10 — Complete Dispute Matrix

Purpose: Every classical juristic dispute in ʿilm al-farāʾiḍ, tabulated with: all known opinions → mathematical-elegance analysis → textual-fidelity analysis → which opinion Path A selects → which opinion Path B selects → source citation → config toggle (where applicable).

Two-Path Reminder: Path A = maximally consistent choice per dispute (may be opinion-unfaithful overall). Path B = choice most directly supported by the source text. Neither path is a madhhab preset; madhhab alignment is incidental.

Master Dispute Table

#	Dispute	Arabic Term	Path A choice	Path B choice	Toggle
D1	ʿAwl — does proportional compression apply?	العَوْل	Jumhūr (ʿawl applies)	Jumhūr (ʿawl applies)	— (agreed)
D2	ʿUmariyyatān — mother’s ⅓ of what?	العُمَرِيَّتَان	Jumhūr (⅓ of remainder)	Jumhūr (⅓ of remainder)	— (derived)
D3	Radd — surplus to blood heirs or Bayt al-Māl?	الرَّدّ	ʿAlī / Ḥanafī (Radd applies)	ʿAlī / Ḥanafī (Radd applies)	— (agreed)
D3a	Radd — spouses included or excluded?	الرَّدّ على الزوجين	ʿUthmān (include — symmetric)	Jumhūr (exclude — strong auth)	`spouse_participates_in_radd`
D4	Grandfather vs. siblings — excludes or coexists?	الجد مع الإخوة	Ḥanafī (grandfather excludes)	Jumhūr (coexist; δ sets floor)	`grandfather_equals_father`
D5	Grandmother excluded by father?	حجب الأب لأمه	Jumhūr 3-school (excluded)	Jumhūr 3-school (excluded)	—
D5a	Grandmother immune from father (Ḥanbalī variant)	حصانة أم الأب	Absent	ε₂ active (Ḥanbalī option)	`grandmother_immune_to_father`
D6	Grandmother priority — proximity vs. side?	أولوية الجدات	Ḥanafī/Ḥanbalī (proximity)	Mālikī/Shāfiʿī (direction)	—
D7	Double-kinship grandmother share	الجدة ذات القرابتين	Equal (Abū Ḥanīfa/Shāfiʿī)	Weighted (Muḥ. ibn al-Ḥasan)	—
D8	Mushtaraka — full brother joins ⅓ pool?	المشتركة / الحمارية	Pre-empted by ε₃′	Output of δ (joins pool)	`use_delta`
D9	Akdariyya — post-ʿawl pooling	الأكدريَّة	Standard (no post-ʿawl pool)	Zayd’s pooling (Jumhūr)	— (open Q)
D10	Number of siblings halving mother	حجب الأم بالإخوة	Jumhūr (2 suffice)	Jumhūr (2 suffice)	— (agreed)
D11	Dhawī al-Arḥām — inherit at all?	توريث ذوي الأرحام	ʿAlī/Ḥanafī (yes)	ʿAlī/Ḥanafī (yes)	—
D12	DhA method — tanzīl or qarāba?	التنزيل أم القرابة	Ḥanafī (qarāba)	Ḥanbalī/Shāfiʿī (tanzīl)	`dharham_method`
D13	Grandmother tiers — how many qualify?	درجات الجدات الوارثات	Shāfiʿī/Ḥanafī (by rule)	Mālikī (2 tiers only)	—
D14	ʿAṣaba jiha count	عدد جهات التعصيب	5 (Ḥanafī)	6–7 (Jumhūr)	driven by D4
D15	ʿAwl of base 3	عَوْل الأصل من ثلاثة	Jumhūr (never)	Jumhūr (never)	— (agreed)
D16	Ḥaml — how many scenarios?	تقادير الحمل	Ḥanafī ($k=2$: male, female)	Shāfiʿī/Ḥanbalī ($k=6$: stillborn, 1m, 1f, 2m, 2f, 1m+1f)	`haml_scenario_set`
D17	Khunthā — aggregation rule	ميراث الخنثى المشكل	Ḥanafī (single-pick: worst-for-khunthā scenario)	Shāfiʿī/clarifiable (component-min + mawqūf); Mālikī/matured (component-mean)	`khunthas_aggregation`
D18	Gharqā — mutual inheritance	توارث الغرقى	Jumhūr — Ḥanafī/Mālikī/Shāfiʿī (no mutual inheritance)	Ḥanbalī (Chain Forest with TILD/ṬARIF cycle-breaking)	`gharqa_mutual_inheritance`

Detailed Analysis

D1 — ʿAwl: Does Proportional Compression Apply?

Scenario: Fard shares sum to > 1. Who absorbs the deficit?

