From 432a7ddae2e71b16d81a1441c26cfc583132cced Mon Sep 17 00:00:00 2001
From: ToikY <27568617+ToikY@users.noreply.github.com>
Date: Fri, 6 Nov 2026 15:51:45 -0800
Subject: [PATCH] research: close all-cap formal A2 st concave tariff

---
 research/a2-st-all-cap-tariff/PROOF.md        |  146 +
 research/a2-st-all-cap-tariff/README.md       |   97 +
 research/a2-st-all-cap-tariff/RESULT.json     | 4432 +++++++++++++++++
 .../a2-st-all-cap-tariff/SOURCE-PINS.json     |   49 +
 research/a2-st-all-cap-tariff/SOURCES.json    |   30 +
 research/a2-st-all-cap-tariff/TASK.json       |   95 +
 research/a2-st-all-cap-tariff/VERIFY.json     |   28 +
 .../predecessor/PR1941-README.md              |  286 ++
 .../predecessor/PR1966-PROOF.md               |  357 ++
 .../predecessor/PR6817-PROOF.md               |  191 +
 .../predecessor/PR6817-README.md              |  154 +
 research/a2-st-all-cap-tariff/verify.py       |  417 ++
 12 files changed, 6282 insertions(+)
 create mode 100644 research/a2-st-all-cap-tariff/PROOF.md
 create mode 100644 research/a2-st-all-cap-tariff/README.md
 create mode 100644 research/a2-st-all-cap-tariff/RESULT.json
 create mode 100644 research/a2-st-all-cap-tariff/SOURCE-PINS.json
 create mode 100644 research/a2-st-all-cap-tariff/SOURCES.json
 create mode 100644 research/a2-st-all-cap-tariff/TASK.json
 create mode 100644 research/a2-st-all-cap-tariff/VERIFY.json
 create mode 100644 research/a2-st-all-cap-tariff/predecessor/PR1941-README.md
 create mode 100644 research/a2-st-all-cap-tariff/predecessor/PR1966-PROOF.md
 create mode 100644 research/a2-st-all-cap-tariff/predecessor/PR6817-PROOF.md
 create mode 100644 research/a2-st-all-cap-tariff/predecessor/PR6817-README.md
 create mode 100644 research/a2-st-all-cap-tariff/verify.py

diff --git a/research/a2-st-all-cap-tariff/PROOF.md b/research/a2-st-all-cap-tariff/PROOF.md
new file mode 100644
index 0000000000..7f2279ec34
--- /dev/null
+++ b/research/a2-st-all-cap-tariff/PROOF.md
@@ -0,0 +1,146 @@
+# Exact capped concave tariff for repeated A2 two-letter targets
+
+This is a formal tariff theorem, not a native implementation or supplier bill.
+The generator-selective cut method is credited to Adam Gall/PR1941 and Ross
+Richardson/PR1966; the model and classical packing lemma follow Benny Zhao /
+zhaobenny's PR6817. The contribution is the complete st price for every
+remainder, not a new cut or packing principle.
+
+## Model and theorem
+
+Let K be any field, G=S3=<s,t | s²=t²=e,(st)³=e>, and d,b,r positive integers
+with finite r≥d. Work coefficientwise in M_r(K[G]). A paid factor is
+
+    F_u(A)=I+([u]−[e])A,   u∈{s,t},   0<rank_K(A)≤b.
+
+The scalar matrix A is arbitrary: it need not be idempotent, and paid factors
+are not assumed invertible. Arbitrary interleaved GL_r(K) matrices with scalar
+[e] coefficients are free. There are no free address maps, resets, discarded
+outputs or clean allocations. The complete endpoint is
+
+    T_d=diag([st],…,[st],[e],…,[e]),
+              d copies
+
+so every spectator has arbitrary input and returns exactly, with no leakage.
+The product convention is [st]=[s][t]; the rightmost factor acts first.
+
+Let f:{0,…,b}→R be nonnegative, nondecreasing, f(0)=0, with nonincreasing
+successive differences. Charge each paid factor f(rank A), identically for
+s and t. Define
+
+    P_(b,f)(d)=floor(d/b)f(b)+f(d mod b).
+
+**The exact minimum cost over finite legal words for T_d is**
+
+    2 P_(b,f)(d).                                      (1)
+
+This includes every field characteristic, every finite number of restored
+dirty spectators, nondivisible d, b=1 and b>r. It asserts neither a unique
+minimizing profile nor a classification/minimum count of calls. Flat or zero
+prices are allowed.
+
+## Two credited generator-selective cuts
+
+Use the left regular action λ of G on its six distinct basis labels over K,
+and tensor with K^r. A formal matrix word therefore induces a linear map on
+V=K[G]⊗K^r, preserving multiplication. This is a lower-bound image; attainment
+below will be checked in the full formal algebra.
+
+For a partition V=V_+⊕V_− put q(X)=rank(X_−+). For arbitrary linear X,Y,
+
+    (XY)_−+=X_−+Y_+++X_−−Y_−+,
+    q(XY)≤q(X)+q(Y).                                  (2)
+
+No factor's invertibility is needed. Choose V_+=span{e,t}⊗K^r, with the other
+four group labels in V_−. Left multiplication by t preserves {e,t}; hence
+q(F_t(A))=0. Left multiplication by s sends its two labels to s,st, both
+outside the pair. Its crossing matrix consists of two disjoint permutation
+entries, tensored with A. Therefore
+
+    q(F_s(A))=2 rank_K(A).
+
+The identity subtraction has no crossing block. Every free scalar GL_r(K)
+also has crossing rank zero: it mixes roles within each group label, not
+labels across the cut. Finally st sends e,t to st,s, both outside the pair.
+The target crosses on precisely its d active roles, giving q(T_d)=2d.
+Applying (2) to the entire word yields
+
+    sum_(s-calls) rank A ≥ d.                          (3)
+
+For the other cut take V_+=span{e,s}⊗K^r. Now s preserves the pair, while t
+sends it to t,ts and st sends it to st,sts, again outside. The same calculation
+gives
+
+    sum_(t-calls) rank A ≥ d.                          (4)
+
+These statements hold even when free maps mix productive and dirty spectator
+roles. In characteristic two, minus equals plus in the paid formula, but
+the six group labels remain distinct and the two crossing entries remain
+independent. The factor 2 is an ordinary dimension count, not a scalar whose
+inverse is used. No semisimplicity, root-sign specialization, augmentation,
+paid projector or nonsingularity hypothesis enters (2)–(4).
+
+## Classical capped packing
+
+For integers 1≤k_j≤b with sum k_j≥d, the standard concave concentration lemma
+gives
+
+    sum_j f(k_j)≥P_(b,f)(d).                           (5)
+
+For completeness, permit zero parts temporarily. If 0<a≤c<b, transfer one
+unit from a to c. The cost change is
+`[f(c+1)−f(c)]−[f(a)−f(a−1)]≤0`. Continue with this pair until one part is
+zero or b, decreasing the number of intermediate parts. Repetition leaves full b-parts and
+at most one remainder. For the original total M this gives P_(b,f)(M).
+This packed function is nondecreasing: increments within a block are those
+of f, and the step into the next full block is f(b)−f(b−1)≥0. Thus M≥d proves
+(5). For b=1 the argument is immediate. This is PR6817's cited classical
+packing argument, not a new optimization theorem.
+
+Apply (5) separately to the s ranks in (3) and t ranks in (4). Their actual
+costs add, proving the lower bound (1), even with singular/nonprojector paid
+couplings and arbitrary free scalar interleavings.
+
+## Full formal attainment and boundary
+
+Partition the first d coordinate roles into floor(d/b) disjoint blocks of
+size b and, if nonempty, one remainder block of size d mod b. Let E_B be a
+block's coordinate projector. The legal factors obey the full algebra identity
+
+    F_s(E_B)F_t(E_B)=I+([st]−[e])E_B.
+
+Indeed they act as [s][t] on that block and as identity on its complement.
+Different blocks have disjoint supports. Their products are exactly T_d,
+including every spectator column and row. No free basis operation, preparation
+or restoration is needed. Each letter's cost is P_(b,f)(d), proving attainment.
+If b>r≥d, there are no full b-blocks; the one rank-d remainder is legal, so
+the apparently unavailable rank b is never invoked. Empty remainders require
+no rank-zero paid call.
+
+PR6817 already covers divisible identical targets, every single-reflection
+remainder and d=3,b=2 for st. Formula (1) completes this particular two-letter
+target's remainder family. It does not extend to the different longest target
+sts, heterogeneous targets, other reflection alphabets or tariffs. In
+particular the combined root-orbit packing floor alone must not replace the
+two separate letter constraints. The result does not reclassify prior rank-one
+st words or claim an unrestricted circuit optimum outside this paid grammar.
+
+## Credit and limits
+
+Credit Adam Gall1941 and Ross Richardson1966 for the established separate-letter
+cut/rank-volume method; David Arias2738 and its retained root/model lineage;
+Benny Zhao/zhaobenny6817 for the all-field arbitrary-coupling model, concave
+packing and existing divisible/single-reflection/three-target st results;
+Nick Petalas5083 for the distinct longest-element batching schedule; and
+ToikY6550 for the earlier all-field rank-one st classification. Standard rank
+subadditivity and tensor-rank multiplication are elementary methods.
+
+Substantial source-aware OpenAI Codex assistance is disclosed. The coordinator
+proposed this recipe during source orientation, before expanded review identified
+the established separate-cut predecessors; this is not a blind derivation or
+literature-priority claim. Finite checks, if provided, corroborate
+the written theorem and full formal endpoints, not all possible words. No
+native rank primitive, physical price for scalar mixing, owned allocation,
+restoration/routing/precision/stock/caller bill, all-size recursion or selected
+supplier is supplied. All five framework assumptions and the frontier remain
+unchanged. No human or proof-kernel acceptance is claimed.
diff --git a/research/a2-st-all-cap-tariff/README.md b/research/a2-st-all-cap-tariff/README.md
new file mode 100644
index 0000000000..65b290f040
--- /dev/null
+++ b/research/a2-st-all-cap-tariff/README.md
@@ -0,0 +1,97 @@
+# All-cap formal tariff for repeated A2 st targets
+
+This answers the repeated `st` part of [PR6817](https://github.com/CrocSwap/integer-mult-bounds/pull/6817)'s
+nondivisible-target question. It applies **established separate-letter cuts**
+to that source's common concave rank price; no new cut principle is claimed.
+
+Let K be any field, d>=1, b>=1, and r>=d any finite number of arbitrary-input
+scalar roles. The only paid address generators are the simple involutions s,t
+of S3. A paid factor is `I+([u]-[e])A`, u in {s,t}, with arbitrary scalar A
+of positive rank at most b. A may be singular or nonidempotent. Interleaved
+scalar GL_r(K) maps are free. The complete endpoint in M_r(K[S3]) is
+`diag([st],...,[st],[e],...,[e])`, with d productive roles and every spectator
+restored, including all off-diagonal group coefficients.
+
+For a common nonnegative, nondecreasing, discretely concave price f on
+0..b with f(0)=0, the exact tariff is
+
+```
+2 * (floor(d/b) * f(b) + f(d mod b)).
+```
+
+The selective regular-representation cuts `{e,t}` and `{e,s}` give separate
+rank-volume demands d. Applying the credited integer-packing lemma to each
+letter gives the lower price; disjoint coordinate blocks of size at most b
+give a matching full formal word. [PROOF.md](PROOF.md) states the argument
+over every field, including characteristic two, with arbitrary restored dirty
+workspace. [RESULT.json](RESULT.json) retains the completed finite
+corroboration. The universal statement is a written proof,
+separate from finite execution, human review and proof-kernel acceptance.
+
+The source's three-target/cap-two result and divisible cases are credited
+positive cases. The new scope is all d and b, including every nondivisible
+remainder for this specific two-letter target. Other multi-reflection or
+heterogeneous targets are outside this result.
+
+## Evidence and reproduction
+
+The new standard-library verifier authenticates the four retained source
+prose files before calculating. It checks the selective cut ranks with
+arbitrary scalar couplings over finite fields, complete signed attaining
+endpoints with dirty-role mixing, formal prices and meaningful semantic
+controls. It imports and runs no predecessor scientific checker, word,
+constructor, census or native program. The retained result covers all 722 scalar 2-by-2 matrices over F2/F3/F5
+(668 nonprojectors), 2,888 one-gate cut checks, 108 complete formal endpoints
+with 378 paid calls (297 mixing dirty roles), 144 exact price cases and seven
+controls. Arbitrary-coupling exhaustion is a one-gate cut test, not all-word
+optimality. Nonconcave/nonmonotone controls refute the packing lemma outside
+its premises; they are not completed legal-word counterexamples. The first
+run took 0.161 seconds / 21.65625 MiB; normal and optimized replays give the
+same 49,624-byte packet. The written proof supplies the universal statement.
