Page 22 of Damsel in Disguise

“And if you’re here because you enjoy being challenged, then you may find this term instructive,” my eyes instinctively swipe over to the youngest student here and find her listening to me with rapt attention—Ariella.

I turn back to the board and begin writing without ceremony, letting the system take shape beneath my hand as though it were nothing more than a routine exercise.

Clean vectors emerge first, followed by familiar operators and transfer functions—structures every senior in this room should’ve seen before in diluted form.

I speak as I write, explaining each component with the ease of repetition, outlining assumptions that sound reasonable enough to pass unchallenged.

“This,” I say, gesturing toward the diagram, “is a closed-loop feedback system under nominal conditions. Input signal, amplification, corrective feedback.”

Then, without emphasis, I add another layer.

A recursive element slips into place, almost polite in its subtlety. The sort of addition that appears harmless until it compounds, until it feeds on itself quietly enough that no one notices the damage being done.

I continue evenly. “It’s a common architecture used everywhere from communication networks to biomedical instrumentation. When tuned correctly, it suppresses instability. When mishandled…” I let the sentence trail off, unfinished by design.

Pens move faster. Screens glow brighter. Heads nod with a confidence that has not yet earned its place.

I proceed as though the system behaves exactly as advertised, defining parameters, assigning values, discussing convergence and signal-to-frequency ratios with the calm assurance of inevitability.

I don’t mention phase drift or correct the assumption about frequency independence. And I certainly don’t acknowledge what happens when an error is allowed to feed back into recursion unchecked.

Because the flaw is already buried there within the structure. It’s a feedback delay mismatched to the recursion depth, which might look trivial on paper but is catastrophic in practice.

Senior students should notice that.

To my disappointment, they don’t.

I can tell by how quickly they write, or from the absence of hesitation, the way they’re accepting the conclusion I’m guiding them toward. They’re following the path I laid out for them, unaware that it leads somewhere unsafe.

I extend the model further, letting the recursive loop run freely, and then ask, almost casually, “What do we expect the output to do after sufficient iterations?”

A few hands begin to rise. I don’t acknowledge them right away because I notice one particular student is looking at me with untamed confidence in her hazel eyes. Ariella.

The moment I look at her, she joins the rest of the students by raising her hand subtly.

I allow my attention to linger on her for a fraction longer than necessary, careful not to betray the observation. She wants only me to notice.

At last, I gesture toward the raised hands. “You,” I say, selecting one at random.

A final-year student straightens immediately, confidence locking into place as he delivers his answer with polished assurance. He speaks, gesturing toward the board as if the system has already resolved itself in his favor.

I let him finish before dismantling the response with careful precision. Then I point out the unverified assumption, alter a single variable, and explain how attenuation becomes amplification once the error term feeds back into the recursion.

His certainty drains visibly as the structure collapses beneath his logic.

Another student follows.

Then another…

Each answer is being delivered confidently. Most incorrect. Some almost correct.

The room grows quieter with every correction, unease replacing certainty as pens slow and eyes flick away from the board in search of reassurance.

“Miss Keinston?” I point at her at last, making her the center of attention once again for the day.

The room stills almost immediately. Several heads turn in her direction, curiosity sharpening into something closer to disbelief.

“Uh—yes,” her eyes flick briefly toward the board before returning to me, already assembling the structure of her response. “The issue isn’t convergence, it’s the assumption that the system remains memoryless once recursion is introduced.”