research: refined-induction ceiling on #4436's direct flow grows with lockstep depth (log-gap g, no new κ)

research/direct-flow-log-gap/README.md

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+# The refined-induction ceiling on #4436's direct flow grows with the lockstep depth
+
+**Claim type:** a first-order argument plus finite floating-point evidence. **No new κ.** It follows
+my correction of [#4459](https://github.com/CrocSwap/integer-mult-bounds/pull/4459). There, leaf exits
+turned out to dominate, so the question becomes what the total exit coefficient of zxv's
+[#4436](https://github.com/CrocSwap/integer-mult-bounds/pull/4436) direct flow can certify under
+#4405's refined induction (`κ(N, z0) ≤ η = z0·m^{z0−1}`).
+
+## First-order picture
+
+- **Balance at the root.** At the unit root `t*`, where `Σ_k W_k k^t = Σ_r P_r r^t` (raw p7 gives
+  `t* = 0.9973135`, `s* = 0.0026865`), exits and sources carry equal `t*`-potential. The verifier
+  checks this to `10⁻⁹` at every rank.
+- **The log-gap.** Define `g(N) = E_S[ln s] − E_X[ln e]`, with `t*`-potential weights. To first order:
+
+      κ(N, z0) ≈ exp(−(z0 − t*)·g),      η(z0) ≈ exp(−(1 − z0)(1 + ln m)).
+
+- **The ceiling.** The test therefore certifies savings up to about
+
+      cap(N) = s*·g / (g + 1 + ln m),     with 1 + ln 121 = 5.796.
+
+  The first-order formula matches the exact κ to `3·10⁻⁵` across the grid.
+
+## Measurements (raw p7)
+
+`g` is a sawtooth in `N`: it drops each time the lockstep depth `G` increases. Its per-level minima
+nevertheless rise steadily, by about 0.23 per level, which is a factor of about 5 in `N`.
+
+| G | rank of minimum | min g | cap at minimum |
+| --- | --- | --- | --- |
+| 1 | 1,000 | 1.977 | 0.000683 |
+| 2 | 3,071 | 2.204 | 0.000740 |
+| 3 | 16,135 | 2.538 | 0.000818 |
+| 4 | 76,886 | 2.751 | 0.000865 |
+| 5 | 384,675 | 2.954 | 0.000907 |
+| 6 | 1,924,603 | 3.207 | 0.000958 (local run) |
+| 7 | 9,629,156 | 3.373 | 0.000990 (local run) |
+
+Rows 1–5 rerun in the verifier, in about 7 seconds. Rows 6–7 come from
+`evidence/gap_large.py 2e7`, whose output is `evidence/LOGGAP-LARGE.txt`, through `N = 1.73·10⁷`.
+
+Consistency checks:
+
+- In the verifier, saving 0.0008 passes at every grid rank `N ≥ 3·10⁵`, and saving 0.001 still
+  fails at some.
+- #4456's effective savings (0.000634 at 10³, 0.000835 at 3·10⁴) sit just under these caps.
+
+## What drives the growth
+
+Decompose the log-depth `E ln(N/·)` at the per-level minima. Columns 2–5 are checked in the verifier;
+columns 6–7 are local.
+
+| G | 2 | 3 | 4 | 5 | 6 | 7 |
+| --- | --- | --- | --- | --- | --- | --- |
+| leaf exits | 5.020 | 5.129 | 5.509 | 5.701 | 6.026 | 6.209 |
