02 — The 4 Irreducible Axioms
02 — The 4 Irreducible Axioms
Section titled “02 — The 4 Irreducible Axioms”Core claim (revised): The distribution logic of Islamic inheritance law is generated by 4 axioms: Intermediary Exclusion ($\alpha_1$), ʿAṣaba Ranking ($\alpha_2$), Entitlement ($\beta$), and Conservation ($\gamma$). A 4th structural axiom, Kinship Monotonicity ($\delta$), captures the anti-zero / anti-inversion principle that the Jumhūr applied as case-by-case corrections.
Key revision: The original model merged two distinct exclusion mechanisms from
faraid/hajb.md(القاعدة الأولى and القاعدة الثانية) into a single axiom $\alpha$. Splitting them resolves Q2 (the Jumhūr hidden principle) and inverts the Path A / Path B exception counts.
Axiom $\alpha_1$ — INTERMEDIARY EXCLUSION (القاعدة الأولى)
Section titled “Axiom $\alpha_1$ — INTERMEDIARY EXCLUSION (القاعدة الأولى)”“Whoever connects to the deceased through an intermediary is excluded by that specific intermediary.”
Statement
Section titled “Statement”$$\vec{h}_k \text{ is excluded} \iff \exists, \vec{h}_i \in H’ \text{ such that } \vec{h}_i \text{ is the direct واسطة (intermediary) of } \vec{h}_k$$
The intermediary is the specific person through whom $\vec{h}_k$ connects to the deceased — not “anyone closer,” not “anyone in a higher jiha.” The Arabic is precise: حجبته تلك الواسطة — “that intermediary excluded them.”
Source:
faraid/hajb.md— «القاعدة الأولى: من أدلى إلى الميت بواسطة حجبته تلك الواسطة»
Examples
Section titled “Examples”| Heir | Intermediary (واسطة) | Excluded? |
|---|---|---|
| Son’s son | Son | Yes, by son |
| Mother’s mother | Mother | Yes, by mother |
| Father’s father | Father | Yes, by father |
| Son’s daughter | Son | Yes, by son |
| Brother’s son | Brother | Yes, by brother |
Listed Exceptions (from the source text itself)
Section titled “Listed Exceptions (from the source text itself)”| Exception | Status | Schools |
|---|---|---|
| $\epsilon_1$: Maternal siblings survive their intermediary (mother) | By ijmāʿ | All |
| $\epsilon_2$: Father’s mother survives her intermediary (father) | Disputed | Ḥanbalī only |
| $\epsilon_{2b}$: Grandfather’s mother survives her intermediary (grandfather) | Disputed | Ḥanbalī only |
Critical for the grandfather-sibling question: The grandfather is not the siblings’ واسطة — the father is. The Jumhūr state this explicitly: «الإخوة إنما حجبوا بالأب لإدلائهم به وهو منتفٍ في الجد» — “the brothers are excluded by the father because they connect through him, and this is absent in the grandfather.” Under $\alpha_1$ alone, the grandfather has no exclusion power over siblings.
Axiom $\alpha_2$ — ʿAṢABA RANKING (القاعدة الثانية)
Section titled “Axiom $\alpha_2$ — ʿAṢABA RANKING (القاعدة الثانية)”“Among ʿaṣaba competing for the residual, priority is jiha → daraja → quwwa.”
Statement
Section titled “Statement”For any two ʿaṣaba heirs $\vec{h}_1, \vec{h}_2$, define priority:
$$\pi(\vec{h}) = (j, ; d, ; q)$$
evaluated lexicographically (lower = higher priority).
The jiha ordering is:
$$\text{Bunuwwa}(j=1) \succ \text{Ubuwwa}(j=2) \succ \text{Ukhuwwa}(j=3) \succ \text{ʿUmūma}(j=4)$$
At the same jiha, closer daraja wins. At the same daraja, stronger quwwa wins.
Source:
faraid/hajb.md— «القاعدة الثانية: إذا اجتمع عاصبان فأكثر… يُقَدَّمُ صاحب الجهة المقدمة»
Relationship to $\alpha_1$
Section titled “Relationship to $\alpha_1$”$\alpha_1$ is the narrow, intermediary-based exclusion rule. $\alpha_2$ is the broader, rank-based competition rule for ʿaṣaba heirs. They are independent mechanisms:
| $\alpha_1$ (Intermediary) | $\alpha_2$ (Ranking) | |
|---|---|---|
| Scope | Any heir type (farḍ or ʿaṣaba) | ʿAṣaba only |
| Mechanism | Blocked by specific person you connect through | Outranked by anyone with superior $(j, d, q)$ |
| Effect | Total exclusion | Takes residual instead of lower-ranked |
Why the Split Matters
Section titled “Why the Split Matters”The Ḥanafī grandfather-excludes-siblings ruling requires an additional assumption not present in either rule: that the grandfather occupies the father’s jiha (جهة الأبوة). This is argued by analogy (Qurʾān calls Ibrahim “father”; ḥadīth says “your father was an archer”), but the Jumhūr reject this as metaphorical. Under $\alpha_1$ + $\alpha_2$ taken literally, grandfather-sibling coexistence is the default.
