From 777b642e85eb81564d1774e4747cb2313f0b54ec Mon Sep 17 00:00:00 2001
From: Nick Petalas <webmasternikos@gmail.com>
Date: Wed, 11 Nov 2026 08:05:33 +0000
Subject: [PATCH] feat(research): certify fixed-profile helper split ceilings

---
 .../helper-split-headroom-ladder/README.md    |  210 ++
 .../helper-split-headroom-ladder/RESULT.json  | 1893 +++++++++++++++++
 .../helper-split-headroom-ladder/SOURCES.json |   30 +
 .../helper-split-headroom-ladder/TASK.json    |  135 ++
 .../helper-split-headroom-ladder/VERIFY.json  |   35 +
 .../helper-split-headroom-ladder/intervals.py |  117 +
 .../helper-split-headroom-ladder/verify.py    |  377 ++++
 7 files changed, 2797 insertions(+)
 create mode 100644 research/helper-split-headroom-ladder/README.md
 create mode 100644 research/helper-split-headroom-ladder/RESULT.json
 create mode 100644 research/helper-split-headroom-ladder/SOURCES.json
 create mode 100644 research/helper-split-headroom-ladder/TASK.json
 create mode 100644 research/helper-split-headroom-ladder/VERIFY.json
 create mode 100644 research/helper-split-headroom-ladder/intervals.py
 create mode 100644 research/helper-split-headroom-ladder/verify.py

diff --git a/research/helper-split-headroom-ladder/README.md b/research/helper-split-headroom-ladder/README.md
new file mode 100644
index 0000000000..7858840be6
--- /dev/null
+++ b/research/helper-split-headroom-ladder/README.md
@@ -0,0 +1,210 @@
+# Fixed-profile helper-split headroom on the #7000 invoice
+
+This package was produced by Nick Petalas.
+
+Claim type: scoped written proof with exact finite certificates. On the pinned
+#7000 invoice, the greatest first crossing obtainable by replacing each helper
+with at most two positive-integer-rank pieces is approximately
+0.004689286593684114. Allowing all rank-one pieces gives approximately
+0.011115356648105199. These are optimistic raw-margin ceilings on
+**0 ≤ κ ≤ 0.02**, with the paid profile and every main-origin target fixed.
+They do not establish a legal split or a new conditional κ.
+
+## Model and exact statement
+
+The source is `research/shared-wire-split-composition/FULL-RESULT.json`,
+SHA256 `d9ba709fa5dcefffe3f3abdcf34600ce188224703cfd743e6db8b58ba4816cb7`.
+[SOURCES.json](SOURCES.json) pins its Git blob and base commit. The source
+construction and its admission are taken as premises, not replayed here.
+
+Let T_r and P_r be that file's complete productive and paid histograms. Define
+
+    F_0(κ) = Σ_r (T_r − P_r) r^(1−κ).
+
+The helper histogram h_r is:
+
+| r | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
+|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|
+| h_r | 13 | 18 | 44 | 215 | 68 | 42 | 18 | 67 | 595 |
+
+The invoice has T_r = 18h_r for r ≤ 8 and T_9 = 18h_9 + 416.
+All targets above rank 9, and the extra 416 rank-nine targets, are fixed.
+There are 1,080 original helpers, with total rank 7,650. The factor 18 counts
+the eighteen LOCAL invocations. One helper split means the same replacement
+in all eighteen invocations.
+
+Replace an original helper of rank r with a partition λ = (a_1,…,a_j),
+a_i ≥ 1 and Σ_i a_i = r. Its change to the complete raw margin is exactly
+
+    Δ_λ(κ) = 18 (Σ_i a_i^(1−κ) − r^(1−κ)).
+
+No new paid jobs are charged in this relaxation. For a piece cap m, let
+b(r,m) be the balanced partition of r into min(r,m) parts. If
+r = qj + s, 0 ≤ s < j, it has s parts q+1 and j−s parts q. Let
+
+    F_m(κ) = F_0(κ) + Σ_r h_r Δ_b(r,m)(κ).
+
+For each m = 1,…,9, F_m has exactly one zero κ_m on [0,0.02]. Among all
+allowed partitions with at most m **final pieces per original helper**, the
+largest first zero on this interval is κ_m, attained algebraically by b(r,m)
+for every helper. A sequence of binary splits with more than two final pieces
+is outside the m=2 row. Any actual split-only continuation satisfying these
+hypotheses has its first zero no later than κ_m.
+
+| Maximum final pieces m | κ_m, approximate |
+|---:|---:|
+| 1 | 0.003630058174688367 |
+| 2 | 0.004689286593684114 |
+| 3 | 0.005653731986780233 |
+| 4 | 0.006587917894714106 |
+| 5 | 0.007413801865724420 |
+| 6 | 0.008144674208796709 |
+| 7 | 0.008982372525120905 |
+| 8 | 0.009988261544163143 |
+| 9 | 0.011115356648105199 |
+
+[RESULT.json](RESULT.json) gives rational open brackets of width 2×10^-45
+for the roots, rather than treating the displayed decimals as exact roots.
+The original source's stored crossing is reproduced numerically as a model
+calibration. This is not an independent verification of its construction.
+
+## Proof
+
+For 0 < κ ≤ 0.02, x^(1−κ) is strictly concave on the positive reals.
+If two integer pieces differ by at least two, moving one rank unit from the
+larger to the smaller strictly increases their total credit. Repeating this
+finite operation balances every fixed-length partition. If fewer than
+min(r,m) pieces are used, splitting a piece of size at least two strictly
+increases credit, because a^p+b^p > (a+b)^p for 0<p<1. Thus b(r,m) maximizes
+credit for every κ in the stated interval. This classical balancing argument
+is also the zero-fee case of #6651; the optimization principle is not new here.
+Summing over helpers gives F_candidate(κ) ≤ F_m(κ) pointwise.
+
+Every partition preserves rank mass. Hence every candidate has
+F_candidate(0) = F_m(0) = 1,520. The verifier proves F_m(0.02)<0 and strict
+monotonicity on [0,0.02]. Consequently each F_m has exactly one zero. At κ_m,
+F_candidate ≤ 0, so continuity gives a candidate zero no later than κ_m.
+Balanced splitting attains κ_m. The ladder is strictly increasing: at every
+κ>0, the 595 rank-nine helpers gain strictly when m rises from 1 through 9.
+
+Here is the numerical part of the uniqueness proof. For coefficients c_r of
+one scenario, put
+
+    G(κ) = −F'(κ) = Σ_r c_r r^(1−κ) ln r,
+    L = Σ_r |c_r| r (ln r)^2.
