Record that the 40 frame-0 monomials are not selected-invoice primitives.

research/p104-frame0-invoice-primitive-names/PROOF.md

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+# The 40 frame-0 monomials are not primitives of the selected invoice
+
+No new κ. The selected record stays `781451/250000000`. This note answers the
+naming step left open by PR #7031: whether the 40 charged monomials can be
+named as primitives on the selected #6030 invoice. They cannot, under the
+naming rules that invoice actually uses. The missing obligation is one total
+map that those rules accept. This is not a comparison with the selected
+profile, and it is not a claim that no future contract could name the
+monomials.
+
+## 1. The 40 monomials
+
+PR #7031 charges the degree-at-least-2 terms of
+
+```text
+(1 + x0)(1 + x1 + x2 + x4)(1 + x5)(1 + x6 + x7)
+```
+
+in the square-free ring `F_2[x0,...,x7]/(x_i^2 = x_i)`. The four factors use
+disjoint variables, so the expansion is the Cartesian product of the summands.
+That domain has `2 · 4 · 2 · 3 = 48` elements, every product is distinct, and
+there is no cancellation. The degrees are 1, 7, 17, 17, and 6 in degrees 0
+through 4. The eight monomials of degree at most 1 are `1, x0, x1, x2, x4,
+x5, x6, x7`. The other 40 are the charged monomials.
+
+The SHA256 of the 48 sorted exponent integers, joined by commas, is
+
+```text
+26803d29aa3141a357da4aa551195fe6378d3f2e1cba0c4291a4dedbce6ded2a
+```
+
+which is the normal-form hash recorded by the #7031 verifier at head
+`531ecca652dde63c6010cd6097cb55a994367ce7`. This note does not replay that
+pull request's hosting census. It takes this polynomial as the charged object.
+
+An exponent is the integer whose bit `i` is the factor `x_i`. The six
+degree-4 exponents are `99, 101, 113, 163, 165, 177`, namely
+
+```text
+x0*x1*x5*x6, x0*x2*x5*x6, x0*x4*x5*x6,
+x0*x1*x5*x7, x0*x2*x5*x7, x0*x4*x5*x7.
+```
+
+## 2. What the selected invoice calls a primitive
+
+The selected record points at
+`research/p104-helper470-twin-rung-removal/RESULT.json`. Its word has schema
+`p104-accumulated-block-composition/2`. A recursive primitive on that invoice
+is a pair `(side, rank)` produced by one of the following constructors, as
+written in `local.py` `replay` and `pricing.py` `assemble`.
+
+- A `MOVE` from frame `a` to frame `b` has local paid rank `|dim b - dim a|`,
+  where the dimension is the length of the stored basis.
+- A non-main public helper has local target rank `dim(final) - dim(initial)`.
+- Eighteen forward `LOCAL` calls repeat both local histograms.
+- Fifteen query shears add paid rank `1`, with count `104` each.
+- For each idle length `L` in `{7, 8}` and each `j = 1, ..., L`, the assembly
+  adds paid rank `8(j+1)` and target rank `1 + 8(j+1)`, with count `208`.
+- One further target of rank `145` is added with count `208`.
+
+Replaying every `MOVE` and every non-main helper of the selected word, then
+applying this finite formula, reproduces `RESULT.json` `local.paid`,
+`local.helpers`, `complete_cost.paid`, and `complete_cost.targets`. The paid
+keys are
+
+```text
+1, 2, 3, 4, 5, 6, 7, 8, 16, 24, 32, 40, 48, 56, 64, 72.
+```
+
+The target keys are
+
+```text
+1, 2, 3, 4, 5, 6, 7, 8, 9, 17, 25, 33, 41, 49, 57, 65, 73, 145.
+```
+
+The ordinary names on the same word are the opcodes that occur: `ADD`,
+`CLEAR`, `COPY`, `INIT`, `MOVE`, and `SCALE`. Each acts on a role index and,
+for a scalar event, a dyadic coefficient. The assembly text and the replay
+text do not mention a chart variable, a polynomial degree, or a monomial.
+
+The native routine in `local.py` builds Clifford recipes from frame charts
+and ports. It is not replayed here. Its inputs are those frames and ports,
+not the 40 exponents, so it does not assign these monomials names either.
+
+## 3. Readings that still do not name all 40
+
