Absorb the approximant base 10 p^2 + p^6 exp(-p^2) (no new κ)

research/gaussian-tensor-approximant/README.md

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+# An approximant norm stays inside the telescoping budget
+
+**Claim type:** an explicit absorption bound for one approximant monomial.
+**No new κ.** The section ceiling `10 p^2` and the section error
+`p^6 exp(-p^2)` stay hypotheses. The selected record remains `23/10000`.
+
+## Question
+
+#4485 absorbs `g (10 p^2)^{g-1} p^6 exp(-p^2)` under the gate
+`p ≥ max(64, 4K + 2C)` and `16 g ceil(log2 p) ≤ p`. If each of the `g - 1`
+factors is an approximant, the ceiling becomes `10 p^2 + p^6 exp(-p^2)`.
+Does
+
+```text
+g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(-p^2)
+```
+
+stay below `2^{-C p}` under that same gate?
+
+## Lemma
+
+Let `p ≥ 2`, `g ≥ 1`, `K ≥ 0` and `C ≥ 0` be integers, and let
+`L = ceil(log2 p)`. If
+
+```text
+p ≥ max(64, 4K + 2C)    and    16 g L ≤ p,
+```
+
+then
+
+```text
+g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(K p - p^2) < 2^{-C p}.
+```
+
+The case `K = 0` is the monomial in the question. A global factor `exp(K p)`
+does not change the conclusion. Whenever `g log p = o(p)`, every fixed pair
+`K, C` satisfies the gate for all large `p`.
+
+The additive section error is below `1/2` for every integer `p ≥ 64`, so
+
+```text
+10 p^2 + p^6 exp(-p^2) < 10 p^2 + 1 < 16 p^2.
+```
+
+The base therefore remains strictly below the power of two already used for
+`(10 p^2)^{g-1}`. No extra bit is spent. At `g = p^2` the same error is
+greater than 1 for every integer `p ≥ 2`, because it is larger than the
+inverse-ceiling monomial, which is already greater than 1.
+
+## Proof
+
+The series `1 + 1 + 1/2 = 5/2 > 2` gives `e > 2`, so `exp(-p^2) < 2^{-p^2}`.
+For `k ≥ 2`, `k! ≥ 2^{k-1}`: the values at 2, 3 and 4 are `2`, `6` and `24`,
+and the step uses `k + 1 ≥ 2`. The tail of `e` is then at most the geometric
+series of ratio `1/2`, so `e < 3 < 4` and `exp(K p) ≤ 2^{2 K p}`, with
+equality only at `K = 0`.
+
+For every integer `p ≥ 64`, `8 L ≤ p`. The base is `8 · 7 = 56 ≤ 64` at
+`p = 2^6`, and `8 ≤ 64`. If `8(m + 1) ≤ 2^m`, then
+`8(m + 2) ≤ 2^m + 8 ≤ 2^{m+1}`. On each dyadic interval the largest ratio of
+`L` to `p` is at the left endpoint. From `8 L ≤ p`, `48 L = 6 · 8 L ≤ 6 p`.
+The quadratic `4 p^2 - 3 p - 4` equals 6 at `p = 2` and increases by `8 p + 1`
+at the next integer, so it stays positive. Equivalently
+`6 p ≤ 8(p^2 - 1)`. Combining these estimates gives `6 L ≤ p^2 - 1`. Since
+`p ≤ 2^L`, `p^6 ≤ 2^{6 L} ≤ 2^{p^2 - 1}`, and therefore
+
+```text
+p^6 exp(-p^2) < p^6 2^{-p^2} ≤ 1/2.
+```
+
+Hence `10 p^2 + p^6 exp(-p^2) < 10 p^2 + 1`. Also `1 < 6 p^2` for every
+integer `p ≥ 2`, so `10 p^2 + 1 < 16 p^2`. The approximant base is strictly
+smaller than `16 p^2`.
+
+At `g = 1` the power `g - 1` is zero. For `g ≥ 2`,
+
+```text
+(10 p^2 + p^6 exp(-p^2))^{g-1} < (16 p^2)^{g-1}
+  = 2^{4(g-1)} p^{2(g-1)}.
+```
+
+In both cases
+
+```text
+g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(K p - p^2)
+  < g 2^{4g-4} p^{2g+4} 2^{2 K p - p^2}.
+```
+
