09 — Open Questions

These problems remain unresolved. Each is precisely stated with what is known, what is unknown, and why it matters.

Q1: Can the Three Eligibility Predicates Be Unified?

Status: ✅ RESOLVED. Directionless unification is provably impossible; a direction-parametric (pivot-based) unification is complete and minimal.

Prior error corrected: An earlier draft (and the original Theorem 1 statement) characterised $c_{\text{asc}}$ as “no male between two females” (no $(F,M,F)$ subsequence). This was wrong. The correct predicate is $M^F^$ — once any female appears on the ascending chain, no male may follow. The difference matters: the maternal-grandfather path $[\text{deceased}, \text{mother}(F), \text{maternal-grandfather}(M)]$ contains no $(F,M,F)$ but violates $M^F^$, and the maternal grandfather is correctly excluded ($c=0$). The implementation in eligibility.ts already enforces the correct $M^F^$ rule.

Impossibility proof (minimal counterexample). There is no function $f([p_1,\ldots,p_n])$ that correctly computes $c$ without the pivot index $\pi$:

Path	$[p_1,p_2]$	Context	Correct $c$
deceased → daughter(F) → granddaughter(F)	$[F,F]$	$j=1$, descendant	0 (female intermediary)
deceased → mother(F) → maternal-grandmother(F)	$[F,F]$	$j=2$, ascendant	1 ($[F,F]\in M^F^$)

Same raw suffix, opposite required outputs. Directionless unification is impossible. $\square$

Complete unified formula. Let $\pi$ be the pivot index (0 for pure descendants, $n$ for pure ascendants). Then:

$$c = 1 \iff [p_1,\ldots,p_\pi] \in M^F^ ;\wedge; p_{\pi+1},\ldots,p_{n-1} \in M^*$$

This single expression specialises correctly to all three axes and is provably minimal — each condition is independently necessary. See Corollary 1.1 in 05-proofs.md for the full proof.

No implementation change required. eligibility.ts already implements the correct rules; only the documentation (Theorem 1 statement) needed correction.

Q2: Is There a Hidden Principle Behind the Jumhūr Exceptions?

Status: ✅ RESOLVED. The hidden principle is Axiom $\delta$ (Kinship Monotonicity).

The original problem: P2 (comments/2.md) hypothesized that the Jumhūr exceptions follow a Pareto optimization or minimax principle: “prevent any close kinship node from collapsing to zero share.”

Resolution: The principle has two components, both stated explicitly in the source text:

Anti-zero: If heir A has strictly stronger kinship than heir B, A cannot receive 0 while B receives > 0. Source: «فكيف يرث الأضعف ويسقط الأقوى؟»
Anti-inversion: If A’s blood-paths are a strict superset of B’s, then share(A) $\ge$ share(B). Source: «فللجدّ الأحظّ من ثلاثة تقديرات» (الأحظّ = literal argmax).

These two components are formalized as Axiom $\delta$ in 02-axioms.md. Under the revised 4-axiom system:

Mushtaraka is explained by anti-zero: the full brother’s kinship is a superset of the uterine siblings’, so he cannot get 0 while they get $\frac{1}{6}$.
Grandfather-sibling sharing is the default under $\alpha_1$ (not an exception), and the floor guarantee (max of three options) follows from anti-inversion.
ʿUmariyyatān is already derived (Theorem 7) but also consistent with anti-inversion: father has superset-kinship of mother.

P2’s assessment (partial credit): P2 was right that a unifying principle exists (anti-zero, max structure) but wrong to claim it was “impossible to formalize” or that the Jumhūr positions resist unification. They do unify — under $\delta$.

What $\delta$ does NOT explain: $\epsilon_1$ (maternal immunity). That remains irreducible.

Q3: What is the Status of the Akdariyya?

Status: Resolved.

Verdict (short): The Akdariyya pooling is a $\delta$-motivated repair applied after ʿawl; it does not require a new independent exception nor a separate pipeline phase. The correct classification is: internal to $\gamma_2$ (post-ʿawl normalization), triggered by the anti-inversion/monotonicity concern expressed by $\delta$.

Details: Operationally the pattern is:

Apply ʿawl (compress farḍ shares).
Observe that an ascendant (GF) ends up with a compressed share less than a lateral farḍ holder (sister).
This violates the juristic monotonicity intent captured by $\delta$ (anti-inversion/anti-zero). The remedy is to pool the GF and sibling shares and redistribute in the 2:1 ratio prescribed by Zayd.

