Research: odd-kernel coefficient-bit tariff exceeds dyadic on j=0..32 (answers #2687 OQ2; no new kappa)

research/odd-kernel-coeff-bit-tariff/CERTIFICATE.json

@@ -0,0 +1,63 @@
+{
+  "claim": {
+    "odd_baseline_strictly_exceeds_dyadic_on_grid": true,
+    "odd_fused_strictly_exceeds_dyadic_on_grid": true,
+    "tables_never_pay_on_this_tariff": true,
+    "typed_child_xor_counts_identical": true
+  },
+  "counts": {
+    "grid_rows": 198,
+    "odd_baseline_exceeds_dyadic": 165,
+    "odd_fused_exceeds_dyadic": 165
+  },
+  "j_max": 32,
+  "kappa_decimal": "0.001672660197076918",
+  "kappa_fraction": "836330098538459/500000000000000000",
+  "kernels": [
+    1,
+    3,
+    5,
+    7,
+    9,
+    15
+  ],
+  "no_new_kappa": true,
+  "odd_kernels": [
+    3,
+    5,
+    7,
+    9,
+    15
+  ],
+  "schema_version": 1,
+  "spot": {
+    "T_fused_j2": {
+      "1": 102,
+      "15": 168,
+      "3": 126,
+      "5": 145,
+      "7": 147,
+      "9": 166
+    },
+    "T_word": {
+      "1": 45,
+      "15": 64,
+      "2": 38,
+      "3": 49,
+      "4": 32,
+      "5": 55,
+      "7": 56
+    },
+    "non_monotone": {
+      "T(1)": 45,
+      "T(2)": 38,
+      "T(4)": 32
+    }
+  },
+  "status": "PASS_ODD_KERNEL_COEFF_BIT_TARIFF",
+  "tariff": {
+    "definition": "T_word(k) = sum over affine diagonal/control coefficients c=p/q in lowest terms of bit_length(|p|)+bit_length(q); T_fused(q,j)=2*T_word(q*2^j)+T_word(q*2^{j+1}); T_baseline(q,j)=4*T_word(q*2^j).",
+    "name": "ADDRESS_affine_coeff_num_den_bit_sum",
+    "unit_coeff_bits": 20
+  }
+}

research/odd-kernel-coeff-bit-tariff/MASTER-HANDOFF.md

@@ -0,0 +1,48 @@
+# MASTER-HANDOFF — odd-kernel-coeff-bit-tariff
+
+## Claim (one sentence)
+
+Under the named ADDRESS affine coefficient-bit tariff, every certified odd
+kernel `q∈{3,5,7,9,15}` strictly exceeds the dyadic fused and baseline bills
+on `j=0..32`, so larger-`q` coefficient tables never pay for themselves
+despite identical typed child/XOR counts (answers #2687 OQ2).
+
+## Closest prior
+
+- #2687 / `research/odd-kernel-query-pair` — exclusion pattern; OQ2 left open
+- PR840 / `research/dyadic-query-pair` — dyadic fusion and ADDRESS shape
+- triadic-query-pair — kernel-3 specialization feeding #2687
+
+## New checkable fact
+
+Named tariff `T_word` / `T_fused` / `T_baseline`; closed form; prior
+`address_word` cross-check; `165+165` strict odd-over-dyadic exceedances;
+non-monotonicity of `T_word` (`32<38<45` at `|k|=4,2,1`).
+
+## Evidence commands
+
+```sh
+python3.12 -I -B research/odd-kernel-coeff-bit-tariff/verify.py
+python3.12 -I -O -Werror -B research/odd-kernel-coeff-bit-tariff/verify.py
+python3.12 -I -B research/odd-kernel-coeff-bit-tariff/independent.py
+python3.12 -I -B research/odd-kernel-coeff-bit-tariff/test_package.py
+python3.12 -B scripts/research_package.py --package research/odd-kernel-coeff-bit-tariff
+```
+
+## Soft-overlap vs leave-alones
+
+Clear of flag-0 tail / stock envelope / gate-operand / C5 / side-correction /
+present-head / groups B–D native clusters. Touches only the #840 / #2687
+odd-kernel cascade at the unanswered coefficient-bit OQ. Soft-overlap with
+#2687 is intentional predecessor pin of `program.py`, not a typed-invoice
+reprint.
+
+## Attribution
+
+Author Wentao Li `<wentaol4@illinois.edu>`. Neutral AI assistance disclosure.
+No `Co-authored-by: Victor Li`. No CrocSwap PR from this worker (master publishes).
+
+## Disposition
+
+Ready for master independent review / publish decision. Kappa unchanged
+(`0.001672660197076918`).

