Research: selected m=73 complex stock shape A=0 under finite-stock-as-C (no new kappa)

research/selected-stock-shape-a-zero/CERTIFICATE.json

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+{
+  "schema_version": 1,
+  "status": "PASS_SELECTED_STOCK_SHAPE_A_ZERO",
+  "producer": "selected-complex-m73",
+  "shape_A": 0,
+  "shape_B_bits": 2642,
+  "rho": "unbound",
+  "finite_stock_as_C_admission": true,
+  "digit_tower_lift_in_complex_cover": false,
+  "ordinary_A_lower_bound": 4096,
+  "zero_slopes": {
+    "1->2": 0.0,
+    "2->8": 0.0,
+    "8->64": 0.0
+  },
+  "composition_under_A_zero": "(43/26)*(1-s)*2642 + rho",
+  "no_new_kappa": true,
+  "kappa_decimal": "0.001672660197076918",
+  "marker": "PASS_SELECTED_STOCK_SHAPE_A_ZERO A=0 B_bits=2642 rho=unbound no_digit_tower_lift no_new_kappa"
+}

research/selected-stock-shape-a-zero/MASTER-HANDOFF.md

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+# Handoff: selected stock shape A = 0 (m=73 complex)
+
+## Result
+
+Under the finite-stock-as-C reading already named by
+`research/selected-coarse-depth-stock/PROOF.md`, the selected m=73 complex
+producer's per-level divisor candidate is independent of atom width `w`, so
+the shape bound `log_2 C(w) <= A*w + B` holds with **A = 0** and **B = 2642**.
+Leaf reserve `rho` remains unbound. Selected κ is unchanged at
+`0.001672660197076918`.
+
+Pinned main: `6f100d4bb5083f8c35d96fd95a539811c62f245c`.
+Branch: `cursor/selected-stock-a-zero-60b9`.
+Scientific package tip: `9b257d8e75bf205d150b4d2a6a8280f5e1d8f76a` (Co-authored-by Victor Li stripped from branch).
+Branch tip: this commit (git rev-parse HEAD on the branch).
+Package: `research/selected-stock-shape-a-zero`.
+
+## Switch
+
+Did not switch cards. Preferred obligation 1 (selected-stock-envelope OQ1:
+bind A or prove A=0) was still open after #3113. Avoided recomputing B=2642 /
+halving. Did not take P104 coefficient work (soft overlap with open
+#3363/#3365/#3371/#3372) or scatter-export schema (would need identical-receipt
+tooling bar).
+
+## Exact commands (expected PASS)
+
+```sh
+python3.12 -I -B research/selected-stock-shape-a-zero/verify.py
+# PASS_SELECTED_STOCK_SHAPE_A_ZERO A=0 B_bits=2642 rho=unbound no_digit_tower_lift no_new_kappa
+
+python3.12 -I -O -B research/selected-stock-shape-a-zero/verify.py
+# same marker
+
+python3.12 -I -B research/selected-stock-shape-a-zero/independent.py
+# PASS_SELECTED_STOCK_SHAPE_A_ZERO A=0 B_bits=2642 rho=unbound no_digit_tower_lift no_new_kappa_INDEPENDENT
+
+python3.12 -I -B research/selected-stock-shape-a-zero/test_package.py
+# PASS_SELECTED_STOCK_SHAPE_A_ZERO_TESTS groups=6
+
+python3.12 -B scripts/research_package.py --package research/selected-stock-shape-a-zero
+# PASS_REGISTERED_RESEARCH_PACKAGE research/selected-stock-shape-a-zero outcome=completed
+```
+
+## Closest priors
+
+- [#3113](https://github.com/CrocSwap/integer-mult-bounds/pull/3113)
+  `research/selected-stock-envelope-m73` — bound B_bits=2642, left A,B,rho unbound.
+- [#2105](https://github.com/CrocSwap/integer-mult-bounds/pull/2105)
+  `research/selected-coarse-depth-stock` — 43/26 depth and shape composition.
+- [#2084](https://github.com/CrocSwap/integer-mult-bounds/pull/2084)
+  `research/selected-bit-parent-reserve` — ordinary BIT A≥4096 contrast.
+
+## Soft-overlap risks
+
+- Do not read this as recomputing #3113's B_bits or halving.
+- Do not read A=0 as a closed sigma envelope (rho and finite-stock-as-C
+  admission remain).
+- Ordinary BIT packages that bind positive A are contrast only.
+
+## What remains open
+
+Bind `rho` (or prove `rho=0`); maintainer adoption of finite stock as uniform
+`C(w)`.
+
+## Credit
+
+Aaron Wang (#3113), Sricharan Vagvala (#2105), Adam Gall (#2084), retained
+cover notes. Notices untouched.
+
+Prepared by Wentao Li with substantial AI assistance.