+A naming, in the sense asked by #7031, would be a total function from the 40
+exponents to this alphabet that the constructors above determine. The
+readings that use only an integer or a string already attached to the
+monomial fail.
+
+**Exponent integer as a histogram key.** Of the 40 exponents, exactly three
+are paid keys: `3, 5, 48`. Exactly seven are target keys: `3, 5, 17, 33, 49,
+65, 145`. The union has eight exponents, so 32 exponents are keys of neither
+side. The two exponents `3` and `5` are keys of both sides, so the integer
+does not choose a unique primitive even in the collisions. All six degree-4
+exponents lie in the 32. This is an exhaustive check of a 40-element set
+against the two finite key sets.
+
+**Printed name as an opcode.** None of the 40 strings `x0*x1`, ... equals
+`ADD`, `CLEAR`, `COPY`, `INIT`, `MOVE`, or `SCALE`.
+
+**Exponent integer as a selected port.** The word stores 104 ports, and the
+local native check identifies the initial frame of each live line with the
+singleton `[port]`. Exactly 14 of the 40 exponents occur in that port list:
+`35, 37, 49, 69, 81, 98, 100, 112, 131, 133, 145, 161, 164, 176`. The other
+26 do not. This reading does not name all 40.
+
+**Degree as rank.** The degrees that occur are 2, 3, and 4, and each of those
+integers is both a paid key and a target key. That numerical fact is not a
+name. Neither constructor reads a degree. Calling these monomials `H_2`,
+`H_3`, and `H_4` would be a new tariff. This note does not adopt that tariff
+and does not claim that every future contract must reject it.
+
+## 4. The chart is not the invoice coordinate system
+
+The same integer can appear in both lists without being the same object.
+The hosting word of #7031 uses ports `11, 21, 38, 140`. Their simple-root
+labels, with the ninth coordinate set, are `282, 284, 270, 454`. None equals
+its port, and none is one of the 40 exponents (every charged exponent is at
+most `177`). The invoice frames are vectors in the port-space `F_2^9`. The
+monomials are functions of the simple-root coordinates. An exponent that
+happens to equal a port, or a frame basis vector, is not that port and not
+that vector.
+
+For the record, 34 of the 40 exponent integers occur somewhere in the stored
+frame bases, and the six that do not are exactly the degree-4 exponents.
+That census is not a naming: a basis vector is not a primitive of the
+invoice, and the two coordinate systems are distinct.
+
+## 5. The obstruction
+
+Under the constructors in section 2, the selected invoice does not determine
+a total function from these 40 exponents to its primitive alphabet. The
+missing obligation is that function, together with a proof that an inherited
+constructor, or a stated extension of one, sends each exponent to the image
+it names. Until the function exists, these 40 monomials are not primitives
+on the selected invoice.
+
+No profile comparison is made. The other twelve frames are not expanded.
+The selected κ is unchanged. This is not a proof that the monomials cannot
+be named by a different contract, and it is not a proof that no function
+into the alphabet exists. Many functions exist. None is fixed by the
+invoice that was asked about.
+
+## Evidence class
+
+The expansion is an exhaustive census of a 48-element product, complete
+because the factors are disjoint and every combination is enumerated. The
+histogram identity is an exhaustive replay of the finite selected word plus
+the finite assembly formula, matched to the selected receipt. The three
+failed readings are exhaustive checks of the same 40 exponents against
+finite sets. The refusal of the degree reading is the observation that the
+constructor texts do not read a degree. Together this is a scoped proof and
+an exhaustive finite census. It is not a heuristic and not an incomplete
+search inside those readings.

