As per my brief e-mail, here are some thoughts.
Certainly what you have includes at least 3 different items/kinds/"species":
1. (your hardcopy p.9- view referenced image) mold of bryozoan (your "phosphatic structure").
2. (your hardcopy p. 8 bottom - view referenced image) juvenile baby brachiopod, possibly a rhynchonellid (of which there are many adults - Lepidocyclus capas for example - in the type Cincinnatian; yourt view is looking directly at the commissure, which is the line along which the two valves/halves/shells come together when the animal closes up.
3. (the rest - your "mystery plates") - I don't know, but read on for possible suggestions.
I see you've tapped a great deal of expertise already. A couble of those were on my mind as soon as I read your initial e-mail, but I see you've already asked them (Carl Brett, Dave Meyer, Richard Davis).
I would recommend that you get - either via interlibrary loan, or better yet by driving to Oxford, Univ. of Miami - that MS thesis by Anthony Martin - (would be quicker by far than interlibrary loan; just be sure to phone ahead so you don't arrive when they're closed for the holidays.) He may have seen and photographed something like these plates; you'll never know until you look at his illustrations yourself.
I showed these also to my advanced doctural student, Paul Zell, who has had a great deal of experience looking at fragmentary fossils here in the Appalachians; he suggested a number of the various possibilities, in addition to the ones I had thought of (later, herein).
I would like to examine the thin-sections of your "plates" which Ivan Sansom made (my lab here doesn't handle such tiny specimens, and it would be nice to see what he was looking at); I'm not sure how practical that might be. However, I'm pretty sure I could recognize if the material is hardened mudstone (the "molds" hypothesis - #3a.) versus one or more of the skeletal/shell/ossicle materials (the "echinoid teeth" idea - #3b).
Before talking about the possibilities for the plates, let me comment about #'s 1 & 2 on the first page here.
#1. Bryozoans - I noticed a number of very small fragments of bryoz in the fossil hash, but they're all preserved as original calcite, & so show the typical appearance of that phylum (in fact, I could - with time and effort - probably track down species names for several of them).
I don't think most of the mystery plates are right for bryo zooid/zooecial molds - wrong size, shape and packing; also, the rear parallelism of the grooves isn't quite like any bryo I know.
The "phospatic structure" (p.9 view referenced image also p. 10 bottom, hardcopy - view referenced image) does look like 5-6 zooecia filled with mud, then the mud phosphatized while the tube-wall/skeletal-wall calcite dissolved away, I think the plates next to them are just random burial together, not related.
And, phosphatized zooecial chambers don't show any grooves/ridges, whereas your plates are quite regularly/evenly ribbed. Nor would zooid fillings fuse together so consistently the same size and shape and groove arrangement.
#2. Baby Rhynchonellid Brachiopod - the very perfect sine-wave shape of the line between the "plates" is exactly what such a fossil would show. Moreover, I saw several tiny brach shells & shell fragments in the fossil hash, so such things are already present here, and so another with different type of preservation would be reasonable to expect. And, you do note (hardcopy pg 8 - view referenced image) that this one is smaller-sized than most), which suggests that it might be different from the others.
#3. For the "plates", there are a number of possibilities, all of which have some difficulties and none of which seem to fit well. I think the two most likely are #'s 3a and 3b, but the others should be considered as well.
(Overall, note that some of the possibilities are impressions/molds/casts, hence the sediment/mudstone, while others are the actual skeletal hard parts, hence calcitic with microstructure).
#3a. Molds (external; "impressions" of outer surface) of tiny/baby rhynchonellid brachiopod shells:
Several Aspects:
There are tiny brach shells & shell fragments in the fossil hash; their ribbing is the same size "corrugations" as the grooves on your mystery plates. Moreover, most of your plates seem to be slightly/shallowly concave on their grooved/ridged surface (as are the brachs), and at least some seem to have the grooves radiating/diverging slightly (rather than strictly parallel) with a "radius of origin" womewhat outside the edges of the plate.
Much of the fossil hash consists of tiny pieces of gray calcareous mudstone; when any lime/carbonate mud lithifies, you get tiny blebs within the mass which have slightly more calcite cement between the original mud grains than the average/normal (just randomness in nature, no other real "cause"). If there's a brach/fragment sitting in the mud, the mud next to it is liable to pick up slightly more calcite being dissolved by the ground water, & hence becomes better cemented right in contact with the shell. With later mechanical disaggregation of the rock, that better-cemented mudstone bit will fall away & look like a loose sand grain with the impression of the shell still on it.
