Bound two-child serial affine COPY at 121 ranks and price the mixed-width gate

research/r437-affine-two-child-bound/PROOF.md

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+# Two serial affine children cannot reach the cheap R437 COPY repair
+
+Chris R / Clear1nsight, with substantial OpenAI Codex assistance. Apache-2.0.
+
+## Contract and the genuinely changed premise
+
+PR4014 excluded affine translations from its one-child dependency proof and
+explicitly left unequal two-child widths open. Here translations are allowed
+and two child widths are arbitrary. The new result concerns a **serial affine
+source pullback**. It does not include general multi-role, address-masked Boolean
+circuits covered by the earlier linear-address one-child theorem.
+
+Let R=Z/q^w for an odd prime q and w>=1. Fix one physical prefix, fresh-row and
+spectator fiber. Its two banks have n=64f address digits each, f>=1; the physical
+tail is reflected. A forward affine map is x -> Lx+t with L invertible lower
+triangular and t arbitrary in R^(2n). This grants every unit diagonal and every
+translation, including translations supplied by fixed header digits, for free.
+All schedule and coefficients are payload-independent. There are no additional
+source payload channels, arbitrary-address readers or payload-controlled global
+branches. The final operation on every arbitrary Boolean source F is exactly
+one pullback F(Ax+t). Cloning a source or retaining dirty helpers is harmless
+only when the output remains this single serial pullback; using masks to combine
+differently routed copies changes the contract.
+
+The standard reversed child W_a swaps coordinate i with 2n-1-i for 0<=i<a,
+and fixes the others. Any serial circuit with two such completed children has
+linear part
+
+                     A = L W_a M W_b R,                 (1)
+
+where L,M,R are invertible lower triangular. Arbitrary affine offsets disappear
+from this identity. Conversely, correctness for all Boolean F forces equality
+of the address maps: a delta source separates distinct source addresses. Thus
+a translation cannot repair a wrong linear part on the full address cube.
+This argument uses the whole map, not PR4014's single-address orbit invariant.
+
+## The two-child preserved-flag lemma
+
+Work over any commutative ring; all diagonal entries below are units. Suppose a
+target T fixes the first and last N coordinates, has no couplings between those
+coordinates and the middle block, and its middle block is not lower triangular.
+If a,b<=N, equation (1) cannot equal T.
+
+For a matrix X and a cut p, write NE_p(X)=X[0:p,p:d], d=2n. For invertible lower
+triangular L,R,
+
+       NE_p(LXR) = L[0:p,0:p] NE_p(X) R[p:d,p:d].          (2)
+
+Consequently NE_p is zero before multiplication exactly when it is zero after.
+Take p=max(a,b). The target has NE_p(T)=0 because p<=N. Set X=W_a M W_b.
+If a>b, select rows of X corresponding to original M rows d-a..d-b-1 and columns of X
+corresponding to the SAME original indices. W_a sends these rows into 0..a-1;
+W_b leaves these columns unchanged. The columns are >p-1, since a<=N<n. Their
+principal M block is triangular with unit diagonal, with only row reversal.
+Thus NE_p(X) contains an invertible (a-b)-square minor. If b>a, use original
+M indices a..b-1: W_a leaves these rows in the first p rows and W_b sends their
+columns into the final b columns. Again a triangular unit minor of size b-a
+appears. Hence NE_p(X) cannot vanish unless a=b=t. Only a lower bound on the
+minor size is asserted; NE rank need not equal |a-b|.
+
+Partition the coordinates as t, d-2t, t, and let J reverse t coordinates. Write
+
+     M = [ A0  0  0 ]        W_t M W_t = [ JFJ  JE  JDJ ]
+         [ B   C  0 ]                      [  0    C   BJ ]
+         [ D   E  F ]                      [  0    0  JA0J].
+
+The target has zero NE blocks at cuts t and d-t. By (2), X has the same zeros,
+so D=E=B=0. Its middle diagonal block is C, which is lower triangular.
+The middle block of LXR is exactly L_middle C R_middle: terms from the first
+block vanish because R is lower triangular; terms from the last block vanish
+because X's upper off-diagonal blocks are zero. This product is lower
+triangular, contradicting the target. This proves the lemma without passing to
+a residue field, division by nonunits, an enumeration of GL or a size limit.
+
+## Actual all-class rank-seven target and the width bound
+
+Let P be any of the nine literal rank-seven split projectors in the retained
+R437 source. Its all-class GL64 contract includes a conjugate Q whose image is
+the last seven head axes, 57..63. PR4014 records the source-bound construction:
+a basis of im(P) followed by a basis of ker(P), then a basis permutation.
