capital ($r^* \cdot C_K$). At equilibrium the marginal miner earns exactly their required return.

Using Q4 2024 parameters ($H$ ~750 EH/s, $R$ = 3.125 BTC, $p_E$ = $0.05/kWh, $\varepsilon$ = 30 J/TH), the implied yield is approximately $1.66 \times 10^{-5}$ BTC/USD. At BTC price of ~$100,000, this implies roughly $1.66 of gross mining revenue per $1 of electricity, before hardware depreciation, facility costs, and curtailment.

The implied equilibrium hash rate under these cost assumptions is approximately 529 EH/s. Observed network hash rate (Q4 2024) is ~750 EH/s—substantially higher than predicted. The discrepancy is material: 42% above the model's prediction. It suggests either that marginal miners operate at lower costs than assumed (in which case the cost inputs are wrong), or that some miners accept negative economic profit while speculating on BTC appreciation (in which case the equilibrium concept is wrong, or at least incomplete). Both explanations are consistent with the model's structure; neither invalidates the floor mechanism. The gap is informative: it measures the degree to which speculative behavior subsidizes network security beyond what competitive equilibrium alone would provide. A 42% calibration gap does, however, imply that forward projections built on this model should carry a wider confidence band than the calibration exercise alone might suggest. The model describes the equilibrium the difficulty adjustment enforces; the path to that equilibrium, and the population of miners along the cost curve, are imprecisely characterized.

Efficient institutional miners operating below marginal cost ($p_E$ = $0.035/kWh, $\varepsilon$ = 25 J/TH, financing at 6%) earn approximately 72% gross return on capital. Converting to BTC-denominated returns with expected BTC appreciation of 15–20% annually, the efficient miner earns approximately 4–6% in BTC terms. This figure represents the risk-adjusted opportunity cost of computational capacity—the hurdle rate against which all productive workloads must compete.

The equilibrium relationship dissolves if Bitcoin's difficulty adjustment mechanism breaks down, whether through prolonged price decline causing miner capitulation faster than difficulty can adjust, or through other protocol-level failures.

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A.2 — The O(N²) Coordination Problem

Consider $N$ autonomous agents making multi-period commitments. Without a common benchmark rate, each pair must negotiate a bespoke credit curve. The number of unique bilateral relationships:

For $N$ = 10,000: approximately 50 million pairwise curves, each requiring $k$ parameters across maturities. Total parameter space: $O(kN^2)$.

With a common benchmark $r_f(\tau)$, each agent quotes spreads: $r_i(\tau) = r_f(\tau) + s_i(\tau)$. The parameter space collapses to $O(kN)$.

Agents ($N$)	Bilateral ($N^2$)	Benchmark ($N$)	Reduction
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 a mathematical prerequisite for market formation.

The coordination reduction holds for any common reference rate. The argument in this section is neutral with respect to which asset denominates that rate; it establishes only that some benchmark is a mathematical prerequisite, not which benchmark. The Bitcoin-specific argument—why the benchmark must be a bearer asset with particular settlement properties, including dilution immunity, permissionless finality, and energy-anchored convertibility—is developed in A.3 and A.4. What A.2 establishes is the mathematical prerequisite for coordination, not its institutional resolution.

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A.3 — Overcollateralized Bonding Mechanics

The Enforcement Problem

Human economic coordination relies on legal recourse, social sanction, and physical coercion. Each assumes the counterparty possesses legal identity, physical presence, and vulnerability to social pressure. Autonomous agents possess none of these. Formally, for enforcement function $F_a$ mapping mechanisms to effectiveness: $F_a(e_i) = 0$ for all traditional mechanisms. Economic coordination therefore requires pre-committed collateral with programmatic release.

Bonding Contract Lifecycle

Define a bonding contract $\mathcal{B}$ between provider $A$ and consumer $B$, with collateral $C_A$ (performance bond) and $C_B$ (payment escrow), service value $V$, slash coefficient $\alpha$, oracle attestation $\sigma \in \{0,1\}$, and service window $\Delta t$.

1. Collateralization: $A$ escrows $C_A$; $B$ escrows $C_B$
2. Execution: Off-chain service within $\Delta t$
3. Attestation: Oracle signs $\sigma$
4. Settlement: If $\sigma = 1$: payment and bond released to $A$. If $\sigma = 0$: $C_A \cdot \alpha$ transferred to $B$; remainder returned; payment refunded.

Incentive Compatibility

For honest behavior to dominate defection, the provider's expected profit from performance must exceed the profit from defection. The provider receives $V$ for honest performance (minus costs). For defection, the provider avoids performance costs but loses $C_A \cdot \alpha$:

If $V > \text{cost}$ (the service is profitable), this condition is satisfied for any $\alpha > 0$. The mechanism is incentive-compatible: even with partial slashing, the provider strictly prefers to perform.

The Overcollateralization Requirement

The consumer must also be made whole if the provider defects. The slashed collateral must cover the consumer's loss $L_B$:

With $\alpha < 1$, the required collateral exceeds the potential loss: $C_A > L_B$.