Opinion	Holder(s)	School(s)	Effect
ʿAwl applies — all fard heirs absorb deficit proportionally	ʿUmar, Zayd ibn Thābit, ʿAlī, Ibn Masʿūd; Jumhūr	Ḥanafī, Mālikī, Shāfiʿī, Ḥanbalī	Each heir’s actual share = fard × (1 / Σfard)
No ʿawl — priority exists; deficit falls only on “shiftable” heirs	Ibn ʿAbbās رضي الله عنهما; Ẓāhiriyyah	Minority	Husband/Wife/Mother paid in full; sister absorbs deficit

Example: {Husband ½, Mother ⅓, Full Sister ½}. Base 6, Σ = 4/3.

Jumhūr: ʿawl to 8. Husband 3/8, Mother 2/8, Sister 3/8.
Ibn ʿAbbās: Husband 3/6, Mother 2/6, Sister 1/6 (remainder).

Mathematical-elegance analysis:
Ibn ʿAbbās is logically prior: fractions summing > 1 is a self-contradiction if the estate is the unit. His priority hierarchy avoids this by defining a canonical reduction order. However, his system requires an external classification of “shiftable vs. non-shiftable” heirs — that classification is itself a non-trivial rule. The Jumhūr’s ʿawl is a single formula (proportional scaling) with no additional classification. For a computational engine, Jumhūr is simpler to implement.

Textual-fidelity analysis:
Ibn ʿAbbās reads the Qurʾān as not having sanctioned conflicting mandates: «إن الله لم يفرض في المال فريضتين يستغرقان أكثر من المال» (reconstructed principle). The Jumhūr apply every Qurʾānic fard at face value and accept the resulting sum.

Path A choice: Jumhūr (ʿawl). The proportional formula is a single rule; no extra classification of heirs is required.
Path B choice: Jumhūr (ʿawl). The source texts treat ʿawl as the established ruling.

Note: Both paths agree. This is not a genuine Path A/B split. However, an engine MUST include an Ibn ʿAbbās mode to be fully opinion-agnostic.

«ألحقوا الفرائض بأهلها، فما بقي فلأولى رجل ذكر» — Ḥadīth; basis for ʿaṣaba + ʿawl sequence
«لوددت أني وهؤلاء … نبتهل فنجعل لعنة الله على الكاذبين» — Ibn ʿAbbās on the ʿawl dispute

Toggle: awl_method: "proportional" | "ibn_abbas_priority" (not in current architecture; may be needed for full opinion-agnosticism)
Source: faraid/awl.md

D2 — ʿUmariyyatān: Mother’s ⅓ of Remainder or Total?

Scenario 1: {Husband ½, Father ʿaṣaba, Mother}.
Scenario 2: {Wife ¼, Father ʿaṣaba, Mother}.

Opinion	Holder(s)	School(s)	Case-1 result	Case-2 result
Mother gets ⅓ of remainder after spouse	ʿUmar ibn al-Khaṭṭāb (endorsed by Zayd); Jumhūr	All four	Mother 1/6, Father 1/3	Mother 1/4, Father 1/2
Mother gets ⅓ of total	Ibn ʿAbbās رضي الله عنهما	Minority	Mother 1/3, Father 1/6	Mother 1/3, Father 5/12

Mathematical-elegance analysis:
The Jumhūr position is the unique solution preserving the constraint father ≥ mother when a spouse is present (proven formally in 05-proofs.md, Theorem 7). It is derived from the axiom system — not an exception. Ibn ʿAbbās’ position is texturally simpler but produces father < mother in Case 1, violating a general principle his own system accepts.

Textual-fidelity analysis:
Ibn ʿAbbās reads ﴿فَلِأُمِّهِ الثُّلُثُ﴾ without qualification. The Jumhūr’s “remainder” reading requires an interpretive step (contextualizing the ⅓ mandate by another mandate that father ≥ mother).

Path A choice: Jumhūr — mathematically forced; the only consistent solution.
Path B choice: Jumhūr — the source texts treat this as the established ruling of ʿUmar and Zayd.

Note: Both paths agree. Ibn ʿAbbās’ position is mathematically inconsistent within the 4-axiom framework — it violates the father ≥ mother entailment. An opinion-agnostic engine would need to explicitly relax that constraint to offer Ibn ʿAbbās mode.

Toggle: None in current architecture. Adding one requires relax_father_ge_mother: bool and replaces Theorem 7 with a fixed ⅓.
Source: faraid/awl.md, findings/05-proofs.md (Theorem 7)

D3 — Radd: Does Surplus Return to Blood Heirs?

Scenario: Fard shares sum to < 1; no ʿaṣaba present.

Opinion	Holder(s)	School(s)	Effect
Radd applies — surplus to blood fard heirs proportionally	ʿAlī, Ibn Masʿūd	Ḥanafī, Ḥanbalī; Shāfiʿī (if Bayt al-Māl dysfunctional)	Blood heirs scaled up
No Radd — surplus to Bayt al-Māl	Zayd ibn Thābit	Mālikī; Shāfiʿī (if Bayt al-Māl functional)	Blood heirs keep fard; surplus goes to state

Mathematical-elegance analysis:
Radd and ʿawl are symmetric operations under Axiom γ (Conservation): ʿawl scales shares down when Σ > 1; Radd scales them up when Σ < 1. Accepting Radd makes γ a bidirectional normalization law. Rejecting Radd breaks the symmetry and requires a conditional (“if Bayt al-Māl functional”).