+
+```sh
+python3.12 -B scripts/research_package.py --package research/a2-st-all-cap-tariff
+python3.12 -B scripts/check_research_commit.py --package research/a2-st-all-cap-tariff
+```
+
+SOURCES.json binds four canonical-main provenance files. SOURCE-PINS.json
+records the exact foreign PR6817 head/path/blob/mode/hash associations and
+authenticates the byte-identical retained prose. That foreign association was
+checked during source acquisition; it is separate from the generic canonical
+Git ancestry check. Caching an open PR's commit does not make it publication
+history. Direct replay authenticates retained bytes, has no external live file
+or network dependency, and does not claim foreign ancestry. Registered normal
+and optimized commands compare the entire typed result. No new dependencies
+are required.
+
+## Open next questions
+
+Can one actual rank-capped native call be implemented with a complete price?
+Start with its native/address layout, arbitrary input and owned/restored fields,
+coupling-rank interpretation, movement and precision before assigning a new
+construction round. No such realization or map from scalar coupling rank to
+recursive child rank is supplied here. Free scalar GL maps are a formal-model
+assumption, not a physical cost waiver. Ownership, restoration, routing,
+all-size compilation, stock, precision, outer bounds and the complete supplier
+invoice remain open; all five selected assumptions and the frontier are unchanged.
+
+## Credit and scope
+
+Prepared for [ToikY](https://github.com/ToikY) with substantial source-aware
+OpenAI Codex proposal, source/overlap, proof/implementation and related-role
+review assistance. The provisional cut recipe arose during orientation before
+the claim; expanded review then identified and credited its prior method.
+Roles saw the sources and proposal; this is not blind independent acceptance.
+
+Credit Adam Gall #1941's separate cuts; Ross Richardson #1966/#1989's odd
+dihedral, separate-letter and symmetric cut work; Benny Zhao / zhaobenny
+#6817's model, concave packing lemma and three-target tariff; David Arias
+#2738's Coxeter framework; Nick Petalas #5083's distinct longest-element
+schedule; own excluded #6550's rank-two shortest-word classification; and
+their retained Adam Gall, Ross Richardson, ywz65, eumemic, Ali Romman and
+Gareth Davidson lineage. Those theorems, classifications and old finite
+replays are not claimed again. Rank subadditivity, regular representations,
+concave concentration and block batching are classical. Source notices are
+retained verbatim; new work is Apache-2.0. The `bit-new-register-model` card
+is explicitly changed to this partial formal tariff application; its native
+conversion and complete-supplier acceptance gate remain unmet.
diff --git a/research/a2-st-all-cap-tariff/RESULT.json b/research/a2-st-all-cap-tariff/RESULT.json
new file mode 100644
index 0000000000..524e8a33e3
--- /dev/null
+++ b/research/a2-st-all-cap-tariff/RESULT.json
@@ -0,0 +1,4432 @@
+{
+  "complete_endpoints": [
+    {
+      "call_labels": [
+        "s",
+        "t"
+      ],
+      "call_ranks": [
+        1,
+        1
+      ],
+      "cap": 1,
+      "complete_endpoint_sha256": "0e8a9fc25eba9db4c27df21c4f396781441ae4ca7bba860ba77877a9dcc835cd",
+      "d": 1,
+      "dirty_coupled_calls": 2,
+      "dirty_spectators": 2,
+      "field": 2,
+      "mixed_interleaved_scalars": 1,
+      "word_sha256": "7a7917b2ce6168fbd36a4b456f2c465338947204a1d8c9b16922d5cffb15be93"
+    },
+    {
+      "call_labels": [
+        "s",
+        "t"
+      ],
+      "call_ranks": [
+        1,
+        1
+      ],
+      "cap": 2,
+      "complete_endpoint_sha256": "0e8a9fc25eba9db4c27df21c4f396781441ae4ca7bba860ba77877a9dcc835cd",
+      "d": 1,
+      "dirty_coupled_calls": 2,
+      "dirty_spectators": 2,
+      "field": 2,
+      "mixed_interleaved_scalars": 1,
+      "word_sha256": "7a7917b2ce6168fbd36a4b456f2c465338947204a1d8c9b16922d5cffb15be93"
+    },
+    {
+      "call_labels": [
+        "s",
+        "t"
+      ],
+      "call_ranks": [
+        1,
+        1
+      ],
+      "cap": 3,
+      "complete_endpoint_sha256": "0e8a9fc25eba9db4c27df21c4f396781441ae4ca7bba860ba77877a9dcc835cd",
+      "d": 1,
+      "dirty_coupled_calls": 2,
+      "dirty_spectators": 2,
+      "field": 2,
+      "mixed_interleaved_scalars": 1,
+      "word_sha256": "7a7917b2ce6168fbd36a4b456f2c465338947204a1d8c9b16922d5cffb15be93"
+    },
+    {
+      "call_labels": [
+        "s",
+        "t"
+      ],
+      "call_ranks": [
+        1,
+        1
+      ],
+      "cap": 4,
+      "complete_endpoint_sha256": "0e8a9fc25eba9db4c27df21c4f396781441ae4ca7bba860ba77877a9dcc835cd",
+      "d": 1,
+      "dirty_coupled_calls": 2,
+      "dirty_spectators": 2,
+      "field": 2,
+      "mixed_interleaved_scalars": 1,
+      "word_sha256": "7a7917b2ce6168fbd36a4b456f2c465338947204a1d8c9b16922d5cffb15be93"
+    },
+    {
+      "call_labels": [
+        "s",
+        "t"
+      ],
+      "call_ranks": [
+        1,
+        1
+      ],
+      "cap": 5,
+      "complete_endpoint_sha256": "0e8a9fc25eba9db4c27df21c4f396781441ae4ca7bba860ba77877a9dcc835cd",
+      "d": 1,
+      "dirty_coupled_calls": 2,
+      "dirty_spectators": 2,
+      "field": 2,
+      "mixed_interleaved_scalars": 1,
+      "word_sha256": "7a7917b2ce6168fbd36a4b456f2c465338947204a1d8c9b16922d5cffb15be93"
+    },
+    {
+      "call_labels": [
+        "s",
+        "t"
+      ],
+      "call_ranks": [
+        1,
+        1
+      ],
+      "cap": 6,
+      "complete_endpoint_sha256": "0e8a9fc25eba9db4c27df21c4f396781441ae4ca7bba860ba77877a9dcc835cd",
+      "d": 1,
+      "dirty_coupled_calls": 2,
+      "dirty_spectators": 2,
+      "field": 2,
+      "mixed_interleaved_scalars": 1,
+      "word_sha256": "7a7917b2ce6168fbd36a4b456f2c465338947204a1d8c9b16922d5cffb15be93"
+    },
+    {
+      "call_labels": [
+        "s",
+        "t",
+        "s",
+        "t"
+      ],
+      "call_ranks": [
+        1,
+        1,
+        1,
+        1
+      ],
+      "cap": 1,
+      "complete_endpoint_sha256": "cfeaadd71196ebb0208f3b3bad30f640303b99f65418815aa5fbde4341602223",
+      "d": 2,
+      "dirty_coupled_calls": 3,
+      "dirty_spectators": 2,
+      "field": 2,
+      "mixed_interleaved_scalars": 3,
+      "word_sha256": "e999908fd3303e22cce1d37c570932590880f6aa89f5f08281c3c2758ed8921b"
+    },
+    {
+      "call_labels": [
+        "s",
+        "t"
+      ],
+      "call_ranks": [
+        2,
+        2
+      ],
+      "cap": 2,
+      "complete_endpoint_sha256": "cfeaadd71196ebb0208f3b3bad30f640303b99f65418815aa5fbde4341602223",
+      "d": 2,
+      "dirty_coupled_calls": 2,
+      "dirty_spectators": 2,
+      "field": 2,
+      "mixed_interleaved_scalars": 1,
+      "word_sha256": "ec23c8b74fe5945d5b2853378f85c3b4bbe4ed0d02079886f87941fc26d7266d"
+    },
+    {
+      "call_labels": [
+        "s",
+        "t"
+      ],
+      "call_ranks": [
+        2,
+        2
+      ],
+      "cap": 3,
+      "complete_endpoint_sha256": "cfeaadd71196ebb0208f3b3bad30f640303b99f65418815aa5fbde4341602223",
+      "d": 2,
+      "dirty_coupled_calls": 2,
+      "dirty_spectators": 2,
+      "field": 2,
+      "mixed_interleaved_scalars": 1,
+      "word_sha256": "ec23c8b74fe5945d5b2853378f85c3b4bbe4ed0d02079886f87941fc26d7266d"
+    },
+    {
+      "call_labels": [
+        "s",
+        "t"
+      ],
+      "call_ranks": [
+        2,
+        2
+      ],
+      "cap": 4,
+      "complete_endpoint_sha256": "cfeaadd71196ebb0208f3b3bad30f640303b99f65418815aa5fbde4341602223",
+      "d": 2,
+      "dirty_coupled_calls": 2,
+      "dirty_spectators": 2,
+      "field": 2,
+      "mixed_interleaved_scalars": 1,
+      "word_sha256": "ec23c8b74fe5945d5b2853378f85c3b4bbe4ed0d02079886f87941fc26d7266d"
+    },
+    {
+      "call_labels": [
+        "s",
+        "t"
+      ],
+      "call_ranks": [
+        2,
+        2
+      ],
+      "cap": 5,
+      "complete_endpoint_sha256": "cfeaadd71196ebb0208f3b3bad30f640303b99f65418815aa5fbde4341602223",
+      "d": 2,
+      "dirty_coupled_calls": 2,
+      "dirty_spectators": 2,
+      "field": 2,
+      "mixed_interleaved_scalars": 1,
+      "word_sha256": "ec23c8b74fe5945d5b2853378f85c3b4bbe4ed0d02079886f87941fc26d7266d"
+    },
+    {
+      "call_labels": [
+        "s",
+        "t"
+      ],
+      "call_ranks": [
+        2,
+        2
+      ],
+      "cap": 6,
+      "complete_endpoint_sha256": "cfeaadd71196ebb0208f3b3bad30f640303b99f65418815aa5fbde4341602223",
+      "d": 2,
+      "dirty_coupled_calls": 2,
+      "dirty_spectators": 2,
+      "field": 2,
+      "mixed_interleaved_scalars": 1,
+      "word_sha256": "ec23c8b74fe5945d5b2853378f85c3b4bbe4ed0d02079886f87941fc26d7266d"
+    },
+    {
+      "call_labels": [
+        "s",
+        "t",
+        "s",
+        "t",
+        "s",
+        "t"
+      ],
+      "call_ranks": [
+        1,
+        1,
+        1,
+        1,
+        1,
+        1
+      ],
+      "cap": 1,
+      "complete_endpoint_sha256": "120b225c36096414fe11abb97bf0992141e55323b7bb9453014167c7238d4e11",
+      "d": 3,
+      "dirty_coupled_calls": 4,
+      "dirty_spectators": 2,
+      "field": 2,
+      "mixed_interleaved_scalars": 5,
+      "word_sha256": "b748d36f4768c3486b7271a71b8a6a8b7f268df0f40b0684f0a8fddfb0d7fde9"
+    },
+    {
+      "call_labels": [
+        "s",
+        "t",
+        "s",
+        "t"
+      ],
+      "call_ranks": [
+        2,
+        2,
+        1,
+        1
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diff --git a/research/a2-st-all-cap-tariff/SOURCE-PINS.json b/research/a2-st-all-cap-tariff/SOURCE-PINS.json
new file mode 100644
index 0000000000..f170157c33
--- /dev/null
+++ b/research/a2-st-all-cap-tariff/SOURCE-PINS.json
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+      "local_path": "predecessor/PR6817-PROOF.md",
+      "source_repository": "zhaobenny/integer-mult-bounds",
+      "source_commit": "747364e7f5dd605fe08fd16e2278647a69cd2fda",
+      "source_path": "research/coxeter-capped-concave-tariff/PROOF.md",
+      "git_blob": "00c633badd2f1848ec34b9893553161012e7a8f3",
+      "sha256": "1ef4cb87aee9ca82b9fe42a114b152e1241b7454d68dd49ea8d260fb91e28353",
+      "bytes": 9422,
+      "mode": "100644",
+      "usage": "retained mathematical provenance; not executable"
+    },
+    {
+      "local_path": "predecessor/PR1941-README.md",
+      "source_repository": "CrocSwap/integer-mult-bounds",
+      "source_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+      "source_path": "research/finite-dihedral-cut-bound/README.md",
+      "git_blob": "7ef17c9f10e4c32292f6cb92ea6e19628b1638f1",
+      "sha256": "d8a94457ecb8e79caa22bd66d34b873556c365d2b5847d936b3ab531cc79c04a",
+      "bytes": 14445,
+      "mode": "100644",
+      "usage": "retained mathematical provenance; not executable"
+    },
+    {
+      "local_path": "predecessor/PR1966-PROOF.md",
+      "source_repository": "CrocSwap/integer-mult-bounds",
+      "source_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+      "source_path": "research/odd-dihedral-joint-tariff/PROOF.md",
+      "git_blob": "cde698569addd76161b3030f90e75940386c51a7",
+      "sha256": "bf56bbf426ca30030ad3868d0df9582b3077d21782460bddb3d2e6ae952b01a3",
+      "bytes": 12982,
+      "mode": "100644",
+      "usage": "retained mathematical provenance; not executable"
+    }
+  ]
+}
diff --git a/research/a2-st-all-cap-tariff/SOURCES.json b/research/a2-st-all-cap-tariff/SOURCES.json
new file mode 100644
index 0000000000..c283682767
--- /dev/null
+++ b/research/a2-st-all-cap-tariff/SOURCES.json
@@ -0,0 +1,30 @@
+{
+  "schema_version": 1,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/finite-dihedral-cut-bound/README.md",
+      "usage": "provenance",
+      "git_blob": "7ef17c9f10e4c32292f6cb92ea6e19628b1638f1",
+      "sha256": "d8a94457ecb8e79caa22bd66d34b873556c365d2b5847d936b3ab531cc79c04a"
+    },
+    {
+      "path": "research/odd-dihedral-joint-tariff/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "cde698569addd76161b3030f90e75940386c51a7",
+      "sha256": "bf56bbf426ca30030ad3868d0df9582b3077d21782460bddb3d2e6ae952b01a3"
+    }
+  ]
+}
diff --git a/research/a2-st-all-cap-tariff/TASK.json b/research/a2-st-all-cap-tariff/TASK.json
new file mode 100644
index 0000000000..822dcd8742
--- /dev/null
+++ b/research/a2-st-all-cap-tariff/TASK.json
@@ -0,0 +1,95 @@
+{
+  "schema_version": 1,
+  "task_id": "bit-new-register-model",
+  "priority_revision": 8,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "original_card": {
+    "id": "bit-new-register-model",
+    "question": "Can an explicitly larger or nonabelian register model escape the scoped three-register obstruction?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/bit-three-register-lower-bound/README.md",
+      "certificates/selected-result.json",
+      "research/bit-gl3-mixed-targets-length10/README.md",
+      "docs/research/community-efficient1000-review.md"
+    ],
+    "methods": [
+      "Construct the smallest exact identity in a newly specified legal operation alphabet.",
+      "Derive a target-specific relaxation bound for that same explicit alphabet."
