Bound one-child COPY echoes with linear body ADDs and price fixed repair

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+# One-child dirty echo boundary and the fixed two-child budget
+
+Chris R / Clear1nsight, with substantial OpenAI Codex assistance. Apache-2.0.
+This is a scoped obstruction and cost decision, not a new multiplication bound.
+
+## 1. Changed proposal and optimistic gate
+
+PR3982 supplies an all-class rank-seven COPY conjugator with 8036 ordered ADD
+slots per column, 196 ordinary interchanges and one reversed 7f child. Its
+ordinary tariff remains a positive exponent-gap term under the available B1/B2
+upper bounds. The new proposal tested here removes ordinary interchanges by
+using coupled dirty payload roles, coordinatewise reversible/Boolean echo,
+and the existing reversed child as the only backward address primitive.
+This is a changed consumer/operation contract, not a smaller permutation route.
+
+The completed old bill is
+
+ T_R(v,7f,w) + 196 b(v,f) + C_q v(8036f+196+3) + Gamma
+
+plus 63 actual dirty XOR readers and outer owner/producer/return costs when
+not already included. All of these are charges; none is a negative credit.
+For a candidate with child widths k_1 f,...,k_t f and no ordinary calls, use
+
+ sum_i T_R(v,k_i f,w) + C_scan(v; clone, coordinatewise gates, ordered ADDs,
+ readers, transfers, releases, disposal, head return) + Gamma_echo.       (1)
+
+Its reserve must sum the whole-run visited maxima on fixed reusable source,
+owned-output, helper, child, coefficient and parking inventories. Erasure does
+not refund previously visited cells, and disjoint occurrence inventories cannot
+be replaced by a maximum. These implementation-dependent scan and reserve terms
+are unknown here. Setting all of them to zero is the optimistic gate below,
+not a physical upper bound.
+
+The source profile has m=64, W=28416, paid rank mass1808352, hence deficit
+64W-1808352=10272. It has2592 COPY executions (nine captures in each of nine
+stages and32 replicas), already using one private transformed stream through
+63 readers. A fixed schedule replacing every one with two rank-seven children
+adds2592*7=18144 mass, giving1826496 and excess7872. Its optimistic moment is
+
+ M_two(c)= (1/W) sum_r n_r(r/64)^(1-c)
+                  + (2592/W)(7/64)^(1-c).
+
+For every c>=0, each term (r/64)^(1-c)>=r/64, so
+
+ M_two(c) >= M_two(0)=1826496/1818624=1+41/9472>1.       (2)
+
+Thus this particular fixed two-rank-seven replacement fails even before the
+remaining positive costs in(1). This does NOT follow for an adaptive schedule
+or for two children of different widths. If only a fraction theta pays one
+extra rank-seven child, the zero-exponent necessary condition is
+ theta<10272/(2592*7)=107/189. At positive c the necessary bound is
+ theta < (1-M_old(c))/((2592/W)(7/64)^(1-c)), when the numerator is positive.
+The obstruction below supplies a worst-case class, not its density. It does
+not license inserting theta=1 into an adaptive original good/fallback scheme.
+
+## 2. Actual address convention and the relaxed operation class
+
+Fix an odd prime q, w>=1, R=Z/q^w and A=q^w. Take the allowed subsequence
+n=64f with any f>=1 (delta=0); a uniform all-size claim must cover it. In one
+fixed physical prefix/fresh-row/spectator fiber, order the two banks as
+
+ head atoms 0,...,n-1; reflected tail atoms n,...,2n-1.
+
+Atom a=if+j in the head is paired with physical tail atom 2n-1-a. The standard
+reversed child W_l swaps a with2n-1-a for0<=a<l and fixes every other address
+coordinate. Its complete source-preserving payload operation is the pullback
+ F'(x)=F(W_l x), exposing one transformed output and no additional payload
+channels; borrowed sources and helpers are restored. A rank-k child in the incumbent notation has l=kf.
+
+An ordered ADD has its source physically before its target. Its address matrix
+is lower unitriangular: x_i <- x_i+a x_j for j<i, with a determined by fixed
+metadata. Products and inverses remain in the group U of lower unitriangular
+matrices over R. We generously allow ALL such matrices on the two banks,
+including cross-bank ordered additions, not just PR3982's same-bank alphabet.
