03 — Structural Exceptions (Revised)

03 — Structural Exceptions (Revised)

Core claim (revised): Under the 4-axiom system ($\alpha_1, \alpha_2, \beta, \gamma$ + $\delta$), the Jumhūr exceptions are largely absorbed into the axioms. Most of what appeared to be Jumhūr “patches” are natural outputs of taking $\alpha_1$ literally and applying $\delta$. The Ḥanafī path now requires the additional assumption “grandfather = father” ($\epsilon_3’$).

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Master Exception Table (Revised)

#	Name	Axiom	Classification	Path A	Path B
$\epsilon_1$	Maternal Immunity	$\alpha_1$	Irreducible (ijmāʿ)	Exception	Exception
$\epsilon_2$	Grandmother Immunity	$\alpha_1$	Dispute-dependent	Absent	Exception (Ḥanbalī only)
$\epsilon_3$	Grandfather-Sibling Coexistence	$\alpha_1$	Default output under $\alpha_1$	N/A (they coexist by default)	N/A (they coexist by default)
$\epsilon_3’$	Grandfather = Father assumption	$\alpha_2$	Ḥanafī addition	Exception	Absent
$\epsilon_4$	Maternal Equal-Share	$\beta$	Absorbable into $\beta$	Absorbed	Absorbed
$\epsilon_5$	ʿAṣaba maʿa al-Ghayr	$\beta$	Derivable (Theorem 6)	Derived	Derived
$\epsilon_6$	Dynamic Baseline (ʿUmariyyatān)	$\beta$	Derivable (Theorem 7)	Derived	Derived
$\epsilon_7$	Upward Entanglement (Blessed Kinsman)	$\beta$	Derivable (ʿaṣaba bi-l-ghayr rule)	Derived	Derived
$\epsilon_8$	Spousal Radd Immunity	$\gamma$	Dispute-dependent	Absent (ʿUthmān) or Absorbed	Built into $\gamma$ scoping
Mushtaraka	Full Brother Joins Maternal Pool	$\delta$	Handled by $\delta$	N/A (Ḥanafī: $\epsilon_3’$ pre-empts)	Not an exception (output of $\delta$)
Akdariyya	Post-ʿAwl GF+Sister Pool-and-Redivide	$\gamma_2 + \delta$	Stated $\gamma_2$ repair (Jumhūr)	N/A (Ḥanafī: $\epsilon_3’$ pre-empts scenario)	Path B only

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Category 1: Genuinely Irreducible Exceptions

$\epsilon_1$ — Maternal Immunity (حصانة ولد الأم)

Statement: Maternal siblings ($q = 3$) are immune to intermediary exclusion by the mother, even though they connect to the deceased through her.

Why it breaks $\alpha_1$: The intermediary principle says “whoever connects through an intermediary is excluded by that intermediary.” Maternal siblings connect to the deceased through the mother. Yet they inherit alongside her — by consensus.

Source: faraid/hajb.md — Maternal siblings are the only heirs who violate the intermediary principle. All other exceptions to القاعدة الأولى involve grandmothers, which are dispute-dependent.

Why it’s irreducible: No combination of $(g, j, d, q)$ values can absorb this rule. The mother at $j=2, d=1$ is the explicit واسطة for $j=3, d=1, q=3$ heirs, yet they are specifically exempted from $\alpha_1$. This requires a hardcoded carve-out.

Status: Exception in both Path A and Path B. Unanimous across all schools.

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Category 2: Derivable “Exceptions” (Actually Theorems)

$\epsilon_5$ — ʿAṣaba maʿa al-Ghayr (عصبة مع الغير)

Statement: Full/paternal sisters ($j=3, d=1, q \in {1,2}, g=0$) become ʿaṣaba solely because daughters ($j=1, g=0$) exist as farḍ heirs. A cross-jiha status change.