+| boundary exits | 1.859 | 1.842 | 1.814 | 1.832 | 1.848 | 1.841 |
+| sources | 2.453 | 2.362 | 2.454 | 2.440 | 2.458 | 2.462 |
+
+Only the leaf exits sink, and they carry about 91% of the exit potential. A leaf exit is `rρ`, with
+`ρ` a product of `G` letters of `P` under the barrier `mρr ≤ N`, weighted by `cnt(ρ)ρ^t`. In log
+scale this is a random walk with tilted step mean `1.6259`, killed above the barrier
+`ln(N/m) ≈ G ln 5`. The relative drift is small and positive: `1.6259 − ln 5 = +0.0165`. A walk with
+positive drift, conditioned to stay below a line of matching slope, behaves like a meander. Its end
+gap grows like `√G`, not linearly. The data cannot separate `a + b√G` (with `b ≈ 0.97`) from linear
+growth of about 0.24 per level.
+
+## Reading
+
+- **The ceiling is not bounded away from `s*`.** If `g → ∞`, the cap tends to `s*`, so #4436's
+  direct exponent gate plausibly closes in the limit.
+- **But extremely slowly.** A saving `s` needs `g ≥ s(1 + ln m)/(s* − s)`. Saving 0.001 needs
+  `g ≈ 3.44`, about level 8 (`N ~ 5·10⁷`). The selected 0.0023 needs `g ≈ 34.5`.
+  - Under linear growth that is roughly 135 more levels, `N ~ 10^{100}`.
+  - Under the meander-motivated `√G` law it is `G ≈ 1,300`, `N ~ 10^{900}`.
+
+  Both are heuristic extrapolations, not bounds.
+- **The tensor/padding route is not superseded.** The direct flow is not a practical substitute for
+  it at κ near 0.0023, unless the induction itself is improved, for example by using several steps
+  of exits.
+
+## Not covered
+
+- A proof that `g → ∞`, or a rate for it.
+- Second-order terms beyond the measured `3·10⁻⁵`.
+- Native premises.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/direct-flow-log-gap --check-only
+python3.12 -B scripts/research_package.py --package research/direct-flow-log-gap
+cd research/direct-flow-log-gap && python3.12 -B evidence/gap_large.py 2e7   # ~2 min, rows 6-7
+```
+
+The verifier prints `PASS_DIRECT_FLOW_LOG_GAP`, using only the standard library.
+
+## Open next questions
+
+1. **Prove the leaf meander law: is `E ln(N/e)` for leaves of order `√G`?**
+   - *Why it matters:* it gives the rate at which #4436's direct saving approaches the root.
+   - *First step:* a ballot or meander estimate for the tilted P-walk below `G ln 5`, with drift
+     `+0.0165`.
+2. **Can a multi-step exit test beat the single-step ceiling?**
+   - *First step:* substitute the exits' own flows once, giving a two-step coefficient, and remeasure.
+
+## Credit
+
+- The direct flow and the gate are zxv's (#4436).
+- The refined induction is 1337isnot1337's (#4405).
+- The lockstep construction is Jacob Alan Sussman's Theorem 1G.
+
+Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.
+Apache-2.0.

