research: price P104 encoded-source return at a retained inverse-shear caller

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research/p104-role-exchange-caller-audit/PROOF.md

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+# Fixed-location return and a bounded inverse-shear caller
+
+## 1. The retained caller and the physical limit of this study
+
+The concrete predecessor is the **two-stage complex bank-exchange caller** in
+`notes/copied-centers-complex.tex`, section "Two-stage phase endpoint and its
+paid correction". Its first forward shear and inverse second shear have the
+scalar composition `(x,y) -> (-y,x+y)`. Its subsequent correction reads the
+first output, applies `T^-1` to a paid copy, adds that copy to the second output,
+and disposes of the copy. None of those later operations is donated here.
+`scripts/copied_centers_network.py` records the separate paid endpoint child.
+
+That is an actual retained caller, but its documented two-factor geometry is
+not a supplied P104 global owner embedding. We bind the caller's **inverse
+shear equation** and give an explicit conservative realization at the P104
+post-channel cut. We do not substitute 104 for the caller's published triple
+count, identify its two tensor axes for free, or claim that our local readout
+is its complete second-stage physical implementation. This distinction is
+essential to the stopping decision.
+
+At this cut, let `R(u,w)=(u-w,w)`, with all 72 dirty helpers unchanged. A legal
+common-chart realization lifts each required `Y_i` from `H_i` to FULL, performs
+`X_i -= Y_i` while both are FULL, and returns `Y_i` to `H_i`. This realization
+costs 104 unit Gaussian additions and 208 rank-one conversions. It preserves
+all returned `w` values. This is an explicit local implementation of the
+retained equation, not a free source read or an outer recursive admission.
+
+For the source-preserving correction `S(x,y)=(x,x+y)`, the composed endpoint is
+`R S(x,y)=(-y,x+y)`. Its return equations are `y=-X_out` and
+`x=Y_out+X_out`; every arbitrary helper returns in its stated final chart.
+
+## 2. PR3882's boundary and why relabelling fails
+
+All algebra is over `D=Z[i,1/2]`, acting coordinatewise on arbitrary finite
+arrays. Write `A=R_root^t R_root/8` in the pinned doubled-root port order.
+For `J=[0,1,3,6,22,31,93,103]`, with complement N of size 96, the retained
+geometry gives
+
+```
+A = [[I, B], [B^t, B^t B]],    nnz(B)=384,
+```
+
+where every nonzero B coefficient is a signed half. PR3882 returns
+
+```
+X_J=q=x_J+B x_N,       Y_J=w_J,
+X_N=w_N,              Y_N=v_N=y_N+B^t q.
+```
+
+`X_J,Y_N` are FULL; `X_N,Y_J` have their port hyperplanes. The 72 inherited
+dirty helpers remain FULL and logically restored. This is a changed location
+and data contract. Even allowing a free logical gather into residual order
+`r=(q,v_N)`, the naive inverse shear returns `(r-w,w)`, whose error from the
+required `(-y,w)` is `r-x`. It is generally nonzero. The verifier includes
+the explicit nonpivot one-source, zero-y witness; zero dirty values are valid
+for this separating witness, while the positive endpoint replay includes all
+280 independent input columns.
+
+The residual-as-source error has rational rank exactly 104: its J rows are
+`B x_N`, and `B B^t=12 I_8` proves their rank eight. Its N rows additionally
+contain the 96 independent `y_N` coordinates with identity coefficient.
+Thus the N rows are independent modulo the J rows. This is a logical
+incompatibility statement, not a transport or recursive-cost lower bound.
+
+PR3882's inverse equations already recover x from `(w,q,v_N)`; those equations
+and its sharp eight-replacement theorem are credited predecessor results.
+The new result below is a literal return schedule and its sharp **fixed
+physical-location** frame tariff. It is not a new residual-minimality theorem.
