research: derive P104 two-axis endpoint intertwiners and cost gate

research/p104-two-axis-compatibility/PREDECESSORS.json

@@ -0,0 +1,47 @@
+{
+  "schema_version": 1,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "pending_work": [
+    {
+      "pr": 3882,
+      "head": "9d56eeedbed5e3e9bbbc8c189d66f14cf15b6067",
+      "publication_head": "d76586303e7036baa74f8deb2155038290174ffd",
+      "credit": "Clear1nsight",
+      "used": "Residual equations, orthogonal pivots and sharp eight-residual theorem; independently reconstructed from canonical roots. No pending executable imported."
+    },
+    {
+      "pr": 3894,
+      "head": "13b397e36d43f2a91b9e6d42b3ca67f0ec50cea7",
+      "credit": "1337isnot1337",
+      "used": "Direct source-preserving correction and 208-H1 invoice, shared with PR3940; new full native replay not claimed."
+    },
+    {
+      "pr": 3940,
+      "head": "51ba88c007cfbcc3d9cf22dd321f5224be3a5bb1",
+      "credit": "Clear1nsight",
+      "used": "Caller-cut study and complete-cost discipline; neither its 1544 threshold nor its free logical caller is treated as the physical two-axis embedding."
+    },
+    {
+      "pr": 3943,
+      "head": "2cba526a4ebe0ef6c3c0143498adcf67dca97941",
+      "credit": "Clear1nsight",
+      "used": "Parallel actual internal cleanup consumer and separate return/paired-chain results. Explicit overlap credit only; not replayed or reimplemented."
+    }
+  ],
+  "canonical_lineage": [
+    "Rohan Arun/PR1972",
+    "Chafik Boukhalfa/PR1991",
+    "Adam Gall/PR1940",
+    "Josie Elliston/PR1071",
+    "Aurel Prosz/Paureel",
+    "Zhihao Chen",
+    "Rohan Arun",
+    "Dominik Scholz",
+    "icekylinx",
+    "Sagar Thakrar/PR1012",
+    "Douglas Colkitt",
+    "Jose Manuel",
+    "Vasek/PR2744"
+  ],
+  "license": "Apache-2.0; canonical and predecessor notices retained"
+}

research/p104-two-axis-compatibility/PROOF.md

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+# Physical two-axis compatibility and the named cost gate
+
+## 1. Scope and source meanings
+
+This is a boundary theorem and conditional rejection for one explicitly described
+two-axis P104 scaffold. It is not an embedding of P104 into the existing h=28
+mixed-center producer, not a new native compiler replay, and not a supplier.
+The retained caller in notes/copied-centers-complex.tex supplies the physical
+two-stage equations and separately paid endpoint correction. PR3882/3940
+supply the residual representation. PR3943 supplies a distinct actual internal
+cleanup consumer; no new claim is made for that cleanup or the shared
+192-transport return floor.
+
+All operators below act on arbitrary Gaussian-dyadic address functions. The
+physical statements are ideal exact identities; signed-floor, finite-width and
+recursive admission do not follow from them.
+
+Order the 104 ports as J followed by N, where
+J=(0,1,3,6,22,31,93,103). Let B=A[J,N] for A=G/2. The pinned roots give
+A[J,J]=I, A[N,N]=B^t B and BB^t=12I. Put
+```
+D = [ I      B          ]       H = diag(0_8,I_96)
+    [ B^t    B^t B - I  ]
+
+D^-1 = [ -11 I   B  ]
+       [ B^t    -I  ].
+```
+Multiplication proves the inverse. With w=u+y, PR3882's residual vector is
+r=(q,v_N)=D u+H w. Thus the common-chart encoding and its inverse are
+```
+E0 = [D H; 0 I],     E0^-1 = [D^-1 -D^-1 H; 0 I].
+```
+Extend every compatibility map by identity on the 72 arbitrary dirty roles.
+This extension preserves all their information; it does not supply the
+unexecuted physical native return of a completed global caller.