+
+For κ≥0, |G'(κ)|≤L. On the eleven grid points κ=j/500, j=0,…,10,
+the verifier proves G(κ)>L/500 by outward intervals. Every point in
+[0,0.02] is within 1/500 of a grid point, so G>0 throughout. Exact sign
+checks at both rational endpoints of each root bracket then certify the
+unique root. This argument is applied to every ladder and single-split
+scenario, and to the four stock-sufficient plans below.
+
+The interval restriction matters. At κ=1 the full-split margin is exactly
+24,040>0, while F_9(0.02)<0. Therefore another zero exists above 0.02;
+this package does not assert global single crossing or a ceiling on [0,1].
+
+## Exchange rates and stock
+
+RESULT contains all 87 nontrivial integer partitions of ranks 2,…,9. Each row
+records Δ_λ at the original crossing, the change in its first root after one
+helper replacement, and the least integer n for which
+
+    F_0(w) + n Δ_λ(w) > 0.
+
+The two comparison prices are the complex wrapper's fixed BIT pin
+w=0.00376669 and the paired BIT ordinary admission w=0.00385248. The latter
+is an accounting comparison point, not a claim that the complex wrapper can
+use that price unchanged. These prices come from the two pinned input files.
+The least count is proved by a negative interval at n−1 and a positive one
+at n. Rows whose counts exceed h_r are explicitly marked unavailable.
+
+At either price the best gain from touching one helper comes from rank nine.
+The verifier certifies this against each rank 2,…,8. Available rank-nine stock
+suffices, so the following minima hold even when helper ranks may be mixed:
+
+| Comparison price | Any partition | At most two final pieces |
+|---|---:|---:|
+| 0.00376669 | 43 helpers, all 9→1+…+1 | 137 helpers, all 9→5+4 |
+| 0.00385248 | 68 helpers, all 9→1+…+1 | 218 helpers, all 9→5+4 |
+
+For the lower bound, no n−1 touched helpers can supply more than n−1 copies
+of the best per-helper gain. For attainability in the relaxation, take n
+rank-nine helpers. The stock is 595. Both sides are exact interval checks.
+For comparison, using only 9→8+1 needs 269 and 427 helpers respectively.
+
+## Certificate arithmetic and verification limits
+
+`intervals.py` uses integers scaled by Q=10^70. To enclose ln r, write
+r=2^a u with 1≤u<2, and use the first 100 terms of
+2 atanh z = 2Σ z^(2j+1)/(2j+1), z=(u−1)/(u+1). The nonnegative remainder
+is at most 2z^201/(201(1−z²)); the same formula supplies ln 2.
+Every conversion to Q is rounded outward.
+
+To enclose r^(1−κ), evaluate r/exp(κ ln r). Here 0≤κ ln r<1/8, since the
+largest rank is 145. Eighty positive exponential terms are evaluated with
+outward integer rounding. Twice the next term bounds the remaining tail,
+since successive ratios are below 1/2. Reciprocal bounds are also outward.
+`Decimal` logarithms and Newton steps only nominate rational root endpoints
+and printed approximations; they cannot make a failing interval check pass.
+All printed receipt approximations have absolute error below 10^-39.
+
+Every rank's power enclosure at κ=1/100 is additionally checked using exact
+integer 100th powers. A separate combinations enumeration checks the partition
+table, and exact majorization checks corroborate the balancing lemma. Five
+negative controls reject a changed helper count, an out-of-range κ, an
+unbalanced purported minimizer, a bracket missing the root and reversed
+monotonicity. A sixth control checks signs around the original crossing.
+
+Power-of-ten scope is the two Python files. Loops are bounded by the pinned
+rank set, ranks 1…9, 100 logarithm terms, 80 exponential terms, 60 Newton
+steps, eleven grid points, and the finite partition list. Python's managed
+allocation is a rule-3 deviation: both inputs are capped at one megabyte,
+there are at most 145 ranks, and the independent enumeration has fewer than
+100,000 candidate tuples. No process, network access, recursion or dynamic
+code generation is used. Static compilation and `-W error` replays cover the
+available Python diagnostics. No shared build configuration is changed.
+
+## Reproduce
+
+From the repository root, with CPython 3.12 and the standard library:
+
+```sh
+~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/helper-split-headroom-ladder/verify.py
+~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/helper-split-headroom-ladder/verify.py
+~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/helper-split-headroom-ladder
+~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/helper-split-headroom-ladder --check-source-history
+```
+
+The first two commands rebuild the hash-bound receipt and compare every field.
+To regenerate it, redirect `verify.py --emit` to a temporary file and compare
+before replacing RESULT.json. The receipt binds both verifier files and both
+scientific inputs. It does not bind its own bytes.
+
+Legality of the candidate splits, their pivot/cut/boundary bases, their
+ADD/SCALE and ordinary costs, native/compiler admission and every outer gate
+remain unproved here. No selected replay, Lean or Docker run is claimed.
+Independent human and formal proof-kernel review are unclaimed.
+
+## Open next questions
+
+- Which rank-nine helpers admit legal 5+4 splits with no added paid rank?
+  A small exact prefix/suffix witness and complete cost ledger would test
+  whether the 137-helper wrapper threshold is relevant to construction.
+  Physical availability and shared correction costs remain the blockers.
+- How does a changed paid profile alter these ceilings? First substitute an
+  exact successor invoice and redo its helper-origin decomposition; the
+  current ladder does not apply automatically to changed callers or tariffs.
+
+## Credit and related work
+
+Jerry Chen / CubeJerry's #7000 supplies the complete invoice and paired BIT
+pin, building on DreamingOfClouds's #6936/#6937, CubeJerry's #6980 caller,
+Ali Romman's helper-splitting method and the retained P104 lineage. The
+current construction chain includes dillona, ToikY and Rohan Arun. zxv's
+#5554 proves a different fixed-local-bill path ceiling. Nick Petalas's #6651
+already records classical balanced-partition optimality in a uniform-fee
+model; that theorem and its thresholds are not claimed here. The new result
+is the source-specific first-crossing ladder, exhaustive exchange table and
+stock-constrained helper minima on the #7000 invoice. New material is
+Apache-2.0; predecessor files and their notices are unchanged.