Exceptions to $\alpha_1$ and $\alpha_2$ are catalogued in 03-exceptions.md.
Axiom $\delta$ — KINSHIP MONOTONICITY (مبدأ الأحظّ)
Section titled “Axiom $\delta$ — KINSHIP MONOTONICITY (مبدأ الأحظّ)”“A stronger heir must not receive less than a weaker heir.”
Statement
Section titled “Statement”If heir $A$‘s set of blood-paths to the deceased is a strict superset of heir $B$‘s, and both are in the survivor set $S$, then:
$$\text{share}(A) \ge \text{share}(B)$$
Source
Section titled “Source”The classical jurists stated this in plain Arabic:
«أصول المواريث موضوعة على تقديم الأقوى… فكيف يرث الأضعف ويسقط الأقوى؟»
“The foundations of inheritance prioritize the stronger… so how can the weaker inherit while the stronger gets nothing?”
The grandfather-sibling chapter uses the term الأحظّ (“the most advantageous”) — a literal max() function:
«فللجدّ الأحظّ من ثلاثة تقديرات» — “The grandfather gets the maximum of three computations.”
What $\delta$ Handles
Section titled “What $\delta$ Handles”| Case | Without $\delta$ | With $\delta$ |
|---|---|---|
| Mushtaraka | Full brother (superset-kinship) gets 0 while uterine siblings get $\frac{1}{6}$ each | Full brother joins the $\frac{1}{3}$ pool — share $\ge$ uterine share |
| Grandfather-siblings | Under $\alpha_1$ alone, grandfather and siblings coexist but shares are unconstrained | Grandfather gets max(Muqāsama, $\frac{1}{3}$, $\frac{1}{6}$) — floor guaranteed |
Status
Section titled “Status”$\delta$ is a post-processing constraint on the output of ${\alpha_1, \alpha_2, \beta, \gamma}$. The classical term for applying it is istiḥsān (juristic preference) — a check-and-correct layer.
Path A (Ḥanafī) does not use $\delta$ — it instead adds the assumption “grandfather = father” to $\alpha_2$, which pre-empts the situations where $\delta$ would trigger.
Path B (Jumhūr) uses $\delta$ as the unified principle behind their case-specific corrections.
Axiom $\beta$ — ENTITLEMENT (Distribution / التقدير)
Section titled “Axiom $\beta$ — ENTITLEMENT (Distribution / التقدير)”“Each surviving vector draws from a capacity-limited channel.”
Axiom $\beta$ has two sub-components:
$\beta_1$ — Halving (The Pressure Trigger / النقصان)
Section titled “$\beta_1$ — Halving (The Pressure Trigger / النقصان)”Share capacities are halved by the presence of dominant vectors:
$$\text{Share} = \frac{\text{Base}}{2^{\text{Pressure}}}$$
The base shares are the Qurʾānic fractions: $\frac{1}{2}, \frac{1}{3}, \frac{2}{3}$.
The pressure is a count of specific conditions:
| Heir | Base | Pressure from | Result |
|---|---|---|---|
| Husband | $\frac{1}{2}$ | Descendant heir exists | $\frac{1}{4}$ |
| Wife | $\frac{1}{4}$ | Descendant heir exists | $\frac{1}{8}$ |
| Mother | $\frac{1}{3}$ | Descendant heir exists OR $\ge 2$ siblings | $\frac{1}{6}$ |
| Daughter (solo) | $\frac{1}{2}$ | — | — |
| Full Sister (solo) | $\frac{1}{2}$ | — | — |
Critical architectural note — Ghost Pressure: Pressure is evaluated on the initial heir set $H’$ (after attribute exclusion but before person-exclusion). Excluded-by-person vectors (المحجوب بشخص) still cast “ghost” pressure. Example: two brothers excluded by the father still reduce the mother from $\frac{1}{3}$ to $\frac{1}{6}$.
