Prove actual P112 invoices at d=1 and d=2 for t=2,3

research/p112-six-seven-supplement/CALLERS.md

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+# Nine literal C1 piece profiles excluded
+
+## Exact source and statement
+
+Use the P112 evaluation code $C$ and binary support weight
+
+$$
+       w(p)=\dim_{\mathbf F_2}
+              \operatorname{span}\{\ell(r):p(r)\ne0\}
+$$
+
+from [PROOF.md](PROOF.md), over an arbitrary characteristic-zero field.
+Let independent pieces $p_1,\ldots,p_n$ span $T$, and put $D=T\cap C$.
+
+For each row below there is no such independent family with the displayed
+weight upper bounds and $\dim D\ge d$.
+The displayed $t$ and slack are the literal source tags, with
+$n=d+t$ and $\text{slack}=\sum W-8d$.
+The theorem uses only the independent-piece and intersection hypotheses;
+it does not require a minimum basis or a decoder realization.
+
+| Source $t$ | $d$ | Slack | Weight bounds $W$ | Small supports remaining after the deletion | Site cap |
+|---:|---:|---:|---|---|---:|
+| 2 | 1 | 7 | $7,6,2$ | one rank-two piece | 2 |
+| 2 | 1 | 8 | $7,6,3$ | one rank-three piece | 4 |
+| 2 | 2 | 6 | $7,7,6,2$ | one rank-two piece | 2 |
+| 2 | 2 | 7 | $7,7,6,3$ | one rank-three piece | 4 |
+| 3 | 1 | 7 | $7,6,1,1$ | two rank-one pieces | 2 |
+| 3 | 1 | 8 | $7,6,2,1$ | rank-two and rank-one pieces | 3 |
+| 3 | 2 | 6 | $7,7,6,1,1$ | two rank-one pieces | 2 |
+| 3 | 2 | 7 | $7,7,6,2,1$ | rank-two and rank-one pieces | 3 |
+| 3 | 2 | 8 | $7,7,6,2,2$ | two rank-two pieces | 4 |
+
+All nine are recorded survivors in DreamingOfClouds's
+[PR3154](https://github.com/CrocSwap/integer-mult-bounds/pull/3154),
+pinned at
+`708f0ad8928c383de8294aea3c126e9461d2066d`.
+Their exact source files are
+
+    research/p112-reduction-to-conjectures/checks/CONJG2/NOTES.md
+    research/p112-reduction-to-conjectures/checks/CONJG2/land_e2_k8.out
+    research/p112-reduction-to-conjectures/checks/CONJG2/land_e3_k8.out.
+
+The two output files are retained byte-for-byte under this package's
+inputs and authenticated by [SOURCES.json](SOURCES.json). The first
+records 35 profiles and the second 280. Those are the source's recorded
+lists, not a claim about the number currently unresolved by every
+subsequent public theorem.
+
+## Dimension-deletion lemma
+
+Suppose $T$ is spanned by independent pieces. Their coefficients identify
+$T$ with $K^n$. If $\dim(T\cap C)\ge d$, imposing zero coefficients
+on any chosen $d-1$ pieces leaves a nonzero codeword.
+
+Indeed, restrict the corresponding $d-1$ coordinate functionals to
+$D=T\cap C$. Their common kernel has dimension at least
+$\dim D-(d-1)\ge1$. A nonzero vector in that kernel is a nonzero
+codeword because the original pieces are independent. This argument
+allows $\dim D>d$; equality is unnecessary.
+
+Each table row has exactly $d$ designated rank-seven pieces.
+For $d=1$, choose any nonzero word in $D$.
+For $d=2$, set the coefficient of one designated rank-seven piece
+to zero by the lemma. In both cases the resulting nonzero word is a
+linear combination of
+
+* one piece of binary support rank at most seven;
+* one piece of binary support rank at most six;
+* the displayed small pieces, whose union has at most four sites.
+
+The site cap follows from the labels' leading one:
+a rank-one, rank-two or rank-three support has at most $1,2,4$ sites.
+Taking linear combinations does not enlarge any piece's support, and
+aggregating the small pieces has support contained in their union.
+Arbitrary overlap and cancellation are permitted.
+
+The nonzero word therefore contradicts the theorem of
+[PROOF.md](PROOF.md). This proves every row of the table.
+There is no quotient-normalization step and no inference from a failed
+support search.
+
+## What the caller theorem establishes
+
+The source CONJG2 section3 uses $T$ as the low-weight subspace,
+$d=\dim(T\cap C)$, $e=\dim T-d=t$, and slack
+$\operatorname{cost}(T)-8d$. A minimum basis of that $T$, if it has
+one of the nine displayed profiles, is an independent family satisfying
+the exact hypotheses just excluded. Thus these nine profiles cannot
+occur as such a minimum basis.
