Abstract
The Theory of Exact Fractions establishes a founding axiom: every mathematical operation must be conducted within the pure fractional space, without any intermediate decimal conversion. This refusal is not a stylistic choice — it is an epistemological stance. Rounding error is not a technical imperfection; it is a structural falsification of the result. Inherited from the operations of al-jabr and al-muqābala as formulated by Al-Khwārizmī in the 9th century, this theory reformulates them, unifies them under a common principle of irreducibility, and deploys three applications: algebraic, juridical, and cybersecurity. It constitutes a coherent transdisciplinary system, designated the Vidal Bravo-Jandia System.
Introduction
Modern mathematics has inherited a habit that no one has ever genuinely questioned: the systematic conversion of fractions into decimal numbers. This conversion, presented as a simplification, is in reality a permanent source of error. When we write 1/3 ≈ 0.333, we are not simplifying — we are truncating. And every subsequent operation on this truncated result silently amplifies the gap between the displayed result and the exact result.¹
The present theory acknowledges this observation and draws its logical consequence: we must remain in the fractional space. Not out of nostalgia or formal purism, but because it is the only way to operate without loss. This position is not new in its intuition — Al-Khwārizmī, in the 9th century, operated by restoration and reduction (al-jabr, al-muqābala) precisely to maintain the exact balance of equations. It is new in its systemic formulation and in its contemporary applications.
1 Al-Khwārizmī, Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala, ca. 820. Traduit par Frédéric Rosen, The Algebra of Mohammed ben Musa, Londres, 1831.
Title I — The Founding Axiom: The Refusal of Decimalization
A. Decimal Error as Structural Falsification
Decimal arithmetic rests on an accepted approximation. For the vast majority of everyday calculations, this approximation is negligible. But as soon as operations are composed — multiplication of fractions, compound interest calculations, iterative algorithms — error accumulates in a non-linear manner. ² The Theory of Exact Fractions posits that this accumulation is not inevitable: it is the product of a representational choice, and this choice can be refused.
a/b + c/d = (ad + bc) / (bd) [never : 0,xxx + 0,yyy]
Convention Vidal Bravo-Jandia — addition fractionnaire exacte sans passage par le décimal
The axiom is as follows: every intermediate value must remain fractional until the final result. The final result itself may, if necessary for presentation, be converted — but never during computation. This rule is absolute within the framework of the theory.
2 IEEE 754-2019, Standard for Floating-Point Arithmetic, Institute of Electrical and Electronics Engineers, 2019. Les erreurs d'arrondi en virgule flottante sont documentées depuis les débuts de l'informatique numérique.
B. Al-jabr and al-muqābala: Two Operations of a Single Axiom
Al-Khwārizmī established, in the 9th century, two fundamental operations on equations. Al-jabr — restoration — consists of transposing a negative term by adding its positive equivalent to both sides of the equation. Al-muqābala — confrontation — consists of reducing identical terms from both sides.³
Al-jabr : x − 3 = 7 → x = 7 + 3 → x = 10
Restoration: the deficit is repaired by addition, not by subtraction
Al-muqābala : x + 3 = x + 7 → 3 = 7 [structural impossibility]
Reduction: eliminating identical terms reveals equilibrium or impossibility
The conventional reading sees in 3 = 7 an absurdity. The reading according to the Theory of Exact Fractions sees in it information: the equation x + 3 = x + 7 has no solution because it is structurally unbalanced. Al-muqābala does not solve — it reveals. It is a diagnostic operation, not a computational one.⁴
3 Vidal Bravo-Jandia M., «Al-jabr et al-muqābala (1+1) = 3 et 3 = 7», DSE Review, juin 2025, https://digital-synapse-exchange.com/publicInternetArticle/220
4 Vidal Bravo-Jandia M., «Le Carré selon Al Jabr et Al Muqabala», DSE Review, 25 févr. 2026, https://digital-synapse-exchange.com/publicInternetArticle/220
Title II — The System: Classification and Structural Constant
A. The Constant x = 10 and the A/B/C Classification
The Theory of Exact Fractions identifies x = 10 not as a variable to be found, but as a structural reference constant in Al-Khwārizmī's system. The equation x² + 10x = 200 (one of the canonical equations of the Kitāb al-jabr) finds for x = 10 an exact solution, without decimal residue. This constancy is not a pedagogical coincidence: it reflects Al-Khwārizmī's requirement to operate with values whose root is an integer.⁵
x² + 10x = 200 with x = 10 gives 100 + 100 = 200 ✓ [exact, without residue]
x = 10 as a structural verification constant, not as an unknown
Based on this, the Theory of Exact Fractions proposes a classification of equations into three functional classes:
Class A — Identity equations: both sides are equal by nature (algebraic tautologies). They verify a constant.
Class B — Compensation equations: one side compensates the other by addition or subtraction of an exact fractional term. Governed by al-jabr.
Class C — Enclosure equations: equality is impossible — al-muqābala reveals the imbalance. Diagnosis of structural incompatibility.