research/selected-stock-shape-a-zero/NOTICE.md

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+# Notice
+
+Copyright 2026 Wentao Li
+
+This directory is a scoped construction/audit of archived research. It is
+licensed under the Apache License, Version 2.0, the same license as the
+surrounding project.
+
+The finite stock allowance B_bits = 2642 and the recorded missing shape lemma
+are Aaron Wang's work from PR #3113
+(`research/selected-stock-envelope-m73`). The coarse depth rate 43/26 and
+conditional ledgers are Sricharan Vagvala's PR #2105
+(`research/selected-coarse-depth-stock`). The ordinary BIT digit-tower contrast
+is Adam Gall's PR #2084 (`research/selected-bit-parent-reserve`). The complex
+and ordinary cover notes are retained project sources.
+
+Those notices and files are unchanged. This package adds an independent
+reading that binds A = 0 for the finite-stock-as-C candidate. Prepared by
+Wentao Li with substantial AI assistance. No independent human or formal-kernel
+review is claimed.

research/selected-stock-shape-a-zero/PROOF.md

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+# Shape constant A = 0 for the selected m=73 complex stock divisor
+
+## Claim
+
+Let C(w) be the per-level divisor candidate given by the selected m=73 complex
+producer's finite stock invoice: the product of the fixed group order
+|Sp(72,2)| and the fixed 14,826 physical families, whose bit length B_bits = 2642
+was already bound by `research/selected-stock-envelope-m73` (PR #3113). Under the
+finite-stock-as-C reading already named in
+`research/selected-coarse-depth-stock/PROOF.md`,
+
+```text
+log_2 C(w) <= A * w + B
+```
+
+holds for every atom width w >= 1 with **A = 0** and **B = 2642**.
+
+The leaf reserve constant rho in
+
+```text
+sigma <= (43/26) * (1 - s) * (A + B) + rho
+```
+
+remains unbound. Selected kappa is unchanged.
+
+## Ordinary contrast (why A is positive for BIT)
+
+For the selected ordinary BIT producer (PR #2084 / `selected-bit-parent-reserve`),
+the level divisor is realized over the local ring:
+
+```text
+|GL_m(Z/q^w Z)| = |GL_m(F_q)| * q^{m^2 (w-1)}.
+```
+
+The factor q^{m^2(w-1)} is an atom-width digit tower. With m = 64 it yields the
+source-bound linear envelope
+
+```text
+log_q W_q(w) < 4096 w + 1 <= 4097 w,
+```
+
+so any shape bound log_q C(w) <= A w + B for that producer requires A >= 4096.
+The same tower appears as W(w) = W_0 q^{m^2(w-1)} in `notes/three-stage-cover-bit.tex`,
+with the m^2(w-1) digits charged in the row reserve.
+
+## Complex case (why A = 0)
+
+1. **No digit-tower cover.** The pinned complex cover note
+   `notes/three-stage-cover-complex.tex` contains neither W(w) nor q^{m^2(w-1)}
+   nor a Z/q^w local-ring realization. Its Sp(U) → O lifts are parametrized by
+   u ∈ U and c ∈ F_2 — binary parameters inside a finite orthogonal group, not
+   atom-width digits.
+2. **Finite role counts retained.** The same note states that the full finite
+   role counts are retained in the scalar guard and row stock, and that group
+   elements are specified by binary matrices.
+3. **Selected stock factors are w-free.** The selected invoice charges a fixed
+   Sp(72,2) order and a fixed family split 14336 + 480 + 8 + 2 = 14826. Neither
+   factor depends on atom width w. Their combined bit length is the already-bound
+   B_bits = 2642.
+4. **Zero slope.** For every pair of atom widths w1 < w2 the charged stock bit
+   length is the same constant 2642, so the discrete slope of log_2 C is zero.
+   Hence A = 0 and B = 2642 satisfy the shape inequality at every probed w.
+
+## Composition consequence
+
+Substituting A = 0 and B = 2642 into the retained composition gives
+
+```text
+sigma <= (43/26) * (1 - s) * 2642 + rho.
+```
+
+This is not a closed envelope: rho is still unbound, and adopting the finite
+stock as uniform C(w) remains the physical/compiler admission named in the
+coarse-depth proof. Historical m=120 coefficient 57604 and external slack 252
+are not substituted.
+
+## What this package does not claim
+
+- It does not recompute |Sp(72,2)| or the 2642-bit decomposition as a new result.
+- It does not bind rho or prove a numerical sigma.
+- It does not change selected kappa, suppliers, frontiers, or cutoffs.
+- It does not identify the ordinary BIT m=64 producer with the complex m=73 producer.

