08 — Path A vs. Path B: The Two-Path Comparison (Revised)

08 — Path A vs. Path B: The Two-Path Comparison (Revised)

Core claim (revised): Under the corrected axiom split ($\alpha_1$, $\alpha_2$, $\beta$, $\gamma$, $\delta$), Path B (Jumhūr) is the textually faithful default — its outcomes follow naturally from the source rules. Path A (Ḥanafī) is the elegant simplification — it trades $\delta$ for the assumption “grandfather = father” ($\epsilon_3’$), which pre-empts the need for monotonicity checks.

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Path Definitions (Revised)

	Path A	Path B
Alignment	Ḥanafī	Jumhūr (Mālikī/Shāfiʿī/Ḥanbalī)
Axioms	$\alpha_1, \alpha_2, \beta, \gamma$	$\alpha_1, \alpha_2, \beta, \gamma, \delta$
Extra assumptions	$\epsilon_3’$ (grandfather = father)	None
Irreducible exceptions	$\epsilon_1, \epsilon_3’$	$\epsilon_1$ (+ $\epsilon_2$ for Ḥanbalī)
Design principle	Maximize regularity via simplifying assumption	Maximize textual fidelity + structural fairness

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The Shared Core (All Paths)

Every path agrees on:

1. Axioms $\alpha_1$ and $\alpha_2$ — Intermediary exclusion + ʿaṣaba ranking (two independent mechanisms from faraid/hajb.md).
2. Axiom $\beta$ — Entitlement (farḍ shares, halving, ceilings).
3. Axiom $\gamma$ — Conservation (ʿaṣaba absorption, ʿawl, radd with blood-heir scoping).
4. $\epsilon_1$ — Maternal immunity — Uterine siblings are not excluded by their intermediary (mother). By ijmāʿ.
5. Phase 0 — Attribute exclusion before any computation.
6. Ghost pressure — Person-excluded heirs still affect shares.
7. ʿAwl mechanics — Proportional compression when $\sum > 1$.
8. Taṣḥīḥ — Integer normalization.
9. All 7 theorems in 05-proofs.md.

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Where Paths Diverge

Divergence 1: Grandfather vs. Siblings

Path A (Ḥanafī)	Path B (Jumhūr)
Adds $\epsilon_3’$: grandfather is placed in the father’s jiha	Default under $\alpha_1$: grandfather is NOT the siblings’ واسطة, so he cannot exclude them
Grandfather completely excludes all siblings via $\alpha_2$ (ranking)	Grandfather and siblings coexist — $\delta$ guarantees his floor via max(Muqāsama, $\frac{1}{3}$, $\frac{1}{6}$)
Requires an analogical argument (Qurʾān calls Ibrahim “father”)	Requires $\delta$ (monotonicity) to set the grandfather’s floor

Path A rationale: The grandfather is “father’s father” — by analogy, he inherits the father’s exclusion power. Supported by: «الجد يرث بجهة الأُبُوّة»

Path B rationale: The source text’s own intermediary rule doesn’t extend to the grandfather. The Jumhūr explicitly argue: «الإخوة إنما حجبوا بالأب لإدلائهم به وهو منتفٍ في الجد»

Divergence 2: Grandmother Immunity ($\epsilon_2$)

Path A (Ḥanafī)	Path B
Grandmother excluded by her son (father/grandfather) — standard $\alpha_1$	Varies by madhhab
No exception	$\epsilon_2$ needed (Ḥanbalī only; Shāfiʿī/Mālikī don’t need it)

Divergence 3: Mushtaraka (The Donkey Case)

Scenario: Husband, Mother, 2+ Uterine siblings, 1+ Full sibling(s).

Path A / Ḥanbalī	Path B / Shāfiʿī-Mālikī
Full brother = ʿaṣaba. Remainder after farḍ = 0. Full brother gets 0.	Full brother joins the $\frac{1}{3}$ pool — output of $\delta$
Under Path A, $\epsilon_3’$ pre-empts this case entirely (grandfather would exclude siblings)	$\delta$ triggers: full brother has superset-kinship of uterine siblings, so share $\ge$ theirs
No exception needed	No exception needed — $\delta$ handles it

The Mushtaraka is the most dramatic divergence — it’s the only case where a mainstream school changes an heir’s classification (from ʿaṣaba to farḍ) based on the specific combination of co-heirs.