Source:
faraid/hajb.md— “أما المحجوب بشخص: فلا يحجب أحداً حرماناً، وقد يحجبه نقصانًا”
$\beta_2$ — The Group Ceiling (السقف)
Section titled “$\beta_2$ — The Group Ceiling (السقف)”Female farḍ groups have a two-state capacity: solo (Base) vs. plural (Ceiling).
| Group | Solo share | Plural ceiling | Completion gap |
|---|---|---|---|
| Daughters | $\frac{1}{2}$ | $\frac{2}{3}$ | $\frac{1}{6}$ |
| Son’s Daughters | $\frac{1}{2}$ | $\frac{2}{3}$ | $\frac{1}{6}$ |
| Full Sisters | $\frac{1}{2}$ | $\frac{2}{3}$ | $\frac{1}{6}$ |
| Paternal Sisters | $\frac{1}{2}$ | $\frac{2}{3}$ | $\frac{1}{6}$ |
| Maternal Siblings | $\frac{1}{6}$ | $\frac{1}{3}$ | $\frac{1}{6}$ |
The completion gap is universally $\frac{1}{6}$ — the same quantum that appears as the minimum parental guarantee. This is not coincidence; see 05-proofs.md, Theorem 4 (the $\frac{2}{3}$ cap is forced).
Source:
faraid/tasil.md— the 7 agreed-upon bases derive from the denominators of exactly these fractions.
Axiom $\gamma$ — CONSERVATION (Accounting / الاستغراق)
Section titled “Axiom $\gamma$ — CONSERVATION (Accounting / الاستغراق)”“The estate is fully distributed. No fraction is created or destroyed.”
$$\sum_{h \in S} \text{share}(h) = 1$$
This is enforced via three mechanisms:
$\gamma_1$ — Residual Absorption (ʿAṣaba / التعصيب)
Section titled “$\gamma_1$ — Residual Absorption (ʿAṣaba / التعصيب)”If $\sum \text{farḍ shares} < 1$ and an ʿaṣaba heir exists, the ʿaṣaba absorbs the remainder:
$$\text{share}(\text{ʿaṣaba}) = 1 - \sum_{f \in \text{farḍ}} \text{share}(f)$$
Among multiple ʿaṣaba in the same class: for males, equal split; for mixed gender, $2:1$ (للذكر مثل حظ الأنثيين).
Source:
faraid/asaba.md— “ألحقوا الفرائض بأهلها، فما بقي فلأولى رجل ذكر”
$\gamma_2$ — Proportional Compression (ʿAwl / العول)
Section titled “$\gamma_2$ — Proportional Compression (ʿAwl / العول)”If $\sum \text{farḍ shares} > 1$ and no ʿaṣaba heir exists:
$$\text{share}(h) = \frac{\text{farḍ}(h)}{\sum_{f \in S} \text{farḍ}(f)} \times 1$$
All shares are reduced proportionally. The new denominator (عول المسألة) replaces the original base.
Source:
faraid/awl.md— only bases with a $\frac{1}{6}$ component can ʿawl: bases 6, 12, 24.
$\gamma_3$ — Proportional Expansion (Radd / الرد)
Section titled “$\gamma_3$ — Proportional Expansion (Radd / الرد)”If $\sum \text{farḍ shares} < 1$, no ʿaṣaba exists, and blood farḍ heirs remain:
$$\text{share}(h) = \frac{\text{farḍ}(h)}{\sum_{b \in B} \text{farḍ}(b)} \times R$$
where $B$ = blood farḍ heirs (excluding spouses), $R = 1 - \text{share}(\text{spouse})$ if a spouse exists, or $R = 1$ otherwise.
Source:
faraid/radd.md— “صرف الباقي عن الفروض على ذوي الفروض النَّسَبِيَّةِ بقدر فروضهم عند عدم عصبة”
Key finding: ʿAwl and Radd are the same formula — see 04-pipeline.md for the unified normalization.
The Fraction Field $\mathbb{Q}_{(2,3)}$
Section titled “The Fraction Field $\mathbb{Q}_{(2,3)}$”Seed Fractions
Section titled “Seed Fractions”All farḍ shares derive from exactly two seeds:
$$\text{Seeds} = \left{\frac{1}{2}, ; \frac{1}{3}\right}$$
Under halving (division by 2) and the ceiling operation (multiplication by $\frac{4}{3}$, i.e., solo → plural):
$$\frac{1}{2} \xrightarrow{\div 2} \frac{1}{4} \xrightarrow{\div 2} \frac{1}{8}$$
$$\frac{1}{3} \xrightarrow{\div 2} \frac{1}{6}$$
$$\frac{1}{2} \xrightarrow{\times \frac{4}{3}} \frac{2}{3}$$
This generates the complete set: $\left{\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{3}, \frac{1}{6}, \frac{2}{3}\right}$
The 7 + 2 Bases
Section titled “The 7 + 2 Bases”The bases (أصول المسائل) are the LCMs of all possible combinations of these denominators:
7 agreed-upon bases: ${2, 3, 4, 6, 8, 12, 24}$
All have the form $2^a \times 3^b$ where $0 \le a \le 3, ; 0 \le b \le 1$.