diff --git a/research/helper-split-headroom-ladder/RESULT.json b/research/helper-split-headroom-ladder/RESULT.json
new file mode 100644
index 0000000000..41e82b360a
--- /dev/null
+++ b/research/helper-split-headroom-ladder/RESULT.json
@@ -0,0 +1,1893 @@
+{
+ "claim_interval": [
+  "0",
+  "0.02"
+ ],
+ "exchange": [
+  {
+   "available": 18,
+   "available_paired": false,
+   "available_pin": false,
+   "gain_at_k0": "0.0904680612098380197772025320730753825889",
+   "need_paired": 963,
+   "need_pin": 605,
+   "parts": [
+    1,
+    1
+   ],
+   "r": 2,
+   "root_bracket": [
+    "0.0036302758759103188281331680459057387006652741645941034221454413836717092245561319",
+    "0.0036302758759103188281331680459057387006652761645941034221454413836717092245561319"
+   ],
+   "single_split_dk": "2.177012219517421995912810088390280E-7"
+  },
+  {
+   "available": 44,
+   "available_paired": false,
+   "available_pin": false,
+   "gain_at_k0": "0.2149245847170514173720008534378509293051",
+   "need_paired": 406,
+   "need_pin": 255,
+   "parts": [
+    1,
+    1,
+    1
+   ],
+   "r": 3,
+   "root_bracket": [
+    "0.0036305754094334052680042440234620759754921576525748688603641392139741149704593299",
+    "0.0036305754094334052680042440234620759754921596525748688603641392139741149704593299"
+   ],
+   "single_split_dk": "5.172347450381820706672585651763028E-7"
+  },
+  {
+   "available": 44,
+   "available_paired": false,
+   "available_pin": false,
+   "gain_at_k0": "0.1244565235072133975947983213647755467161",
+   "need_paired": 700,
+   "need_pin": 440,
+   "parts": [
+    2,
+    1
+   ],
+   "r": 3,
+   "root_bracket": [
+    "0.0036303576720839125919258221730957494567489899424240687300300967978547151843126388",
+    "0.0036303576720839125919258221730957494567489919424240687300300967978547151843126388"
+   ],
+   "single_split_dk": "2.994973955455059922454081988497841E-7"
+  },
+  {
+   "available": 215,
+   "available_paired": false,
+   "available_pin": true,
+   "gain_at_k0": "0.3614175520560705791871057769197515013931",
+   "need_paired": 241,
+   "need_pin": 152,
+   "parts": [
+    1,
+    1,
+    1,
+    1
+   ],
+   "r": 4,
+   "root_bracket": [
+    "0.0036309280423660913103233023071819101888556913771190937504217073206079550146099625",
+    "0.0036309280423660913103233023071819101888556933771190937504217073206079550146099625"
+   ],
+   "single_split_dk": "8.698676777242243897255422850105162E-7"
+  },
+  {
+   "available": 215,
+   "available_paired": false,
+   "available_pin": true,
+   "gain_at_k0": "0.2709494908462325594099032448466761188042",
+   "need_paired": 322,
+   "need_pin": 202,
+   "parts": [
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+   "root_bracket": [
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+    "0.0036307102624965992280940897569851996971086006567073506681216806734459144576043274"
+   ],
+   "single_split_dk": "6.520878082321421605129920883000244E-7"
+  },
+  {
+   "available": 215,
+   "available_paired": false,
+   "available_pin": false,
+   "gain_at_k0": "0.1804814296363945396327007127736007362152",
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+   "need_pin": 304,
+   "parts": [
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+   ],
+   "r": 4,
+   "root_bracket": [
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+    "0.0036304925089136074965059336419637865922915290099655082690699649717121247072894698"
+   ],
+   "single_split_dk": "4.343342252404105723568770668869196E-7"
+  },
+  {
+   "available": 215,
+   "available_paired": false,
+   "available_pin": false,
+   "gain_at_k0": "0.1464929673390191618151049234819005720880",
+   "need_paired": 595,
+   "need_pin": 374,
+   "parts": [
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+   "root_bracket": [
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+    "0.0036304107066560455194213648563977956527777659485396789809643215412942962857214802"
+   ],
+   "single_split_dk": "3.525319676784334877880915008959801E-7"
+  },
+  {
+   "available": 68,
+   "available_paired": false,
+   "available_pin": false,
+   "gain_at_k0": "0.5242787903436374051223036565058894113746",
+   "need_paired": 167,
+   "need_pin": 105,
+   "parts": [
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+    1,
+    1
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+    "0.0036313201571356087770777635346821848256516349710901972047050317576774472097992959"
+   ],
+   "single_split_dk": "0.0000012619824472416911441867697852851530"
+  },
+  {
+   "available": 68,
+   "available_paired": false,
+   "available_pin": false,
+   "gain_at_k0": "0.4338107291337993853451011244328140287856",
+   "need_paired": 201,
+   "need_pin": 127,
+   "parts": [
+    2,
+    1,
+    1,
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+   ],
+   "r": 5,
+   "root_bracket": [
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+    "0.0036311023299931297889039321430714127463893083448250376431137150908970940278296343"
+   ],
+   "single_split_dk": "0.0000010441553047627029703553781745130737"
+  },
+  {
+   "available": 68,
+   "available_paired": false,
+   "available_pin": false,
+   "gain_at_k0": "0.3433426679239613655678985923597386461967",
+   "need_paired": 254,
+   "need_pin": 160,
+   "parts": [
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+    2,
+    1
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+   "r": 5,
+   "root_bracket": [
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+    "0.0036308845291457525000071360094727393846747565606520611073600555206938008728304198"
+   ],
+   "single_split_dk": "8.263544573854140735592445758397120E-7"
+  },
+  {
+   "available": 68,
+   "available_paired": false,
+   "available_pin": false,
+   "gain_at_k0": "0.3093542056265859877503028030680384820695",
+   "need_paired": 282,
+   "need_pin": 177,
+   "parts": [
+    3,
+    1,
+    1
+   ],
+   "r": 5,
+   "root_bracket": [
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+    "0.0036308027091744660731526713290892599179175717925572211898001693383853229135323274"
+   ],
+   "single_split_dk": "7.445344860989872190945641923602452E-7"
+  },
+  {
+   "available": 68,
+   "available_paired": false,
+   "available_pin": false,
+   "gain_at_k0": "0.2188861444167479679731002709949630994806",
+   "need_paired": 398,
+   "need_pin": 250,
+   "parts": [
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+    "0.0036305849444704963514191085775384692310959603742536101194405600989697874801670342"
+   ],
+   "single_split_dk": "5.267697821292654855318126415695584E-7"
+  },
+  {
+   "available": 68,
+   "available_paired": false,
+   "available_pin": false,
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+   "need_paired": 535,
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+    "0.0036304501007729833525473642540064493184380870903883033292966469494140612820296293"
+   ],
+   "single_split_dk": "3.919260846162666137874891095496458E-7"
+  },
+  {
+   "available": 42,
+   "available_paired": false,
+   "available_pin": false,
+   "gain_at_k0": "0.7001731413699679371519753970232503972578",
+   "need_paired": 125,
+   "need_pin": 79,
+   "parts": [
+    1,
+    1,
+    1,
+    1,
+    1,
+    1
+   ],
+   "r": 6,
+   "root_bracket": [
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+    "0.0036317437466421347055746639732801521903227695681616145162374901545055500781408632"
+   ],
+   "single_split_dk": "0.0000016855719537676196410872083832525176"
+  },
+  {
+   "available": 42,
+   "available_paired": false,
+   "available_pin": false,