Textual-fidelity analysis:
Zayd’s position is based on the principle that designated fard shares are maxima, not minima. ʿAlī’s position is based on keeping inheritance within the family. Both have Companion authority; neither has a Qurʾānic text directly on point.

Path A choice: ʿAlī/Ḥanafī (Radd applies) — preserves the γ symmetry.
Path B choice: ʿAlī/Ḥanafī (Radd applies) — the source texts favor this for the engine’s use case.

Note: Both paths agree on Radd applying. See D3a for the spousal sub-dispute.
Source: faraid/radd.md

D3a — Radd: Are Spouses Included?

Opinion	Holder(s)	School(s)	Effect
Spouses excluded from Radd	Jumhūr	All four schools	Blood heirs pool = P_blood; spouses locked at fard
Spouses included in Radd	ʿUthmān ibn ʿAffān رضي الله عنه (weak narration); adopted by Ibn Taymiyya, al-Saʿdī	Minority	Radd pool = all heirs; spouses scaled up with everyone

Mathematical-elegance analysis:
Including spouses makes ʿawl and Radd apply identically (same pool, same formula). The normalization step in γ becomes:

$$\forall i,; s_i \leftarrow s_i \cdot \frac{1}{\sum_j s_j}$$

No pool-partitioning rule is needed. This is the most elegant formulation.

Textual-fidelity analysis:
The Jumhūr’s exclusion has near-consensus authority: «أجمعوا أن لا يُرَدّ على زوج ولا زوجة، إلا شيء روي عن عثمان لا يصح» (Ibn ʿAbd al-Barr). The ʿUthmān narration is classified as weak (لا يصح). Mathematical elegance conflicts with authentication strength here.

Path A choice: ʿUthmān (include spouses) — maximally symmetric γ formula; no pool-splitting needed.
Path B choice: Jumhūr (exclude spouses) — strong consensus; pool-partitioning is absorbed into γ’s definition.

Toggle: spouse_participates_in_radd: bool (false = Jumhūr default)
Source: faraid/radd.md

D4 — Grandfather vs. Siblings: Excludes or Coexists?

Scenario: Paternal grandfather present alongside full brothers, paternal brothers, or sisters. Father is deceased.

Opinion	Holder(s)	School(s)	Effect
Grandfather excludes all siblings	Abū Bakr al-Ṣiddīq, Ibn ʿAbbās, ʿĀʾisha, Muʿādh, Ubayy, Abū al-Dardāʾ, Abū Hurayra, Ibn al-Zubayr, Abū Mūsā	Ḥanafī (fatwa); one Aḥmad riwāya	Grandfather inherits alone (ʿaṣaba or with fard heirs)
Grandfather coexists with siblings (Zayd’s three-option method)	ʿAlī, Ibn Masʿūd, Zayd ibn Thābit	Mālikī, Shāfiʿī, Ḥanbalī	Grandfather takes best of: Muqāsama / ⅓ / ⅙; siblings share remainder

Zayd’s three options for grandfather’s share (no other fard holders):

$$s_{\text{jadd}} = \max!\left(\frac{1}{n+1},; \frac{1}{3},; \frac{1}{6}\right)$$

where $n$ = number of competing siblings (counting each sibling as 1 regardless of gender; sisters count as ½ in Muqāsama calculation).

Example cases:

{Grandfather, 1 Sister}: Muqāsama → Grandfather ⅔, Sister ⅓.
{Grandfather, 2 Brothers}: Muqāsama = ⅓ = floor(⅓). Grandfather ⅓, Brothers share ⅔.
{Grandfather, 5 Brothers}: Muqāsama = ⅙ = floor(⅙). Grandfather ⅙, Brothers share 5/6.

Mathematical-elegance analysis:
The Ḥanafī position (grandfather = father, encoded as ε₃′) is simpler: no three-option max formula is needed. The grandfather’s rank in the ʿaṣaba cascade is identical to the father’s. The Jumhūr’s max formula requires a non-trivial optimization step, but it is the natural output of $\alpha_1$ without the Ḥanafī analogical extension — no additional assumption is required.

Textual-fidelity analysis:
The source text states the argument directly: «الإخوة إنما حجبوا بالأب لإدلائهم به وهو منتفٍ في الجد» (the siblings are only excluded by the father because they connect through him; that intermediary relationship is absent for the grandfather). This is $\alpha_1$ applied literally. The Ḥanafī position requires an extra analogical step: «الجد يرث بجهة الأبوة» (grandfather inherits by the direction of paternity) — accepted by the Ḥanafī school but rejected by the Jumhūr as «إطلاق مجازي لا يقتضي تسوية الجد بالأب من جميع الوجوه».