+Optional invertible diagonal factors would not change the argument.
+
+The macro circuit is finite and its event schedule and address permutations
+are independent of payload values. It may have arbitrarily many payload roles,
+clones, dirty helpers and fresh constants. At any full physical address it may
+apply arbitrary Boolean functions of roles at that SAME address, including
+address-dependent masks, not merely XOR or linear mixing. It may apply any
+number of U pullbacks on a role, and at most one completed W_l pullback event,
+where l<n. Owned release/reset and required restoration are allowed. Every
+primitive preserves the prefix/fresh-row/spectator fiber.
+
+This is an address-dependency class, not a claim every Boolean gate has an
+admitted physical tariff. Allowing them for free only strengthens a negative
+result. A global payload-dependent branch that reads one address to control
+another is excluded: it adds cross-address dependency edges. A differently
+routed physical scan, table lookup, gather/scatter or arbitrary-address copy
+is also not silently counted as a coordinatewise operation. Such primitives
+change the hypothesis and must appear in(1). Fixed metadata may depend on q,w,
+header and column; it must not convey source payload information.
+
+Linearity of the body-address alphabet is essential. An affine translation
+z -> z+(y-x) reaches the separating y from x, so admitting arbitrary address
+translations destroys this particular orbit obstruction even with no child.
+An addition from a fixed nonzero header/address digit into a body digit may
+also act affinely on the fiber and is outside U here. No claim excludes every
+legal physical scan or proves that such an enlarged alphabet is expensive;
+a complete translated/affine mechanism would need its own full cost gate.
+
+## 3. The dependency-orbit theorem
+
+Define lead(z) as the first nonzero coordinate of z. If L is lower
+unitriangular, lead(Lz)=lead(z), with the leading value unchanged. This holds
+over R including nonunits and high carries: before the leading coordinate all
+summands are zero, and at it the diagonal entry is1.
+
+Let x=e_n, the first physical tail coordinate. For every V in U, Vx has zeros
+before n and value1 at n. If l<n, W_l fixes coordinate n. It moves any nonzero
+among the last l coordinates to the first l positions. Therefore
+
+ lead(W_l Vx) is either <l or n.
+
+Applying another U preserves that lead. Consequently the set
+
+ O_1(x) = Ux union {L W_l Vx : L,V in U}                  (3)
+
+contains no vector with lead n-1, because l<=n-1. This is an exact all-ring,
+all-f statement; it does not enumerate the huge address cube.
+
+Now unroll the macro circuit in time. A dependency path from a final output
+bit backwards to an initial payload bit stays at its address through a
+coordinatewise gate and changes address through the pullback's matrix at an
+address operation. A path can cross the ONE chronological child execution at
+most once, even if the role graph reuses roles or has apparent cycles. Collapse
+all its ordered-ADD matrices on either side to U. Every input address that can
+influence output at x is in(3). Fanout and Boolean cancellations cannot create
+an input dependency absent from this union. This proof covers arbitrary dirty
+helper contents: compare two runs with identical initial helpers and metadata,
+changing only one source bit outside(3). Their final output at x is identical.
+No restoration assumption is needed for the lower bound, so requiring all
+helpers and the borrowed source to be restored cannot invalidate it.
+
+## 4. A legal all-class rank-seven COPY outside the orbit
+
+Let Q=diag(0,...,0,1,...,1) on64 coordinates, with ones precisely at57,...,63.
+Let H=diag(h_i) be arbitrary ancestor units (possibly depending on column and
+fixed header). The required two-bank map in paired head/tail coordinates is
+
+ D_H(Q)=diag(H,I) [[I-Q,Q],[Q,I-Q]] diag(H^-1,I).
+
+Apply the same Q on each of f columns, and reflect the physical tail as above.
+At x=e_n only the tail atom paired to head n-1 is nonzero. Thus
+
+ D_H(Q)x = h_(63,f-1) e_(n-1) = y.
+
+The unit h makes lead(y)=n-1, so y is absent from(3). The two arbitrary Boolean
+sources zero and delta_y require different COPY-output bits at x. The circuit
+class in§2 produces the same bit, a contradiction. Thus at least two proper-
+width reversed-child events, or another address-mixing operation, are needed
+for a uniform all-class COPY in this class. A single wider child of width42f
+still fails; width n leaves this obstruction but is not a smaller recursive
+child at delta=0. No sufficiency claim is made for two events or for width n.
