> For the complete documentation index, see [llms.txt](https://axon-finance.gitbook.io/axon-finance-docs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://axon-finance.gitbook.io/axon-finance-docs/yellowpaper-appendices/appendix-notation.md).

# Appendix I · Notation

Formal notation conventions used throughout

The formal notation used throughout is defined uniformly here for reference by each section.

## System & Consensus

| Symbol                            | Meaning                                                            | Source                                                                                                                                                                                      |
| --------------------------------- | ------------------------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| $\mathcal{V} = {v\_1,\dots,v\_n}$ | Validator set                                                      | [A.1.1](/axon-finance-docs/yellowpaper-a-system-model-and-cryptography/a1-system-model.md)                                                                                                  |
| $s\_i$                            | Effective stake of validator $v\_i$ (self + delegated)             | [A.1.1](/axon-finance-docs/yellowpaper-a-system-model-and-cryptography/a1-system-model.md)·[F.2.1](/axon-finance-docs/yellowpaper-f-protocol-economics-and-security/f2-staking-slashing.md) |
| $S = \sum\_i s\_i$                | Total stake                                                        | [A.1.1](/axon-finance-docs/yellowpaper-a-system-model-and-cryptography/a1-system-model.md)                                                                                                  |
| $S\_f$                            | Byzantine-node stake, assumed $S\_f < \tfrac13 S$                  | [A.1.3](/axon-finance-docs/yellowpaper-a-system-model-and-cryptography/a1-system-model.md)                                                                                                  |
| $\Delta,\ \mathrm{GST}$           | Message-delay bound, global stabilization time (partial synchrony) | [A.1.2](/axon-finance-docs/yellowpaper-a-system-model-and-cryptography/a1-system-model.md)                                                                                                  |
| $r$                               | View (consensus round)                                             | [B.1.2](/axon-finance-docs/yellowpaper-b-consensus-and-ledger/b1-consensus.md)                                                                                                              |
| $\mathsf{QC}(b)$                  | Quorum certificate of block $b$ (stake $>\tfrac23 S$)              | [B.1.2](/axon-finance-docs/yellowpaper-b-consensus-and-ledger/b1-consensus.md)                                                                                                              |
| $\mathsf{Leader}(r)$              | Leader of view $r$ (VRF-derived)                                   | [B.2.3](/axon-finance-docs/yellowpaper-b-consensus-and-ledger/b2-validators.md)                                                                                                             |
| $\mathsf{commit}(b,h)$            | Block $b$ finally committed at height $h$                          | [A.1.4](/axon-finance-docs/yellowpaper-a-system-model-and-cryptography/a1-system-model.md)                                                                                                  |
| $T\_{\text{final}}$               | Finality latency                                                   | [B.1.7](/axon-finance-docs/yellowpaper-b-consensus-and-ledger/b1-consensus.md)·[G.2.2](/axon-finance-docs/yellowpaper-g-networking-and-performance/g2-performance.md)                       |

## Cryptography

| Symbol                                                | Meaning                           | Source                                                                                                                                                                |
| ----------------------------------------------------- | --------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| $H(\cdot)$                                            | Collision-resistant hash (BLAKE3) | [A.2.1](/axon-finance-docs/yellowpaper-a-system-model-and-cryptography/a2-cryptography.md)                                                                            |
| $\Sigma=(\mathsf{KeyGen},\mathsf{Sign},\mathsf{Vrf})$ | Signature scheme (Ed25519)        | [A.2.2](/axon-finance-docs/yellowpaper-a-system-model-and-cryptography/a2-cryptography.md)                                                                            |
| $\sigma\_{\mathsf{agg}}$                              | BLS aggregate signature           | [A.2.3](/axon-finance-docs/yellowpaper-a-system-model-and-cryptography/a2-cryptography.md)                                                                            |
| $\mathsf{root}(\mathsf{st})$                          | State commitment root             | [A.2.4](/axon-finance-docs/yellowpaper-a-system-model-and-cryptography/a2-cryptography.md)·[B.3.3](/axon-finance-docs/yellowpaper-b-consensus-and-ledger/b3-state.md) |
| $\mathsf{VRF}\_{sk}(x)\to(y,\pi)$                     | Verifiable random function        | [A.2.5](/axon-finance-docs/yellowpaper-a-system-model-and-cryptography/a2-cryptography.md)                                                                            |

## State & Execution

| Symbol                                         | Meaning                                                | Source                                                                          |
| ---------------------------------------------- | ------------------------------------------------------ | ------------------------------------------------------------------------------- |
| $\mathsf{st}:\mathsf{Addr}\to\mathsf{Account}$ | Global state                                           | [B.3.1](/axon-finance-docs/yellowpaper-b-consensus-and-ledger/b3-state.md)      |
| $\delta,\ \delta\_{\mathsf{tx}}$               | State-transition function (block / single transaction) | [B.3.2](/axon-finance-docs/yellowpaper-b-consensus-and-ledger/b3-state.md)      |
| $\mathsf{seqNo}(\mathsf{tx})$                  | Global monotonic sequence number                       | [B.4.1](/axon-finance-docs/yellowpaper-b-consensus-and-ledger/b4-sequencing.md) |
| $\mathsf{Log},\ e\_k,\ h\_k$                   | Entry-Log, log entry, hash chain                       | [B.4.2](/axon-finance-docs/yellowpaper-b-consensus-and-ledger/b4-sequencing.md) |
| $\mathsf{Replay}(\mathsf{st}\_0,\mathsf{Log})$ | Replay function                                        | [B.4.3](/axon-finance-docs/yellowpaper-b-consensus-and-ledger/b4-sequencing.md) |

