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Econ-92 · economics · computer science · Computational economics

Finite exact cake-cutting protocols for mixed-resource EFM

LLM status Open unverified

Statement

ECON-009 | P | Explicit published open question, at source date

Definitions and formal target

Let $n\geq3$, let $M$ contain $m$ indivisible goods with known additive values $v_{ig}\geq0$, and let $V_i(D)=\int_D f_i(x)\,dx$ for nonnegative Lebesgue-integrable densities $f_i$ on $C=[0,1]$. An allocation consists of an item partition $(O_i)$ and a complete measurable cake partition $(D_i)$. Write $u_i(O,D)=\sum_{g\in O}v_{ig}+V_i(D)$.

Use the weak EFM convention of [S1], Definition 4.3: for every pair $i,j$, if $V_i(D_j)>0$ require $u_i(O_i,D_i)\geq u_i(O_j,D_j)$; otherwise require either no envy or some $g\in O_j$ with \[ u_i(O_i,D_i)\geq u_i(O_j\setminus\{g\},D_j). \] The Robertson--Webb (RW) interface permits exact queries \[ \operatorname{Eval}_i(a,b)=V_i([a,b]),\qquad \operatorname{Cut}_i(a,z)=\min\{b\in[a,1]:V_i([a,b])=z\}, \] for $0\leq a\leq b\leq1$ and $0\leq z\leq V_i([a,1])$. As in the survey, the cut answer is the leftmost valid endpoint; requests outside the admissible value range return failure. Is there a protocol outputting an exact complete EFM allocation using finitely many RW queries on every input? Is there a uniform bound $Q(n,m)$ independent of the measures?

Clarifications and required assumptions

The output cake pieces are finite unions of intervals with endpoints specified by queried cuts and the original endpoints. Endpoints carry zero value under atomlessness and can be assigned to make the partition literally complete. A protocol having access to an additional oracle for a perfect cake partition has more power than the RW model. Instancewise finiteness means termination separately for every input under the stated leftmost-cut convention; boundedness means one finite $Q(n,m)$ works for all of them.

Variants and changes of scope

Physical-cake EFM$_0$. Require exact envy-freeness whenever $D_j$ has positive physical measure, even when $V_i(D_j)=0$. Stronger fallback. The item-only fallback used in ECON-008 is another convention and must be named separately. Approximation. After normalizing total resource value to one for each nonzero agent, allow additive envy error $\varepsilon>0$; zero-total agents need no normalization. Connected cake. Restricting each $D_i$ to one interval is an additional feasibility restriction. The two-agent case is not the unresolved general frontier. Oracle/measure changes. Allowing any valid cut answer instead of the leftmost one requires correctness against all permitted answers. Allowing singular atomless measures enlarges the cake domain beyond the survey's integrable-density model. Both are editorial extensions requiring separate status checks.

Audit conclusion and source relationship

[S1], Open Question 4, printed p. 22645, genuinely asks about finite/bounded RW protocols. Definitions 4.1 and 4.3 distinguish EFM$_0$ and weak EFM. The query question is retained with that nomenclature made explicit; abstract existence with a perfect-partition oracle is not a finite RW protocol.

References and exact source roles

[S1] Shengxin Liu, Xinhang Lu, Mashbat Suzuki, and Toby Walsh (2024). Mixed Fair Division: A Survey. Proceedings of AAAI 38, 22641–22649.

Verified locator / source role: Open Question 4, p. 22645: exact EFM and finite/bounded protocols.

https://ojs.aaai.org/index.php/AAAI/article/view/30274/32268

Common conventions: Additional-source status and mathematical conventions

Of the 100 supplied ECON entries, 65 distinct problems appear here; 32 close matches refine earlier entries, and three duplicate questions map to existing entries. Appendix app:audited-crosswalk lists every destination. The attachment's P/R/S/U distinctions and source audits are retained. Its precise mathematical formalizations are editorial unless the individual audit says otherwise. Candidate subsidy targets ECON-004--006 retain their U flags; the resolved approval-core question ECON-007 updates OP-0202 above. The following supplied conventions govern both this part and the attributed ECON refinements elsewhere in the catalogue, subject to local restrictions.