research/direct-flow-log-gap/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"
+    }
+  ]
+}

research/direct-flow-log-gap/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "independent-moment",
+  "priority_revision": 7,
+  "base_commit": "2a13eb1827784206202db90e5dc8327cf8a250db",
+  "original_card": {
+    "id": "independent-moment",
+    "question": "Does a new paid-profile or mixture-ceiling claim have independent source-bound coverage of its actual changed ledger?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/staged544-mixed-fourth-entry/README.md",
+      "research/staged-fourth-mixture-ceiling/README.md",
+      "research/r437-two-window-carrier/README.md"
+    ],
+    "methods": [
+      "Independently reconstruct one changed paid ledger or pooled dual with exact source and model binding.",
+      "Check a genuinely uncovered moment implication with a different enclosure method and semantic controls."
+    ],
+    "acceptance_gate": "Demonstrate independence, source binding, positive cases and semantic failure controls. Name the existing checks and the missing coverage. More digits or another equivalent evaluation of an unchanged root is not by itself new verification coverage.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Choose one uncovered implication in #4088 literal mixed fill, #4091 pooled dual/model binding or #4067 changed frame/host bill. #4091 pins an earlier #4088 head: validate word, capacity and stock equivalence before applying that ceiling to the later fill.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Independently reconstruct one changed ledger or pooled certificate with a positive case and a semantic failure control. Compare exact sources and name what existing checks miss. Keep this separate from supplier-boundary-audit (ordinary/outer dispatch) and uniform-transfer-audit (all-size proof). Stop after the chosen implication; more digits of an unchanged root are not additional mechanism coverage.",
+    "bottleneck_link": "Independent pricing prevents unsupported frontier claims and can reveal genuine accounting slack.",
+    "allowed_variations": [
+      "Reconstruct either a source ledger or a numeric enclosure with a different method."
+    ],
+    "scope_change_triggers": [
+      "Importing the primary function and calling its output independent verification.",
+      "Calling an aggregate histogram or source archive an independently accepted physical supplier."
+    ],
+    "overlap_terms": [
+      "independent moment",
+      "rank histogram",
+      "exact enclosure",
+      "2979",
+      "rank7 joint endframes",
+      "paired extraction",
+      "joint lowering",
+      "4088",
+      "4091",
+      "4067",
+      "staged544-mixed-fourth-entry",
+      "staged-fourth-mixture-ceiling",
+      "r437-two-window-carrier"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4088: mixed local/owner/phase ledger and moment reported.",
+        "evidence": "research/staged544-mixed-fourth-entry/README.md",
+        "remaining": "Independent binding of changed construction to paid histogram, distinct from full admission.",
+        "status": "author-checked",
+        "source_commit": "72d7efc181edb21806a878f174847c96dd410ab0",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4091: sharp pooled ceiling proposed for fixed THREE/FOUR words and inventory.",
+        "evidence": "research/staged-fourth-mixture-ceiling/README.md",
+        "remaining": "Validate dual, exhaustive pattern scope and exact source equivalence before applying to newer fills.",
+        "status": "author-checked",
+        "source_commit": "a29bdd8f56cd18b805917775cdec09ff935eddc4",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4067: paid (2,2) to (1,3) exchange and fixed OQ4 host reported.",
+        "evidence": "research/r437-two-window-carrier/README.md",
+        "remaining": "Independent full cost delta; remaining caller/precision/stock obligations are separate.",
+        "status": "author-checked",
+        "source_commit": "a95f11c52e3ca1e4c5171b3c20d8f4b0815aee2e",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Ceiling of #4405's refined-induction test on #4436's direct flow via the t*-weighted log-gap g(N).",
+  "scope_change_reason": "Follow-up to the #4459 correction (leaf exits dominate).",
+  "bottleneck_link": "Quantifies how far a direct-flow exponent (no tensor/padding) is from the selected saving.",
+  "budget_minutes": 45,
+  "stop_rule": "Stop at measurements to N ~ 1.7e7 and the first-order cap.",
+  "outcome": "completed",
+  "evidence": "Balance at t*; kappa ~ exp(-(z0-t*)g) within 3e-5; per-level minima of g: 1.98, 2.20, 2.54, 2.75, 2.95 (verifier), 3.21, 3.37 (local to 1.7e7); first-order cap 0.00068 -> 0.00099. Saving 0.0023 would need g ~ 34.5 (heuristic N ~ 1e100). Growth comes from leaf exits alone (leaf log-depth 5.02->6.21; boundary ~1.84 and sources ~2.45 fixed); tilted P-step mean exceeds ln 5 by 0.0165, suggesting a sqrt(G) meander law.",
+  "author": "Ryanson Jonathan, with Claude Code (Claude Opus 5.5) assistance",
+  "ai_assistance": "Substantial Claude Code (Claude Opus 5.5) assistance; no independent human review.",
+  "open_questions": [
+    {
+      "question": "Is the leaf log-depth of order sqrt(G) (meander law)?",
+      "why_it_matters": "Gives the rate at which the direct saving approaches the root.",
+      "first_step": "Ballot/meander estimate for the tilted P-walk below G ln 5 with drift +0.0165.",
+      "blocked_by": "None."
+    }
+  ]
+}

research/direct-flow-log-gap/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "#4436 direct p7 flow, 5% grid 1000..5e5: t*-potential balance (1e-9), first-order kappa ~ exp(-(z0-t*)g) within 3e-5, per-level minima of the log-gap g increase with G (levels 2..5), saving 0.0008 passes and 0.001 fails at N>=3e5. Leaf-only growth: boundary/source log-depths stable within 0.1, leaf log-depth increasing; tilted P-step mean > ln 5."
+    }
+  ]
+}