This is exactly the procedure already implemented in phase4 as the Akdariyya branch (gated by config.useDelta): the condition compares the post-ʿawl GF share vs. its muqāsama entitlement and triggers pooling when necessary. Thus it is a $\delta$-motivated redistribution inside $\gamma_2$, not a freestanding $\epsilon$ nor a new phase.

Actionable note: Mark Q3 resolved; document that phase4.ts’s Akdariyya block is the canonical implementation location and that no Phase 4.5 is required (the behavior remains toggleable via config.useDelta).

Q4: Can Maternal Immunity ($\epsilon_1$) Be Proven Irreducible?

Status: Claimed irreducible, not formally proven.

The problem: We assert that $\epsilon_1$ (uterine siblings are not excluded by the mother, their intermediary) is the only universally irreducible exception. But “irreducible” means “not derivable from axioms $\alpha_1, \alpha_2, \beta, \gamma, \delta$.”

What we know:

$\epsilon_1$ contradicts $\alpha_1$ (intermediary exclusion): the mother at $j=2, d=1$ is the explicit واسطة for $j=3, d=1, q=3$ heirs, yet they are immune.
The immunity has Qurʾānic basis (Sūrat al-Nisāʾ specifies uterine siblings’ shares in a context that implies they inherit alongside parents).
$\delta$ (kinship monotonicity) does not help: the mother’s kinship is not a subset of the uterine siblings’ kinship, so $\delta$ does not trigger.

What we need: A formal proof that no composition of $\alpha_1, \alpha_2, \beta, \gamma, \delta$ (without $\epsilon_1$) produces the same outcome. This requires showing there is no alternative axiom encoding where uterine siblings’ immunity is emergent.

Q5: Is the $q$ Encoding Optimal?

Status: ✅ Largely resolved.

The encoding: The 5-tuple uses $q \in {1, 2, 3, 9}$:

$q$	Meaning	Arabic
1	Full	شقيق
2	Paternal	لأب
3	Maternal	لأم
9	N/A	— (spouses, direct ancestors/descendants)

Why this encoding is correct:

The source text explicitly restricts quwwa to collaterals: «ولا يتصور التقديم بالقوة إلا في جهة فروع الأبوة». Quwwa is structurally inapplicable to descendants ($j=1$), ascendants ($j=2$), and spouses ($j=0$). The “N/A” sentinel is the only honest encoding for non-collaterals.
$q$ is used only for ordering (α₂: lower $q$ wins) and identity testing (ε₄: $q=3$ → 1:1 ratio; eligibility: sons of brothers with $q=3$ get $c=0$). It is never used in arithmetic. Therefore the specific numerical values are irrelevant — any 3-element ordered set + sentinel works.
${1, 2, 3}$ are the simplest possible ordinals. $9$ is a distant sentinel that sorts last, ensuring non-collaterals never interfere with the $q$-comparison tiebreaker.

Note on an earlier alternative: An alternative encoding $q \in {1, 3, 6, 9}$ was explored in which $q=1$ meant “Direct” (parent/child) and $q=9$ meant “Maternal.” This was abandoned because: (a) it contradicts the source text’s restriction of quwwa to collaterals, (b) it makes $q$ redundant with $j$ for non-collateral heirs, and (c) it forces a $q=0$ patch for spouses since $q=9$ is no longer available as a sentinel.

Remaining open question: Could there be a non-trivial encoding where the numerical values of $q$ carry algebraic meaning (e.g., the gender-ratio rule $q=3 \to 1{:}1$ vs $q \ne 3 \to 2{:}1$ being expressible as a function of $q$ and the prime structure ${2,3}$)? This is minor and unlikely to have practical consequences.

Q6: Path-Sequence Storage Optimization

Status: Engineering problem, not mathematical.

The problem: The BFS resolver must output full gender sequences for grandmother paths (to compute $c_{\text{asc}}$). Storing a variable-length path for each heir is more complex than storing a scalar $d$.

Options:

Full path: Store the complete gender sequence $(g_1, g_2, \ldots, g_d)$. Most general, most expensive.
Pattern flag: For grandmothers only, store a boolean “has male-between-two-females.” Cheaper but ad hoc.
Compressed encoding: Store the path as a binary string (M=0, F=1), which is at most ~5 bits for realistic depths. Compact and general.