+   "gain_at_k0": "0.6097050801601299173747728649501750146689",
+   "need_paired": 143,
+   "need_pin": 90,
+   "parts": [
+    2,
+    1,
+    1,
+    1,
+    1
+   ],
+   "r": 6,
+   "root_bracket": [
+    "0.0036315258684368739665475810229415366533816091366572565284063071250862277691688597",
+    "0.0036315258684368739665475810229415366533816111366572565284063071250862277691688597"
+   ],
+   "single_split_dk": "0.0000014676937485068806140042580446369807"
+  },
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+ ],
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+  "5": 68,
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+ },
+ "inputs": {
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+  "bit-admission.json": "a46b6c6040f75184cab5467656811ad988c216e3bc8e44d595a5015b6e49620b"
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+ "kappa0": "0.0036300581746883670859335767648968996727",
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+ ],
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+   ]
+  },
+  {
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+   "kappa": "0.0099882615441631426112874665151275098172",
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+    "0.0099882615441631426112874665151275098171914757838907518750478224969230227395738845"
+   ]
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+  {
+   "gain": "0.0074852984734168317239908490046282095066",
+   "kappa": "0.0111153566481051988099244257695251091792",
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+    "0.011115356648105198809924425769525109179247988145785192105376298125264263714240760"
+   ]
+  }
+ ],
+ "lemma_partition_checks": 436,
+ "min_touched": {
+  "paired_any": {
+   "helpers": 68,
+   "plan": [
+    [
+     9,
+     68
+    ]
+   ]
+  },
+  "paired_binary": {
+   "helpers": 218,
+   "plan": [
+    [
+     9,
+     218
+    ]
+   ]
+  },
+  "pin_any": {
+   "helpers": 43,
+   "plan": [
+    [
+     9,
+     43
+    ]
+   ]
+  },
+  "pin_binary": {
+   "helpers": 137,
+   "plan": [
+    [
+     9,
+     137
+    ]
+   ]
+  }
+ },
+ "numeric_error_bound": "1e-39",
+ "verifier_sha256": {
+  "intervals.py": "3d876b3e0a8589cdb8eb16e749e531b9ece04d6c1387df6eb7f27730846ac1f4",
+  "verify.py": "e37cff87b70c52df45f59465f0607c59d6a94704c5a4f61b9699d3fbfbf2ff53"
+ },
+ "wall_margin_F": {
+  "paired": "-92.3927683626391439562416693993238103048527",
+  "pin": "-56.7661715024831636841231888993000674098476"
+ },
+ "walls": {
+  "paired": "0.0038524800000000000000000000000000000000",
+  "pin": "0.0037666900000000000000000000000000000000"
+ }
+}
diff --git a/research/helper-split-headroom-ladder/SOURCES.json b/research/helper-split-headroom-ladder/SOURCES.json
new file mode 100644
index 0000000000..dac252d781
--- /dev/null
+++ b/research/helper-split-headroom-ladder/SOURCES.json
@@ -0,0 +1,30 @@
+{
+  "schema_version": 1,
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "75d77b328ea02ded535e7f23aee5b1e55d0bb1a8",
+      "sha256": "154c3d30455d778fd3343b6b4851f0fa4665f0f7625b0adb3675e2bafbbf6bf1"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/shared-wire-split-composition/FULL-RESULT.json",
+      "usage": "input",
+      "git_blob": "ff3256f0d3ffbfc56b062f9ec0fdbbc7b30e76d5",
+      "sha256": "d9ba709fa5dcefffe3f3abdcf34600ce188224703cfd743e6db8b58ba4816cb7"
+    },
+    {
+      "path": "research/shared-wire-split-composition/data/bit-admission.json",
+      "usage": "input",
+      "git_blob": "575f76487f9999ecc3e7b1b70f9f71bc582ef8a8",
+      "sha256": "a46b6c6040f75184cab5467656811ad988c216e3bc8e44d595a5015b6e49620b"
+    }
+  ]
+}
diff --git a/research/helper-split-headroom-ladder/TASK.json b/research/helper-split-headroom-ladder/TASK.json
new file mode 100644
index 0000000000..42987b469a
--- /dev/null
+++ b/research/helper-split-headroom-ladder/TASK.json
@@ -0,0 +1,135 @@
+{
+  "schema_version": 1,
+  "task_id": "selected-packing",
+  "priority_revision": 8,
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "original_card": {
+    "id": "selected-packing",
+    "question": "Is there meaningful uncovered packing headroom for an exact current word and boundary contract?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+      "docs/research/recursive-invoice-preflight.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "Check live overlap and exact model/source binding first. Distinguish a chronological construction from a pooled optimum. State the applicable fixed-inventory ceiling and remaining headroom; changed words or interfaces require a new fully charged model. Do not launch a residual search without an explicit scope and time cap.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Derive or authenticate a ceiling for the exact current paid/target inventory before searching. Old #1240/#4091 caps do not constrain #5599 automatically. Require an unclaimed domain, time cap and useful complete margin. Stop at the applicable ceiling; change the word or boundary instead of adding seeds.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "T16",
+      "owner exchange",
+      "packing",
+      "4091",
+      "4088",
+      "staged-fourth-mixture-ceiling",
+      "staged544-mixed-fourth-entry"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4091: pooled optimum at one-third staging for pinned fixed words; not a chronological fill.",
+        "evidence": "research/staged-fourth-mixture-ceiling/README.md",
+        "remaining": "Independent scope/source check; any residual optimization is explicitly bounded and optional.",
+        "status": "author-checked",
+        "source_commit": "a29bdd8f56cd18b805917775cdec09ff935eddc4",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4088: later 1090-call literal fill improves the earlier 1080-call witness.",
+        "evidence": "research/staged544-mixed-fourth-entry/README.md",
+        "remaining": "Complete admission remains separate; old residual-bin instructions must not be copied blindly.",
+        "status": "author-checked",
+        "source_commit": "72d7efc181edb21806a878f174847c96dd410ab0",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Exact first-crossing ceilings and helper-split exchange rates for the fixed paid and main-origin target profiles of selected #7000, on 0 <= kappa <= 0.02.",
+  "scope_change_reason": "The assigned packing card is narrowed to an optimistic accounting theorem. Legal split construction and complete supplier admission are outside this result.",
+  "bottleneck_link": "Quantify the rank-splitting credit available before searching for legal helper cuts; compare the exact selected wrapper price and the paired BIT ordinary price without treating either as an admitted complex improvement.",
+  "budget_minutes": 60,
+  "stop_rule": "Ship the fixed-profile ladder, all rank-2..9 single-helper partitions and the stock-constrained thresholds after exact interval verification. Do not extend to new words or paid-cost models.",
+  "outcome": "completed",
+  "evidence": "README.md written proof; RESULT.json hash-bound root brackets and finite exchange/minimum certificates; verify.py and intervals.py.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Which rank-nine helpers admit legal 5+4 splits with no added paid rank?",
+      "why_it_matters": "The relaxed threshold for the fixed complex wrapper price is 137 touched helpers.",
+      "first_step": "Give one exact prefix/suffix and boundary witness with a complete cost ledger.",
+      "blocked_by": "Literal pivot availability, restoration and any shared correction costs."