Path A choice: Ḥanafī (grandfather excludes) — ε₃′ added; simpler cascade; no three-option formula.
Path B choice: Jumhūr (coexist) — $\delta$ sets the grandfather’s floor; ε₃′ absent; $\alpha_1$ applied literally.

Toggle: grandfather_equals_father: bool (false = Jumhūr default)
Source: faraid/jaddma'aikhwah.md, findings/03-exceptions.md (ε₃, ε₃′)

D5 — Grandmother Excluded by Father?

Scenario: Father alive. His mother (أم الأب) also alive. Do they coexist?

Opinion	Holder(s)	School(s)	Effect
Father does exclude his mother	Three-school majority	Ḥanafī, Mālikī, Shāfiʿī	أم الأب gets 0; standard $\alpha_1$ applies
Father does NOT exclude his mother	Ibn Masʿūd (narration: «إنها أول جدة أطعمها رسول الله ﷺ سدسًا مع ابنها»); Ḥanbalī	Ḥanbalī	أم الأب gets ⅙ alongside her son the father

Mathematical-elegance analysis:
The three-school position is more consistent with $\alpha_1$: the father is the explicit واسطة for his mother, so she should be excluded. Permitting the exception (ε₂) requires a hardcoded carve-out that the intermediary rule does not apply to this specific relationship.

Textual-fidelity analysis:
The Ḥanbalī position rests on a specific narration that the Prophet ﷺ gave ⅙ to the first grandmother alongside her living son. This is a direct Prophetic precedent and is textually compelling.

Path A choice: Three-school Jumhūr (father excludes) — no exception needed; consistent with $\alpha_1$.
Path B choice: Jumhūr 3-school default, with ε₂ as an optional Ḥanbalī variant.

Toggle: grandmother_immune_to_father: bool (false = three-school default; true = Ḥanbalī)
Source: faraid/jaddah.md, findings/03-exceptions.md (ε₂)

D6 — Grandmother Priority: Proximity vs. Side

Scenario: Two grandmothers at different degrees — one closer from father’s side, one more distant from mother’s side.

Opinion	Holder(s)	School(s)	Rule
Closer wins regardless of side	Ḥanafī, Ḥanbalī	—	أم الأب (degree 2) excludes أم أم أم (degree 3) even if the former is from father’s side
Side preserved — if paternal-side grandmother is closer, she takes ⅙ alone; if equal degree, both share	Mālikī, Shāfiʿī	—	Each side maintains independent claim
If equal degree, share equally	All schools (agreed)	—	No dispute at equal degree

Specific case (unequal degree, father’s side closer):

{أم الأب, degree 2} vs {أم أم أم, degree 3}:
- Ḥanafī/Ḥanbalī: أم الأب takes ⅙ alone (proximity wins).
- Mālikī/Shāfiʿī: Both share ⅙ equally (½ each).

Mathematical-elegance analysis:
The proximity-only rule (Ḥanafī/Ḥanbalī) is simpler: one criterion (degree) governs all priority. No directional weighting is needed. The Mālikī/Shāfiʿī rule requires a two-dimensional tie-breaking (degree first, then side).

Path A choice: Ḥanafī/Ḥanbalī (proximity only) — single criterion.
Path B choice: Mālikī/Shāfiʿī (direction preserved) — source texts distinguish maternal and paternal lines independently.

Toggle: grandmother_priority: "proximity" | "direction" (not in current architecture)
Source: faraid/jaddah.md

D7 — Double-Kinship Grandmother

Scenario: A grandmother who is both أم أم الأم AND أم أبي الأب (counts twice) competes with a single-kinship grandmother.

Opinion	Holder(s)	School(s)	Share
Both share ⅙ equally	Abū Ḥanīfa, Abū Yūsuf, Shāfiʿī, Mālik (by analogy)	—	⅛ each (½ × ⅙)
Double-kinship gets ⅔ of the ⅙; single gets ⅓	Muḥammad ibn al-Ḥasan al-Shaybānī; Ḥanbalī (muʿtamad)	—	Double: 2/18; Single: 1/18

Mathematical-elegance analysis:
Equal sharing (⅙ ÷ 2) requires no additional weighting formula. The weighted approach (2:1 for double:single kinship) is more consistent with the general 2:1 gender/strength principle of agnatic inheritance, but introduces complexity.

Path A choice: Equal sharing — simpler; no kinship-count weighting mechanism needed.
Path B choice: Weighted (Ḥanbalī/Muḥammad) — more consistent with the general kinship-strength principle.

Source: faraid/jaddah.md

D8 — Mushtaraka: Does Full Brother Join the ⅓ Pool?

Scenario: {Husband ½, Mother ⅙, 2+ Uterine Siblings ⅓, Full Brother}. Standard ʿaṣaba logic gives Full Brother remainder = 0. But uterine siblings (weaker kinship) receive ⅓.

