diff --git a/research/dyadic-horner-direction/CLAIM.json b/research/dyadic-horner-direction/CLAIM.json
new file mode 100644
index 0000000000..e00855109b
--- /dev/null
+++ b/research/dyadic-horner-direction/CLAIM.json
@@ -0,0 +1,141 @@
+{
+  "assignment_seed": 39,
+  "drawn_at_utc": "2026-10-13T06:23:11.445790+00:00",
+  "claimed_at_utc": "2026-10-13T06:33:36.779096+00:00",
+  "track": "complex supplier",
+  "base_commit": "7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261",
+  "base_tree": "2bb27541ee289e7a650eb0559abc3df09754059e",
+  "candidate_selection": "39 modulo 3 = 0",
+  "candidates": [
+    {
+      "object": "Each dyadic coefficient in #626 CK/JP8 and current #741 SW4",
+      "method": "Compare binary and NAF Horner in both digit directions with the original per-macro private boundary",
+      "chosen": true
+    },
+    {
+      "object": "Unequal-numerator common-source transitions",
+      "method": "One-read affine updates",
+      "chosen": false,
+      "reason": "#739/#811 already study coefficient routes; avoid repeating their search"
+    },
+    {
+      "object": "Restored target panels for CK",
+      "method": "Signed target-basis forest",
+      "chosen": false,
+      "reason": "#816 already supplies a construction and exact class floor"
+    }
+  ],
+  "scope": "Construct and exactly replay bidirectional per-coefficient Horner words. Six finite local scalar programs; complete primitive invoices, arbitrary private input, fresh positive prefix bounds. No cross-macro sharing, production integration, or new kappa.",
+  "expected_deliverable": "Focused finite circuit construction, proof, exact certificate, source pins, independent arithmetic crosscheck and negative controls",
+  "expected_finish": "Within approximately 90 minutes of the claim; bounded to the six pinned programs and a finite coefficient crosscheck",
+  "coordination": "Only open non-PR issue #650 requests Discussions; no sprint coordination issue was available. Local claim under gist instructions.",
+  "open_pr_snapshot_count": 633,
+  "related_prs": {
+    "816": {
+      "title": "Research: signed CK target forests save 66,520 primitives with an exact class floor (no new kappa)",
+      "head": "133735a2cc2319c12ba5d786214b2217e083e7da",
+      "author": "dd-hemant",
+      "final_head": "133735a2cc2319c12ba5d786214b2217e083e7da"
+    },
+    "815": {
+      "title": "Research: exact frame-only search shows #626 providers are frame-optimal (negative result)",
+      "head": "52175dd28867aa3766730e02a4e253b887a3a2ab",
+      "author": "Rohansguliani",
+      "final_head": "d8c8b40ab0a95b6d77389e6726d86dddb60fb53c"
+    },
+    "811": {
+      "title": "Research: JP8 additive coefficient routes save 2,784 beyond rescaling (no new kappa)",
+      "head": "b5c244c916dd0b30957332edad7ae7ae22b0eb9d",
+      "author": "Ryan-Haines",
+      "final_head": "b5c244c916dd0b30957332edad7ae7ae22b0eb9d"
+    },
+    "807": {
+      "title": "Research: restored CK helper pairs save 35,872 primitives per orientation (no new \u03ba)",
+      "head": "b1a5c533a22c2665600102d872f6a84e0664a70d",
+      "author": "davidawad",
+      "final_head": "b1a5c533a22c2665600102d872f6a84e0664a70d"
+    },
+    "793": {
+      "title": "Research: batch JP8 compensation, saving 80,150 / 81,464 primitives (no new kappa)",
+      "head": "f9383f324ee86793c5ae7b73f8e39d5827a8aa52",
+      "author": "Varun-Mittal-1001",
+      "final_head": "f9383f324ee86793c5ae7b73f8e39d5827a8aa52"
+    },
+    "787": {
+      "title": "Research: compose complex coefficient lowering and cleanup sharing",
+      "head": "93ceb65cfbf339d7ca743614d4756a3af1440cd4",
+      "author": "bitplane",
+      "final_head": "93ceb65cfbf339d7ca743614d4756a3af1440cd4"
+    },
+    "769": {
+      "title": "Research: CK dyadic ladders save 13,770 / 67,146 primitives (no new kappa)",
+      "head": "e891b2b19e3185ec91bcf001e255af87f42389ce",
+      "author": "24kHandsome1201",
+      "final_head": "e891b2b19e3185ec91bcf001e255af87f42389ce"
+    },
+    "751": {
+      "title": "Research: save 19,688 JP8 backward primitives with reverse Horner (no new kappa)",
+      "head": "7a1a3a319da359ec8b6e83230a7fcf7733fac8e5",
+      "author": "thakrarsagar",
+      "final_head": "7a1a3a319da359ec8b6e83230a7fcf7733fac8e5"
+    },
+    "739": {
+      "title": "Research construction: rescale JP8 coefficient work across magnitudes (no new kappa)",
+      "head": "abb9825202dba6f9bc95c7df8d12d86088fdff1a",
+      "author": "namredips",
+      "final_head": "abb9825202dba6f9bc95c7df8d12d86088fdff1a"
+    },
+    "734": {
+      "title": "Research: prefix-bounded signed-Horner circuits for CK and JP8 (no new kappa)",
+      "head": "ba76ba3321d000c8996b0b254bcf0d7260e1bbb6",
+      "author": "AllStars101-sudo",
+      "final_head": "ba76ba3321d000c8996b0b254bcf0d7260e1bbb6"
+    },
+    "723": {
+      "title": "Research: save 3,276 JP8 scalar primitives per orientation with signed digits",
+      "head": "d3bc41b87fc7d23f50d064a0f06f1a89e3ab02a7",
+      "author": "thakrarsagar",
+      "final_head": "d3bc41b87fc7d23f50d064a0f06f1a89e3ab02a7"
+    },
+    "696": {
+      "title": "Research: lower selected complex scalar primitives by 22\u201325% (no new kappa)",
+      "head": "5a3f983715027c4fd83f50bc9001d4b18463bed1",
+      "author": "thakrarsagar",
+      "final_head": "5a3f983715027c4fd83f50bc9001d4b18463bed1"
+    },
+    "830": {
+      "title": "Research: transposed JP8 work saves 32,810 backward primitives (no new kappa)",
+      "head": "6a0b8d55bb8026f8b6d708ae07740453b41c872f",
+      "author": "Ryan-Haines",
+      "relation": "Transposed cross-macro #811 work. This contribution preserves each individual coefficient boundary; no combined saving claimed."
+    }
+  },
+  "status": "submitted",
+  "final_overlap_check": {
+    "checked_at_utc": "2026-10-13T06:46:00.251632+00:00",
+    "open_pr_count": 646,
+    "latest_number": 833,
+    "upstream_head": "7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261"
+  },
+  "result": {
+    "extra_real_saved_per_orientation": {
+      "ck": 928,
+      "jp8": 810,
+      "sw4": 312
+    },
+    "programs": 6,
+    "coefficients": 962,
+    "negative_controls": 14,
+    "independent_grid_inputs": 13286,
+    "kappa": "unchanged",
+    "pr_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/837"
+  },
+  "validated_at_utc": "2026-10-13T06:47:36.745728+00:00",
+  "publication_refresh": {
+    "upstream_head": "f567884aeb8851b6e794af1184119a3bfbe1e631",
+    "new_selected_pr": 799,
+    "new_selected_kappa": "1661634653669567/1000000000000000000",
+    "scope": "#741 has become a checkpoint during submission. This six-program result retains its pinned #741/#626 scope; no transport to #799 is claimed.",
+    "record_relocation": "The identical pinned #741 record is accepted at its new official checkpoint path; its hash and manifest binding are still enforced."
