Reduce all-class rank-seven COPY from 41832 to 196 ordinary adapters

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+# Parabolic COPY with batched pivot and graph routing
+
+Chris R / Clear1nsight, with substantial OpenAI Codex assistance. Apache-2.0.
+
+## Result and boundary
+
+For m=64, r=7 and every split rank-seven projector over Z/q^w, the construction
+below replaces PR3950's alternate generic SL64 word by **8,036 ordered scalar
+ADD slots per column and 196 ordinary width-f interchanges**, surrounding the
+same single reversed child of width 7f. PR3950's alternate has 41,832 of each.
+All slots, including zero coefficients, are charged. This is a uniform algebraic
+construction and a **conditional relative caller/cost theorem**; an encoded
+physical compiler has not been implemented. The original R437 good/fallback
+implementation is a different construction and its tariff is not altered.
+
+The change combines the parabolic stabilizer of a projector with small fixed
+permutation networks. It removes unnecessary internal image/kernel basis work
+and shares routing across commuting graph passes and mutually exclusive pivot
+stars. Generic conjugacy, local-ring split projectors and elementary matrices
+are credited predecessor mathematics, not new claims of this package.
+
+Even the reduced positive ordinary term is not absorbable from the pinned B1/B2
+upper tariffs at the R437 target scaling. The optimistic decision is therefore:
+retain this smaller constructive relative bill, then stop supplier expansion
+until a cheaper complete ordinary adapter or a different consumer is supplied.
+This does not lower-bound actual runtime or exclude the original algorithm.
+
+## 1. Domain and a principal chart that always exists
+
+Fix an odd prime q, w>=1, R=Z/q^w, A=q^w, m>=2 and 0<r<m. For the application,
+m=64 and r=7. Let P be a split rank-r projector and g invertible over R, and
+put Q=gPg^-1. Let H=diag(h_i) consist of arbitrary units; h_i may depend on
+physical header and column, but not on body addresses or fresh unused rows.
+The requested two-bank address map, with tail coordinates reflected, is
+
+    D_H(Q)=diag(H,I) [[I-Q,Q],[Q,I-Q]] diag(H^-1,I).
+
+Every Boolean source function is allowed, including arbitrary correlated dirty
+captures. Equal field-header values at distinct physical prefix ordinals remain
+distinct. There is no SOURCE projection or clean-data assumption.
+
+A principal r-by-r minor of Q is a unit. Indeed a split idempotent is similar
+to E=diag(I_r,0), so its characteristic polynomial is t^(m-r)(t-1)^r. Its
+coefficient of t^(m-r) is (-1)^r times both 1 and the sum of all principal
+r-minors. Reducing modulo q, their sum is 1. At least one is nonzero. This
+argument also works when q divides r; trace is not used as a rank certificate.
+
+Choose a sorted subset S=(s_0,...,s_(r-1)) with unit principal minor, so
+ i<=s_i<=m-r+i. One effective deterministic rule is the first unit subset in
+lexicographic order. This fixed-m search can be very expensive: at m=64,r=7
+there are binom(64,7) subsets. It is charged in gamma below, not called free
+or a practical large-instance implementation. No enumeration of GL64 is needed.
+The same S selected modulo q works at every high lift and precision.
+
+## 2. Fixed-support pivot stars
+
+For i=0,...,r-1 and every k in i+1,...,m-r+i, let e_ik=1 if s_i=k and 0
+otherwise. Use three stars, in chronological left-action order:
+
+    all E_(i,k)(e_ik), then all E_(k,i)(-e_ik),
+    then all E_(i,k)(e_ik).
+
+At most one flag is nonzero in this entire i-stage. Consequently this is either
+identity (s_i=i) or the signed swap (x_i,x_s)->(x_s,-x_i). Grouping the triples
+into stars is legal precisely because of that mutually exclusive flag property;
+it would be false with two active candidates. Earlier stages cannot move s_i,
+which is strictly larger than every earlier s_j. The product J therefore moves
+the original S to the first r coordinates and has determinant one.