Opinion	Holder(s)	School(s)	Effect
Full brother gets 0	ʿAlī, Ibn Masʿūd, Abū Mūsā; ʿUmar’s first ruling	Ḥanafī, Ḥanbalī	Strict ʿaṣaba rule; no classification change
Full brother joins ⅓ pool equally with uterine siblings	ʿUthmān, Zayd; ʿUmar’s second ruling (reversed)	Mālikī, Shāfiʿī	Full brother reclassified from ʿaṣaba to fard for this case

Famous exchange: When ʿUmar initially gave full brothers zero, they protested: «هَبْ أن أبانا كان حمارًا، أليست أمنا واحدة؟» — hence “Ḥimāriyya” (the Donkey Case). Zayd’s justification: «ما زادهم الأبُ إلا قربًا، وأشرك بينهم في الثلث».

Mathematical-elegance analysis:
The Ḥanafī position is more consistent with the ʿaṣaba definition: the full brother is ʿaṣaba; ʿaṣaba takes remainder; remainder is zero. No classification change needed. The Mālikī/Shāfiʿī position requires temporarily reclassifying the full brother’s jiha for this specific co-heir configuration — which is the anti-zero component of Axiom δ.

Textual-fidelity analysis:
The anti-zero principle «كيف يرث الأضعف ويسقط الأقوى؟» (How can the weaker inherit while the stronger gets nothing?) is stated explicitly in the source text and is the justification for δ. The Mālikī/Shāfiʿī position is therefore the direct output of applying δ to this case — it is not a patch.

Interaction with D4: Under Path A (grandfather_equals_father = true), the grandfather would have excluded all siblings before the Mushtaraka scenario can arise. D8 is only reachable under Path B.

Path A choice: D8 is pre-empted (ε₃′ eliminates the scenario).
Path B choice: Mālikī/Shāfiʿī (full brother joins pool) — output of δ; not a separate exception.

Toggle: use_delta: bool — when true, δ handles Mushtaraka automatically. No separate toggle for Mushtaraka.
Source: faraid/hajb.md, findings/03-exceptions.md (Mushtaraka section)

D9 — Akdariyya: Post-ʿAwl Pooling Step

Scenario: {Husband ½, Mother ⅓, Grandfather, Full Sister}. This produces ʿawl. Then what?

Opinion	Holder(s)	School(s)	Effect
Zayd’s method: Apply grandfather-sibling rules (sister takes ½ fard), ʿawl the estate to base 9, then pool grandfather’s share + sister’s post-ʿawl share and redistribute 2:1	Zayd ibn Thābit	Mālikī, Shāfiʿī, Ḥanbalī	Husband 3/9, Mother 2/9, Grandfather 4/9×⅔ = 8/27 (?), Sister remainder after pooling
Standard grandfather rule, no post-ʿawl redistribution	Not explicitly attributed	Mechanical reading	Sister gets 0 (ʿaṣaba after ʿawl exhausts remainder)

Note on final shares (Zayd’s method, simplified): The estate is first ʿawled; Grandfather + Sister’s combined post-ʿawl portion is then split 2:1 internally. The classical books resolve this to base 27:

Husband: 9/27, Mother: 6/27, Grandfather: 8/27, Sister: 4/27.

Mathematical-elegance analysis:
The post-ʿawl pooling is an ad hoc step with no parallels elsewhere in the system. It is unclear whether it can be derived from δ (which might prevent the sister from receiving less than a δ-forced floor), or whether it is an independent exception (candidate ε₉). This question is flagged in 09-open-questions.md (Q3) and is not yet resolved.

Path A choice: N/A — D4’s grandfather_equals_father = true removes the grandfather-sibling coexistence that creates this case.
Path B choice: Zayd’s pooling (Jumhūr) — default in the source texts; classification pending δ analysis.

Toggle: None; this is handled by the grandfather-sibling coexistence path. Under Path A it never arises.
Source: faraid/awl.md, faraid/asaba.md, findings/09-open-questions.md (Q3)

D10 — Number of Siblings Required to Halve Mother

Scenario: Mother is present with siblings. Normally mother takes ⅓; if siblings are present, she is reduced to ⅙. How many siblings trigger this?

Opinion	Holder(s)	School(s)	Threshold
2 or more siblings (any gender) suffice	Jumhūr, all Companions except Ibn ʿAbbās and Muʿādh	Ḥanafī, Mālikī, Shāfiʿī, Ḥanbalī	2 siblings = mother halved
1 sibling suffices	Ibn ʿAbbās	Minority	1 sibling = mother halved
Males only (or mixed groups) trigger halving; two females alone do not	Muʿādh ibn Jabal	Isolated	All-female sibling groups don’t reduce mother

Path A choice: Jumhūr (2 suffice, any gender) — most consistent and most supported.
Path B choice: Jumhūr (2 suffice, any gender) — both paths agree.

Note: Muʿādh’s position produces base-3 ʿawl (D15), which all schools reject.
Source: faraid/awl.md, faraid/hajb.md

D11 — Dhawī al-Arḥām: Do They Inherit at All?