I note, too, that the abundance/number of tiny brach shells in the hash sample you sent is on the same order of magnitude as the grooved plates, which would be consistent with this "baby brach shell impression" hypothesis.
Some of the specimens photographed on the edge (hardcopy pg. 7, upper right - view referenced image) look like they have the grooves/ridges on both sides; such would be expected if mud got inside the shell before it finally closed, so that eventually you'd end up with an impression of each half/valve on the two sides of the now-lithified mud filling.
I have rarely seen such impressions of shell surfaces at such small sizes, but I do recall some of ostracods, forams, tiny brachs, & tiny (baby) pelecypods in micro-hashes from various ages & places. And, one often meets with larger/macroscopic-sized shell impressions on outcrops where the mudstone weathers out as/in rounded lumps.
#3b. The tiny teeth from the mouth area of an echinoid ("sea-urchin"). Echinoids have just gotten started back in the Ordovician, and so this would fit into their time range. If you cross-section their body:
Your "several-part" plates look like such a combination fallen apart after death. Each tooth originally would have been calcite; it would help to see Ivan Sansom's thin-sections to check how much that's the case, or perhaps how much the fossilization/diagenetic processes might have altered their mineralogic composition from a crystalline calcite beginning.
Now, the remaining possibilities (#3cff) are just listed in taxonomic order, since I can't assess how likely each is, in comparison with the others.
#3c. Molds/impressions of outer surfaces of sides of gastropod whorls.
#3d. Gastropod opercula fragments.
#3e. The interlocking teeth on the hinge of pelecypod ("bivalve") shells. Several of your 3- & 4-plate specimens look a lot like some of those, such as on trigoniids & heterodonts. If so, these would likely be molds/impressions/casts (of those structures), because most (though not all) pelecypods' shells are aragonite, and so would have dissolved away or inverted to calcite long ago.
#3f. Jaw parts/elements/components of worms (annelid - scolecodont-like); not the protruding "pincer" or "tooth", but small plates to the rear and making up part of the entire apparatus. Some worms produce chitinous jaws, others phosphatic hard parts.
#3g. Small jaw parts of some arthropod, perhaps trilobite, perhaps eurypterid, perhaps other obscure groups; remember how many little pieces go into an insect's or a crab's mouth parts - and how little we know about such small bits in any of the fossil groups.
#3h. Impressions/molds of tiny bivalved-arthropod shells known as estheriids; some have ornamentation, others are smooth; the shells themselves are thin & chitinous, so rarely are preserved themselves, though their impressions sometimes are.
#3i. Echinoderm tiny plates (such as from the crinoids' anal sac, or even molds/impressions of those, which are so rare that little is know about their possible shape variations).
#3j. Crinoid columnal from top of stem, right under the calyx; in some species, that piece is very thin and fragile and so could break into fragments, or make impressions on the mudstone sediment.
#3k. (Larger ossicles from crinoids or other echinoderms seem unlikely, if there's no stereom inside.)
#3l. Certain starfishes have tiny calcite plates of various odd shapes around the edges of their mouths; perhaps yours are these??
#3m. Likewise (same as worm-scolecodont) inner/rear small elements in a conodont jaw apparatus.
#3n. Impressions of the top of the flat part of a platform-type conodont (or, if phosphatic, a broken fragment of that wide part of a conodont).
#3o. Impressions of tiny fish "scales" (bone plates embedded in their skin); this would be early for vertebrates, but still within the known time range for the most primitive fishes. (However, if one of the others (Sansom?) has experience with fish, and thinks not, then less likely.)
Now a couple of closing comments:
I couldn't find where you'd said what your mystery plates were in composition, but my impression was "phosphatic", - that must not be right, though, since I sacrificed 1-2 to check in acid, & they fizzed vigorously (ie. are calcareous). Phosphatic fossil debris (conodonts, teeth, etc) dissolves so slowly that the standard procedure for extracting it is to throw the limestone block into a beaker/bowl of hydrochloric acid, & dissolve away the carbonate, thus leaving the phosphatic debris as an insoluble residue at the bottom.
Parenthetically, your fossils come from the "middle" Arnheim - OK, but as I recall from doing our type-Cincinnatian guidebook, there are just two members in the Arnheim, the Sunset & Oregonia, which'd be upper vs. lower & separated by some sort of surface reflecting non-deposition if not also minor erosion. (I'll check - I'm writing this while giving a final exam, so don't have copy nearby.) Which of those two would your hash/plates be from, I wonder? (from our email printout of Dec 10: "from middle & upper Arnheim" - I suspect that means the upperhalf??
I hope all of these comments & possibilities will help you a bit.
All best regards,
Roger (Cuffey)