16 — Uncertainty and Forest Extensions: The Unified $(\mathcal{S}, w, \mathcal{A})$ Schema

Core claim: All six extensions of the Farāʾiḍ engine (Munāsakhat, Mafqūd, Dhawī al-Arḥām, Ḥaml, Khunthā, Gharqā) are instances of a single mathematical structure parameterized by a scenario set $\mathcal{S}$, an estate-weight function $w : \mathcal{S} \to \mathbb{Q}_{\ge 0}$, and an aggregation rule $\mathcal{A}$. The 4-axiom core ($\alpha_1, \alpha_2, \beta, \gamma, \delta$) is not changed by any of the three new source files (faraid/haml.md, faraid/khuntsa.md, faraid/gharqa.md). All three are extension-layer phenomena.

Sources: faraid/haml.md (ḥaml scenarios), faraid/khuntsa.md (khunthā aggregation), faraid/gharqa.md (Chain Forest), plus the existing formalism in findings/11-extension-formalism.md.

1. Motivation

The three new source files were not part of the original corpus that produced findings/01–15. After reading them:

faraid/haml.md (إرث الحمل): the unborn heir is a “missing entity” whose gender and count are unknown. The juristic method is to compute the estate under multiple gender×count scenarios and apply a min-for-known / max-for-unborn aggregation, holding the maximum as mawqūf (suspended) until birth.
faraid/khuntsa.md (ميراث الخنثى المشكل): the hermaphrodite heir has indeterminate gender. The juristic method is to compute two scenarios (male and female) and aggregate per a school-specific rule (single-pick, mean, or component-min + mawqūf).
faraid/gharqa.md (الغرقى ومن في حكمهم): when multiple mutual heirs die in an accident and the order is unknown, the Ḥanbalī school allows mutual inheritance by running the Munāsakhat algorithm independently for each assumed-first-deceased, with a cycle-breaking rule (TILD/ṬARIF) preventing each deceased from inheriting what they themselves bequeathed.

All three fit one schema. The classical algorithms (multi-scenario tables, jāmiʿa computation, mawqūf distribution) are integer-normalization strategies for evaluating this schema without decimals — exactly as the Munāsakhat chapter’s 80-page table is an integer-normalization strategy for the Estate Flow Tree formula $\phi(\ell) = \sum_\pi \prod \omega$.

2. The $(\mathcal{S}, w, \mathcal{A})$ Schema

Definition. An extension instance is a triple $(\mathcal{S}, w, \mathcal{A})$ where:

Component	Type	Meaning
$\mathcal{S}$	finite set	Scenario set: alternative instantiations of the unknown heir(s) or alternative orderings of deceased
$w : \mathcal{S} \to \mathbb{Q}_{\ge 0}$ with $\sum_s w(s) = 1$	weight function	Fraction of total estate handled by each scenario (uniform for uncertainty extensions; estate-ratio for Chain Forest)
$\mathcal{A} : (\mathcal{S} \to \text{ShareMap}) \to \text{ShareMap}$	aggregation rule	How per-scenario shares are combined into a final share for each heir

For each scenario $s \in \mathcal{S}$, the core pipeline $F$ is invoked on the instantiated heir set $H_s$:

$$F(H_s) = (B_s, \vec{s}_s)$$

Scale to common base $J = \text{lcm}_{s} B_s$; let $\hat{s}_s(h) = s_s(h) \cdot J / B_s$.

The final share: $\text{sahm}(h) = \mathcal{A}\bigl({s \mapsto \hat{s}s(h)}{s \in \mathcal{S}}\bigr)$.

The mawqūf: $\text{mawqūf} = J - \sum_h \text{sahm}(h)$.

3. Conservation Theorem

Theorem (Conservation under $(\mathcal{S}, w, \mathcal{A})$). The final distribution conserves ($\sum_h \text{sahm}(h) + \text{mawqūf} = J$) by construction (the mawqūf absorbs the gap). But full conservation without mawqūf ($\text{mawqūf} = 0$) holds if and only if $\mathcal{A}$ is $\mathcal{S}$-componentwise affine with non-negative coefficients summing to 1:

$$\text{mawqūf} = 0 \iff \mathcal{A}(f) = \sum_{s \in \mathcal{S}} \lambda_s \cdot f(s) \text{ with } \lambda_s \ge 0 \text{ and } \sum_s \lambda_s = 1$$

Proof. If $\mathcal{A}$ is convex:

$$\sum_h \text{sahm}(h) = \sum_h \sum_s \lambda_s \hat{s}_s(h) = \sum_s \lambda_s \underbrace{\sum_h \hat{s}s(h)}{=J} = J \cdot \sum_s \lambda_s = J ;\checkmark$$

For component-min, the inequality $\min \le$ any convex combination means $\sum_h \min_s \hat{s}_s(h) \le J$, with the gap being the mawqūf. $\square$

Corollary. The following aggregation rules conserve exactly (mawqūf = 0): component-mean, single-pick, weighted-sum (Chain Forest). The following leave mawqūf $\ge 0$: component-min (Mafqūd, Ḥaml, Khunthā Shāfiʿī).