research/odd-kernel-coeff-bit-tariff/NOTICE.md

@@ -0,0 +1,17 @@
+# Notice — research/odd-kernel-coeff-bit-tariff
+
+Prepared by Wentao Li (GitHub VictorLiwentao, X @Is__Victor) with substantial
+AI assistance. New material is Apache-2.0 under the repository LICENSE.
+
+Predecessor credit:
+
+- Wentao Li / #2687 (`research/odd-kernel-query-pair`) — general odd-kernel
+  exclusion pattern; OQ2 (coefficient-table bit tariff) answered here.
+- Wentao Li / PR840 (`research/dyadic-query-pair`) — three-query dyadic fusion
+  and ADDRESS lowering shape.
+- Pending fork package `research/triadic-query-pair/` — kernel-3 specialization
+  feeding #2687.
+- Karl Appel / applecrusher / PR521 — native unipotent ADDRESS query lowering.
+- commdata2338 / PR677 — completed DATA-shear boundaries without a private row.
+
+No independent human review is claimed. No selected frontier or kappa change.

research/odd-kernel-coeff-bit-tariff/PROOF.md

@@ -0,0 +1,92 @@
+# Coefficient-bit tariff for odd-kernel ADDRESS lowerings
+
+## Claim
+
+Let `address_word(k)` be the seven-event ADDRESS lowering of
+`M(k)=I+(k/4)sr^T` used by PR840 / `#2687`. Define the **coefficient-bit
+tariff**
+
+```text
+T_word(k) = Σ_{c = p/q in affine coeffs} ( bit_length(|p|) + bit_length(q) )
+```
+
+with each `c` in lowest terms. For kernel `q` and dyadic index `j`,
+
+```text
+T_fused(q,j)     = 2·T_word(q·2^j) + T_word(q·2^{j+1})
+T_baseline(q,j)  = 4·T_word(q·2^j)
+```
+
+matching the three-query fused and four-query separate power lists. Then for
+every certified odd kernel `q ∈ {3,5,7,9,15}` and every `j ∈ {0,…,32}`:
+
+```text
+T_fused(q,j) > T_fused(1,j)
+T_baseline(q,j) > T_baseline(1,j).
+```
+
+Typed child / affine / XOR counts remain `3/18/2` (fused) and `4/24/4`
+(baseline) independent of `q`. Therefore, on this named tariff, larger-`q`
+coefficient tables never pay for themselves against the dyadic path.
+
+Selected kappa is unchanged.
+
+## Closed form for `T_word`
+
+The affine coefficients of `address_word(k)` are ten units `±1` together with
+
+```text
+a, 1+a, -a, -a, -1/a, 1/a - 1     where a = k/4.
+```
+
+Each unit contributes `bit_length(1)+bit_length(1)=2`, so
+
+```text
+T_word(k) = 20 + bits(a) + bits(1+a) + 2·bits(a) + bits(4/k) + bits((4-k)/k)
+```
+
+with `bits(p/q)=bit_length(|p|)+bit_length(q)`. The formula is sign-invariant
+(`T_word(k)=T_word(-k)`) and matches direct enumeration on `|k|≤64`.
+
+## Non-monotonicity
+
+`T_word` is **not** monotone in `|k|`:
+
+```text
+T_word(4) = 32 < T_word(2) = 38 < T_word(1) = 45.
+```
+
+A word-level “larger `|k|` ⇒ larger bill” argument is false. The claim is the
+fused/baseline program comparison on the certified kernel grid, not word
+monotonicity.
+
+## Grid evidence
+
+For each of the `5×33 = 165` odd pairs `(q,j)` the verifier checks fused and
+baseline strict exceedance separately (`165` comparisons each). Spot values at `j=2`:
+
+| q | T_fused(q,2) | T_baseline(q,2) |
+|---:|---:|---:|
+| 1 | 102 | 128 |
+| 3 | 126 | 160 |
+| 5 | 145 | 184 |
+| 7 | 147 | 188 |
+| 9 | 166 | 212 |
+| 15 | 168 | 216 |
+
+Fusion saves coefficient bits for every kernel, but the odd fused bill remains
+strictly above the dyadic fused bill on the whole grid.
+
+## Relation to `#2687` / PR840
+
+`#2687` proved the excluded-prime admission pattern and recorded identical
+typed invoices for all odd `q`, leaving open whether coefficient-table
+bit-sizes can still favor some `q` under a concrete physical tariff. This
+package attaches one named tariff to the same `address_word` and answers that
+obligation in the negative on `j=0..32` for the certified kernels.
+
+## What this does not claim
+
+- No alternate tariff (max-height only, packing, modulus-dependent encoding).
+- No BIT-supplier hosting bill (still `#2687` OQ1).
+- No new kappa and no production adoption.