+The gate forces `g ≤ p ≤ 2^L`, and `L ≥ 1`. The product is strictly less than
+
+```text
+2^{2 g L + 4 g + 5 L - 4 + 2 K p - p^2}.
+```
+
+Clear the eighths exactly as for the inverse ceiling. The four pieces scale
+as `8 · 2 g L = 16 g L ≤ p`, `8 · 4 g = 2 · 16 g ≤ 2 p`,
+`8 · 5 L = 5 · 8 L ≤ 5 p`, and `8 (2 K + C) p ≤ 4 p^2` once
+`p ≥ 4 K + 2 C`, because `8 (2 K + C) = 4 (4 K + 2 C)`. The coefficients of
+`p` are `1 + 2 + 5 = 8`. Doubling the exponent plus `C p` leaves at most
+`-(p^2 - 2 p + 8)`. That polynomial equals 8 at `p = 2` and increases by
+`2 p - 1`, so the doubled quantity is at most `-8` and the exponent is at
+most `-C p - 4`.
+
+If `g log p = o(p)`, then `g log2 p ≤ p/32` for large `p`. For `p ≥ 3`,
+`L ≤ 2 log2 p`, so the gate holds eventually, and every fixed `K` and `C`
+satisfy `p ≥ 4 K + 2 C` as well.
+
+At `g = p^2` the approximant monomial is larger than
+`g (10 p^2)^{g-1} p^6 exp(-p^2)`, since the added section error is positive
+and `g ≥ 2`. The smaller monomial equals
+`g (10 p^2)^g exp(-p^2) · p^4 / 10`. Here
+`(10 p^2)^{p^2} > 4^{p^2} > exp(p^2)` because `10 p^2 ≥ 40 > 4` and `e < 4`,
+while `p^6 / 10 > 1` because `p^6 ≥ 64 > 10`. That product is greater than 1,
+and so is the approximant monomial.
+
+## Not covered
+
+- A proof that a section inverse is at most `10 p^2`, or that a section
+  error is less than `p^6 exp(-p^2)`. The approximant ceiling is their sum.
+- The factor `2g` in place of `g`.
+- Editing the #4100 notes. They still say `g p^{10 g} exp(-p^2)`.
+- Any new κ.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/gaussian-tensor-approximant --check-only
+python3.12 -B scripts/research_package.py --package research/gaussian-tensor-approximant
+```
+
+Base `2a13eb1827784206202db90e5dc8327cf8a250db`, priority revision 7, selection
+refresh κ = `23/10000`.
+
+Overlap on 2026-10-22 (open PRs fetched 2026-10-22T01:09:00Z, 339 records;
+discussions fetched 2026-10-22T01:09:44Z, 3364 records) names this draft, #4485
+(the inverse-ceiling fold, which leaves this base open), #4484, discussion
+comment 19078965 (a factor `2g` on `g p^{10 g}`), the locality audit that
+keeps `g p^{10 g} exp(-p^2)`, and `research/gaussian-expansion-headroom`.
+`research/weighted-fourier-layout` uses the word approximant for a resampling
+mask. No other package assigns the base `10 p^2 + p^6 exp(-p^2)`. A search
+miss is not a novelty certificate. #4485 is the predecessor fold. HALO is the
+error sentence.
+
+## Open next questions
+
+1. **Does the face factor 2g still fit?** HALO partitions boundary outputs
+   into `2g` rectangles, and an earlier review note asks for `2g` in front of
+   the core error. The first step is to charge that extra factor of 2 against
+   the slack of 4 in the exponent above.
+
+## Credit
+
+- The tensor sentence and the gate `g log p = o(p)` are #4100's.
+- The ceiling `10 p^2` is Ryanson Jonathan's #4462, as carried through #4481.
+- The section error `p^6 exp(-p^2)` is the within-cell bound of #4452.
+- The inverse-ceiling fold is #4485. This package replaces the base.
+
+Prepared by James Reed with substantial Cursor Grok 4.7 assistance. Apache-2.0.