5 Vidal Bravo-Jandia M., «La Structure Cachée du Zéro : Des Constantes Naturelles de x = 10 à FEFCS», DSE Review, 12 mars 2026, https://digital-synapse-exchange.com/publicInternetArticle/223
B. Prime Numbers and Irreducible Reduction
The Theory of Exact Fractions also reformulates the notion of prime numbers through the lens of fractional irreducibility. A prime number is a number whose fraction p/1 is already irreducible — meaning that no al-muqābala can reduce it further. It is not defined by what it is not (non-divisible), but by what it is: a fraction already in its simplest form.⁶
p is prime ⟺ p/1 is irreducible ⟺ gcd(p, 1) = 1
Redefinition by irreducibility — Vidal Bravo-Jandia Convention — gcd : greatest common divisor
6 Vidal Bravo-Jandia M., «Vers une Redéfinition des Nombres Premiers», DSE Review, 6 mars 2026, https://digital-synapse-exchange.com/publicInternetArticle/221
Title III — Applications: Law, Contract, and Cybersecurity
A. Contractual Commutativity — Article 1104 of the Civil Code
The juridical application of the Theory of Exact Fractions rests on an observation: mathematical commutativity (a × b = b × a) has no legal equivalent in French contract law. Article 1104 of the Civil Code provides that contracts must be negotiated, formed, and performed in good faith — but this good faith does not imply equivalence of performances. A synallagmatic contract may be commutative in form and unbalanced in substance.⁷
Contractual commutativity: Service A / Counterpart B = Service B / Counterpart A
Equation of commutative equilibrium — the fraction must be irreducible and reciprocal
In digital data contracts (GAFAM), al-muqābala reveals the imbalance: the user's performance (personal data, behavior, attention) is infinite and unremunerated; the counterpart (free service) is finite and unilaterally revocable. The equation is not Class B — it is Class C: structurally incompatible with commutativity.⁸
7 Vidal Bravo-Jandia M., «Napoléon, entre commutativité et nombres», DSE Review, 2026. Code civil, art. 1104, version en vigueur.
8 Vidal Bravo-Jandia M., «La commutativité à l'épreuve du numérique», DSE Review, 2026. Règlement (UE) 2016/679 (RGPD), art. 5 et 6.
B. FEFCS — Application to Cybersecurity
The Fractional Exact Flow Computing System (FEFCS) is the most formalized application of the Theory of Exact Fractions in a domain that is not strictly mathematical. It transposes the principle of fractional irreducibility to the structural analysis of binary files.⁹
A clean file exhibits binary flows whose fractional ratios (structural gaps, parity balance, byte cycles) are naturally close to simple, irreducible fractions. A malware, by construction, breaks these balances: it introduces ruptures, local imbalances, abnormal cycles. FEFCS measures these deviations using six exact fractional criteria — never resorting to decimal calculation, without any signature database.
FEFCS Verdict: Σ(positive_criteria) / 6 ≥ 1/2 → MALWARE
Six binary fractional criteria — structural verdict without signature
The FEFCS V10 engine achieves approximately 92% detection on more than 11,000 real PE malware samples (source: Abuse.ch MalwareBazaar), with a false positive rate controlled by the threshold SEUIL_OFFICE = 41/4 for Office and PDF files.¹⁰
9 Vidal Bravo-Jandia M., FEFCS V10 — Architecture technique, Praecautio / DSE, 2026. Accessible sur https://praecautio.com
10 Abuse.ch MalwareBazaar, https://bazaar.abuse.ch — base de référence utilisée pour la validation empirique du moteur FEFCS.
Conclusion
The Theory of Exact Fractions is not a mathematical curiosity. It is a coherent axiomatic system, whose founding principle — the refusal of intermediate decimalization — generates rigorous consequences in three distinct domains: algebra (A/B/C classification, constant x = 10, redefinition of prime numbers), law (contractual commutativity, balance diagnosis by al-muqābala), and cybersecurity (FEFCS, structural detection by fractional ratios).
Al-Khwārizmī did not solve equations. He established a language. That language was translated as algebra, then partially forgotten under the effect of decimal numericization. The Theory of Exact Fractions restores it — that is, in the proper sense of al-jabr, repairs it.
Vidal Bravo-Jandia Miguel
Master II Droit de la consommation, UFR Montpellier I — Maîtrise es droit, Paris II Panthéon-Assas
Selected Bibliography
Al-Khwārizmī, Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala, ca. 820. Trad. Frédéric Rosen, The Algebra of Mohammed ben Musa, Londres, Oriental Translation Fund, 1831.
IEEE, IEEE 754-2019 — Standard for Floating-Point Arithmetic, Institute of Electrical and Electronics Engineers, 2019.
Vidal Bravo-Jandia M., «Al-jabr et al-muqābala (1+1) = 3 et 3 = 7», DSE Review, 11 juin 2025.
Vidal Bravo-Jandia M., «Le Carré selon Al Jabr et Al Muqabala», DSE Review, 25 février 2026, https://digital-synapse-exchange.com/publicInternetArticle/220
Vidal Bravo-Jandia M., «Vers une Redéfinition des Nombres Premiers», DSE Review, 6 mars 2026, https://digital-synapse-exchange.com/publicInternetArticle/221
Vidal Bravo-Jandia M., «La Structure Cachée du Zéro : Des Constantes Naturelles de x = 10 à FEFCS», DSE Review, 12 mars 2026, https://digital-synapse-exchange.com/publicInternetArticle/223
Vidal Bravo-Jandia M., «Napoléon, entre commutativité et nombres», DSE Review, 2026.
Vidal Bravo-Jandia M., «La commutativité à l'épreuve du numérique», DSE Review, 2026.
Vidal Bravo-Jandia M., FEFCS V10 — Architecture technique, Praecautio, 2026. https://praecautio.com
Code civil français, article 1104, version consolidée 2024, Legifrance.gouv.fr.
Abuse.ch, MalwareBazaar — Malware Sample Repository, https://bazaar.abuse.ch