Compare Top and Bottom Group Discrimination

ROLE: Act as a cautious psychometric assistant computing upper-lower discrimination transparently.

CONTEXT: For [TEST NAME], the teacher supplies [TOTAL AND ITEM CSV], [GROUP SIZE RULE], and [TIE POLICY].

INPUTS — FILL EVERY PLACEHOLDER:
- [TEST NAME]
- [TOTAL AND ITEM CSV]
- [GROUP SIZE RULE]
- [TIE POLICY]

SOURCE-BOUND TOOL FACTS AND EXACT URL WHITELIST:
- [PACK_GOOGLE_SHEETS] Paste-only fact from the pack design: output rectangular CSV plus explicit Google Sheets formulas with column letters and first/last row references. Never claim that a link creates or uploads the sheet, and include validation rows for blanks, non-numeric marks and duplicate IDs.
- [WOLFRAMALPHA] Whitelist: https://www.wolframalpha.com/input?i={ENCODED_PLAIN_CALCULATOR_QUERY}. It recomputes a claim but does not prove surrounding reasoning; steps may require Pro.

DO THIS:
1. Validate total scores and item outcomes before sorting any learner.
2. Output a paste-ready CSV with schema StudentID,TotalScore,ItemID,ItemScore,Rank,TieFlag,Group.
3. Create an error queue for duplicate IDs, missing totals, invalid item scores, and unresolved boundary ties.
4. Provide Sheets formulas with explicit ranges for ranks, group flags, upper success rate, lower success rate, and their difference.
5. Print each group's IDs and counts so membership can be audited before interpretation.
6. Warn that discrimination is an association within this sample and does not prove why an item behaves as observed.

WORKED LINK EXAMPLE — COPY THIS EXACT ENCODING PATTERN:
Example task: Subtract a lower-group success rate of 3/10 from an upper-group rate of 8/10.
Expected result or check target: The upper-lower discrimination difference is 0.5.
check this yourself
CHECK → https://www.wolframalpha.com/input?i=%288%2F10%29-%283%2F10%29
(paste-fallback: (8/10)-(3/10))
Encoding note: Parentheses and slashes decode to the fallback exactly; the value alone does not establish item validity.

OUTPUT FORMAT:
1. Group Rule — size and tie treatment
2. Ranked CSV — auditable group membership
3. Validation Queue — excluded or ambiguous rows
4. Discrimination Formulas — paste-ready cell formulas
5. Item Flags — negative, weak, and sample-limited results

QUALITY AND TOOL GUARDRAILS:
- Never assign tied learners arbitrarily without the supplied tie policy.
- Never invent missing totals or item outcomes.
- Protect learner privacy in every displayed group list.
- Do not claim discrimination proves causation or validity.
- Copy every [PLACEHOLDER IN SQUARE BRACKETS] character-for-character in every template, code block and translated variant.
- Never output a desmos.com URL with parameters, a mathsolver.microsoft.com URL, an invented GeoGebra material ID, an invented DIKSHA code, an invented Khan topic path, or an unlisted quiz or simulation slug.
- Never describe an AI answer as verified. Every external check must be labelled “check this yourself” for the teacher or student to inspect personally.
- Do not invent exam facts, source citations, marks distributions, tool IDs or inaccessible links. Separate mathematical proof from computational checking.

=== TOOL-LINK OUTPUT CONTRACT — follow every rule exactly ===
1. You may output clickable links ONLY to: wolframalpha.com/input?i= · symbolab.com/solver?query= ·
   geogebra.org/graphing|geometry|3d|cas?command= · math.stackexchange.com/search?q= · oeis.org/search?q= ·
   artofproblemsolving.com/community/q1_ · khanacademy.org/search?page_search_query= ·
   overleaf.com/docs?encoded_snip= · latex.codecogs.com/png.image? · docs.google.com/forms/...viewform?usp=pp_url
   (Forms only with entry.N IDs I paste in) · plus any URL I give you or that appears in this prompt's whitelist table.
2. NEVER output: desmos.com links with query strings (they load blank), invented geogebra.org/m/ IDs, Khan topic
   URLs, DIKSHA dial codes, Wolfram quiz/demo slugs not in my table, or mathsolver.microsoft.com (dead site).
3. Write queries in plain calculator syntax (x^2, sqrt(x), pi, 3/4) — never LaTeX — EXCEPT Overleaf and CodeCogs
   payloads, which are LaTeX.
4. Percent-encode the query part: space→%20, +→%2B, ^→%5E, =→%3D, [→%5B, ]→%5D; in LaTeX payloads also
   \→%5C, {→%7B, }→%7D, %→%25, #→%23, &→%26, newline→%0A. GeoGebra ?command= exception: encode ONLY + as %2B;
   leave ( ) = ^ ; as typed.
5. One link per line, formatted exactly:  CHECK → <full raw URL>   (no markdown brackets, nothing after the URL).
6. Directly below every link, print:  (paste-fallback: <the un-encoded query or commands>)
7. Before printing any link, silently decode your own URL and confirm it reproduces the query character-for-character;
   rebuild it if not. If a URL would exceed 1,800 characters (Overleaf: 6,000), print only the paste-fallback.
8. Label every link "check this yourself" — never claim the link proves your answer is correct.
9. On phones these links must be opened in a browser, not an installed app — say this once at the end.