+  }
+}
diff --git a/research/dyadic-horner-direction/EXPECTED.json b/research/dyadic-horner-direction/EXPECTED.json
new file mode 100644
index 0000000000..3ba3527e51
--- /dev/null
+++ b/research/dyadic-horner-direction/EXPECTED.json
@@ -0,0 +1,18386 @@
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+        "32768": 3,
+        "32769": 1,
+        "33/4": 2,
+        "3552": 1,
+        "3584": 1,
+        "36": 1,
+        "3839/4": 1,
+        "384": 2,
+        "39/4": 1,
+        "4": 129,
+        "4064": 3,
+        "4080": 2,
+        "4096": 3,
+        "40992": 1,
+        "4113": 1,
+        "4113/2": 1,
+        "4128": 1,
+        "496": 2,
+        "5": 4,
+        "5/2": 9,
+        "5/4": 6,
+        "510": 2,
+        "512": 9,
+        "517/2": 1,
+        "517/4": 1,
+        "576": 1,
+        "6": 5,
+        "62": 1,
+        "63": 1,
+        "63488": 1,
+        "64": 13,
+        "640": 1,
+        "65504": 2,
+        "65534": 3,
+        "65536": 5,
+        "65539/4": 1,
+        "65552": 1,
+        "65568": 2,
+        "65823": 1,
+        "69696": 1,
+        "7": 2,
+        "7/2": 7,
+        "72": 1,
+        "768": 3,
+        "8": 35,
+        "8160": 1,
+        "8192": 5,
+        "8195/2": 1,
+        "8196": 1,
+        "8220": 1,
+        "8221/2": 1,
+        "8256": 1,
+        "896": 1,
+        "9/2": 12,
+        "9/4": 2,
+        "96": 1,
+        "98289": 1,
+        "992": 1
+      },
+      "dirty_helpers_restored": 504,
+      "events": 24403,
+      "improved_by_reverse": 148,
+      "logical_columns": 728,
+      "lowered_stream_sha256": "8d17f62d1e049aec1e3504c6b64768dceacc8b63660bfe16a7e443cf9b7225c9",
+      "macros": 19197,
+      "new_positive_prefix": "24712511549002215298927/8",
+      "new_prefix_bits": 72,
+      "original_positive_prefix": "48299247699832632898063/16",
+      "paid_profile": {
+        "1": 2556,
+        "2": 334,
+        "3": 430,
+        "4": 194,
+        "5": 56,
+        "8": 8
+      },
+      "primitive_kinds_per_real_component": {
+        "add": 9455,
+        "clear": 18364,
+        "copy": 11928,
+        "double": 6883,
+        "halve": 14233,
+        "negate": 648,
+        "subtract": 9609
+      },
+      "private_columns": 9,
+      "private_streams_cleared": true,
+      "real_primitives": {
+        "bidirectional": 142240,
+        "copy_binary": 143674,
+        "forward_best": 142552,
+        "original": 193210
+      },
+      "saved_beyond_forward": 312
+    },
+    "sw4/forward": {
+      "charged_payload_bits": 192,
+      "chosen_methods": {
+        "binary-forward": 18939,
+        "binary-reverse": 148,
+        "naf-forward": 110
+      },
+      "coefficient_occurrences": {
+        "-1": 2764,
+        "-1/1024": 1,
+        "-1/16": 1,
+        "-1/2": 2780,
+        "-1/4": 1925,
+        "-1/8": 3,
+        "-1023/4": 1,
+        "-1024": 2,
+        "-1025/4": 1,
+        "-111": 1,
+        "-111/2": 1,
+        "-12": 4,
+        "-123": 1,
+        "-123/2": 1,
+        "-125": 1,
+        "-125/2": 1,
+        "-126": 1,
+        "-126976": 1,
+        "-128": 13,
+        "-1280": 3,
+        "-129": 1,
+        "-129/2": 1,
+        "-13/2": 4,
+        "-131/2": 1,
+        "-131/4": 1,
+        "-131008": 2,
+        "-131068": 2,
+        "-131072": 4,
+        "-131076": 1,
+        "-131104": 1,
+        "-131136": 2,
+        "-131646": 1,
+        "-139392": 1,
+        "-14": 1,
+        "-14336": 1,
+        "-1535/4": 1,
+        "-1541/4": 1,
+        "-16": 16,
+        "-16384": 4,
+        "-17": 2,
+        "-17/2": 3,
+        "-1776": 1,
+        "-1792": 1,
+        "-1793/4": 1,
+        "-19": 1,
+        "-19/2": 2,
+        "-19/4": 1,
+        "-192": 2,
+        "-1920": 1,
+        "-196578": 1,
+        "-2": 754,
+        "-2032": 3,
+        "-2040": 1,
+        "-2048": 1,
+        "-20496": 1,
+        "-24": 3,
+        "-248": 1,
+        "-251/2": 1,
+        "-251/4": 1,
+        "-255": 3,
+        "-255/2": 1,
+        "-256": 11,
+        "-2560": 2,
+        "-257/4": 1,
+        "-261/2": 1,
+        "-261/4": 1,
+        "-262141/4": 1,
+        "-262143/2": 1,
+        "-262144": 2,
+        "-262152": 1,
+        "-28": 1,
+        "-28672": 1,
+        "-288": 1,
+        "-29/2": 1,
+        "-3": 25,
+        "-3/2": 36,
+        "-3/4": 28,
+        "-31": 1,
+        "-31/4": 1,
+        "-32": 12,
+        "-32767": 1,
+        "-32768": 3,
+        "-32769": 1,
+        "-33/4": 2,
+        "-3552": 1,
+        "-3584": 1,
+        "-36": 1,
+        "-3839/4": 1,
+        "-384": 2,
+        "-39/4": 1,
+        "-4": 129,
+        "-4064": 3,
+        "-4080": 2,
+        "-4096": 3,
+        "-40992": 1,
+        "-4113": 1,
+        "-4113/2": 1,
+        "-4128": 1,
+        "-496": 2,
+        "-5": 4,
+        "-5/2": 9,
+        "-5/4": 6,
+        "-510": 2,
+        "-512": 9,
+        "-517/2": 1,
+        "-517/4": 1,
+        "-576": 1,
+        "-6": 5,
+        "-62": 1,
+        "-63": 1,
+        "-63488": 1,
+        "-64": 13,
+        "-640": 1,
+        "-65504": 2,
+        "-65534": 3,
+        "-65536": 5,
+        "-65539/4": 1,
+        "-65552": 1,
+        "-65568": 2,
+        "-65823": 1,
+        "-69696": 1,
+        "-7": 2,
+        "-7/2": 7,
+        "-72": 1,
+        "-768": 3,
+        "-8": 35,
+        "-8160": 1,
+        "-8192": 5,
+        "-8195/2": 1,
+        "-8196": 1,
+        "-8220": 1,
+        "-8221/2": 1,
+        "-8256": 1,
+        "-896": 1,
+        "-9/2": 12,
+        "-9/4": 2,
+        "-96": 1,
+        "-98289": 1,
+        "-992": 1,
+        "1": 3648,
+        "1/16": 1,
+        "1/2": 2734,
+        "1/4": 2108,
+        "1/8": 1,
+        "10": 1,
+        "1020": 1,
+        "1024": 3,
+        "1025/4": 1,
+        "1056": 1,
+        "11": 1,
+        "11/2": 3,
+        "11/4": 1,
+        "12": 3,
+        "12224": 1,
+        "12256": 1,
+        "12352": 1,
+        "124": 1,
+        "128": 15,
+        "1280": 1,
+        "13/4": 1,
+        "130626": 1,
+        "131040": 1,
+        "131055": 1,
+        "131059": 1,
+        "131068": 1,
+        "131072": 1,
+        "131073/2": 1,
+        "131076": 1,
+        "132352": 1,
+        "136": 1,
+        "14": 1,
+        "15": 2,
+        "1536": 3,
+        "1552": 1,
+        "16": 21,
+        "16352": 1,
+        "16382": 1,
+        "16384": 1,
+        "1680": 1,
+        "17": 1,
+        "192": 1,
+        "196646": 1,
+        "2": 508,
+        "20": 2,
+        "2048": 3,
+        "21/2": 1,
+        "2112": 1,
+        "24": 2,
+        "2448": 1,
+        "248": 1,
+        "255": 1,
+        "256": 16,
+        "257": 2,
+        "257/2": 1,
+        "262111/4": 1,
+        "262115/4": 1,
+        "272": 1,
+        "28": 2,
+        "28416": 1,
+        "3": 14,
+        "3/2": 30,
+        "3/4": 41,
+        "30": 2,
+        "31/4": 1,
+        "3104": 1,
+        "31505": 1,
+        "32": 12,
+        "32015": 1,
+        "32144": 1,
+        "32257": 1,
+        "32752": 1,
+        "32764": 1,
+        "32768": 3,
+        "32769": 1,
+        "33/4": 1,
+        "33008": 1,
+        "33423": 1,
+        "33535": 1,
+        "33535/2": 1,
+        "3360": 1,
+        "34": 1,
+        "35/2": 1,
+        "36": 1,
+        "3616": 1,
+        "36831": 1,
+        "36831/2": 1,
+        "384": 2,
+        "4": 164,
+        "40": 1,
+        "4096": 3,
+        "4112": 2,
+        "4896": 1,
+        "5": 4,
+        "5/2": 7,
+        "5/4": 2,
+        "510": 2,
+        "512": 8,
+        "514": 1,
+        "56": 2,
+        "56832": 1,
+        "6": 10,
+        "6112": 1,
+        "6128": 1,
+        "6176": 1,
+        "63010": 1,
+        "64": 8,
+        "640": 1,
+        "64030": 1,
+        "64288": 1,
+        "64514": 1,
+        "65313": 1,
+        "65504": 1,
+        "65520": 1,
+        "65534": 1,
+        "65536": 2,
+        "65538": 2,
+        "66016": 1,
+        "66176": 1,
+        "66848": 1,
+        "7": 5,
+        "7/2": 14,
+        "7232": 1,
+        "768": 4,
+        "8": 46,
+        "80": 1,
+        "8156": 1,
+        "8157/2": 1,
+        "8176": 1,
+        "8192": 1,
+        "8195/2": 1,
+        "8196": 1,
+        "8224": 2,
+        "9": 1,
+        "9/2": 9,
+        "9/4": 1,
+        "98323": 1
+      },
+      "dirty_helpers_restored": 504,
+      "events": 24403,
+      "improved_by_reverse": 148,
+      "logical_columns": 728,
+      "lowered_stream_sha256": "bf021866a6de4ede1ae652e91ad4f8de603415dc71d4b24b10969f2ae0c993ac",
+      "macros": 19197,
+      "new_positive_prefix": "27605459502654407221/1024",
+      "new_prefix_bits": 55,
+      "original_positive_prefix": "6812058044968180837/256",
+      "paid_profile": {
+        "1": 2556,
+        "2": 334,
+        "3": 430,
+        "4": 194,
+        "5": 56,
+        "8": 8
+      },
+      "primitive_kinds_per_real_component": {
+        "add": 10141,
+        "clear": 18356,
+        "copy": 11936,
+        "double": 6883,
+        "halve": 14233,
+        "negate": 648,
+        "subtract": 8915
+      },
+      "private_columns": 9,
+      "private_streams_cleared": true,
+      "real_primitives": {
+        "bidirectional": 142224,
+        "copy_binary": 143658,
+        "forward_best": 142536,
+        "original": 193162
+      },
+      "saved_beyond_forward": 312
+    }
+  },
+  "status": "finite coefficient circuit construction; no new kappa"
+}
diff --git a/research/dyadic-horner-direction/README.md b/research/dyadic-horner-direction/README.md
new file mode 100644
index 0000000000..60ab2b4065
--- /dev/null
+++ b/research/dyadic-horner-direction/README.md
@@ -0,0 +1,184 @@
+# Bidirectional Horner for individual dyadic coefficients
+
+**Finite complex-supplier circuit construction; no new κ.** Building a
+fractional coefficient from its low digits can avoid the numerator doublings
+followed by denominator halvings in conventional Horner lowering. This package
+chooses the cheaper direction separately for each unchanged scalar macro.