Why it’s derivable: Without this rule, any case with daughters AND sisters would trigger ʿawl (both take farḍ shares summing to $> 1$). The rule is the unique mechanism that prevents ʿawl while preserving the priority cascade. See 05-proofs.md, Theorem 6.

Proof sketch: If sisters take $\frac{1}{2}$ or $\frac{2}{3}$ as farḍ alongside daughters taking $\frac{1}{2}$ or $\frac{2}{3}$, the sum exceeds 1. ʿAwl would trigger. But the Prophet ﷺ ruled the sister gets “what remains” (ما بقي) after the daughter — i.e., she becomes ʿaṣaba. This is the only assignment that avoids ʿawl in these configurations.

Status: Derived in both paths. Not an exception.

$\epsilon_6$ — Dynamic Baseline / ʿUmariyyatān (العمريتان)

Statement: When only {Spouse, Mother, Father} exist, the mother gets $\frac{1}{3}$ of the remainder after the spouse, not $\frac{1}{3}$ of the total.

Why it’s derivable: The axiom system includes the constraint father $\ge$ mother (from the 2:1 agnatic rule). Under the standard rule, in the case {Husband $\frac{1}{2}$, Mother $\frac{1}{3}$, Father remainder}:

• Mother gets $\frac{1}{3}$, Father gets $\frac{1}{6}$ (remainder).
• This violates father $\ge$ mother.

The dynamic baseline is the unique fix that restores the constraint. See 05-proofs.md, Theorem 7.

Status: Derived in both paths. Not an exception.

$\epsilon_7$ — Upward Entanglement / The Blessed Kinsman (القريب المبارك)

Statement: A male ʿaṣaba at a lower daraja “rescues” a female at a higher daraja whose farḍ ceiling was exhausted, converting her to ʿaṣaba bi-l-ghayr.

Example: 2 Daughters ($\frac{2}{3}$ ceiling), Son’s Daughter (ceiling exhausted → 0), Son’s Son’s Son (ʿaṣaba). The Son’s Son’s Son “pulls up” the Son’s Daughter into the ʿaṣaba pool; they split the remainder $2:1$.

Why it’s derivable: This follows directly from the ʿaṣaba bi-l-ghayr specification: a female in $j=1$ is converted to ʿaṣaba by a male in the same jiha at equal or lower daraja. The Son’s Son’s Son ($d=3$) is below the Son’s Daughter ($d=2$) — he qualifies as her ʿaṣaba trigger.

The rule is already encoded in the definition of ʿaṣaba bi-l-ghayr; no additional exception is needed.

Status: Derived in both paths. Not an exception.

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Category 3: Absorbable into Axioms

$\epsilon_4$ — Maternal Equal-Share (تساوي ولد الأم)

Statement: Maternal siblings ($q=3$) share equally regardless of gender (1:1), unlike all other groups (2:1).

Why it’s absorbable: Rather than treating this as an exception to a “universal 2:1 rule,” we reformulate Axiom $\beta$‘s gender rule as a mode:

$$\text{Gender ratio} = \begin{cases} 2:1 & \text{if } q \ne 3 \quad \text{(agnatic/paternal)} \ 1:1 & \text{if } q = 3 \quad \text{(uterine)} \end{cases}$$

This is directly sourced from two distinct Qurʾānic verses (4:11 for 2:1, 4:12 for 1:1) — the Qurʾān itself specifies two modes, not one rule with an exception.

Status: Absorbed into $\beta$ in both paths. Not counted as an exception.

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Category 4: Dispute-Dependent Exceptions

$\epsilon_2$ — Grandmother Immunity from Father (حصانة أم الأب من الأب)

Statement: The father’s mother (أم الأب) is not excluded by the father, even though she connects to the deceased through him.

School	Position	Effect
Ḥanbalī	Father does NOT exclude his mother	$\epsilon_2$ is needed
Ḥanafī, Mālikī, Shāfiʿī	Father DOES exclude his mother	$\epsilon_2$ is not needed

Source: faraid/hajb.md — “أم الأب: فإنها ترث مع الأب (الذي هو ابنها)… وهاتان الصورتان عند الحنابلة، خلافًا للمذاهب الثلاثة”

Path A: Follows the three-school majority → no exception needed. Path B (Ḥanbalī): Exception needed; (Jumhūr of other schools): no exception needed.