+That split-projector premise and the all-GL64 quantifier are inherited inputs;
+this package does not infer production admission from finite small-prime tests.
+
+For an arbitrary diagonal ancestor-unit matrix H=diag(h_i) on the head, the required map is
+
+ D_H(Q) = diag(H,I) [[I-Q,Q],[Q,I-Q]] diag(H^-1,I),
+
+with physical tail reflected and all f columns included. It fixes the first
+and last N=57f physical coordinates and swaps the central 7f head/tail pairs
+with unit weights. Its middle block has nonzero entries above the diagonal.
+The lemma forces p=max(a,b)>N=57f. There is a stronger constraint when
+both children are proper, a,b<n. Reduce the linear maps modulo q. For the
+central target, NE_p(T) has rank max(0,p-N) whenever p<n: exactly the p-N
+active head rows present in that prefix have independent weighted tail columns.
+The same unequal-width unit minor used above survives modulo q, so (2) gives
+
+                  |a-b| <= rank NE_p(T) = p-N.
+
+Hence min(a,b)=p-|a-b|>=N. If recursive widths are a=k_1 f,b=k_2 f with
+positive integer ranks, then
+
+           min(k_1,k_2)>=57, max(k_1,k_2)>=58,
+                         k_1+k_2>=115.                  (3)
+
+This is necessary, not a construction attaining115. For arbitrary positive
+integer address widths the conclusion is a+b>=114f+1; it is wrong to infer
+115f without the integer-rank premise. In particular7f+1f,7f+7f,42f+42f,
+and58f+1f cannot realize this target in the serial affine domain. The
+rank inequality explains why a single very large child plus a cheap child
+also fails; this is stronger than merely requiring one child wider than57f.
+
+## Quotient refinement: total width at least121f
+
+There is a further necessary bound. We know a,b>=N. Factor W_a=W_N V_s and
+W_b=V_t W_N, where s=a-N,t=b-N and V_s,V_t are outer reversed swaps wholly
+inside the middle2r coordinates, r=7f. They commute with W_N. Partition M into
+N,2r,N blocks. In X=W_a M W_b, the two outer target-zero cuts force its
+cross blocks to vanish. Specifically those blocks are invertible permutation
+multiples of the lower off-diagonal blocks B,D,E of M, so B=D=E=0.
+The induced middle equation is therefore
+
+          L_mid V_s C V_t R_mid = weighted W_r,
+
+where all three non-permutation middle factors are lower triangular. Reduce
+modulo q. In NE_r(V_s C V_t), the first s rows have rank at most s. Each
+remaining top-half row comes from a top-half row of the lower-triangular C;
+after V_t its right-half support lies within the t columns moved from the
+first t positions. These rows have rank at most t. The target has rank r
+across that cut, so s+t>=r. Consequently
+
+                    a+b >= 2N+r =121f.                 (4)
+
+With positive integer recursive ranks and proper widths k_i<=63, this gives
+k_1+k_2>=121; the boundary pairs are58+63,59+62,60+61 and reversals. These are
+necessary candidates, not asserted constructions. Without the proper-child
+restriction a full-width child changes the admissibility question.
+A separate F2 example with n=3,r=2,N=1 and two width2 children attains2N+r=4;
+it prevents extrapolating the bound into a claim that all proper pairs fail.
+The independently authored checker retains the literal factors.
+
+An all-class implementation with a fixed pair of child widths must handle this
+class. An adaptive implementation may choose a different pair for this class;
+we have not established its density among actual GL64 classes or lifts. The
+result neither excludes three or more children nor masked multi-role synthesis.
+
+## Favorable complete-profile gate before construction
+
+The retained current pooled profile from PR4010 is
+{1:380160,2:400877,3:22176,4:26592,5:29376,6:17375,7:26496,14:1,42:403}.
+It has W=28416, rank mass1808352, deficit64W-mass=10272, and2592 COPY calls.
+The earlier unpooled OQ4 profile has the same mass; profiles are kept distinct.
+
+Replacing the incumbent single rank-seven COPY by any fixed serial affine
+pair satisfying (4) adds at least114 ranks per COPY. Even optimistically charging
+one rank58 and one rank63 and assigning ZERO cost to all other new work gives
+
+ M_min(c) = (sum_r n_r(r/64)^(1-c)
+             +2592[(58/64)^(1-c)+(63/64)^(1-c)-(7/64)^(1-c)])/28416.