+It preserves clearing after every addition, including arbitrary incoming work.
+
+The new saving, beyond copy/unit and forward signed-digit lowering, is **928
+real primitives per CK orientation, 810 per JP8 orientation, and 312 per SW4
+orientation**. SW4 was the #741 selected supplier at the starting base; CK and
+JP8 belong to the preserved #626 checkpoint. These are circuit counts, not
+runtime measurements. Earlier improvements receive separate columns:
+
+| Program | Original | Copy/unit (#696) | Best forward binary/NAF | Bidirectional | New saving |
+|---|---:|---:|---:|---:|---:|
+| CK forward | 224,938 | 169,618 | 169,416 | 168,488 | 928 |
+| CK backward | 224,986 | 169,634 | 169,432 | 168,504 | 928 |
+| JP8 forward | 257,232 | 200,380 | 197,104 | 196,294 | 810 |
+| JP8 backward | 257,280 | 200,396 | 197,120 | 196,310 | 810 |
+| SW4 forward | 193,162 | 143,658 | 142,536 | 142,224 | 312 |
+| SW4 backward | 193,210 | 143,674 | 142,552 | 142,240 | 312 |
+
+The first two reference columns are inherited methods, freshly applied here
+to the #741 scalar word as well as the checkpoint words. Forward NAF is the
+credited #723/#787 mechanism. This package makes no claim of discovering those
+savings. Cross-macro sharing or target/helper basis changes are not composed
+into this table.
+
+## Reproduce
+
+Python 3.11+ and the standard library; offline, no writes by default:
+
+```sh
+python3 -B -I research/dyadic-horner-direction/verify.py
+python3 -B -I research/dyadic-horner-direction/crosscheck.py
+python3 -B -I research/dyadic-horner-direction/test_verify.py
+```
+
+Expected first two final lines:
+
+```text
+PASS_DYADIC_HORNER_DIRECTION programs=6 coefficients=962 extra_saved_per_orientation=928,810,312 controls=14 kappa=unchanged
+PASS_INTEGER_HORNER_CROSSCHECK coefficients=962 grid_cases=13286 input_columns=3 costs_and_prefixes=exact
+```
+
+The last command runs eight tests. `verify.py --write /new/path.json`
+reconstructs the receipt instead of comparing it, and refuses an existing
+destination. The default command checks every committed proof-relevant field
+in [EXPECTED.json](EXPECTED.json), including integer versus floating JSON
+representations. [VALIDATION.md](VALIDATION.md) records executed checks.
+
+## Construction and exact cost
+
+Consider `t += q*s`, with distinct target, source and work. Write the nonzero
+dyadic as `q = sign * a * 2^e`, with positive odd `a`. Let
+`a = sum(d_i * 2^i, i=0..h)`, using binary or non-adjacent signed digits.
+Then `d_0` is ±1. Initialize `w = s` by a charged copy. For increasing
+`i=1..h`, halve `w`, then add `d_0*d_i*s` when `d_i` is nonzero. After step i,
+
+```text
+w = d_0 * (sum(d_j * 2^j, j=0..i)) / 2^i * s.
+```
+
+The assertion holds initially since `d_0^2=1`; a halving and the next signed
+source addition establish the induction. Scale by `2^(h+e)` using charged
+doublings or halvings. Accumulate `sign*d_0*w` into t, then clear w. The
+result is exactly `(t+q*s, s, 0)` for arbitrary initial `(t,s,w)`. No other
+register is touched. The sign correction is necessary: reverse NAF for 11/8
+starts with -1, so its final accumulation subtracts work for a positive q.
+Fourteen adverse controls include deleting that correction.
+
+If the expansion has m nonzero digits, the reverse word costs
+
+```text
+1 copy + h halvings + (m-1) signed reads
+       + |h+e| shifts + 1 target accumulation + 1 clear
+  = 2 + h + m + |h+e| real primitives.
+```
+
+For a reduced positive numerator A and denominator `2^k`, forward Horner
+with expansion height H and weight M costs `2+H+M+k`. For odd A this shows
+why reverse processing can help: it substitutes `|H-k|` for k. For example,
+3/2 uses `copy; halve; add source; add target; clear`, five primitives rather
+than six under forward copy-Horner. Integral even numerators are reduced to
+their odd part only for the reverse construction; their extracted exponent
+is charged in the final scaling. Unit coefficients keep #696's two-gate
+accumulation/clear circuit.
+
+The four candidates are forward binary, forward NAF, reverse binary and
+reverse NAF. The first shortest word in that order wins. Consequently this
+finite menu cannot raise the scalar primitive count relative to the credited
+forward baseline. No shortest-chain or general circuit optimality is claimed.
+Every real-component gate is charged; the complex bill is twice that count.
+No additional private register, logical helper, frame transition, wrapper or
+recursive child is required. All emitted gates retain their macro's frame.
+
+## Evidence and intermediate bounds
+
+[verify.py](verify.py) executes every primitive in all six fixed programs,
+using exact sparse rational rows. The #626 words have 716 arbitrary logical
+inputs, including 492 dirty helpers. SW4 has 728, including 504 dirty helpers.
+Each replay also starts all eight private centres and the coefficient stream
+with nine independent arbitrary inputs. Every macro boundary agrees, every
+logical endpoint agrees, every helper is restored and every private stream
+is erased. The unchanged paid event histograms and original invoices agree
+with their pinned SPECs.
+
+The checker independently propagates positive triangle majorants through
+every primitive; it does not borrow a receipt for the changed words. The new
+forward/backward bounds are 40/62 bits for CK, 64/101 for JP8, and 55/72 for
+SW4. All are below the inherited 192-bit local allowance. Exact rational
+envelopes, operation histograms and framed stream digests are in the receipt.
+Signed cancellation is never used to understate those majorants.
+
+All 962 distinct coefficients receive arbitrary three-column checks for
+**every** available candidate, not just the chosen one. Reverse words win on
+36 distinct coefficients, appearing in 415 CK, 393 JP8 and 148 SW4 additions
+per orientation. Some coefficient sets overlap. The 14 controls cover signs,
+initialization, source integrity, halving, erasure, inadmissible coefficients,
+aliasing, cost undercharging and an endpoint-neutral prefix excursion.
+
+[crosscheck.py](crosscheck.py) uses dense integer vectors with binary
+denominators instead of the primary rational-row interpreter. It derives
+NAF height/weight from shifted bits of `3A` and A, independently of the
+lowerer's modular digit recurrence, and checks the closed cost formulas.
+It agrees on all receipt coefficients and the fixed 13,286-input grid
+`q=±n/2^k`, `1<=n<=511`, `0<=k<=12`, including reducible inputs. This is a
+completed finite crosscheck, not a universal optimality search. Both
+implementations were written by one AI agent, not separate human reviewers.
+
+## Pins, predecessors and scope
+
+Exact base: **7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261**. Selected #741 head:
+`e516b1463606840a15f80aac1995d64ce0092e89`. Historical #626 head:
+`660de893d27bb7abd1d5b01b1640207528a5c851`, from #622 at
+`f69477f4f6d3c7eb1f740b166881bffed4ffa9f6`.
+[SOURCES.json](SOURCES.json) binds both reviewed records to their manifests,
+SPECs and consumed program payloads. SW4 inputs are read directly from the
+committed maintainer replay archive without extraction or code execution.
+Every consumed gzip member is compared after decompression. The original
+archive container hash is retained as provenance; harmless repackaging is
+accepted. This does not cold-rebuild the supplier construction chain.