Parameterized questions. A problem family lists inputs that must be fixed before an instance is evaluated: population, horizon, policies, information, data law, model class and welfare weights. These are required choices, not quantities the citations are assumed to specify. Undefined applications should not be treated as identified estimands.

Indices and integrability. Sets of agents, firms and regions are finite unless a population measure is explicitly used. $i,j$ index units, $r$ regions, $t$ dates and $h$ horizons. $\mathbb E$ and $\Pr$ refer to the declared population and law; $\mathbf1\{A\}$ is an indicator. Every displayed expectation and discounted sum needs existence in the stated domain. Infinite horizons use a discount in $(0,1)$ and a convergence condition; finite horizons may use discount one.

Optimization and existence. A displayed $\max$, $\min$, or $\arg\max$ is conditional on attainment. For finite feasible menus this is immediate; general spaces need continuity and compactness/coercivity or another existence argument. Without attainment, the target is the supremum/infimum and arbitrarily near-optimal feasible choices. $\inf\varnothing=+\infty$. Empty feasibility and an unidentified target are different issues.

Potential outcomes. $Y_i(z)$ is the outcome under a specified intervention, not the observed conditional mean among people choosing $z$. In binary comparisons $\Delta Y_i=Y_i(1)-Y_i(0)$. Specify assignment, treatment versions, compliance, information and observation horizon. Interference is allowed under a full policy vector; compressed exposure notation needs a justified mapping. No interference, consistency and overlap are substantive assumptions, not automatic consequences of notation.

Populations and observation. Fix baseline eligibility and population weights. Conditioning on post-policy employment, survival, relocation or program uptake can change the population being compared. Death is not ordinary loss to follow-up; outcomes truncated by death need a composite convention, joint survival target, or explicitly defined survivor stratum. Fertility-changing interventions need a population functional rather than an unexplained fixed descendant cohort. Measurement and missingness models must be supplied.

Identification. A structural model $m\in\mathcal M$ implies observable law $P_m$ and target $\theta(m)$. Identification means \[ P_{m_1}=P_{m_2}\ \Longrightarrow\ \theta(m_1)=\theta(m_2) \quad\text{for all }m_1,m_2\in\mathcal M . \] Without identification, the target is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Writing $\theta(P)$ before establishing identification can hide incompatible latent models. Confidence sets additionally need a sampling law, confidence level, asymptotic sequence and uniformity class. Point identification, coverage and informative precision are separate properties.

Mediation. Total effects of randomized assignment are distinct from cross-world natural indirect effects. Belief/effort-channel effects require a meaningful mediator intervention and additional measurement, exclusion or mediation assumptions. Randomization of the initial policy alone does not supply those assumptions. Report bounds or interventional alternatives when the stronger target is unsupported.

Equilibrium. A counterfactual $W(z)$, $p(z)$, or $\mu_t^z$ requires individual optimization, feasibility, government/resource budgets, market clearing and state-transition laws. State equilibrium existence and selection, or report ranges over compatible equilibria. Initial-state integration includes subsequent endogenous transitions; current-state integration must use the policy-dependent distribution. Reduced-form invariance after policy is not assumed.

Welfare accounts. Scores, utility, health, dollars and ecological quantities require explicit conversion before addition. Fees, taxes, subsidies, wages and extortion payments are transfers between parties; distributional weights can make them welfare relevant, but their face value is not automatically a resource loss. State ownership, geographic boundaries, foreign-income weights and fiscal finance. Do not add profits to household welfare already including dividends, or count land capitalization and the underlying benefit twice. A benefit-cost expression must distinguish gross benefits from net welfare and subtract every real cost exactly once.

Derivatives and risk. Log derivatives require positive arguments; local responses require admissible perturbations and specified equilibrium branches. Differentiation under expectations needs regularity/domination, and boundaries may allow only one-sided derivatives. Risk uses a specified law; ambiguity uses a declared nonempty law set on a common history space. Adaptive policies must be nonanticipative. A robust criterion is an editorial choice unless attributed explicitly.

Scope overlaps. Allocation, collective choice, contracts, household bargaining and coordinated policy overlap economics with optimization and game theory. They are retained because they appeared in the original catalogue; no claim of a strictly game-theory-free selection is made.

