Research construction: phase-typed two-source order-four echo (no new kappa)

research/z4-distinct-source-seed55/NOTICE.md

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+# Source notices
+
+This package is distributed under the repository's Apache-2.0 license.
+
+The following unmodified predecessor snapshots are included for exact replay and attribution:
+
+- `inputs/PR1391/CERTIFICATE.json` and `inputs/PR1391/verify.py`, from `CrocSwap/integer-mult-bounds` commit `6ce8017e4c27c8b220b6c3a27a53bb18f138212a`, PR #1391. SHA256 values are recorded in `SOURCES.json` and checked by `verify.py`.
+- `inputs/PR1410-README.md`, from commit `5b1c00b9f2770aabaa0085539405757f7818a14a`, PR #1410. Its SHA256 and Git blob are checked by `verify.py`.
+
+PR #1391 and PR #1410 credit the order-three frame and paid output-sign-frame predecessors, PR #1287 by Ross Richardson and PR #1227 by Sagar Thakrar. This package retains those attributions and does not modify their snapshots.

research/z4-distinct-source-seed55/README.md

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+# Phase-indexed order-four echo across distinct sources
+
+**Scoped recursive-interface construction; no new κ or supplier admission.** This package extends the order-four five-child interface of PR #1391 and the same-source, two-root composition of PR #1410. It composes two five-child calls on independent source banks while preserving an arbitrary dirty output and restoring both sources and workspace.
+
+Research base: `4040a0f46d67b41c36a472a7cd4af791fd4c36cc`. Exact PR #1391 and PR #1410 dependency heads and content pins are recorded in [SOURCES.json](SOURCES.json).
+
+The main construction indexes each child's output reader by the carried workspace phase. If the original child reads through `R_i` and advances workspace by `P`, define
+
+```text
+C_(i,t)(D,W) = (D + Q^(-t) R_i W, P W),    t in Z/4Z.
+```
+
+When `W = P^t w`, the child contributes the original `R_i w`, since `Q^(-t) R_i P^t = R_i`. The typed reader is an eight-row signed selector: it changes which output register each ADD targets while retaining the child's eight ADDs and its 24-coordinate `P` update. This is an explicit change to the child contract.
+
+## Exact boundary recurrence
+
+The pinned predecessor has 24 workspace coordinates. The complete `P^r` gate vectors are
+
+| `r` | ADD | NEG | Total |
+|---:|---:|---:|---:|
+| 0 | 0 | 0 | 0 |
+| 1 | 54 | 6 | 60 |
+| 2 | 36 | 12 | 48 |
+| 3 | 54 | 6 | 60 |
+
+Let `A(r)` be that vector. With starting workspace phase `t`, the entry charge is `A(t)`. After a child at phase `t`, the workspace has advanced once under `P`; reaching the next phase `s` costs `A((s-t-1) mod 4)`. The exact recurrence is
+
+```text
+F_0(t)       = A(t)
+F_(j+1)(s)   = min_t [F_j(t) + A((s-t-1) mod 4)]
+g(k)         = F_k(0)
+```
+
+The verifier compares this recurrence with every path for the two fixed sizes: all 1,024 five-child paths and all 1,048,576 ten-child paths. The exact minima are:
+
+- `g(5) = (54,6)`, total 60, on phases `[0,0,1,2,3,0]`.
+- `g(10) = (36,12)`, total 48, on phases `[0,1,0,1,2,3,0,1,2,3,0]`.
+
+Let `T_i=(62,6)` be the complete forward cost of child `i`, including its eight data ADDs and 24-coordinate workspace update. The paired child body and literal inverse contribute `2 sum_i T_i`. For one source, the source copy and reset cost `2d` ADDs; for two sources, the entry copy, two-sided handoff, and exit reset cost `4d` ADDs, where `d=24`. The complete cost recurrences are
+
+```text
+E_1(k)       = 2 sum_(i=1..k) T_i       + (2d,0) + 2 g(k)
+E_2(k1,k2)   = 2 sum_(i=1..k1+k2) T_i   + (4d,0) + 2 g(k1+k2)
+```
+
+The inverse uses the same exact phase word in reverse and has the same ADD/NEG vector. These equations include every child, source update, phase change, and restoration charged by this finite interface.
+
+For the two-source call, the ten-child path is at phase 3 after the first five children. At that point the workspace contains `P^3(W+X)`. The handoff applies `P^3(Y-X)`, using two ADDs per each of 24 coordinates, for 48 ADDs. The second five children then read `Y`. The complete word is
+
+```text
+W += X
+phase-typed 10-child forward body, with W += P^3(Y-X) after child 5
+W -= Y
+literal inverse of the no-handoff 10-child body
+```
+
+The resulting map is
+
+```text
+(D,X,Y,W) -> (D + sum_(i=0..4) R_i X + sum_(i=5..9) R_i Y, X, Y, W).
+```
+
+All 80 basis columns (8 output, 24 coordinates in each of `X`, `Y`, and `W`) are checked exactly. The two source banks are read-only, and arbitrary initial values in all four banks are allowed.
+
+## Complete invoice
+
+The joint typed invoice is
+
+```text
+20*(62,6) + (96,0) + 2*(36,12) = (1408,144)
+```
+
+for 1,552 primitives and 2,960 data reads. Two separate calls using the same typed child family cost
+
+```text