+
+During publication, main advanced to
+`f567884aeb8851b6e794af1184119a3bfbe1e631` and selected #799 at
+κ = 0.001661634653669567. The six programs here retain their #741/#626 scope;
+no transport to #799 is claimed. The unchanged #741 record is now archived at
+`certificates/checkpoints/frontier741.json`. The verifier accepts either
+official location only with the same exact pinned bytes and manifest binding.
+Its complete receipt also passes using the actual inputs from refreshed main.
+
+Sagar Thakrar's **#696** copy/unit lowering at
+`5a3f983715027c4fd83f50bc9001d4b18463bed1` is vendored verbatim with its notice.
+Its Git blob is `a2c2889e5be41a0fc30d4dfa5dfcebb03d6d47f8`. Credit his
+**#723** for forward signed digits, Chris Pagolu's **#734** for a bounded
+forward-Horner class and **our #787** for the replay/NAF implementation
+adapted here (`93ceb65cfbf339d7ca743614d4756a3af1440cd4`).
+
+Sagar's **#751** uses reverse Horner to collect different sources at a shared
+target across macros. This contribution reverses the digit direction *within
+one coefficient macro* and retains its original clearing boundary. **#739,
+#769 and #811** reuse coefficients across reads; **#793/#807/#816** alter
+source, helper or target panels. Their savings are not assumed additive.
+The submission refresh also inspected Ryan Haines's **#830**, which
+transposes #811's cross-macro work into backward JP8 batches; that different
+boundary contract and its savings are not combined here.
+[CLAIM.json](CLAIM.json) records the inspected heads, three candidates, OS
+seed **39**, finite scope and local claim in the absence of a sprint issue.
+
+Credit icekylinx's primitive/emitter lineage, DreamingOfClouds, Jeremy Kothe,
+Chafik Boukhalfa, Rohan Arun, eumemic and the retained contributors to #622,
+#626 and the #741 chain. Their source and Apache-2.0 notices are retained.
+Prepared for **Gareth Davidson (@bitplane) with substantial OpenAI Codex
+assistance**. No independent human review is claimed.
+
+This contribution does not change the selected κ. Production lowering,
+global invoices/admission and any frontier promotion are separate work.
+Native geometry, ordered phase/wrapper identities and global ownership are
+inherited from the pinned construction; their expensive checks are not rerun
+here. Uniform compilation, common-label/mutable-source realization, physical
+hosting, recursive interfaces, precision/tape and analytic transfer remain
+assumptions. This exact finite scalar replay is not a proof of the complete
+multiplication theorem. No selected artifact or `upstream/` file is edited.
diff --git a/research/dyadic-horner-direction/SOURCES.json b/research/dyadic-horner-direction/SOURCES.json
new file mode 100644
index 0000000000..9825196b85
--- /dev/null
+++ b/research/dyadic-horner-direction/SOURCES.json
@@ -0,0 +1,90 @@
+{
+  "base_commit": "7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261",
+  "base_tree": "2bb27541ee289e7a650eb0559abc3df09754059e",
+  "pr626_head": "660de893d27bb7abd1d5b01b1640207528a5c851",
+  "pr622_head": "f69477f4f6d3c7eb1f740b166881bffed4ffa9f6",
+  "pr741_head": "e516b1463606840a15f80aac1995d64ce0092e89",
+  "pr696_head": "5a3f983715027c4fd83f50bc9001d4b18463bed1",
+  "pr696_lower_git_blob": "a2c2889e5be41a0fc30d4dfa5dfcebb03d6d47f8",
+  "pr787_replay_origin": "93ceb65cfbf339d7ca743614d4756a3af1440cd4",
+  "files": [
+    {
+      "path": "research/complex-k492-owner-repair/data/SPEC.json",
+      "encoding": "file",
+      "sha256": "2a2577ac526535826fed9d5f7f83f6e2e0ec058a146c64f7286720b995fa2323"
+    },
+    {
+      "path": "research/complex-k492-owner-repair/code/hpe_emit_local.py",
+      "encoding": "file",
+      "sha256": "7e4a6fbc38c8228157b6311457d1496bf460291393043932d2c365f004f77805"
+    },
+    {
+      "path": "research/complex-k492-owner-repair/code/hpe_audit_local.py",
+      "encoding": "file",
+      "sha256": "6bd9a1e279c9c9e4f3767d9f82e537b559292906feb2ddab02f380fb6523a810"
+    },
+    {
+      "path": "research/complex-k492-owner-repair/code/invoice_allcold.py",
+      "encoding": "file",
+      "sha256": "141bd2fd58bbfb5b8d0fc8f56f61c45586d5608a5af3487c6b883dd987f998ec"
+    },
+    {
+      "path": "research/community-k492-audit/pr626/replay/mixed-global/base/providers/ck/LOCAL-PROGRAMS.json.gz",
+      "encoding": "gzip-payload",
+      "sha256": "854f17274092263dc25e60367f4cd0d29f60a8446becd17cc1883ebed7a97900"
+    },
+    {
+      "path": "research/community-k492-audit/pr626/replay/mixed-global/base/providers/ck/CANONICAL-WORD.json.gz",
+      "encoding": "gzip-payload",
+      "sha256": "9bbc75dede090cfd7a0a0252424ea9d9f6211e2350d2a801d4e4f2f0784167c7"
+    },
+    {
+      "path": "research/community-k492-audit/pr626/replay/mixed-global/base/providers/jp8/LOCAL-PROGRAMS.json.gz",
+      "encoding": "gzip-payload",
+      "sha256": "efe45ce7fb025a6f53cc9eb54e951ac2211a5ed927e689a049e4f68ca680654b"
+    },
+    {
+      "path": "research/community-k492-audit/pr626/replay/mixed-global/base/providers/jp8/CANONICAL-WORD.json.gz",
+      "encoding": "gzip-payload",
+      "sha256": "fb4a8d14ed2a8048698bea6ae16f86163a1aa6c80f2eb622222eee5cdbce9b3b"
+    },
+    {
+      "path": "research/community-k492-audit/SELECTED-SOURCE.json",
+      "encoding": "file",
+      "sha256": "36e9e421230c4cb7b6a537d76a4bdfb6f667730d383e56062d4e45957484ab9e"
+    },
+    {
+      "path": "certificates/checkpoints/k492626.json",
+      "encoding": "file",
+      "sha256": "b3bf402f31a69bde0629936a9b569bed48281568d7b16d5fba120b5311ea1f53"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "encoding": "file",
+      "sha256": "8f2bfbf11609194654d4da145866223d4c546c5418727cae056bfb9968cabcac",
+      "archived_paths": [
+        "certificates/checkpoints/frontier741.json"
+      ]
+    },
+    {
+      "path": "research/community-frontier741-audit/SELECTED-SOURCE.json",
+      "encoding": "file",
+      "sha256": "8bf379a8a3d1a34c76fd4c868e1d125e78360f02066553a8d1a30d3bc96f6d8b"
+    },
+    {
+      "path": "research/complex-sw4-frame8/data/SPEC.json",
+      "encoding": "file",
+      "sha256": "7fc2bd2e3c0ac13729c1fa1f0f327a4eb7c72373ba58e5ed3954b30e23c803be"
+    },
+    {
+      "path": "research/dyadic-horner-direction/vendor/pr696_lower.py",
+      "encoding": "file",
+      "sha256": "eb37d00063dc09323e9aca041d23da749ce17fbd71dade9aaf1c059cab0b9753"
+    }
+  ],
+  "archive": {
+    "path": "research/community-frontier741-audit/composed-replay-evidence.tar.gz",
+    "committed_container_sha256": "27516447e242c264030ba427dabc5e616f9a4aacbff9e0623db4d00c058e1a7b",
+    "comparison": "Only the two consumed member payloads are gated, against the source-bound SPEC. The immutable committed container digest is recorded for provenance. Repackaging is accepted."
+  }
+}
diff --git a/research/dyadic-horner-direction/VALIDATION.md b/research/dyadic-horner-direction/VALIDATION.md
new file mode 100644
index 0000000000..2add5f4551
--- /dev/null
+++ b/research/dyadic-horner-direction/VALIDATION.md
@@ -0,0 +1,76 @@
+# Validation record
+
+Executed on Linux on 2026-10-13, from base
+`7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261`. This record describes local
+execution. Hosted GitHub checks have not been claimed as passing.
+
+| Check | Result |
+|---|---|
+| CPython 3.14.4, isolated normal primary verifier | Six programs, 962 coefficients, 14 controls pass; deterministic receipt reconstructed |
+| CPython 3.14.4, `-B -I -O` primary verifier | Complete committed receipt matches; all guards remain effective |
+| CPython 3.14.4, isolated optimized integer checker | All 962 coefficient records and 13,286 grid inputs agree |
+| CPython 3.14.4 focused tests | Eight tests pass, normally and with optimization |
+| `/usr/bin/python3.12`, 3.12.3, `-B -I -S` | Primary receipt, integer crosscheck and eight tests pass without site packages |
+| CPython 3.12.3 minimal relocation, `-B -I -O` | Complete primary verifier passes with only the package and pinned inputs, from a path containing spaces |
+| Source corruption | Appending whitespace to pinned SW4 SPEC fails the source digest check |
+| Receipt corruption | Adding two to a primitive invoice fails complete receipt comparison |
+| Typed receipt corruption | Replacing an integer event count with its equal-valued JSON float fails comparison |
+| Actual circuit mutation | Changing the reverse halving into doubling fails a scalar boundary check |
+| Gzip and tar repackaging | Different compression level/timestamps for CK programs and both consumed SW4 archive members passes |
+| Existing output file | `--write` fails before work; sentinel content is preserved |
+| `make selected-record-check` | Selected κ, original source inventory and proof supplement pass |
+
+The packaging mutations were applied one at a time to the disposable minimal
+checkout and restored between checks. Only the described source, circuit and
+receipt checks were deliberately corrupted. No production sources were edited.
