Does the word "spectrum" (as in a progressing array of values) necessarily imply separate extremes?

Actual electromagnetic spectra do not have endpoints, except in that spectrographs have observational limits -- a prism will not generate a spectrum that's far outside visible light frequencies. And of course if one is investigating spectral lines, there will always be a shortest and a longest in frequency on any spectrum. But it's just the orderly continuous series of values that counts as a spectrum.

In metaphoric use, these properties seem to be constant. A spectrum of opinions will have extreme cases, but only because there is a gradient, and all finite ordered sets have maximum and minimum elements by virtue of the trichotomy law. Similarly for a spectrum of spicy dishes, a spectrum of musical notes, a spectrum of travel alternatives, or whatever one is metaphorizing into a spectrum.

But that doesn't mean that the scale stops at the highest and lowest instances; the scale they're arranged on -- the cline, the ordering -- goes on, unless it happens that the set is closed and the maximum and minimum points actually occur at the maximum and minimum possible values of the scale. (Ordered sets can be half-closed at either end.)

And there's no reason why the scale can't loop back. It's just a case of modulo arithmetic; each note gets repeated in a higher key the same way 8:30 loops around from morning to evening every day. It's still 8:30, but it's a different 8:30, the same way High C is still C, but a different C.

The word "spectrum" in science started with light, which just like sound can have a frequency that is as low or as high as you can wish. In practice, there will be physical limits, but there is no set maximum frequency of light - it just gets harder and harder to generate.

From light, the term spread by analogy to other areas, and you could almost say that the common thread going through all of them is precisely that they do not loop, but extend (usually smoothly) over a range. And the range might be infinite.

Having said that, though, it's your metaphor. Sound has a very legitimate spectrum in the physical sense with a close analogy to light, and, from a physics point of view, 440 Hz is a totally different frequency than 880. So it doesn't loop. But perceptually it kind of does.

So I would say, No, the ends of a spectrum cannot come together, but the ends of an octave do line up in a special way that make the phrase "looping spectrum" not entirely unreasonable.

There is no looping that happens in a spectrum. A tiny portion of the electromagnetic spectrum is in the form of visible light.

The extremes of visible light range from red to violet. Below red is infrared and above violet is ultraviolet, but these are not visible to the human eye.

The color wheel portrays primary and secondary colors as a looping circle. While red and violet can be adjacent in some of these color wheels, there is no loop in real life: red has infrared next to it (which humans can't see) and violet has ultraviolet next to it (which humans can't see). The first visible-light spectrum pictured above is better conceptually than the color wheel because there is a red doesn't loop back to violet; it just keeps on going.

On a pitch pipe, one can see that the notes continuously circle. But are A and G# extremes of a looping spectrum? A little reflection will reveal that the answer is no. The notes on the pitch pipe are A4 and G#4. Even a piano keyboard has octaves 1 to 7, with normal note going from A1 to C8. If you sang a rising scale (or played a violin) starting at C8 your voice or violin wouldn't suddenly jump to an A1! In other words, the notes are conceptually adjacent but not actually adjacent.

In music there is a circle of fifths which can be really handy for understanding key signatures and Brahms A German Requiem. But again, the looping circle is illusory. The circle of fifths might make it seem that Gb and F# the same keys. (They are not.) And while well-tempered instruments (say a piano) make no distinction between an Gb and an F#, musicians on unfretted instruments (who are skilled with just intonation) can (and do) make a distinction between Gb and F#.

Sorry to be pedantic, but these spectra don't loop. They go shorter than the shortest you can perceive and longer than the longest you can perceive, and there are still shorter and longer frequencies that cannot be perceived.

All that to say:

Does the word “spectrum” (as in a progressing array of values) necessarily imply separate extremes?

Yes, the extremes of spectra are separate.