This document is substrate-specific. It maps the neutral theory onto JAM only where there is an explicit Gray Paper or ELVES referent.
Tags:
- [GP] stated in the Gray Paper;
- [ELVES] stated/proved in Burdges et al. (2024), ePrint 2024/961;
- [extension] derived here by modifying ELVES assumptions;
- [absent] not specified by those sources.
ELVES Corollary 1 sets collateral per party to
[ C=\nu M/n, ]
where (M) bounds the adversary's gain from acceptance of an invalid block. The rational-adversary soundness condition is
[ \frac1\varepsilon>\frac n\nu+1. ]
Therefore the strict upper boundary is
[ \boxed{ \varepsilon_*=\frac{\nu}{n+\nu}. } ]
For (n=1023) and (\nu=0.05),
[ \varepsilon_*=1/20461\approx4.887\times10^{-5}. ]
Thus the neutral theory's probability threshold (\varepsilon) does have a possible JAM referent, but it is economically derived rather than freely chosen. The Gray Paper does not fix (\nu).
[ELVES] Let (s_0) be expected initial committee size and (s_\delta) the expected tranche size opened by a no-show. Under the pessimistic adaptive-crash analysis, an honest tranche has Poisson mean
[ \lambda=(1-\gamma)s_\delta, ]
where (\gamma<1/3) bounds the adversarial share. The extinction probability (q) solves
[ q=e^{-\lambda(1-q)}. ]
Security requires
[ \boxed{\lambda>1} ]
and the soundness failure probability is bounded by
[ \boxed{ P_{\rm accept}\le q^{s_0/s_\delta}. } ]
For scalability, faulty/no-show-prone validators have reproduction number
[ A=(\alpha+\beta)s_\delta, ]
which must satisfy
[ \boxed{A<1}. ]
In the static bound, expected final committee size is at most
[ \boxed{ s_f\le\frac{s_0}{1-A}. } ]
Together these imply
[ \frac1{1-\gamma}<s_\delta<\frac1\gamma. ]
At (\gamma=1/3), this is (1.5<s_\delta<3). ELVES reports an optimum near 2.0794; JAM uses (F=s_\delta=2).
Using (s_0=30), (s_\delta=2), (n=1023):
| adversarial share (\gamma) | (\lambda) | branching bound |
|---|---|---|
| 1/3 | 1.333 | (1.13\times10^{-4}) |
| 0.20 | 1.600 | (2.04\times10^{-7}) |
| 0.10 | 1.800 | (2.58\times10^{-9}) |
| 0.05 | 1.900 | (3.19\times10^{-10}) |
At the worst-case boundary (\gamma\to1/3), the bound is above (\varepsilon_*) for (\nu=0.05). The same expected-profit inequality would require
\frac{nP}{1-P} \approx0.1156 ]
when one collateral unit is lost.
If two equal guarantor collaterals are both lost, the analogous scenario calculation halves this to about 0.0578. That two-collateral calculation is a JAM mapping extension, not ELVES Corollary 1 itself.
ELVES assumes honest auditors who audit an invalid block detect it. Correlated honest client failure is therefore outside the ELVES model.
[extension] Let (f) be the fraction of otherwise-honest validators whose client accepts one particular crafted invalid report. The effective honest branching mean becomes
[ \boxed{ \lambda_f=(1-\gamma)(1-f)s_\delta. } ]
The same pessimistic extinction calculation then gives
[ P_{\rm accept}(f)\le q_f^{s_0/s_\delta}, \qquad q_f=e^{-\lambda_f(1-q_f)}. ]
Honest escalation ceases to be supercritical at
[ \lambda_f=1, ]
hence
1-\frac1{(1-\gamma)s_\delta}. } ]
For JAM (s_\delta=2):
| (\gamma) | (f_{\rm crit}) |
|---|---|
| 1/3 | 0.250 |
| 0.20 | 0.375 |
| 0.05 | 0.474 |
This is a branching-process threshold, not a Gray Paper or ELVES theorem.
For a target (\varepsilon), define
[ q_*=\varepsilon^{s_\delta/s_0}. ]
The required branching mean is
-\frac{\ln q_}{1-q_}. ]
Therefore the largest shared-client fraction certified by this branching bound is
1- \frac{\lambda_*}{(1-\gamma)s_\delta}, } ]
provided the target is already met at (f=0).
For JAM with (\nu=0.05), hence (\varepsilon_*=1/20461):
- at (\gamma=1/3), the target is already missed at (f=0);
- at (\gamma=0.20), (f_{\max}\approx0.1456);
- at (\gamma=0.05), (f_{\max}\approx0.2805).
This is a JAM-native candidate certified viability margin: the largest shared-client fraction for which this pessimistic bound certifies the chosen target. It is not an exact protocol security frontier.
[ELVES] The rational-adversary argument charges the attacker's/backer's collateral when an invalid block is caught.
[extension] If a crafted bug causes otherwise-honest guarantors to sign the invalid report, those guarantors may be the parties slashed. In that scenario the specific ELVES expected-profit argument does not automatically assign the collateral loss to the actor benefiting from the bug. This does not prove that an attacker has zero total cost; it means Corollary 1 no longer supplies that cost without an additional economic mapping.
The remaining protection is then the probability of detection/final rejection, evaluated per exploitable bug.
The shared-client extension above concerns whether a negative judgment is found. JAM has a second layer after detection: the Gray Paper dispute/verdict rules.