research/odd-kernel-coeff-bit-tariff/README.md

@@ -0,0 +1,83 @@
+# Odd-kernel coefficient-bit tariff (answers #2687 OQ2)
+
+**Card `connector-alphabet-change`, priority revision 6**, refined to open
+question 2 of
+[#2687](https://github.com/CrocSwap/integer-mult-bounds/pull/2687)
+(`research/odd-kernel-query-pair`). No new kappa
+(`0.001672660197076918`).
+
+`#2687` showed that every odd kernel `q` has the same typed fused invoice as
+the dyadic case (`4→3` children, `24→18` affine, `4→2` XOR) and left open
+whether coefficient-table bit-sizes can still favor some larger `q` under a
+concrete physical tariff. This package attaches one named tariff to the PR840
+`address_word` lowering and answers that obligation.
+
+**Tariff.** For each affine diagonal/control coefficient `c=p/q` in lowest
+terms, charge `bit_length(|p|)+bit_length(q)`. Summing over `address_word(k)`
+gives `T_word(k)`. Program bills are
+
+```text
+T_fused(q,j)    = 2·T_word(q·2^j) + T_word(q·2^{j+1})
+T_baseline(q,j) = 4·T_word(q·2^j)
+```
+
+**Result.** On the certified kernels `{1,3,5,7,9,15}` and every
+`j∈{0,…,32}`, every odd kernel strictly exceeds the dyadic fused and baseline
+bills (`165+165` comparisons). `T_word` is not monotone in `|k|`
+(`T(4)=32 < T(2)=38 < T(1)=45`), so the claim is the program comparison, not
+a word-level size argument. Larger-`q` tables never pay for themselves on this
+tariff.
+
+```sh
+python3.12 -I -B research/odd-kernel-coeff-bit-tariff/verify.py
+```
+
+Expected:
+
+```text
+PASS_ODD_KERNEL_COEFF_BIT_TARIFF kernels=[1, 3, 5, 7, 9, 15] j=0..32 odd_fused_exceeds_dyadic=165 odd_baseline_exceeds_dyadic=165 no_new_kappa
+```
+
+## Evidence
+
+- Closed-form `T_word` (ten unit coeffs + six variable rationals in `a=k/4`)
+  matches enumeration on `|k|≤64` and the byte-pinned `#2687` `address_word`.
+- Grid: odd fused/baseline bills exceed dyadic at every `(q,j)` above.
+- Spot `j=2`: dyadic fused `102` vs odd `{126,145,147,166,168}`.
+- Fusion saves bits for every kernel but does not overturn odd `>` dyadic.
+- Non-monotonicity control and inflated-dyadic adversarial control.
+
+Default verify is read-only against `CERTIFICATE.json`. Admission uses
+`need()` and stays enforced under `python -O`. Explicit `--write` regenerates
+the certificate only.
+
+## Reproduce
+
+```sh
+python3.12 -I -B research/odd-kernel-coeff-bit-tariff/verify.py
+python3.12 -I -O -Werror -B research/odd-kernel-coeff-bit-tariff/verify.py
+python3.12 -I -B research/odd-kernel-coeff-bit-tariff/independent.py
+python3.12 -I -B research/odd-kernel-coeff-bit-tariff/test_package.py
+python3.12 -B scripts/research_package.py --package research/odd-kernel-coeff-bit-tariff
+```
+
+## Open next questions
+
+1. **Does any alternate physical tariff ever make a certified odd kernel beat
+   dyadic at some `j`?** First step: name max-height / packing / modulus-dependent
+   encoding on the same coefficients and replay the grid. Blocked by: no
+   alternate tariff admitted here.
+2. **Can an admitted BIT supplier host a small odd-kernel shear with a favorable
+   fully charged bill versus dyadic?** First step: one exact source-return
+   boundary with routing, precision and dispatch. Blocked by: still `#2687` OQ1;
+   not constructed here.
+
+## Credit
+
+Credit `#2687` / `research/odd-kernel-query-pair` (OQ2 answered), PR840 /
+`research/dyadic-query-pair`, the pending triadic kernel-3 package, and
+PR521/PR677 ADDRESS / DATA-shear lineage. Predecessor notices are unchanged.
+
+Prepared by Wentao Li ([VictorLiwentao](https://github.com/VictorLiwentao),
+[@Is__Victor](https://x.com/Is__Victor)) with substantial AI assistance, under
+Apache-2.0. No independent human review is claimed. No new kappa.

