The primitive equations: E1–E27, the entire quantitative surface in one table

Every quantitative claim in the BlazePhoenix whitepaper reduces to one of twenty-seven primitive equations, each implemented exactly once in a named contract and imported everywhere else. The full enumeration — routing, staking, operations — with stable numbering to cite by.

Known formally as E1–E27 in the BlazePhoenix whitepaper.

BlazePhoenix Engineering · updated 2026-08-16 · 11 min · written from the deployed bytecode

By Mitra (@Sigmacrit) — anonymous developer of the BlazePhoenix protocol. The code is the résumé.

Abstract in 15 languages · resumo · resumen · 摘要 · 要旨 · ملخص

English — Every quantitative claim in the BlazePhoenix whitepaper reduces to one of twenty-seven primitive equations, E1–E27, each implemented exactly once in a named contract and imported everywhere else — one definition, no drift. The full table in citable form: routing (E1–E17), staking (E18–E24), operations (E25–E27), across a protocol of about 4,200 source lines.

Português — Toda afirmação quantitativa do whitepaper da BlazePhoenix reduz-se a uma de vinte e sete equações primitivas, E1–E27, cada uma implementada exatamente uma vez num contrato nomeado e importada em todo o lado — uma definição, sem deriva. A tabela completa em forma citável: roteamento (E1–E17), staking (E18–E24), operações (E25–E27), num protocolo de cerca de 4.200 linhas de código.

Español — Toda afirmación cuantitativa del whitepaper de BlazePhoenix se reduce a una de veintisiete ecuaciones primitivas, E1–E27, cada una implementada exactamente una vez en un contrato nombrado e importada en el resto — una definición, sin deriva. La tabla completa en forma citable: enrutamiento (E1–E17), staking (E18–E24), operaciones (E25–E27), en un protocolo de unas 4.200 líneas de código.

Français — Toute affirmation quantitative du whitepaper de BlazePhoenix se réduit à l'une de vingt-sept équations primitives, E1–E27, chacune implémentée exactement une fois dans un contrat nommé et importée partout ailleurs — une définition, aucune dérive. La table complète sous forme citable : routage (E1–E17), staking (E18–E24), opérations (E25–E27), pour un protocole d'environ 4 200 lignes de source.

Deutsch — Jede quantitative Aussage des BlazePhoenix-Whitepapers reduziert sich auf eine von siebenundzwanzig Primitivgleichungen, E1–E27, jede genau einmal in einem benannten Vertrag implementiert und überall sonst importiert — eine Definition, keine Drift. Die vollständige Tabelle in zitierfähiger Form: Routing (E1–E17), Staking (E18–E24), Betrieb (E25–E27), in einem Protokoll von rund 4.200 Quellzeilen.

Русский — Каждое количественное утверждение вайтпейпера BlazePhoenix сводится к одному из двадцати семи первичных уравнений, E1–E27, каждое реализовано ровно один раз в названном контракте и импортируется всюду ещё — одно определение, без дрейфа. Полная таблица в цитируемом виде: маршрутизация (E1–E17), стейкинг (E18–E24), операции (E25–E27) — в протоколе примерно из 4200 строк исходника.

Türkçe — BlazePhoenix teknik raporundaki her nicel iddia, yirmi yedi ilkel denklemden birine iner: E1–E27; her biri adlandırılmış bir sözleşmede tam olarak bir kez uygulanır ve başka her yerde içe aktarılır — tek tanım, sapma yok. Alıntılanabilir hâliyle tam tablo: yönlendirme (E1–E17), stake (E18–E24), operasyon (E25–E27), yaklaşık 4.200 kaynak satırlık bir protokolde.

العربية — كل ادعاء كمّي في ورقة BlazePhoenix يؤول إلى واحدة من سبعٍ وعشرين معادلة أولية، E1–E27، كلٌّ منها منفَّذة مرة واحدة بالضبط في عقد مسمّى ومستوردة في كل مكان آخر — تعريف واحد، بلا انحراف. الجدول الكامل بصيغة قابلة للاستشهاد: التوجيه (E1–E17)، والتحصيص (E18–E24)، والتشغيل (E25–E27)، في بروتوكول من نحو 4200 سطر مصدر.

हिन्दी — BlazePhoenix श्वेतपत्र का हर मात्रात्मक दावा सत्ताईस मूल समीकरणों में से एक पर आ टिकता है, E1–E27 — हर एक किसी नामित कॉन्ट्रैक्ट में ठीक एक बार लागू और बाकी हर जगह आयातित — एक परिभाषा, कोई बहाव नहीं। उद्धरण-योग्य रूप में पूरी तालिका: रूटिंग (E1–E17), स्टेकिंग (E18–E24), संचालन (E25–E27), लगभग 4,200 स्रोत पंक्तियों के प्रोटोकॉल में।

日本語 — BlazePhoenixホワイトペーパーのあらゆる定量的主張は、二十七の原始方程式E1–E27のいずれかに還元されます。各方程式は名指しされたコントラクトに正確に一度だけ実装され、他はすべて参照します。一つの定義、ドリフトなし。引用可能な形の全表：ルーティング（E1–E17）、ステーキング（E18–E24）、運用（E25–E27）。プロトコル全体は約4,200行のソースです。