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Econ-92-A2 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-92

Finite exact cake-cutting protocols for mixed-resource EFM: Bounded Robertson--Webb protocols with a common cake measure

The unrestricted target is an exact finite Robertson--Webb protocol for additive indivisible goods and heterogeneous cake densities. Weak EFM requires exact envy-freeness towards a bundle with positively valued cake; towards a zero-valued cake bundle it permits deleting one of that bundle's goods: \[ V_i(D_j)>0\ \Longrightarrow\ u_i(O_i,D_i)\geq u_i(O_j,D_j). \] This note removes the query obstacle when every agent has the same cake measure, even if that common density is unknown and irregular. Computational enumeration is allowed: the question asks for a finite query bound, not polynomial total computation.

Theorem (Finite exact search for a common cake measure): Suppose all agents have the same cake measure $V$, induced by a nonnegative integrable density, and have known additive nonnegative item values. There is an exact weak-EFM protocol using one evaluation query and at most $n-1$ cut queries. Between queries it solves finitely many linear programs with the known item values and the returned total cake value as coefficients. With rational coefficients these programs use ordinary exact rational arithmetic; in the usual exact-real RW model they use exact arithmetic and comparisons on returned values.
Proof: Query $T=V([0,1])$. Enumerate every complete item partition $O$ and every subset $J\subseteq[n]$ proposed to receive positive cake value. For fixed $O$ write \[ a_{ij}=v_i(O_j)-v_i(O_i),\qquad h_{ij}=\max\left(\{v_{ig}:g\in O_j\}\cup\{0\}\right). \] Seek cake masses $t_i\geq0$ with $\sum_i t_i=T$. Set $t_j=0$ for $j\notin J$. For each $j\in J$ impose \[ t_i-t_j\geq a_{ij}\quad\text{for all }i. \] For each $j\notin J$ impose \[ t_i\geq a_{ij}-h_{ij}\quad\text{for all }i. \] These are exactly the weak-EFM inequalities for this support. In the zero-cake case, removing the most valuable item of $O_j$ is the best deletion for $i$; if $O_j$ is empty, the displayed inequality is simply no envy. Allowing no envy as a separate branch gives the same condition because $h_{ij}\geq0$.

If $J$ is nonempty, introduce $0\leq e\leq T$ and $t_j\geq e$ for all $j\in J$, and maximize $e$. Accept this case precisely when its optimum is strictly positive. This tests strict positivity exactly: a feasible finite list of positive masses has a positive minimum, and conversely $e>0$ forces every declared positive mass to be positive. The feasible region is bounded and closed, so a feasible program attains its optimum. If $J$ is empty, test the displayed system directly, necessarily with $T=0$.

This finite enumeration finds masses whenever a weak-EFM allocation exists, because every actual allocation has one enumerated item partition and a cake-value support $J$. Existence in the present additive-goods domain is the established EFM existence theorem of Bei et al. [bei]; their stronger physical-cake version implies the weak condition used here. This existence input is not a new theorem of this note.

Realize the accepted masses with sequential cuts. For each $i<n$, cut a piece of value $t_i$ from the current left endpoint, skipping zero requests if convenient. Assign the remainder to agent $n$. Each request is admissible because remaining mass equals the sum of the remaining $t_i$. Continuity of the cumulative density integral guarantees a cut, and the prescribed leftmost answer still has exactly the requested value. Null tails and endpoints may be included in the final piece. Thus the returned intervals form a complete partition, and the linear inequalities certify weak EFM. There are at most $n-1$ cut queries after the initial evaluation. ∎

Proposition (A direct construction for identical item values): If the common cake assumption is supplemented by identical nonnegative item values $v_{ig}=w_g$, an allocation with the same query bound can be constructed without the EFM existence theorem or exhaustive enumeration.
Proof: Assign items sequentially to a currently minimum-load bundle, and denote final item loads by $\ell_i$. This allocation is EF1. To see this, fix $i,j$ and take the last item $g$ assigned to $j$. Immediately before that assignment, $j$ had minimum load, at most the then load of $i$, which is at most $i$'s final load by nonnegativity. Thus $\ell_j-w_g\leq\ell_i$. Empty compared bundles cause no envy.

