Normalize eight center chains and construct a conditional two-call consumer (no new K)

research/center-helper-normal-form/CONSUMER.md

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+# An exact two-call controller for a homogeneous three-step query chain
+
+**Conditional controller identity; no physical native realization or higher K.**
+This is an exact consumer for a specified three-field offer. The normalized
+PR4694 scalar chains alone do not prove that offer's physical hypothesis.
+
+Let Q be a symmetric invertible operator with Q^4=I and let P=Q^2 be an
+available monomial involution. A tensor Gaussian C^tensor(f) has these
+properties under the retained phase convention. On three arbitrary independent
+fields assume an oracle
+
+    U = [[I,Q,0],[0,I,Q],[0,0,I]].
+
+Its actual raw inverse transpose is
+
+    U^-T = [[I,0,0],[-Q,I,0],[P,-Q,I]].
+
+The bottom-left term is required. Define free matrices with only ordinary
+scalar and Pauli entries:
+
+    D = diag(P,I,P),             E = I - P E_31,
+    L = [[I,0,0],[0,-P,0],[I,0,P]],
+    R = [[0,I,I],[I,0,0],[0,0,P]].
+
+Then the exact identity is
+
+    L U E D U^-T D R = diag(Q,Q,I).
+
+The chronological program is R, D, the raw-dual call, D, E, the forward
+call, L. Every boundary is invertible and uses only scalar ADDs, signs,
+field exchanges and P. For example an ADD with coefficient P can conjugate
+its source by P, perform the ordinary ADD, then restore the source by P;
+no Gaussian query or zero-only input is hidden in the notation.
+
+To prove the identity, use
+
+    E D U^-T D = I - Q^-1 (E_21+E_32).
+
+Indeed the D conjugation changes -Q on the two adjacent lower edges to
+-Q^-1 and leaves the P long-root term. E removes that term exactly. Multiplying
+by U gives
+
+    [[0,Q,0],[-Q^-1,0,Q],[0,-Q^-1,I]].
+
+Direct multiplication by R and L gives diag(Q,Q,I), using P^2=I and
+P Q^-1=Q. These are block-operator identities at every width, not numerical
+specializations or tensor powers of a sum. Symmetry Q^T=Q is needed to identify
+the raw dual as written.
+
+consumer.py verifies every coefficient in the exact cyclic group ring
+Z[q]/(q^4-1), checks the actual raw inverse transpose, and includes separate
+dense rational specializations. It rejects a wrong long-root sign, an omitted
+correction, an omitted raw-dual quadratic term, missing Pauli conjugation,
+and dropping the order-four hypothesis.
+
+## Conditional transverse and price ledger
+
+Suppose the actual forward provider also multiplies all three fields by the
+same fresh rank-b Gaussian D_s, commuting with Q and these boundaries. Suppose
+the actual raw-dual provider has the corresponding identical inverse Gaussian
+on all three fields, converted to a positive one by its true Pauli normalizer.
+Two disjoint transverse stages then give targets
+
+    two rank-(2b+1) jobs and one rank-2b job.
+
+The third field has its central query restored; it still receives the
+transverse Gaussian. If each full invocation has autonomous helper profile H
+and complete paid profile P_local, the complete moment for this gadget is
+
+    2 H_t + 2 (2b+1)^t + (2b)^t - 2 P_local,t,   t=1-C.
+
+Writing the local rank deficit relative to three rank-b main jobs as
+E=P_local,1-H_1-3b, its rank surplus at C=0 is 2-2E. This is a conditional
+bookkeeping formula, not an invoice assigned to PR4694 or a new exponent.
+
+## The actual missing interface
+
+The real retained-center word has different charts on its helper, center
+and Y directions. Its source helper has rank nine, its retained center moves
+from an eight-dimensional chart to the empty chart, and the terminal Y
+combination spans different eight-dimensional charts. A scalar Jordan chain
+therefore does not supply U(Q) with a common commuting transverse multiplier.
+
+The next concrete test is a costed embedding of one of the eight recorded
+chains into this homogeneous offer, with every chart conversion and the
+remaining R_X data terms retained. Only after that succeeds may the formula
+above be used with actual paid and target profiles. The package does not
+claim a free full Jordan basis, independent center credit, a profitable native
+consumer or a higher K.
+
+Prepared for Cinna Davis with substantial OpenAI Codex assistance; Apache-2.0.
+Credit the PR4694 coupled interface, Cinna Davis's controller lineage, Cao Yifan,
+zxv, Jacob Alan Sussman and retained native/phase/precision authors. No priority
+claim is made for elementary block elimination identities.