+    ],
+    "acceptance_gate": "Name the changed hypothesis, all counted roles and a uniform identity; finite column tests alone do not certify it. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Resolve the current selected review, separate complex/ordinary BIT/admitted BIT/outer budgets, and compare the card against live handoffs before starting. Historical ceilings require exact source/model equivalence.",
+      "scope": "Source-binding orientation only; not a scientific replay or all-size proof. Materialize selected inputs if this checkout is sparse."
+    },
+    "first_experiment": "First compare the alphabet, role count and length with the fully replayed #1974 census and #2027 mixed-root tariff. Identify a changed premise and a legal native implementation; then price one smallest completed instance before searching larger words.",
+    "bottleneck_link": "A small-width completed saving changes the bit primitive; a finite-address identity alone is insufficient.",
+    "allowed_variations": [
+      "Choose a fixed larger register count or a specified nonabelian operation family."
+    ],
+    "scope_change_triggers": [
+      "Remaining inside the already-excluded three-register abelian model.",
+      "Repeating the two-register GL3 census or a covered three-register length without changing a hypothesis; a formal group-algebra word without native conversion is a partial result."
+    ],
+    "overlap_terms": [
+      "three-register",
+      "nonabelian",
+      "group algebra"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Exact common concave formal tariff for every positive productive count d and paid rank cap b for identical A2 st targets, over every coefficient field with arbitrary finite restored dirty spectators, free scalar GL mixing and arbitrary rank-capped scalar couplings.",
+  "scope_change_reason": "Taking PR6817 explicit general nondivisible-target handoff. Separate-letter cuts are established1941/1966 methods; the new contribution is their all-field/arbitrary-coupling all-cap st packed-price consequence. Native conversion and the original register-model acceptance gate remain unmet; this is a partial formal price result.",
+  "bottleneck_link": "The source6817 combined root floor can miss the exact formal batched price. A sharp declared tariff distinguishes supported local costs before any native implementation or full supplier expansion.",
+  "budget_minutes": 120,
+  "stop_rule": "Stop after this exact A2 st price application and bounded new finite corroboration. No other target/alphabet, native construction or kappa claim; preserve at least30actual verification minutes after science freeze.",
+  "outcome": "partial",
+  "evidence": "Written all-field all-cap A2 st tariff proof; new finite corroboration PASS:722 scalar matrices,2888 one-gate cut checks,108 full endpoints/378 paid calls,144 price cases,7 controls; first0.161s/21.65625MiB. Normal/optimized full49624-byte packet equality. Native conversion and complete supplier unknown.",
+  "author": "ToikY, with substantial OpenAI Codex assistance",
+  "ai_assistance": "Source-aware OpenAI Codex proposal, source/overlap, proof/implementation and related-role challenge. Roles see sources and provisional recipe; no blind, independent human or proof-kernel acceptance is claimed.",
+  "selection": {
+    "kind": "explicit_handoff",
+    "source_pr": 6817,
+    "source_head": "747364e7f5dd605fe08fd16e2278647a69cd2fda",
+    "investigation": "https://github.com/CrocSwap/integer-mult-bounds/discussions/6814#discussioncomment-19451022",
+    "picker_draw": null,
+    "reason": "Concrete open nondivisible multi-reflection target request; no suitable priced native contract in inspected6812/6819 leads."
+  },
+  "exposure": {
+    "preclaim_proposal": "Root proposed the regular S3 cuts{e,t}/{e,s} during source orientation before claim; unaccepted and no new execution before claim. Expanded prior review identified and credited established separate-letter cuts; no new invariant priority claimed.",
+    "source_programs_executed": false,
+    "old_results_excluded": [
+      "ToikY6550 shortest rank-two rank-one classification",
+      "NickPetalas5083 longest-element schedule",
+      "BennyZhao6817 d3/cap2 and packing theorem"
+    ],
+    "pre_first_metadata_fix": "Removed generic foreign-head provenance rows before first science: cached source objects are not publication ancestry or host availability. Exact foreign pin/body association retained in SOURCE-PINS and source custody, separate from generic canonical source-history checks."
+  },
+  "physical_obligations": {
+    "native_rank_capped_call": null,
+    "coupling_rank_to_recursive_child_rank": null,
+    "scalar_GL_physical_cost": null,
+    "ownership_layout_and_dirty_restoration_implementation": null,
+    "complete_supplier_invoice": null
+  },
+  "open_questions": [
+    {
+      "question": "Can one exact rank-capped call in this formal alphabet be physically realized and priced?",
+      "why_it_matters": "A formal scalar-coupling tariff is not a selected native or recursive bill.",
+      "first_step": "Specify one native call, arbitrary input/owned/restored fields, frame/address layout, movement and precision, and its complete physical invoice before extending construction.",
+      "blocked_by": "No actual native call or rank-to-recursive-child price map is supplied."
+    },
+    {
+      "question": "Which other nondivisible multi-reflection or heterogeneous targets have a sharp price?",
+      "why_it_matters": "This result is restricted to identical A2 st; the longest-element schedule and arbitrary heterogeneous targets have different chronology.",
+      "first_step": "Choose a genuinely uncovered target and exact alphabet, inspect existing separate/joint cuts and complete signed endpoint before another search.",
+      "blocked_by": "No claim or follow-on reservation; distinct target-specific proof and source overlap are required."
+    }
+  ]
+}
diff --git a/research/a2-st-all-cap-tariff/VERIFY.json b/research/a2-st-all-cap-tariff/VERIFY.json
new file mode 100644
index 0000000000..0e314cf386
--- /dev/null
+++ b/research/a2-st-all-cap-tariff/VERIFY.json
@@ -0,0 +1,28 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "New finite corroboration of credited separate-letter cut application, full formal st endpoints, packed formal tariffs and semantic controls; no predecessor scientific code or native program."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Same new scoped finite corroboration with explicit optimized-mode guards and typed full receipt binding."
+    }
+  ]
+}
diff --git a/research/a2-st-all-cap-tariff/predecessor/PR1941-README.md b/research/a2-st-all-cap-tariff/predecessor/PR1941-README.md
new file mode 100644
index 0000000000..7ef17c9f10
--- /dev/null
+++ b/research/a2-st-all-cap-tariff/predecessor/PR1941-README.md
@@ -0,0 +1,286 @@
+# Even finite dihedral groups: sharp reflection tariffs from address cuts
+
+**Scoped all-register theorem; no new kappa.** Put
+`G = D_{2n} = <r,x | r^n=x^2=1, xrx=r^-1>`, where the rotation period
+`n=2m` is even, and set `y=r^-1 x`, so `r=xy`. For any finite number
+`R>=W` of arbitrary-input scalar registers, the complete formal endpoint
+
+```
+diag([r^k_1 x^e_1], ..., [r^k_W x^e_W], 1, ..., 1)
+```
+
+in `M_R(F2[G])` has the exact separate paid-call bills
+
+```
+A = sum_i d_n(k_i+e_i)       x calls
+B = sum_i d_n(k_i)           y calls,
+d_n(t) = min(t mod n, n-(t mod n)),   e_i in {0,1}.
+```
+
+Both minima are attained **simultaneously**. Thus any nonnegative per-letter
+prices `alpha,beta` give the sharp total `alpha*A+beta*B`. No scalar mixing,
+interleaving, or extra restored dirty workspace can lower that bill.
+For example, the D16 central pair costs 8 x calls and 8 y calls for **every**
+workspace count, not just the two-register census of PR1821.
+
+This answers the finite-torsion question left explicitly open by
+[PR1911](https://github.com/CrocSwap/integer-mult-bounds/pull/1911).
+Its infinite-dihedral Laurent-degree proof does not descend through `r^n=1`.
+The new proof uses two finite permutation representations and a cut rank,
+which does survive that relation. It is not an enlarged finite word search.
+D8 central targets with arbitrary workspace were already settled by PR1573;
+that overlapping special case is credited, not claimed anew.
+
+## Exact model and fully paid boundary
+
+The only paid address labels are x and y. A unit call is
+`D_N(g)=I+([g]+1)N`, with `N=u v^T` over F2 and `v^T u=1`.
+All binary rank-one idempotents are allowed, not only coordinate projectors.
+Free scalar `GL_R(F2)` changes may occur anywhere. They do not change address
+coordinates. There are no other free address maps, resets, erasures, COPYs or
+clean inputs. Every input, including every workspace input, is arbitrary;
+the full boundary restores each spectator and has no residual couplings.
+Every paid preparation/restoration call is counted.
+
+This replaces the rational abelian address group in the retained three-register
+model by an explicitly nonabelian finite group (abelian only at the degenerate
+boundary n=2). It does not reduce rational frame entries modulo two. Scalar
+mixing is granted free as an **optimistic relaxation**, not asserted free in a
+physical supplier. No physical compiler, hosting, frame, precision/tape or
+analytic bridge is provided. Those adoption obligations remain unestablished.
+
+## Uniform lower-bound proof
+
+### 1. A subadditive crossing rank
+
+Split a finite vector space into address halves H and K, including all scalar
+roles in each half. For any linear operator M, let `q(M)=rank_F2 M[H,K]`.
+Block multiplication gives
+
+```
+(AB)[H,K] = A[H,H] B[H,K] + A[H,K] B[K,K],
+q(AB) <= q(A)+q(B).
+```
+
+No invertibility is needed for this inequality. A scalar register map acts
+independently at each address, so its crossing block is zero. Invertible scalar
+changes, even interleaved, therefore contribute zero to the bound.