+    },
+    {
+      "question": "What ceilings hold for an exact successor paid profile and caller?",
+      "why_it_matters": "The present theorem is source-specific and does not apply automatically to newer construction proposals.",
+      "first_step": "Extract a source-bound helper-origin decomposition from a successor invoice before reusing the optimization.",
+      "blocked_by": "A changed paid/target invoice and its precise split contract."
+    }
+  ],
+  "claim_type": "scoped written proof with exact finite certificates",
+  "claim_interval": [
+    "0",
+    "0.02"
+  ],
+  "environment": {
+    "interpreter": "3.12.15 (main, Oct 14 2026, 00:55:37) [Clang 23.1.3 ]",
+    "reproduction_interpreter": "~/git/integer-mult-bounds/.venv/bin/python3.12",
+    "dependencies": "Python standard library"
+  },
+  "upstream_checkpoint": {
+    "commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+    "priority_revision": 8,
+    "checked_at": "2026-11-11T08:03:18.681948+00:00"
+  },
+  "overlap": {
+    "open_pr_snapshot_records": 468,
+    "discussion_snapshot_records": 7089,
+    "complete_remote_pagination": true,
+    "local_sparse_coverage": "Generic helper omitted 10001 HEAD note paths; additional git grep searched all main README blobs for distinctive formulas and terms. No matches is not proof of novelty.",
+    "related_general_theorems": [
+      "#5554 fixed-local-bill path ceiling",
+      "#6651 classical balanced-partition optimum with a uniform fee"
+    ],
+    "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19530156",
+    "own_recent_checked": [
+      7168,
+      7167,
+      7135,
+      7134,
+      7133,
+      7132
+    ]
+  },
+  "progress_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19537529"
+}
diff --git a/research/helper-split-headroom-ladder/VERIFY.json b/research/helper-split-headroom-ladder/VERIFY.json
new file mode 100644
index 0000000000..15778ccf23
--- /dev/null
+++ b/research/helper-split-headroom-ladder/VERIFY.json
@@ -0,0 +1,35 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/shared-wire-split-composition/FULL-RESULT.json",
+    "research/shared-wire-split-composition/data/bit-admission.json"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exact outward root/slope certificates; all 87 nontrivial partitions; independent partition enumeration and integer-power oracle; minimal wall counts and six controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-W",
+        "error",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "The same scientific certificates and controls with assertions disabled."
+    }
+  ]
+}
diff --git a/research/helper-split-headroom-ladder/intervals.py b/research/helper-split-headroom-ladder/intervals.py
new file mode 100644
index 0000000000..898ace38af
--- /dev/null
+++ b/research/helper-split-headroom-ladder/intervals.py
@@ -0,0 +1,117 @@
+"""Outward integer intervals, scaled by 10**70, for ranks at most 145.
+
+No transcendental library call participates in a certificate. Decimal values
+used by the caller only nominate rational points for these checks.
+"""
+from fractions import Fraction
+
+Q = 10**70
+LOG_TERMS = 100
+EXP_TERMS = 80
+
+
+def require(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def ceiling(a, b):
+    return -(-a // b)
+
+
+def rational(x):
+    value = Fraction(x)
+    return value.numerator * Q // value.denominator, ceiling(value.numerator * Q, value.denominator)
+
+
+def add(a, b):
+    return a[0] + b[0], a[1] + b[1]
+
+
+def scale(a, n):
+    return (a[0] * n, a[1] * n) if n >= 0 else (a[1] * n, a[0] * n)
+
+
+def multiply(a, b):
+    require(0 <= a[0] <= a[1] and 0 <= b[0] <= b[1], "nonnegative factors required")
+    return a[0] * b[0] // Q, ceiling(a[1] * b[1], Q)
+
+
+def log_unit(value):
+    """2 atanh z, z=(value-1)/(value+1), 1 <= value <= 2."""
+    require(1 <= value <= 2, "log range reduction")
+    z = (value - 1) / (value + 1)
+    total, power = Fraction(0), z
+    for j in range(LOG_TERMS):
+        total += 2 * power / (2 * j + 1)
+        power *= z * z
+    tail = 2 * power / ((2 * LOG_TERMS + 1) * (1 - z * z))
+    return rational(total)[0], rational(total + tail)[1]
+
+
+def log_rank(rank):
+    require(isinstance(rank, int) and 1 <= rank <= 145, "rank outside 1..145")
+    shift = rank.bit_length() - 1
+    return add(scale(log_unit(Fraction(2)), shift), log_unit(Fraction(rank, 2**shift)))
+
+
+def exp_small(x):
+    """exp(x) for 0 <= x <= 1/8, with an explicit geometric tail."""
+    require(0 <= x[0] <= x[1] <= Q // 8, "exponential argument outside 0..1/8")
+    term, total = (Q, Q), (Q, Q)
+    for j in range(1, EXP_TERMS + 1):
+        term = term[0] * x[0] // (Q * j), ceiling(term[1] * x[1], Q * j)
+        total = add(total, term)
+    next_upper = ceiling(term[1] * x[1], Q * (EXP_TERMS + 1))
+    # All subsequent term ratios are <= 1/(8*82) < 1/2.
+    return total[0], total[1] + 2 * next_upper
+
+
+def power(rank, log, k):
+    """rank**(1-k) = rank/exp(k*log(rank))."""
