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This document proves both directions of the detector-placement iff
(FORMALIZATION.md §6, Conjectures B and A), after an honest repair that the
proof itself forced. The proofs are elementary once the model is right; per
repo convention, we say so — the content is in the definitions and in the
quantifier structure, not in analytic difficulty.
What is proved here:
Lemma 1 (coupling): inside an observation defect, the detector's gain-capture
rate and its fatal-miss rate are the same number.
Theorem 1 (observation defects): the specialization premium collapses to
zero, unconditionally — transductive access does not help.
Theorem 1′ (value formula): the exact within-defect value, with the
removability coefficient σ₀ (residual separability) governing the collapse.
Theorem 2 (capacity defects): the premium collapses in the inductive regime,
and the inductive condition is exactly a minimax gap — transduction closes
it, matching Goldwasser et al. (2020).
Theorem 2′ (capacity value formula, §7): for any detector class, the
premium is the support function of the class's ROC set at an economic price
vector, with removability coefficient the zero-leak capture capacity
$\bar\sigma_0(H)$. This subsumes Theorem 1′ — observation and capacity
defects obey one value formula — and relocates the split entirely to
joint realizability across confusable directions (the transduction gap).
The deficit characterized (§8): the joint-realizability deficit is the
limiting transduction gap in premium units; it decomposes into exactly two
obstructions (cross-leak, non-closure); exact formulas for monotone classes
("global paranoia") and halfspaces (convex non-separability); and Theorem R
bounds it by a direction router's confusion mass at the same two prices —
the router is itself a detector, the question recurses, and the recursion
is the observation/capacity split in its third formulation.
Theorem M (mixed obstructions, §9): the k-direction game, with per-task
randomization, collapses by LP duality to a single Theorem 2′ instance at
a least-favorable mixture of directions; the transduction gap splits
additively into irreducible cross-leak plus the closure deficit — a coin
buys back the gluing failure, never the cross-leak. XOR witnesses both
halves at exactly $A/2$ each.
Corollary 3 (growth form): under multiplicative dynamics, survival and
specialization gain cannot coexist inside the defect.
Theorem A (achievability, §11): a detector outside the defect earns a
worst-case premium bounded below by an explicit $\Pi^* > 0$ under the
advantage condition; the premium identity has three per-unit-priced terms.
A meets B exactly at the $L \to \infty$ boundary, and the two give the
full iff (Corollary A+B).
Theorem E/E′ (end-to-end, §12): the detector learned from stratified
samples of a declared cell cover, with certified rates. The rare-event form
certifies the fatal side at a one-time training bill linear in severity
$L$ (the agnostic route costs $L^2$); amortized, learning preserves the
advantage condition. Default-deny gating makes the certificate fail-safe
against cover incompleteness; the residual risk is mis-declaration.
What the proof forced us to admit: the drafted definition of "removable"
was vacuous, and the drafted Conjecture B was false under it (§2). The
repaired notion is profitable removability.
Still open (§14): the iterated adversary, the deterministic max-min's
complexity, iterated routing, and beyond-mixture uncertainty classes.
1. Model
Task space $(X, \Sigma)$, one task per period. Competence set $C \in \Sigma$;
fatal exposure $E := F \setminus C \in \Sigma$, disjoint from $C$. Per-task
payoffs for the two actions:
$x \in C$
$x \in E$
act
$+g$
$-L$
escalate
$-p$
$-p$
with $g, p > 0$ and $L$ the fatal loss. A detector is a measurable
$d : X \to {a, e}$; the agent's per-period expected payoff under task
distribution $P$ is $G_P(d) = \mathbb{E}_P[\text{payoff}]$. The detector rent
$c_d$ is a constant shift within any fixed detector class and is dropped. A
third action decline (payoff $0$, task not done) changes nothing below
except replacing the forfeit value $-p$ by $0$; we note where.
Throughout, fix $\bar\varepsilon \in (0,, g/(g+p))$ — the fatal-mass budget —
and define the critical loss
Observation defect. The agent perceives tasks only through a measurable
map $\varphi : X \to Z$; its detector must be $\sigma(\varphi)$-measurable.
Write $\mathcal{D}_R$ for this class. (Coarsened observation channel;
covers pruned/quantized perception.)
Capacity defect. The agent perceives $x$ fully but its detector must
come from a restricted class $H \subseteq {a,e}^X$ containing the two
constant maps. (Restricted function class.)
Adversarial families. For probability measures $\nu_0$ (supported in $C$)
and $\nu_1$ (supported in $E$), define the mixture family
All results are stated for uncertainty classes $\mathcal{P}$ containing such a
family; protection claims are always relative to the declared class, never to
nature (FORMALIZATION.md §7's honesty rule).
2. The definition was vacuous — profitable removability
FORMALIZATION.md defined a defect removable if some policy keeps $G$
bounded below uniformly over $\mathcal{P}$. Writing the proof exposed the
flaw: the constant detector $d \equiv e$ (always escalate) lies in every
detector class — constants are measurable for any $\sigma(\varphi)$ and were
assumed in $H$ — and bounds $G \geq -p$ regardless of the defect. Under
the drafted definition every defect is removable and Conjecture B was
vacuously false, independently of the Goldwasser escape already recorded in
LITERATURE.md. Survival was never the question; paid-for survival is.
Definition (specialization premium). For a detector class $\mathcal{D}$
and family $\mathcal{P}$,
the worst-case value above the always-escalate baseline. (With decline
available, the baseline is $0$; all premium statements are unchanged.)
Definition (profitably removable). The defect is profitably removable
on $\mathcal{P}$ if $\Pi(\mathcal{D}_{\text{defect}}) > 0$: some within-defect
policy retains part of the specialization gain while avoiding ruin.
The theorems below say: with a confounded detector the premium is zero (or
$\to 0$), while an out-of-defect detector on the same family earns premium
$(1-\bar\varepsilon)(g+p)$. That is the honest, non-vacuous form of "the
defect must not include the detector."
3. Lemma 1 — the coupling lemma
Lemma 1. Let $d \in \mathcal{D}R$ (observation defect) and let
$\nu_0, \nu_1$ satisfy $\varphi* \nu_0 = \varphi_* \nu_1$ (equal
pushforwards: the coarsened view cannot distinguish the safe ensemble from
the fatal one). Then
Proof. By Doob–Dynkin, $\sigma(\varphi)$-measurability of $d$ gives a
measurable $d' : Z \to {a,e}$ with $d = d' \circ \varphi$. Then
$\nu_i(d = a) = \nu_i(\varphi^{-1}(d'{=}a)) = (\varphi_*\nu_i)(d'{=}a)$, and
the two pushforwards are equal. $\square$
Reading. Inside the defect, the miss rate is the capture rate: one
scalar $q := q_0 = q_1$ prices both. Every unit of specialization gain the
detector lets through carries a proportional unit of fatal exposure — the
two cannot be decoupled by any cleverness downstream of $\varphi$, because
they are literally the same number. Outside the defect they decouple
completely ($q_0 = 1$, $q_1 = 0$ is available). This is the formal content of
"shared difficulty function ⇒ correlated failure" (Eckhardt–Lee;
Littlewood–Rushby), upgraded from a correlation model to an identity.
