Research: bound D8 inverse pair with third role

research/d8-inverse-pair-third-role-seeddf1e67/CERTIFICATE.json

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+{
+  "schema_version": 1,
+  "certificate_type": "complete finite exact word census plus displacement-rank lower bound",
+  "field": "F2",
+  "group_indexing": "index i+4e denotes r^i*s^e, i in Z/4 and e in Z/2",
+  "group_law": "(i,e)(j,f)=(i+(-1)^e*j mod 4,e+f mod 2)",
+  "elements": {
+    "identity": 0,
+    "r": 1,
+    "r_inverse": 3,
+    "s": 4,
+    "t=r_inverse*s": 7
+  },
+  "generator_family": {
+    "factor": "F_g(N)=I+([g]+1)N",
+    "labels": [4, 7],
+    "rank_one_idempotents": 28,
+    "paid_generators": 56
+  },
+  "endpoint": [
+    {"role": 0, "group_element": 1},
+    {"role": 1, "group_element": 3},
+    {"role": 2, "group_element": 0}
+  ],
+  "lower_bound": {
+    "left_regular_dimension": 8,
+    "full_role_space_dimension": 24,
+    "rank_each_factor_minus_identity": 4,
+    "rank_endpoint_minus_identity": 12,
+    "excluded_lengths": [0, 1, 2]
+  },
+  "finite_search": {
+    "word_length": 3,
+    "ordered_words": 175616,
+    "hits": 0,
+    "complete": true
+  },
+  "direct_word": [
+    {"label": "s", "projector": "E11"},
+    {"label": "t", "projector": "E11"},
+    {"label": "t", "projector": "E22"},
+    {"label": "s", "projector": "E22"}
+  ],
+  "exact_conclusion_in_named_model": "minimum paid word length four"
+}

research/d8-inverse-pair-third-role-seeddf1e67/OVERLAP.json

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+{
+  "schema_version": 1,
+  "checked_at_utc": "2026-10-14T20:29:48Z",
+  "coverage": {
+    "open_prs": 391,
+    "discussions": 802,
+    "complete_pagination": true,
+    "warnings": []
+  },
+  "exact_all_term_query": {
+    "terms": ["D8", "inverse rotation", "third role", "rank-one"],
+    "matches": 0
+  },
+  "predecessors_and_neighbors": [
+    {
+      "item": "PR #1881",
+      "commit": "5a51deaaab2746d42eddb454bbb968bda3af02cc",
+      "scope": "two roles, D8 reflection labels, noncentral diag([r],[r^-1]), exact minimum four",
+      "relation": "same group and endpoint, one fewer dirty role"
+    },
+    {
+      "item": "Discussion #1738 / PR #1742",
+      "commit": "679f3624a0a41ed14942825dd7f413132412d1e3",
+      "scope": "three roles, D8/UT3 central endpoint diag([z],[z],[1]), exact minimum eight",
+      "relation": "same group and role count, different central endpoint and paid labels"
+    },
+    {
+      "item": "PR #1443",
+      "scope": "two roles, D8 central endpoint diag([r^2],[r^2])",
+      "relation": "different role count and endpoint"
+    }
+  ],
+  "active_exact_scope_claim_found": false,
+  "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1738#discussioncomment-18933540",
+  "hub_pointer_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1167#discussioncomment-18933546"
+}