+
+The ordered (r,w) notation is not the literal register occupancy. PR3882
+places r_J in X_J and r_N in Y_N, and w_J in Y_J and w_N in X_N. Put
+J0=I-H and Pi=[[J0,H],[H,J0]]. Pi swaps the two slots exactly at the 96 N
+ports; Pi^-1=Pi. The literal slot encoding is Pi E0, not E0. Pi is an
+explicit bookkeeping permutation in the equations below, NOT a free physical
+move or a proved compatible owner reassignment. Its realization, chart
+alignment and fixed return locations belong in the complete boundary bill.
+
+## 2. Logical inverse shear and the physical caller
+
+The original inverse shear R=[I -I;0 I] on (u,w) has encoded form
+```
+R_encoded = E0 R E0^-1 = [I -D;0 I].
+```
+Hence an encoded source may be carried by the exact update r <- r-Dw, with
+w unchanged. D has 5,000 nonzero entries: eight unit diagonal entries and
+4,992 signed half entries. A direct common-FULL-chart implementation applies
+the eight unit terms, halves all 104 temporarily writable w donors once,
+performs the 4,992 signed unit updates, and doubles all w back. This is
+5,000 unit array ADD/SUBs plus 104 halves and 104 restoring doubles. It is
+a constructive common-chart scalar invoice, not a scalar optimum and not a
+physical implementation with free chart/owner changes. It is not the
+source-preserving 1,736-ADD fused schedule of PR3940.
+
+For each physical data pair use a nondegenerate tensor line U=P tensor Q,
+T=C_U, F=C_I, and E=F T^-1. Let bold T be the diagonal family over the ports.
+For the proposed P104 ports we explicitly choose T=C_u with the positive
+rank-one operator formula in section 3, and define E=F T^-1. This specifies
+the physical operators rather than assuming every ambient-projector or
+historical chart orientation has the same phase. Any orientation/monomial
+adjustment needed to embed these choices into a producer belongs in the
+additional boundary bill; no free identification with the h=28 producer is
+asserted. A different orientation must rebind the finite phase census.
+The retained caller has source charts (T,I) and terminal charts (F,E).
+Writing u=T^-1 x, its two stages are exactly
+```
+(A,B) = diag(F,E) R S diag(T^-1,I) (x,y)
+      = (-F y, F T^-2 x + E y),
+S=[I 0;I I].
+```
+This is the actual retained endpoint equation, not merely the scalar
+(-y,x+y) equation. The paid correction is
+P=[I 0;T^-1 I], so P(A,B)=(A,F T^-2 x).
+
+In ordered (r,w) coordinates at the terminal charts the encoding is not E0. Conjugating by
+G=diag(F,E) gives the physical encoding
+```
+L_T = G E0 G^-1 = [D H T;0 I].
+```
+F is common and commutes with the rank-one C_U operators; H is diagonal
+and commutes with T. D need not commute with T.
+In the literal register order the physical encoding is A_T=Pi L_T.
+
+To preserve this explicitly fixed physical encoding on both sides of the
+paid correction, the exact operator is
+```
+P_encoded = L_T P L_T^-1
+ = [ I+H D^-1      -H D^-1 H T       ]
+   [ T^-1 D^-1    I-T^-1 D^-1 H T   ].
+```
+Equivalently, delta=D^-1(r-H T B), followed by
+r'=r+H delta and B'=B+T^-1 delta. This is an operator identity with the
+displayed order; it grants neither matrix reads nor chart transport for free.
+Its r-to-r increment H D^-1 has rational rank 96, or 96 times the address-space
+dimension on arrays. That is an information/linear-map rank, NOT a lower bound
+of 96 native tariff units.
+
+The correction in literal slots is therefore
+P_literal_encoded=Pi P_encoded Pi, rather than P_encoded alone. Keeping the
+permutation symbolic exposes the required ownership/return obligation without
+silently donating moves between the actual X/Y banks.
+
+Applying the old correction unchanged to encoded fields is wrong:
+```
+P L_T - L_T P = [-H             0]
+                  [T^-1(D-I)    H].
+```
+For B=0 and A a nonzero N-port input, the first-bank error is -A.
+This counterexample survives even homogeneous T. A corrected physical caller
+must implement the conjugated correction, choose a different explicitly
+intertwined output encoding, or decode. None is priced as zero.