+
+The schedule has 3r(m-r) scalar factors with support independent of S, q, w,
+header and column. No conditional physical permutation or low-class regrouping
+is assumed: only scalar coefficients carry the pivot flags. All zero factors
+remain emitted and paid. This addresses a source-dependent routing issue that
+merely choosing a matrix basis would leave unresolved.
+
+## 3. Quotient out the stabilizer
+
+Write Q'=JQJ^-1 in blocks [[A0,B],[C,D]], with A0 an invertible r-by-r matrix.
+Put X=C A0^-1 and Z=-B. Define
+
+    L_X=[[I,0],[X,I]], U_Z=[[I,Z],[0,I]], T0=L_X U_Z.
+
+Then Q'=T0 E T0^-1. Here is a proof over the local ring, without a probabilistic
+chart hypothesis. The first r columns of Q' are independent modulo q; since
+im(Q') is a free direct summand of rank r, they form an R-basis of that image.
+Thus D=C A0^-1 B=XB. Idempotence gives A0^2+BC=A0, hence A0+BX=I. Multiplying
+out T0 E T0^-1 gives blocks I-BX, B, X(I-BX), XB, exactly Q'. Both L_X and
+U_Z have determinant one; each is r(m-r) commuting elementary shears.
+Set T=J^-1 T0, so Q=TET^-1 and det(T)=1.
+
+This uses only the image/kernel splitting, rather than a full arbitrary GL64
+basis. The stabilizer degrees of freedom do not need to be implemented.
+
+## 4. Ancestor units and kernel-only normalization
+
+Let H'=J H J^-1=diag(d_i). This is diagonal because J is a signed
+permutation. Define the determinant-one diagonal
+
+    D*=diag(d_0,...,d_(r-1), (product_(i<r) d_i)^-1, 1,...,1),
+    Z_h=H'^-1 D*.
+
+Z_h is identity on every image coordinate; its arbitrary unit kernel entries
+are allowed. The desired head and tail conjugators are
+
+    U_t=T=J^-1 T0,
+    U_h=H T Z_h
+       =J^-1 (H' L_X H'^-1)(H' U_Z H'^-1) D*.
+
+Both have determinant one. The conjugated graph matrices have the same supports
+and their coefficients are multiplied by d_target/d_source. Kernel ancestor
+weights remain present in these coefficients; they have not been discarded.
+The right compensation Z_h fixes every image coordinate, so diag(Z_h,I)
+commutes with D(E). Therefore
+
+    K=diag(U_h,U_t),   K D(E) K^-1=D_H(Q).
+
+Normalizing an image column independently would generally change the weighted
+swap. Here only the kernel compensation is free. This image-only diagonal
+choice emerged in the coordinating agent's cross-review of the first all-axis
+normalization; it removes unnecessary kernel diagonal factors. The result and
+all checks remain in this same assisted research session.
+
+For any unit a, diag(a,a^-1) on coordinates (i,k) is four shears:
+
+    U(1) L(a-1) U(-a^-1) L(-a(a-1)).
+
+This displayed product uses ordinary matrix multiplication; chronological left
+actions run right to left. Its off-diagonal entries vanish and its diagonal is
+(a,a^-1). Pair each image coordinate i<r with the single kernel hub k=r.
+Their pair diagonals multiply to D*, giving 4r scalar factors, with two in each
+orientation. This identity is checked separately over all units of several
+local rings. Unlike implementing an entire arbitrary SL_m diagonal, no factors
+are emitted for the other kernel coordinates.
+
+## 5. Fixed permutation networks for ordered ADD
+
+An ADD is assumed available only when the source block precedes its target
+physically. Blocks contain f atoms; the coefficient for each column comes from
+unchanged saved metadata. Ordinary block interchange is a **completed**, same-q,
+source-preserving primitive that preserves the entire intervening spectator,
+returns its heads and restores owned work. Its full cost is b(v,f).