4. Complete Extension Instantiation Table

Extension	$\mathcal{S}$	$w$	$\mathcal{A}$	mawqūf	Source
Munāsakhat (Chain)	Tree of deceased; ordered sequence	Path products $\omega_{v \to w}$	Sum of path products (estate flow formula)	0	`faraid/munasakhat.md`
Gharqā Ḥanbalī (Chain Forest)	Forest: one tree per co-deceased root	$E_k / E_{\text{total}}$	Weighted sum across roots	0	`faraid/gharqa.md:240–282`
Gharqā Jumhūr	No scenarios (each estate independent)	1 per estate	Independent $F$-calls, no merge	0	`faraid/gharqa.md:232–236`
Mafqūd (Min)	$2^{n}$ alive/dead masks	Uniform $1/2^n$ (for $w$; but aggregation is min not mean)	component-min	$\ge 0$	`faraid/mafqud.md`
Ḥaml Jumhūr	6 gender×count: {still, 1m, 1f, 2m, 2f, 1m+1f}	Uniform	min for known; max (as mawqūf) for ḥaml	$\ge 0$	`faraid/haml.md:441–500`
Ḥaml Ḥanafī	2 gender: {1m, 1f}	Uniform	min for known; max (as mawqūf) for ḥaml	$\ge 0$	`faraid/haml.md:431–439`
Khunthā Shāfiʿī	${m, f}$	Uniform	component-min (all heirs incl. khunthā)	$\ge 0$	`faraid/khuntsa.md:511–525`
Khunthā Mālikī	${m, f}$	Uniform	component-mean $\tfrac{1}{2}(s_m + s_f)$	0	`faraid/khuntsa.md:344–346`
Khunthā Ḥanafī	${m, f}$	$\delta_{s^*}$ (point mass at $\arg\min_{s}(s_{\text{khunthā}})$)	single-pick: use $F(s^*)$ for everyone	0	`faraid/khuntsa.md:336–342`
Khunthā Ḥanbalī	${m, f}$	Uniform	case-split: Shāfiʿī if clarifiable, Mālikī otherwise	varies	`faraid/khuntsa.md:352–358`
DhA Tanzīl (Project)	Depth-2 tree: root → virtual heirs → DhA leaves	Per-path products	Sub-distribution products	0	`faraid/dzawilarham.md`
DhA Qarāba (Project)	Depth-1: root → DhA leaves (priority-filtered)	Uniform (filtered)	Priority-weighted split	0	`faraid/dzawilarham.md`

5. The Khunthā’s 5-Tuple: No Change Needed

Claim: The 5-tuple $\vec{h} = (g, j, d, q, c)$ with $g \in {0, 1}$ is sufficient for Khunthā cases. No third value of $g$ is needed.

Proof: In the scenario expansion framework, the khunthā’s $g$ is parameterized across scenarios — it equals 0 in the female scenario and 1 in the male scenario. Within each individual scenario, $g$ is definitively 0 or 1. Theorem 2 (5-tuple completeness) applies within each scenario.

The “khunthā” status is a case-input metadata flag that triggers scenario expansion at the extension layer. It is not a 5-tuple attribute and does not affect the core pipeline. $\square$

Khunthā jiha restriction: From faraid/khuntsa.md:113–129:

$j = 0$ (spouse): impossible — marriage requires definite gender.
$j = 2$ (ascendant): impossible — biological parenthood implies definite gender.
$j \in {1, 3, 4, 5}$: possible.

This restriction emerges from biological facts and marriage law; it is not an axiom and requires no change to the eligibility predicates.

6. The Gharqā Forest: How TILD/ṬARIF Relates to Chain

The Munāsakhat Chain (§4 of 11-extension-formalism.md) processes deaths in a fixed temporal order — tree rooted at the first deceased, with subsequent deceased becoming internal nodes.

The Gharqā Chain Forest has no fixed global ordering. Instead:

For each assumed-first deceased $d_k$: run Chain as if $d_k$ died first.
The TILD/ṬARIF rule emerges naturally: in the sub-tree where $d_k$ is root, the other co-deceased $d_l$ receives a share of $d_k$‘s estate (ṭarīf). When $d_l$ subsequently propagates this share to $d_l$‘s heirs, $d_k$ is excluded (since $d_k$ is the root — the “first to die” in this assumed ordering, so $d_k$ cannot be $d_l$‘s heir).