+
+## Mathematical coverage
+
+The primary verifier freshly checks all scalar primitives in six orientations,
+including 716+9 or 728+9 arbitrary columns, complete endpoints, scratch erasure,
+unchanged paid-event histograms, original and replacement bills and positive
+prefix envelopes. All four coefficient candidates are checked wherever
+applicable. The integer implementation has a separate dense state model,
+binary-denominator arithmetic and closed-form cost computation. Its agreement
+is implementation diversity, not independent authorship or a formal theorem.
+
+The 14 primary controls and eight tests include arbitrary dirty work, the
+negative initial NAF digit, forbidden aliases, omitted reads/halves/clears,
+wrong signs, understated cost and an endpoint-neutral excessive prefix. The
+grid's stopping rule is exactly `1<=n<=511`, `0<=k<=12`, both signs; all
+13,286 inputs were completed. The main receipt covers the full coefficient
+census of the pinned programs separately from that grid.
+
+## State and overlap refresh
+
+The initial queue contained 633 open PRs through #820. Before packaging,
+646 open PRs through #833 were checked. Searches for fractional/bidirectional
+Horner, reversed digit order and related coefficient work found the known
+cross-macro #751 and new #830. Their scope differs as recorded in README.md.
+The #815 head changed to include a current-word frame search; it is neither
+imported nor used to establish this circuit result. Source and relation pins
+are recorded in CLAIM.json. Main still pointed to the exact base above.
+
+While publishing, upstream advanced to
+`f567884aeb8851b6e794af1184119a3bfbe1e631`, selecting #799. The complete
+verifier was rerun with the actual consumed files from that commit and its
+new selected pointer. It matches the original receipt. The #741 record moved
+to `certificates/checkpoints/frontier741.json` with exactly identical bytes;
+the source check now accepts that official archive path under the original
+digest. A corrupted archive record is rejected. This is a #741/#626 result,
+not a claim to optimize #799. [PR #837](https://github.com/CrocSwap/integer-mult-bounds/pull/837)
+is the single submission for this pass.
+
+Only issue #650 (requesting Discussions) was open outside PRs; no sprint
+coordination issue was available, so the scope/result claim is local.
+
+## Checks outside scope
+
+No full repository `make verify`, supplier cold regeneration, global
+admission/assembly, native phase or wrapper regeneration, Linux hosted CI or
+Lean run is claimed. These unchanged construction obligations remain inherited.
+The selected record and `upstream/` are unchanged. The result neither promotes
+a supplier nor claims a new κ or a practical runtime improvement.
+
+Prepared for Gareth Davidson (@bitplane) with substantial OpenAI Codex
+assistance. Apache-2.0.
diff --git a/research/dyadic-horner-direction/crosscheck.py b/research/dyadic-horner-direction/crosscheck.py
new file mode 100644
index 0000000000..e19fdef69a
--- /dev/null
+++ b/research/dyadic-horner-direction/crosscheck.py
@@ -0,0 +1,146 @@
+"""Independent integer-scaled three-column and cost-formula crosscheck.
+
+No import of the rational replay, its interpreter, or its digit routines.
+The emitted words are the object under test. Gareth Davidson (@bitplane),
+substantial OpenAI Codex assistance; Apache-2.0. Not independent authorship.
+"""
+
+import importlib.util
+import json
+from math import gcd
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+
+
+def require(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def normal(nums, exponent):
+    nums = tuple(nums)
+    while exponent and all(x % 2 == 0 for x in nums):
+        nums = tuple(x // 2 for x in nums)
+        exponent -= 1
+    return nums, exponent
+
+
+def add(a, b, sign=1):
+    av, ae = a
+    bv, be = b
+    e = max(ae, be)
+    return normal([x * 2**(e-ae) + sign * y * 2**(e-be) for x, y in zip(av, bv)], e)
+
+
+def scale(a, double):
+    nums, e = a
+    return normal([2*x for x in nums], e) if double else normal(nums, e+1)
+
+
+def greater(a, b):
+    return a[0][0] * 2**b[1] > b[0][0] * 2**a[1]
+
+
+def scalar_text(value):
+    n, e = value[0][0], value[1]
+    return str(n) if e == 0 else str(n) + "/" + str(2**e)
+
+
+def execute(word, numerator, denominator):
+    """Three dense integer columns, each with a shared binary denominator."""
+    t, s, w = ("T", 0), ("S", 0), ("W", 0)
+    state = {t: ((1, 0, 0), 0), s: ((0, 1, 0), 0), w: ((0, 0, 1), 0)}
+    majorant = dict.fromkeys(state, ((1,), 0))
+    peak = ((1,), 0)
+    for step in word:
+        tag, dest = step[:2]
+        require(dest in (t, w), "foreign/source write")
+        if tag == "clear":
+            state[dest], majorant[dest] = ((0, 0, 0), 0), ((0,), 0)
+        elif tag == "copy":
+            state[dest], majorant[dest] = state[step[2]], majorant[step[2]]
+        elif tag in ("add", "subtract"):
+            src = step[2]
+            state[dest] = add(state[dest], state[src], 1 if tag == "add" else -1)
+            majorant[dest] = add(majorant[dest], majorant[src])
+        elif tag in ("double", "halve"):
+            state[dest] = scale(state[dest], tag == "double")
+            majorant[dest] = scale(majorant[dest], tag == "double")
+        else:
+            raise ValueError("unexpected primitive")
+        if greater(majorant[dest], peak):
+            peak = majorant[dest]
+    expected = normal((denominator, numerator, 0), denominator.bit_length()-1)
+    require(state[t] == expected, "dense target endpoint mismatch")
+    require(state[s] == ((0, 1, 0), 0), "dense source modified")
+    require(state[w] == ((0, 0, 0), 0), "dense dirty work survives")
+    return scalar_text(peak)
+
+
+def statistics(n, signed):
+    if not signed:
+        return n.bit_length()-1, n.bit_count()
+    # NAF digits from the difference of shifted bits of 3n and n.
+    # This uses neither the lowerer's modular recurrence nor its digit list.
+    ds = [((3*n >> i) & 1) - ((n >> i) & 1) for i in range(1, (3*n).bit_length())]
+    require(sum(d * 2**i for i, d in enumerate(ds)) == n, "independent NAF identity")
+    return len(ds)-1, sum(d != 0 for d in ds)
+
+
+def formula(n, denominator):
+    n = abs(n)
+    if n == denominator:
+        return {"binary-forward": 2}
+    k = denominator.bit_length()-1
+    h, weight = statistics(n, False)
+    hn, wn = statistics(n, True)
+    out = {"binary-forward": 2+h+weight+k, "naf-forward": 2+hn+wn+k}
+    trailing = (n & -n).bit_length()-1
+    odd = n >> trailing
+    for signed, name in ((False, "binary-reverse"), (True, "naf-reverse")):
+        height, weight = statistics(odd, signed)
+        out[name] = 2 + height + weight + abs(height+trailing-k)
+    return out
+
+
+def check(lowering, n, d):
+    common = gcd(abs(n), d)
+    n, d = n//common, d//common
+    op = ["add", ["T", 0], ["S", 0], n, d]
+    words = lowering.candidates(op)
+    expected = formula(n, d)
+    costs = {name: len(word) for name, word in words.items()}
+    require(costs == expected, "closed-form cost mismatch")
+    peaks = {name: execute(word, n, d) for name, word in words.items()}
+    chosen = min(expected, key=expected.get)
+    require(lowering.choose(op)[0] == chosen, "tie/choice mismatch")
+    return costs, peaks, chosen
+
+
+def main():
+    spec = importlib.util.spec_from_file_location("tested_lowering", HERE / "lower.py")
+    lowering = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(lowering)
+    receipt = json.loads((HERE / "EXPECTED.json").read_bytes())
+    records = receipt["coefficient_certificates"]
+    for record in records:
+        pieces = record["coefficient"].split("/")
+        n, d = int(pieces[0]), int(pieces[1]) if len(pieces) == 2 else 1
+        costs, peaks, chosen = check(lowering, n, d)
+        require(costs == record["costs"] and peaks == record["positive_prefixes"] and
+                chosen == record["chosen"], "rational/integer receipt disagreement")
+    # A fixed domain includes both signs, reducible inputs, integral powers,
+    # fractions above/below one and negative leading reverse NAF digits.
