slug: cao-chen-miller-egg-drop-conjecture-one bibkey: caochenmiller2025eggdrop doi: 10.48550/arXiv.2511.18330 url: https://arxiv.org/abs/2511.18330 triage: theorem motivation_gids:
- D5/S3/Observer/Budget/CaoChenMillerEggDropRefutation.result
Cao-Chen-Miller Conjecture 1: the multidimensional egg-drop bound
Problem
Cao, Chen, and Miller, Egg Drop Problems: They Are All They Are Cracked Up To Be!, arXiv:2511.18330v2, printed page 10, Conjecture 1:
In a d-dimensional setting with sides N_1, N_2, …, N_d and k eggs, the minimum number of drops required in the worst-case scenario under the same strategy satisfies
for k >= d.
The subscripted names and inequality above transcribe the printed notation.
Preregistration issue #8448 writes the positive integer domains explicitly.
The hidden point ranges over the product of the integer intervals
{1, ..., N_i}. A drop at a returns intact exactly when a_i < x_i for
every coordinate; otherwise it breaks one egg. The objective is exact recovery.
The formal claim weakens success of the paper’s named strategy to existence
of any deterministic adaptive strategy with those outcomes and egg semantics:
forall (d k : Nat) (N : Fin d -> Nat),
1 <= d -> d <= k -> (forall i, 0 < N i) ->
exists s : EggStrategy N k (paperBound d k N), Correct s
Refuting this weaker existence assertion refutes the named-strategy assertion.
Motivation
At d = 4, k = 5, and sides (5,5,5,5), the proposed budget is nine,
but there are 625 possible hidden points and only 512 nine-bit words.
This instance tests the conjecture without assuming any particular strategy.
Gap
The arXiv identifier and egg-drop counterexample queries returned no resolution.
Crossref identifies a 2026 Fibonacci Quarterly publication of the same title,
DOI 10.1080/00150517.2026.2681644, with zero recorded citations. OpenAlex
work W4416935832 and its exact cites query returned zero citing works;
the Semantic Scholar DOI citations endpoint returned an empty array.
These bounded index readings do not establish historical priority.
WorstCaseDepthInformationLowerBound assumes transcript injectivity. The
missing input for this model is that exact recovery implies injectivity even
when a strategy stops before exhausting its budget.
Route
Represent hidden coordinate x_i by x_i - 1 in Fin (N i) and physical
query coordinates by Fin (N i + 1). The comparison is then
query_i <= hidden_i. Strategy nodes consume one drop and, on the broken
branch, one egg. Stop nodes are allowed under unused budget.
Pad each terminal transcript with zeroes to the full drop budget. Structural
induction on the strategy proves that equal padded transcripts reach equal
terminal predictions. Correctness makes the transcript map injective.
At the displayed instance, the real-power ceiling is exactly nine, so
Fintype.card_le_of_injective would give 625 <= 512, a contradiction.
Falsifier
A correct strategy for all 625 points within nine drops would falsify the
counterexample. In the formal model it would contradict result : Not claim.
The source-to-model implication depends on the integer critical-point and
single binary-outcome semantics stated in the paper.
Evidence
D5/S3/Observer/Budget/CaoChenMillerEggDropRefutation.lean proves
result : Not claim. The proof establishes paperBound 4 5 (fun _ => 5) = 9,
transcript injectivity for the alleged correct strategy, and the impossible
cardinality inequality. Its axiom closure is propext, Classical.choice,
and Quot.sound.
Triage
theorem: the universal bound in Conjecture 1 is refuted by the specified
four-dimensional instance. No other statement from the paper is resolved here.
ASSUMED-UNVERIFIED
The literature search is not exhaustive. The publisher page returned HTTP 403, and Semantic Scholar’s separate metadata endpoint returned HTTP 429; neither is counted as a completed content check. The target statement is the arXiv v2 text, not an independently checked version-of-record statement. No historical priority claim follows from the zero citation or search counts.