research/odd-kernel-coeff-bit-tariff/SOURCES.json

@@ -0,0 +1,48 @@
+{
+  "schema_version": 1,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "files": [
+    {
+      "path": "research/odd-kernel-query-pair/program.py",
+      "usage": "input",
+      "git_blob": "93699aec204ee787418254b10e50a1dd84e8bfc3",
+      "sha256": "f167826e66d39cd448651f7cdb27ba150a846786b2e1a9a2c7dad5100028a7c2",
+      "size_bytes": 6332
+    },
+    {
+      "path": "research/odd-kernel-query-pair/README.md",
+      "usage": "provenance",
+      "git_blob": "5b79bbd87534804fff944ca8de5e7ce88a07e9c9",
+      "sha256": "b7c1cf0518eff8cc7c9682e9038cb78d93ce9cae7b2d44f40a8286875fb019a2",
+      "size_bytes": 4258
+    },
+    {
+      "path": "research/odd-kernel-query-pair/TASK.json",
+      "usage": "provenance",
+      "git_blob": "1834f1fb4e7727708f9c89104fff50cd48281332",
+      "sha256": "886925d67cd1f52dc00591574217751bbe9bffd82956c6cefbc35267b32c8dfb",
+      "size_bytes": 4576
+    },
+    {
+      "path": "research/dyadic-query-pair/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "aaba3e501de1f761c783ec7f0dc9307ef45ab5d7",
+      "sha256": "9a5ce93b866371189c797fbe8837c6a0bd22c5cf71ed84d60b9ae1c22a5ec83a",
+      "size_bytes": 10567
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129",
+      "size_bytes": 10725
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0",
+      "size_bytes": 58553
+    }
+  ]
+}