Scenario: No fard heirs, no ʿaṣaba. Distant relatives (maternal uncle, aunt, daughter’s son, etc.) exist. Who gets the estate?

Opinion	Holder(s)	School(s)	Effect
DhA inherit (before Bayt al-Māl)	ʿAlī, Ibn Masʿūd, ʿUmar رضي الله عنهم	Ḥanafī, Ḥanbalī; Shāfiʿī (if Bayt al-Māl dysfunctional)	Estate distributed to distant relatives
No DhA inheritance — surplus to Bayt al-Māl	Zayd ibn Thābit	Mālikī; Shāfiʿī (if Bayt al-Māl functional)	Estate to public treasury

Key narration: «الخال وارث من لا وارث له» (The maternal uncle inherits from one who has no heir). Authenticity is disputed.

Path A choice: ʿAlī/Ḥanafī (DhA inherit) — avoids the conditional on Bayt al-Māl functionality; simpler branching.
Path B choice: ʿAlī/Ḥanafī (DhA inherit) — both paths agree for the engine’s use case.

Source: faraid/dzawilarham.md

D12 — DhA Method: Tanzīl or Qarāba?

Scenario: Among those who accept DhA inheritance (D11 = yes), how are shares computed?

Method	Holder(s)	School(s)	Algorithm
Tanzīl (تنزيل — “placement”)	ʿUmar, ʿAlī, Ibn Masʿūd (narrated); Ḥanbalī, Shāfiʿī	Ḥanbalī	Each dhū al-raḥim is placed in the seat of their deceased intermediary heir; that intermediary’s share is then distributed to them
Qarāba (قرابة — “kinship priority”)	Ḥanafī	Ḥanafī	DhA compete by proximity: jiha → degree → quwwa → connection strength; closer excludes farther

Example — {Daughter’s Son, Maternal Uncle’s Son}:

Tanzīl: Daughter’s son → placed in “daughter’s seat” → takes daughter’s share (½ or portion thereof). Maternal uncle’s son → placed in “maternal uncle’s seat” → takes remainder.
Qarāba: Both assessed on degree; daughter’s son is degree 3 (daughter→son), maternal uncle’s son is degree 4 (mother→maternal uncle→son) → daughter’s son is closer → takes entire estate (in some views).

Mathematical-elegance analysis:
Tanzīl reuses the entire farḍ computation engine recursively — it is more consistent with the core mechanics. Qarāba extends the ʿaṣaba ranking algorithm naturally (one priority cascade governs all) and is simpler to implement as a single-pass algorithm.

Path A choice: Qarāba (Ḥanafī) — single-pass priority algorithm; no recursive call needed.
Path B choice: Tanzīl (Ḥanbalī/Shāfiʿī) — consistent with core mechanics; engine reuse.

Toggle: dharham_method: "qaraba" | "tanzil" (both in current architecture)
Source: faraid/dzawilarham.md

D13 — How Many Grandmother Tiers Qualify?

Scenario: How many generations back, and from which paths, do grandmothers inherit?

Position	Holder(s)	School(s)	Rule
Unlimited — any grandmother who connects through an eligible intermediary (أدلت بوارث)	Ḥanafī, Shāfiʿī	—	Upward female chains (with eligible intermediaries) can be of any length
Three standard grandmothers + upward unbroken female chains	Ḥanbalī	—	أم أم, أم أب, أم أبي الأب, and their mothers upward
Two tiers only — أم أم and أم أب (and their mothers upward in unbroken female chains)	Mālikī	Most restrictive	No third-tier paternal grandmother

Mathematical-elegance analysis:
The Ḥanafī/Shāfiʿī rule-based approach (any grandmother who passes the eligibility test) is more general and requires no enumerated list. The Mālikī cap at two tiers avoids edge cases at the cost of a fixed limit.

Path A choice: Ḥanafī/Shāfiʿī (rule-based, unlimited) — single eligibility criterion replaces enumerated list.
Path B choice: Ḥanbalī (three standard tiers) — most textually supported.

Source: faraid/jaddah.md

D14 — ʿAṣaba Jiha Count (5 vs. 6 vs. 7)

Count	Holder(s)	School(s)	Key differences
5 directions	Ḥanafī	Ḥanafī	Grandfather merged into Ubuwwa direction (follows from ε₃′ in D4); Radd used instead of Bayt al-Māl
6 directions	Abū Yūsuf, Muḥammad ibn al-Ḥasan; Ḥanbalī	Ḥanbalī	Grandfather in separate direction; Radd used; no Bayt al-Māl
7 directions	Mālikī, Shāfiʿī	Mālikī, Shāfiʿī	Grandfather separate; Bayt al-Māl is the 7th direction

Note: This dispute is largely a consequence of D4 (grandfather) and D3 (Radd vs. Bayt al-Māl). Setting those toggles automatically resolves the jiha count.