research/selected-stock-shape-a-zero/README.md

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+# Selected stock shape: A = 0 at the m=73 complex producer
+
+**Card `selected-stock-envelope`, scope refined** to the missing shape constant
+A left open by [`research/selected-stock-envelope-m73`](../selected-stock-envelope-m73/)
+(PR [#3113](https://github.com/CrocSwap/integer-mult-bounds/pull/3113)). That
+package already bound the finite per-level stock allowance `B_bits = 2642` and
+the halving guard; this package does not recompute those. It proves that the
+retained finite-stock candidate for the per-level divisor `C(w)` is independent
+of atom width `w`, so
+
+```text
+log_2 C(w) <= A * w + B
+```
+
+holds with **A = 0** and **B = 2642**. The leaf constant `rho` stays unbound.
+No new kappa (selected decimal `0.001672660197076918`).
+
+```sh
+python3.12 -I -B research/selected-stock-shape-a-zero/verify.py
+```
+
+Expected:
+
+```text
+PASS_SELECTED_STOCK_SHAPE_A_ZERO A=0 B_bits=2642 rho=unbound no_digit_tower_lift no_new_kappa
+```
+
+Python 3.12, standard library only. Default verify is read-only: it re-hashes
+six pinned inputs, admits against `CERTIFICATE.json`, and rejects mutation
+controls. Admission uses `need()` and stays enforced under `python -O`.
+
+## Why A is zero here and positive for ordinary BIT
+
+| Producer | Level cover | Atom-width factor | Shape A |
+|---|---|---|---:|
+| Ordinary BIT m=64 | `GL_m(Z/q^w Z)` | `q^{m^2(w-1)}` digit tower | ≥ 4096 |
+| Complex m=73 | finite `Sp(72,2)` / `O(·,2)` over `F_2` | none | **0** |
+
+The ordinary digit tower and the bound `log_q W_q(w) < 4096 w + 1` are cited
+from [`research/selected-bit-parent-reserve`](../selected-bit-parent-reserve/)
+(PR [#2084](https://github.com/CrocSwap/integer-mult-bounds/pull/2084)) and
+`notes/three-stage-cover-bit.tex`. The complex cover note parametrizes Sp→O
+lifts by `u ∈ U`, `c ∈ F_2` and retains finite role counts; it has no `W(w)`
+and no `q^{m^2(w-1)}`. The selected invoice charges fixed `|Sp(72,2)|` and
+fixed 14,826 families — already reduced to `B_bits = 2642` by #3113 — with
+zero discrete slope across atom widths.
+
+## Composition under A = 0
+
+```text
+sigma <= (43/26) * (1 - s) * 2642 + rho
+```
+
+This collapses the `(A+B)` term but is not a closed envelope: `rho` remains
+open, and adopting the finite stock as uniform `C(w)` is still the admission
+named in `research/selected-coarse-depth-stock/PROOF.md`. Historical `57604`
+and slack `252` are not substituted.
+
+## Reproduce
+
+```sh
+python3.12 -I -B research/selected-stock-shape-a-zero/verify.py
+python3.12 -I -O -B research/selected-stock-shape-a-zero/verify.py
+python3.12 -I -B research/selected-stock-shape-a-zero/independent.py
+python3.12 -I -B research/selected-stock-shape-a-zero/test_package.py
+python3.12 -B scripts/research_package.py --package research/selected-stock-shape-a-zero
+```
+
+`test_package.py` rejects duplicate JSON keys, a temp-corrupted certificate
+(`shape_A` flipped to 1), hash failures, and a fake `w` injected into the Sp
+formula, and checks that package files are unchanged after replay.
+
+## Open next questions
+
+1. **Bind `rho` (or prove `rho = 0`).** With A = 0 the composition is
+   `(43/26)*(1-s)*2642 + rho`. First step: locate a complex analogue of the
+   ordinary completed-leaf hypothesis. Blocked by: no per-leaf recurrence in
+   the retained complex sources.
+2. **Adopt finite stock as uniform `C(w)`.** A = 0 is for that candidate.
+   First step: exhibit one selected residual that consumes a fresh complete
+   physical-role packet whose cardinality equals the finite stock at every w.
+
+## Credit
+
+Aaron Wang / #3113 bound `B_bits = 2642` and recorded the missing shape lemma.
+Sricharan Vagvala / #2105 supplied the coarse `43/26` depth and conditional
+ledgers. Adam Gall / #2084 bound the ordinary BIT parent envelope that supplies
+the positive-A contrast. The complex cover note and selected Sp(72,2) invoice
+lineage are retained predecessors. Their notices are unchanged.
+
+Prepared by Wentao Li with substantial AI assistance.

research/selected-stock-shape-a-zero/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "6f100d4bb5083f8c35d96fd95a539811c62f245c",
+  "files": [
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+    },
+    {
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+    },
+    {
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+      "git_blob": "f05225d21f92cf8cef21b6e12601d5b4d61991f0",
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+      "sha256": "1c5b638aa9b75365b8addec9c8eeee6bc5e499722901079b94c65fce033fde8a"
+    },
+    {
+      "path": "docs/research/selected-bootstrap-scope-map.md",
+      "usage": "input",
+      "git_blob": "af8f232cd2385f88a21095dfbbf73cebc20b6df1",
+      "sha256": "b9196c339ed06c749780749d30c93f2deba0ebb489368a1eb4505e461a1c3062"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    }
+  ]
+}