+
+### 2. Two different valid representations, including n=2 mod 4
+
+For **each** integer c, represent G on addresses `a in Z/nZ` by
+
+```
+rho_c(r)(a) = a+1,
+rho_c(x)(a) = c-a,
+rho_c(y)(a) = c-1-a.
+```
+
+Both reflections square to the identity; their product x after y is `a+1`;
+that translation has order n; and x conjugates it to its inverse. These are
+therefore representations of the actual finite presentation. Extend them
+linearly to F2[G] and entrywise to register matrices.
+
+Use the balanced interval `H={0,...,m-1}`, `K={m,...,2m-1}`. For an interval
+translated by t on this 2m-cycle,
+
+```
+| (H+t) intersect K | = d_n(t).                         (1)
+```
+
+For residues `0<=t<=m` precisely t points cross; replace t by `n-t` for the
+other half of the residues. This proves (1) for all integers, not by sampling.
+
+Take two representations:
+
+* `c=m`: y preserves H, while x sends H to H+1. Thus y has no crossing
+  entries and x has exactly one entry in each directed crossing block.
+* `c=m-1`: x preserves H, while y sends H to H-1. Thus x has no crossing
+  entries and y has exactly one.
+
+The reflection offset **must change between the representations**. In
+particular this handles n=6,10,...: we do not assume the fixed-point-free
+reflection `a -> -a-1` preserves an odd-cardinality half. The verifier includes
+n=2,6,10 and rejects that incorrect fixed-offset shortcut.
+
+For arbitrary N, the crossing block of a represented paid factor is
+`N tensor E`, where E is the indicated address crossing block; the identity
+part contributes zero. Hence its rank is exactly `rank(N)` for the active
+letter and zero for the other. In the rank-one model the two call charges are
+therefore `(q_m(x),q_m(y))=(1,0)` and
+`(q_(m-1)(x),q_(m-1)(y))=(0,1)`. This is why the argument includes every
+non-coordinate scalar projector and any number of dirty roles.
+
+### 3. Exact endpoint crossing ranks
+
+A target `r^k` translates H by k. A target `r^k x` maps H to the interval
+
+```
+H + (k+c-m+1).
+```
+
+A permutation's crossing submatrix has one 1 in each participating row and
+column; its rank equals its number of crossing addresses. By (1), the target's
+crossing ranks in the two representations are
+
+```
+c=m:    d_n(k+e),
+c=m-1:  d_n(k).
+```
+
+The endpoint is diagonal in scalar roles. Its crossing blocks are direct sums,
+so ranks add over roles; identity spectators contribute zero. Subadditivity
+now yields `#x>=A` and `#y>=B` for every finite R and every word length.
+Formal equality implies represented equality; we never use the converse.
+Indeed a negative control below explicitly demonstrates the common kernel of
+these proof representations.
+
+The same reasoning also bounds **total scalar rank** per label if paid calls
+are allowed arbitrary-rank idempotents: x rank mass is at least A and y rank
+mass at least B. This is a rank-linear corollary, not a sharp invocation-count
+claim for batched calls or a statement about recursive child-rank pricing.
+
+## Uniform attaining identity and complete invoice
+
+Choose the representative `k' in [-m,m-1]` of each k. For a rotation, freely
+reduce `(xy)^k'`; for a reflection, freely reduce `(xy)^k' x`. Negative powers
+mean repetitions of yx. The resulting letter counts are
+
+```
+x: |k'+e| = d_n(k+e),    y: |k'| = d_n(k).
+```
+
+For e=1 the interval `-m<=k'<=m-1` is precisely what ensures both equalities;
+this simultaneous choice is not automatic for odd n.
+
+On role i use its coordinate projector E_i. Direct multiplication in the full
+formal group algebra gives
+`D_E(g) D_E(h) = D_E(gh)`, and calls on different coordinate roles commute.
+Consequently the above literal words attain the complete diagonal boundary.
+The rightmost factor acts first; the implementation executes each displayed
+word right-to-left. No finite representation is used to certify this identity.
+
+| Resource | Complete attaining implementation |
+|---|---|
+| Target roles | W, arbitrary dirty inputs |
+| Extra dirty roles | Arbitrarily many allowed; zero needed |
+| x / y paid calls | A / B, including all endpoint work |
+| Scalar basis changes | Zero needed by attaining word |
+| Clean allocations, COPYs, resets | None |
+| Extra restoration, free frame/address changes | None in this formal implementation |
+| Native supplier / physical realization | Not supplied or priced as free |
+
+On any concrete n-address action `rho_c`, each reflection can be implemented
+by disjoint swaps. For even n a reflection `a -> c-a` has two fixed addresses
+if c is even and none if c is odd. Thus x uses `(n-2)/2` or `n/2` swaps per
+call, and y has the opposite parity. A three-XOR swap needs no clean data bit;
+this gives an explicit implementation of the witness, not an optimal physical
+algorithm or an admitted recursive supplier tariff. Free scalar mixing in the
+lower-bound relaxation only strengthens the no-compression result.
+
+## Exact executable evidence and reproduction
+
+```
+uv run --python 3.12 --no-project python -I -B research/finite-dihedral-cut-bound/verify.py
+uv run --python 3.12 --no-project python scripts/research_package.py --package research/finite-dihedral-cut-bound
+```
+
+The verifier is standard-library-only. It distinguishes three engines:
+
+1. Full formal F2[G] coefficient multiplication using dihedral normal forms.
+2. Independently applied literal permutations on the **faithful regular** 2n
+   addresses, including every dirty basis column and non-coordinate calls.
+3. The two n-address proof representations, with binary Gaussian elimination
+   of their actual crossing submatrices rather than substituting the theorem's
+   claimed rank formula.
+
+Finite checks corroborate implementations; the universal result is the written
+proof above. The proof is not Lean-checked and no independent human review is
+claimed. The retained source-binding working example is orientation only, not
+selected cold replay. See the generated receipt and validation record for
+actually executed domains, counters and environment.
+
+Observed on macOS arm64, CPython **3.12.12**: the direct isolated replay and
+registered optimized replay both passed. The receipt records **1,027** complete
+forward/inverse dirty fixtures, **134,864** regular basis columns, **9,420**
+mixed words, **47,872** representation-product checks, **3,720** rank-one call
+cut checks over all **155** projectors through four roles, and **348** checks
+over all **174** invertible scalar maps through three roles. All **9,249**
+single-call omissions and **13** additional semantic/model controls reject.
+A deliberately incomplete receipt also rejects under `-O` with
+`receipt differs from exact recomputation`. Exact commands and environment are
+in [VALIDATION.json](VALIDATION.json); the repeatable counts and digest are in
+[RECEIPT.json](RECEIPT.json). No native, hosted CI or selected cold replay was run.
+
+Controls include omitted paid calls, wrong letter/order, residual dirty
+workspace, illegal projectors/labels, odd-period simultaneous-attainment failure,
+and the wrong parity-insensitive reflection offset. An invertible off-diagonal
+perturbation by `Omega=sum_{g in G}[g]` is invisible in **both** n-point proof
+representations: each permutation coefficient there occurs twice. It remains
+nonzero in the formal algebra and faithful regular action and is rejected by
+the endpoint verifier. Thus represented lower-bound checks are not mistakenly
+used as sufficient endpoint certificates.
+
+## Provenance, overlap and credit
+
+Base `0b1cccf86e2f5132ed286a26ef0a681a495740ea`; independent OS nonce
+`63537661f5ee8acaa486b70f82726911`, track `bit-mechanisms`, emitted card
+`bit-new-register-model`, method target-specific relaxation. Complete chooser
+output is retained in `SELECTION.json`; original source byte/blob pins and
+catalogue/selected-record provenance are in `SOURCES.json`. The selected record
+remains unchanged at SHA256
+`10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.
+
+Closest inspected live predecessors:
+
+* Adam Gall, [PR1911](https://github.com/CrocSwap/integer-mult-bounds/pull/1911),
+  head `712593f82739203639d0a169b88c6bcac6f0590c`: infinite-dihedral sharp
+  tariffs and the precise finite-torsion open question. No source code copied.
+* Cinna Davis, [PR1821](https://github.com/CrocSwap/integer-mult-bounds/pull/1821),
+  head `a986127fa82b680a5bbb29f49fec4c8c3a944bde`: D16 central pair at R=2.
+  The present result replaces that bounded-register gap with a uniform proof.
+* steph4n-gh, [PR1573](https://github.com/CrocSwap/integer-mult-bounds/pull/1573),
+  head `0b8c5a69297c794f2de22c04494c83e41300ae36`: arbitrary-register D8
+  central targets and separate tariffs, already established. The natural
+  representation/zero-projector-sum argument is a predecessor, not this proof.
+* Arham Tahir, [PR1919](https://github.com/CrocSwap/integer-mult-bounds/pull/1919),
+  head `f5610483cab0b5547be778a4e8983faf78aefa69`: rank-bounded D8 central
+  block calls and their invocation optimum. We do not claim that optimum.
+* [PR1792](https://github.com/CrocSwap/integer-mult-bounds/pull/1792), head
+  `cc3a4adbae23fb07687803d1361bb7a1bcd449b0`: D8/UT3 scope audit identifying
+  the already-completed arbitrary-workspace result and the danger of stale
+  next-question text. Its navigation warning is respected here.
+* woklerx, retained PR1443 D8 census; eumemic PR288 register model;
+  Ali Romman PR468 three-register obstruction; Gareth Davidson PR1333
+  nonabelian affine direction; their attribution remains unchanged.
+
+Initial shared overlap snapshot: 579 open PRs and 547 discussion records,
+completely paginated at 2026-10-14T14:10:13Z / 14:11:43Z. The final offline
+recheck at 14:30:45Z used the refreshed **584 open PRs** (14:27:47Z) and
+**565 discussion records** including comments/replies (14:29:09Z), both
+complete. The only new exact dihedral/cut matches were this package and this
+round's own claims. Targeted live predecessor heads were unchanged. A live
+all-state dihedral PR search and local README/ledger search also included the
+merged PR1443 predecessor and the PR1792 D8 scope audit.
+
+The helper's closed-PR coverage remains only local notes and the ledger;
+arbitrary code diffs are not searched, and cached records may miss live changes.
+No-match results are not novelty certificates. The mathematical change is
+finite torsion and a new balanced-cut invariant, not the seed or a larger census.
+[Precise pre-computation claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1929#discussioncomment-18925900).
+
+Prepared for **Adam Gall** with substantial **OpenAI Codex GPT-6 Astra**
+assistance in proof, implementation, verification and writing. All engines were
+written in the same AI-assisted investigation; algorithmic independence is not
+independent human authorship. Apache-2.0. No supplier, frontier or kappa change.
+
+## Open next questions
+
+1. **Odd periods.** The even half-interval proof and simultaneous choice of
+   representatives do not automatically extend. At n=3 the farthest reflection
+   has separate cyclic distances (1,1) but no word with those counts. Determine
+   the missing joint tariff constraint before attempting an odd-period theorem.
+2. **Physical admission.** Specify and fully price an all-size native reflection
+   child, including frame/compiler, hosting, restoration, precision and tape.
+   Until such a bridge exists, even the exact formal theorem is only a
+   mechanism-screening side result, not an exponent result.
diff --git a/research/a2-st-all-cap-tariff/predecessor/PR1966-PROOF.md b/research/a2-st-all-cap-tariff/predecessor/PR1966-PROOF.md
new file mode 100644
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--- /dev/null
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+# Odd finite dihedral groups: separate cuts and a sharp joint tariff
+
+This is an all-parameter written proof for the formal operation model below.
+Finite executable checks corroborate its implementation; they do not replace
+the proof, and no formal-kernel verification is asserted. This result supplies
+no physical BIT supplier, recursive child profile, precision estimate, or new
+multiplication exponent.
+
+## Statement and exact operation model
+
+Let `n=2m+1`, where `m>=1`, and let
+
+```
+G = D_(2n) = <r,x | r^n=x^2=1, xrx=r^-1>,
+y = r^-1 x, so that xy=r.
+```
+
+The coefficient ring is the full, generally noncommutative group algebra
+`F2[G]`. There are any finite number `R>=W` of scalar register roles. A paid
+unit call has the form
+
+```
+D_N(g) = I_R + ([g]+1) N,
+g in {x,y},
+N = u v^T over F2, with v^T u = 1.
+```
+
+Thus `N` is any binary rank-one idempotent, including non-coordinate ones.
+Arbitrary scalar `GL_R(F2)` maps are free and may be interleaved anywhere.