+    denominator = exp_small(multiply(k, log))
+    return rank * Q * Q // denominator[1], ceiling(rank * Q * Q, denominator[0])
+
+
+class Oracle:
+    def __init__(self, ranks):
+        require(len(ranks) <= 145, "too many ranks")
+        self.logs = {r: log_rank(r) for r in ranks}
+
+    def powers(self, low, high=None):
+        lo, hi = rational(low), rational(low if high is None else high)
+        require(0 <= lo[0] <= hi[1] <= Q // 50, "k outside 0..0.02")
+        k = lo[0], hi[1]
+        return {r: power(r, log, k) for r, log in self.logs.items()}
+
+    def moment(self, coefficients, low, high=None, derivative=False):
+        powers, total = self.powers(low, high), (0, 0)
+        for r, n in coefficients.items():
+            value = multiply(powers[r], self.logs[r]) if derivative else powers[r]
+            total = add(total, scale(value, n))
+        return total
+
+    def certify(self, coefficients):
+        """Certify -F' > 0 by a Lipschitz grid on [0,1/50]."""
+        bound = sum(abs(n) * r * multiply(log, log)[1]
+                    for r, n in coefficients.items() for log in [self.logs[r]])
+        step = Fraction(1, 500)
+        error = ceiling(bound, 500)
+        for j in range(11):
+            require(self.moment(coefficients, j * step, derivative=True)[0] > error,
+                    "slope interval does not prove monotonicity")
+        require(self.moment(coefficients, 0)[0] > 0, "initial margin not positive")
+        require(self.moment(coefficients, Fraction(1, 50))[1] < 0, "final margin not negative")
+
+    def bracket(self, coefficients, low, high):
+        require(Fraction(low) < Fraction(high), "reversed root bracket")
+        require(self.moment(coefficients, low)[0] > 0, "lower root endpoint not positive")
+        require(self.moment(coefficients, high)[1] < 0, "upper root endpoint not negative")
+
+
+def close(interval, approximation):
+    """The printed approximation has absolute error less than 10**-39."""
+    middle = rational(approximation)
+    error = Q // 10**39
+    require(middle[1] - error < interval[0] <= interval[1] < middle[0] + error,
+            "approximation outside certified error bound")
diff --git a/research/helper-split-headroom-ladder/verify.py b/research/helper-split-headroom-ladder/verify.py
new file mode 100644
index 0000000000..840467be64
--- /dev/null
+++ b/research/helper-split-headroom-ladder/verify.py
@@ -0,0 +1,377 @@
+"""Helper-split headroom ladder on the selected #7000 invoice (standard library only).
+
+Model.  For productive-target counts T_r and paid-child counts P_r by rank r,
+the raw margin is F(k) = sum_r (T_r - P_r) r^(1-k), and the raw crossing is
+its first root in k.  The pinned invoice stores that crossing, and this file
+recomputes it from T and P (a calibration of the model, not a review of the
+source construction).
+
+Helper structure read from the pinned data: every helper of rank r supplies
+18 productive fields of rank r (T_r = 18 h_r for r <= 8 and T_9 = 18 h_9 + 416).
+The 416 further rank-9 fields and every field of rank >= 16 are held fixed.
+Splitting one helper into parts (a_i) changes F by 18 (sum a_i^p - r^p), p = 1 - k.
+
+Usage:  verify.py          check everything and compare with RESULT.json
+        verify.py --emit   print the generated receipt (redirect to RESULT.json)
+"""
+import hashlib
+import json
+import pathlib
+import sys
+from decimal import Decimal, getcontext
+from fractions import Fraction
+from itertools import combinations_with_replacement
+from intervals import Oracle, Q, close
+
+HERE = pathlib.Path(__file__).resolve().parent
+SRC = HERE.parent / "shared-wire-split-composition"
+FULL = SRC / "FULL-RESULT.json"
+BIT = SRC / "data" / "bit-admission.json"
+RESULT = HERE / "RESULT.json"
+FULL_SHA = "d9ba709fa5dcefffe3f3abdcf34600ce188224703cfd743e6db8b58ba4816cb7"
+BIT_SHA = "a46b6c6040f75184cab5467656811ad988c216e3bc8e44d595a5015b6e49620b"
+
+getcontext().prec = 80
+FIELDS = 18            # productive fields per helper (checked against the data)
+KMAX = Decimal("0.02")  # all certificates live on [0, KMAX]
+EPS = Decimal("1e-45")  # half-width of every root bracket
+NEWTON_CAP = 60
+RANK_CAP = 9            # helpers have ranks 1..9
+DIGITS = Decimal("1e-40")
+
+
+def fail(msg):
+    raise SystemExit("FAIL: " + msg)
+
+
+def check(cond, msg):
+    if not cond:
+        fail(msg)
+
+
+# ---------------------------------------------------------------- inputs
+def read_pinned(path, sha):
+    check(path.stat().st_size <= 1000000, "input exceeds the one-megabyte limit")
+    raw = path.read_bytes()
+    check(hashlib.sha256(raw).hexdigest() == sha, "pinned input hash: " + path.name)
+    return json.loads(raw)
+
+
+def int_map(d):
+    return {int(k): int(v) for k, v in d.items()}
+
+
+def load_invoice(full):
+    """T, P, helper counts, with the 18-field structure checked."""
+    inv = full["invoice"]
+    T, P, H = int_map(inv["T"]), int_map(inv["P"]), int_map(full["native"]["helpers"])
+    check(sorted(H) == list(range(1, RANK_CAP + 1)), "helper ranks are 1..9")
+    for r in range(1, RANK_CAP):
+        check(T[r] == FIELDS * H[r], "18 fields per helper at rank %d" % r)
+    extra = T[RANK_CAP] - FIELDS * H[RANK_CAP]
+    check(extra == 416, "416 rank-9 fields beyond the helpers")
+    check(all(r > RANK_CAP for r in T if r not in H), "fixed fields have rank > 9")
+    mass = sum(r * n for r, n in T.items()) - sum(r * n for r, n in P.items())
+    check(mass == int(inv["S"]) == 1520, "surplus S = sum r (T_r - P_r)")
+    return T, P, H, extra
+
+
+def frac(text):
+    num, _, den = text.partition("/")
+    return Fraction(int(num), int(den or 1))
+
+
+# ------------------------------------------------------- margin arithmetic
+class Model:
+    """Logarithm table for a fixed rank set; everything else is a pure function."""