Quantitative version. Without equal pushforwards, let
$\tau := \mathrm{TV}(\varphi_*\nu_0, \varphi_*\nu_1)$. Then for every
$d \in \mathcal{D}_R$: $;q_0 - q_1 \leq \tau$ (immediate from the definition
of total variation). The retained distinguishability $\tau$ is the only
wedge the restricted detector can drive between capture and miss.
Theorem 1. Suppose $\varphi_*\nu_0 = \varphi_*\nu_1$ and
$\mathcal{P} \supseteq \mathcal{F}(\nu_0, \nu_1)$. Then for every
$L \geq L^*$:
(Collapse.) $\displaystyle \sup_{d \in \mathcal{D}R} \inf{\lambda} G_{P_\lambda}(d) = -p$, attained only by policies with $q = 0$ (total forfeit on the family). Hence $\Pi(\mathcal{D}_R) = 0$.
(Transduction is powerless.) Minimax equality holds:
$\displaystyle \inf_{\lambda} \sup_{d \in \mathcal{D}R} G{P_\lambda}(d) = -p$ as well. Even a detector chosen after seeing the deployment distribution earns zero premium.
(Separation.) The unrestricted detector $d^*= \mathbf{1}[x \in C]\cdot a$ achieves $\inf_\lambda G_{P_\lambda}(d^*) = (1-\bar\varepsilon)g - \bar\varepsilon p > 0$, i.e. premium $(1-\bar\varepsilon)(g+p)$. This is exactly optimal: $\sup_{d \text{ } \Sigma\text{-meas.}} \inf_\lambda G_{P_\lambda}(d) = (1-\bar\varepsilon)g - \bar\varepsilon p$.
Proof. Let $d \in \mathcal{D}_R$ with common rate $q$ (Lemma 1). Direct
computation:
(1) The expression is linear in $q$ and decreasing in $\lambda$ for $q>0$, so
$\inf_\lambda$ sits at $\lambda = \bar\varepsilon$, giving
$q\big[(1-\bar\varepsilon)g - \bar\varepsilon L\big] - (1-q)p$. For
$L \geq L^*$ the bracket is $\leq -p$, so the whole expression is maximized
at $q = 0$ with value $-p$; any $q > 0$ does strictly worse (and $\to -\infty$
as $L \to \infty$).
(2) For fixed $\lambda$, $\sup_{d}$ over $\mathcal{D}R$ is the same linear
program in $q \in [0,1]$ (both endpoints are realized by the constant
detectors, which lie in $\mathcal{D}R$):
$\sup_d G{P\lambda} = \max\big(-p,; (1-\lambda)g - \lambda L\big)$. This is
decreasing in $\lambda$; the adversary plays $\lambda = \bar\varepsilon$ and
for $L \geq L^*$ the max is $-p$.
(3) For arbitrary $\Sigma$-measurable $d$ let $q_0 := \nu_0(d{=}a)$,
$q_1 := \nu_1(d{=}a)$ — now free to differ. $G_{P_\lambda}(d) = (1-\lambda)[q_0 g - (1-q_0)p] + \lambda[-q_1 L - (1-q_1)p]$
is maximized by $q_1 = 0$, $q_0 = 1$, i.e. by $d^*$, giving
$(1-\lambda)g - \lambda p$, whose infimum over the family is at
$\lambda = \bar\varepsilon$. $\square$
Remarks.
Point (2) is stronger than the repaired Conjecture B asked for: for
observation defects the inductive-regime condition is unnecessary.
Transduction tells you that fatal mass is present ($\lambda$), never
which task is fatal — the information was destroyed per-task by
$\varphi$, before any distributional knowledge can act. Knowing $\lambda$,
the optimal within-defect play is precisely a Chow rule on the coarsened
posterior — and at the confounded points that posterior equals $\lambda$
itself, forcing all-or-nothing behavior: acting everywhere or escalating
everywhere. Chow's rule executes the collapse; it cannot prevent it.
The gap between inside and outside is $(1-\bar\varepsilon)(g+p)$ — the
entire specialization premium, independent of $L$ beyond the threshold
$L^*$.
5. Theorem 1′ — exact value and the removability coefficient
Drop the equal-pushforward assumption. Write $m_i := \varphi_* \nu_i$ and
define the residual separability
$$\sigma_0 ;:=; \sup{, m_0(S) ;:; S \subseteq Z \text{ measurable},; m_1(S) = 0 ,}$$
— the share of the safe ensemble recoverable at zero coarsened fatal risk.
Theorem 1′. On the family $\mathcal{F}(\nu_0,\nu_1)$, within-defect
detectors correspond exactly to measurable act-regions $S \subseteq Z$, and
Proof. The correspondence $d \leftrightarrow S = {d' = a}$ is Lemma 1's
factorization; the displayed value is the direct computation of
$\inf_\lambda G$ (infimum again at $\lambda = \bar\varepsilon$ for any $S$
with $m_1(S) \geq 0$, since the $\lambda$-coefficient is negative whenever
$S$ has any mass). For $L \to \infty$, any $S$ with $m_1(S) > 0$ has value
$\to -\infty$, so the supremum concentrates on ${S : m_1(S) = 0}$, over
which the value is increasing in $m_0(S)$ with supremum
$(1-\bar\varepsilon)[\sigma_0 g - (1-\sigma_0)p] - \bar\varepsilon p = -p + \sigma_0(1-\bar\varepsilon)(g+p)$. $\square$
Reading.$\sigma_0$ is the fraction of the specialization premium that
survives the coarsening: $\sigma_0 = 1$ recovers the full premium,
$\sigma_0 = 0$ is Theorem 1's collapse. For observation defects on mixture
families this settles the "iff" direction of the slogan: the defect is
profitably removable in the large-$L$ limit iff the restriction retains a
region where safe mass lives at zero fatal mass — iff the detector-relevant
distinction survives outside the defect. Removability is not a yes/no after
all (QUESTIONS.md #6): it is the coefficient $\sigma_0$.