+2 * [10*(62,6) + (48,0) + 2*(54,6)] = (1552,144)
+```
+
+for 1,696 primitives and 3,248 reads. The joint typed call saves **144 primitives and 288 reads**. Its exact prefix row-L1 peak is 21, compared with 16 for the two separate typed calls; this is a coefficient-width cost that could affect fixed-precision admission.
+
+For comparison, the unchanged-reader interface from PR #1410 also composes across two distinct sources when charged with the same 48-ADD handoff. Its exact joint invoice is 1,952 primitives versus 2,016 for two separate original calls, saving 64 primitives and 136 reads. Its prefix row-L1 peaks are 42 joint and 30 separate. This is a matched same-contract construction; it is not included in the typed-call saving above.
+
+## Reproduction and evidence
+
+From the repository root, with CPython 3.12:
+
+```sh
+python3.12 scripts/research_package.py --package research/z4-distinct-source-seed55
+```
+
+Expected verifier marker:
+
+```text
+PASS_Z4_DISTINCT_SOURCE_PHASE_TYPED_ECHO g5=60 g10=48 separate=1696 joint=1552 saved=144 reads_saved=288 paths=1024+1048576 basis=80 controls=5
+```
+
+The registered replay checks the base and predecessor pins, all twelve `(phase, reader)` identities, unchanged child ADD/NEG counts, both complete 80-column maps, every phase path at both sizes, exact ADD/NEG/read invoices, prefix row-L1 peaks, and five negative controls. The exact evidence is frozen in [RESULT.json](RESULT.json).
+
+## Scope and limits
+
+The phase recurrence is exact for the stated phase-indexed order-four child alphabet. It is not a lower bound for arbitrary ADD/NEG programs. Output-row reindexing is encoded by the static target of each ADD and costs no additional scalar ADD/NEG instruction in this word model. Address calculation, physical row routing, source hosting, tape travel, fixed-precision admission, recursive assembly into a selected supplier, and analytic transfer remain unproved. The row-L1 increase makes precision a live obligation. The selected result at the pinned base remains κ = `836330098538459/500000000000000000`; this package does not change it or claim a new κ.
+
+## Overlap check
+
+The live refresh found several adjacent connector results. PR #1158 gives an exact readout-price census for a four-source/four-output Hadamard connector. PR #1438 and Cinna Davis's Discussion #1412 study a second arbitrary-dirty root and paid full-row transfers for universal five-source masks. PR #1439 studies a persistent `z = z_initial + Bx` boundary across a paid source mutation `x += Cu`, correcting it by `z += BCu`; PR #1442 proves a weighted query optimum under chronological output-delivery cuts. Here the interface keeps one 24-coordinate dirty workspace and changes between two read-only source banks by the explicit 48-ADD delta `P^t(Y-X)` inside a fixed five-child echo. Its separate evidence is the order-four phase-indexed reader identity, exhaustive five- and ten-child recurrence, and complete arbitrary-dirty two-call map/invoice, not a four-source Hadamard census, two-root transfer bound, general mutable-source connector theorem, or delivery-cut optimum. We make no priority claim for those connector mechanisms and retain their distinct scopes.
+
+## Open next questions
+
+Does a named selected supplier realize all four phase-indexed signed target rows with the charged handoff and no binding routing or precision cost? This matters before the operation saving can alter a multiplication profile. First inspect one retained #1094 child matrix and test `Q^(-t) R_i P^t = R_i` for every phase, then price target routing and the prefix envelope. The blocker is that no physical host, tape schedule, or precision-admitted supplier for this paired interface has been identified.
+
+## Credit and license
+
+This work builds on Cinna Davis's PR #1391 order-four child family and PR #1410 same-source persistent boundary, and retains their attribution. Those results extend Ross Richardson's PR #1287 and Sagar Thakrar's PR #1227. The selected-result context is Rohan Arun's PR #1094 on Jeremy Kothe's #1087 construction. The exact PR #1391 certificate and verifier snapshots are preserved under `inputs/PR1391/`; the direct PR #1410 README snapshot is pinned under `inputs/`. See [NOTICE.md](NOTICE.md) for source heads and snapshot hashes.
+
+The overlap review also credits Cinna Davis's PR #1158 for an adjacent four-source Hadamard connector census, woklerx's PRs #1438, #1439, and #1442, and Cinna Davis's concurrent Discussion #1412 for distinct two-root, mutable-source, and chronological-delivery models.
+
+Prepared by **Cinna Davis** with substantial **OpenAI Codex assistance** in derivation, implementation, exact verification, source pinning, and prose. No independent human review or proof-assistant formalization is claimed. New material is Apache-2.0; the project license is retained.