+
+There is also a direct literal-slot counterexample, without granting a gather:
+with K=T^-1,
+```
+P A_T - A_T P = [-H K                   0]
+                  [K J0(D-I)-H          K H].
+```
+The first-bank defect on an N-port A input is -K A, again nonzero and of
+rank 96 times the address dimension. The separate noncommutative checker
+verifies Pi^2=I, both literal-slot identities and the ordered-coordinate
+identities. No rank here is identified with a native conversion tariff.
+
+After the correction, the retained phase/sign/exchange wrappers finish the
+unencoded caller. In the fixed literal A_T contract they must be preceded by
+A_T^-1=L_T^-1 Pi (or by L_T^-1 after a separately realized ordered gather),
+or replaced jointly by a proved and priced encoded endpoint. This study does
+not assert that every possible co-designed final encoding must decode there.
+
+## 3. Explicit two-axis physical lines and a noncommuting read
+
+Independently reconstruct the doubled E8 roots, the rank-nine binary pairing
+and an explicit orthonormal isometry. Check all 120^2 pairings, then remove the
+two pinned frames to obtain 104 odd distinct physical labels p_i in F2^9.
+The receipt records the complete isometry and ordered labels. This is an
+explicit coordinate gauge; it need not spell the historical compiler's gauge.
+
+For the proposed replacement scaffold take V=F2^9 tensor F2^9, m=81, and
+U_ij=span(p_i tensor p_j). All 10,816 lines are distinct and nondegenerate.
+For a fixed Q=p_j the first-stage active space is F2^9 tensor Q; the
+second-stage active space is P tensor F2^9. This supplies actual binary
+subspaces for the boundary calculation, not the chronological global owner
+map or the old triple-center producer. Opposite-orientation P104 execution
+in this scaffold is deliberately granted only as a favorable cost hypothesis.
+
+The native-boundary proof pins
+C_u=((1+i)I+(1-i)X_u)/2. Consequently
+```
+C_u^-1-C_v^-1 = (1+i)(X_u-X_v)/2.
+```
+For distinct nonzero binary u,v, X_u-X_v has rank 2^(m-1), since exactly
+half the Fourier characters separate their parities. A nonzero B entry
+therefore cannot move unchanged through the port-dependent T^-1 family:
+[D,T^-1]_ij=D_ij(T_j^-1-T_i^-1) is nonzero.
+
+The receipt gives original ports 0 and 2 as one explicit half-coefficient
+edge, along with their physical labels, tensor vectors and a separating
+frequency. On the four-address span {0,u,v,u+v}, a 4-delta input produces
+opposite nonzero Gaussian values +(1+i) and -(1+i), up to the sign of B.
+The separate checker reconstructs the literal four-address translation
+matrices and their rank two difference.
+
+For a fixed nonzero Q, contraction of an 81-bit Fourier character with Q
+ranges over all 512 vectors in F2^9, each equally often. The complete
+512-pattern check finds that D commutes with T^-1 only for the homogeneous
+patterns z=0 and z=511; the other 510 fail. This is exact for this finite
+P104 encoding and gauge. It is not a no-intertwiner theorem for changed
+encodings, paid cross-port operators or other geometries.
+
+Positive control: any common completed operator C on every role commutes with
+E0, since its entries are scalar bank coefficients. Thus a matched entrance
+and exit encoding can pass through an already completed uniform tensor call.
+This is why the present boundary obstruction does not close every jointly
+designed encoded caller.
+
+## 4. Port-dependent endpoint normalization
+
+For any binary u, C_u^2=X_u and the retained normalization generalizes to
+```
+i^wt(u) Z_u (F C_u^-2) Z_u = F.
+```
+On Fourier character z the exponent difference is
+wt(u)+wt(z xor u)+2*(z xor u).u-wt(z), a multiple of four.
+This proves the identity on all address functions. The checker additionally
+evaluates 5,537,792 explicit (tensor line, character) pairs.
+
+The old triple generators have weight nine. In this explicit P104 product
+gauge, 5,440 lines have weight 1 mod 4 and 5,376 have weight 3 mod 4.