6. Theorem 2 — capacity defects: collapse in the inductive regime, and the minimax gap
Now the detector sees $x$ fully but is confined to
$H \subseteq {a,e}^X$ (constants included). Fix distinct pairs
$(x_0^i, x_1^i){i=1}^k$ with $x_0^i \in C$, $x_1^i \in E$, and let
$\mathcal{P} = \bigcup_i \mathcal{F}(\delta{x_0^i}, \delta_{x_1^i})$.
Confounding condition: every $h \in H$ pools some pair:
$\forall h, \exists i : h(x_0^i) = h(x_1^i)$.
Resolution condition: every pair is resolved by some member:
$\forall i, \exists h_i \in H : h_i(x_0^i) = a,; h_i(x_1^i) = e$.
Theorem 2. Under the confounding condition, for $L \geq L^*$:
(Inductive collapse.)$\displaystyle \sup_{h \in H} \inf_{P \in \mathcal{P}} G_P(h) = -p$; hence $\Pi(H) = 0$ in the inductive regime.
(Transductive escape.) Under the resolution condition additionally,
$\displaystyle \inf_{P \in \mathcal{P}} \sup_{h \in H} G_P(h) \geq (1-\bar\varepsilon)g - \bar\varepsilon p$, so the minimax gap is at least $(1-\bar\varepsilon)(g+p)$ — the entire premium.
Proof. (1) Fix $h$; pick a pooled pair $i = i(h)$. On
$P^i_{\bar\varepsilon}$: if $h$ acts on both members,
$G = (1-\bar\varepsilon)g - \bar\varepsilon L \leq -p$ for $L \geq L^*$; if
it escalates both, $G = -p$. Either way
$\inf_P G_P(h) \leq -p$, and the constant $e$ attains $-p$. (2) For any
$P^i_\lambda$, play the resolving $h_i$:
$G = (1-\lambda)g - \lambda p \geq (1-\bar\varepsilon)g - \bar\varepsilon p$. $\square$
Remarks.
The inductive condition of the repaired Conjecture B is now identified
formally: it is the order of quantifiers. Inductive = detector committed
before the deployment distribution ($\sup_h \inf_P$); transductive =
detector chosen with knowledge of it ($\inf_P \sup_h$). For capacity
defects the gap between the two is the whole premium — this is the toy form
of exactly the mechanism by which Goldwasser–Kalai–Kalai–Montasser (2020)
escape: unlabeled deployment data moves the system from the first value to
(toward) the second. For observation defects, Theorem 1(2) says the gap is
zero. Backlog #6's "one concept or three" receives its first hard
evidence: observation and capacity defects are provably different along the
transduction axis.
Realizability of the conditions is generic, not exotic. Example: $H$ =
threshold detectors on a one-dimensional confidence score $s$, with
safe/fatal points interleaved in $s$ (scores $1, 2, 3, 4$ for
$x_0^1, x_1^1, x_0^2, x_1^2$): every threshold pools one of the two pairs,
yet each pair is resolved by some threshold. Interleaved confidence is the
norm for restricted scores — this is Ulmer–Cinà's ReLU-confidence failure
as a two-pair combinatorial fact.
7. Theorem 2′ — the capacity value formula, and the unification
Theorem 2 gave the capacity-defect collapse as an inequality under a
combinatorial condition. Here is the exact value, for an arbitrary class — and
it turns out to be the same formula as Theorem 1′, which relocates the
observation/capacity split.
Setup. Detector class $H \subseteq {a,e}^X$ (constants included),
ensemble pair $\nu_0$ (supported in $C$), $\nu_1$ (supported in $E$), family
$\mathcal{F}(\nu_0,\nu_1)$. Each $h \in H$ has an operating point$\big(q_0(h), q_1(h)\big) := \big(\nu_0(h{=}a),, \nu_1(h{=}a)\big)$ —
capture rate and leak rate; note $q_1$is the miss rate $\mu$ on the fatal
ensemble and $1 - q_0$ the false-alarm rate on the safe one. Call
$\mathrm{ROC}(H) := {(q_0(h), q_1(h)) : h \in H} \subseteq [0,1]^2$ the
class's ROC set against the pair. Write the price vector
$$A := (1-\bar\varepsilon)(g+p) \quad (\text{per unit captured}), \qquad B := \bar\varepsilon,(L-p) \quad (\text{per unit leaked}).$$
Lemma 2 (operating-point reduction). For every $h \in H$, the worst-case
premium over the family depends on $h$ only through its operating point:
$$\Pi(h) ;=; A,q_0(h) ;-; B,q_1(h).$$
Proof.$G_{P_\lambda}(h) = (1-\lambda)[q_0 g - (1-q_0)p] + \lambda[-q_1 L -
(1-q_1)p]$, which is nonincreasing in $\lambda$ (strictly, unless
$q_0 = q_1 = 0$), so the infimum sits at $\lambda = \bar\varepsilon$;
subtracting the baseline $-p$ and collecting terms gives the display. $\square$
Theorem 2′ (capacity value formula). For any $H$ containing the
constants,
— the support function of the class's ROC set at the price vector
$(A, -B)$. Consequently:
The supremum is approached on the Pareto frontier of $\mathrm{ROC}(H)$ —
the class-restricted ROC curve — and the within-class optimal detector is
the class-restricted Neyman–Pearson point at slope $B/A$: Chow's rule
again, executed inside the class.
As $L \to \infty$, $;\Pi(H) \to A \cdot \bar\sigma_0(H)$, where
$$\bar\sigma_0(H) ;:=; \lim_{t \downarrow 0}; \sup{, q_0(h) : h \in H,; q_1(h) \leq t ,}$$
is the zero-leak capture capacity of the class (the limit exists by
monotonicity in $t$).
Proof. The formula is Lemma 2 plus taking the supremum; $(0,0) \in
\mathrm{ROC}(H)$ via the constant $e$, so $\Pi(H) \geq 0$ automatically. For
(2): upper bound — for any $h$, $\Pi(h) \leq \min\big(A,q_0(h),; A -
B,q_1(h)\big)$, so any $h$ competing with the limit must have $q_1(h) \leq
A/B \to 0$ and hence $q_0(h) \leq \sup{q_0 : q_1 \leq A/B} \to
\bar\sigma_0(H)$. Lower bound — for any $t$, pick $h$ with $q_1(h) \leq t$,
$q_0(h)$ within $\eta$ of the sup: $\Pi(h) \geq A(q_0) - Bt$, and let
$t \downarrow 0$ after $L \to \infty$ along the definition of the limit.
$\square$
Remark (closure).$\bar\sigma_0(H) \geq \sigma_0(H) := \sup{q_0 : q_1 =
0}$, with equality when the ROC set is closed — finite classes, compact
operating sets — and possibly strict otherwise (a class whose every
non-constant member leaks a little, with no zero-leak limit point, has
$\sigma_0 = 0 < \bar\sigma_0$). For σ-algebra classes the two coincide: from
$S_n$ with $m_1(S_n) \to 0$ summably and $m_0(S_n) \geq c$, the set $S =
\limsup S_n$ has $m_1(S) = 0$ and $m_0(S) \geq c$ by reverse Fatou. So
Theorem 1′'s $\sigma_0$ needed no correction.