research/gaussian-tensor-approximant/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "2a13eb1827784206202db90e5dc8327cf8a250db",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ed33caef5c5827693e9572d252a66f99cbb73d40",
+      "sha256": "5732707cc1e460b090b53f2463162749ad860de878a18f7408938671d295a9d9"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "64bfc5af6709ed9811171e0a2108f05e1219cb65",
+      "sha256": "352a6184713b8843d96fd2c2071eb7aeaa655906b7c51d45ad444912666c8a96"
+    },
+    {
+      "path": "research/unconditional-batched-crt/HALO.md",
+      "usage": "input",
+      "git_blob": "5f601b3c8ac368aa79aa1007effe5933a360237b",
+      "sha256": "a2adcf18614983e9255def6d3f5e59be0018a392e418d65e94323722838289cc"
+    }
+  ]
+}

research/gaussian-tensor-approximant/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 7,
+  "base_commit": "2a13eb1827784206202db90e5dc8327cf8a250db",
+  "original_card": {
+    "id": "uniform-transfer-audit",
+    "question": "Does one quantified step of the proposed #4100 all-size construction hold at its actual caller?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/unconditional-batched-crt/README.md"
+    ],
+    "methods": [
+      "Prove one missing implication with explicit quantified hypotheses.",
+      "Find an exact counterexample at the claimed interface boundary."
+    ],
+    "acceptance_gate": "Identify the proposition and what would change downstream; do not equate a finite replay with a universal proof.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read PROOF.md and AUDIT.md in #4100. Select one distinct obligation: CRT total repair, native/free-algebra lifting, precision/conditioning, or complete fixed-tape cost. This is an author-submitted proof, not an accepted unconditional theorem.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Check live handoffs; name one proposition, its caller, quantified domain and dependency closure. Independently prove the implication or exhibit a precise gap/counterexample. Prioritize CRT.md total repair or COMPLEX.md free-algebra lift only if unclaimed. Stop at the chosen obligation; do not substitute finite samples or root arithmetic for an all-size proof, or launch the complete package by default.",
+    "bottleneck_link": "Finite supplier improvements are useful only under the retained transfer obligations.",
+    "allowed_variations": [
+      "One CRT repair, native lift, precision/conditioning or tape-cost proposition, with its exact caller."
+    ],
+    "scope_change_triggers": [
+      "Replacing a quantified obligation with another finite smoke test.",
+      "Treating archive intake, an author PASS receipt or a finite composed fixture as discharge of the universal hypotheses."
+    ],
+    "overlap_terms": [
+      "uniform transfer",
+      "all-size",
+      "precision",
+      "4100",
+      "unconditional-batched-crt"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4100: proposed unconditional construction and finite checks, without independent acceptance.",
+        "evidence": "research/unconditional-batched-crt/README.md",
+        "remaining": "Independent proof audit of one caller-bound all-size implication; all other premises remain explicit.",
+        "status": "author-checked",
+        "source_commit": "337d04939571a637aeab54a0f58f54fde811663a",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Decide whether g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(-p^2), times a global factor exp(K p), still falls below 2^{-C p} under the same gate as the inverse-ceiling fold.",
+  "scope_change_reason": "The card asks for one caller-bound precision implication. This package only checks that the added section error keeps the approximant base below 16 p^2. It does not prove the section ceiling or the section error, and it does not claim a new kappa.",
+  "bottleneck_link": "The selected frontier is unchanged. This is the approximant base left open by the telescoping absorption.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once the approximant monomial is compared with 2^{-C p} under the explicit gate. Do not replace g by 2g, do not edit the #4100 notes, and do not claim a new kappa.",
+  "outcome": "completed",
+  "evidence": "python3.12 -B scripts/research_package.py --package research/gaussian-tensor-approximant",
+  "author": "James Reed",
+  "ai_assistance": "Substantial help from Cursor Grok 4.7 on the absorption bound and the package text.",
+  "open_questions": [
+    {
+      "question": "Does 2g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(-p^2) stay below 2^{-C p} under the same gate?",
+      "why_it_matters": "HALO partitions boundary outputs into 2g rectangles, and a review note asks for a factor 2g on the core error.",
+      "first_step": "Charge the extra factor 2 against the slack of 4 in the exponent proved here.",
+      "blocked_by": "The section ceiling 10 p^2 and the section error p^6 exp(-p^2) remain conditional on the ideal gaps."
+    }
+  ]
+}

research/gaussian-tensor-approximant/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "certificates/selected-result.json",
+    "research/unconditional-batched-crt/HALO.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Under p >= max(64, 4K+2C) and 16 g ceil(log2 p) <= p, the approximant monomial g (10 p^2 + p^6 exp(-p^2))^{g-1} p^6 exp(Kp - p^2) is below 2^{-C p}; g = p^2 leaves that error above 1."
+    }
+  ]
+}