research/z4-distinct-source-seed55/RESULT.json

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+{
+  "assignment_seed": 55,
+  "assumptions_and_limits": [
+    "Output-row reindexing is part of the typed ADD target choice and has no additional scalar ADD/NEG instruction in this word model.",
+    "Address, tape, source-hosting, precision admission, recursive assembly into a selected supplier and analytic transfer are not charged or proved.",
+    "The finite state recurrence is exact only for the declared order-four phase-indexed child alphabet; it is not a lower bound for arbitrary words.",
+    "No new kappa or selected-result change is claimed."
+  ],
+  "base_commit": "4040a0f46d67b41c36a472a7cd4af791fd4c36cc",
+  "complete_paid_cost_recurrence": {
+    "child_forward_ADD_NEG": {
+      "ADD": 62,
+      "NEG": 6
+    },
+    "d": 24,
+    "literal_inverse": "counted by the factor 2 on every child and phase boundary; inverse leaves preserve ADD/NEG counts",
+    "single_source": "2*sum_i(T_i) + (2d,0) + 2*g(k)",
+    "two_source_split": "2*sum_i(T_i) + (4d,0) + 2*g(k1+k2)",
+    "wrapper_accounting": "single: W+=X and W-=X; two-source: W+=X, two ADDs per coordinate for P^t(Y-X), and W-=Y"
+  },
+  "controls": {
+    "fixed_reader_midpoint_carry": "PASS",
+    "missing_final_source_restore": "REJECTED",
+    "missing_literal_inverse": "REJECTED",
+    "omitted_source_handoff": "REJECTED",
+    "typed_child_phase_identity_checks": "PASS for 12 (phase,reader) pairs",
+    "wrong_handoff_phase": "REJECTED",
+    "wrong_reader_type": "REJECTED"
+  },
+  "dimensions": {
+    "D": 8,
+    "W": 24,
+    "X": 24,
+    "Y": 24,
+    "total_basis_columns": 80
+  },
+  "handoff": {
+    "ADDs": 48,
+    "formula": "W += P^t (Y-X) at the 5|5 boundary, t=phase[5]",
+    "selected_phase": 3
+  },
+  "joint_distinct_sources": {
+    "ADD_NEG": {
+      "ADD": 1408,
+      "NEG": 144
+    },
+    "map": "PASS on all 80 basis columns",
+    "prefix_row_l1_peak": 21,
+    "primitives": 1552,
+    "reads": 2960
+  },
+  "phase_reader": "C_(i,t): D += Q^(-t) R_i W; W += (P-I)W; type t has same ADD/NEG count as C_i.",
+  "predecessors": {
+    "PR1391_certificate_sha256": "2a6ac306ddc6f58514b89f680d8ff3027b4195e2754574f8f04ac1f17785c9d3",
+    "PR1391_head": "6ce8017e4c27c8b220b6c3a27a53bb18f138212a",
+    "PR1391_verifier_sha256": "c3d8e76dad9b3cc159affa5e56dbf73612eab0c0ce4a4464f523d4cf47669c05",
+    "PR1410_head": "5b1c00b9f2770aabaa0085539405757f7818a14a",
+    "PR1410_readme_sha256": "6dda7d5b83b41c71d73711401edbc146271a546a518ef6b038b0bdbb6578d405"
+  },
+  "recurrence": {
+    "F0(t)": "A(t)",
+    "Fnext(s)": "min_t F(t)+A((s-t-1) mod 4)",
+    "g10_ADD_NEG": {
+      "ADD": 36,
+      "NEG": 12
+    },
+    "g10_exhaustive_paths": 1048576,
+    "g10_path": [
+      0,
+      1,
+      0,
+      1,
+      2,
+      3,
+      0,
+      1,
+      2,
+      3,
+      0
+    ],
+    "g10_total": 48,
+    "g5_ADD_NEG": {
+      "ADD": 54,
+      "NEG": 6
+    },
+    "g5_exhaustive_paths": 1024,
+    "g5_path": [
+      0,
+      0,
+      1,
+      2,
+      3,
+      0
+    ],
+    "g5_total": 60,
+    "terminal_phase": 0,
+    "workspace_phase_costs_ADD_NEG": [
+      {
+        "ADD": 0,
+        "NEG": 0
+      },
+      {
+        "ADD": 54,
+        "NEG": 6
+      },
+      {
+        "ADD": 36,
+        "NEG": 12
+      },
+      {
+        "ADD": 54,
+        "NEG": 6
+      }
+    ]
+  },
+  "same_reader_control": {
+    "joint_ADD_NEG": {
+      "ADD": 1768,
+      "NEG": 184
+    },
+    "joint_prefix_row_l1_peak": 42,
+    "joint_primitives": 1952,
+    "joint_reads": 3720,
+    "map": "PASS on all 80 basis columns",
+    "phase_path": [
+      0,
+      1,
+      2,
+      3,
+      0,
+      1,
+      2,
+      3,
+      0,
+      1,
+      0
+    ],
+    "saving_primitives": 64,
+    "separate_ADD_NEG": {
+      "ADD": 1840,
+      "NEG": 176
+    },
+    "separate_prefix_row_l1_peak": 30,
+    "separate_primitives": 2016,
+    "separate_reads": 3856
+  },
+  "saving": {
+    "primitives": 144,
+    "reads": 288
+  },
+  "scope": "Exact phase-indexed typed child family for two distinct source banks and the pinned PR1391 five-child reader sequence; not an admitted selected supplier.",
+  "separate_calls": {
+    "ADD_NEG": {
+      "ADD": 1552,
+      "NEG": 144
+    },
+    "map": "PASS on all 80 basis columns",
+    "prefix_row_l1_peak": 16,
+    "primitives": 1696,
+    "reads": 3248
+  },
+  "status": "PASS_Z4_DISTINCT_SOURCE_PHASE_TYPED_ECHO"
+}