2 disputed bases: ${18, 36}$
These arise only in grandfather-with-siblings cases (under the Jumhūr who allow siblings to inherit with the grandfather). Under the Ḥanafī view (grandfather excludes siblings), these bases never occur.
$$18 = 2 \times 3^2, \quad 36 = 2^2 \times 3^2$$
Source:
faraid/tasil.md— “الأصول المتفق عليها سبعة، هي: (٢، ٣، ٤، ٦، ٨، ١٢، ٢٤)” and “الأصول المختلف فيها اثنان، هي: (١٨، ٣٦)“
Which Bases ʿAwl
Section titled “Which Bases ʿAwl”Only bases with a $\frac{1}{6}$ component — those divisible by both 2 and 3 — can ʿawl:
| Base | Can ʿAwl? | ʿAwl values | Source |
|---|---|---|---|
| 2 | No | — | |
| 3 | Disputed | → 4 (Muʿādh only) | |
| 4 | No | — | |
| 6 | Yes | → 7, 8, 9, 10 | faraid/awl.md |
| 8 | No | — | |
| 12 | Yes | → 13, 15, 17 | faraid/awl.md |
| 24 | Yes | → 27 only | faraid/awl.md (“الأصل البخيل”) |
The structural reason: ʿAwl requires at least two distinct prime factors in the denominator to create fraction sums exceeding 1. Bases that are pure powers of 2 (4, 8) cannot ʿawl because all their fractions are nested ($\frac{1}{8} \subset \frac{1}{4} \subset \frac{1}{2}$).
The Gender Rule Within ʿAṣaba
Section titled “The Gender Rule Within ʿAṣaba”When males and females co-inherit as ʿaṣaba:
$$\text{Male share} = 2 \times \text{Female share}$$
Source: Qurʾān 4:11 — “لِلذَّكَرِ مِثْلُ حَظِّ الأُنثَيَيْنِ”
Exception: Maternal siblings ($q = 3$) share equally regardless of gender:
$$\text{share}(q=3, g=1) = \text{share}(q=3, g=0)$$
Source: Qurʾān 4:12 — “فَهُمْ شُرَكَاءُ فِي الثُّلُثِ” (with no gender distinction).
This exception is absorbed into Axiom $\beta$ as a mode parameter: uterine ($q=3$) uses 1:1 ratio; agnatic ($q \ne 3$) uses 2:1 ratio. See 03-exceptions.md for analysis of whether this constitutes a true exception.
Summary
Section titled “Summary”| Axiom | Mechanism | Implements |
|---|---|---|
| $\alpha_1$ | Intermediary-based exclusion | Ḥajb al-ḥirmān (narrow: your واسطة blocks you) |
| $\alpha_2$ | Lexicographic priority $\pi = (j, d, q)$ | ʿAṣaba ranking (broad: best JDQ wins residual) |
| $\beta_1$ | $\text{Share} = \text{Base} / 2^{\text{Pressure}}$ | Farḍ share assignment |
| $\beta_2$ | Solo/Plural capacity with $\frac{1}{6}$ gap | Group ceiling |
| $\gamma_1$ | Residual absorption | ʿAṣaba |
| $\gamma_2$ | Proportional compression | ʿAwl |
| $\gamma_3$ | Proportional expansion | Radd |
| $\delta$ | $\text{superset-kinship} \implies \text{share} \ge$ | Anti-zero / anti-inversion (الأحظّ) |
Generating structure: 2 prime seeds ${2, 3}$, 2 fraction seeds ${\frac{1}{2}, \frac{1}{3}}$, 4 axioms (Path B) or 3 + assumption (Path A), and a finite set of exceptions (see 03-exceptions.md).
References
Section titled “References”- Axiom definitions:
my findings/faraid_axioms.md - Base computation:
faraid/tasil.md - ʿAwl mechanics:
faraid/awl.md - Radd mechanics:
faraid/radd.md - ʿAṣaba ordering:
faraid/asaba.md - Exclusion rules:
faraid/hajb.md