Corollary (unification). Theorem 1′ is the special case $H = $ the
$\sigma(\varphi)$-measurable detectors: by Lemma 1's factorization,
$\mathrm{ROC}(H_\varphi) = {(m_0(S), m_1(S)) : S \subseteq Z}$, the support
function reproduces Theorem 1′'s value, and $\bar\sigma_0 = \sigma_0$ by the
closure remark. Observation and capacity defects obey one value formula:
premium $=$ support function of the detector class's ROC set at the economic
price vector; removability coefficient $=$ zero-leak capture capacity. What
distinguishes the two defect types is only how the ROC set arises — the image
of a coarsened σ-algebra versus the trace of a function class.
Where the split actually lives. On a single mixture family the
transduction gap vanishes for every class: each $h$'s infimum over
$\lambda$ sits at $\bar\varepsilon$, so
$\inf_\lambda \sup_h G \leq \sup_h G_{P_{\bar\varepsilon}} = \sup_h
\inf_\lambda G$, and the minimax inequality closes the other side. The gap of
Theorem 2 is therefore a multi-direction phenomenon: with a union of
confusable families the inductive value is the joint max-min
$\sup_h \min_j [A,q_0^j(h) - B,q_1^j(h)]$, the transductive value is the
per-direction $\min_j \sup_h$, and the gap measures joint-realizability
deficit — whether one member of the class can serve all confusable
directions at once. Theorem 2's confounding/resolution conditions are the
extreme case (each direction resolvable, no member resolves all). This
sharpens the taxonomy result: the observation/capacity split is not about
how a detector is priced — the formula above is common — but about whether
deployment knowledge lets the class pick its member per direction. The
limiting coefficient for a union family is accordingly
$\bar\sigma_0(H; {\nu^j}) = \lim_{t \downarrow 0} \sup{\min_j q_0^j(h) :
h \in H,; \max_j q_1^j(h) \leq t}$, which recovers Theorem 2(1)'s collapse
as $\bar\sigma_0 = 0$ under confounding.
8. The joint-realizability deficit, characterized
Theorem 2′ located the transduction gap in joint realizability. This section
makes that quantitative: what the deficit is, which two obstructions
produce it, exact formulas for the natural classes, and a general bound that
turns out to be the theory applied to itself.
Setup. Directions $j = 1, \dots, k$: ensemble pairs $(\nu_0^j, \nu_1^j)$,
union family $\mathcal{P} = \bigcup_j \mathcal{F}(\nu_0^j, \nu_1^j)$. Write
$\bar\sigma^j(H)$ for the per-direction zero-leak capture capacity and
$\bar\sigma^{\mathrm{jt}}(H) := \lim_{t \downarrow 0} \sup{\min_j q_0^j(h) :
h \in H,\ \max_j q_1^j(h) \leq t}$ for the joint one (§7). Define the
joint-realizability deficit
(nonnegative since any jointly feasible $h$ is feasible per direction). By
Theorem 2′'s limits applied per-direction and jointly, the transduction gap
satisfies $\Gamma_L := \Pi^{\mathrm{tr}}_L - \Pi^{\mathrm{ind}}_L \to
A \cdot \Delta(H)$ as $L \to \infty$: the deficit is the limiting gap in
premium units.
Proposition 4 (the two obstructions). If for each $j$ and every $t > 0$
there is $h_j \in H$ with $q_0^j(h_j) \geq \bar\sigma^j - \eta$ and
$q_1^i(h_j) \leq t$ for all$i$ (cross-safe local solutions), and $H$
contains a member whose act region contains the union of the $h_j$'s act
regions (union closure over them), then $\Delta(H) = 0$.
Proof. The union member captures $q_0^j \geq \bar\sigma^j - \eta$ on each
$j$ and leaks at most $kt$ on each $i$; let $t, \eta \downarrow 0$. $\square$
So every positive deficit is attributable to cross-leak (direction $i$'s
solution acts on direction $j$'s fatal mass) or non-closure (the class
cannot glue its local solutions), and the two are separated by the examples
below: thresholds fail only by cross-leak, halfspaces only by non-closure.
Proposition 5 (monotone classes — exact formula, "global paranoia").
Let $H = { \text{act iff } s(x) \geq \theta }$ for a score $s$ (act regions
nested, hence union-closed). Let $T_j := \inf{\theta : \nu_1^j(s \geq
\theta) = 0}$ be direction $j$'s fatal ceiling and $T := \max_j T_j$ the
global one. Then
(with $>$ relaxed to $\geq$ at ceilings carrying no fatal atom). The deficit
is the safe mass trapped between each direction's own fatal ceiling and the
global one: a single threshold must be paranoid enough for the worst
direction and pays that paranoia in every direction. Point-pair corollary:$\Delta \in {0,1}$, and $\Delta = 1$ iff the pairs are correctly ordered
but interleaved — every $s(x_0^j) > s(x_1^j)$, yet some safe point scores
below another direction's fatal point. (Theorem 2's threshold example, now
exact; Ulmer–Cinà's interleaved-confidence failure is this formula.)
Proof. Nested act regions make every supremum one-dimensional in $\theta$;
zero leak on $j$ forces $\theta > T_j$ (all $i$: $\theta > T$), and
${s \geq \theta} \downarrow {s > T}$ as $\theta \downarrow T$ gives the
right-limits. $\square$
Proposition 6 (halfspaces — the deficit is non-separability). Let $H$ =
halfspace act regions in $\mathbb{R}^n$, directions given by point pairs.
Each pair is always individually resolvable (two points), so
$\min_j \bar\sigma^j = 1$; and $\bar\sigma^{\mathrm{jt}} = 1$ iff the safe
points ${x_0^j}$ are strictly linearly separable from the fatal points
${x_1^j}$, else $0$. Hence $\Delta \in {0,1}$ with
The minimal witness is the XOR configuration ($k = 2$, $n \geq 2$).
Halfspaces are cross-safe here (a pair's separator can be chosen missing the
other points) but not union-closed — the complementary failure mode to
Proposition 5. $\square$
Remark (no single capacity number). If $H$ shatters the $2k$ direction
points, $\Delta = 0$ trivially — so a positive deficit certifies that the
directions witness a non-shattered set. But the converse fails badly:
thresholds (VC 1), intervals (VC 2), and halfspaces (VC $n{+}1$) all admit
$\Delta = 1$ at $k = 2$ under adversarial placement. The deficit is a
function of the geometry of the directions relative to the class, not of
class capacity alone; shattering gives sufficiency, never a characterization.