+
+This implication does not assume the source's N8 correction cap, the
+completeness of the 35/280 lists, or its global peeling and branch-coverage
+arguments. It is valid for each displayed piece family in isolation.
+It also does not show that an arbitrary actual decoder basis is adapted
+to $T$. A basis-surplus unit can change the number of actual low and
+high members; no argument about those members' restricted dual rows is
+needed here.
+
+Consequently this caller argument supplies nine literal T-level exclusions
+with a uniform geometric mechanism. [D1.md](D1.md) separately proves the
+complete actual d=1 marked/unmarked invoices and stated singleton consumer
+using the additional retained matched-cut geometry. [D2.md](D2.md) extends the actual invoice to d=2. Complete C1, actual
+d>=3 invoices, global normalization and supplier admission remain outside
+the package result.
+It does not certify a present global survivor count.
+
+## Boundary and further work
+
+The sharp partition in [PROOF.md](PROOF.md) has piece ranks $7,6,4$.
+Splitting its five-site remainder into four sites and the root singleton
+gives ranks $7,6,3,1$, independent pieces, code intersection dimension
+one, minimum basis cost seventeen, and slack nine. Minimum cost follows
+from the disjoint-support determinant matching argument in that proof.
+Hence the primary obstruction cannot simply be
+extended from four small sites to five, or used to discard that nearby
+T-level boundary. That fixture is not an actual marked decoder.
+
+The actual d=1 argument in D1.md retains all basis-surplus scenarios and
+closes the cost-seventeen invoice without excluding its feasible T-space.
+The remaining higher-d profiles require further shared-cover or coefficient
+arguments. This caller inventory does not certify a current global survivor count.
+
+Credit DreamingOfClouds for the source reduction, notation, filters and
+recorded profiles; credit PR2816 and the retained E8/P112 lineage for the
+piece model and earlier circuit/quotient methods. The new dimension-deletion
+application is by conorholds with substantial OpenAI Codex assistance.
+
+## Appendix: inherited four-six pencil composition
+
+There is also a useful direct composition of the earlier coordinate-kernel
+and root-pencil method. Credit Nick Petalas's PR3171, head
+`95b683196950efea6f3d7ac66745e00dba73425e`,
+`research/p112-quotient-parity/PROOF.md`, and DreamingOfClouds's PR2816,
+head `4c32ac3b88d63ad06d2b30e526aa07a12929db36`, for the retained
+root-rigidity and circuit/pencil lineage. This appendix records a caller
+composition, not a new geometric mechanism.
+
+**Four-six-plus-nine lemma.** Suppose an independent piece family has four
+designated pieces of binary support rank at most six. Suppose every other
+piece is supported in one common set E of at most nine live sites. Its span
+cannot meet C in dimension at least two.
+
+Choose a two-dimensional subspace D of that intersection, and restrict the
+four designated piece-coordinate functionals to D. For any two of these
+functionals, their joint kernel is zero: a nonzero vector there would be a
+codeword supported on the other two rank-six pieces and E, hence on at most
+`20+20+9=49<53` sites. Each single functional is consequently nonzero and
+has a one-dimensional kernel, and the four kernel lines are distinct.
+On each kernel line a nonzero codeword is supported on three rank-six
+pieces and E, hence on at most `3*20+9=69<70` sites. Root-direction
+rigidity identifies each with an E8 root line, including frame roots.
+Injectivity of root evaluation puts four distinct root lines in the same
+two-dimensional normal space. This is impossible: a rank-two subsystem of
+the simply laced E8 root system is A1+A1 or A2 and has at most three
+projective root lines. The argument works over any characteristic-zero
+field by the same rational-line transfer as the retained pencil method.
+
+More generally, for d>=2, if the original code intersection has dimension at least d,
+first kill any d-2 chosen piece coordinates. The dimension-deletion lemma
+leaves an intersection of dimension at least two. The preceding lemma then
+applies whenever the remaining pieces consist of four rank-at-most-six
+pieces and pieces supported in a common set of at most nine sites.
+
+For the pinned PR3154 outputs, deleting the heaviest d-2 pieces and applying
+this criterion selects two literal t2 rows and 83 literal t3 rows. In t2
+the four remaining pieces all have rank at most six. In t3 the five
+remaining pieces include four of rank at most six and one of rank at most
+four; the latter has support at most eight, so can serve as E. These are
+implications for the literal recorded profiles, not 85 currently open
+cases or 85 first closures. They neither establish completeness of those
+historical lists nor bypass the actual-basis and normalization obligations
+for higher-d invoices.