research/d8-inverse-pair-third-role-seeddf1e67/PR-BODY.md

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+## Result
+
+This package settles one exact three-role formal group-algebra question. For
+`D8=<r,s | r^4=s^2=1, srs=r^-1>`, `t=r^-1s`, and paid rank-one factors
+`F_g(N)=I+([g]+1)N` with `g in {s,t}`, the endpoint
+`diag([r],[r^-1],1)` has minimum paid length four on three arbitrary dirty
+roles, even with interspersed free `GL3(F2)` scalar maps. It does not give a
+supplier or `kappa` improvement.
+
+## Evidence
+
+- The regular-representation displacement rank is 12 for the endpoint and 4
+  per paid factor, so lengths zero through two are impossible.
+- The complete exact length-three census checks all 175,616 ordered words over
+  the 56 generators and finds zero hits.
+- The explicit four-call word reaches the endpoint. A separate tuple-valued
+  verifier replays the whole census, positive control, omitted-call control,
+  and rank control.
+
+## Reproduction
+
+```sh
+python3 -I -B research/d8-inverse-pair-third-role-seeddf1e67/search.py
+python3 -I -B research/d8-inverse-pair-third-role-seeddf1e67/verify.py
+python3 scripts/research_package.py --package research/d8-inverse-pair-third-role-seeddf1e67 --check-only
+```
+
+The scientific base is `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`.
+Predecessors are the two-role inverse-pair result (#1881, commit
+`5a51deaaab2746d42eddb454bbb968bda3af02cc`) and the distinct three-role
+central-pair result (#1742). The model excludes native third-role hosting,
+recursive charges and supplier admission; no selected frontier changes.
+
+Substantial OpenAI Codex assistance in derivation, implementation, verification,
+and packaging. No independent human review is claimed.

research/d8-inverse-pair-third-role-seeddf1e67/README.md

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+# A third dirty role does not reduce the D8 inverse-pair bill
+
+## Result
+
+In the formal group algebra `F2[D8]`, the three-role endpoint
+
+```text
+diag([r], [r^-1], 1)
+```
+
+has exact paid-call minimum four when the paid labels are the two reflections
+`s,t` and arbitrary free scalar `GL3(F2)` maps are allowed between calls.
+The direct four-call word remains optimal after adding a third arbitrary dirty
+role. An exact rank bound excludes lengths zero through two; a complete census
+of the 56^3 ordered three-call words finds no endpoint hit.
+
+This is a scoped finite-algebra result. It does not improve the selected bit
+supplier, price physical hosting of the third role, or change `kappa`.
+
+## Model and complete paid bill
+
+Index `D8=<r,s | r^4=s^2=1, srs=r^-1>` by `r^i s^e`, with `i mod 4` and
+`e in {0,1}`. Define `t=r^-1s`; then `s,t` are involutions and `st=r`,
+`ts=r^-1`. Work in the full group algebra `F2[D8]`, not a finite address
+quotient. A paid factor on three scalar roles is
+
+```text
+F_g(N) = I + ([g] + 1)N,  g in {s,t},
+```
+
+where `N=u v^T` is any rank-one idempotent over `F2`, so
+`u,v in F2^3 \ {0}` and `v^T u=1`. There are 28 such projectors and hence
+56 distinct paid generators. All three input roles are arbitrary. The exact
+diagonal endpoint restores the third role, and every paid factor is charged
+once. No additional role, COPY, discarded output, or address operation is
+available. Arbitrary interspersed scalar `GL3(F2)` maps are included: push them
+to the right by conjugating each projector, then apply augmentation
+`[g] -> 1`; because the target augments to `I`, the residual scalar output map
+must be `I`.
+
+The direct word is
+
+```text
+F_s(E11) F_t(E11) F_t(E22) F_s(E22)
+  = diag([s][t], [t][s], 1)
+  = diag([r], [r^-1], 1).
+```
+
+Its complete formal bill is four calls. A three-call word would save one
+factor (25%) in this block, so the best-case budget justified searching exactly
+three calls.
+
+## Lower bound for the first three lengths
+
+Use the left regular representation of `F2[D8]`; each role becomes an
+eight-dimensional block, for a total dimension of 24. Write `L_g` for left
+multiplication by `g`. Since each reflection is an involution, `L_s+I` and
+`L_t+I` have rank four: each is a product of four disjoint transpositions over
+`F2`. Thus the difference from identity of one paid factor is
+
+```text
+(L_g + I) tensor N
+```
+
+and has rank four. Multiplication by `r` or `r^-1` consists of two cycles of
+length four on `D8`, so each of `L_r+I` and `L_(r^-1)+I` has rank six. The
+endpoint difference is block diagonal with ranks `6,6,0`, hence rank 12.
+For any matrices `A,B`, `rank(AB-I) <= rank(A-I)+rank(B-I)`. Therefore at
+least three paid factors are necessary.
+
+This makes length three the only unresolved length below the direct word. The
+packed search checks every ordered triple from the 56 gates using exact
+eight-coefficient group-algebra multiplication. A separate verifier repeats
+the entire census with tuple-valued coefficients and direct group-law
+convolution. Both report zero hits among 175,616 triples. Together with the
+rank lower bound and direct word, this proves minimum four in the stated model.
+
+## Reproduction
+
+```sh
+python3 -I -B research/d8-inverse-pair-third-role-seeddf1e67/search.py
+# D8 inverse-pair W3 exact search: generators=56 triples=175616 matches=0 direct_calls=4
+
+python3 -I -B research/d8-inverse-pair-third-role-seeddf1e67/verify.py
+# PASS_D8_INVERSE_PAIR_W3 rank_floor=12/4 generators=56 triples=175616 matches=0 minimum=4 direct_control=pass omission_control=pass altered_rank_control=pass
+
+python3 scripts/research_package.py --package research/d8-inverse-pair-third-role-seeddf1e67 --check-only
+```
+
+The registered replay requests Python 3.12. Local direct runs used CPython
+3.14.7; the registered 3.12 replay is not claimed unless available in the
+submission environment.
+
+## Relation to prior work and limits
+
+- PR #1881 (`5a51deaaab2746d42eddb454bbb968bda3af02cc`) establishes the
+  two-role `diag([r],[r^-1])` minimum of four. This result adds a third
+  arbitrary restored role and keeps the same noncentral endpoint.
+- Discussion #1738 / PR #1742 establish an eight-call minimum for the
+  three-role central endpoint `diag([z],[z],[1])` with `z=r^2`. The target,
+  paid labels and word-length question here differ.
+- PR #1443 concerns a two-role central endpoint `diag([r^2],[r^2])`.
+
+The exact minimum concerns only the finite named formal alphabet. It does not
+include native implementation/hosting of the third role or its recursive
+charges, so it is not a bit-supplier construction or a new `kappa`. The task
+record pins the selected frontier, current task catalogue and predecessors.
+Prepared with substantial OpenAI Codex assistance; no independent human
+review is claimed.
+
+## Open next questions
+
+Can a fourth arbitrary dirty role lower the four-call minimum for the same
+inverse-pair endpoint and reflection alphabet? The dimension-five projector
+count grows sharply; first apply the regular-representation rank floor and
+estimate the complete length-three search. Any formal saving would still need
+an added-role hosting and recursive-cost invoice.