Theorem R (router bound — the theory applied to itself). Let
$r : X \to [k]$ be any direction router with confusion mass
$\rho := \max_j \max\big(\nu_0^j(r \neq j),\ \nu_1^j(r \neq j)\big)$, and let
the gated composite $h_r(x) := h_{r(x)}(x)$ be admissible (i.e. play in the
augmented class $R \ltimes H^k$). Then at every$L$:
$$\Gamma_L(R \ltimes H^k) ;\leq; \rho \cdot (A + B).$$
Proof. Pick per-direction detectors $h_j$ within $\eta$ of the transductive
optimum at the current $L$. On direction $j$ the composite's capture drops by
at most $\nu_0^j(r \neq j) \leq \rho$ and its leak rises by at most
$\nu_1^j(r \neq j) \leq \rho$, so its premium is within
$\rho(A + B) + \eta$ of $\Pi^{\mathrm{tr}}_L$. $\square$
Three consequences:
The deficit is priced by the same two prices — router confusion on
safe mass costs $A$ per unit, on fatal mass $B$ per unit. Since
$B = \bar\varepsilon(L - p)$ grows with $L$, a useful bound at large $L$
needs router confusion $O(1/L)$ on the fatal ensembles — exactly
Theorem A's law for the primary detector, at the same
$O(\log L)$ hypothesis-testing rent. The router is a detector — of
directions instead of fatality — and obeys detector economics.
The escape-route question recurses. "Can the class serve all
directions at once?" reduces to "can the direction be detected per task?"
— a second-order instance of the original problem. The recursion
terminates in one of two ways, and this is the observation/capacity split
in its third and sharpest formulation: for a capacity defect the
recursion bottoms out in a cheap router (directions distinguishable from
the task; deficit $\leq \rho(A{+}B)$, bought at log-rent); for an
observation defect the router would need exactly the information the
coarsening destroyed — its confusion is bounded below by the coupling
lemma — and the recursion halts at level one with nothing to buy.
Fleet corollary (QUESTIONS.md #5): a menu of $k$ specialists plus a
router is one generalist detector, and $\rho(A + B)$ is precisely what the
fleet loses to the ideal; the composition question inherits the whole
pricing apparatus.
The bound is stated for the augmented class: it characterizes what
composition closure buys, while Propositions 5–6 give the exact deficit of
the raw classes. The mixed case — cross-leak and non-closure together, on
ensemble directions — is §9.
9. Theorem M — the mixed-obstruction deficit: duality and the price of determinism
Where §8 characterized the deficit with one obstruction acting alone, this
section gives the general case. The conjectured shape (a Neyman–Pearson
problem over the joint ROC set) is correct, and its dual is the theorem:
the k-direction game collapses to a single-direction Theorem 2′ instance at a
least-favorable mixture of directions — and the duality separates the two
obstructions exactly: randomization eliminates non-closure and can never
touch cross-leak.
Lemma 3 (vector operating-point reduction). Each $h$ has a joint
operating point $Q(h) := (q_0^j(h), q_1^j(h))_{j \leq k} \in [0,1]^{2k}$;
write $\mathrm{ROC}_k(H) := {Q(h) : h \in H}$. On the union family, the
deterministic inductive premium is
— the max-min of the price functional over the joint ROC set, whose Pareto
frontier does not factorize per direction. Proof: per direction, Lemma 2's
computation (worst $\lambda$ at $\bar\varepsilon$); the adversary's choice of
direction is the min. $\square$
Randomized play. Let the agent draw its detector per task from a finite
lottery $\pi$ over $H$. The operating point of a lottery is the mixture of
its atoms' points, so lotteries realize exactly
$\mathrm{conv},\mathrm{ROC}_k(H)$ (Carathéodory: at most $2k+1$ atoms
suffice — the optimal randomized detector is a small menu with a coin).
Write $\Pi^{\mathrm{rand}}_L(H)$ for the inductive premium over lotteries.
Theorem M (least-favorable mixture duality). For any $H$ (constants
included), any $k$ ensemble directions, and every $L$:
where $\Pi^{2'}_L(H; \nu_0, \nu_1) = \sup_h [A,\nu_0(h{=}a) -
B,\nu_1(h{=}a)]$ is Theorem 2′'s single-pair value. The minimizer $\mu^*$
is the least-favorable direction mixture; the optimal lottery is the
class-restricted Neyman–Pearson solution against the mixed ensembles
$(\bar\nu_0^{\mu^*}, \bar\nu_1^{\mu^*})$; and $\mu^*$ is supported on the
binding directions (complementary slackness).
Proof. $\Pi^{\mathrm{rand}}L = \sup{q \in \mathrm{conv},\mathrm{ROC}k}
\min_j \langle c_j, q\rangle$ with $c_j$ the $j$-th price functional. The
objective is concave (min of linear) and unchanged by passing to the closed
convex hull; the adversary's set is finite; von Neumann/LP duality gives
$\sup_q \min_j = \min{\mu \in \Delta_k} \sup_q \langle \sum_j \mu_j c_j,
q\rangle$, with the min attained on the compact simplex. Finally
$\sum_j \mu_j [A q_0^j(h) - B q_1^j(h)] = A,\bar\nu_0^\mu(h{=}a) -
B,\bar\nu_1^\mu(h{=}a)$ because operating points are linear in the measure —
so the inner sup is a Theorem 2′ instance. $\square$
Corollary M1 (the obstructions, exactly separated). For every $L$,
The first bracket is the cross-leak against the least-favorable mixture — it
survives any amount of randomization, because Theorem M already plays the
convex hull. The second is the convexity gap of $\mathrm{ROC}_k(H)$ at the
optimal price functional — the price of determinism, and it is all of
Prop 4's non-closure obstruction: a coin buys back the gluing failure,
never the cross-leak. Union-closed classes (Prop 5) have zero closure
deficit already; for them randomization is worthless, as the exact formula
showed.
Worked example (XOR, the numbers). Halfspaces on the XOR directions
(§8, Prop 6), $L \to \infty$, in units of $A$: deterministic $0$, randomized
$1/2$ (the coin over the two corner halfspaces ${x{+}y < \tfrac12}$,
${x{+}y > \tfrac32}$; dual check: at $\mu^* = (\tfrac12, \tfrac12)$ no
halfspace excludes both fatal points while capturing more than half the
mixed safe mass), transductive $1$. Both obstructions are worth exactly
$1/2$ here: the three values separate, and the decomposition of M1 is
witnessed with all brackets positive.