+They act on roles, not on group addresses. There are no additional free
+address operations, clean inputs, resets, erasures, or COPY operations.
+All inputs, including all spectator inputs, are arbitrary. The required
+complete formal endpoint is
+
+```
+T = diag([r^k_1 x^e_1], ... , [r^k_W x^e_W], 1, ... , 1),
+e_i in {0,1}.
+```
+
+In particular, every spectator is restored and every off-diagonal role
+coupling vanishes. Every paid call used for preparation or restoration is
+included in the bill.
+
+Define, for integers `k`,
+
+```
+d_n(k) = min(k mod n, n-(k mod n)),
+A = sum_i d_n(k_i+e_i),
+B = sum_i d_n(k_i),
+t = #{i : e_i=1 and k_i=m mod n}.
+```
+
+The unique exceptional group element is the longest reflection `r^m x`.
+If a circuit has `X` paid x calls and `Y` paid y calls, then
+
+```
+X >= A,
+Y >= B,
+X+Y >= A+B+t.                                      (1)
+```
+
+For every integer `j` with `0<=j<=t`, the pair
+
+```
+(X,Y) = (A+j, B+t-j)                               (2)
+```
+
+is attained by a literal coordinate-projector circuit with no extra roles
+and no scalar basis changes. Every feasible count pair componentwise
+dominates at least one pair in (2). These statements do not characterize
+every feasible pair away from that boundary.
+
+Consequently, for arbitrary nonnegative real prices `alpha,beta`, the exact
+minimum weighted cost is
+
+```
+alpha*A + beta*B + t*min(alpha,beta).                (3)
+```
+
+This includes zero prices. The exact minimum unweighted number of calls is
+`A+B+t`. The bounds hold uniformly over all finite `R`, all word lengths, all
+allowed rank-one idempotents, and all interleavings with free scalar maps.
+
+## The common cut-rank argument
+
+Given a permutation action of `G` on a finite address set `Omega`, extend it
+linearly to `F2[G]` and entrywise to register matrices. Choose a partition
+`Omega=H disjoint_union K`, and include every scalar role in each side. For
+an operator `M` on the represented space define
+
+```
+q_H(M) = rank_F2 M[K,H].
+```
+
+Here the block takes inputs supported on `H` to outputs supported on `K`.
+Block multiplication gives
+
+```
+(UV)[K,H] = U[K,H] V[H,H] + U[K,K] V[K,H],
+q_H(UV) <= q_H(U) + q_H(V).                         (4)
+```
+
+No invertibility is needed for this inequality. A free scalar map is
+`S tensor I_Omega` and has crossing rank zero. For a paid call the crossing
+block is
+
+```
+N tensor rho(g)[K,H],
+```
+
+so its rank is `rank(N)*q_H(rho(g))=q_H(rho(g))`.
+For a permutation `g`, the crossing block is a partial permutation matrix,
+and its rank is the number of elements of `H` mapped into `K`.
+
+The endpoint is diagonal in roles, so its crossing blocks are direct sums
+of the target crossing blocks. Their ranks add; identity spectators
+contribute zero. Formal equality of circuits implies equality under every
+representation used here. The proof never takes represented equality as a
+sufficient certificate of formal equality.
+
+This directly handles interleaved free scalar operations. Equivalently,
+one can normalize them by
+
+```
+S D_N(g) S^-1 = D_(SNS^-1)(g).
+```
+
+Conjugation preserves binary rank-one idempotence. Applying augmentation
+`[g] -> 1` to the full formal endpoint shows that the product of all free
+scalar factors must be `I_R`. Thus normalization leaves no unpriced scalar
+endpoint. This observation is consistent with, but not needed for, (4).
+
+## The two separate-letter cuts
+
+The finite permutation representations and separate-letter cut idea are
+inherited from Adam Gall's even-period result in
+[PR1941](https://github.com/CrocSwap/integer-mult-bounds/pull/1941), pinned at
+`5938ce3e91bbb8b5c2227af7976d444b2fbac7d4`. The argument is checked here for
+the changed odd period and its unequal sides.
+
+On addresses `a in Z/nZ`, for any integer `c`, put
+
+```
+rho_c(r)(a) = a+1,
+rho_c(x)(a) = c-a,
+rho_c(y)(a) = c-1-a.
+```
+
+These permutations satisfy the displayed presentation of the actual finite
+group. Use `H={0,...,m-1}` of size `m`; its complement has size `m+1`.
+For any interval of length `m` or `m+1` on the `(2m+1)`-cycle, translation
+by `k` sends exactly `d_n(k)` of its points outside the interval. To see
+this, it suffices to take `0<=k<=m`, for which precisely `k` points leave;
+negative translations give the same count and exhaust the other residues.
+
+For `c=m`, the letter y preserves H while x sends H to `H+1`. The per-call
+crossing ranks are therefore `(1,0)` for `(x,y)`. For `c=m-1`, x preserves
+H while y sends it to `H-1`, giving `(0,1)`. A rotation translates H by k.
+A reflection `r^k x` sends H to
+
+```
+H + (k+c-m+1).
+```
+
+The endpoint crossing ranks are therefore
+
+```
+c=m:    sum_i d_n(k_i+e_i) = A,
+c=m-1:  sum_i d_n(k_i)     = B.
+```
+
+Applying (4) proves the first two inequalities in (1). These unequal cuts
+alone miss one call for each longest reflection: its two ranks are `(m,m)`.
+The next cut supplies the joint constraint without a parity relaxation.
+
+## A regular-action cut gives the full alternating-word length
+
+Use the faithful left regular action on the `2n` addresses
+
+```
+(a,e) <-> r^a x^e,  a in Z/nZ, e in {0,1}.
+```
+
+In these coordinates,
+
+```
+r(a,e)       = (a+1,e),
+x(a,e)       = (-a,1-e),
+y(a,e)       = (-a-1,1-e),
+(r^k x)(a,e) = (k-a,1-e).
+```
+
+Let `I={-m,...,0}` and `J={0,...,m-1}`, interpreted modulo n. Choose the
+balanced regular-address cut
+
+```
+H = (I x {0}) union (J x {1}),
+K = (G-addresses) minus H.
+```
+
+Both sides have size `n`. Write `q(g)` for the crossing rank of the left
+regular permutation of g at this cut.
+
+For a rotation, each layer is simply translated. The two intervals have
+lengths `m+1` and `m`, respectively, so the interval calculation above yields
+
+```
+q(r^k) = 2 d_n(k).                                 (5)
+```
+
+A reflection interchanges the layers, and
+
+```
+(r^k x) H = ((k-J) x {0}) union ((k-I) x {1}).
+```
+
+The two intersection sizes `|I intersect (k-J)|` and
+`|J intersect (k-I)|` are equal under `a -> k-a`. Since H has n points,
+
+```
+q(r^k x) = n - 2 |I intersect (k-J)|.               (6)
+```
+
+Choose the unique representative `k' in {-m,...,m}`. For `0<=k'<=m`, the
+intersection in (6) has `m-k'` points: it is the part of
+`{k'-m+1,...,k'}` lying in `{-m,...,0}`. No additional modular intersection
+occurs because the positive endpoints are at most m. For `k'=-j` with
+`1<=j<=m`, the intersection is `{-m,...,-j}`, of size `m-j+1`. The part
+below `-m` wraps to positive residues at most m and adds no points of I.
+It follows that
+
+```
+q(r^k x) = 2k'+1,   if 0<=k'<=m,
+            -2k'-1, if -m<=k'<=-1.                 (7)
+```
+
+In particular, `q(x)=1` and `q(y)=1`. At the longest reflection `k'=m`,
+the rank is `2m+1=n`: it interchanges H and K completely. For every other
+reflection it is `d_n(k+1)+d_n(k)`. Equations (5)--(7) can thus be written
+uniformly as
+
+```
+q(r^k x^e) = d_n(k+e)+d_n(k)
+             + 1_{e=1 and k=m mod n}.              (8)
+```
+
+The quantities in (5)--(7) also equal the lengths of the shortest
+alternating x/y words. This equality does not need an independent word
+metric theorem: subadditivity and `q(x)=q(y)=1` give a lower bound on any
+literal word, and the attaining words below have exactly these lengths.
+
+By additivity over roles, the endpoint has regular-cut crossing rank
+`A+B+t`. Each paid call has crossing rank one at this cut, and each free
+scalar operation has rank zero. Equation (4) proves
+
+```
+X+Y >= A+B+t,
+```
+
+the remaining inequality in (1). It applies to arbitrarily many longest
+reflections simultaneously, without limiting the number of restored dirty
+spectators.
+
+## Attaining formal circuit and complete invoice
+
+Let `E_i` be the coordinate projector onto role i. In the full formal group
+algebra, directly multiplying the two factors gives
+
+```
+D_(E_i)(g) D_(E_i)(h) = D_(E_i)(gh).                (9)
+```
+
+Indeed `E_i^2=E_i`, and in characteristic two the coefficient of `E_i`
+reduces to `[gh]+1`. No permutation representation is used to certify (9).
+Calls on distinct coordinate roles commute.
+
+For rotations, take the representative `k' in {-m,...,m}` and the literal
+word `(xy)^k'`, interpreting negative powers as repetitions of `yx`. Its
+counts are `(d_n(k),d_n(k))`.
+
+For a reflection other than `r^m x`, choose `k' in {-m,...,m-1}` and freely
+reduce `(xy)^k' x`. If `k'>=0`, its counts are `(k'+1,k')`; if `k'=-j<0`,
+they are `(j-1,j)`. In either case the counts are exactly
+`(d_n(k+1),d_n(k))`.
+
+The longest reflection has the two formal words
+
+```
+(xy)^m x = r^m x,
+(yx)^m y = r^(-m-1) x = r^m x,
+```
+
+with counts `(m+1,m)` and `(m,m+1)`. Choose the first word on exactly j of
+the t longest-reflection roles and the second on the remainder. Equations
+(9) and the group relation `r^n=1` prove the complete formal endpoint with
+the counts in (2). With column-vector conventions, the rightmost displayed
+factor acts first, so each displayed product is executed right to left.
+
+For any feasible counts write `X=A+u`, `Y=B+v`. By (1), the integers u,v
+are nonnegative and `u+v>=t`. There is an integer
+
+```
+max(0,t-v) <= j <= min(t,u),
+```
+
+so `(X,Y)` componentwise dominates `(A+j,B+t-j)`. Nonnegative weights
+therefore have minimum (3), attained by selecting the cheaper extra letter
+on each longest-reflection role. Equal weights permit every choice of j.
+
+| Resource | Complete attaining implementation |
+| --- | --- |
+| Arbitrary-input target roles | W |
+| Extra roles required | Zero; any added spectators remain untouched |
+| Paid x/y calls | `(A+j,B+t-j)` for chosen `0<=j<=t` |
+| Scalar changes of basis | Zero |
+| Clean inputs, COPYs, resets, erasures | None |
+| Extra preparation or inverse restoration | None hidden or deferred |
+| Endpoint verification | Equality in the full formal group algebra |
+| Physical supplier or recursive adoption | Not supplied |
+
+The lower-bound argument also applies to arbitrary-rank binary idempotent
+calls if each call is charged its scalar rank: (1) then bounds x rank mass,
+y rank mass, and their sum. This is not an invocation-count optimum for
+batched calls and does not price recursive children by their rank.
+
+## Scope, predecessor credit, and remaining obligations
+
+The separate-letter construction and the explicit odd-period gap come from
+Adam Gall's [PR1941](https://github.com/CrocSwap/integer-mult-bounds/pull/1941),
+at the pinned head above. Adam Gall's
+[PR1911](https://github.com/CrocSwap/integer-mult-bounds/pull/1911), head
+`712593f82739203639d0a169b88c6bcac6f0590c`, supplies the infinite-dihedral
+predecessor and its finite-torsion question. The new mathematical step here
+is the regular-action cut (6)--(8) that closes the joint odd-period tariff;
+the general subadditive cut-rank principle is not claimed as new.
+
+Relevant overlapping evidence also includes the coordinate/CNOT setting of
+[PR1414](https://github.com/CrocSwap/integer-mult-bounds/pull/1414) and
+Cinna Davis's [PR1953](https://github.com/CrocSwap/integer-mult-bounds/pull/1953),
+head `4f3bdf7c3bdd7033180c9d8c41245023cc402bc6`, with its finite D10
+inverse-rotation special case. Those special cases are credited, not claimed
+as previously unsettled instances. This proof allows every binary rank-one
+idempotent, arbitrary free scalar mixing, and any finite number of restored
+dirty spectators. Overlap review and publication status belong to the package
+metadata; this mathematical proof is not a novelty certificate.
