Choose a finite ordered grid \(0<C_1<\cdots <C_m\) and evaluate each architecture at every declared grid point under the same task and protocol. The resulting values \(\widehat V_A(C_i)\) are observations of the restricted frontier, not its values between grid points.

For an observed score \(s_A(C_i)\) define the discrete upper envelope \(U_A(C_i)=\max _{j\leq i}s_A(C_j)\). It is a descriptive monotone summary of the measured points; it is not evidence that unmeasured costs attain the envelope or that extra cost cannot reduce score.

Attach a predeclared uncertainty interval \(I_A(C_i)\) to each score and carry those intervals through score differences and threshold crossings. A finite band describes sampling and measurement variation; it does not cover unmeasured training procedures or architectures.

Fix a quality target \(q\in \mathbb R\), tolerance and cost increment \(\varepsilon ,\delta \in (0,+\infty )\), and a budget \(C\in \mathbb R_{\geq 0}\). Require both \(V_A(C)\) and \(V_A(C+\delta )\) to be finite real values. A study calls an architecture \(\varepsilon \)-saturated at \(C\) only relative to its allowed intervention class when the restricted frontier has certified gain \(V_A(C+\delta )-V_A(C)\leq \varepsilon \).

Take finite real \(U\), \(C\geq 0\), \(\delta \geq 0\), and \(\varepsilon \geq 0\), with the nondecreasing frontier of Definition [ftip-00JJ]. If a justified upper bound gives \(V_A(C+\delta )\leq U\) and a justified lower bound gives \(V_A(C)\geq U-\varepsilon \), then \(U-\varepsilon \leq V_A(C)\leq V_A(C+\delta )\leq U\). Both endpoint values are therefore finite real, and \(0\leq V_A(C+\delta )-V_A(C)\leq \varepsilon \). This is a conditional bound for the named frontier; it proves no universal saturation of intelligence.

For the finite controller class with hard resource admission, Corollary [ftip-00MC] obtains the required bounds from a feasible controller and a uniform Bellman certificate.

Let \(S\subset \mathbb R_{\geq 0}\) be a finite set of measured costs, and choose \(C_0>0\) larger than every element of \(S\). Suppose the exact frontier value observed at each cost in \(S\) is zero. For finite budgets \(C\geq 0\), suppose the declared class of possible frontiers permits both \[ V_0(C)=0, \qquad V_1(C)= \begin {cases} 0,&0\leq C<C_0,\\ 1,&C\geq C_0. \end {cases} \] These nondecreasing frontiers with scores in \([0,1]\) agree on every observed cost and differ at \(C_0\). The observations alone do not distinguish them.

Both possibilities have finite realizations in the framework of Definition [ftip-00JJ]: allow two interventions with costs \(0\) and \(C_0\). Give the first score zero and the second score \(\theta \in \{0,1\}\). For example, on a single deterministic evaluation task with utility equal to the output bit, let the two resulting protocols return \(0\) and \(\theta \). The two possible choices of \(\theta \) give \(V_0\) and \(V_1\), respectively, under the same intervention and cost specification. Every score is finite, and the feasible score maximum is attained at each budget.

This example does not supply a strictly better agreeing frontier for every possible data set or every admissible class. If a proved global score bound is \(1\) and an intervention attains it at a finite cost \(C_*\geq 0\), monotonicity forces \(V_A(C)=1\) for every \(C\geq C_*\); no higher value is admissible. A singleton class of possible frontiers can also identify the frontier without such an alternative. A conditional certificate such as Theorem 5 therefore requires its stated upper-bound evidence; that evidence does not follow merely from the absence of an observed improvement.

The primary cost \(C\) is a declared scalarization of the recorded cost vector. A frontier statement is conditional on its units and weights; the same paired measurements may produce a different ordering under a different scalarization.

Two architectures may alternate in the observed ordering across costs: one can score higher at small \(C\) while the other catches up at larger \(C\). Thus a single comparison point cannot establish a global ordering or a common ceiling.

A Kimi Delta Attention versus Transformer study should publish the cost grid, paired scores, uncertainty intervals, scalarization weights, and the exact identities and configurations of the compared models. The report may then state which measured points are Pareto or iso-quality comparisons under that protocol.

The conditional frontier is indexed by architecture, intervention class, task family, evaluation law, and cost scalarization. Changing any of these coordinates creates a new estimand; no ordering transfers automatically to a new checkpoint, optimizer, hardware stack, or task family.