Path A choice: 5 directions — follows from D4 (grandfather_equals_father = true) + D3 (Radd applies).
Path B choice: 6 directions — follows from D4 (grandfather coexists) + D3 (Radd applies).

Source: faraid/asaba.md

D15 — ʿAwl of Base 3

Scenario: Does the fraction base of 3 ever ʿawl to 4?

Opinion	Holder(s)	School(s)
Base 3 never ʿawls	Jumhūr (all four schools)	All
Base 3 can ʿawl to 4	Muʿādh ibn Jabal (isolated; follows from his position that all-female sibling groups don’t reduce the mother)	Rejected minority

Note: Muʿādh’s position on base-3 ʿawl is a downstream consequence of his position on D10 (female-only groups don’t halve mother). Since both paths choose the Jumhūr position on D10, base-3 ʿawl is rejected in all configurations.

Path A choice: Never (base-3 never ʿawls).
Path B choice: Never (same).

Source: faraid/awl.md

Dispute-to-Toggle Mapping (Engine Config Summary)

config = {
    // D3a — Spousal Radd
    spouse_participates_in_radd: bool,          // false = Jumhūr (Path B default)
                                                // true  = ʿUthmān (Path A choice)

    // D4 — Grandfather vs. Siblings
    grandfather_equals_father: bool,            // false = Jumhūr (Path B default)
                                                // true  = Ḥanafī / ε₃′ (Path A choice)

    // D5a — Grandmother Immunity from Father
    grandmother_immune_to_father: bool,         // false = Jumhūr 3-school (both paths default)
                                                // true  = Ḥanbalī variant (optional)

    // D12 — Dhawī al-Arḥām Method
    dharham_method: "tanzil" | "qaraba",        // "tanzil" = Ḥanbalī (Path B)
                                                // "qaraba" = Ḥanafī (Path A)

    // D4 activates/deactivates D8 and D9 automatically
    use_delta: bool,                            // true  = Path B (δ active; Mushtaraka handled)
                                                // false = Path A (ε₃′ pre-empts δ-dependent cases)

    // FUTURE — if full opinion-agnosticism is needed:
    awl_method: "proportional" | "ibn_abbas_priority",  // D1
    relax_father_ge_mother: bool,                        // D2 (Ibn ʿAbbās mode)
    grandmother_priority: "proximity" | "direction",     // D6
}

Preset configurations:

Config Name	`grandfather_equals_father`	`use_delta`	`spouse_radd`	`dharham_method`	`grandmother_immune`
Path A	`true`	`false`	`true`	`"qaraba"`	`false`
Path B / Shāfiʿī-Mālikī	`false`	`true`	`false`	`"tanzil"`	`false`
Path B / Ḥanbalī	`false`	`true`	`false`	`"tanzil"`	`true`
Ḥanafī (strict)	`true`	`false`	`false`	`"qaraba"`	`false`

Disputes That Do NOT Split the Paths

The following disputes have the same answer in both Path A and Path B:

Dispute	Shared choice	Reason
D1 ʿAwl	Jumhūr (proportional)	Ibn ʿAbbās requires `relax_father_ge_mother` which breaks the axiom system
D2 ʿUmariyyatān	Jumhūr (⅓ of remainder)	Proven as the unique consistent solution (Theorem 7)
D3 Radd applies	ʿAlī/Ḥanafī (yes)	Bayt al-Māl conditionality adds complexity with no mathematical gain
D10 Sibling count	Jumhūr (2 suffice)	Ibn ʿAbbās / Muʿādh positions are inconsistent with the broader system
D11 DhA inherits	Yes	Conditional on Bayt al-Māl adds complexity
D15 Base-3 ʿawl	Never	Muʿādh’s position depends on a rejected D10 position

Disputes Where Path A Differs from Path B

#	Dispute	Path A	Path B	Key trade-off
D3a	Spousal Radd	Include (ʿUthmān)	Exclude (Jumhūr)	Elegance (symmetric γ) vs. authentication strength
D4	Grandfather-siblings	Excludes (Ḥanafī + ε₃′)	Coexists (Jumhūr + δ)	Fewer rules vs. literal $\alpha_1$
D5a	Grandmother immunity	Absent	Optional (Ḥanbalī)	Consistency vs. Prophetic narration
D8	Mushtaraka	Pre-empted	Handled by δ	Cascade simplicity vs. kinship fairness
D12	DhA method	Qarāba (Ḥanafī)	Tanzīl (Ḥanbalī/Shāfiʿī)	Single-pass vs. recursive engine reuse
D16	Ḥaml scenarios	$k=2$ (Ḥanafī)	$k=6$ (Jumhūr)	Simplicity vs. coverage of rare births
D17	Khunthā rule	Single-pick (Ḥanafī)	Component-min+mawqūf (Shāfiʿī) / mean (Mālikī)	Decisiveness vs. fairness under uncertainty
D18	Gharqā mutual	No mutual (Jumhūr)	Chain Forest (Ḥanbalī)	Simplicity vs. presumption of survival

D16 — Ḥaml: Scenario Count

Source: faraid/haml.md:431–453

The scholars differ on how many gender×count hypotheses to compute for an unborn heir:

Opinion	School	Scenarios
$k = 2$: male or female	Ḥanafī	`{m, f}`
$k = 6$: all realistic outcomes	Shāfiʿī / Ḥanbalī	`{stillborn, 1m, 1f, 2m, 2f, 1m+1f}`

For each scenario, run the 6-phase pipeline on the heir set with the unborn instantiated to that gender/count. Known heirs receive الأضر (min across scenarios). The unborn’s entitlement is الأحظ (max), held as mawqūf until birth.

