Bound no-idle query roots and paired productive stages (no new κ)

research/noidle-root-defect/NOTICE

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+No-idle rank-two caller defect bound and exact verifier.
+Jerry Chen / @CubeJerry, with substantial OpenAI GPT-6 Astra Pro assistance.
+Apache-2.0.
+
+The header rank identity is rederived from DreamingOfClouds's
+CrocSwap/integer-mult-bounds#6832 (research/p104-query-block-bound).
+The shortest-header comparison is CrocSwap/integer-mult-bounds#6842.
+Earlier scoped rank/star arguments and the native monotonicity bound are
+CrocSwap/integer-mult-bounds#6840 and CrocSwap/integer-mult-bounds#6839.
+
+retained/pricing.py is unchanged from CrocSwap/integer-mult-bounds#6833,
+head 51f54e3c31f2f3a3e056a483b3ed66ae916ad06a. Its original header credits
+DreamingOfClouds, Jerry Chen / CubeJerry, zxv, Cao Yifan, Jacob Alan Sussman,
+and the binary/caller/native authors; the accompanying original NOTICE is
+retained as retained/NOTICE-pr6833.txt. The source is parsed for literal
+census constants only and is never imported by this package's verifier.
+
+The new graph/defect and complete-invoice result is a scoped obstruction.
+It does not claim a stronger kappa or change the selected frontier.