research/d8-inverse-pair-third-role-seeddf1e67/RESULT.json

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+{
+  "schema_version": 1,
+  "status": "exact scoped negative result",
+  "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
+  "seed": "df1e67db401c155561180b9d7f1a77f8",
+  "model": {
+    "field": "F2",
+    "group": "D8 = <r,s | r^4=s^2=1, srs=r^-1>",
+    "paid_labels": ["s", "t=r^-1*s"],
+    "roles": 3,
+    "paid_factor": "I + ([g]+1)N",
+    "projector_domain": "all 28 rank-one idempotents N=u*v^T in M3(F2), v^T*u=1",
+    "endpoint": "diag([r],[r^-1],1)",
+    "free_scalar_maps": "arbitrary interspersed GL3(F2), pushed into the complete projector alphabet; augmentation forces residual output map I"
+  },
+  "rank_certificate": {
+    "regular_rep_dimension": 24,
+    "target_displacement_rank": 12,
+    "each_paid_factor_displacement_rank": 4,
+    "lower_bound_paid_calls": 3
+  },
+  "length_three_census": {
+    "distinct_paid_generators": 56,
+    "ordered_words_checked": 175616,
+    "target_hits": 0,
+    "independent_full_coefficient_replay": "PASS"
+  },
+  "positive_control": {
+    "word": ["F_s(E11)", "F_t(E11)", "F_t(E22)", "F_s(E22)"],
+    "product": "diag([s][t],[t][s],1)=diag([r],[r^-1],1)",
+    "paid_calls": 4,
+    "exact_replay": "PASS"
+  },
+  "conclusion": "The exact minimum is four paid factors in this formal D8, reflection-only, three-role model. No improvement over the direct four-call word; no native supplier realization or kappa claim."
+}

research/d8-inverse-pair-third-role-seeddf1e67/SOURCES.json

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