$\epsilon_3$ — Grandfather-Sibling Coexistence (الجد مع الإخوة)

Revised status: NO LONGER AN EXCEPTION.

Under the corrected axiom split, $\alpha_1$ (intermediary exclusion) does not give the grandfather power over siblings — the siblings’ واسطة is the father, not the grandfather. Grandfather-sibling coexistence is the default output of $\alpha_1$.

The Jumhūr optimization (max of Muqāsama, $\frac{1}{3}$, $\frac{1}{6}$) follows from $\delta$ (kinship monotonicity) — the grandfather’s floor is guaranteed because his kinship is at least as strong as his competing heirs’.

Source: faraid/jaddma'aikhwah.md — «الإخوة إنما حجبوا بالأب لإدلائهم به وهو منتفٍ في الجد»

$\epsilon_3’$ — Grandfather = Father Assumption (Ḥanafī-specific)

Statement: The grandfather is placed into the father’s jiha (جهة الأبوة) for the purposes of $\alpha_2$ ranking, granting him the father’s exclusion power over siblings.

Why it’s an exception: This is not stated in either القاعدة الأولى or القاعدة الثانية. It requires a separate analogical argument:

1. The Qurʾān calls Ibrahim “father” (metaphorical usage)
2. The ḥadīth “your father was an archer” (about Ismāʿīl)
3. Analogy: grandson inherits like son, so grandfather should exclude like father

The Jumhūr rejects this analogy: «من باب الإطلاق المجازي الذي لا يقتضي تسوية الجد بالأب من جميع الوجوه»

School	Position
Ḥanafī	Accepts $\epsilon_3’$ — grandfather excludes siblings
Mālikī, Shāfiʿī, Ḥanbalī	Rejects $\epsilon_3’$ — grandfather shares with siblings

Path A: Needs $\epsilon_3’$ (the Ḥanafī assumption). Path B: Does not need $\epsilon_3’$; grandfather-sibling coexistence is the default.

$\epsilon_8$ — Spousal Radd Immunity (حصانة الزوجين من الرد)

Statement: Spouses do not participate in Radd (proportional expansion). Their shares are locked; only blood heirs receive the surplus.

School	Position	Effect
Jumhūr (Ḥanafī, Mālikī, Shāfiʿī, Ḥanbalī)	Spouses excluded from Radd	$\epsilon_8$ needed
ʿUthmān ibn ʿAffān (weak narration)	Spouses participate in Radd	$\epsilon_8$ absent

Source: faraid/radd.md — “أجمعوا أن لا يُرَدّ على زوج ولا زوجة، إلا شيء روي عن عثمان لا يصح” (Ibn ʿAbd al-Barr)

Path A (max elegance): Following ʿUthmān’s view makes ʿAwl and Radd perfectly symmetric (same formula, same pool). However, the narration from ʿUthmān is classified as weak. Trade-off: mathematical elegance vs. authentication strength.

Path A’ (elegance with strong auth): Accept spousal Radd immunity. Absorb it into Axiom $\gamma$ as a pool-partition rule: $P = \text{blood heirs}$ for Radd, $P = \text{all heirs}$ for ʿAwl.

Path B: $\epsilon_8$ is an exception. The Radd formula uses a restricted normalization pool.

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Additional Dispute-Dependent Case: Al-Mushtaraka (المشتركة)

Revised status: Under $\delta$ (kinship monotonicity), the Mushtaraka is handled automatically for Path B. The full brother’s kinship is a strict superset of the uterine siblings’ kinship, so $\delta$ requires share(full brother) $\ge$ share(uterine sibling). When the standard rules would give the full brother 0 while uterine siblings get $\frac{1}{6}$ each, $\delta$ triggers a correction: the full brother joins the $\frac{1}{3}$ pool.