+
+    def __init__(self, ranks):
+        self.ln = {r: Decimal(r).ln() for r in ranks}
+        self.oracle = Oracle(ranks)
+
+    def margin(self, c, k):
+        p = 1 - k
+        return sum(n * (p * self.ln[r]).exp() for r, n in c.items() if n)
+
+    def slope(self, c, k):
+        """G(k) = -F'(k) = sum c_r r^(1-k) ln r."""
+        p = 1 - k
+        return sum(n * (p * self.ln[r]).exp() * self.ln[r] for r, n in c.items() if n)
+
+    def certify(self, c):
+        """F strictly decreasing on [0, KMAX], F(0) > 0 > F(KMAX): unique root."""
+        self.oracle.certify(c)
+
+    def root(self, c):
+        """Certified root of F on [0, KMAX] by Newton with a sign bracket."""
+        self.certify(c)
+        k = Decimal("0.004")
+        for _ in range(NEWTON_CAP):
+            step = self.margin(c, k) / self.slope(c, k)
+            k += step
+            if abs(step) < Decimal("1e-60"):
+                break
+        check(0 < k < KMAX, "Newton left the interval")
+        self.oracle.bracket(c, k - EPS, k + EPS)
+        return k
+
+
+def invoice_diff(T, P):
+    c = {r: T.get(r, 0) - P.get(r, 0) for r in set(T) | set(P)}
+    return {r: n for r, n in c.items() if n}
+
+
+# ------------------------------------------------------------- partitions
+def partitions_table(nmax):
+    """All integer partitions of n <= nmax (coin-change enumeration, no recursion)."""
+    table = [[()] if n == 0 else [] for n in range(nmax + 1)]
+    for s in range(1, nmax + 1):
+        for n in range(s, nmax + 1):
+            table[n] = table[n] + [q + (s,) for q in table[n - s]]
+    return [[tuple(sorted(q, reverse=True)) for q in row] for row in table]
+
+
+def balanced(r, m):
+    """Balanced partition of r into min(m, r) parts."""
+    parts = min(m, r)
+    q, rem = divmod(r, parts)
+    return tuple([q + 1] * rem + [q] * (parts - rem))
+
+
+def majorizes(lam, mu, width):
+    """Exact integer test: sorted lam majorizes sorted mu (zero padded)."""
+    a = list(lam) + [0] * (width - len(lam))
+    b = list(mu) + [0] * (width - len(mu))
+    sa = sb = 0
+    for x, y in zip(a, b):
+        sa, sb = sa + x, sb + y
+        if sa < sb:
+            return False
+    return sa == sb
+
+
+def check_balanced_lemma(table):
+    """Every partition with <= m parts majorizes the balanced m-part one."""
+    count = 0
+    for r in range(1, RANK_CAP + 1):
+        independent = {tuple(reversed(parts))
+                       for length in range(1, r + 1)
+                       for parts in combinations_with_replacement(range(1, r + 1), length)
+                       if sum(parts) == r}
+        check(len(table[r]) == len(set(table[r])) and set(table[r]) == independent,
+              "partition enumeration agrees with independent combinations")
+    for r in range(2, RANK_CAP + 1):
+        for m in range(1, r + 1):
+            bal = balanced(r, m)
+            for lam in table[r]:
+                if len(lam) <= m:
+                    check(majorizes(lam, bal, m), "balanced partition is minimal")
+                    count += 1
+    return count
+
+
+# ----------------------------------------------------------- scenarios
+def apply_split(T, r, lam, times):
+    """Split `times` helpers of rank r into the parts lam (18 fields each)."""
+    out = dict(T)
+    out[r] -= FIELDS * times
+    for a in lam:
+        out[a] = out.get(a, 0) + FIELDS * times
+    return out
+
+
+def split_all(T, H, m):
+    """Every helper of rank >= 2 into its balanced m-part partition."""
+    out = dict(T)
+    for r in range(2, RANK_CAP + 1):
+        out = apply_split(out, r, balanced(r, m), H[r])
+    return out
+
+
+def gain(lam, r, p, model):
+    value = FIELDS * (sum((p * model.ln[a]).exp() for a in lam) - (p * model.ln[r]).exp())
+    coefficients = {r: -FIELDS}
+    for a in lam:
+        coefficients[a] = coefficients.get(a, 0) + FIELDS
+    close(model.oracle.moment(coefficients, 1 - p - EPS, 1 - p + EPS), q(value))
+    return value
+
+
+def ceil_div(num, den):
+    q = num / den
+    n = int(q)
+    return n if n >= q else n + 1
+
+
+# ------------------------------------------------------------- receipt
+def q(x):
+    return str(x.quantize(DIGITS))
+
+
+def build(full, bit):
+    T, P, H, extra = load_invoice(full)
+    ranks = sorted(set(T) | set(P) | set(range(1, RANK_CAP + 1)))
+    model = Model(ranks)
+    # Independent integer-power test of every rank's enclosure at k=1/100.