Corollary M2 ($L \to \infty$). $\Pi^{\mathrm{rand}}L / A$ decreases to
the randomized joint coefficient, which under a closed joint ROC set equals
$\min{\mu} \bar\sigma_0\big(H;\ \bar\nu_0^\mu, \bar\nu_1^\mu\big)$ — the
zero-leak capture capacity at the least-favorable mixture. Without
closedness the limit sits between the $t{=}0$ and right-limit versions,
the same caveat as §7's closure remark; we do not pretend otherwise.
Remarks.
The lineage is now verified (LITERATURE.md, second round). The duality
technology is fully classical: Wald (1945; 1950) for minimax-equals-Bayes
against a least favorable prior with randomized rules; Huber–Strassen
(1973) for the exact collapse move — composite minimax testing reduces
to one NP test against a least favorable pair; Fillatre (2017) for the
finite-LP/dual-LP computation. Even the coin mechanism is in print:
randomized rules realize the ROC convex hull (NP lemma, Lehmann–Romano
Thm 3.2.1; Provost–Fawcett 2001; Fauß–Zoubir–Poor §V-D on point-mass LRs
forcing randomization). We cite all of it and claim only the application —
and must distinguish Managoli–Sahasranand–Prabhakaran (arXiv:2501.12938),
who already pair robust testing with abstention (asymptotic exponents,
unrestricted detectors): M's claim sits at the restricted-class premium
game, the closure-deficit identification, and the obstruction split.
Theorem R is now the constructive counterpart of M: blind randomization
(route without looking) is the $\rho = 1 - 1/k$ corner of the router bound,
and M says the best blind play is not gating but the NP solution against
$\mu^*$; a router with per-task information interpolates between M's value
and the transductive one.
What M does not deliver: the deterministic value itself. The max-min
over a non-convex $\mathrm{ROC}_k(H)$ (Lemma 3) is exact but not, in
general, simplifiable — computing it is plausibly hard (the nearby
classifier-rejector hardness of Mozannar et al. 2023 suggests so, but we
have not proved it), and the closure deficit is exactly what it costs. That
is now the honest residual.
10. Corollary 3 — the growth form
Corollary 3. Under multiplicative dynamics (wealth factor $1+r$ per task:
$r = \tilde g$ on safe acts, $r = -\tilde L$ on missed fatals, $r = -\tilde p$
on escalations, i.i.d. tasks), with a confounded detector class: every
within-defect policy with positive specialization premium has, on some
$P \in \mathcal{P}$, expected log-growth $\to -\infty$ as $\tilde L \to 1$;
every within-defect policy with bounded log-growth has premium $0$.
Proof. Positive premium requires positive act-rate $q$ on the confounded
family (Theorems 1, 2), and $\mathbb{E}{P{\bar\varepsilon}}[\log(1+r)]$
contains the term $\bar\varepsilon, q \log(1 - \tilde L) \to -\infty$.
Policies avoiding this have $q = 0$, hence premium $0$ by the same theorems. $\square$
Survival and specialization gain cannot coexist inside the defect. This
is Proposition 3 (FORMALIZATION.md §5) with its missing half: there, ruin
forces $\mu = 0$; here, inside the defect, $\mu = 0$ forces $q_0 = 0$
(Lemma 1) — no gain. The two halves together are the theorem the founding
statement needed.
11. Theorem A — the achievability half (the converse's partner)
The converse (§4–§7) says a confounded detector earns zero premium. The
partner statement: a detector placed outside the defect earns a premium
bounded below by an explicit constant, so removal is not merely possible but
profitable — closing the "if" direction of the iff. Where §4–§7 fixed the
detector class and varied $P$ adversarially, here we fix a good detector and
lower-bound its worst-case premium over the class.
Hypotheses.
Structured class with a competence floor.$\mathcal{P} \subseteq {P : P(E) \leq \bar\varepsilon,; P(C) \geq \underline{c}}$
with $\underline{c} + \bar\varepsilon \leq 1$. The floor $\underline c > 0$
is necessary: if the world may contain no in-competence mass, no policy can
guarantee positive premium (there is nothing to specialize on).
Out-of-defect detector. There is a detector $d$ achieving, uniformly
over $\mathcal{P}$, miss rate $\sup_P P(d{=}a \mid E) \leq \delta$ and
false-alarm rate $\sup_P P(d{=}e \mid C) \leq \alpha_0$, at rent $c_d$, and
escalating the benign region ($P(d{=}a \mid N) = 0$ — the conservative
choice; acting on benign only helps, by $(p-\ell) \geq 0$ per unit). "Out of
the defect" is exactly what makes both bounds simultaneously attainable:
Lemma 1 forbids it inside an observation defect (there capture rate $=$
miss rate), and the repaired Conjecture C (FORMALIZATION.md §7) constructs
such a $d$ for structured $\mathcal{P}$ (finite-VC families,
$f$-divergence balls). A inherits C's structural restriction — no more, no
less.
The premium identity. Against the always-escalate baseline $B_0 = -p$,
for any detector and any $P$ (writing $\alpha_P, \mu_P$ for the realized
rates, benign escalated):
Derivation. Add $p = p,[P(C)+P(E)+P(N)]$ to $G_P(d)$ region by region. On
$C$: $P(C)[(1-\alpha_P)g - \alpha_P p] + pP(C) = P(C)(1-\alpha_P)(g+p)$. On
$E$: $-P(E)p - P(E)\mu_P(L-p) + pP(E) = -P(E)\mu_P(L-p)$ (baseline exactly
cancels the escalate cost on $E$). On $N$ with benign escalated: $-pP(N) +
pP(N) = 0$. Subtract rent $c_d$. $\square$
Three terms, each a per-unit price: captured competence earns $(g+p)$ per
unit (gain $g$ where the baseline paid $p$); leaked fatal costs $(L-p)$
per unit (loss $L$ where the baseline paid $p$); rent is flat.
Proof. The identity gives $\Pi_P(d) = P(C)(1-\alpha_P)(g+p) - P(E)\mu_P(L-p)
c_d$, which is nondecreasing in $P(C)$ and nonincreasing in each of
$\alpha_P, \mu_P, P(E)$. Apply the uniform caps $\alpha_P \leq \alpha_0$,
$\mu_P \leq \delta$ and the class bounds $P(C) \geq \underline c$, $P(E) \leq
\bar\varepsilon$ — all valid for every $P \in \mathcal{P}$ — to get
$\Pi_P(d) \geq \Pi^*$ pointwise, hence for the infimum. $\square$
This is Proposition 1 / the advantage condition (FORMALIZATION.md §3),
re-derived as the achievability threshold with the detector's real
operating characteristics substituted for the abstract "price of
compensation": the escalation cost enters as the $(1-\alpha_0)$ haircut on
captured competence, and the missed-fatal cost as $\bar\varepsilon\delta(L-p)$.