+For one parallel column, using the old constant i^9 on the latter gives the
+wrong global sign. For f parallel columns the correct scalar is i^(f*wt(u));
+this particular sign error disappears when f is even. The corrected phase is
+explicit and introduces no positive-rank child, but its
+phase/address passes remain paid physical work. These counts are
+gauge-specific; no obstruction to choosing a better physical gauge is claimed.
+
+## 5. Complete conditional cost gate for this scaffold
+
+Enumerate n=104 row fibers and n column fibers. The persistent owner inventory
+is two n-by-n data banks plus two DISJOINT banks of 72 helpers per fiber:
+```
+N=n^2=10816,   W=2n^2+2n*72=36608,   m=9*9=81.
+```
+Each data pair begins with source chart C_U of rank one and target identity.
+All other persistent entrances are identity. The retained completed all-role
+contract is full F on every one of the W physical roles. Therefore required
+persistent progress is P_total=81W-N=2,954,432.
+
+The rejection has the following explicit hypotheses:
+(a) 2n=208 whole local calls, each granted the same favorable corrected P104
+profile of mass M=2576, even in the unproved opposite orientation;
+(b) each local block supplies at most P=2312 of that persistent progress;
+(c) disjoint persistent helpers as enumerated, the above entrance/exit contract,
+and no uncharged frame preparation;
+(d) all remaining native work has rank tariff at least its net persistent
+dimension progress; all child ranks are at most 81;
+(e) the retained equal physical-role volume factor 1/W.
+These are a named proposed replacement model, not a proved global P104 embedding.
+
+The original 4,058-event witness is used only to recount mass 2368 and
+persistent progress 2312; adding the retained 208-H1 correction gives 2576.
+Neither its full endpoint nor its native recipes are freshly replayed here.
+The local I-G/2 correction does not earn the same progress a second time.
+The most favorable remaining rank is consequently
+```
+P_total - 208*2312 = 2473536,
+lower mass = 208*2576 + 2473536 = 3009344,
+capacity   = 81*36608            = 2965248,
+excess     = 44096 > 0.
+```
+This deliberately most favorable bound permits a new joint implementation to
+absorb the old physical endpoint correction. If instead one retains the
+literal decode / old-P / encode route, the caller's separately paid rank-one
+copy transforms contribute another N=10,816 units with no persistent progress:
+the lower mass is then 3,020,160 and the excess is 54,912, before decoding,
+encoding, scalar copy/add/disposal or any other new work. This second number
+is a bound for that literal route, not a lower bound on arbitrary resynthesis.
+For 0<=a<1, (r/81)^(1-a)>=r/81. Hence the complete favorable moment is
+at least 3,009,344/2,965,248 > 1 before any new encoded boundary work.
+Adding b fully paid persistent banks adds 81b to both bill and capacity,
+preserving the 44,096 excess. This is a different scaffold from PR3940's
+width-73 nine-call law and PR3943's hypothetical paired chain. Their 1,544,
+425/426 and other thresholds are not transferred here.
+
+The full gate is
+```
+M_complete(a) >= [sum_local_children (r/81)^(1-a)
+                  + sum_remaining_children (r/81)^(1-a)] / W
+                 + A_intertwiners(a)+A_scalar(a)+A_physical(a) < 1.
+```
+The A terms include acquisition/encoding/decoding, the explicit Pi occupancy
+map and fixed-slot return, both changed-bank updates,
+endpoint work not already counted in the chosen native route, new wrappers,
+routing/holding/control/tape, finite
+width/precision/recovery and recursive admission as applicable. They are
+unknown nonnegative costs, not zero. No numerical physical total is supplied.
+Even granting them all zero, the native lower bill already fails. Under this
+unchanged invoice at least 44,097 net rank units of improvement are necessary
+at a=0, and are not sufficient at positive a or with added costs.
+A changed owner inventory, coverage law, net progress or local profile needs
+a newly derived gate.
+
+Decision: NARROW. Publish the compatibility equations, explicit counterexamples
+and conditional complete-cost rejection. Do not expand native/supplier
+construction for this scaffold. A genuinely different joint encoding or owner/
+coverage contract remains open.