+    cases = 0
+    for n in range(1, 512):
+        for k in range(13):
+            for sign in (-1, 1):
+                check(lowering, sign*n, 2**k)
+                cases += 1
+    print("PASS_INTEGER_HORNER_CROSSCHECK coefficients=" + str(len(records)) +
+          " grid_cases=" + str(cases) + " input_columns=3 costs_and_prefixes=exact")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/research/dyadic-horner-direction/lower.py b/research/dyadic-horner-direction/lower.py
new file mode 100644
index 0000000000..a7855d2a87
--- /dev/null
+++ b/research/dyadic-horner-direction/lower.py
@@ -0,0 +1,92 @@
+"""Both Horner directions for a single nonzero dyadic scalar macro.
+
+Gareth Davidson (@bitplane), with substantial OpenAI Codex assistance.
+Apache-2.0. Binary copy/unit lowering is Sagar Thakrar's verbatim PR696
+module. Forward NAF follows PR723 and our PR787; see README.md for pins.
+"""
+
+import importlib.util
+from fractions import Fraction as Q
+from pathlib import Path
+
+
+def load_reference():
+    spec = importlib.util.spec_from_file_location(
+        "pr696", Path(__file__).resolve().parent / "vendor/pr696_lower.py")
+    module = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(module)
+    return module
+
+
+reference = load_reference()
+
+
+def digits(n, signed):
+    """Least-significant-first binary or non-adjacent digits."""
+    if type(n) is not int or n <= 0:
+        raise ValueError("positive integer required")
+    result = []
+    while n:
+        d = (2 - n % 4 if signed else 1) if n % 2 else 0
+        result.append(d)
+        n = (n - d) // 2
+    return result
+
+
+def candidates(op, work=("W", 0)):
+    """Emit four circuits in stable tie order; each clears work on exit.
+
+    Non-addition macros are inherited unchanged. No circuit writes source,
+    borrows a logical register, or crosses the original macro boundary.
+    """
+    binary = reference.lower(op, work)  # Also validates dyadics and aliases.
+    result = {"binary-forward": binary}
+    if op[0] != "add" or abs(Q(op[3], op[4])) == 1:
+        return result
+    target, source = tuple(op[1]), tuple(op[2])
+    q = Q(op[3], op[4])
+    a, k = abs(q.numerator), q.denominator.bit_length() - 1
+    sign = 1 if q > 0 else -1
+
+    forward = [("copy", work, source)]
+    for d in reversed(digits(a, True)[:-1]):
+        forward.append(("double", work))
+        if d:
+            forward.append(("add" if d > 0 else "subtract", work, source))
+    forward.extend([("halve", work)] * k)
+    forward.extend([("add" if sign > 0 else "subtract", target, work), ("clear", work)])
+    result["naf-forward"] = forward
+
+    # Remove numerator trailing zeros so the initial reverse digit is +/-1.
+    odd, exponent = a, -k
+    while odd % 2 == 0:
+        odd //= 2
+        exponent += 1
+    for signed in (False, True):
+        ds = digits(odd, signed)
+        initial_sign = ds[0]
+        reverse = [("copy", work, source)]
+        for d in ds[1:]:
+            reverse.append(("halve", work))
+            if d:
+                reverse.append(("add" if d * initial_sign > 0 else "subtract", work, source))
+        shift = len(ds) - 1 + exponent
+        reverse.extend([("double" if shift > 0 else "halve", work)] * abs(shift))
+        reverse.extend([
+            ("add" if sign * initial_sign > 0 else "subtract", target, work),
+            ("clear", work),
+        ])
+        result["naf-reverse" if signed else "binary-reverse"] = reverse
+    return result
+
+
+def choose(op, work=("W", 0), reverse=True):
+    words = candidates(op, work)
+    if not reverse:
+        words = {k: v for k, v in words.items() if k.endswith("forward")}
+    name = min(words, key=lambda key: len(words[key]))
+    return name, words[name]
+
+
+def lower(op, work=("W", 0)):
+    return choose(op, work)[1]
diff --git a/research/dyadic-horner-direction/test_verify.py b/research/dyadic-horner-direction/test_verify.py
new file mode 100644
index 0000000000..dabfceac32
--- /dev/null
+++ b/research/dyadic-horner-direction/test_verify.py
@@ -0,0 +1,91 @@
+"""Focused mathematical and verifier failure-mode tests; Apache-2.0.
+
+Gareth Davidson (@bitplane), substantial OpenAI Codex assistance.
+"""
+
+import importlib.util
+import unittest
+from fractions import Fraction as Q
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+
+
+def load(name):
+    spec = importlib.util.spec_from_file_location(name, HERE / (name + ".py"))
+    module = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(module)
+    return module
+
+
+v, lower, independent = load("verify"), load("lower"), load("crosscheck")
+
+
+class DirectionTests(unittest.TestCase):
+    def test_elementary_fraction(self):
+        # 1 -> 1/2 -> 3/2, then accumulate and clear: five primitives.
+        op = ["add", ["T", 0], ["S", 0], 3, 2]
+        name, word = lower.choose(op)
+        self.assertEqual(name, "binary-reverse")
+        self.assertEqual(len(word), 5)
+        self.assertEqual(v.coefficient_check(Q(3, 2), word, 5), "5/2")
+        self.assertEqual(independent.execute(word, 3, 2), "5/2")
+
+    def test_negative_initial_naf_and_sign(self):
+        for sign in (-1, 1):
+            op = ["add", ["T", 0], ["S", 0], sign*11, 8]
+            word = lower.candidates(op)["naf-reverse"]
+            self.assertEqual(word[-2][0], "subtract" if sign > 0 else "add")
+            self.assertEqual(v.coefficient_check(Q(sign*11, 8), word, len(word)),
+                             independent.execute(word, sign*11, 8))
+
+    def test_reducible_fraction_and_even_integer(self):
+        for numerator, denominator in [(6, 4), (24, 1), (-8, 32), (-2, 2)]:
+            costs, _, _ = independent.check(lower, numerator, denominator)
+            self.assertTrue(costs)
+        a = lower.lower(["add", ["T", 0], ["S", 0], 6, 4])
+        b = lower.lower(["add", ["T", 0], ["S", 0], 3, 2])
+        self.assertEqual(a, b)
+
+    def test_dirty_work_is_an_independent_input(self):
+        op = ["add", ["T", 0], ["S", 0], 3, 2]
+        word = lower.lower(op)[1:]
+        with self.assertRaisesRegex(ValueError, "endpoint"):
+            independent.execute(word, 3, 2)
+        with self.assertRaisesRegex(ValueError, "endpoint"):
+            v.coefficient_check(Q(3, 2), word, 100)
+
+    def test_contract_controls_survive_optimization(self):
+        names = v.negative_controls(lower)
+        self.assertEqual(len(names), 14)
+        self.assertIn("endpoint_neutral_prefix_growth", names)
+        self.assertIn("reverse_naf_initial_sign", names)
+
+    def test_endpoint_neutral_prefix_growth(self):
+        op = ["add", ["T", 0], ["S", 0], 3, 4]
+        word = lower.lower(op)
+        peak = Q(v.coefficient_check(Q(3, 4), word, len(word)))
+        mutated = word[:1] + [("double", v.WORK), ("halve", v.WORK)] + word[1:]
+        # Endpoint and scratch checks alone accept this exact neutral pair.
+        self.assertGreater(Q(v.coefficient_check(Q(3, 4), mutated, len(mutated))), peak)
+        with self.assertRaisesRegex(ValueError, "prefix"):
+            v.coefficient_check(Q(3, 4), mutated, len(mutated), peak)
+
+    def test_coefficient_cost_control_is_independent_of_map(self):
+        op = ["add", ["T", 0], ["S", 0], 3, 2]
+        word = lower.lower(op)
+        self.assertEqual(independent.execute(word, 3, 2), "5/2")
+        with self.assertRaisesRegex(ValueError, "cost"):
+            v.coefficient_check(Q(3, 2), word, len(word)-1)
+
+    def test_unit_coefficients_preserve_arbitrary_private_input_contract(self):
+        for sign in (-1, 1):
+            word = lower.lower(["add", ["T", 0], ["S", 0], sign, 1])
+            self.assertEqual(len(word), 2)
+            self.assertEqual(independent.execute(word, sign, 1), "2")
+            with self.assertRaisesRegex(ValueError, "work"):
+                independent.execute(word[:-1], sign, 1)
+
+
+if __name__ == "__main__":
+    unittest.main()
diff --git a/research/dyadic-horner-direction/vendor/pr696_lower.py b/research/dyadic-horner-direction/vendor/pr696_lower.py
new file mode 100644
index 0000000000..a2c2889e5b
--- /dev/null
+++ b/research/dyadic-horner-direction/vendor/pr696_lower.py
@@ -0,0 +1,56 @@
+"""Smaller exact dyadic circuits for the fixed PR626 complex scalar words.
+
+Sagar Thakrar, with substantial OpenAI Codex assistance. Apache-2.0.
+Primitive vocabulary and the original Horner circuit follow icekylinx's
+PR425/433/531 lineage, retained by Jeremy Kothe's PR622 supplier.
+"""
+from fractions import Fraction
+
+
+def power_of_two(n):
+    return n > 0 and n & (n - 1) == 0
+
+
+def lower(op, work=("W", 0)):
+    """Return real-component primitives; work may be dirty on entry.