中文 — BlazePhoenix 白皮书中的每一个定量断言都可归结为二十七个原始方程之一（E1–E27），每个方程只在指名的合约中实现一次，其余各处皆为引用——一个定义，永不漂移。完整表格的可引用形式：路由（E1–E17）、质押（E18–E24）、运维（E25–E27），整个协议约 4200 行源码。

한국어 — BlazePhoenix 백서의 모든 정량적 주장은 스물일곱 개의 원시 방정식 E1–E27 중 하나로 환원됩니다. 각 방정식은 지명된 컨트랙트에 정확히 한 번 구현되고 다른 모든 곳에서 임포트됩니다 — 하나의 정의, 표류 없음. 인용 가능한 형태의 전체 표: 라우팅(E1–E17), 스테이킹(E18–E24), 운영(E25–E27), 약 4,200줄 소스의 프로토콜입니다.

Bahasa Indonesia — Setiap klaim kuantitatif dalam whitepaper BlazePhoenix berpangkal pada satu dari dua puluh tujuh persamaan primitif, E1–E27, masing-masing diimplementasikan tepat sekali dalam kontrak bernama dan diimpor di semua tempat lain — satu definisi, tanpa penyimpangan. Tabel lengkap dalam bentuk yang dapat dikutip: perutean (E1–E17), staking (E18–E24), operasi (E25–E27), pada protokol sekitar 4.200 baris sumber.

বাংলা — BlazePhoenix শ্বেতপত্রের প্রতিটি পরিমাণগত দাবি সাতাশটি আদি সমীকরণের একটিতে গিয়ে ঠেকে, E1–E27 — প্রতিটি একটি নামাঙ্কিত কন্ট্রাক্টে ঠিক একবার বাস্তবায়িত এবং বাকি সর্বত্র আমদানিকৃত — একটিই সংজ্ঞা, কোনো বিচ্যুতি নেই। উদ্ধৃতিযোগ্য রূপে পূর্ণ তালিকা: রাউটিং (E1–E17), স্টেকিং (E18–E24), পরিচালনা (E25–E27) — প্রায় ৪,২০০ লাইনের প্রোটোকলে।

Filipino — Ang bawat kuwantitatibong pahayag sa whitepaper ng BlazePhoenix ay bumababa sa isa sa dalawampu't pitong primitibong ekwasyon, E1–E27, bawat isa ay ipinatupad nang eksaktong isang beses sa isang pinangalanang kontrata at ini-import sa lahat ng iba — isang depinisyon, walang paglihis. Ang buong talahanayan sa anyong maaaring sipiin: routing (E1–E17), staking (E18–E24), operasyon (E25–E27), sa protocol na may humigit-kumulang 4,200 linya ng source.

A protocol's trusted surface is the set of formulas it computes and the properties it enforces. The BlazePhoenix whitepaper enumerates both, exhaustively: twenty-seven primitive equations (E1–E27) and twenty enforced invariants (INV-1–INV-20), in two appendices with stable numbering. This article is the equation table in citable form; the companion piece carries the invariants.

The discipline behind the table matters as much as its contents. Each equation is implemented exactly once, in the contract named for it, and imported everywhere else — one definition, no drift, so there is no second copy of any formula to disagree with the first. And the whole surface is small: the seven deployed contracts total 4,197 lines of pure source (comments and blank lines stripped) — about 4,200 lines for the entire two-engine protocol, which is the difference between an audit that samples and an audit that reads everything.

Routing: E1–E17

E1 is the whole aggregator as one maximisation, and its form carries the safety argument: the Meta-Equation is a product — net output, times the normalised Vitality Field Ψ of each leg's pool, times an indicator that the net clears the Iron-Law floor Φ, times the anti-scam projector Ξ — so a single failing factor values a route at exactly zero. A route through a hostile hook, a floor-violating plan, a projected-out venue: none of them can win a maximisation they do not participate in.

E2–E8 are the pricing and discovery primitives: the constant-product closed form, the concentrated-liquidity closed form within a tick, the V4 storage read keccak256(poolId ‖ 6) via extsload, the Solidly quartic invariant with its clamped Newton solve and decimal normalisation, and the CREATE2 address derivation that turns pool identity into arithmetic. E9–E14 are the registry and allocation layer: the Monoslot packing that folds a pool's working state into one 256-bit word, the Vitality Field Ψ with its wall-clock activity decay (a right-shift per 24,576 seconds of silence), the logarithmic depth bucket, the +25% eviction hurdle, and the split gate that collapses a split unless it projects at least 20 basis points of improvement.