+
+A reverse individual shear can be implemented by swapping its two whole blocks,
+performing the ordered ADD on the relabelled coordinates, and swapping back.
+This is used for the diagonal factors and costs two ordinary calls per reverse
+factor. It is never identified with the central reversed child.
+
+A whole reverse pivot star has centre i and arms in i+1,...,m-r+i. Swap i with
+its largest arm, execute the entire relabelled star, and swap back. All arms
+then precede the centre (or the opposite relation in the reflected bank).
+There are exactly two ordinary calls per reverse star, regardless of the
+number of arms. The routing permutation is fixed, even if every coefficient
+is zero. The one-hot property is needed for the algebraic regrouping, not
+for the routing operation itself.
+
+For a reverse graph pass, put t=min(r,m-r). Swap i with m-t+i for i=0,...,t-1,
+execute its relabelled shears, then undo these t swaps. The images now lie
+entirely after the kernels (or entirely before them in reflected order), so
+every ADD is ordered. This costs 2t ordinary calls per reverse graph pass.
+It preserves the original layout at the pass boundary and acts pointwise on
+all external spectators, fresh rows and physical prefix ordinals.
+
+Within this permutation-only route-and-return model, t swaps per direction
+are optimal for reversing the complete image/kernel order: the smaller group
+must all leave its original side and a transposition moves at most one of its
+members to the other side. This narrow claim does not lower-bound all COPY
+implementations. Exhaustive permutations through dimension seven check it.
+
+Head/tail physical orders are opposite. Across both banks and both directions,
+there are 6r reverse pivot stars and four reverse graph passes. The diagonal
+appears only on the head, in both directions, and has 4r reverse scalar
+slots in total. Hence the exact fixed emitted counts are
+
+    scalar ADD factors = 4[3r(m-r)+2r(m-r)]+2[4r]
+                       =20r(m-r)+8r,
+    ordinary calls     =20r+8 min(r,m-r).
+
+At (m,r)=(64,7), these are **8,036** and **196**. Breakdown of ordinary calls:
+84 pivot, 56 graph, 56 diagonal. There is one reversed child, not one per
+prefix. At fixed header/column some coefficients can vanish; the invoice does
+not omit them. The interpreter checks every emitted ADD's physical direction
+and executes every ordinary swap, including routing back.
+
+## 6. Full caller, consumer and relative bill
+
+Use the complete packet shape and ownership contract of PR3950:
+
+    [Z physical prefixes][U unused fresh rows][B][A^n head][G][A^n tail][S],
+    n=64f+delta, 0<=delta<64, v=Z U B G S A^(2n).
+
+Clone the borrowed complete arbitrary source. Apply the complete address
+permutation using the two conjugation words and one reversed 7f child, return
+the owned result to the actual COPY private port, and preserve it through all
+63 literal XOR readers. Release every intermediate only after its last reader.
+At the equality guard, clear the private output and reset its port.
+
+Matrix words are chronological left actions on address vectors. Pullback
+implementation reverses the entire vector-event order. That reversal preserves
+the physical direction and coefficient of each emitted ADD; it is not the
+operation of negating a factor. Inverse conjugators separately reverse and
+negate their scalar words. The finite payload checker executes these primitive
+pullbacks and all 63 dirty consumers, rather than checking only matrix rank.
+
+The central child's spectators are B'=B, G'=G A^(114f+2delta), S'=S, with
+fresh rows U and all Z physical prefixes. Their volume equals v exactly and
+keeps the peeled atoms/external gap in their original coordinate order. The
+child is one complete call outside any prefix loop. For the inherited stock
+K0=269241600 |GL64(F_q)| q^(4096(w-1)) and
+D(n)=0 below64, otherwise1+D(42 floor(n/64)), the standard parent stock
+ t K0^D(n) leaves U=t*9475*K0^(D(n)-1). Since 7f<=42f<n, child divisibility
+follows. This is reuse of the stock implication; it does not prove the original
+producer's stock coefficient or authorize replacing its source program.