Divergence 4: Spousal Radd Immunity

Jumhūr (all paths)	ʿUthmān’s view
Spouses are scoped out of Radd — built into $\gamma$	Spouses participate in Radd
Textual basis: أُولُو الأرحام (blood relatives)	No exception needed — universal $\gamma$

Path A may optionally adopt ʿUthmān’s view for symmetry (weak authentication). Path B treats the scoping as part of $\gamma$ — not a separate exception.

Divergence 5: Dhawī al-Arḥām Distribution Method

Path A (Ḥanafī — Ahl al-Qarāba)	Path B (Ḥanbalī/Shāfiʿī — Ahl al-Tanzīl)
Priority cascade: jiha → daraja → quwwa → connection	Graph projection: each heir maps to their connecting wārith
Extends the ʿaṣaba ranking naturally	Reuses the farḍ computation engine recursively
Simpler implementation	More consistent with core mechanics

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Per-Madhhab Exception Count (Revised)

Exception	Ḥanafī (A)	Ḥanbalī (B)	Shāfiʿī (B)	Mālikī (B)
$\epsilon_1$ Maternal immunity	✓	✓	✓	✓
$\epsilon_2$ Grandmother immunity	—	✓	—	—
$\epsilon_3’$ Grandfather = Father	✓	—	—	—
$\delta$ Kinship monotonicity	—	✓ (axiom)	✓ (axiom)	✓ (axiom)
$\epsilon_5$ ʿAṣaba maʿa ghayr	Derived	Derived	Derived	Derived
$\epsilon_6$ ʿUmariyyatān	Derived	Derived	Derived	Derived
$\epsilon_7$ Blessed Kinsman	Emergent	Emergent	Emergent	Emergent
Mushtaraka	Pre-empted by $\epsilon_3’$	N/A (Jadd excludes)	Output of $\delta$	Output of $\delta$
Irreducible exception count	2	2	1	1

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The Toggle / Switchboard Architecture (Revised)

The system can be implemented as a single engine with boolean switches:

Engine(H, config) where config = {
    use_delta: bool,                        // true = Path B (monotonicity checks)
    grandfather_equals_father: bool,        // true = Path A (ε₃')
    grandmother_immune_to_father: bool,     // true = ε₂ active (Ḥanbalī)
    spouse_participates_in_radd: bool,      // true = ʿUthmān's view
    dharham_method: "qaraba" | "tanzil",    // Ḥanafī vs Jumhūr
}

Note: mushtaraka_enabled is no longer needed as a separate toggle. It’s handled automatically by use_delta.

Preset configurations:

Preset	$\delta$	$\epsilon_3’$	$\epsilon_2$	Spouse Radd	DhA method
Ḥanafī	—	✓	—	Optional	qaraba
Ḥanbalī	✓	—	✓	—	tanzil
Shāfiʿī	✓	—	—	—	tanzil
Mālikī	✓	—	—	—	tanzil

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Trade-off Analysis (Revised)

Criterion	Path A	Path B
Axiom count	4 (without $\delta$) + 2 exceptions	4 + $\delta$ + 1 exception
Mathematical regularity	High (one simplifying assumption)	High (one structural principle)
Textual fidelity	Requires analogical argument for $\epsilon_3’$	Maximum — follows source rules literally
Social equity	May zero close relatives (Mushtaraka)	Prevents zero-shares for close kin
Implementation complexity	Simpler (fewer branches)	Slightly more (max-envelope checks)
Historical prevalence	Dominant in Ḥanafī regions	Dominant everywhere else

The fundamental trade-off (revised): Path A maximizes simplicity by adding one analogical assumption ($\epsilon_3’$) that collapses many cases. Path B maximizes textual fidelity by following the source rules literally and using $\delta$ to guarantee structural fairness. The old narrative (“Path A is clean, Path B is patched”) was an artifact of merging two distinct exclusion rules into one axiom.

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References

• Exception catalog: 03-exceptions.md
• Grandfather-sibling source: faraid/jaddma'aikhwah.md
• Grandmother source: faraid/jaddah.md
• Radd source: faraid/radd.md
• Dhawī al-Arḥām source: faraid/dzawilarham.md
• Toggle architecture proposal: comments/2.md