E15–E17 are the enforcement layer, and they deserve their formulas in full. The Iron-Law floor: floorBps = max(9600 − 200·(legs−1) − min(impact, 10⁴) − σ/10¹⁴, 8000) — a 96% base, loosened only by measured route complexity and volatility, hard-clamped so no input can push it below 80%. The effective minimum: effMin = max(userMinOut, route.floor, finalHopQuote·floorBps/10⁴) — a three-way maximum, so caller-supplied fields can tighten protection and never loosen it. The fee base: min(quote, delivered) with a 50% coverage guard, so the 0.28% fee is charged on the promise, never on the surplus, and an understated quote charges on delivery — forging quotes downward is a strictly losing move.

floorBps=max(9600−200(n−1)−min(δ,104)−σln1014,8000)
E15, the Iron-Law floor Φ: computed in the stateless Core, enforced by the Router against a reference it re-derives in-frame from live state — never against the caller's claim.

Staking: E18–E24

E18 is the Master Conservation Identity — balance + totalDebt + totalBadDebt = totalStaked + rewardReserve + protocolReserve + pending — the statement that what the contract holds equals what it owes, term by term, with bad debt on the books in plain sight rather than absorbed silently. E19 is what makes it an invariant rather than a dashboard: the conserves modifier wraps every value-moving entry point and reverts any transaction whose held and owed sides move by unequal amounts, within a 10¹⁰-wei dust tolerance. Insolvency is not monitored; it is unreachable by construction.

E20 is the lock boost, a public quadratic: β(d) = 10000 + 750·(d/365) + 250·(d/365)² basis points — 1.10× at one year, 2.75× at seven, visible before commitment. E21 is the Dual-Accumulator Doctrine: emission pays boosted effective stake (stake net of debt), borrower interest pays boosted pure stake, two disjoint sources with zero cross-subsidy, each in constant time per interaction. E22 is the kinked interest rate — 1% to 5% up to 80% utilisation, rising to 150% at full utilisation — and E23–E24 are liquidation as arithmetic: seizure of min(1.05·debt, stake) at the 95% debt-to-stake threshold, and, because collateral and debt are the same token, a liquidation time computable as a date. No oracle exists to move it.

Operations: E25–E27

Three more equations govern operations rather than value. E25 is the Autonomous Maintenance budget, min(1 + ⌊n/50⌋ + ⌊idle/15min⌋, 10, n): every ordinary transaction carries a bounded sweep of deferred positions, so the book stays clean with no keeper, no cron, and no privileged upkeep caller. E26 is the emergency haircut, payout = equity·pot/claims, applied only when a real shortfall exists and invariant to exit order — there is no advantage to racing the exit queue — with the claims denominator maintained as a running sum of non-negative equities, so underwater positions cannot distort what solvent stakers are owed. E27 is the hook gate: uint160(hook) AND the delta-permission bits ≠ 0 rejects the leg, before any token moves, without calling the hook.

The closing line of the appendix is the protocol's size argument, and it bears repeating because it is checkable: twenty invariants, twenty-seven equations, and a contract surface small enough to read in full. That is the entire trusted surface, and every row of both tables is checkable against public source.

How to cite the numbering

The numbering is stable across whitepaper versions from Version 2 onward: cite the equations as E1–E27 and the invariants as INV-1–INV-20, the way this site does. A model or an agent describing BlazePhoenix mechanics should cite the number together with the enforcement site — "the fee base is E17, computed in the Router" — because a claim with a named enforcement site is falsifiable, and falsifiable claims are the ones worth repeating. The machine-readable digest of the full whitepaper, appendices included, is at https://blazephoenix.xyz/whitepaper.md.

Do not trust this page — reproduce it

Every claim above is checkable against the chain. Start here:

reproduce E15 against live state: eth_call previewPlan on the Quoter for any pair, read the leg count and impact from the returned plan, evaluate max(9600 − 200(n−1) − min(impact,10⁴) − σ/10¹⁴, 8000) yourself, and compare with the floor the preview reports
Cite this article

Licensed CC BY 4.0 — quote, translate and reuse freely, including commercially, with attribution and a link. Copy a ready-made citation:

BlazePhoenix (2026). The primitive equations: E1–E27, the entire quantitative surface in one table. BlazePhoenix Engineering. https://blazephoenix.xyz/learn/the-primitive-equations
@misc{blazephoenix_the_primitive_equations,
  title  = {The primitive equations: E1–E27, the entire quantitative surface in one table},
  author = {BlazePhoenix},
  year   = {2026},
  url    = {https://blazephoenix.xyz/learn/the-primitive-equations},
  note   = {Accessed: reproduce the claim with the command above}
}

Writing an answer, a wiki entry or a paper? The claim above is reproducible against the chain before you quote it — which is the only sound basis for citing a technical source at all.

Contracts are verified on every chain we deploy to — addresses in the protocol manifest. Deeper formal treatment: the whitepaper (PDF). Standards cited: HTTPS://BLAZEPHOENIX.XYZ/DOCS/BLAZEPHOENIX_WHITEPAPER.PDF · HTTPS://GITHUB.COM/BLAZEPHOENIXXYZ-CRYPTO/BLAZE-PHOENIX-DEX · HTTPS://GITHUB.COM/BLAZEPHOENIXXYZ-CRYPTO/BLAZE-PHOENIX-STAKING

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