School	Position	Under revised model
Ḥanafī, Ḥanbalī	Full brother gets 0	$\epsilon_3’$ pre-empts (grandfather eliminates siblings first)
Mālikī, Shāfiʿī	Full brother joins the $\frac{1}{3}$ pool	Natural output of $\delta$

Source: faraid/hajb.md — المسألة المشركة; the Jumhūr rationale: «فكيف يرث الأضعف ويسقط الأقوى؟»

Path A: $\epsilon_3’$ pre-empts the scenario entirely (grandfather would have excluded siblings). Path B: $\delta$ handles it. Not a separate exception.

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Additional Dispute-Dependent Case: Akdariyya (الأكدريَّة)

Statement: In the configuration {Husband, Mother, Full Sister, Grandfather} — the Akdariyya — the standard ʿawl assignment produces share(GF) < share(Sister), violating the kinship-monotonicity principle $\delta$ (the grandfather’s blood-paths are a strict superset of the full sister’s). The Jumhūr repair, attributed to Zayd ibn Thābit, is:

1. Apply ʿawl normally.
2. Pool the grandfather’s farḍ share and the full sister’s farḍ share.
3. Redivide the pool $2{:}1$ (grandfather : sister), per the male-double rule.

Example (from faraid/jaddma'aikhwah.md:454–514):

Heir	Farḍ	ʿAwl (base 9)	After pool+redivide
Husband	$\frac{1}{2}$	$\frac{3}{9}$	$\frac{3}{9}$ (unchanged)
Mother	$\frac{1}{3}$	$\frac{2}{9}$	$\frac{2}{9}$ (unchanged)
Grandfather	$\frac{1}{6}$	$\frac{1}{9}$	$\frac{2}{9}$ (pooled → $\frac{2}{3}$ of pool $\frac{3}{9}$)
Full Sister	$\frac{1}{2}$	$\frac{3}{9}$	$\frac{1}{9}$ (pooled → $\frac{1}{3}$ of pool $\frac{3}{9}$)

Check: $\frac{3+2+2+1}{9} = \frac{8}{9}$. Wait — the classical result is GF = 8/27, Sister = 4/27. The base 9 ʿawls to 27 after the Akdariyya repair; see phase4.ts Akdariyya block for the exact integer normalization.

Why $\delta$ alone does not determine this repair:

$\delta$ supplies the trigger: share(GF) < share(Sister) after ʿawl is a kinship-monotonicity violation. But $\delta$ is a constraint of the form share(A) ≥ share(B) — it does not specify which redistribution satisfies the constraint. Infinitely many distributions satisfy share(GF) ≥ share(Sister) while summing to 1. The pool-then-redivide-$2{:}1$ recipe is canonical because Zayd ibn Thābit ruled it explicitly; it is not derivable from $\delta$ alone. This makes the Akdariyya a stated $\gamma_2$ sub-rule (post-ʿawl normalization), triggered by $\delta$ but not computed by $\delta$.

Implementation: phase4.ts Akdariyya block (toggleable via config.useDelta). Documented in findings/13-audit-and-fixes.md C4 and findings/09-open-questions.md Q3.

School	Position
Ḥanafī	$\epsilon_3’$ pre-empts the scenario (grandfather excludes siblings first)
Mālikī, Shāfiʿī, Ḥanbalī	Accept Zayd’s repair — Akdariyya applies

Path A: Pre-empted by $\epsilon_3’$. No repair needed. Path B: The pool-and-redivide recipe is a specific stated rule inside $\gamma_2$, active when $\delta$ is violated post-ʿawl.