research/p104-frame0-invoice-primitive-names/README.md

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+# The 40 frame-0 monomials are not selected-invoice primitives
+
+**No new κ.** The selected frontier stays `781451/250000000`. This package
+asks whether the 40 monomials charged by open PR #7031 can be named as
+primitives on the selected #6030 invoice. Under that invoice's constructors,
+they cannot. The missing map is stated in [PROOF.md](PROOF.md). Naming them
+would not by itself be a κ change, and this obstruction is not one either.
+
+## Result
+
+The charged polynomial is
+
+```text
+(1 + x0)(1 + x1 + x2 + x4)(1 + x5)(1 + x6 + x7).
+```
+
+Its 40 degree-at-least-2 monomials are the object #7031 asked to place on
+the invoice. A recursive primitive of the selected invoice is a pair
+`(side, rank)` built from frame-dimension gaps, eighteen local copies,
+fifteen query shears, the idle lengths 7 and 8, and one rank-145 target.
+Ordinary names on the same word are the opcodes `ADD`, `CLEAR`, `COPY`,
+`INIT`, `MOVE`, and `SCALE`. Those constructors do not read a chart monomial.
+
+Against that alphabet:
+
+- 32 of the 40 exponent integers are keys of neither histogram. The six
+  degree-4 masks `99, 101, 113, 163, 165, 177` are among them.
+- The eight integers that do collide are not unique names. Masks `3` and `5`
+  are both paid keys and target keys.
+- No printed monomial equals an opcode.
+- 14 exponent integers occur in the selected port list, and 26 do not.
+- Degrees 2, 3, and 4 are histogram keys, but the constructors do not set
+  rank equal to degree. That reading is not adopted.
+
+The chart is a different coordinate system from the invoice. The simple-root
+labels of ports `11, 21, 38, 140` are `282, 284, 270, 454`, not the ports
+themselves. An integer collision with a port or a frame vector is not an
+identification.
+
+## Evidence class
+
+The product expansion is an exhaustive 48-element census. The invoice
+histograms are an exhaustive replay of the selected word, matched to
+`RESULT.json`. The failed readings are exhaustive checks of the 40
+exponents. The degree reading is refused because the constructor texts do
+not mention it. This is a scoped proof together with those censuses. It is
+not a heuristic, not a search over arbitrary maps, and not a proof that
+another contract could not name the monomials.
+
+## Scope
+
+Catalogue card `complex-rank-shape-decoder`, priority revision 8, pinned at
+`17469fff78c22941280f04d39dc5a1faee09c840`. The card asks whether a decoder
+or coframe change improves the signed moment. Scope is **changed** to the
+naming step in open question 3 of
+[#7031](https://github.com/CrocSwap/integer-mult-bounds/pull/7031). The
+connection to the bottleneck is in [TASK.json](TASK.json): a thirteen-row
+comparison with the selected profile is not available until each charged
+monomial has a primitive name, and this package stops when that name is
+shown to be missing.
+
+The selected record, `upstream/`, and the #6030 word are inputs, not edits.
+Open #7031 already hosted the row and charged the 40 monomials; this package
+does not redo that hosting. It does not host frame 1, and it does not ask
+whether a sum of letters can use fewer than 40 monomials.
+
+Overlap searched on 2026-11-10: the open-PR search for this naming returned
+only #7031, whose discussion finding
+[19514167](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19514167)
+leaves the naming open. No fork branch had taken the question.
+
+## Reproduce
+
+From the repository root, Python 3.12, standard library only:
+
+```sh
+python3 -B research/p104-frame0-invoice-primitive-names/verify.py
+python3 -I -O -B research/p104-frame0-invoice-primitive-names/verify.py
+python3 -B scripts/research_package.py --package research/p104-frame0-invoice-primitive-names
+python3 -B scripts/check_research_commit.py --package research/p104-frame0-invoice-primitive-names --check-source-history
+```
+
+The verifier prints `PASS_P104_FRAME0_INVOICE_NAMES` with `monomials=40`,
+`outside_both_histograms=32`, `degree4_outside=6`, and the unchanged κ.
+Failure controls drop the factor `x0`, accept mask `99` as a paid rank, and
+replace the 18 local copies by 17.
+
+Not run: `make frontier6030-verify`, the native Clifford replay, Lean, and
+any expansion of frames 1–12.
+
+## Open next questions
+
+1. Give a total map from these 40 exponents to the inherited primitive
+   alphabet that one of the selected constructors determines. The
+   thirteen-row comparison stays blocked until that map exists. The first
+   step is to make one constructor read one of these exponents, or to show
+   that a constructor already does, and then check the image of all 40.
+   The present constructors read frame dimensions, idle lengths, and a
+   diagonal popcount only.
+
+The thirteen-row profile comparison is not opened. Frames 1–12 remain the
+separate hosting question already stated by #7031, and a missing name for
+these 40 blocks that comparison on its own.
+
+## Credit
+
+The 40 monomials and the factored word: unpaidmoon, PR #7031. The selected
+invoice and word: DreamingOfClouds, PR #6030, on the PR #5827 verifier chain.
+Ports and the odd-graph line map: PR #6933. Prepared by unpaidmoon with
+substantial Cursor Grok assistance. New work is Apache-2.0.
+
+Workflow note: https://github.com/CrocSwap/integer-mult-bounds/discussions/1200

research/p104-frame0-invoice-primitive-names/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "17469fff78c22941280f04d39dc5a1faee09c840",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "input",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/p104-helper470-twin-rung-removal/RESULT.json",
+      "usage": "input",
+      "git_blob": "e4a03118d585fb80c9ac932281b528ac394ccc5f",
+      "sha256": "84fcbe2a0b64307b5703c363408042bdd91555b0419509aa50daa43b3b1924ad"
+    },
+    {
+      "path": "research/p104-helper470-twin-rung-removal/WORD.json.gz",
+      "usage": "input",
+      "git_blob": "22f9ab02d1b73ea0aa4b07ce445a14cc2602c760",
+      "sha256": "7834725d9e34ee846a9ccc615221ea81e0201613938afa185d2eda0c9a6c232f"
+    },
+    {
+      "path": "research/p104-helper470-twin-rung-removal/pricing.py",
+      "usage": "input",
+      "git_blob": "65bb723f8558dc60ff92764db1a43d8bf438aa09",
+      "sha256": "3c6cf2966ccfe5a5fb6c90f89f46ac224ea455127e35dd34975683c68c36d710"
+    },
+    {
+      "path": "research/p104-port-window-haemers/data/e8_line_map.json",
+      "usage": "input",
+      "git_blob": "19ea2d553aff60e53af50dd9582ab70730e93450",
+      "sha256": "b37530e8e9b2b3bd5be418d5ed3060a42ceb66e60ed3789f8ee79f29dabde437"
+    },
+    {
+      "path": "research/p104-unread-path-resynthesis/local.py",
+      "usage": "input",
+      "git_blob": "b7be846b10c4c54a4e1558d7b563bc8c3a170223",
+      "sha256": "7201df8e32f5c99f7403150e8664042322037906088bc754c67293a241368597"
+    }
+  ]
+}