| paper | d4-conf-atal-2015-0271 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | yes |
Theorem 13 is a named stability theorem, but it gives a sufficient condition for core emptiness rather than an algorithm, complexity result, or computational formulation. The proposed one-type population limit is a coherent stability question, but it cannot meet the required computational anchor, so bit (a) fails.
fails bit a — no named computational result to mirror
The proposed mirror covers Theorem 13's symmetric standard-core Miner Game; it leaves the coalition-structure stability results, the other miner-game theorems, and the simulations aside.
There is an anchor problem: the paper contains no named result classifying a computational problem as tractable, hard, or parameterized. Theorem 13 is the closest positive anchor, but it proves a stability property rather than a complexity classification. So it cannot meet the anchor rule strictly. The best case for a mirror rests on it as a structural result.
Theorem 13, proved in this paper, says that in its Miner Game with \(n>3\) miners of equal computational power, the core is empty if
\[
d\le \frac{D(n-3)}{2(n+1)(1+D\beta)}.
\]
Here \(D\) is the delay between pools, \(d\) the delay within a pool, and \(\beta\) the target rate of longest-chain growth. The game sets singleton values to \(1/n\), the grand coalition’s value to \(1\), and estimates other coalitions’ values using the paper’s two-miner approximation to their share of accepted blocks. The theorem says that, under the stated bound, every division of the grand coalition’s reward is blocked by some coalition. Its proof is in this paper; it uses the two-miner reward-share formula from Theorem 4, attributed there to Sompolinsky and Zohar.
The natural regime is a large population of similar small miners: for example, many operators using the same hardware and subject to the same communication assumptions. Each is indistinguishable in the model’s relevant respects, so the type set has one element, while the number of agents is large. Let \(\Omega=[0,1]\) represent this population, with Lebesgue measure; a set \(A\subseteq\Omega\) has mass \(|A|\), its fraction of the population. Since the miners are homogeneous, that is also their fraction of total hash power. In the limit, a mass \(x\) of miners contributes hash share \(x\). The finite theorem’s delay threshold tends to \(D/[2(1+D\beta)]\) as \(n\) grows.
I would ask the following continuous version of Theorem 13:
One-type continuous mining-core existence. Given \(D,d,\beta>0\), use the paper’s Section 6 characteristic-value construction with \(|C|/n\) replaced by \(x\in[0,1]\). Thus a deviating set of mass \(x\) pools, the remaining mass stays solo, and the resulting pool value \(\nu_{D,d,\beta}(x)\) is calculated using the paper’s effective-rate construction and Equation (2), retargeted to \(\beta\). The instance’s coalitions are all measurable sets \(A\subseteq\Omega\), with value \(\nu_{D,d,\beta}(|A|)\).
A solution decides whether there is a measurable reward density \(r\ge0\) such that \(\int_\Omega r=1\) and
\[
\int_A r\ge \nu_{D,d,\beta}(|A|)
\quad\text{for every measurable }A\subseteq\Omega.
\]
If one exists, output \(r\); if the core is empty, output a mass \(x\) with \(\nu_{D,d,\beta}(x)>x\). Such an \(x\) certifies emptiness: any reward density totaling \(1\) leaves some set of mass \(x\) with reward at most \(x\). Conversely, if \(\nu(x)\le x\) for every \(x\), the uniform allocation \(r=1\) is stable. So this one-type mirror reduces core existence to maximizing the scalar function \(\nu(x)-x\). I expect this version to be tractable by continuous optimization, though the paper does not establish that classification. The finite threshold suggests a no-core region below \(d<D/[2(1+D\beta)]\); proving that requires showing the characteristic values converge appropriately, and I would not claim it from Theorem 13 alone.
This is close to the authors’ problem: the object remains the stability of reward sharing against miners pooling together, and only the population has become divisible. It does not continuize the reward outcome. Further questions include whether a multi-type population with heterogeneous hash capacities and delays admits an efficient separation method, and whether the same boundary holds under the full network dynamics rather than the paper’s Section 6 approximation.
The mirror covers only Theorem 13’s symmetric, standard-core Miner Game. It does not claim to mirror the paper’s coalition-structure stability results or its simulations. Its weakest point is also its main qualification: Theorem 13 is not a computational-complexity result, and the continuous decision problem is a new formulation. In addition, its proposed boundary depends on extending an approximate characteristic function to the continuum. The positive case is therefore a plausible continuation of the paper’s stability question, not a computational result already present in it.
The strongest objection is that Theorem 13 is not a computational result. It proves a sufficient condition for the core to be empty; it gives no algorithm, complexity classification, or computational task whose difficulty comes from the number of miners. The population parameter \(n\) appears in a mathematical bound, not as an input to an algorithm. That makes this a weak anchor for a programme about complexity landscapes.
The proponent’s one-type mirror is nevertheless coherent. In the symmetric setting, if a coalition’s value depends only on its mass \(x\), write it as \(\nu(x)\). Under the proposed density-based core definition, the core exists exactly when \(\nu(x)\le x\) for every \(x\in[0,1]\): the uniform density \(r=1\) works in that case, and if \(\nu(x)>x\), every density of total reward \(1\) leaves some coalition of mass \(x\) with reward at most \(x\). So the proposed computation reduces to maximizing the scalar function \(\nu(x)-x\). That may yield a useful stability boundary, but it does not expose a high-multiplicity algorithmic challenge like pricing over exponentially many alternatives. Nor does Theorem 13 establish the proposed boundary: it gives a sufficient finite-\(n\) condition, and taking the limit of that condition does not prove an exact continuous threshold.
A richer mirror could use several miner types and ask which measurable coalitions block a reward allocation. But the paper’s concrete game in Section 6 assumes equal computational power and estimates coalition values with a two-miner approximation. To make heterogeneous types computationally meaningful, one must specify how hash power, delays, and network position determine coalition values under retargeting. That could become an interesting new model, but the paper’s results do not supply that machinery or a complexity problem to continuize.
This is a marginal-value objection, not a proof that no worthwhile mirror exists. Equal-power miners are a plausible high-multiplicity regime, and the mass-based core question is a legitimate continuous stability question. The honest negative case is therefore limited: this paper offers little evidence for a *computational-complexity* mirror, but Theorem 13’s population limit remains a plausible structural mirror, so I cannot defend the universal claim that no worthwhile continuous mirror survives.
The adversarial triple: the proponent anchors on up to three named results; the opponent sees that case and must defeat every anchor; the judge decides which case convinced it. These are the pipeline’s own outputs, generated by tools/triple_run.py — no human edited them. The paper’s own text is not reproduced here beyond the quoted statement above.