# Elementary Bitcoin

> Elementary Bitcoin is a free mathematics textbook on Bitcoin by Melvin
> Carvalho, licensed CC BY-SA 4.0, at https://elementarybitcoin.org.
> "Elementary" in the mathematician's sense: self-contained, not simple.
> 40 chapters and 4 appendices of numbered definitions, theorems, proofs,
> and exercises; every result is proved before it is used.

## Structure

Five volumes, graded by epistemic status:

- Volume I, Mathematical Foundations (Ch 1-8): proved mathematics.
  Groups, finite fields, elliptic curves over R and F_p, secp256k1,
  hash functions, ECDSA, Schnorr signatures.
- Volume II, Protocol Architecture (Ch 9-17): verifiable protocol fact,
  checkable against running code and the cited BIPs. Keys and addresses,
  transactions, script, Merkle trees, blocks and the validation function
  V(C, B) (Definition 13.18), proof of work, consensus parameters, soft
  forks, SPV theory.
- Volume III, Scaling and Verification (Ch 18-25): verifiable protocol
  fact. Bloom filters, compact block filters (BIP-157/158), light clients
  and the validation gap, node optimizations, client-side validation,
  payment channels, Lightning, claims and misconceptions.
- Volume IV, Forks and Futures (Ch 26-34): the contested present,
  positions attributed and dated. Fork theory (forks as relations between
  accept-sets), historical forks, SegWit and Taproot as case studies,
  covenants, sidechains, post-Lightning protocols, governance, scaling
  verification.
- Volume V, The Path to a Sustainable Future (Ch 35-40): disciplined
  speculation, labeled as such. Consensus cleanup, security threats and
  defenses, the security budget problem, quantum resistance, monetary
  futures.

Appendices: A notation, B references, C subject index (310 terms),
D consensus validation rule catalog.

## Key results and treatments

- The validation predicate V(C, B): Bitcoin's consensus rules as a total,
  publicly computable function (Definition 13.18; Appendix D catalogs it).
- Fork theory: soft/hard forks classified by accept-set containment
  (Chapter 26), with the Nakamoto double-spend formula derived via
  gambler's ruin (Theorem 14.2, Theorem 26.3).
- Proposition 34.1 (activation game): miners converge on a rule change
  exactly when the enforcing economic fraction e exceeds 1/2, and e is
  bounded by the set of actors who can afford to compute V.
- Issuance without an issuer: proof of work as the only known mechanism
  that distributes a digital asset with no allocator position
  (Remark 14.3); Bitcoin's fair launch (Remark 15.4).
- Rough consensus precisely: RFC 7282's machinery versus Bitcoin's usage,
  "deployment is the declaration, adoption is the minutes, and the fork
  is the appeal" (Definition 33.4).

## Pages

- Contents: https://elementarybitcoin.org/
- Chapters: https://elementarybitcoin.org/chapters/01-groups.html through
  40-monetary-future.html, plus appendix-a-notation.html,
  appendix-b-references.html, appendix-c-index.html,
  appendix-d-validation-rules.html
- Source and errata: https://github.com/elementarybitcoin/elementarybitcoin.github.io

## Citation

Carvalho, M. (2026). Elementary Bitcoin: A Mathematical Introduction from
First Principles. https://elementarybitcoin.org. CC BY-SA 4.0.

Corrections and contributions are welcome via GitHub issues; an error
found at any level is a contribution.