+
+## 3. A sharp 192-rank fixed-location restoration adapter
+
+Demand the same physical source and target locations as PR3894: X contains x
+in FULL, Y contains w in `H_i`, and all helpers retain their values and frames.
+Lift each `X_N` to FULL. In chronological order perform
+
+```
+Y_N -= X_N                       # v_N-w_N
+Y_N -= B^t X_J                   # -x_N
+X_J += B Y_N                     # x_J
+for each i in N:
+    X_i -= Y_i                   # w_i+x_i
+    Y_i += X_i                   # w_i
+    X_i -= Y_i                   # x_i
+```
+
+Then descend `Y_N` to its target hyperplanes. The other eight sums `Y_J` and
+the dirty helpers are untouched. Every scalar source and target is FULL.
+No payload swap, sign operation, clean bank or private erasure is hidden in
+the three displayed unit additions. The explicit schedule uses 1,152 unit
+Gaussian ADD/SUBs, 104 halves and 104 restoring doubles, 96 lifts and 96
+descents. It contains 1,552 formal events.
+
+For each fixed physical register, the sum of absolute frame-dimension changes
+is at least the absolute difference of its final and initial dimensions.
+Here 96 X registers must go from dimension eight to nine, and 96 Y registers
+must go from nine to eight. Thus **192 added rank units are necessary** under
+the inherited nested dimension-gap tariff. The construction attains 192.
+This bound permits arbitrary scalar resynthesis; it does not permit free
+physical role renaming, changing the required return locations, or replacing
+the caller's frame contract. It is not a scalar-operation or universal decoder
+lower bound. Additional temporary banks cannot remove the fixed registers'
+required endpoint dimension changes without changing this model.
+
+Consequently the literal PR3882 tail followed by this fixed-location adapter
+has 400 added H1 conversions above the original channel, versus PR3894's 208.
+The inherited channel mass is 2,368, so the respective totals are 2,768 and
+2,576 before the next reader. Appending the separate inverse reader gives
+2,976 and 2,784. These are constructive local invoices and a scoped adapter
+minimum, not optimum complete caller costs.
+
+## 4. Fuse the return instead of paying an intermediate encoded boundary
+
+At the original channel exit X=x, Y=v=y+A x. Lift all 104 Y fields to FULL.
+Use the eight original source pivots as temporary accumulators:
+
+```
+X_J += B X_N
+Y_J -= B X_N
+Y_N -= B^t X_J
+X_J -= B X_N
+Y_N += X_N
+```
+
+After the first two operations, `X_J=q` and `Y_J=w_J`. The third gives
+`Y_N=v_N-B^t q=y_N`. The fourth restores the original `x_J`; the last gives
+`Y_N=w_N`. Every `X_N` source is unchanged. The result is exactly S, including
+all source, target and helper rows, so a caller which accepts PR3894's physical
+endpoint accepts the same endpoint here **conditional on the same native
+chart semantics**. The derivation uses the same source-pivot geometry as
+PR3882; source-consuming and in-place pivot ideas are credited, not invented.
+
+There are four B traversals and 96 direct additions: `4*384+96=1632` unit
+Gaussian ADD/SUBs. For a literal dyadic lowering, each of the two X_N gather
+passes halves and restores each of 96 donors; the scatter halves and restores
+eight pivots. Thus there are 200 halves and 200 doubles. The final 104 Y
+descents and initial 104 lifts give exactly the same **208 H1** profile as
+PR3894. This source-preserving suffix contains 2,240 formal events.
+
+To consume S with the bounded inverse reader, perform all `X_i -= Y_i`
+**before** the Y descents. The result is `(-y,w)` with 1,736 unit additions,
+200 halves, 200 doubles and 208 conversions: 2,344 suffix events. No X source
+is consumed until every corrected Y sum exists. This chronology matters.
+
+For a fair control, retime PR3894's direct correction into the same all-Y-FULL
+window. Halve each of the 104 original X sources once, apply its 4,992 signed
+unit correction additions, restore every X by doubling, then perform the same
+104 inverse-reader additions and descend Y. This uses 5,096 unit additions,
+104 halves, 104 doubles and the same 208 conversions: 5,512 suffix events.