## Authorization & AI

| Symbol                                                      | Meaning                                                        | Source                                                                                                       |
| ----------------------------------------------------------- | -------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------ |
| $\mathsf{Verify}(\mathsf{Account},\mathsf{Tx})$             | Programmable verifier                                          | [C.1.2](/axon-finance-docs/yellowpaper-c-account-abstraction-and-ai-authorization/c1-account-abstraction.md) |
| $P={c\_1,\dots,c\_k}$                                       | Authorization policy (constraint set)                          | [C.2.2](/axon-finance-docs/yellowpaper-c-account-abstraction-and-ai-authorization/c2-session-keys.md)        |
| $\mathsf{Auth}\_P(\mathsf{tx},\mathsf{ctx})=\bigwedge\_c c$ | Authorization predicate                                        | [C.2.3](/axon-finance-docs/yellowpaper-c-account-abstraction-and-ai-authorization/c2-session-keys.md)        |
| $L\_{\text{total}},L\_{\text{tx}},W,\mathcal{S},\rho$       | Total limit / per-tx cap / allowlist / asset scope / frequency | [C.2.2](/axon-finance-docs/yellowpaper-c-account-abstraction-and-ai-authorization/c2-session-keys.md)        |
| $\mathsf{spent},\mathsf{count}$                             | Session cumulative spend / window count                        | [C.2.3](/axon-finance-docs/yellowpaper-c-account-abstraction-and-ai-authorization/c2-session-keys.md)        |

## PayFi & Economics

| Symbol                                          | Meaning                                                        | Source                                                                                                                                                                       |
| ----------------------------------------------- | -------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| $U = B/D$                                       | Utilization (borrowed / supplied)                              | [E.1.2](/axon-finance-docs/yellowpaper-e-payfi-protocol-mechanisms/e1-money-market.md)                                                                                       |
| $r\_{\text{borrow}}(U),\ r\_{\text{supply}}(U)$ | Borrow / supply interest-rate curves                           | [E.1.3](/axon-finance-docs/yellowpaper-e-payfi-protocol-mechanisms/e1-money-market.md)                                                                                       |
| $U^{\*}$                                        | Optimal-utilization kink                                       | [E.1.3](/axon-finance-docs/yellowpaper-e-payfi-protocol-mechanisms/e1-money-market.md)                                                                                       |
| $H=\sum C\_i\lambda\_i/\text{Debt}$             | Health factor                                                  | [E.1.5](/axon-finance-docs/yellowpaper-e-payfi-protocol-mechanisms/e1-money-market.md)                                                                                       |
| $\lambda\_i$                                    | Liquidation threshold of collateral $i$                        | [E.1.5](/axon-finance-docs/yellowpaper-e-payfi-protocol-mechanisms/e1-money-market.md)                                                                                       |
| $\phi$                                          | Reserve factor                                                 | [E.1.3](/axon-finance-docs/yellowpaper-e-payfi-protocol-mechanisms/e1-money-market.md)·[E.2.4](/axon-finance-docs/yellowpaper-e-payfi-protocol-mechanisms/e2-liquidation.md) |
| $\hat{p},\ \mathrm{MAD},\ z\_i$                 | Aggregated price / median absolute deviation / deviation score | [D.2](/axon-finance-docs/yellowpaper-d-settlement-oracle-compliance/d2-oracle.md)                                                                                            |
| $g$                                             | Copy-trading guaranteed floor ($\in\[1%,3%]$)                  | [E.3.5](/axon-finance-docs/yellowpaper-e-payfi-protocol-mechanisms/e3-copy-trading.md)                                                                                       |
| $\chi$                                          | Post-hedge max loss ratio per round                            | [E.4.2](/axon-finance-docs/yellowpaper-e-payfi-protocol-mechanisms/e4-reserve-risk.md)                                                                                       |
| $R\_{ct}$                                       | Copy-trading reserve pool                                      | [E.4.3](/axon-finance-docs/yellowpaper-e-payfi-protocol-mechanisms/e4-reserve-risk.md)                                                                                       |
| $\Xi=R\_{ct}/\sum\text{MaxLoss}$                | Copy-trading reserve coverage ($\geq\Xi\_{\min}=150%$)         | [E.4.3](/axon-finance-docs/yellowpaper-e-payfi-protocol-mechanisms/e4-reserve-risk.md)                                                                                       |
| $\phi\_{ct}$                                    | Copy-trading reserve injection rate                            | [E.4.3](/axon-finance-docs/yellowpaper-e-payfi-protocol-mechanisms/e4-reserve-risk.md)                                                                                       |
| $\Delta\_{\text{chal}}$                         | Settlement challenge window                                    | [E.3.5](/axon-finance-docs/yellowpaper-e-payfi-protocol-mechanisms/e3-copy-trading.md)                                                                                       |
| $\mathsf{gas}(\mathsf{tx})$                     | Transaction gas                                                | [F.1.1](/axon-finance-docs/yellowpaper-f-protocol-economics-and-security/f1-gas-fees.md)                                                                                     |
| $T\_{\text{unbond}}$                            | Unbonding period                                               | [F.2.4](/axon-finance-docs/yellowpaper-f-protocol-economics-and-security/f2-staking-slashing.md)                                                                             |
| $\Theta,\ \text{fee}$                           | Throughput / per-transaction fee                               | [G.2](/axon-finance-docs/yellowpaper-g-networking-and-performance/g2-performance.md)                                                                                         |

***

*Next:* [*Appendix II · Protocol Parameters*](/axon-finance-docs/yellowpaper-appendices/appendix-parameters.md)


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