+    for r, (low, high) in model.oracle.powers(Fraction(1, 100)).items():
+        exact = r**99 * Q**100
+        check(low**100 <= exact <= high**100, "rational-exponent control")
+    c0 = invoice_diff(T, P)
+    k0 = model.root(c0)
+    stored = Decimal(full["invoice"]["raw_crossing"])
+    check(abs(stored - k0) < Decimal("1e-60"), "stored raw crossing reproduced")
+    pin = frac(full["gate"]["BIT_pin"])
+    paired = frac(bit["ordinary_admitted"])
+    check(pin == Fraction(376669, 10 ** 8) and paired == Fraction(12039, 3125000), "wall values")
+    walls = {"pin": Decimal(pin.numerator) / Decimal(pin.denominator),
+             "paired": Decimal(paired.numerator) / Decimal(paired.denominator)}
+    for w in walls.values():
+        close(model.oracle.moment(c0, w), q(model.margin(c0, w)))
+    check(k0 < walls["pin"] < walls["paired"], "walls above the selected crossing")
+    table = partitions_table(RANK_CAP)
+    lemma = check_balanced_lemma(table)
+    ladder = ladder_rows(T, P, H, model, k0)
+    exch = exchange_rows(T, P, H, model, c0, k0, walls, table)
+    mins = touched_rows(T, P, H, model, c0, walls)
+    at_one = sum(split_all(T, H, RANK_CAP).values()) - sum(P.values())
+    check(at_one == 24040, "full-split endpoint explains the interval restriction")
+    return {
+        "verifier_sha256": {name: hashlib.sha256((HERE / name).read_bytes()).hexdigest()
+                            for name in ("verify.py", "intervals.py")},
+        "inputs": {"FULL-RESULT.json": FULL_SHA, "bit-admission.json": BIT_SHA},
+        "helpers": {str(r): H[r] for r in sorted(H)},
+        "fixed_rank9_fields": extra,
+        "kappa0": q(k0),
+        "kappa0_bracket": [str(k0 - EPS), str(k0 + EPS)],
+        "walls": {k: q(v) for k, v in walls.items()},
+        "wall_margin_F": {k: q(model.margin(c0, v)) for k, v in walls.items()},
+        "lemma_partition_checks": lemma,
+        "ladder": ladder,
+        "exchange": exch,
+        "min_touched": mins,
+        "claim_interval": ["0", "0.02"],
+        "numeric_error_bound": "1e-39",
+        "full_split_margin_at_one": at_one,
+    }
+
+
+def ladder_rows(T, P, H, model, k0):
+    rows, prev = [], k0 - 1
+    for m in range(1, RANK_CAP + 1):
+        km = model.root(invoice_diff(split_all(T, H, m), P))
+        check(km > prev or m == 1, "ladder strictly increasing")
+        prev = km
+        rows.append({"m": m, "kappa": q(km), "gain": q(km - k0),
+                     "root_bracket": [str(km - EPS), str(km + EPS)]})
+    check(rows[0]["kappa"] == q(k0), "m = 1 is the selected invoice")
+    return rows
+
+
+def exchange_rows(T, P, H, model, c0, k0, walls, table):
+    p0, rows = 1 - k0, []
+    for r in range(2, RANK_CAP + 1):
+        for lam in table[r]:
+            if len(lam) < 2:
+                continue
+            one = model.root(invoice_diff(apply_split(T, r, lam, 1), P))
+            row = {"r": r, "parts": list(lam), "gain_at_k0": q(gain(lam, r, p0, model)),
+                   "single_split_dk": q(one - k0), "available": H[r],
+                   "root_bracket": [str(one - EPS), str(one + EPS)]}
+            for name, w in walls.items():
+                g = gain(lam, r, 1 - w, model)
+                count = ceil_div(-model.margin(c0, w), g)
+                # These are algebraic counts; stock may be insufficient.
+                for number, sign in ((count - 1, -1), (count, 1)):
+                    c = invoice_diff(apply_split(T, r, lam, number), P)
+                    interval = model.oracle.moment(c, w)
+                    check(interval[1] < 0 if sign < 0 else interval[0] > 0,
+                          "wall count is not minimal")
+                row["need_" + name] = count
+                row["available_" + name] = count <= H[r]
+            rows.append(row)
+    return rows
+
+
+def touched_rows(T, P, H, model, c0, walls):
+    out = {}
+    for name, w in walls.items():
+        for label, pieces in (("any", RANK_CAP), ("binary", 2)):
+            lam = balanced(9, pieces)
+            best = gain(lam, 9, 1 - w, model)
+            used = ceil_div(-model.margin(c0, w), best)
+            check(0 < used <= H[9], "rank-nine stock suffices")
+            for r in range(2, RANK_CAP):
+                other = gain(balanced(r, pieces), r, 1 - w, model)
+                check(Decimal(q(best)) - Decimal(q(other)) > Decimal("2e-39"),
+                      "rank nine has the greatest certified gain")
+            # Exact interval proof: no used-1 helpers can suffice even if all
+            # receive the largest per-helper gain, while used rank-nine do.
+            model.certify(invoice_diff(apply_split(T, 9, lam, used), P))
+            for number, positive in ((used - 1, False), (used, True)):
+                c = invoice_diff(apply_split(T, 9, lam, number), P)
+                interval = model.oracle.moment(c, w)
+                check(interval[0] > 0 if positive else interval[1] < 0,
+                      "minimal touched-helper certificate")
+            out[name + "_" + label] = {"helpers": used,
+                                       "plan": [[9, used]]}
+    return out
+
+
+# ------------------------------------------------------------ controls
+def run_controls(full, receipt):
+    T, P, _, _ = load_invoice(full)
+    model = Model(sorted(set(T) | set(P)))
+    bad = json.loads(json.dumps(full))
+    bad["native"]["helpers"]["4"] += 1
+    expect_fail(lambda: load_invoice(bad), "helper count off by one")
+    expect_fail(lambda: model.oracle.powers(Fraction(9, 10)), "outside certified range")
+    c0 = invoice_diff(T, P)
+    k0 = Decimal(receipt["kappa0"])
+    check(model.margin(c0, k0 - Decimal("1e-6")) > 0 > model.margin(c0, k0 + Decimal("1e-6")),
+          "wrong-root control brackets the real root")
+    expect_fail(lambda: check_not_balanced((3, 1), 2, 4), "unbalanced partition is not minimal")
+    expect_fail(lambda: model.oracle.bracket(c0, Decimal("0.004"), Decimal("0.005")),
+                "a bracket missing the root")
+    expect_fail(lambda: model.certify({r: -n for r, n in c0.items()}), "reversed monotonicity")
+    return 6
+
+
+def check_not_balanced(lam, m, r):
+    check(majorizes(balanced(r, m), lam, m), "unbalanced partition is not minimal")
+
+
+def expect_fail(fn, label):
+    try:
+        fn()
+    except (SystemExit, ValueError):
+        return
+    fail("control did not reject: " + label)
+
+
+def summary(receipt):
+    lad = {row["m"]: row["kappa"] for row in receipt["ladder"]}
+    return ("PASS_HELPER_SPLIT_HEADROOM_LADDER kappa0=%s m2=%s m9=%s "
+            "pin_any=%d pin_binary=%d paired_any=%d paired_binary=%d"
+            % (receipt["kappa0"][:14], lad[2][:12], lad[9][:12],
+               receipt["min_touched"]["pin_any"]["helpers"],
+               receipt["min_touched"]["pin_binary"]["helpers"],
+               receipt["min_touched"]["paired_any"]["helpers"],
+               receipt["min_touched"]["paired_binary"]["helpers"]))
+
+
+def main(argv):
+    check(argv in ([], ["--emit"]), "usage: verify.py [--emit]")
+    full = read_pinned(FULL, FULL_SHA)
+    bit = read_pinned(BIT, BIT_SHA)
+    receipt = build(full, bit)
+    text = json.dumps(receipt, indent=1, sort_keys=True) + "\n"
+    if "--emit" in argv:
+        sys.stdout.write(text)
+        return
+    check(RESULT.read_text() == text, "RESULT.json differs from the recomputed receipt")
+    controls = run_controls(full, receipt)
+    print(summary(receipt), "controls=%d" % controls)
+
+
+if __name__ == "__main__":
+    main(sys.argv[1:])