+The new scalar schedule saves 3,360 unit additions but adds 96 halves and 96
+doubles. A separate copy/cleanup operation, less precision, faster runtime or
+lower recursive-child profile does not follow from the addition count.
+
+At 512 complex words, multiply a Gaussian array addition/halving/doubling by
+1,024 real component operations. The exact real-addition saving is 3,440,640;
+each extra halving/doubling category has 98,304 real component operations.
+
+The normalized scalar prefixes have sharp maximum coefficient denominator
+four in the pivot schedule and two in the retimed baseline; both have maximum
+row one-norm 26 including their channel-exit entrance. The direct bound is
+`26H` per computed data component when original normalized x,y components are
+bounded by H. Untouched helpers retain their own input bounds. These are
+**tail-only common-chart** certificates. The original channel
+interior, physical native entrance, literal signed-floor divisibility, native
+scratch, accumulated rounding and all-depth numerical formats are not proved.
+
+The all-Y-FULL schedules have 2,520 persistent frame coordinates at peak,
+including 72 FULL dirty helpers; PR3894's target-major suffix peaks at 2,417.
+The receipt also records after-event frame-coordinate sums with an explicit
+convention. These finite occupancy diagnostics omit native internal scratch
+and are not a recursive owner/idle tariff.
+
+## 5. Complete-cost decision and precise exclusions
+
+Retiming either baseline or pivot suffix into the reader window leaves the
+profile `(1282,58,86,104,48,24,8,8)` and mass 2,576. The unchanged local
+width-nine, 280-role bare screen is `2576/2520=46/45>1`. Every positive child
+rank is at most nine, so for `0<=a<1`, `(r/9)^(1-a)>=r/9`. Nonnegative omitted
+costs cannot rescue this particular unchanged-interface screen. This is not a
+universal obstruction to a caller with a different coverage law.
+
+For the retained **source-preserving nine-call paired-chain template**, use
+width 73, 832 families, one unpaid coordinate per live family, 2,312 persistent
+progress units per local call, and no extra progress from the closed terminal
+excursion. The remaining collective correction/helper/residual/idle coverage
+is `72*832-9*2312=39096`. The favorable whole rank bill for the unchanged
+profile is therefore
+
+```
+9*2576 + 39096 = 62280 > 60736 = 73*832,
+M(0) >= 7785/7592 > 1.
+```
+
+This is the inherited complete-coverage rejection, now applied to an explicit
+source-preserving resynthesis of the role-exchange idea. It is not a new
+coverage theorem and does not transplant the separate PR3609/3620 dual's
+425/426 threshold or six/seven/six format. Fully paid auxiliary banks add 73
+to both coverage and capacity, preserving the positive 1,544-rank deficit for
+every finite number. The already charged terminal mass is not subtracted from
+idle coverage as a free credit.
+
+The complete **physical** price retains native wrappers/arithmetic, inherited
+private COPY/disposal, source availability, original and changed bank owners,
+all holding/routing/address/control work, actual numerical formats, paid
+endpoint copy/correction/cleanup, recursive backend, all-size transfer and
+the outer caller's actual coverage law. Those quantities are unknown or
+inherited as stated, not zero. The rejection uses a favorable lower bound,
+so no additional implementation is needed to reject this unchanged budget.
+
+Decision **NARROW**: retain the exact fixed-location adapter, its sharp tariff,
+the all-input caller-cut compositions, the scalar resynthesis and the fair
+baseline comparison. No native expansion or supplier claim is justified.
+The older 1,352-add chronological snapshot correction and 2,376-add dirty-root
+factorization remain credited alternatives; this is not a record for all
+source-preserving correction words. All native recipes, global ownership,
+physical caller adoption, formal-kernel acceptance and independent human
+review remain separate.