First-Principles Derivation of Islamic Inheritance Law (ʿIlm al-Farāʾiḍ)

First-Principles Derivation of Islamic Inheritance Law (ʿIlm al-Farāʾiḍ)

Status: Synthesized findings from independent derivation, peer review, and source verification.

Source corpus: Talkhīṣ Fiqh al-Farāʾiḍ (Ibn ʿUthaymīn), Ḥanbalī madhhab primary reference.

Method: Two parallel paths maintained throughout — Path A (maximum mathematical elegance, choosing the most consistent opinion in each dispute) and Path B (Jumhūr/majority consensus).

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Table of Contents

#	Document	Summary
01	The 5-Tuple and Family Graph	Heir identification: the vector $(g, j, d, q, c)$, DAG architecture, BFS path resolution, eligibility predicates, path-sequence requirement for grandmothers.
02	The 4 Irreducible Axioms	Axiom $\alpha_1$ (Intermediary Exclusion), Axiom $\alpha_2$ (ʿAṣaba Ranking), Axiom $\beta$ (Entitlement), Axiom $\gamma$ (Conservation), Axiom $\delta$ (Kinship Monotonicity). The fraction field $\mathbb{Q}_{(2,3)}$, the 7+2 bases, halving/ceiling mechanics, ghost pressure.
03	Structural Exceptions	All exceptions reclassified under 4-axiom system: $\epsilon_3$ is now a default (not an exception), $\epsilon_3’$ (grandfather=father) is Ḥanafī-specific, Mushtaraka handled by $\delta$, $\epsilon_8$ absorbed into $\gamma$. Akdariyya formalized as a stated $\gamma_2$ sub-rule (triggered by $\delta$, recipe from Zayd). Path A: 2 exceptions. Path B: 1 exception + Akdariyya $\gamma_2$ rule.
04	The Compiler Pipeline	The 6-phase execution order (Phase 0–5), ghost pressure paradox resolution, the unified ʿAwl/Radd normalization formula.
05	Formal Theorems and Proofs	7 theorems with proof sketches: fraction generation, $\frac{2}{3}$ cap forced, ʿaṣaba maʿa al-ghayr as optimization, ʿUmariyyatān as forced, and more.
06	Core vs Extensions	The single-pass/multi-pass boundary. Radd is core (closed-form formula). Six extensions: Chain (Munāsakhat), Min (Mafqūd), Project (Dhawī al-Arḥām), Ḥaml, Khunthā, Gharqā. All are instances of the $(\mathcal{S}, w, \mathcal{A})$ schema.
07	Professor Adjudication	All claims from 3 independent reviewers, adjudicated with true/false/withdrawn verdicts and source citations. Result: 2 genuine corrections to the pipeline.
08	Path A vs Path B	Side-by-side comparison — revised: Path B (Jumhūr) is the textually faithful default with fewer exceptions. Path A (Ḥanafī) trades $\delta$ for $\epsilon_3’$. Toggle/switchboard architecture for parameterization.
09	Open Questions	Remaining unsolved problems: single eligibility predicate, Akdariyya status, maternal immunity’s irreducibility. Q2 (“hidden Jumhūr principle”) now RESOLVED — the principle is $\delta$.
11	Extension Formalism: The Estate Flow Tree	Unified mathematical structure for all three extensions. Core result: inheritance with multiple deaths/missing/DhA heirs is evaluation of a weighted rooted tree over $\mathbb{Q}$. Exact formulas, pseudocode, complexity bounds, and composition rules.
12	Walā (Patronage Inheritance)	Walā is a degenerate projection: one new jiha value ($j = 5$), one pre-processing resolution function, zero new axioms/exceptions/pipeline phases. The formalism is self-similar — walā resolution reuses the same $\alpha_2$ priority cascade in the muʿtiq’s reference frame.
13	Audit and Fixes	Comprehensive source verification against Talkhīṣ Fiqh al-Farāʾiḍ; two bugs discovered and fixed in core pipeline.
14	Scenario Generator	Pure algorithmic heir-space sampler for property-based testing; uniform random cases with stratified flag coverage and retry limits.
15	DAG-Based Munāsakhat Implementation	Automatic re-expression via family DAG; per-step trace capture; الجدول المعلق (hanging table) renderer; solveChainFromDAG API and POST /chain/dag endpoint.
16	Uncertainty and Forest Extensions	Unified $(\mathcal{S}, w, \mathcal{A})$ schema subsuming all six extensions (Munāsakhat, Mafqūd, DhA, Ḥaml, Khunthā, Gharqā). Chain Forest for simultaneous deaths. Generalized Min for ḥaml/khunthā. Conservation theorem. All three new faraid/ files (haml.md, khuntsa.md, gharqa.md) are extension-layer; 4-axiom core unchanged.

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Key Results

• Heir Identification: Every heir is uniquely determined by a 5-dimensional vector $\vec{h} = (g, j, d, q, c)$ computed via BFS on a kinship DAG.
• Axiom Count: The entire system reduces to 4 irreducible axioms ($\alpha_1$: intermediary exclusion, $\alpha_2$: ʿaṣaba ranking, $\beta$: entitlement, $\gamma$: conservation) + 1 structural principle ($\delta$: kinship monotonicity, used by Jumhūr).
• Exception Count: Path A (Ḥanafī): 2 irreducible exceptions ($\epsilon_1, \epsilon_3’$). Path B (Jumhūr): 1 irreducible exception ($\epsilon_1$), or 2 for Ḥanbalī ($\epsilon_1, \epsilon_2$).
• The Inversion: Under the revised axiom split, Path B (Jumhūr) is the cleaner formulation — the old narrative that Path A was minimal was an artifact of merging two distinct exclusion rules.
• ʿAwl and Radd are the same operation — a single normalization formula with different domains.
• Core/Extension factorization: $\text{Farāʾiḍ} = F(\text{core}) + {\text{Chain}, \text{Min}, \text{Project}, \text{Ḥaml}, \text{Khunthā}, \text{Gharqā}}(\text{extensions})$
• Unified Extension Schema: All six extensions are instances of $(\mathcal{S}, w, \mathcal{A})$ — a scenario set, estate-weight function, and aggregation rule. Conservation holds (mawqūf = 0) iff $\mathcal{A}$ is affine-convex over scenarios. Min-type aggregations leave mawqūf $\ge 0$ as the “uncertainty premium.” See 16-uncertainty-and-forest-extensions.md.
• Automatic re-expression: The DAG-based API (solveChainFromDAG) eliminates manual kinship re-expression for Munāsakhat; the classical الجدول المعلق table is a structured presentation of per-step trace data.
• New faraid/ files do not change the axioms: faraid/haml.md, faraid/khuntsa.md, and faraid/gharqa.md are all extension-layer phenomena. The 4-axiom system, 6-phase pipeline, and 5-tuple encoding are unchanged.

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Source References

Directory	Contents
faraid/	Primary Arabic source texts (16 files covering all inheritance topics, including haml.md, khuntsa.md, gharqa.md)
my findings/	Original independent axiom derivation and architecture design (v3)
comments/	Peer review from 3 independent professors/mathematicians
tests/	Test case data for validation

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Reference: Talkhīṣ Fiqh al-Farāʾiḍ, Muḥammad ibn Ṣāliḥ al-ʿUthaymīn (Ḥanbalī madhhab)