+
+For c>=0 and child widths<64, every rank term of the replacement profile is
+at least its c=0 value. Thus M_min(c)>=M_min(0)>1. For the intended0<=c<=1 regime, concavity and monotonicity of x^(1-c)
+make58+63 the favorable proper integer pair among those satisfying(4).
+The verifier computes the
+exact value21915/18944 from the retained profile; rank mass becomes2103840,
+exceeding64W by285216. A width>=64 cannot be a proper
+smaller recursive child and cannot rescue this gate. Larger proper children
+only increase the c=0 mass; for c>1 the simple profile monotonicity still gives
+its own c=0 lower bound, so no optimizing-width claim is needed there.
+
+If only a fraction theta of COPY calls incurs this minimum replacement and all
+others optimistically retain rank7, a necessary zero-exponent condition is
+ theta <10272/(2592*114)=107/3078. At positive c, writing
+ Delta(c)=(58/64)^(1-c)+(63/64)^(1-c)-(7/64)^(1-c), the necessary condition is
+ theta < [1-M_old(c)] / [(2592/28416)Delta(c)] when Delta(c)>0.
+This is a **hypothetical fixed58+63 invoice**, not a proved achievable algorithm
+or a frequency estimate; actual larger child choices require their own Delta.
+
+Every real bill also includes affine/header preparation, admitted source
+production, coefficient evaluation, clone/mask/ADD scans, source preservation,
+child CALL/RETURN, output transfer, all dirty readers, reset, erasure and head
+return. Whole-run visited backing counts output, helper, child, coefficient and
+parking inventories; erasure does not refund visited cells. These nonnegative
+terms and supplier stock/precision bounds are unresolved. Since even the zero
+version fails for the fixed schedule, no physical implementation is warranted.
+
+## A constructive masked translation escape, with its actual count
+
+There is no affine impossibility claim for general circuits. For any linear T
+on a finite address group, let D=im(T-I). The exact identity
+
+ F(Tx) = XOR_{d in D} [ (T-I)x=d ] AND F(x+d)             (4)
+
+holds for every Boolean F: exactly one mask is active. Thus translations plus
+address masks implement the target with zero children. For the weighted split
+swap above, T-I has free image rank7f: each swapped pair contributes one free
+coordinate even with arbitrary unit weights. Therefore |D|=q^(7fw).
+For translation-only branches with fixed translations, this count is minimal:
+each branch can supply F(Tx) only where its translation equals (T-I)x, and every
+displacement in D occurs. Arbitrary source delta functions rule out borrowing
+one displacement's bit for another. This branch-count lower bound does not
+apply to affine lower-triangular branch maps, which have different linear parts.
+
+A direct realization uses one zeroed output and one translated temporary:
+produce F(x+d), scan and mask-XOR into output, erase temporary, repeat. The
+source is borrowed unchanged; arbitrary dirty helpers can remain untouched;
+the output is then shared with the incumbent readers. For V physical bits this
+literal full-scan realization performs |D| full mask-XOR scans, processes
+|D|V mask/source positions and erases |D|V temporary bits, in addition to the
+|D| translation-production invoices and final head/owner return. Source,
+output and reusable temporary occupy3V payload cells excluding metadata,
+translation scratch, child inventories and numerical/parking storage. It is
+not an O(Vn) realization: the branch count is exponential in fw for fixed q.
+No unproved physical tariff for a translation is supplied. Sparse or fused
+routing could evade the full-scan accounting but must specify and price its
+new mechanism. This exact positive control explains the serial scope boundary.
+
+## Evidence and stopping decision
+
+The checker enumerates complete small lower-unitriangular middle factors and
+uses north-east rank tables, not just the proof's zero predicate, to exclude
+membership in the target lower-triangular double coset. Additional randomized
+unit-diagonal local-ring cases check the explicit unit minors and block lemma.
+An exhaustive affine translation/mask replay checks (4), arbitrary source and
+unchanged dirty helpers; missing masks, missing unit weights and a corrupted profile receipt are failure controls. An
+independent bit-packed enumeration checks all64^3 F2 three-factor words, finds
+16960 distinct outputs and excludes the middle swap. A literal two-full-width
+child identity is a positive scope escape. F2 is an algebra control outside
+odd-prime supplier admission; the written theorem covers the admitted rings.
+
+Stop the fixed serial affine two-child route, including its7+1 candidate.
+Retain masked circuits, three-child words and proven adaptive density as open
+changed hypotheses. No improved kappa, encoded supplier, full physical caller,
+formal-kernel proof or selected-frontier change is asserted.