This is exactly Munāsakhat Case 2 applied once per assumed ordering. The Ḥanbalī method runs Case 2 for each of the $n$ co-deceased roots, then sums the per-root share contributions weighted by estate ratios.

Structural comparison:

Feature	Munāsakhat (Chain)	Gharqā Ḥanbalī (Chain Forest)
Root count	1	$n$ (one per co-deceased)
Temporal order	Fixed (input given)	Assumed per-root
Edge semantics	Sequential inheritance	TILD (tilād) only; no ṭarīf back-edges
Aggregation	Sum of path products (single tree)	Weighted sum across $n$ trees
Implementation	`resolveChainFromDAG` (existing)	Extend to multi-root with cycle detection

7. Composition Rules

The $(\mathcal{S}, w, \mathcal{A})$ extensions compose with each other and with Chain:

Composition	Meaning	Resolution
Mafqūd ∘ Munāsakhat	Missing heir whose estate is later chained	Run Min first; for each scenario where mafqūd is alive, their estate enters Chain if they later die before division
Ḥaml ∘ Munāsakhat	Unborn heir whose estate is chained after birth (if they die)	Resolve ḥaml scenarios first; surviving ḥaml may trigger Chain if they die before division
Gharqā ∘ Mafqūd	A co-deceased is also missing	Run Min across alive/dead scenarios for the mafqūd; for the alive-mafqūd scenario, include them in the Gharqā forest
Khunthā ∘ Gharqā	A khunthā is among the co-deceased	Outer scenario expansion for khunthā gender × inner Gharqā forest per ordering

Composition is well-defined because $F$ is a pure function — each scenario produces an independent $F$-call. The extension operators $(\mathcal{S}, w, \mathcal{A})$ are closed under composition: the composed operator has scenario set $\mathcal{S}_1 \times \mathcal{S}_2$, weight function $w_1 \otimes w_2$, and the appropriate aggregation hierarchy.

8. Audit Impact: What Did NOT Change

The following are unaffected by the three new source files:

Finding	Status
4-axiom system ($\alpha_1, \alpha_2, \beta, \gamma, \delta$)	Unchanged
6-phase pipeline (Phase 0–5)	Unchanged
5-tuple $(g, j, d, q, c)$ with $g \in {0,1}$	Unchanged (scenario expansion, not tuple change)
Theorems 1–7 in `findings/05-proofs.md`	Unchanged
Exception catalog $\epsilon_1$–$\epsilon_8$, $\epsilon_3’$	Unchanged (Akdariyya clarified in `findings/03-exceptions.md`)
Fraction field $\mathbb{Q}_{(2,3)}$, 7+2 bases	Unchanged
Ghost-pressure mechanics (Phase 1)	Unchanged (each scenario applies Phase 1 independently)

9. Open Engineering Questions (Not Addressed Here)

This document defines the mathematical schema for ḥaml, khunthā, and gharqā extensions. Engineering implementation is deferred to a separate document.

Question	Notes
How does `scenarioExtension()` integrate with the existing `packages/core/` API?	Extension layer in `packages/extensions/`, analogous to existing `chain.ts`, `min.ts`.
How does the Chain Forest’s TILD/ṬARIF rule interface with `resolveChainFromDAG`?	Requires multi-root support + cycle-detection in the DAG resolver.
What toggle config keys govern D16/D17/D18?	`haml_scenario_set`, `khunthas_aggregation`, `gharqa_mutual_inheritance` — see 10-dispute-matrix.md.
What test cases cover the new extensions?	Ḥaml: `faraid/haml.md:596–700`; Khunthā: `faraid/khuntsa.md:546–612`; Gharqā: `faraid/gharqa.md:299–360`.

References

Source	Content
`faraid/haml.md`	Ḥaml: conditions, scenario count, algorithm, worked examples
`faraid/khuntsa.md`	Khunthā: 4 schools, jiha restrictions, aggregation rules, worked examples
`faraid/gharqa.md`	Gharqā: 5 cases, Ḥanbalī TILD/ṬARIF method, worked examples
`faraid/mafqud.md`	Original Min-extension source
`findings/11-extension-formalism.md`	§4 Chain, §4.2 Chain Forest, §5 Generalized Min — detailed formulas and pseudocode
`findings/06-core-vs-extensions.md`	Updated summary table including new extensions
`findings/10-dispute-matrix.md`	D16/D17/D18 — dispute analysis for haml/khunthā/gharqā
`findings/03-exceptions.md`	Akdariyya formalized as stated $\gamma_2$ rule
`findings/09-open-questions.md`	Q7 closed — ḥaml/khunthā now formally in corpus as extensions