research/center-helper-normal-form/PROOF.md

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+# A legal helper basis and exact three-step chains
+
+Prepared for Cinna Davis with substantial OpenAI Codex assistance. Apache-2.0.
+
+## Source interface and real common charts
+
+Assume the exact PR4694 interface M on 640 arbitrary inputs. Let x=(X,Y,h),
+with 408 original helpers, and let z contain the eight retained centers:
+
+    x_out = A x + B z,       z_out = R_X X + R_H h + z,
+    A(X,Y,h)=(X,Y+X,h).
+
+B has rank eight and is supported only on Y. All 106 helper columns used by
+R_H have birth chart 0 and complete output chart 1. The complete cold pool
+contains 326 helpers. The frozen source has no endpoint adapter on these
+helpers, so the charts used by the boundary circuit are actual common charts.
+The source supplies their identical exact operator F=C^tensor(9f), for each
+positive column width f, under its retained native/all-width hypotheses.
+
+## Exact sparse basis
+
+Let J be the eight pivot roles in RESULT.json and let N be the other cold
+roles. Set L=R_H[:,J]. Exact rational elimination gives det(L)=64 and pivots
+2,2,1,4,1,2,1,2. In particular every division used in the emitted algorithm
+is a dyadic unit. Define h=P u by
+
+    h_N = u_N,
+    h_J = L^-1 (u_J - R_H[:,N] u_N).
+
+Then R_H P u=u_J. This is an invertible change, not a restriction or a clean
+input promise: all 408 original helper inputs remain arbitrary. The verifier
+emits the gathering subtractions and the eight-row elimination as 379 actual
+ADD/SCALE gates. Row exchanges use three ADDs and a sign scale. All operands
+are cold helpers at chart 0. Reverse the gates and invert each to emit P^-1
+in 379 gates at output chart 1. Their coefficients and both inverse identities
+are checked exactly.
+
+Because P acts on field labels and F is the same operator on each cold field,
+
+    P^-1 diag(F,...,F) P = diag(F,...,F)
+
+at every width f. This is ordinary commutation of a scalar matrix with an
+identical direct-sum operator, not tensoring a sum or assuming equal ranks
+imply equal physical jobs. Other helper jobs are untouched. All monomial
+normalizers and source physical phase conventions remain the source's.
+The entry and exit circuits add no recursive Gaussian request. Fixed scalar,
+route and storage work is real finite work and is not claimed to cost zero.
+
+Conjugating M only on these helper coordinates therefore preserves A, B and
+all helper targets, while replacing R_H by the coordinate selector E_J.
+Data observations R_X and all center input columns are exactly retained.
+The native replay checks this full identity on all 640 columns and also on
+the true gatewise raw inverse transpose. It retains every paid source call.
+
+## Eight actual chains and the scalar Jordan type
+
+Put K=M_new-I. On the three sectors (old fields, centers),
+
+    K = [[S,B],[R_new,0]],   S=A-I,
+    K^2 = [[B R_new,0],[0,0]],   K^3=0.
+
+S maps the 112 X coordinates bijectively to Y. B is supported on Y, and
+R_new reads only X and the eight distinct pivot helper inputs. Thus the image
+of K contains all 112 Y directions (choose X and cancel R_X X with the pivot
+helpers) and all eight center directions (choose pivot helpers). It is
+contained in their sum. Hence rank(K)=120.
+
+R_new is surjective because of its helper selector. B is injective, so
+rank(K^2)=rank(B R_new)=8. For a nilpotent operator of index at most three,
+the number of size-three blocks is rank(K^2), and the number of size-two
+blocks is rank(K)-2 rank(K^2). The remaining dimension gives 408 size-one
+blocks. This proves the stated Jordan type over Q, not only numerically.
+
+More concretely, for each pivot helper e_hj,
+
+    K e_hj = e_zj,       K e_zj = B e_j,       K B e_j = 0.
+
+Their 24 vectors are independent: the first eight lie in the helper sector,
+the next eight in the center sector, and the last eight are the independent
+Y columns of B. The verifier checks those exact sparse columns. It does not
+claim that converting all Y/data fields to a full Jordan basis is physically
+free; those output charts differ.
+
+## Verification and remaining bottleneck
+
+The frozen source engine is reused unchanged for actual native replay; it is
+not relabeled an independently written evaluator. Separate sparse matrix
+composition checks the basis inverses, conjugation and both operator inverses.
+Every native scalar chart and paid rank is checked. Complete positive scalar
+envelopes for primal and raw dual remain below 2^192, without a new rounded
+recursive induction claim. Controls reject a changed header sign, an omitted
+output inverse and an output circuit moved to the wrong chart.
+
+The eight center outputs still contain R_X X, and incoming z still enters Y.
+Thus eight normalized observations are not eight independent Gaussian jobs.
+No contraction, complete controller, uniform compiler, new BIT endpoint,
+outer saving or multiplication theorem is established by this basis change.