+
+    Every addition leaves work zero. All operations use the macro's frame.
+    The complex circuit applies this word separately to real and imaginary parts.
+    """
+    tag, target = op[0], tuple(op[1])
+    if target == work:
+        raise ValueError("coefficient work aliases a scalar register")
+    if tag == "clear":
+        return [("clear", target)]
+    if tag == "copy":
+        source = tuple(op[2])
+        if source == work:
+            raise ValueError("coefficient work used as a scalar source")
+        return [("copy", target, source)]
+    if tag == "scale":
+        q = Fraction(op[2], op[3])
+        if not power_of_two(abs(q.numerator)) or not power_of_two(q.denominator):
+            raise ValueError("scale is not a signed power of two")
+        shift = abs(q.numerator).bit_length() - q.denominator.bit_length()
+        result = [("double" if shift > 0 else "halve", target)] * abs(shift)
+        if q < 0:
+            result.append(("negate", target))
+        return result
+    if tag != "add":
+        raise ValueError("unknown scalar macro")
+    source = tuple(op[2])
+    q = Fraction(op[3], op[4])
+    if source in (target, work) or not q or not power_of_two(q.denominator):
+        raise ValueError("addition requires distinct registers and a nonzero dyadic")
+    accumulate = "add" if q > 0 else "subtract"
+    if abs(q) == 1:
+        return [(accumulate, target, source), ("clear", work)]
+    # Copy replaces clear + addition of the leading binary digit.
+    result = [("copy", work, source)]
+    for bit in bin(abs(q.numerator))[3:]:
+        result.append(("double", work))
+        if bit == "1":
+            result.append(("add", work, source))
+    result.extend([("halve", work)] * (q.denominator.bit_length() - 1))
+    result.extend([(accumulate, target, work), ("clear", work)])
+    return result
diff --git a/research/dyadic-horner-direction/verify.py b/research/dyadic-horner-direction/verify.py
new file mode 100644
index 0000000000..6c362e026d
--- /dev/null
+++ b/research/dyadic-horner-direction/verify.py
@@ -0,0 +1,347 @@
+"""Exact six-program replay for bidirectional dyadic Horner circuits.
+
+Gareth Davidson (@bitplane), substantial OpenAI Codex assistance, Apache-2.0.
+Sparse rational replay adapts our PR787 (93ceb65c); no production imports.
+"""
+
+import argparse
+import gzip
+import hashlib
+import importlib.util
+import json
+import tarfile
+from collections import Counter
+from fractions import Fraction as Q
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+WORK = ("W", 0)
+OLD_SPEC = "research/complex-k492-owner-repair/data/SPEC.json"
+NEW_SPEC = "research/complex-sw4-frame8/data/SPEC.json"
+PROVIDERS = "research/community-k492-audit/pr626/replay/mixed-global/base/providers"
+ARCHIVE = "research/community-frontier741-audit/composed-replay-evidence.tar.gz"
+
+
+def need(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def canonical(value):
+    return json.dumps(value, sort_keys=True, separators=(",", ":"))
+
+
+def digest(value):
+    return hashlib.sha256(canonical(value).encode()).hexdigest()
+
+
+def payload(path):
+    raw = path.read_bytes()
+    return gzip.decompress(raw) if path.suffix == ".gz" else raw
+
+
+def read(path):
+    return json.loads(payload(path))
+
+
+def load_lower():
+    spec = importlib.util.spec_from_file_location("direction_lower", HERE / "lower.py")
+    module = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(module)
+    return module
+
+
+def source_check(root=ROOT):
+    pins = read(HERE / "SOURCES.json")
+    resolved = {}
+    for item in pins["files"]:
+        for name in [item["path"]] + item.get("archived_paths", []):
+            path = root / name
+            if not path.is_file():
+                continue
+            data = payload(path) if item["encoding"] == "gzip-payload" else path.read_bytes()
+            if hashlib.sha256(data).hexdigest() == item["sha256"]:
+                resolved[item["path"]] = path
+                break
+        need(item["path"] in resolved, "source hash: " + item["path"])
+    for record, head, spec in [
+        ("certificates/checkpoints/k492626.json", pins["pr626_head"], OLD_SPEC),
+        ("certificates/selected-result.json", pins["pr741_head"], NEW_SPEC),
+    ]:
+        selected = read(resolved[record])
+        need(selected["audited_source_head"] == head, "record head mismatch")
+        manifest = read(root / selected["source_manifest"])
+        need(manifest["files"][spec] == hashlib.sha256((root / spec).read_bytes()).hexdigest(),
+             "record manifest does not bind SPEC")
+    return pins
+
+
+def inputs(root=ROOT):
+    spec = read(root / OLD_SPEC)
+    for provider in ("ck", "jp8"):
+        directory = root / PROVIDERS / provider
+        parts = {n: payload(directory / n) for n in ("CANONICAL-WORD.json.gz", "LOCAL-PROGRAMS.json.gz")}
+        yield provider, spec["providers"][provider], parts
+    # No extraction or execution of archive contents. Container metadata is
+    # irrelevant; every consumed, decompressed member is bound by the SPEC.
+    with tarfile.open(root / ARCHIVE) as archive:
+        parts = {}
+        for name in ("CANONICAL-WORD.json.gz", "LOCAL-PROGRAMS.json.gz"):
+            matches = [m for m in archive.getmembers() if m.name == "complex/local/" + name]
+            need(len(matches) == 1 and matches[0].isfile(), "ambiguous archive member")
+            parts[name] = gzip.decompress(archive.extractfile(matches[0]).read())
+    yield "sw4", read(root / NEW_SPEC)["providers"]["gauge44"], parts
+
+
+def plus(left, right, q):
+    result = dict(left)
+    for k, value in right.items():
+        value = result.get(k, Q(0)) + q * value
+        if value:
+            result[k] = value
+        else:
+            result.pop(k, None)
+    return result
+
+
+def primitive(step, rows, norms):
+    """Sparse exact rows plus cancellation-free positive prefix bounds."""
+    tag, target = step[:2]
+    if tag == "clear":
+        rows[target], norms[target] = {}, Q(0)
+    elif tag == "copy":
+        source = step[2]
+        rows[target], norms[target] = dict(rows[source]), norms[source]
+    elif tag in ("double", "halve", "negate"):
+        factor = {"double": Q(2), "halve": Q(1, 2), "negate": Q(-1)}[tag]
+        rows[target] = {k: factor * value for k, value in rows[target].items()}
+        norms[target] *= abs(factor)
+    elif tag in ("add", "subtract"):
+        source = step[2]
+        rows[target] = plus(rows[target], rows[source], Q(1 if tag == "add" else -1))
+        norms[target] += norms[source]
+    else:
+        raise ValueError("unknown primitive: " + tag)
+    return norms[target]
+
+
+def macro_row(op, rows):
+    tag, target = op[0], tuple(op[1])
+    if tag == "clear":
+        return {}
+    if tag == "copy":
+        return dict(rows[tuple(op[2])])
+    if tag == "scale":
+        return {k: Q(op[2], op[3]) * v for k, v in rows[target].items()}
+    need(tag == "add", "unknown scalar macro")
+    return plus(rows[target], rows[tuple(op[2])], Q(op[3], op[4]))
+
+
+def original_cost(op):
+    """Independent recount of the inherited clear/add binary invoice."""
+    if op[0] in ("clear", "copy"):
+        return 1
+    q = Q(op[3], op[4]) if op[0] == "add" else Q(op[2], op[3])
+    a, b = abs(q.numerator), q.denominator
+    if op[0] == "scale":
+        return abs(a.bit_length() - b.bit_length()) + int(q < 0)
+    return 3 + a.bit_length() - 1 + a.bit_count() + b.bit_length() - 1
+
+
+def coefficient_check(q, chain, budget, prefix_limit=None):
+    """Check three arbitrary inputs, source immutability, work erasure and cost."""