The collateral ratio $\rho = C_A/V$ varies with counterparty history ($\rho(n) = \rho_0 \cdot \gamma^n$ where $\gamma < 1$ is a reputation discount), oracle reliability ($\rho(\epsilon) = \rho_0/(1-\epsilon)$), and contract duration ($\rho(\Delta t) = \rho_0 \cdot e^{\lambda \Delta t}$).

Asset Requirements for Collateral

The collateral asset must satisfy three properties:

Dilution immunity: The supply schedule must be deterministic and known: $dM/dt = f(t)$ where $f$ is deterministic. Bitcoin satisfies this with $f(t)$ converging to zero. Fiat currencies, stablecoins, and alternative L1 tokens with modifiable issuance schedules do not.

Permissionless finality: Any agent must be able to post and receive collateral without identity verification. Bitcoin satisfies this; stablecoins with blacklist capability and traditional assets requiring legal identity do not.

Energy-anchored convertibility: The asset must be directly acquirable through physical work without counterparty risk. Bitcoin satisfies this through proof-of-work mining; all other digital assets require exchange with existing holders.

Scaling Properties

As agent networks mature, reputation accumulation drives collateral ratios toward a floor:

where $\rho_{floor} \approx 1.5\text{–}2.0$ empirically and $T$ is the network maturation time constant. System throughput is $(\text{Total Collateral}/\bar{\rho}) \times \text{Turnover}$.

This mechanism becomes unnecessary if reputation systems emerge that reliably substitute for collateral. 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

Capital allocation requires a discount curve mapping time horizons to interest rates. For machine-scale coordination, the curve must be immune to monetary dilution, credit default, and settlement censorship. The US Treasury curve fails the third condition for agents without legal identity. Bitcoin satisfies all three by construction; the challenge is operational.

Sources of Bitcoin Yield

Bitcoin's native protocol does not generate yield. Yield emerges from three external sources:

1. Mining opportunity cost: From A.1, $r_{BTC}^{spot} = R/(H \cdot \tau \cdot p_J \cdot \varepsilon)$. This is a spot rate, not a term structure.
2. Lending markets: Overcollateralized BTC lending (A.3) eliminates credit risk; the residual rate approximates risk-free borrowing cost for a given tenor.
3. Time-lock opportunity cost: BTC locked in a covenant cannot be deployed elsewhere: $r_{lock}(\tau) \geq r_{BTC}^{spot}$.

Instruments

Time-Lock Notes (TLNs): Zero-coupon instruments where principal $P$ is locked via OP_CHECKLOCKTIMEVERIFY until maturity $T$. Price: $P \cdot e^{-r(T) \cdot T}$.

Secured Coupon Deposits (SCDs): Coupon-bearing instruments for longer maturities, priced at par when $P = \sum c \cdot e^{-r(t_i) \cdot t_i} + P \cdot e^{-r(T) \cdot T}$.

Hash-Rate Forwards: Synthetic exposure via delta-neutral spot/futures positions. Implied repo rate: $r_{repo}(T) = (1/T) \ln(F(0,T)/S_0)$.

Curve Construction

The curve is extracted through standard fixed-income bootstrapping: short-end from futures ($\tau \leq 3$ months), mid-curve from TLN prices (3 months to 2 years), long-end from SCD yields ($\tau > 2$ years). Forward rates must satisfy non-negativity: $f(t_1, t_2) = (r(t_2) \cdot t_2 - r(t_1) \cdot t_1)/(t_2 - t_1) \geq 0$. Violations trigger constrained re-optimization.

Numerical Example

Using hypothetical Year 3 market observations:

Maturity	Instrument	Zero Rate
1 week	Futures	4.68%
1 month	Futures	4.55%
3 months	TLN	4.91%
1 year	TLN	5.00%
2 years	SCD (5% coupon)	4.76%

The slight inversion at the long end could reflect expectations of declining mining profitability post-halving, or elevated short-term demand for locked collateral.

Forward rates are extracted as $f(t_1, t_2) = (r(t_2) \cdot t_2 - r(t_1) \cdot t_1)/(t_2 - t_1)$. For example, the 1-year rate 1 year forward: $f(1,2) = (0.0476 \times 2 - 0.05 \times 1)/1 = 4.52\%$.

Falsifiability

The construction depends on: (1) sufficient instrument liquidity—bid-ask spreads exceeding 50 basis points would indicate failure; (2) arbitrage efficiency—persistent negative forward rates over 30 days would indicate mechanism failure; (3) collateral fungibility—persistent basis exceeding 100 basis points between TLN yields and futures-implied rates would indicate market segmentation.

The term structure $r(\tau)$ constructed here becomes the risk-free benchmark for A.5.

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

Adaptation of Classical CAPM

The Capital Asset Pricing Model relates expected return to systematic risk: $\mathbb{E}[R_i] = r_f + \beta_i (\mathbb{E}[R_M] - r_f)$. Adapting to autonomous agents requires reexamining its assumptions.

