GA: Cumulative reasoning and systems assessment

A changed summand needs a changed invariant

Use $0\le i\le n$ and $s=i^2$. Initially $i=s=0$. Assuming it holds with $i<n$, the new sum is $i^2+2i+1=(i+1)^2$, and the new index is $i+1\le n$. At exit, $i\ge n$ combined with $i\le n$ gives $i=n$ and $s=n^2$. The nonnegative integer variant $n-i$ decreases by one each iteration, proving termination in the ideal model. Bounded machine arithmetic would require a separate overflow bound.

An inner name is not the outer storage slot

Outer $x=4$; the inner initializer uses it to compute 8; the inner body returns 8; the final outer occurrence is still 4. The result is 12. Allocate distinct slots, such as outer slot 0 and inner slot 1, and map names lexically during compilation. Overwriting one shared slot for both names makes the final occurrence read 8 and returns 16. A test with no shadowing would not reveal this bug.

Matching primal flow and cut bounds

Send 2 along $s\to a\to t$, 1 along $s\to a\to b\to t$, and 2 along $s\to b\to t$. Every edge respects capacity and intermediate flow is conserved; the value is 5. The cut $S=\{s\}$ has outgoing capacity $3+2=5$. Any feasible flow value cannot exceed any cut capacity, so this flow attains the upper bound and is maximum. The argument does not rely on which augmenting paths the implementation happened to choose.

The data perturbation lies in the sensitive direction

The first solution is $(1,1)$ and the changed solution is $(1,2)$. The relative data perturbation is $10^{-4}/\sqrt{1+10^{-8}}$, while relative solution change is $1/\sqrt2$. The Euclidean condition number is $10^4$. The exact diagonal arithmetic already displays sensitivity; blaming numerical instability would misidentify the issue. A residual check alone may not communicate coefficient uncertainty.

A bounded state trace answers the specified queue

The job at 0 runs from 0 to 1. The job at $.2$ waits and runs from 1 to 2. At $.4$ there are already two admitted unfinished jobs, so reject it. At time 1 the first job has departed; the new arrival is accepted and runs from 2 to 3. The finite capacity and deterministic arrivals/service differ from an infinite-buffer stationary stochastic model. Keep rejection and waiting definitions explicit.

Intersection of completed quorums is not a whole register protocol

Any size-two read and size-two completed-write sets among three replicas intersect. However, let an incomplete write place new version 1 only at replica A while B and C retain version 0. A read of A,B returns version 1. Without writing it back, a later read of B,C can return version 0. Thus the intersection arithmetic alone does not prove linearizable behavior for the simple maximum-observed-value rule under incomplete writes. A proper register protocol needs additional rules and assumptions; consensus is not supplied merely by choosing majority sizes.