+    target, source = ("T", 0), ("S", 0)
+    rows = {target: {0: Q(1)}, source: {1: Q(1)}, WORK: {2: Q(1)}}
+    norms = dict.fromkeys(rows, Q(1))
+    peak = Q(1)
+    need(len(chain) <= budget, "coefficient cost increased")
+    for step in chain:
+        need(step[1] in (target, WORK), "source/foreign register write")
+        need(len(step) == 2 or step[2] in rows, "foreign primitive source")
+        peak = max(peak, primitive(step, rows, norms))
+    need(rows[target] == {0: Q(1), 1: q}, "wrong coefficient endpoint")
+    need(rows[source] == {1: Q(1)}, "source changed")
+    need(rows[WORK] == {}, "coefficient work not cleared")
+    if prefix_limit is not None:
+        need(peak <= prefix_limit, "coefficient prefix exceeded")
+    return str(peak)
+
+
+def replay(events, word, spec, orientation, lowering):
+    n = 2 * word["v"] + word["R"]
+    rows = {("L", i): {i: Q(1)} for i in range(n)}
+    for i in range(len(word["centers"])):
+        rows["C", i] = {n + i: Q(1)}
+    rows[WORK] = {n + len(word["centers"]): Q(1)}
+    norms = dict.fromkeys(rows, Q(1))
+    old_norms = dict(norms)
+    peak = old_peak = Q(1)
+    paid, kinds, costs, chosen = Counter(), Counter(), Counter(), Counter()
+    changed = macros = 0
+    stream = hashlib.sha256()
+    coefficients = Counter()
+    for event in events:
+        if event[0] in ("climb", "center"):
+            paid[event[5] if event[0] == "climb" else event[4]] += 1
+        if event[0] != "scalar":
+            continue
+        _, op, frame = event
+        target = tuple(op[1])
+        expected = macro_row(op, rows)
+        binary = lowering.reference.lower(op)
+        forward = lowering.choose(op, reverse=False)[1]
+        name, chain = lowering.choose(op)
+        need(len(chain) <= len(forward) <= len(binary), "cost regression")
+        changed += len(chain) < len(forward)
+        chosen[name] += 1
+        macros += 1
+        costs["original"] += 2 * original_cost(op)
+        costs["copy_binary"] += 2 * len(binary)
+        costs["forward_best"] += 2 * len(forward)
+        costs["bidirectional"] += 2 * len(chain)
+        if op[0] == "add":
+            source, q = tuple(op[2]), Q(op[3], op[4])
+            coefficients[q] += 1
+            old_peak = max(old_peak, abs(q.numerator) * old_norms[source])
+            old_norms[target] += abs(q) * old_norms[source]
+        elif op[0] == "scale":
+            old_norms[target] *= abs(Q(op[2], op[3]))
+        elif op[0] == "clear":
+            old_norms[target] = Q(0)
+        elif op[0] == "copy":
+            old_norms[target] = old_norms[tuple(op[2])]
+        old_peak = max(old_peak, old_norms[target])
+        stream.update(canonical([frame, chain]).encode() + b"\n")
+        for step in chain:
+            need(step[1] in (target, WORK), "foreign write")
+            kinds[step[0]] += 1
+            peak = max(peak, primitive(step, rows, norms))
+        need(rows[target] == expected, "scalar macro endpoint mismatch")
+        if op[0] == "add":
+            need(not rows[WORK], "work survives macro boundary")
+    for i in range(n):
+        expected = {i: Q(1)}
+        if orientation == "forward" and word["v"] <= i < 2 * word["v"]:
+            expected[i - word["v"]] = Q(1)
+        elif orientation == "backward" and i < word["v"]:
+            expected[i + word["v"]] = Q(-1)
+        need(rows["L", i] == expected, "full arbitrary-input endpoint mismatch")
+    need(all(not row for reg, row in rows.items() if reg[0] != "L"), "private stream survives")
+    need(dict(paid) == {int(k): v for k, v in spec["expected_profile"].items()}, "paid profile mismatch")
+    prior = spec["expected_local"][orientation] if "expected_local" in spec else spec["expected_local." + orientation]
+    need(costs["original"] == prior["real_component_primitives"], "original invoice mismatch")
+    need(macros == prior["macros"] and len(events) == prior["events"], "event/macro count mismatch")
+    need(old_peak == Q(prior["prefix_l1"]), "original prefix mismatch")
+    need(peak < 2**192, "192-bit local prefix allowance exceeded")
+    return {
+        "events": len(events), "macros": macros, "improved_by_reverse": changed,
+        "logical_columns": n, "private_columns": len(rows) - n,
+        "dirty_helpers_restored": word["R"], "private_streams_cleared": True,
+        "real_primitives": dict(costs), "saved_beyond_forward": costs["forward_best"] - costs["bidirectional"],
+        "chosen_methods": dict(chosen), "primitive_kinds_per_real_component": dict(kinds),
+        "original_positive_prefix": str(old_peak), "new_positive_prefix": str(peak),
+        "new_prefix_bits": peak.numerator.bit_length() - peak.denominator.bit_length() + 1,
+        "charged_payload_bits": 192, "paid_profile": {str(k): v for k, v in sorted(paid.items())},
+        "lowered_stream_sha256": stream.hexdigest(),
+        "coefficient_occurrences": {str(q): count for q, count in sorted(coefficients.items())},
+    }, set(coefficients)
+
+
+def negative_controls(lowering):
+    controls = []
+
+    def rejects(name, action):
+        try:
+            action()
+        except (ValueError, ZeroDivisionError):
+            controls.append(name)
+        else:
+            raise ValueError("negative control accepted: " + name)
+
+    op = ["add", ["T", 0], ["S", 0], 3, 4]
+    word = lowering.lower(op)
+    budget = len(lowering.choose(op, reverse=False)[1])
+    peak = Q(coefficient_check(Q(3, 4), word, budget))
+    for name, mutant in {
+        "missing_initial_copy": word[1:],
+        "missing_first_half": word[:1] + word[2:],
+        "wrong_reverse_digit": word[:2] + [("subtract", WORK, ("S", 0))] + word[3:],
+        "wrong_target_sign": word[:-2] + [("subtract", ("T", 0), WORK)] + word[-1:],
+        "missing_final_clear": word[:-1],
+        "source_write": word + [("clear", ("S", 0))],
+        "cost_understatement": word,
+        "endpoint_neutral_prefix_growth": word[:1] + [("double", WORK), ("halve", WORK)] + word[1:],
+    }.items():
+        # The excursion gets enough cost budget, isolating the prefix check.
+        cap = len(mutant) if name == "endpoint_neutral_prefix_growth" else budget
+        if name == "cost_understatement":
+            cap = len(word) - 1
+        rejects(name, lambda w=mutant, b=cap: coefficient_check(Q(3, 4), w, b, peak))
+    # For 11/8 the reverse NAF starts with -1. Omitting its sign correction
+    # gives -11/8 even though the positive binary example above still passes.
+    signed_op = ["add", ["T", 0], ["S", 0], 11, 8]
+    signed = lowering.candidates(signed_op)["naf-reverse"]
+    coefficient_check(Q(11, 8), signed, len(signed))
+    mutant = signed[:-2] + [("add", ("T", 0), WORK)] + signed[-1:]
+    rejects("reverse_naf_initial_sign", lambda: coefficient_check(Q(11, 8), mutant, len(mutant)))
+    for name, bad in [
+        ("aliased_data", ["add", ["T", 0], ["T", 0], 3, 4]),
+        ("work_as_source", ["add", ["T", 0], list(WORK), 3, 4]),
+        ("work_as_target", ["add", list(WORK), ["S", 0], 3, 4]),
+        ("non_dyadic", ["add", ["T", 0], ["S", 0], 3, 5]),
+        ("zero_coefficient", ["add", ["T", 0], ["S", 0], 0, 1]),
+    ]:
+        rejects(name, lambda o=bad: lowering.lower(o))
+    return controls
+
+
+def verify(root=ROOT):
+    pins = source_check(root)
+    lowering = load_lower()
+    programs, coefficients = {}, set()
+    for provider, spec, parts in inputs(root):
+        for name, raw in parts.items():
+            need(hashlib.sha256(raw).hexdigest() == spec["artifact_semantic_sha256"][name],
+                 "SPEC payload binding: " + provider + "/" + name)
+        word = json.loads(parts["CANONICAL-WORD.json.gz"])
+        for orientation, events in sorted(json.loads(parts["LOCAL-PROGRAMS.json.gz"]).items()):
+            need(digest(events) == spec["program_hashes"][orientation], "event hash mismatch")
+            result, qs = replay(events, word, spec, orientation, lowering)
+            programs[provider + "/" + orientation] = result
+            coefficients.update(qs)
+    certificates = []
+    for q in sorted(coefficients):
+        op = ["add", ["T", 0], ["S", 0], q.numerator, q.denominator]
+        words = lowering.candidates(op)
+        peaks = {k: coefficient_check(q, v, len(v)) for k, v in words.items()}
+        name, chosen = lowering.choose(op)
+        certificates.append({"coefficient": str(q), "chosen": name,
+                             "costs": {k: len(v) for k, v in words.items()}, "positive_prefixes": peaks,
+                             "word_sha256": digest(chosen)})
+    return {"status": "finite coefficient circuit construction; no new kappa",
+            "base_commit": pins["base_commit"], "programs": programs,
+            "coefficient_certificates": certificates, "negative_controls": negative_controls(lowering)}
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--write", type=Path, help="write verified receipt; destination must not exist")
+    args = parser.parse_args()
+    if args.write:
+        need(not args.write.exists(), "output already exists")
+    result = verify()
+    if args.write:
+        with args.write.open("x") as out:
+            json.dump(result, out, indent=2, sort_keys=True)
+            out.write("\n")
+    else:
+        need(canonical(result) == canonical(read(HERE / "EXPECTED.json")), "certificate mismatch")
+    savings = [result["programs"][p + "/forward"]["saved_beyond_forward"] for p in ("ck", "jp8", "sw4")]
+    print("PASS_DYADIC_HORNER_DIRECTION programs=6 coefficients=" + str(len(result["coefficient_certificates"])) +
+          " extra_saved_per_orientation=" + ",".join(map(str, savings)) +
+          " controls=" + str(len(result["negative_controls"])) + " kappa=unchanged")
+
+
+if __name__ == "__main__":
+    main()