+
+The covered domain is odd `n>=3`, any finite `R>=W`, and complete diagonal
+formal endpoints of the displayed kind. The proof uses no integer lift,
+analytic approximation, or precision assumption: the historical scalar
+mixing permission is an optimistic formal relaxation. Moving to a physical
+supplier would require independently admitting and pricing the native group
+actions, all frame/compiler and hosting work, precision/tape behavior, and
+the full recursive profile. None of those obligations follows from the
+formal tariff. Other free address maps, other paid labels, partial endpoint
+contracts, clean ancillas, or nonlinear operations are outside this theorem.
+
+Prepared with substantial OpenAI Codex assistance. Apache-2.0.
diff --git a/research/a2-st-all-cap-tariff/predecessor/PR6817-PROOF.md b/research/a2-st-all-cap-tariff/predecessor/PR6817-PROOF.md
new file mode 100644
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@@ -0,0 +1,191 @@
+# Packed-rank bounds and sharp divisible batch tariffs
+
+## Model and inherited input
+
+Let K be any field and let W,S be a Coxeter system with finitely many simple
+generators. W may be infinite. There are r finite scalar roles, all with
+arbitrary input. Work coefficientwise in M_r(K[W]). A paid factor is
+
+    F_s(A) = I + ([s]-[e]) A,   0 < rank_K(A) <= b.
+
+A need not be a projector. This relaxation even permits singular factors.
+Arbitrary interleaved GL_r(K) scalar maps are free. There are no free address
+maps, resets, clean allocations or discarded outputs. The complete endpoint is
+
+    T = diag([w_1], ..., [w_d], 1, ..., 1).
+
+Every spectator returns its input and all off-diagonal entries vanish.
+For each root orbit O, a call with simple root in O costs f_O(rank A).
+Assume f_O on {0,...,b} is nonnegative, nondecreasing, f_O(0)=0, with
+nonincreasing successive differences. Thus the theorem includes the sampled
+values of every nondecreasing concave real function vanishing at zero.
+Prices must be constant on root orbits, as in the predecessor.
+
+We cite David Arias's PR2738, retained at the source pins in SOURCES.json,
+for the root facts and root-cut method. Its PROOF.md sections 2 through 5
+provide the geometric root action, finite inversion sets, root-orbit reduced
+letter counts and the full formal coordinate construction. Those facts and
+the rank inequality are inherited. Its binary rank-linear tariff does not
+price the batches here. No predecessor executable is a dependency.
+
+## 1. Integer packing lemma
+
+For an integer M>=0 define
+
+    P_(b,f)(M) = floor(M/b) f(b) + f(M mod b).
+
+If 1<=k_j<=b and sum_j k_j>=M, then
+
+    sum_j f(k_j) >= P_(b,f)(M).                         (1)
+
+To prove it, include zero parts temporarily. If two parts satisfy
+0<a<=c<b, transfer delta=min(a,b-c) from a to c. Each unit transferred
+replaces f(a)-f(a-1) by f(c+1)-f(c), which is no larger by concavity.
+The sum of f never increases; one part becomes zero or b. Repeat until
+at most one part lies strictly between zero and b. The resulting cost is
+P_(b,f)(sum k_j). Finally P_(b,f) is nondecreasing. Between multiples of b
+this follows from monotonicity of f; the step to a multiple of b uses
+f(b)>=f(b-1). This proves (1), including b=1 and zero demands.
+
+This is a standard concave concentration argument, not a new optimization
+principle. The contribution below applies it to the complete root-cut bill.
+
+## 2. General orbitwise lower bound, over every field
+
+Write i_O(w) for the number of positive roots in orbit O that w sends
+negative, and M_O=sum_j i_O(w_j). Then every legal word costs at least
+
+    sum_O P_(b,f_O)(M_O).                              (2)
+
+Here is why the binary predecessor's root cuts apply over K. Use the
+algebraic direct sum of K-basis vectors indexed by the real roots in O.
+Positive and negative roots remain distinct labels in every characteristic.
+The W action permutes these labels. Tensor with K^r. Let q_O(U) be the rank
+of U's block from positive-root inputs to negative-root outputs. For two
+operators the block identity
+
+    (UV)_-+ = U_-+ V_++ + U_-- V_-+
+
+gives q_O(UV)<=q_O(U)+q_O(V). Each block has finite rank: every formal
+entry is a finite sum of group elements, each with finitely many inversions.
+This also justifies the argument for infinite W without truncation.
+
+A free scalar map has crossing rank zero. A simple s crosses just its own
+positive simple root, so F_s(A) has crossing rank rank_K(A) in O(s), and
+zero in the other orbits. The minus identity term does not cross the cut.
+The target has crossing rank M_O. Consequently the sum of paid ranks in
+orbit O is at least M_O. Apply (1) to that orbit. Sum its actual price
+bound over the finitely many orbits to prove (2). No idempotence,
+invertibility of paid factors, binary parity or faithful algebra
+representation was used. Endpoint equality is required in the full algebra;
+passing the root-action image alone does not establish it.
+
+## 3. Sharp identical-target theorem
+
+Suppose all d productive targets equal w and b divides d. With any finite
+number of restored dirty spectators, the exact cost is
+
+    (d/b) sum_O i_O(w) f_O(b).                         (3)
+
+Indeed M_O=d i_O(w) is divisible by b, so (2) equals (3). For attainment,
+partition the productive roles into d/b blocks of size b, and let E_B be
+the coordinate projector of one block. For all g,h in W,
+
+    [I+([g]-1)E_B] [I+([h]-1)E_B]
+        = I+([gh]-1)E_B.
+
+A reduced expression for w, applied on that block, uses rank-b legal
+projectors, with i_O(w) calls in orbit O. Repeat on the disjoint blocks.
+The formal product is exactly T; spectators are untouched. All calls,
+including preparation or restoration if used, are subject to the same cap
+and tariff. The upper construction needs neither. For power prices
+f_O(k)=c_O k^theta, 0<theta<=1, (3) is
+
+    d b^(theta-1) sum_O c_O i_O(w).
+
+Thus batching saves against separate rank-one calls when the price is
+strictly concave, and arbitrary dirty workspace cannot improve that saving
+under this explicit rank cap. This is a formal tariff, not a recursive
+invoice or native implementation result.
+
+## 4. Nondivisible single-reflection case and bounds elsewhere
+
+If w=s is a single simple generator, (2) is P_(b,f_O(s))(d), and batching
+into floor(d/b) full blocks and one remainder attains it for every d.
+For a general identical w without b|d, the same construction supplies only
+
+    sum_O P_(b,f_O)(d i_O(w)) <= optimum
+        <= sum_O i_O(w) P_(b,f_O)(d).                  (4)
+
+These bounds can differ without indicating an open optimum. Nick Petalas's
+retained `research/rank2-swap-batching/README.md` already gives the five-call
+profile (2,2,1,2,2) for d=3, b=2 and the A2 longest element. It attains the
+packed lower bound f(1)+4f(2). Its diagonal supports also work over every
+field. We credit that schedule and its binary cap-two tariff; they are not
+new here. The following different remainder target exhibits a genuine
+chronological cost beyond the packed root bound.
+
+## 5. Exact chronological remainder tariff for the A2 two-letter target
+
+Let W=A2=S3 with simple generators s,t, let w=st, d=3 and b=2. In the
+same all-field model with any finite restored dirty spectators, write
+c1=f(1), c2=f(2), so 0<=c1<=c2<=2c1. The exact cost is
+
+    2c1+2c2.                                          (5)
+
+The root bound gives only 3c2. Formula (5) is strictly larger when c2<2c1.
+For f(k)=k/(k+1), the exact cost is 7/3, above the root floor 2.
+
+First normalize the free scalar maps by conjugation and augmentation, as
+in the predecessor. This preserves each paid rank and leaves no final
+scalar residue. Any three-call word has labels among the eight words on
+s,t. Denote its scalar couplings A,B,C. Its coefficient at [st] is
+
+    sss: 0                 tss: 0
+    sst: (A+B-2AB)C         tst: (I-A)BC
+    sts: AB(I-C)           tts: 0
+    stt: A(B+C-2BC)         ttt: 0.
+
+Each displayed coefficient has rank at most two, since it factors through
+one of A,B,C. All shorter words also have [st] coefficient of rank at most
+two; the only two-letter possibility is st with coefficient AB. But the
+required coefficient is diag(I_3,0), of rank three. At least four calls
+are necessary. This proof permits non-idempotent and singular couplings,
+and does not specialize the scalar field or bound spectator count.
+
+The inherited root rank floor gives q1+2q2>=6. Combine it with q1+q2>=4.
+If q2<=2, the cost is at least
+6c1-(2c1-c2)q2 >= 2c1+2c2. If q2=3, at least one rank-one call is needed
+and 3c2+c1>=2c1+2c2. If q2>=4, the cost is at least
+4c2>=2c1+2c2. These cases include zero prices. For attainment, apply s on
+the first two roles and then the third, and do the same for t, in written
+product order. The rank profile is (2,1,2,1) and the formal endpoint is
+exactly diag([st],[st],[st],1,...,1). This proves (5).
+
+This is a chronological remainder obstruction. It concerns st, not the
+predecessor's longest element sts, for which changing the role supports
+attains the packed bound. It does not classify other nondivisible targets.
+
+The rank cap is part of the theorem. With d=b=2 and a single s,
+f(k)=k/(k+1), the sharp cost is 2/3, whereas two rank-one calls cost 1.
+The predecessor's rank-linear cost cannot be reused at this concave price.
+If d=4 and a declared cap b=2 is silently relaxed to 4, one rank-four
+call costs 4/5 rather than the capped optimum 4/3. Restored spectator count
+and allowed paid rank are different resources. Formula (3) does not settle
+an unbounded-rank primitive whose rank cap grows freely with workspace.
+
+## Scope and credit
+
+The new result is the packed orbitwise lower bound and its exact divisible
+and single-reflection consequences under concave rank prices, together with
+the exact chronological two-letter remainder tariff (5). Credit the
+Coxeter root and rank-linear result to David Arias, retaining Adam Gall,
+Ross Richardson, ywz65 and the eumemic/Ali Romman/Gareth Davidson model lineage
+credited there. PR6613 by Nick Petalas changes the monomial reflection
+alphabet and retains unit prices; it is related context, not a dependency.
+The concentration argument and field-independent rank inequalities are
+standard. No literature-priority, independent human review or proof-kernel
+acceptance is claimed.
+
+Produced with OpenAI GPT-6.1 Sol at high reasoning effort through Codex.
+New work is Apache-2.0, prepared for Benny Zhao / zhaobenny.
diff --git a/research/a2-st-all-cap-tariff/predecessor/PR6817-README.md b/research/a2-st-all-cap-tariff/predecessor/PR6817-README.md
new file mode 100644
index 0000000000..a9098b3da0
--- /dev/null
+++ b/research/a2-st-all-cap-tariff/predecessor/PR6817-README.md
@@ -0,0 +1,154 @@
+# Exact capped concave Coxeter batch tariffs
+
+A scoped written theorem with exact finite corroboration. Identical Coxeter
+targets on d productive roles, with paid rank at most b dividing d, have exact
+cost
+
+    (d/b) sum_O i_O(w) f_O(b).
+
+Here O ranges over root orbits and each f_O is a nondecreasing concave rank
+price vanishing at zero. The result permits every coefficient field, arbitrary
+finite restored dirty spectators, free scalar GL mixing, and even arbitrary
+rank-capped scalar couplings. A single simple-reflection target has exact cost
+`floor(d/b) f(b) + f(d mod b)` without divisibility. See [PROOF.md](PROOF.md)
+for the complete formal endpoint and proof.
+
+There is also an exact chronological remainder result. For three A2 targets
+`st`, rank cap two, and prices `0<=c1<=c2<=2c1`, the exact cost is
+`2c1+2c2`. The packed root floor is only `3c2`. At f(k)=k/(k+1), this is
+`7/3` versus a floor of `2`. The proof rules out every three-call word by
+its full signed [st] coefficient, over every field and any spectator count.
+
+The new step is a packed-rank lower bound for every orbit, followed by a sharp
+batch construction in these cases. David Arias's [PR2738](https://github.com/CrocSwap/integer-mult-bounds/pull/2738)
+already supplies the root-cut method and binary rank-linear tariff. Its proof
+explicitly leaves concave invocation prices outside scope. The concentration
+lemma and field-independent rank inequalities are standard. No literature
+priority claim is made.
+
+This resolves one formal batch-price question without a larger word census.