For $T>0$, choose the unique level $z>\min_i\ell_i$ satisfying $\sum_i(z-\ell_i)_+=T$, and set $t_i=(z-\ell_i)_+$. For $T=0$ set all masses to zero. Every positive-cake recipient has final value $z$, while every other agent has final value at least $z$. Hence no agent envies a positive-cake recipient. For a zero-cake recipient, the original EF1 deletion witness remains valid because the envying agent's own utility has only increased. This is weak EFM. Realize the masses by the same sequential cuts as in the theorem. ∎

Source relationship and remaining gap.
The survey [survey] asks for finite or bounded RW protocols; Bei et al. [bei] prove existence using stronger cake-partition access. The present finite-dimensional conversion relies on one common cake measure: a piece is described completely by one number $t_j$. With heterogeneous densities, the vector $(V_i(D_j))_i$ cannot generally be specified and realized by cuts from one measure. Neither the linear programs nor the water-filling construction supplies a finite protocol for that unrestricted domain.

References

Completion Estimate

25%

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Econ-92-A1 · Exploration · LLM-generated · gpt 6 astra pro ·

GPT 6 Astra Pro attempt on Econ-92

{Finite exact Robertson--Webb protocols for mixed EFM}

D63-90.tex

Research attempt and adversarial audit --- 9 October 2026

Tools. Web browsing: YES. Computation: YES (Python, exact integer/rational checks, and explicitly identified numerical diagnostics). Status convention. A full proof of a restricted theorem does not resolve the full input. Literature results are marked as imported; no novelty or exhaustive current-literature certification is claimed. Computational searches certify only their stated finite domains.

FORMAL RESTATEMENT

Let $n\ge3$, $M$ contain $m$ indivisible goods, and $v_i(O)=\sum_{g\in O}a_{ig}$ with known $a_{ig}\ge0$. Cake valuations are $V_i(D)=\int_Df_i$ for nonnegative integrable $f_i$ on $[0,1]$. The complete output is an item partition $O$ and cake pieces $D_i$ that are finite unions of intervals with endpoints among queried cuts and 0,1; endpoints have zero value and are assigned exactly once.

Use weak EFM: if $V_i(D_j)>0$, require $v_i(O_i)+V_i(D_i)\ge v_i(O_j)+V_i(D_j)$. If $V_i(D_j)=0$, require either no envy or a $g\in O_j$ such that \[ v_i(O_i)+V_i(D_i)\ge v_i(O_j\setminus\{g\})+V_i(D_j). \] The Robertson--Webb interface gives exact evaluations of intervals and $\operatorname{Cut}_i(a,z)$, the leftmost $b\in[a,1]$ with $V_i([a,b])=z$; inadmissible requests fail. The questions are, separately, \[ \exists\mathcal P\ \forall\text{ instances }I:\ \mathcal P(I)\text{ terminates exactly after finitely many queries}, \] and whether such a protocol has a uniform finite bound $Q(n,m)$ independent of the measures. Quantifier order matters: instancewise finite termination is not a uniform bound.

Variants requiring no envy toward any positive-physical-length cake piece (EFM$_0$), an item-only fallback, approximate envy, connected cake pieces, arbitrary valid cut answers, or singular atomless measures are not silently equated with this problem.

QUICK LITERATURE/CONTEXT CHECK

The AAAI survey's Open Question 4 asks the finite/bounded RW question [survey]; the longer arXiv version has different numbering. Bei et al. establish mixed-resource EFM existence and distinguish exact special-case computation from approximation [bei]. Pure-cake bounded protocols are established by Aziz--Mackenzie [am]; the September 2026 preprint of Ye--Bai reports a single-exponential pure-cake query bound [yb]. Neither pure-cake statement by itself allocates indivisible goods. Stromquist's connected-piece impossibility [strom] resolves a stricter named variant, not the finite-union main question.

