Mister-Meeseeks · Oct 15, 2026 · Oct 15, 2026 · Oct 15, 2026 · Oct 15, 2026 · Oct 15, 2026
diff --git a/research/p25-root-pair-bound-seed632639/CERTIFICATE.json b/research/p25-root-pair-bound-seed632639/CERTIFICATE.json
@@ -0,0 +1,126 @@
+{
+  "conclusion": "The complete root-pair family at p=25 misses the selected local-word ceiling gate even with zero added implementation charges.",
+  "copy_bound": {
+    "exceptional_term_floor_if_used_on_both_sides_of_same_dual_term": 5,
+    "exceptional_term_floor_if_used_on_one_side": 6,
+    "negative_controls": {
+      "exceptional_pair_bound": 5,
+      "regular_seven_term_floor": 49,
+      "total_floor_recomputed": true
+    },
+    "number_of_terms_below_7": "at most one",
+    "ordinary_term_floor": 7,
+    "pairing_rank_inequality": "rank(X,Y) >= dim(X)+dim(Y)-9",
+    "reason": "one p=25 puncture mask lowers at most one root-normal support; distinctness and duality place it in at most one term",
+    "scope": "exact eight-term all-pairs decoders whose row and column normals are E8 roots, at exactly 25 punctures",
+    "total_copy_floor": 54,
+    "unique_exceptional_root_span": 7
+  },
+  "optimistic_price": {
+    "a_ceiling_interval": [
+      "307709262007/200000000000000",
+      "384636577509/250000000000000"
+    ],
+    "a_ceiling_upper_decimal": "0.001538546310036",
+    "additional_ports_from_selected_P112": 17,
+    "center_charge": 54,
+    "copy_floor": 54,
+    "extra_setup_frame_hosting_precision_restoration_charges": 0,
+    "families_per_replica": 760,
+    "five_percent_gate": "35491136290869/20000000000000000",
+    "idle_mass": "830",
+    "idle_rank_cap": 51,
+    "kappa_translation_cap_only": "384636577509/250384636577509",
+    "live_ports": 95,
+    "most_favorable_center_rank_profile": {
+      "5": 1,
+      "7": 7
+    },
+    "negative_controls": {
+      "floor_upper_below_five_percent_gate": true,
+      "floor_upper_below_selected": true,
+      "rank_55_profiles": {
+        "exceptional_rank_6": {
+          "a_ceiling_interval": [
+            "1488030253437/1000000000000000",
+            "744015126719/500000000000000"
+          ],
+          "center_charge": 55,
+          "profile": {
+            "1": 2293,
+            "2": 418,
+            "3": 432,
+            "4": 125,
+            "5": 65,
+            "6": 1,
+            "7": 7
+          }
+        },
+        "one_regular_rank_8": {
+          "a_ceiling_interval": [
+            "1488053610999/1000000000000000",
+            "1488053611/1000000000000"
+          ],
+          "center_charge": 55,
+          "profile": {
+            "1": 2293,
+            "2": 418,
+            "3": 432,
+            "4": 125,
+            "5": 66,
+            "7": 6,
+            "8": 1
+          }
+        }
+      }
+    },
+    "p_total": 25,
+    "per_call_local_profile": {
+      "1": 2293,
+      "2": 418,
+      "3": 432,
+      "4": 125,
+      "5": 66,
+      "7": 7
+    },
+    "relative_to_selected_upper_percent": "-8.964671500054",
+    "selected_a_ceiling_upper": "1690054109089/1000000000000000",
+    "stages": 9
+  },
+  "pins": {
+    "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
+    "review_sha256": "7e76d2c38e5fce30587835b7251df1430488ae4dd372611ef69a3f03f476b6cb",
+    "selected_certificate_sha256": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045",
+    "selected_record_sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32",
+    "source_manifest_sha256": "79b3de5db3dbcb9829789ba58e9e45758926031e2a7d80667a08429d369be53b"
+  },
+  "status": "PASS_P25_ROOT_PAIR_BOUND",
+  "support_census": {
+    "all_extremal_live_spans": [
+      7
+    ],
+    "candidate_functional_pairs_per_root": 130305,
+    "extremal_mask_root_pairs_sha256": "ffda2df06beb08f68652d04030ebbaf46e9e855b1425d5c4348d508c20bb5da2",
+    "global_unique_size_25_masks": 7560,
+    "label_pairing_rank": 9,
+    "maximum_root_normals_per_mask": 1,
+    "minimum_punctures_for_live_label_span_at_most_7": 25,
+    "negative_controls": {
+      "24_puncture_near_miss_span_8": 8,
+      "all_120_roots_first_drop_at_25": true,
+      "same_puncture_other_root_span_at_least_8": true
+    },
+    "pair_representations_per_mask": 3,
+    "positive_fixture": {
+      "live_span": 7,
+      "minimum_other_root_span": 8,
+      "one_puncture_restored_span": 8,
+      "puncture_mask_hex": "0xffff000000000003c03d",
+      "root_index": 0
+    },
+    "root_lines": 120,
+    "root_normal_count": 120,
+    "size_25_unordered_functional_pairs_per_root": 189,
+    "unique_size_25_masks_per_root": 63
+  }
+}
diff --git a/research/p25-root-pair-bound-seed632639/README.md b/research/p25-root-pair-bound-seed632639/README.md
@@ -0,0 +1,151 @@
+# The p=25 root-pair support boundary misses the local-word gate
+
+**Scoped structural obstruction; no supplier or new κ.** For eight-term exact
+all-pairs decoders whose row and column normals are E8 roots, an exact 25-line
+puncture can lower at most one root-normal support to a seven-dimensional
+label span. The resulting separate-copy bill is at least **54**. Its most
+favorable profile under the pinned selected-word relaxation has ceiling
+`a < 0.001538546310036`, about **8.965% below** the selected local-word ceiling.