Remarks — where A meets B
1. The $L$-dependence, and the exact meeting with the converse.$\Pi^*$
carries $L$ only through the leakage $\bar\varepsilon\delta(L-p)$. With
$\delta > 0$fixed, $\Pi^* \to -\infty$ as $L \to \infty$: a detector of
any fixed positive miss rate is eventually ruined by a severe enough fatal
event. This is not a weakness of the proof — it is A shaking hands with
Conjecture B, which drove $G \to -\infty$ precisely when $\mu$ stayed bounded
away from $0$. The two theorems partition the $(\delta, L)$ plane along the
same curve. Profitable removal survives arbitrary $L$ only if the miss rate
shrinks with severity,
2. Two independent routes to a bound uniform in $L$. Either
(i) sharpen the detector, $\delta = O(1/L)$, so leakage stays $\leq
\bar\varepsilon\kappa$ for a constant $\kappa$; or (ii) cap the blast
radius — a spend ceiling / bounded compensation making realized loss
$\leq \bar L$ regardless of the true $L$ (FORMALIZATION.md §2, point 2), after
which any fixed $\delta$ satisfying the advantage condition at $\bar L$ works.
The architecture uses both; either alone suffices for a constant lower bound.
3. Route (i) is cheap — the tie to detector economics. By the
hypothesis-testing bound (Conjecture D, FORMALIZATION.md §8), miss rate
$\delta$ costs rent $c_d \sim \log(1/\delta)$, so $\delta = O(1/L)$ costs only
$c_d \sim \log L$. Leakage is held constant at the price of logarithmic
rent — the detector's cost grows exponentially slower than the severity it
insures against. Profitability then holds up to a finite but exponentially
large ceiling $L \leq \exp\big(O(\text{margin})\big)$, where the margin is
the specialization premium net of constant leakage. Cheap detection is what
makes the whole architecture affordable, quantified.
4. Graceful degradation recovers the converse's witness. At the ideal
operating point $\alpha_0 = \delta = 0$ (perfect detector) with the fullest
class $\underline c = 1-\bar\varepsilon$, $\Pi^* = (1-\bar\varepsilon)(g+p)$
— exactly the separation witness of Theorem 1(3) and the outside-defect
value against which the converse measured the collapse. A's bound degrades
continuously from that ideal as $(\alpha_0, \delta, c_d)$ rise, so Theorems 1′
and A are two ends of one continuum: $\sigma_0$ measures how much of the
premium a confined detector keeps; $(\alpha_0, \delta)$ measure how much a
free but imperfect detector keeps.
Corollary A+B — the full iff
Combining Theorem A with the converse (§4–§7), over structured classes and
with either $L$ capped or $\delta = O(1/L)$:
A defect is profitably removable iff a detector achieving small
miss and false-alarm rates exists outside it — equivalently, iff the
detector-relevant distinction survives the restriction (zero-leak capture
capacity $\bar\sigma_0 > 0$, one coefficient for both defect types by
Theorem 2′) and the advantage condition holds.
Necessity is B (inside the defect the distinction is gone, premium $= 0$);
sufficiency is A (outside it, the explicit $\Pi^* > 0$). Placement supplies
the possibility; the advantage condition supplies the profit. This is the
theorem the founding statement asked for, in both directions.
12. Theorem E — end-to-end achievability: the detector learned, not assumed
Theorem A takes the uniform rate caps $(\alpha_0, \delta)$ as given. Here the
detector is selected from data, and the caps are replaced by certificates.
The concentration is standard and we use it as such (textbook constants, not
optimized); the content is where the union bound lands and what the
certificate costs as severity grows.
Setup (declared cover; the repaired-C structure made concrete). The
agent declares a finite cover: safe-cell conditionals
$\nu_0^1, \dots, \nu_0^r$ (supported in $C$) and fatal-cell conditionals
$\nu_1^1, \dots, \nu_1^m$ (supported in $E$) — fixed measures, the
Mondrian/group-conditional structure of repaired Conjecture C in its simplest
form. The uncertainty class $\mathcal{P}$ reweights cells arbitrarily subject
to the budgets $P(E) \leq \bar\varepsilon$, $P(C) \geq \underline c$; within
cells, conditionals are fixed. Worst-case rates then reduce to cell maxima:
$\mu(d) := \max_i \nu_1^i(d{=}a)$, $\alpha(d) := \max_l \nu_0^l(d{=}e)$, and
Theorem A's bound holds with these: $\inf_P \Pi_P(d) \geq \Pi^{\mathrm{lb}}(d)
:= \underline c,(1-\alpha(d))(g+p) - \bar\varepsilon,\mu(d)(L-p) - c_d$.
The detector class $D$ (VC dimension $v$, outside the defect — that is what
$D$'s freedom means) is gated by the cover: every $d \in D$ escalates
off the covered cells. Data: $n$ i.i.d. samples from each cell conditional
(stratified — test the candidate detector against known-fatal history per
category).
Lemma 4 (uniform cell concentration; standard). With probability
$\geq 1-\theta$, simultaneously for all $d \in D$ and all $m + r$ cells, the
empirical rates satisfy $|\hat\mu_i(d) - \mu_i(d)| \leq \epsilon_n$ and
$|\hat\alpha_l(d) - \alpha_l(d)| \leq \epsilon_n$ with
$$\epsilon_n = O!\Big(\sqrt{\tfrac{v \log n + \log((m+r)/\theta)}{n}}\Big).$$
Proof: two-sided VC uniform convergence per cell (Vapnik–Chervonenkis
1971), union bound over the $m+r$ cells — the promised "union bound over
$\mathcal{P}$'s structure," costing only $\log(m+r)$. $\square$
Theorem E (agnostic). Let $\hat d$ maximize the empirical bound
$\hat\Pi^{\mathrm{lb}}$. On Lemma 4's event,
Proof.$\Pi^{\mathrm{lb}}$ is $\kappa$-Lipschitz in the sup-norm of the
rate vector (cell-max is 1-Lipschitz), so $|\Pi^{\mathrm{lb}} -
\hat\Pi^{\mathrm{lb}}| \leq \kappa\epsilon_n$ uniformly; sandwich the ERM.