ATTACK PLAN

The proof track makes the common-value water-filling construction into actual RW queries and isolates finite verification of a proposed answer. The disproof track uses the pure-cake connected-piece restriction and checks that a perfect-partition oracle is not being hidden inside a finite-query argument. The main heterogeneous finite-union question remains separate from both tracks' partial successes.
ToolRole
Pairwise deletion witnessesKeeps the precise EFM fallback intact.
Fixed-sign marginal inductionAllows nonadditive common item valuations.
Water fillingAllocates common cake to the lowest item-value bundles.
Continuity of cumulative measuresEnsures exact cuts and handles zero-density plateaus.
Payment-to-cake conversionIdentifies a sufficient cake-budget condition.
Robertson--Webb query accountingDistinguishes existence from an implementable finite protocol.
Finite allocation verificationChecks known interval endpoints without a perfect-partition oracle.
Definition-separation examplesPrevents substituting weak EFM for strong EFM.

WORK

A complete item-only construction for a common doubly monotone valuation

Lemma: Suppose all agents have the same normalized doubly monotone item valuation $v$, with common fixed-sign partition $M=G\sqcup B$. No additivity or marginal-size bound is needed. Starting from empty bundles, assign each good to a bundle of minimum current value and each chore to a bundle of maximum current value. The resulting complete allocation is EF1 with the item-only either-side deletion convention.
Proof: Prove the assertion after each item assignment. It holds for empty bundles. Suppose a good $g$ is added to a minimum-valued bundle $A_j$. Every other agent $i$ has $v(A_i)\ge v(A_j\setminus\{g\})$, so any new envy of $j$ is repaired by deleting $g$. For comparisons from $j$ to another agent, an existing no-envy inequality survives because its own value does not fall. If an old deletion witness $h$ was in its own bundle, the left value after deleting $h$ does not fall when $g$ is added, by the fixed positive marginal sign on every context. If $h$ was in the other bundle, the left value again does not fall. Comparisons between other agents are unchanged.

If a chore $g$ is added to a maximum-valued bundle $A_j$, every new envy from $j$ to another agent is repaired by deleting $g$, returning to a value at least as large as that other bundle. For comparisons from another agent to $j$, the right value only falls. If an old deletion witness lies in $j$'s bundle, the right value after that deletion still only falls when $g$ is added, by the fixed negative marginal sign in every context. The other witness and no-envy cases are preserved for the same reason. This exhausts all ordered pairs and establishes the induction. Finitely many steps assign every item. ∎

Cake as a common-value equalization resource

Lemma (Water filling with exact cuts): Let $\ell_1,\ldots,\ell_n$ be real item values and let all agents have the same finite nonnegative atomless cake measure $V$, of total $C$. If $C>0$, there is a unique $H>\min_i\ell_i$ satisfying \[ \sum_i(H-\ell_i)_+=C. \] There is a complete interval cake partition with $V(D_i)=c_i=(H-\ell_i)_+$, assigning empty pieces to agents with $c_i=0$. All agents' resulting total values are $\max\{\ell_i,H\}$.
Proof: The left side is a continuous piecewise-linear function of $H$, equals zero at $\min_i\ell_i$, is strictly increasing above that point, and tends to infinity. Existence and uniqueness follow from the intermediate value theorem and strict increase. Order the agents with $c_i>0$. Cut off a piece of value $c_i$ successively for all but the last such agent and give the last the remainder. The cumulative function $x\mapsto V([0,x])$ is continuous: continuity of a finite measure from above and below shows its jump at $x$ is exactly $V(\{x\})=0$. Thus each required cut exists, including across intervals of zero density. Nonnegative requested values sum to $C$, so no query asks for more value than remains. Endpoints have zero value and can be assigned once. The final value identity is immediate from $\ell_i+(H-\ell_i)_+=\max\{\ell_i,H\}$. ∎
Theorem (Strong EFM for common preferences): If all agents share a normalized doubly monotone item valuation $v$ and a nonnegative finite atomless cake measure $V$, then a complete strong EFM allocation exists. Each nonempty cake piece can be chosen to be an interval.
Proof: Take the EF1 item partition from Lemma Lemma, and put $\ell_i=v(O_i)$. If $C>0$, apply Lemma Lemma (Water filling with exact cuts). A recipient $j$ of positive cake value has final utility $H$, while every agent's own final utility is at least $H$. Since all valuations are common, nobody envies such a recipient. If $j$ receives zero cake value and an agent envies it, retain the original item-only EF1 witness; the strong fallback requires only that item comparison and zero cake value for the envied bundle, both of which hold.