+
+Let t_R(v,7f,w) be the full completed reversed-child cost. Let gamma include
+principal-chart search, matrix/weight coefficient generation and recomputation,
+Counts, source/program authentication, numerical work erasure and metadata
+preparation not included in the completed primitives. Let C_scan cover actual
+linear copying, scalar ADD, transfers, rewinds, releases and disposal. Then
+
+    C_residual <= t_R(v,7f,w) + 196 b(v,f)
+                     + C_scan,q v(8036f+196+3) + gamma.          (1)
+
+There are at most 8036f+196+1 intermediate releases; this upper bound is paid in
+the scan term. Borrowed source is never cleared. For the literal COPY episode,
+append 63 X(v) for the completed XOR readers and the actual outer owner gather,
+route/scatter, directory producer and protected return charges if those have
+not already been charged by the caller. No reader recomputes the residual;
+no reader's arbitrary incoming target is assumed zero.
+
+Conditional on fixed reusable tape inventories, a conservative visited-cell
+reserve is
+
+    2v + S_R(v,7f,w) + S_B(v,f) + S_ADD(v) + M_coeff + F_q,
+
+where each backend reserve is a whole-run lifetime maximum on its fixed
+reusable inventory, including owned output, numerical work, parking and
+blank-but-visited cells. The two volume buffers must likewise use fixed reusable
+inventories; M_coeff and F_q include whole-run coefficient and finite-control
+backing. Per-call live occupancy does not establish these bounds. If a backend
+allocates disjoint tapes across occurrences, their lifetime backing must be
+summed instead. Sequential cleanup does not justify taking a maximum
+across disjoint tape inventories. There is no stored length-f array of matrices;
+coefficients can be recomputed from header and column, with recomputation time
+included in gamma. Successful cleanup is covered by the named contracts;
+interruption recovery is not proved.
+
+No hidden absorption is made in (1). Under the additionally admitted backend
+and root/history metadata premises one may substitute
+b(v,f)<=C_B v(fw)^p+E_B and absorb the justified scan/preparation terms. However
+the 196 C_B v(fw)^p contribution remains, as does 196 E_B until separately
+absorbed. This package does not prove those physical premises or implement a
+source-bound alternative SelectedBinding.
+
+## 7. Cost gate, evidence and remaining obligations
+
+For c_R=1736646042038446578551/10^24, n approximately e and w approximately
+e^c_R, the retained B1 and B2 rates give (1+c_R)(1-a_B)-1>0. Multiplying their
+upper bound by 196 instead of 41832 does not change its exponent. Thus this
+new constructive constant improvement still cannot infer an O_q(v(n+w^p))
+local allowance from those upper bounds. It is not a runtime lower bound.
+
+The source-bound finite evidence reconstructs all nine actual capture projectors
+from R437's root rows at a24f9ff, covering45 contexts at q=3,5,7,11 with high
+lifts and unit ancestors,192 GL2/weight contexts, an ambient64 nonzero graph
+with singular leading chart, complete small arbitrary payload/63 readers,
+and semantic failures. Separate principal-minor/chart and routing checks do not
+import the compiler. Tests at small q establish their stated algebraic cases;
+large-prime supplier cost theorems still require their admitted q domain.
+
+Still open: an encoded coefficient producer and physical primitive realization
+for this alternate schedule; a genuinely cheaper complete ordinary tariff or
+an actual jointly charged consumer; uniform source/return/refinement and visited
+reserve admission; full native/router/precision/recovery obligations. No selected
+exponent changes. The written proof has not received independent human or
+formal-kernel review; separate code is still from this assisted session.