$\square$
The penalty is priced by the same $\kappa \approx B$ for large $L$: under the
square-root rate, holding the estimation penalty constant costs $n \sim L^2$
samples. The agnostic training bill is quadratic in severity — too
expensive. The rare-event form fixes this:
Theorem E′ (rare-event certificate — the linear bill). Suppose the fatal
side is realizable: some $d^* \in D$ has $\mu_i(d^*) = 0$ on every declared
fatal cell, with $\alpha(d^*) \leq \alpha^*$. Select $\hat d$ among
${d : \hat\mu_i(d) = 0\ \forall i}$ (nonempty surely — a true-zero
detector cannot miss a sample) minimizing $\max_l \hat\alpha_l$. With
probability $\geq 1-\theta$:
(one-sided realizable version-space bound — BEHW 1989, Thm 2.1 — union over
fatal cells; two-sided only on the safe side; E′'s problem statement is
KKM positive-reliable learning and its setup is NP classification at
$\alpha = 0$, see §13), hence
Holding the fatal-side penalty below a constant requires
$\epsilon^{01}_n = O(1/L)$, i.e. $n = O\big(L \cdot (v\log + \log(m/\theta))\big)$
per fatal cell — linear in severity, because certifying a near-zero rate
is a one-sided rare-event problem, not a mean estimation. The safe side needs
only $O(1/\sqrt{n})$, independent of $L$. $\square$
Lifecycle economics. Sustaining severity $L$ end-to-end now has two
bills: a one-time training bill of $O(L \cdot v \log)$ labeled fatal
examples per declared category (amortized over deployment), and the
per-period rent$c_d = O(\log L)$ from §11's Remark 3. Amortized, both
grow slower than the per-event loss they insure against, so the advantage
condition survives learning whenever it held with known rates, up to the
explicit slack above. One economic observation rides along: the scarce input
is labeled catastrophes — precisely what a well-run system stops producing
(it escalates them). The detector's training data must come from history,
simulation, or other agents' failures; in market terms, labeled fatal
examples are a tradable good, and Theorem E′ prices demand for them at
$O(L)$ per category.
Coverage: fail-safe against incompleteness, exposed to mis-declaration.
Because $D$ is gated by the cover, a fatal category outside the declared
cells is escalated by default — incompleteness costs foregone premium (an
uncovered safe cell earns nothing), never uncertified fatal exposure. The
certificate's true residual risk is mis-declaration: fatal mass inside a
declared-safe conditional's support, where the gate admits and the samples
mislead. That is model misspecification, not sample complexity, and it is
mitigated structurally (the blast-radius cap on $L$), not statistically —
FORMALIZATION.md §7's honesty rule, surviving to the end of the pipeline.
This also discharges Theorem A's delegation: existence of the out-of-defect
detector is no longer assumed but reduced to realizability of $D$ on the
declared cells, with everything else certified from data.
13. Positioning
vs. Fang et al. (2022): their impossibility is statistical
(non-learnability of OOD detection in restricted spaces); ours is
decision-theoretic (value collapse for an actor–detector pair sharing one
restriction), with the adversarial-distribution move imported as
technology, as planned.
vs. Goldwasser et al. (2020): not a contradiction but a corner of the
theory — their escape is Theorem 2's minimax gap, and Theorem 1(2) marks
its boundary: transduction rescues capacity defects, never observation
defects.
vs. Littlewood–Rushby / common-cause failure: Lemma 1 is the
common-cause principle upgraded from a correlation model to an identity
($q_0 = q_1$), with the quantitative wedge $\tau$ recovering the graded
version.
vs. Chow: within the defect, optimal play is Chow's rule on the
coarsened posterior (Theorem 1′ is a Neyman–Pearson problem) — and that is
precisely why optimal within-defect play cannot help. The failure is not in
the decision rule but in what reaches it.
vs. least-favorable priors (verified): Theorem M's structure is the
classical minimax-testing duality — Wald (1945/1950), Huber (1965),
Huber–Strassen (1973) for the collapse-to-one-NP-test move, Fillatre
(2017) for the finite-LP form; randomization-realizes-the-ROC-hull is
NP-lemma classical (Lehmann–Romano Thm 3.2.1; Provost–Fawcett 2001;
Fauß–Zoubir–Poor §V-D). We claim the application only, and distinguish
Managoli et al. (arXiv:2501.12938) — robust testing with abstention
exists as of 2025, at asymptotic exponents over unrestricted detectors;
the restricted-class premium game and the obstruction split are ours.
Affirmative evidence the classical robust-detection canon skipped the
reject option: the Fauß–Zoubir–Poor survey contains no abstention action.
vs. NP classification and reliable learning (verified): Theorem E is
standard technology (VC 1971; BEHW 1989 Thm 2.1 for the version-space
bound; textbooks Anthony–Bartlett, Shalev-Shwartz–Ben-David). Theorem E′'s
problem is Kalai–Kanade–Mansour's positive-reliable learning and its
setup is Neyman–Pearson classification at $\alpha = 0$ (Cannon et al.
2002; Scott–Nowak 2005; Rigollet–Tong 2011) — cite both frames. Neither
contains the zero-empirical-miss selection with the one-sided fast rate
(KKM's reduction pays agnostic $\varepsilon^2$ rates; all NP papers run
two-sided $\sqrt{}$ at $\alpha > 0$), nor the linear-vs-quadratic-in-$L$
training bill — which we present as an elementary corollary of the tight
realizable/agnostic $1/\varepsilon$ vs $1/\varepsilon^2$ dichotomy (EHKV
lower bound), not a new phenomenon. Must engage: Casacuberta–Kanade
(NeurIPS 2025), group-wise reliable abstention at agnostic rates — owns
the group-wise frame; lacks zero-miss certification, reweighting
robustness, and the severity asymmetry.
The slogan, final form:a defect is profitably removable exactly to
the extent that the detector's distinction survives outside it —
coefficient $\sigma_0$, not a yes/no.
14. Open
Iterated adversary (QUESTIONS.md #3, second half): the adversary here
chooses $P$ once; the repeated game with a learning detector is untouched.
Beyond mixture families: the theorems quantify over
$\mathcal{F}(\nu_0,\nu_1)$ and unions thereof; characterizing collapse for
arbitrary $\mathcal{P}$ (not built from confusable ensemble pairs) is open,
though any $\mathcal{P}$containing such a family inherits the collapse.
The deterministic max-min. Theorem M resolves the mixed-obstruction
deficit for randomized play; the deterministic value (Lemma 3's max-min
over the non-convex joint ROC set) is exact but unsimplified, and its
computational complexity is unproved — the nearby classifier-rejector
hardness (Mozannar et al. 2023) suggests NP-hardness for halfspace-type
classes, but we have not shown it. Equivalently open: bounding the closure
deficit by a structural property of $H$ (something Shapley–Folkman-shaped
as $k$ grows).
The iterated router. Theorem R's recursion is one level deep. A
hierarchy of directions (directions about directions) would iterate it;
whether the total rent telescopes (each level $O(\log L)$) or compounds
is open, and bears directly on QUESTIONS.md #5's fleet economics.