[GP] For a verdict validator-set size (N), define
[ K=\left\lfloor\frac{2N}{3}\right\rfloor+1, \qquad W=\left\lfloor\frac{N}{3}\right\rfloor. ]
Each verdict contains exactly (K) judgments. It is:
- good when all (K) are positive;
- bad when zero are positive;
- wonky when exactly (W) are positive.
Bad and wonky verdicts are non-positive and remove the report from the availability path. The ordinary culprit/fault offense paths refer to good or bad reports; a wonky report is recorded in the wonky set but does not, through those rules, add guarantors or judges to the punish set. Whether an external staking layer separately acts on a wonky record is [absent] from this audited Gray Paper layer.
Let (P) be the number of validators that would judge the crafted invalid report as valid. Then a verdict of each type can be assembled iff:
[ \boxed{\text{bad possible}\iff N-P\ge K,} ]
[ \boxed{\text{good possible}\iff P\ge K,} ]
and
[
\boxed{
\text{wonky possible}
\iff
P\ge W
\ \land
N-P\ge K-W.
}
]
For (N=1023),
[ K=683,\qquad W=341,\qquad K-W=342. ]
Hence bad is possible for (P\le340), wonky for (341\le P\le681), and good for (P\ge683).
Under the literal three-count rule, (P=682) is a one-vote edge case for which no 683-judgment subset has an allowed positive count. This is a specification question, not a claimed protocol flaw.
[extension] Let
- (A) validators be adversarial;
- (B_x) otherwise-honest validators share fault (x) and judge the invalid report as valid;
- (C_x=N-A-B_x) independently judge it as invalid.
If the adversary votes positive, the maximum positive population is (A+B_x). A bad verdict remains constructible regardless of adversarial voting iff
[ \boxed{C_x\ge K.} ]
Writing
[ \gamma=A/N, \qquad f_x=B_x/(N-A), ]
the maximum wrong-positive share is
[ p_x=\gamma+(1-\gamma)f_x. ]
For large (N), robust availability of the bad-verdict/offender-attribution path therefore requires
[ \boxed{ \gamma+(1-\gamma)f_x<\frac13. } ]
Equivalently,
[ \boxed{ f_x< \frac{1/3-\gamma}{1-\gamma}. } ]
At (\gamma=0.20), this is approximately
[ f_x<1/6\approx0.167. ]
This boundary binds before the (F=2) branching threshold (f_{\rm crit}=0.375).
The same wrong-positive share (p_x) appears in three distinct conditions:
[ \boxed{ \begin{array}{rcl} p_x<1/3 &:& \text{bad-verdict attribution robust to adversarial voting}, \[2pt] p_x<1/2 &:& \text{fault-specific ELVES escalation supercritical for }F=2, \[2pt] p_x<2/3 &:& \text{false-good verdict not yet eligible}. \end{array}} ]
Thus attribution can disappear while audit escalation is still supercritical.
When the adversarial and buggy-positive bloc is large enough to prevent a bad verdict but not large enough to form a good verdict, a wonky verdict may remain constructible. The report is stopped, but the Gray Paper's ordinary culprit/fault paths do not create offender records from that wonky verdict.
This does not establish that an attacker has zero total cost. A staking subsystem could act on wonky records. The precise claim is only:
[ \boxed{ \text{report rejection} \neq \text{ordinary offender attribution}. } ]
If (o) is a disjoint offline fraction and (f_x) is the buggy share among the remaining non-adversarial online validators, the independently correct online share is
[ (1-\gamma-o)(1-f_x). ]
Robust bad-verdict attribution requires
[ \boxed{ (1-\gamma-o)(1-f_x)\ge K/N, } ]
approaching (>2/3) for large (N).
This is distinct from ELVES no-shows: a validator that never announces is not the same event as an announced auditor failing to judge.
The neutral Experiment 3 remains unchanged.
Its generic object is a declared viability margin (M(S)). A JAM instantiation can now use, for example,
f_{\max}, ]
or a security-strength quantity such as
[ -\ln P_{\rm accept}. ]
A validator can affect at least two distinct JAM-native reproduction numbers:
- effective honest correction, through implementation diversity and therefore (\lambda_f);
- fault/no-show amplification, through reliability and therefore (A).
This already indicates that a faithful JAM contribution measure is multi-component: safety and liveness/scalability need not move in the same direction.
ELVES treats (\alpha) and (\beta) as exogenous adversarial budgets. A later [extension] may ask what happens if ordinary load also causes otherwise-honest validators to become no-shows.
Then (A) becomes state-dependent and the static committee-size relation
[ s_f=\frac{s_0}{1-A} ]
becomes an implicit feedback equation
[ \boxed{ s_f=\frac{s_0}{1-A(s_f)}. } ]
Multiple solutions are possible only if a specified load-to-no-show response is strong enough. This is a grounded candidate for revisiting the generic closed-loop/fold mathematics, but no JAM fold is claimed yet.
A second channel is non-announcement under overload. Because escalation is triggered by announced auditors that fail to judge, failure to announce can shrink effective committee size without opening a compensating tranche.
- Gavin Wood, JAM: Join-Accumulate Machine, Gray Paper draft 0.8.0, 3 June 2026.
- Jeff Burdges et al., Efficient Execution Auditing for Blockchains under Byzantine Assumptions, IACR ePrint 2024/961, 14 June 2024.