+The support transition at the first puncture beyond the p≤24 floor is exact,
+but it cannot supply the task's 5% gain inside this root-pair family.
+
+## Exact support result
+
+Use the 120 E8 root lines and the binary pairing
+
+```text
+B[i,j] = ( <r_i,r_j> + 1 ) mod 2.
+```
+
+For a root normal `a`, let `C(a)` be the lines with nonzero real inner product
+with `a`. The live support label span drops to dimension at most seven exactly
+when two independent functionals of the nine-dimensional label space vanish
+on every live label. The corresponding puncture mask is the union of their
+violators, intersected with `C(a)`.
+
+The exact enumeration checks all 130,305 distinct functional pairs for each of
+the 120 root normals. For every root, the least such union has 25 lines. There
+are 189 pair representatives and 63 distinct masks of that size per root; each
+mask has three pair representatives and leaves span exactly seven. Across all
+roots, the 7,560 masks are distinct: one mask never lowers two root-normal
+supports. Reinstating any one punctured line in a positive fixture returns its
+support span to eight.
+
+For a root-pair decoder, the row and column normals in each term are roots and
+the two eight-element normal lists are dual bases. The F2 rank inequality gives
+
+```text
+copy(term) >= dim(row labels) + dim(column labels) - 9.
+```
+
+Every term without the unique seven-span root has bill at least seven. Duality
+forces the unique exceptional root, if used on both sides, to occur in the same
+term. That term costs at least five when used on both sides, or six when used
+on only one. Thus the total copy bill is at least `5 + 7*7 = 54` (and at
+least 55 if the exceptional root is used on only one side). If no support drops,
+the total is at least 56. The bound covers every 25-line puncture in this
+root-pair family. It does not cover arbitrary real or complex dual bases,
+masked identities, or nine or more terms.
+
+## Optimistic price
+
+The selected #1094 local profile and ceiling calculator are pinned in
+[SOURCES.json](SOURCES.json). Moving from its 112 ports to 95 ports removes 17
+ports under PR717's favorable per-port model. The support floor 54 is priced
+with the most favorable compatible rank histogram, one rank-five center child
+and seven rank-seven center children. Added setup, frame, hosting, precision,
+transition, and restoration costs are all set to zero. Raising any one center
+rank raises its child moment and its center charge by the same amount, leaving
+idle mass fixed; therefore this minimum rank vector upper-bounds the ceiling
+of every more expensive root-pair invoice. The verifier also screens two
+one-unit rank increases as controls.
+
+| Quantity | Value |
+|---|---:|
+| Total punctures / live ports | 25 / 95 |
+| Added port reductions from P112 | 17 |
+| Separate-copy floor | 54 |
+| Per-call local profile | `{1:2293, 2:418, 3:432, 4:125, 5:66, 7:7}` |
+| Families / stages / center charge | `760 / 9 / 54` |
+| Idle rank cap / idle mass | `51 / 830` |
+| Certified `a` interval | `[0.001538546310035, 0.001538546310036]` |
+| Selected `a` ceiling upper endpoint | `0.001690054109089` |
+| Five-percent gate | `0.00177455681454345` |
+
+The exact upper endpoint is below the selected ceiling itself, so no allocation
+of idle work or additional physical overhead can make this p=25 root-pair route
+pass the card's gate under the stated coverage law and halving guard. The
+`kappa` translation recorded in [CERTIFICATE.json](CERTIFICATE.json) is only a
+planning cap; it is not a new supplier or exponent result.