research/z4-distinct-source-seed55/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "4040a0f46d67b41c36a472a7cd4af791fd4c36cc",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "c05fafaa25ba95b2d5af86b1800427198e3d20eb",
+      "sha256": "89b5928bc92c81e2647b6d702a7ad85a6795b8f44e2eb267b2ff3145d8f2b2ce"
+    },
+    {
+      "path": "docs/research/community-batch1099-review.md",
+      "usage": "provenance",
+      "git_blob": "6cb6ec061a071a06a2d18cd4dc3e9a2b05d28099",
+      "sha256": "7e76d2c38e5fce30587835b7251df1430488ae4dd372611ef69a3f03f476b6cb"
+    },
+    {
+      "path": "research/community-frontier1094-audit/SELECTED-SOURCE.json",
+      "usage": "provenance",
+      "git_blob": "6a75493790b4eff9c4a98051ef8b01d6aad0d5a8",
+      "sha256": "79b3de5db3dbcb9829789ba58e9e45758926031e2a7d80667a08429d369be53b"
+    },
+    {
+      "path": "research/community-frontier1094-audit/ASSEMBLY.json",
+      "usage": "provenance",
+      "git_blob": "5fd676df060f3b450782fdf5ff5f6beb63a5eeb0",
+      "sha256": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045"
+    }
+  ],
+  "external_sources": [
+    {
+      "repository": "CrocSwap/integer-mult-bounds",
+      "commit": "6ce8017e4c27c8b220b6c3a27a53bb18f138212a",
+      "path": "research/z4-paired-frame-restoration-seed56/CERTIFICATE.json",
+      "local_path": "research/z4-distinct-source-seed55/inputs/PR1391/CERTIFICATE.json",
+      "git_blob": "0132b117d90f1a7a97c40af3e38503c117604e00",
+      "sha256": "2a6ac306ddc6f58514b89f680d8ff3027b4195e2754574f8f04ac1f17785c9d3",
+      "role": "exact predecessor child and recurrence certificate"
+    },
+    {
+      "repository": "CrocSwap/integer-mult-bounds",
+      "commit": "6ce8017e4c27c8b220b6c3a27a53bb18f138212a",
+      "path": "research/z4-paired-frame-restoration-seed56/verify.py",
+      "local_path": "research/z4-distinct-source-seed55/inputs/PR1391/verify.py",
+      "git_blob": "9d866bbf57a6d1f012a21687bde2e231c90efd14",
+      "sha256": "c3d8e76dad9b3cc159affa5e56dbf73612eab0c0ce4a4464f523d4cf47669c05",
+      "role": "pinned signed-ADD child word and basis replay"
+    },
+    {
+      "repository": "CrocSwap/integer-mult-bounds",
+      "commit": "5b1c00b9f2770aabaa0085539405757f7818a14a",
+      "path": "research/z4-persistent-pair-seed57/README.md",
+      "local_path": "research/z4-distinct-source-seed55/inputs/PR1410-README.md",
+      "git_blob": "001929a89cd64f4bbb0d4194c6bde96f45b7ded2",
+      "sha256": "6dda7d5b83b41c71d73711401edbc146271a546a518ef6b038b0bdbb6578d405",
+      "role": "direct same-source predecessor and matched baseline context"
+    }
+  ]
+}