research/center-helper-normal-form/README.md

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+# Concentrate the retained center observations into eight real helpers
+
+**Constructive physical boundary reduction; no higher K.** The new PR4694
+640-field interface observes 106 different helper inputs in its eight center
+outputs. This package changes the basis of actual cold helpers, with all
+operand charts checked, so each center observes **exactly one helper**.
+There are 379 scalar gates at entry and 379 at exit, with **no new Gaussian
+request**. All original helper Gaussian jobs and the complete paid profile
+remain unchanged.
+
+The eight pivot roles are 224, 225, 227, 229, 240, 241, 256 and 257. Their
+actual observation minor has determinant 64; the emitted inverse uses only
+dyadic ADDs and unit dyadic scales. The 106 observed helpers all belong to the
+326-field pool with common empty birth chart and complete nine-axis output
+chart. Therefore the boundary change is physically legal, rather than an
+arbitrary Jordan basis assumed free across different charts.
+
+For each center j the normalized endpoint is now
+
+    z_j_out = z_j_in + h_j_in + (R_X X)_j.
+
+Every data observation is retained: each row of R_X still has 56 terms.
+Incoming centers still contaminate all 112 Y outputs. The centers are coupled
+outputs, not eight independently productive Gaussian jobs.
+
+Writing K=M-I, exact rank(K)=120 and rank(K^2)=8, with K^3=0. Thus the scalar
+Jordan type is eight blocks of size 3, 104 of size 2 and 408 of size 1.
+Eight chains are explicit: pivot helper -> its center -> the actual Y scatter
+column. A full physical change to a Jordan basis is not supplied or priced.
+These chains give a smaller concrete target for a changed controller; they do
+not establish a positive recursive moment or a new K.
+
+## Conditional two-call consumer
+
+CONSUMER.md and consumer.py give the exact identity
+`L U E D U^(-T) D R = diag(Q,Q,I)` for a homogeneous three-field query
+chain with symmetric Q, Q^4=I and monomial P=Q^2. It uses two oracle calls
+and actual scalar/Pauli boundary gates; the raw dual's quadratic term is
+retained and corrected. A conditional complete moment is derived there.
+
+This makes the chain a specific consumer target rather than only a Jordan
+classification. The real source charts do not yet realize that homogeneous
+offer. The physical embedding, its full cost and all R_X terms remain open;
+this conditional identity is not applied to the native invoice.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/center-helper-normal-form
+```
+
+The standard-library replay reconstructs the PR4694 source word and the new
+word from the original PR4610 round-three payload. It checks all 640 arbitrary
+columns in the primal and actual raw inverse transpose, every common chart,
+all paid ranks, both basis inverses, both operator inverses and the explicit
+chains. Both full positive scalar envelopes remain below the retained
+192-bit allowance. Wrong coefficients, an omitted inverse and a wrong physical
+footer chart are rejected. Normal replay leaves files unchanged; --write is
+an authoring command for RESULT.json.
+
+## Sources and scope
+
+PR4694's cache continuation at head 2add42d1edf14a263abe90a3bcde67fc1ba30768
+already records that all 106 observed helpers are cold. That observation is
+credited prior work. The new contribution here is the physically emitted
+basis and inverse, preservation of all helper jobs, and the explicit chains.
+Its direct cache acquisition and scalar-span failures remain in force; this
+normalization does not claim to overcome them. ABI.json and the native payload
+are byte-identical between that head and the frozen source below.
+
+Frozen input bytes are identical to PR4694 head
+039a3e4f8906400cd05908ed86b8f9c6866e6113; inputs/ORIGINS.json records the
+original paths, Git blobs and hashes. That exact history is retained by a
+local tree-preserving provenance merge. Only this package changes relative
+to canonical main. The upstream PR is not merged or modified.
+
+The PR4694 emitter and native checker are copied unchanged as actual replay
+inputs, with their source credit. Its source all-width/native/phase contracts
+remain premises; there is no new physical phase audit or independent theorem
+verification. The new basis commutes with the identical exact nine-axis
+Gaussian on the cold helper pool at every positive column multiplicity.
+PROOF.md states why this preserves all existing helper target jobs.
+
+Prepared for Cinna Davis with substantial OpenAI Codex assistance; Apache-2.0.
+Credit Cinna Davis's PR4694/4440/4441/4447/4470 lineage, Cao Yifan's PR4610,
+zxv, Jacob Alan Sussman and the retained scalar, E8, Clifford, precision,
+compiler and outer contributors. No subagents or independent human review.
+
+## Open next questions
+
+Construct a physical controller using the eight explicit length-three chains.
+The first useful test must retain R_X X, every incoming-center-to-Y term and
+all actual boundary charts. The remaining obstacle is independent Gaussian
+output recovery at a complete paid profit, followed by compiler, BIT and outer
+integration. This normalization does not authorize center target credit or a
+larger exponent.