+It is a changed-scope side result for `bit-new-register-model`, not a native
+BIT construction or a new kappa. All five selected framework assumptions remain.
+
+Nick Petalas's retained [rank-two swap batching](../rank2-swap-batching/README.md)
+already settles the cap-two three-target A2 longest element with a five-call
+schedule. We credit that construction and price theorem. The new remainder
+result concerns the different target st; it shows why a packed rank floor
+can be strict even though the longest-element schedule attains it.
+
+## Evidence and reproduction
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/coxeter-capped-concave-tariff
+python3.12 -I -B research/coxeter-capped-concave-tariff/verify.py
+python3.12 -I -O -B research/coxeter-capped-concave-tariff/verify.py
+```
+
+Direct commands are read-only, import no predecessor code and need no Git,
+network or external input. They print `PASS_COXETER_CAPPED_CONCAVE_TARIFF`
+in a JSON receipt. The registered replay runs ordinary and optimized modes and
+`test_mutations.py`, each with a 120-second cap. The mutation suite reruns the
+three changed verifiers and requires their specific mathematical failures. Source pins are historical provenance;
+metadata validation authenticates them separately from the standalone math.
+
+Finite corroboration covers:
+
+- 14,010 exact packing optima, computed by dynamic programming over demands
+  1 through 30. Caps are 1 through 6. Prices include every decreasing integer
+  slope sequence with slopes in 0 through 4 and the rational tariff k/(k+1).
+- 4,872 arbitrary-coupling gate checks on A2 and B2 root actions. These use
+  every binary 3 by 3 matrix and every ternary and quinary 2 by 2 matrix,
+  including zero, nilpotent and singular couplings.
+- 1,440 chronological product-cut inequalities over F2, F3 and F5.
+- 672 full formal attaining endpoints for all six A2 and eight B2 elements,
+  d=1 through 6, each dividing rank cap, plus nondivisible single-reflection
+  cases. Each has one dirty spectator. Interleaved invertible shears genuinely
+  couple it to the 2,655 paid calls and restore the exact endpoint.
+- Eight signed noncommutative three-call coefficient expansions for the new
+  st obstruction, three full formal four-call st endpoints, and 21 exact
+  price-profile optima with the chronological call-count constraint.
+- 588 omitted-call mutants, all rejected. Boundary checks retain a nonzero
+  dirty leak invisible to augmentation, odd-characteristic non-involutions,
+  and the difference between generic nondivisible lower and upper bounds. Packing controls show failure
+  under nonconcavity or nonmonotonicity and a real improvement when the paid
+  rank cap is increased.
+
+These are exact finite checks of new formulas and explicit formal words,
+not an enumeration of all scalar circuits. The universal lower bound depends
+on the written proof. The finite group-action images are lower-bound tools;
+upper endpoints are checked coefficientwise in the full formal algebra.
+
+Local validation used CPython 3.12.9 on Linux x86_64,
+`6.18.33.2-microsoft-standard-WSL2`. Direct normal and optimized CPython
+3.10.12 runs also passed with identical counts. Registered replay, source pins,
+committed bytes/modes, source-history ancestry and an official package export
+passed. A Git-free read-only export passed both direct modes and the mutation
+suite, preserving every exported byte. Reproduce publication checks after
+committing with:
+
+```sh
+python3.12 -B scripts/check_research_commit.py --package research/coxeter-capped-concave-tariff --check-source-history --check-modes
+python3.12 -B scripts/export_research_package.py export --package research/coxeter-capped-concave-tariff --output /tmp/capped.tar.gz
+python3.12 -B scripts/export_research_package.py check /tmp/capped.tar.gz --check-inputs
+```
+
+Hosted CI is separate from this local evidence and from scientific acceptance.
+No local mathematical check was left for CI.
+
+## Scope, freshness and attribution
+
+Base `7c49fe32c55de02c2c873655c804b16e4c3fef0a`, priority revision 8,
+selected #6030. The first selected-binding orientation passed; it is not a
+selected cold replay. The chooser nonce was
+`673ec751f0c36bc214acd72a41770f58`. The scientific scope and stop rule are in
+[TASK.json](TASK.json), and exact predecessor blobs and SHA256 pins are in
+[SOURCES.json](SOURCES.json). Frozen sources and selected records are unchanged.
+PR6811 and PR6812 are not resumed or repackaged.
+
+[Canonical investigation and claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/6814).
+Entry overlap checked live handoffs #1166, start-here #1163, active discussions
+and the card's open PR search. The completed snapshot covered 762 open PRs at
+20:35:08 UTC and paginated discussions at 20:37:30 UTC on November 6, 2026,
+with all 8,917 tracked local research notes present. Separate discussion
+snapshots returned 6,884 and 6,858 records; these are non-atomic observations,
+not a stable board census. No competing capped-concave Coxeter tariff was
+identified within that coverage. Closed PRs are represented through retained
+notes and the ledger only; arbitrary diffs and unpublished work are outside it.
+The final refresh covered 762 open PRs at 20:42:55 UTC and 6,887 Discussion
+records at 20:44:28 UTC; canonical main remained unchanged. Inspection of all
+Coxeter matches found the cap-two longest-element predecessor, which is now
+pinned and credited rather than treated as a new question. Targeted remainder
+searches found no competing st tariff within that corpus.
+PR6613 at `d2704e235f3b9e8ac01cd794d96db0ece1be4672` changes the monomial
+reflection alphabet while retaining unit rank-one prices. Its concave native
+primitive question remains open; no source or executable from it is used.
+
+Credit Nick Petalas for the retained cap-two longest-element schedule, and
+David Arias and the Adam Gall, Ross Richardson, ywz65,
+eumemic, Ali Romman and Gareth Davidson lineage retained in PR2738.
+The root theory cited there remains mathematical input. New work is
+Apache-2.0, prepared for Benny Zhao / zhaobenny.
+
+Produced with OpenAI GPT-6.1 Sol at high reasoning effort through Codex.
+No subagents, independent human review or formal proof-kernel review were used.
+
+## Open next questions
+
+For general nondivisible d and a multi-reflection w, the packed lower bound
+need not be sharp. The new three-role st theorem proves a strict correction;
+the credited longest-element schedule attains the packed floor. Which other
+small targets have a chronological correction? First derive candidate rank
+profiles and inspect full signed coefficients. The present theorem does not
+classify other remainder cases or heterogeneous targets.
+
+The rank cap is a real primitive restriction. Additional dirty workspace does
+not grant larger-rank calls for free. Native address realization, scalar-map
+charges, owned/borrowed storage, routing, restoration, precision, hosting,
+all-size recursion and a complete supplier invoice are unknown here. Start
+with one physically realized rank-two call and its complete bill before
+assigning further implementation work. The formal price is not that bill.
+Stop this family round at the exact tariff above. No supplier admission,
+selected-frontier change, Lean, Docker, predecessor scientific replay or full
+multiplication theorem is supplied. Structured next questions are in TASK.json.
diff --git a/research/a2-st-all-cap-tariff/verify.py b/research/a2-st-all-cap-tariff/verify.py
new file mode 100644
index 0000000000..201db79d4f
--- /dev/null
+++ b/research/a2-st-all-cap-tariff/verify.py
@@ -0,0 +1,417 @@
+"""New finite formal-algebra corroboration; no predecessor executable runs.
+
+Products are in written order: F_s * F_t has formal label st. Finite fields
+and dimensions corroborate, but do not prove, the all-field tariff theorem.
+"""
+from fractions import Fraction as F
+from itertools import product
+import hashlib
+import json
+from pathlib import Path
+import sys
+
+PACKAGE = Path(__file__).resolve().parent
+MANIFEST_SHA = '13071f312dc31569a121853f4b42d24084617f967937edca5a0cdbf5823b0705'
+PATHS = {'predecessor/PR6817-README.md', 'predecessor/PR6817-PROOF.md',
+         'predecessor/PR1941-README.md', 'predecessor/PR1966-PROOF.md'}
+LABELS = ('e', 's', 't', 'st', 'ts', 'sts')
+
+
+def require(ok, reason):
+    if not ok:
+        raise ValueError(reason)
+
+
+def unique(pairs):
+    result = {}
+    for key, value in pairs:
+        require(key not in result, 'duplicate JSON key')
+        result[key] = value
+    return result
+
+
+def no_number(text):
+    raise ValueError('noninteger JSON number')
+
+
+def load(raw):
+    return json.loads(raw, object_pairs_hook=unique, parse_float=no_number, parse_constant=no_number)
+
+
+def canonical(value):
+    return json.dumps(value, sort_keys=True, separators=(',', ':'), ensure_ascii=True, allow_nan=False).encode()
+
+
+def digest(value):
+    return hashlib.sha256(canonical(value)).hexdigest()
+
+
+def authenticate():
+    raw = (PACKAGE / 'SOURCE-PINS.json').read_bytes()
+    require(hashlib.sha256(raw).hexdigest() == MANIFEST_SHA, 'source manifest bytes')
+    manifest = load(raw)
+    require(type(manifest['schema_version']) is int and manifest['schema_version'] == 1, 'source version')
+    require(len(manifest['files']) == 4 and {x['local_path'] for x in manifest['files']} == PATHS,
+            'source inventory')
+    identities = {}
+    for row in manifest['files']:
+        path = PACKAGE / row['local_path']
+        require(path.is_file() and not path.is_symlink(), 'source regular file')
+        data = path.read_bytes()
+        blob = hashlib.sha1(b'blob ' + str(len(data)).encode() + b'\0' + data).hexdigest()
+        require((len(data), hashlib.sha256(data).hexdigest(), blob) ==
+                (row['bytes'], row['sha256'], row['git_blob']) and row['mode'] == '100644', 'source bytes')
+        identities[row['local_path']] = row['sha256']
+    return identities
+
+
+def group_table():
+    compose = lambda a, b: tuple(a[b[i]] for i in range(3))
+    e, s, t = (0, 1, 2), (1, 0, 2), (0, 2, 1)
+    basis = (e, s, t, compose(s, t), compose(t, s), compose(s, compose(t, s)))
+    require(len(set(basis)) == 6, 'six distinct S3 labels')
+    table = [[basis.index(compose(a, b)) for b in basis] for a in basis]
+    require(table[1][2] == 3 and table[2][1] == 4 and table[1][1] == table[2][2] == 0,
+            'S3 product orientation')
+    return table
+
+
+def zero(n):
+    return [[0] * n for _ in range(n)]
+
+
+def identity(n):
+    return [[int(i == j) for j in range(n)] for i in range(n)]
+
+
+def madd(a, b, field, sign=1):
+    return [[(x + sign*y) % field for x, y in zip(ar, br)] for ar, br in zip(a, b)]
+
+
+def mmul(a, b, field):
+    n, out = len(a), zero(len(a))
+    for i in range(n):
+        for k, value in enumerate(a[i]):
+            if value:
+                for j, other in enumerate(b[k]):
+                    if other:
+                        out[i][j] = (out[i][j] + value * other) % field
+    return out
+
+
+def rank(matrix, field):
+    work = [row[:] for row in matrix]
+    height, width, pivot_row = len(work), len(work[0]) if work else 0, 0
+    for column in range(width):
+        pivot = next((i for i in range(pivot_row, height) if work[i][column]), None)
+        if pivot is None:
+            continue
+        work[pivot_row], work[pivot] = work[pivot], work[pivot_row]
+        inverse = pow(work[pivot_row][column], field - 2, field)
+        work[pivot_row] = [(x * inverse) % field for x in work[pivot_row]]
+        for i in range(height):
+            if i != pivot_row and work[i][column]:
+                scalar = work[i][column]
+                work[i] = [(x - scalar*y) % field for x, y in zip(work[i], work[pivot_row])]
+        pivot_row += 1
+        if pivot_row == height:
+            break
+    return pivot_row
+
+
+def inverse(matrix, field):
+    n = len(matrix)
+    work = [row[:] + tail for row, tail in zip(matrix, identity(n))]
+    for column in range(n):
+        pivot = next((i for i in range(column, n) if work[i][column]), None)
+        require(pivot is not None, 'singular free scalar map')
+        work[column], work[pivot] = work[pivot], work[column]
+        scalar = pow(work[column][column], field - 2, field)
+        work[column] = [(x * scalar) % field for x in work[column]]
+        for i in range(n):
+            if i != column and work[i][column]:
+                scalar = work[i][column]