The term "autonomous services" here denotes configurations instantiated on demand, not persistent entities. Beta attaches to the specification—the model checkpoint, system prompt, tool bindings, and collateral requirements—rather than to any particular runtime invocation. A specification's beta captures how its cash flow covaries with aggregate demand for agent-mediated economic activity.

BTC value maximization follows from the hurdle rate (A.1): agents that fail to clear the BTC-denominated floor are systematically eliminated through competitive selection. The assumption emerges from thermodynamic economics rather than being imposed by fiat.

Risk-free borrowing/lending is provided by the term structure (A.4). Frictionless markets fails partially but frictions can be incorporated as additive spreads. Homogeneous expectations holds more strongly for algorithmic actors processing identical inputs than for humans. Risk-free asset is provided by Time-Lock Notes.

The Agent-CAPM Equation

For $N$ services generating stochastic BTC-denominated cash flows, with market portfolio $R_M$ comprising all services weighted by BTC-denominated market capitalization:

where $\beta_i = \text{Cov}(R_i, R_M)/\text{Var}(R_M)$ and $\pi_{risk} = \mathbb{E}[R_M] - r_{BTC}(\tau_M)$.

High-beta services ($\beta > 1$) correlate with agent economy expansion (general inference, cross-agent coordination, speculative trading). Low-beta services ($\beta < 1$) have steadier demand (infrastructure maintenance, security auditing, data archival). Negative-beta services ($\beta < 0$) perform better during contraction (liquidation, dispute resolution, hedging).

From Expected Return to Fee Pricing

For a service requiring collateral $C_i$, operating costs $K_i$, and tenor $\tau_i$:

Calibration Example

With $r_{BTC}(90d) = 5\%$, $\pi_{risk} = 4\%$, $\rho = 3.0$, for a 90-day inference contract valued at 1,000,000 sats with $\beta = 1.2$ and operating costs of 50,000 sats:

• Time-value: $3{,}000{,}000 \times 0.05 \times 0.25 = 37{,}500$ sats
• Risk: $3{,}000{,}000 \times 1.2 \times 0.04 \times 0.25 = 36{,}000$ sats
• Operating: $50{,}000$ sats
• Minimum fee: 123,500 sats (12.35% of contract value)

Comparative Statics

Parameter	Partial Derivative	Sign
Risk-free rate $r_{BTC}$	$C_i \cdot \tau_i$	+
Beta $\beta_i$	$C_i \cdot \pi_{risk} \cdot \tau_i$	+
Collateral $C_i$	$r_{BTC} \cdot \tau_i + \beta_i \cdot \pi_{risk} \cdot \tau_i$	+
Risk premium $\pi_{risk}$	$C_i \cdot \beta_i \cdot \tau_i$	sign($\beta_i$)

The Security Market Line

The Agent-CAPM implies a linear relationship between expected return and beta: $\mathbb{E}[R_i] = r_{BTC} + \beta_i \cdot \pi_{risk}$. Services above the line are underpriced and attract capital; services below are overpriced and lose capital. The SML establishes equilibrium pricing across the agent economy.

Estimating Beta

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

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:

Fundamental beta (for new services without history): Estimate from service characteristics: $\beta_i^{fundamental} = \beta_{sector} \cdot (1 + D_i/E_i \cdot (1-t)) \cdot \text{adj}_{operating}$, where $\beta_{sector}$ is the average beta for similar services and $D_i/E_i$ is the collateral leverage ratio.

Bayesian updating: Combine both 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

The single-factor model may underperform multi-factor extensions (analogous to Fama-French). Beta may vary over time, particularly across regime transitions in the agent economy's growth rate. The fat-tail problem deserves particular attention. Slashing events create discontinuous left-tail exposure: the return distribution is not a smooth bell curve but a mixture of normal operating returns and a binary slash/no-slash outcome with potentially total loss of posted collateral. CAPM assumes mean-variance optimization, which is adequate when return distributions are approximately Gaussian. It is inadequate for binary outcomes. A CVaR (Conditional Value at Risk) or LPM (Lower Partial Moment) framework would change the pricing equation materially, because the left tail carries weight that variance alone does not capture. The single-factor model presented here is a starting point, not a terminal framework. The direction of error is identifiable: Agent-CAPM as specified underprices tail risk, because the variance of a distribution with binary slash outcomes understates the expected loss conditional on being in the tail. A more complete treatment would price the slash probability separately from the systematic risk factor, treating slashing as a jump process overlaid on the continuous return distribution.

Falsifiability

Agent-CAPM generates four testable predictions: (1) expected returns increase linearly with beta; (2) alpha equals zero in equilibrium; (3) idiosyncratic risk is not priced; (4) the market risk premium is positive. Consistent empirical rejection of any of these, particularly if high-beta services systematically underperform low-beta services, would require revision of the framework.

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End of Appendix A

The five sections build a self-contained argument from energy-compute equivalence through to service pricing, each derivation checkable from first principles. A.1 establishes the floor, A.2 motivates the benchmark, A.3 provides enforcement, A.4 extends pricing across maturities, and A.5 prices systematic risk. The next question is whether the assumptions hold.