If $C=0$, give the entire cake to any one agent and use the item-only EF1 partition. Every cake piece has value zero to every agent, so each pair either has no envy or has its original item-only deletion witness. Both constructions are complete. The measure argument uses only finite atomlessness, so the common-preference subcase also covers singular atomless Borel measures, not merely densities. ∎

A uniform query bound for common preferences

Theorem: For the subdomain in which all item valuations are the same additive nonnegative function and all cake valuations are the same measure, a deterministic protocol uses at most one evaluation and $n-1$ cut queries and returns exact weak EFM. It can simultaneously satisfy the item-only strong fallback, EFM$_0$, and connected cake pieces.
Proof: Compute the item allocation of Lemma Lemma from the known values; all items are goods. Make $\operatorname{Eval}_1(0,1)$ to get $C$. If $C>0$, compute the unique level and shares in Lemma Lemma (Water filling with exact cuts) by sorting the $n$ item loads and checking their finitely many linear intervals. For the agents with positive shares, use sequential cuts by agent 1 for all but the last, and give the last the remaining interval. Every query requests no more than the common value of the residue; at most $n-1$ are needed. Agents with zero shares receive empty cake pieces. Each cake recipient has final common value $H$, no larger than any agent's own value. Hence there is no envy toward any nonempty cake piece, which is stronger than both the positive-value and positive-physical-measure clauses. Remaining envy pairs retain their item-only deletion witnesses. Because items are nonnegative, if strict envy remains its useful witness can be chosen in the envied bundle: removing an item from the envier's own bundle cannot increase its value.

If $C=0$, give the whole cake to an agent with minimum item value and give all others empty cake. No agent envies the cake recipient under common item values. The item EF1 witnesses handle all other envy. Thus even the physical-cake requirement holds in this zero-value case. The pieces are intervals or empty, all endpoints can be assigned consistently, and no perfect-division oracle is used. ∎

The same proof works if a cut query returns any valid endpoint rather than the leftmost one: only its exact value and nondecreasing position are used. The common-measure proof also covers finite nonnegative singular atomless Borel measures, although the main input assumes densities.

Proposition (A sufficient cake budget for exact envy-freeness): Suppose all agents have the same cake measure of total $C$, but their item valuations may differ. If a complete item partition $O$ is envy-freeable with nonnegative payments $p$ of total at most $C$, then an exactly envy-free complete mixed allocation exists using that partition.
Proof: Give agent $i$ cake value $c_i=p_i+(C-\sum_jp_j)/n$. These are nonnegative and sum to $C$, so they can be realized by the sequential-cut construction in Lemma Lemma (Water filling with exact cuts). The added common amount cancels in every comparison, leaving the assumed inequalities $v_i(O_i)+p_i\ge v_i(O_j)+p_j$. Exact envy-freeness implies both EFM conventions. ∎
The sufficient condition is not automatically satisfied when the available cake value is small, and heterogeneous cake measures cannot be treated as a common transferable monetary unit.

A finite verification certificate does not imply finite search

Proposition: Given a complete item partition and cake pieces described by a total of $r$ intervals with allowed endpoints, exact weak EFM can be verified with at most $nr$ evaluation queries and finite arithmetic on the known item values and query answers.
Proof: For each agent evaluate every interval and sum the evaluations belonging to each recipient's piece. Endpoints contribute zero and the intervals are disjoint up to endpoints, so these sums equal all $V_i(D_j)$. For each pair, first test whether $V_i(D_j)>0$ and then the required inequality. When it is zero and there is envy, test each of the finitely many candidate goods in $O_j$ using the known additive values. Also check disjointness and coverage of the listed intervals and the item partition. These tests are exactly the definition. ∎
There are uncountably many possible interval endpoints before queries. Enumerating rational endpoints would neither find arbitrary exact cuts nor satisfy the specified output model. The verifier therefore supplies no bound on a search for a certificate.