Note: Jumhūr further establish that comparing only the {2m, 2f} taqdirai suffices to find الأحظ and الأضر; the other scenarios are computed for post-birth recalculation convenience. Source: faraid/haml.md:517–527.

Path A: $k=2$ (Ḥanafī) — minimum computation, handles the statistically dominant cases. Path B: $k=6$ (Jumhūr) — full coverage. Toggle: haml_scenario_set: "hanafi_2" | "jumhur_6" (default Path B).

D17 — Khunthā Mushkil: Aggregation Rule

Source: faraid/khuntsa.md:332–358

When a khunthā’s (hermaphrodite’s) share differs between the male and female scenarios, the four schools differ on how to aggregate:

School	Rule	mawqūf?
Ḥanafī	Single-pick: compute both scenarios; use the one giving khunthā the less (الأضر للخنثى); this entire scenario applies to all heirs	No
Mālikī	Component-mean: each heir gets ½(share_male + share_female)	No
Shāfiʿī	Component-min + mawqūf: each heir (including khunthā) gets the minimum across the two scenarios; remainder is suspended until status clarifies or heirs settle	Yes
Ḥanbalī	Case-split: if khunthā’s status may still clarify (living minor) → Shāfiʿī method; if not (matured/died) → Mālikī method	Depends

The khunthā can only appear in $j \in {1, 3, 4, 5}$ — never in $j = 0$ (spouses) or $j = 2$ (ascendants), because marriage requires definite gender and biological ascendant status requires procreation. Source: faraid/khuntsa.md:113–129.

Toggle: khunthas_aggregation: "hanafi" | "maliki" | "shafii" | "hanbali" (default Path B → Shāfiʿī for clarifiable; Mālikī for matured).

D18 — Gharqā: Mutual Inheritance Between Co-Deceased

Source: faraid/gharqa.md:22–82, 228–282

When a group of mutual heirs die in an accident and the order is unknown, the schools differ:

School	Ruling	Mathematical structure
Ḥanafī, Mālikī, Shāfiʿī	No mutual inheritance. Each estate distributed to surviving heirs only (co-deceased dropped from each other’s heir sets).	$n$ independent $F$-calls
Ḥanbalī	Mutual inheritance with two conditions: (1) each inherits only from the other’s tilād (pre-existing wealth), not their ṭarīf (what they received from the other); (2) heirs agree on the unknown order.	Chain Forest: $n$ trees, one per assumed-first deceased, with TILD/ṬARIF edge-deletion

The Ḥanbalī condition (1) prevents inheritance cycles: «لأنه لو ورّث أحدهما الآخر من طريف ماله للزم منه الدور، وهو أن يرث الإنسان نفسه، وهو ممتنع» (faraid/gharqa.md:164–168).

Toggle: gharqa_mutual_inheritance: false | "hanbali" (default Path B → false, per Jumhūr majority).

References

Source	Content
`faraid/awl.md`	ʿAwl opinions, Ibn ʿAbbās muʿāhala, ʿUmariyyatān, Muʿādh base-3
`faraid/radd.md`	Radd opinions, ʿUthmān narration, Ibn ʿAbd al-Barr consensus claim
`faraid/hajb.md`	Mushtaraka, القاعدة الأولى, القاعدة الثانية, grandmother passages
`faraid/jaddah.md`	Grandmother disputes (D4–D7, D10, D13)
`faraid/jaddma'aikhwah.md`	Grandfather-sibling dispute (D4), Akdariyya (D9)
`faraid/asaba.md`	ʿAṣaba jiha count (D14)
`faraid/dzawilarham.md`	DhA disputes (D11, D12)
`faraid/haml.md`	Ḥaml scenario count (D16)
`faraid/khuntsa.md`	Khunthā aggregation rule (D17)
`faraid/gharqa.md`	Gharqā mutual inheritance (D18)
`findings/03-exceptions.md`	ε-classification of all disputes
`findings/05-proofs.md`	Theorem 7 (ʿUmariyyatān uniqueness)
`findings/08-path-comparison.md`	Toggle architecture, path presets
`findings/09-open-questions.md`	Akdariyya open question (Q3)
`findings/16-uncertainty-and-forest-extensions.md`	Unified $(\mathcal{S}, w, \mathcal{A})$ schema for D16–D18