+
+This Q is NOT asserted to be one of the nine literal capture matrices P.
+Every literal rank-seven capture is a split projector over the admitted ring;
+a basis of im(P) followed by a basis of ker(P) gives P=T E T^-1. A permutation
+S sends the first seven basis axes to57,...,63. Then g=S T^-1 is invertible
+and g P g^-1=Q. The all-class contract ranges over precisely such GL64 low
+classes and lifts. The verifier reconstructs the actual capture frames from
+the pinned R437 word and builds these conjugacies in finite local-ring cases.
+The proof uses the quantified split-projector premise, not small-q admission.
+
+Choose any physical prefix, unused fresh rows and spectator address, hold it
+fixed, and put delta_y only there. The contradiction is a full-volume source
+counterexample with every other bit zero. Unequal physical prefixes are not
+merged. Adding untouched spectators, or the63 downstream dirty readers, does
+not create the missing source-to-private-output dependency.
+
+## 5. Why fanout and pooling alone do not pay the old term
+
+The inherited private result is already computed once and retained through all
+63 readers. Dividing its invoice by63 would only amortize a constant. If a
+new contract really supplies K identical requests from one unchanged source,
+source sharing can replace K evaluations by one, but it still costs
+ T_R+196 b+K X+Gamma+cleanup, with the shared result preserved until last use.
+For the available bound b(v,f)<=C v(fw)^p+E, write n~e^alpha,w~e^beta
+with f proportional to n. The ratio of its leading term to K v n scales as
+e^delta/K, delta=(alpha+beta)p-alpha. At a proposed R437 target selector
+alpha=1-c_R,beta=c_R and p=1-a_B, this is delta=c_R-a_B>0 for both
+pinned B1/B2 rates. This is a conditional target-scaling calculation; it does
+not establish a retuned R437 selector or its admission. The retained ordinary
+selector pattern is n=floor((e-k)/w), with k=o(e), as in PR3562 head
+abb93be8b9ceca07a2b17f3a97db7bb6b074271d, PROOF lines110–165 (there c=1/577). This explicitly distinguishes original ordinary width e
+from reduced width n. A fixed K
+(including63) cannot remove this upper-tariff gap. To infer absorption from
+that tariff would require growing K at least on the e^delta scale, while
+also paying readers, storage, source compatibility and preparation. This is
+not a lower bound on actual adapter runtime.
+
+For different independent source payloads, packing K fibers enlarges volume
+from v to Kv; the same volume-linear tariff gives196 C Kv(fw)^p, so dividing
+by K gives no reduction at all. Identical field labels are not identical
+physical sources. Different Q maps likewise need not yield the same private
+stream: delta sources separate distinct address permutations. No new equality
+of the actual R437 captures is asserted. PR3995's pooled owner IR still needs
+its replica-indexed physical contract; its finite profile does not establish
+source equality or free transport. PR3989 similarly demonstrates why a finite
+coupled identity's lifted transfer cannot be priced without its full consumer.
+
+## 6. Decision, evidence, and limits
+
+Stop this fixed one-child dirty-echo route: it fails the full-cube dependency
+contract even after relaxing pointwise operations. Stop the fixed two-rank-
+seven-per-COPY repair: its optimistic moment already exceeds1. These are two
+separate conclusions, not a universal lower bound of fourteen recursive ranks.
+A 7f+1f or other mixed-width construction, an adaptive schedule with an actual
+class-frequency proof, a legal backward address operation with a cheaper full
+tariff, or a changed producer/consumer can reopen the question. Each needs an
+exact complete bill and the appropriate whole-profile moment before expansion.
+
+Finite checks authenticate source bytes, all nine capture classes and high-lift
+conjugacies, reachable address orbits in small cubes, a separating arbitrary
+payload bit and arbitrary fixed dirty helpers, and exact profile/tariff
+arithmetic. Controls break the premise deliberately (a full-width child or
+an extra child permits the formerly absent dependency), detect wrong target
+orientation/weight, and reject invalid source/invoice inputs. The mathematical
+proof, finite checks and same-session AI cross-review are distinct evidence.
+No encoded supplier, formal-kernel proof, full native execution, selected cold
+replay, improved kappa or frontier change is claimed.