The connected-piece variant is impossible in general

Theorem (Reduction from an explicitly imported impossibility): No deterministic instancewise-finite RW protocol solves the connected-piece variant for every $n\ge3$ and every permitted input, even with $m=0$.
Proof: Use Stromquist's theorem [strom]: for three agents with strictly positive bounded cake densities, no protocol using finitely many exact evaluation/cut queries on every input always returns a complete envy-free allocation with one interval per agent. Its hard densities are strictly between $\sqrt2/2$ and $\sqrt2$, so they lie inside the present integrable-density domain and make cuts unique. The paper also permits marks calculated from previously obtained information, so allowing arbitrary evaluation endpoints does not evade its query obstruction.

A protocol for the stated variant could be run at $n=3,m=0$ on these densities. With no goods available for deletion, weak EFM requires exact envy-freeness for every pair. Its output would be a complete connected envy-free cake partition after finitely many queries, contradicting that theorem. This proves the reduction and the variant's impossibility. It imposes no restriction on the main problem's finite unions of intervals. ∎

This is a theorem-based disproof of a variant; the imported impossibility proof is not represented as newly established here.

VERIFICATION

The exact rational tests in the accompanying script checked 200 common-preference, three/four-agent water-filling instances with nonadditive fixed-sign items, a larger item domain than needed by Theorem Theorem. All passed the strong item-only fallback. These tests check allocated values; the cut implementation is justified by the measure-continuity proof, not by pretending a density discretization is an exact RW simulation. 
\begin{lstlisting} O = common-value greedy item allocation C = Eval_1(0,1) if C == 0: give all cake to a minimum-item-value agent else:     compute shares c_i = max(0,H-item_value[i]) summing to C     for all but the last positive-share agent i:         b = Cut_1(current_left_endpoint,c_i)         give [current_left_endpoint,b] to i     give the residue to the last positive-share agent \end{lstlisting}
For $n=1$ the complete allocation is immediate; the main target starts at three. For $m=0$, the known pure-cake theorem applies only when disconnected pieces are allowed, consistently with Theorem Theorem (Reduction from an explicitly imported impossibility). When $C>0$, zero-valued fringes remain attached only to positive-share recipients; for $C=0$ the minimum-load recipient prevents physical-cake envy. An approximation theorem with error $\varepsilon>0$ is not an exact theorem obtained by setting $\varepsilon=0$. Failure of one construction using a perfect-partition oracle is not an impossibility result for all RW protocols.

FINAL

LABEL: UNRESOLVED for the main heterogeneous, finite-union exact RW question. FULL SOLUTION / FULL PROOF for a restricted subcase: the common-preference protocol uses at most $n$ queries and handles the stronger named fairness conventions as stated. FULL SOLUTION / DISPROOF for a named variant, using an imported theorem: a general connected-piece protocol is impossible by Theorem Theorem (Reduction from an explicitly imported impossibility). These labels are not applied to the main question.

Exact first gap. No instancewise-finite exact RW replacement is supplied for the heterogeneous mixed-allocation step, without assuming an additional perfect-partition oracle.

Three next moves. (1) Prove a finite-query way to allocate cake to a selected envy-closed agent set while preserving all outside comparisons. (2) Establish a discrete progress invariant that forces termination independently of arbitrarily small utility gaps. (3) Seek an adversarial transcript argument tailored to mixed EFM with disconnected pieces, rather than reusing connected-cake hardness.

Minimal unresolved case. Three heterogeneous agents, positive-value goods and heterogeneous cake densities already lie beyond the common-value theorem; the obstruction, if any, must survive the freedom to output finitely many disconnected intervals. Approximation and random protocols with weaker termination guarantees are not settled by this report.

Completion Estimate

25%

The common-preference exact protocol and several of its variants are proved, and the connected-piece variant is ruled out through a verified theorem. The central heterogeneous finite-union protocol question remains unresolved.

This is a subjective estimate of the fraction of the whole requested mathematical scope addressed by this report, including the named variants. It is not a probability of correctness, an objectively measured fraction of a proof, or an estimate that the remaining work takes the complementary fraction of the time. Only the eleven supplied files are assessed; no missing rank-1--200 statements are inferred.

References

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