research/noidle-root-defect/PAIRED-DEFECT-ADDENDUM.md

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+# Paired query blocks cannot supply six minimal commuting stages at defect two
+
+**Scoped obstruction; no new κ.** This addendum tests a proposed use of
+synchronized pairs of native copies sharing one fresh body variable. The
+aggregate query has rank two and its proposed productive stage has rank four.
+The following theorem permits arbitrary other queries, ordinary changes of
+basis, and arbitrary mixing within the four-dimensional productive stages.
+
+## Statement
+
+Let the ground field have characteristic different from two. On n independent
+coordinates, let
+
+\[
+R_i=X_iY_i,\qquad Y_iX_i=0,\qquad
+\operatorname{rank}X_i=\operatorname{rank}Y_i=r_i,
+\]
+
+and suppose the complete chronological query word satisfies
+
+\[
+(I-2R_s)\cdots(I-2R_0)=-I_n.
+\]
+
+Define the **rank defect** by
+
+\[
+\delta=\sum_{i=0}^{s}r_i-n.
+\]
+
+Assume δ=2. Select full positions with r_i=2, each supplied with a rank-four
+idempotent P_i satisfying P_iR_i=R_iP_i=R_i. If these P_i commute pairwise,
+there are at most five selected full positions.
+
+No actual-prefix commutation is needed for this theorem. Consequently it
+also applies to the stronger caller contract requiring that commutation.
+In particular, nine rank-two full blocks formed from eighteen native copies
+cannot have commuting rank-four productive stages at rank defect two,
+regardless of the number or placement of thin rank-one queries.
+
+This concerns the explicit square-zero block model. It does not classify
+arbitrary shared-variable calls whose aggregate query is not square-zero,
+higher rank defect, paid higher-rank stages, or noncommuting outputs. The
+bound five is not claimed sharp.
+
+## 1. Defect two fixes the cross-Gram blocks
+
+Stack X by block columns and Y by block rows. Put L=Σr_i, G=YX, and
+
+\[
+A=I_L+2\operatorname{lower}_{\mathrm{block}}(G).
+\]
+
+The same ordered-product expansion as the scalar query-defect identity gives
+product(I−2R_i)=I_n−2XA⁻¹Y. Thus XA⁻¹Y=I_n. The matrix Q=A⁻¹YX is an
+idempotent of rank n, so
+
+\[
+H=A-G=A(I_L-Q),\qquad \operatorname{rank}H=L-n=2.
+\]
+
+Its diagonal blocks are I_(r_i), and H_ij=sign(i−j)G_ij off the diagonal.
+Factor H=UV through a two-dimensional space. The block factors obey
+U_iV_i=I_(r_i). At every selected rank-two position, U_i and V_i are
+inverse square matrices. The factor gauge X_i↦X_iU_i and Y_i↦U_i⁻¹Y_i
+leaves R_i unchanged and makes both of its defect factors equal to I₂.
+Therefore selected positions have
+
+\[
+Y_iX_j=\operatorname{sign}(i-j)I_2\quad(i\ne j),
+\qquad Y_iX_i=0.
+\tag{1}
+\]
+
+This step includes every other query in A, G and the rank defect. It does
+not omit the cost or rank of thin queries. The block extension retains the
+query-defect mechanism of CrocSwap/integer-mult-bounds#6832; no new claim of
+priority is made for that mechanism.
+
+## 2. Six selected queries require twelve dimensions
+
+Take six selected positions in chronological order. Their Gram is S₆⊗I₂,
+where S₆[i,j]=sign(i−j). This matrix has rank twelve over every field: the
+integer inverse of S₆ has entries (−1)^(i+j)sign(i−j), using zero-based
+indices. Hence the twelve query columns span twelve independent dimensions.
+Every supporting stage union must contain them.
+
+Commuting idempotents admit a common coordinate basis over the ground field
+by successively splitting into images and kernels. In this basis each P_i
+has a four-element support C_i. The support equations imply P_iX_i=X_i
+and Y_iP_i=Y_i because the query factors have full rank. Consequently the
+rank-two product in (1) passes through C_i∩C_j, giving
+
+\[
+|C_i\cap C_j|\ge2\qquad(i\ne j).
+\tag{2}
+\]
+
+We show that six such supports satisfying (1) have union of size at most
+ten, contradicting the twelve-dimensional requirement.
+
+## 3. The set-family bound away from a common pair
+
+Consider m four-element supports satisfying (2), allowing repetitions. If
+every pair intersects in at least three elements, fixing one support shows
+that each later one adds at most one element. The union has size at most m+3.
+
+Otherwise choose C₁=A∪B and C₂=A∪C, where A, B and C are disjoint pairs.
+If every support contains A, proceed to Section4. Otherwise choose E missing
+at least one element of A. Meeting C₁ and C₂ twice forces E to contain at
+least three elements of their six-element union, or all four if E misses
+both elements of A. Thus T=C₁∪C₂∪E has size at most seven.
+
+Any further support D adds at most one element to T. If it added two or
+more, its at most two old elements would have to be exactly A to meet C₁
+and C₂ twice. It would then meet E at most once, contradicting (2). Therefore
+
+\[
+\left|\bigcup_i C_i\right|\le 7+(m-3)=m+4.
+\tag{3}
+\]
+
+## 4. The ordered signs bound the common-pair case
+
+Suppose every support contains a fixed pair A, and write E_i=C_i∖A. The
+two-element outside sets E_i form the edges of a graph.
+
+That graph cannot contain three pairwise disjoint edges E_i, E_j, E_k.
+Order their query indices i<j<k. Their stage intersections are exactly A.
+Let α_l=X_l[A,:] and β_l=Y_l[:,A]. These square two-by-two matrices are all
+invertible, since their cross-products in (1) are invertible. The equations
+
+\[
+\beta_i\alpha_j=\beta_i\alpha_k=-I_2,\qquad
+\beta_j\alpha_i=\beta_k\alpha_i=I_2
+\]
+
+force α_j=α_k and β_j=β_k. But then β_jα_k=β_kα_j, whereas (1) requires
+the first product to be −I₂ and the second I₂. Characteristic different
+from two rules this out.
+
+Thus the outside graph has matching number at most two. If it has v incident
+vertices, e distinct edges and c connected components, then v≤e+c: every
+component has at least its vertex count minus one edges. Selecting one edge
+in every component gives c≤2. Since e≤m, including repeated supports,
+
+\[
+\left|\bigcup_i C_i\right|=2+v\le2+e+2\le m+4.
+\tag{4}
+\]
+
+For m=6, both cases give union size at most ten. This contradicts the rank
+twelve Gram and proves the theorem.
+
+## Exact controls and the limits they check
+
+Run `python3.12 -I -B verify_paired_defect.py` or its optimized `-O` variant.
+Both return `PASS_PAIRED_DEFECT_TWO_STAGE_CAP`. The verifier uses only the
+Python standard library and exact rational or finite-field arithmetic.
+
+It verifies the integer sign-matrix inverse, 671 normalized set-extension
+cases, extremal ten-coordinate families in both set-family cases, and
+repeated supports. A three-full-block defect-two root with stages I₄ is a
+positive control, showing the hypotheses are nonempty.
+
+The omission controls are mathematical boundary examples. A literal
+seven-block defect-two root on twelve coordinates admits six rank-four
+supporting projectors with eleven noncommuting pairs; the commutation gate
+rejects it. A characteristic-two sunflower realizes six rank-two queries
+with the stipulated Gram and commuting rank-four supports on fourteen
+coordinates. This latter example concerns the isolated Gram/support lemma,
+not a native caller: it shows why the sign contradiction needs its stated
+characteristic restriction. An altered sign-matrix inverse is also rejected.
+
+The universal theorem is proved above; these finite checks are controls of
+its premises and algebra, not a finite-search impossibility claim. The
+optimistic positive cost margin for nine rank-four stages cannot yield a
+caller because those stages do not exist in this model. No κ is changed.
+
+Prepared for @CubeJerry with substantial OpenAI GPT-6 Astra Pro assistance.
+Canonical base: `7c49fe32c55de02c2c873655c804b16e4c3fef0a`.
+Credit the retained query/controller lineage and the defect identity in
+CrocSwap/integer-mult-bounds#6832. Related coupled geometry in
+CrocSwap/integer-mult-bounds#6773 and #6781 concerns different four-field
+stage budgets. The set-family/sign obstruction above is the present addition.