+
+## Provenance and overlap
+
+- Exact base: `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`.
+- Accepted assignment: seed `632639f1d387026a8b953fa82c286d82`,
+  `complex-mechanisms / complex-rank-shape-decoder`, priority revision 4.
+- Catalogue SHA-256: `d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7`.
+- Selected-record SHA-256: `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.
+- Selected certificate SHA-256: `1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045`.
+- Selected review SHA-256: `7e76d2c38e5fce30587835b7251df1430488ae4dd372611ef69a3f03f476b6cb`.
+- Selected source-manifest SHA-256: `79b3de5db3dbcb9829789ba58e9e45758926031e2a7d80667a08429d369be53b`.
+- The selected-record reproduction command is `make frontier1094-verify`; it was not rerun because this package does not alter the selected source chain.
+
+PR717 head `4a97676294f9fa413b65986e9984fddeda71e4cc` and PR968 head
+`6a99e7240b961ac5e115ae6aa0301ec7c4a45d77` establish the exact eight-term
+copy-floor lineage through 24 punctures, retained in
+[the frame-pair package](../complex-frame-pair-decoders/PROOF.md). PR1619 head
+`6c96bc1a0e374a713e9ae493228247f49be49dd5` prices that range. PR1624 head
+`712a9b9aa8802d0135a578c0debee7b56c8e492f` and
+[Discussion #1570](https://github.com/CrocSwap/integer-mult-bounds/discussions/1570)
+handle one fixed p=26 paired-normal object, with PR1020 head
+`bf4ee95df4ee6e3101a88f5d015c8aa85b688b6d` credited for its puncture and
+baseline decoder. This result covers the p=25 root-pair family as a whole; it
+does not audit or extend that fixed p=26 completion. The pre-submission refresh
+also found open PR #2232 (head `3391f27dd0f226f0e1ead8b2b829eb6d3ec567d3`), an
+E6 rank-four projected-Gram screen with an adapted pairing. It uses 36 E6 lines
+and four-term projections, not the E8 p=25 root-pair support family here. Open
+PR #2228 (head `ad1757e319225f3a5b86e3d3fa72897bd9f6db40`) gives a general dense
+root-code floor and lists E8, P112 and P104 cases; its broader framework is
+adjacent, while this package contributes the exact p=25 extremal-mask census
+and an eight-term dual-root COPY bound. No claim is made that this result
+exceeds the broader root-code theorem. The active ten-term P112 gauge, cyclic
+mask and rank-six schedule claims use different objects.
+
+The scope claim and handoff are tracked at
+[Discussion #1167 comment 18937692](https://github.com/CrocSwap/integer-mult-bounds/discussions/1167#discussioncomment-18937692).
+The focused package is submitted for review as
+[PR #2237](https://github.com/CrocSwap/integer-mult-bounds/pull/2237).
+
+## Reproduce
+
+From the repository root:
+
+```sh
+python3 -I -B research/p25-root-pair-bound-seed632639/verify.py
+python3 scripts/research_package.py --package research/p25-root-pair-bound-seed632639 --check-only
+python3.12 scripts/research_package.py --package research/p25-root-pair-bound-seed632639
+```
+
+To regenerate the deterministic certificate, run the verifier with `--write`.
+The implementation uses exact integer and rational arithmetic, checks input
+hashes and Git object pins, enumerates the complete functional-pair domain, and
+recomputes the moment bracket with the pinned standard-library ceiling engine.
+
+## Open next questions
+
+Can a non-root dual basis or an overcomplete/masked decoder at p=25 avoid the
+one-exceptional-root restriction and pass the fully charged local moment gate?
+That would test a genuinely broader support mechanism. The first step is to
+choose one explicit non-root support alphabet, derive its exact support-rank
+floor, and screen the complete profile before building a local network. No
+source-aligned frame, hosting, precision, or restoration invoice for such a
+family is presently identified.
+
+## Credit
+
+The E8 root and label model, p≤24 argument, selected ceiling, and p=26 fixed
+completion retain their original PR and author credit in the pinned sources.
+Prepared by Cinna Davis with substantial OpenAI Codex assistance in scope
+selection, exact enumeration, cost screening, verifier implementation, and
+writing. No independent human or proof-assistant review is claimed. New work
+is Apache-2.0.
diff --git a/research/p25-root-pair-bound-seed632639/SOURCES.json b/research/p25-root-pair-bound-seed632639/SOURCES.json
@@ -0,0 +1,96 @@
+{
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