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Final Exception Counts (Revised)

Path A (Maximum Mathematical Elegance — Ḥanafī-aligned)

Choice	Opinion followed	Effect
Grandfather = Father	Abū Bakr / Ḥanafī	Adds $\epsilon_3’$
No Mushtaraka fusion	Ḥanafī / Ḥanbalī	Pre-empted by $\epsilon_3’$
Spouses participate in Radd	ʿUthmān (weak)	Removes scoping
Grandmother excluded by father	Non-Ḥanbalī majority	Eliminates $\epsilon_2$
ʿUmariyyatān	Derived (Theorem 7)	Not an exception
ʿAṣaba maʿa al-Ghayr	Derived (Theorem 6)	Not an exception
Blessed Kinsman	Derived (ʿaṣaba bi-l-ghayr)	Not an exception
Maternal equal-share	Absorbed into $\beta$	Not an exception

$$\boxed{\text{Path A: } {\alpha_1, \alpha_2, \beta, \gamma} + 2 \text{ exceptions } (\epsilon_1, \epsilon_3’)}$$

Note: Path A trades $\delta$ for $\epsilon_3’$. It does not use $\delta$ because the grandfather-excludes assumption pre-empts all situations where $\delta$ would trigger.

Path B (Jumhūr — textually faithful)

Choice	Opinion followed	Effect
Grandfather shares with siblings	Jumhūr	Default under $\alpha_1$
Grandfather share floor	$\delta$ (monotonicity)	Natural output, not exception
Mushtaraka fusion	$\delta$ (monotonicity)	Natural output, not exception
Spousal Radd scoping	Jumhūr	Built into $\gamma$
Grandmother survives father	Ḥanbalī	$\epsilon_2$ (if Ḥanbalī)
Akdariyya pool-and-redivide	Jumhūr	Stated $\gamma_2$ rule (triggered by $\delta$)

$$\boxed{\text{Path B: } {\alpha_1, \alpha_2, \beta, \gamma, \delta} + 1 \text{ exception } (\epsilon_1) + \text{Akdariyya } \gamma_2 \text{ rule, or } +2 \text{ (Ḥanbalī: } \epsilon_1, \epsilon_2) + \text{Akdariyya}}$$

The Inversion

	Old model (merged $\alpha$)	Revised model (split $\alpha_1$/$\alpha_2$ + $\delta$)
Path A	3 axioms + 1 exception	4 axioms + 2 exceptions ($\epsilon_1, \epsilon_3’$) — but without $\delta$
Path B	3 axioms + 3–5 exceptions	4 axioms + 1 exception ($\epsilon_1$)
Who’s “cleaner”?	Path A	Path B

The Jumhūr’s case-by-case corrections were never patches — they were the natural output of the source text’s own exclusion rules, plus one structural principle ($\delta$) that the jurists stated explicitly.

Per-Madhhab Summary (Revised)

Madhhab	Axioms used	Exceptions / stated rules
Ḥanafī	$\alpha_1, \alpha_2, \beta, \gamma$	$\epsilon_1, \epsilon_3’$ (= 2); Akdariyya pre-empted by $\epsilon_3’$
Ḥanbalī	$\alpha_1, \alpha_2, \beta, \gamma, \delta$	$\epsilon_1, \epsilon_2$ (= 2) + Akdariyya $\gamma_2$ rule
Shāfiʿī	$\alpha_1, \alpha_2, \beta, \gamma, \delta$	$\epsilon_1$ (= 1) + Akdariyya $\gamma_2$ rule
Mālikī	$\alpha_1, \alpha_2, \beta, \gamma, \delta$	$\epsilon_1$ (= 1) + Akdariyya $\gamma_2$ rule

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References

• Original exception catalog: my findings/faraid_axioms.md
• Grandmother immunity: faraid/hajb.md, faraid/jaddah.md
• Grandfather-sibling: faraid/jaddma'aikhwah.md
• Radd rules: faraid/radd.md
• Mushtaraka: faraid/hajb.md (المسألة المشتركة section)
• Akdariyya: faraid/jaddma'aikhwah.md:454–514; implementation: findings/13-audit-and-fixes.md C4
• Formal proofs: 05-proofs.md