+                work[i] = [(x - scalar*y) % field for x, y in zip(work[i], work[column])]
+    result = [row[n:] for row in work]
+    require(mmul(matrix, result, field) == mmul(result, matrix, field) == identity(n), 'free inverse')
+    return result
+
+
+def clean(poly):
+    return {label: matrix for label, matrix in poly.items() if any(any(row) for row in matrix)}
+
+
+def paid(coupling, label, cap, field, edge=False):
+    amount = rank(coupling, field)
+    if not edge:
+        require(amount > 0, 'zero paid rank')
+        require(amount <= cap, 'paid rank cap')
+    return clean({0: madd(identity(len(coupling)), coupling, field, -1), label: coupling}), amount
+
+
+def pmul(left, right, table, field):
+    result = {}
+    for a, matrix in left.items():
+        for b, other in right.items():
+            label, value = table[a][b], mmul(matrix, other, field)
+            result[label] = madd(result.get(label, zero(len(matrix))), value, field)
+    return clean(result)
+
+
+def target(d, n):
+    productive, spectator = zero(n), zero(n)
+    for i in range(n):
+        (productive if i < d else spectator)[i][i] = 1
+    return clean({3: productive, 0: spectator})
+
+
+def augmentation(poly, n, field):
+    result = zero(n)
+    for matrix in poly.values():
+        result = madd(result, matrix, field)
+    return result
+
+
+def cut_rank(poly, n, positive, table, field):
+    negative = [g for g in range(6) if g not in positive]
+    block = []
+    for i in range(n):
+        for output in negative:
+            row = []
+            for j in range(n):
+                for incoming in positive:
+                    row.append(sum(matrix[i][j] for label, matrix in poly.items()
+                                   if table[label][incoming] == output) % field)
+            block.append(row)
+    return rank(block, field)
+
+
+def cut_checks(table):
+    rows = []
+    for field in (2, 3, 5):
+        ranks, nonprojectors, checks = {}, 0, 0
+        for cells in product(range(field), repeat=4):
+            coupling = [list(cells[:2]), list(cells[2:])]
+            amount = rank(coupling, field)
+            ranks[amount] = ranks.get(amount, 0) + 1
+            nonprojectors += int(mmul(coupling, coupling, field) != coupling)
+            for name, positive, isolated in (('s', (0, 2), 1), ('t', (0, 1), 2)):
+                for label in (1, 2):
+                    factor, measured = paid(coupling, label, 2, field, edge=True)
+                    actual = cut_rank(factor, 2, positive, table, field)
+                    require(measured == amount and actual == (2*amount if label == isolated else 0),
+                            'selective cut rank')
+                    checks += 1
+        scalar = [[1, 1], [1, 0]]
+        require(rank(scalar, field) == 2, 'free scalar example')
+        for positive in ((0, 2), (0, 1)):
+            require(cut_rank({0: scalar}, 2, positive, table, field) == 0, 'free scalar cut')
+        require(ranks.get(0) == 1 and nonprojectors > 0, 'arbitrary coupling domain')
+        rows.append(dict(field=field, matrices=field**4, rank_histogram=ranks,
+                         nonprojectors=nonprojectors, gate_cut_checks=checks,
+                         zero_coupling_is_edge_not_paid=True))
+    return rows
+
+
+def mixed(matrix, productive):
+    return any(matrix[i][j] for i in range(len(matrix)) for j in range(len(matrix))
+               if (i < productive) != (j < productive))
+
+
+def word(d, cap, field, order=(1, 2)):
+    n = d + 2
+    left, right = identity(n), identity(n)
+    left[0][d], right[d+1][0] = 1, 1
+    gauge = mmul(left, right, field)
+    instructions, calls = [['scalar', gauge]], []
+    for start in range(0, d, cap):
+        projector = zero(n)
+        for i in range(start, min(start + cap, d)):
+            projector[i][i] = 1
+        calls += [(label, projector, start) for label in order]
+    for index, (label, projector, start) in enumerate(calls):
+        coupling = mmul(mmul(inverse(gauge, field), projector, field), gauge, field)
+        instructions.append(['paid', label, coupling])
+        if index + 1 == len(calls):
+            scalar = inverse(gauge, field)
+        else:
+            scalar = identity(n)
+            a, b = (start, d) if index % 2 == 0 else (d+1, start)
+            scalar[a][b] = 1
+        instructions.append(['scalar', scalar])
+        gauge = mmul(gauge, scalar, field)
+    require(gauge == identity(n), 'gauge return')
+    return instructions
+
+
+def execute(instructions, d, cap, field, table):
+    n, result = d + 2, {0: identity(d + 2)}
+    ranks, labels, coupled, interleaved = [], [], 0, 0
+    for index, instruction in enumerate(instructions):
+        if instruction[0] == 'scalar':
+            matrix = instruction[1]
+            require(rank(matrix, field) == n, 'free scalar rank')
+            if 0 < index < len(instructions)-1:
+                interleaved += int(mixed(matrix, d))
+            factor = {0: matrix}
+        else:
+            require(instruction[0] == 'paid', 'instruction kind')
+            factor, amount = paid(instruction[2], instruction[1], cap, field)
+            ranks.append(amount)
+            labels.append(LABELS[instruction[1]])
+            coupled += int(mixed(instruction[2], d))
+        result = pmul(result, factor, table, field)
+    return result, ranks, labels, coupled, interleaved
+
+
+def price(name, cap):
+    return [F(0)] + [F(0) if name == 'zero' else F(k) if name == 'linear'
+                    else F(1) if name == 'flat' else F(k, k+1) for k in range(1, cap+1)]
+
+
+def packed(demand, cap, prices):
+    return (demand // cap) * prices[cap] + prices[demand % cap]
+
+
+def packing_dp(demand, cap, prices):
+    values, witnesses = [F(0)], [[]]
+    for amount in range(1, demand+1):
+        choices = [(prices[k] + values[max(0, amount-k)],
+                    [k] + witnesses[max(0, amount-k)]) for k in range(1, cap+1)]
+        value, witness = min(choices, key=lambda x: (x[0], x[1]))
+        values.append(value)
+        witnesses.append(witness)
+    return values[-1], witnesses[-1]
+
+
+def rational(value):
+    return [value.numerator, value.denominator]
+
+
+def price_checks():
+    rows = []
+    for cap in range(1, 7):
+        for d in range(1, 7):
+            for name in ('zero', 'linear', 'flat', 'hyperbolic'):
+                prices = price(name, cap)
+                optimum, witness = packing_dp(d, cap, prices)
+                expected = packed(d, cap, prices)
+                require(optimum == expected, 'packing tariff')
+                rows.append(dict(d=d, cap=cap, price=name, exact_st_tariff=rational(2*expected),
+                                 packing_witness=witness))
+    sample = price('hyperbolic', 2)
+    require(2*packed(3, 2, sample) == F(7, 3), 'credited three-role control')
+    require(2*packed(5, 2, sample) == F(11, 3) and packed(10, 2, sample) == F(10, 3),
+            'five-role tariff distinction')
+    return rows
+
+
+def endpoint_rows(table):
+    rows = []
+    for field in (2, 3, 5):
+        for d in range(1, 7):
+            for cap in range(1, 7):
+                instructions = word(d, cap, field)
+                actual, ranks, labels, coupled, interleaved = execute(instructions, d, cap, field, table)
+                expected = target(d, d+2)
+                require(actual == expected, 'full formal endpoint')
+                require(coupled > 0 and interleaved > 0, 'genuine dirty mixing')
+                for positive in ((0, 2), (0, 1)):
+                    require(cut_rank(actual, d+2, positive, table, field) == 2*d, 'target cut rank')
+                for name in ('zero', 'linear', 'flat', 'hyperbolic'):
+                    prices = price(name, cap)
+                    require(sum((prices[k] for k in ranks), F(0)) == 2*packed(d, cap, prices),
+                            'actual word tariff')
+                rows.append(dict(field=field, d=d, cap=cap, dirty_spectators=2,
+                                 call_ranks=ranks, call_labels=labels, dirty_coupled_calls=coupled,
+                                 mixed_interleaved_scalars=interleaved, word_sha256=digest(instructions),
+                                 complete_endpoint_sha256=digest(sorted(actual.items()))))
+    require(len(rows) == 108, 'complete finite endpoint domain')
+    return rows
+
+
+def witness(actual, expected, n):
+    for label in range(6):
+        a, e = actual.get(label, zero(n)), expected.get(label, zero(n))
+        for i in range(n):
+            for j in range(n):
+                if a[i][j] != e[i][j]:
+                    return dict(label=LABELS[label], output_role=i, input_role=j,
+                                actual=a[i][j], expected=e[i][j])
+    raise ValueError('missing differing coefficient')
+
+
+def bad_endpoint(name, instructions, table):
+    actual, _, _, _, _ = execute(instructions, 5, 2, 5, table)
+    expected = target(5, 7)
+    require(actual != expected, 'endpoint mutation accepted: ' + name)
+    return dict(name=name, outcome='DETECTED_FULL_FORMAL_DIFFERENCE',
+                witness=witness(actual, expected, 7), augmentation_identity=augmentation(actual, 7, 5) == identity(7))
+
+
+def controls(table):
+    original = word(5, 2, 5)
+    omitted = original[:]
+    del omitted[next(i for i, ins in enumerate(omitted) if ins[0] == 'paid')]
+    rows = [bad_endpoint('omitted_paid_call', omitted, table),
+            bad_endpoint('wrong_st_order', word(5, 2, 5, (2, 1)), table)]
+    leak = zero(7)
+    leak[5][0] = 1
+    row = bad_endpoint('dirty_leak_invisible_to_augmentation', original + [['paid', 1, leak]], table)
+    require(row['augmentation_identity'] and row['witness']['output_role'] >= 5, 'dirty leak witness')
+    rows.append(row)
+    larger = word(5, 5, 5)
+    actual, ranks, _, _, _ = execute(larger, 5, 5, 5, table)
+    require(actual == target(5, 7) and ranks == [5, 5], 'larger-cap actual endpoint')
+    larger_cost = sum((price('hyperbolic', 5)[k] for k in ranks), F(0))
+    require(larger_cost < 2*packed(5, 2, price('hyperbolic', 2)), 'rank-cap price distinction')
+    try:
+        execute(larger, 5, 2, 5, table)
+    except ValueError as error:
+        require(str(error) == 'paid rank cap', 'wrong cap-control reason')
+        rows.append(dict(name='larger_rank_not_free_workspace', outcome='REJECTED', reason=str(error),
+                         admitted_cap=2, attempted_ranks=ranks, legal_cap5_cost=rational(larger_cost)))
+    else:
+        raise ValueError('rank-cap mutant accepted')
+    for name, prices, demand in (('nonconcave_price', [F(0), F(1), F(4)], 2),
+                                ('nonmonotone_price', [F(0), F(2), F(1)], 1)):
+        best, chosen = packing_dp(demand, 2, prices)
+        floor = packed(demand, 2, prices)
+        require(best < floor, 'price-hypothesis witness')
+        rows.append(dict(name=name, outcome='PACKING_LEMMA_FAILS_OUTSIDE_PREMISE',
+                         demand=demand, ranks=chosen, actual_cost=rational(best), false_bound=rational(floor)))
+    try:
+        paid(zero(2), 1, 2, 5)
+    except ValueError as error:
+        require(str(error) == 'zero paid rank', 'wrong zero-rank reason')
+        rows.append(dict(name='zero_matrix_not_a_paid_call', outcome='REJECTED', reason=str(error)))
+    else:
+        raise ValueError('zero paid rank accepted')
+    return rows
+
+
+def calculate():
+    sources = authenticate()
+    table = group_table()
+    return dict(status='PASS_A2_ST_ALL_CAP_TARIFF', schema_version=1,
+                source_sha256=sources, source_programs_executed=False, group_labels=LABELS,
+                group_multiplication=table, selective_cut_checks=cut_checks(table),
+                complete_endpoints=endpoint_rows(table), price_cases=price_checks(), controls=controls(table),
+                scope='Finite all-cap st formal tariff corroboration; native/physical bill unprovided.')
+
+
+def main():
+    require(sys.argv[1:] in ([], ['--emit']), 'only optional --emit is supported')
+    packet = calculate()
+    if not sys.argv[1:]:
+        require(canonical(load((PACKAGE / 'RESULT.json').read_bytes())) == canonical(packet),
+                'retained complete result mismatch')
+    print(canonical(packet).decode())
+
+
+if __name__ == '__main__':
+    try:
+        main()
+    except (ValueError, KeyError, TypeError, OSError, json.JSONDecodeError) as error:
+        print('FAIL_A2_ST_ALL_CAP_TARIFF: ' + str(error), file=sys.stderr)
+        raise SystemExit(1)