Current position: Option 3 (compressed binary) is recommended. But the question of whether any other axis might need path-sequence information in edge cases (e.g., deep collateral paths) has not been exhaustively checked.

Q7: Completeness — Are There Undiscovered Exceptions?

Status: ✅ RESOLVED (partial — core exceptions are closed; extension-layer gaps are now formalized).

The problem: We have identified 8 original exception candidates ($\epsilon_1$–$\epsilon_8$) plus $\epsilon_3’$ (Ḥanafī grandfather=father), of which 1 is universally irreducible ($\epsilon_1$), 3 are derived ($\epsilon_5, \epsilon_6, \epsilon_7$), 1 is absorbed ($\epsilon_4$), 1 is reclassified as default ($\epsilon_3$), 1 is absorbed into $\gamma$ ($\epsilon_8$), and 2 are dispute-dependent ($\epsilon_2, \epsilon_3’$). Mushtaraka is handled by $\delta$. Is this list complete?

Method used: Exhaustive reading of all 16 source files in faraid/ (including the newly added haml.md, gharqa.md, khuntsa.md). Cross-referencing with 3 professor reviews. Verification against the classical heir table (25 types in faraid/reference.txt).

Resolution of previously flagged gaps:

Al-Ḥaml (unborn heir): faraid/haml.md is now in the corpus. Ḥaml introduces no core-level exception and no change to the 4-axiom system or 6-phase pipeline. It is a generalized Min extension with scenario set $\mathcal{S} = {$stillborn, 1m, 1f, 2m, 2f, 1m+1f$}$ (Jumhūr) or ${$m, f$}$ (Ḥanafī), and asymmetric aggregation (min for known heirs, max for the unborn). The 5-tuple completeness theorem (Theorem 2) holds within each scenario; gender ambiguity is resolved by scenario expansion, not by adding a third value to $g$. See 16-uncertainty-and-forest-extensions.md.
Hermaphrodite (al-Khunthā al-Mushkil): faraid/khuntsa.md is now in the corpus. Khunthā introduces no core-level exception and no change to the axiom system. It is another generalized Min extension with $\mathcal{S} = {$male, female$}$ and four madhhab-specific aggregation rules (Ḥanafī: single-pick; Mālikī: component-mean; Shāfiʿī: component-min + mawqūf; Ḥanbalī: case-split). The khunthā can only appear in $j \in {1, 3, 4, 5}$ — never in $j \in {0, 2}$ (spouses and ascendants have biologically-determined gender by the time they enter a case). See 16-uncertainty-and-forest-extensions.md.
Acknowledgment cases (al-Iqrār): Procedural, not structural — outside the mathematical model. Still not in the corpus.

Additionally resolved: The Akdariyya pool-and-redivide rule was not formally stated as a $\gamma_2$ rule — only called “δ-motivated” in Q3. This has now been corrected in 03-exceptions.md: the rule is classified as a stated $\gamma_2$ sub-rule triggered by $\delta$, with the pool-then-redivide-$2{:}1$ recipe explicitly attributed to Zayd ibn Thābit.

Core exception list status: Complete for the Ibn ʿUthaymīn corpus. The 4-axiom system is not threatened by the new files. Haml, Khunthā, and Gharqā (simultaneous deaths — see faraid/gharqa.md) are all extension-layer phenomena, formalized under the unified $(\mathcal{S}, w, \mathcal{A})$ schema in 16-uncertainty-and-forest-extensions.md.

Summary

#	Question	Type	Difficulty
Q1	Unified eligibility predicate	Mathematical	RESOLVED (→ Corollary 1.1)
Q2	Jumhūr hidden principle	Mathematical / Game Theory	RESOLVED (→ $\delta$)
Q3	Akdariyya status	Classification	RESOLVED (→ $\gamma_2$-internal $\delta$-repair)
Q4	$\epsilon_1$ irreducibility proof	Mathematical	Medium
Q5	$q$ encoding optimality	Design	Largely resolved
Q6	Path-sequence storage	Engineering	Easy
Q7	Exception completeness	Verification	RESOLVED (core closed; extension-layer formalized in `16-*.md`)

References

Eligibility predicates: 05-proofs.md, Theorem 1
Jumhūr hypothesis: comments/2.md
Akdariyya: faraid/jaddma'aikhwah.md
Source texts: all files in faraid/
Exception catalog: 03-exceptions.md