Price an actual centre-chain terminal and certify its scoped cost loss (no new κ)

research/center-chain-terminal-feasibility/OBLIGATIONS.json

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+{
+  "compiler_route": "PR4610 retained. PR4483/4544 not substituted.",
+  "milestones": [
+    {
+      "name": "priced physical chart word",
+      "status": "scoped-primitive-model-witness",
+      "evidence": "Centre0 all640-input endpoint/raw dual; actual added Gaussian and scalar bills. Remaining seven centres remain coupled. Tape and rounded execution are inherited/unproved."
+    },
+    {
+      "name": "favourable complete caller moment",
+      "status": "not-achieved",
+      "evidence": "Defined terminal policy has negative incremental moment throughout C in [0,1]. No complete profitable caller emitted."
+    },
+    {
+      "name": "funded kappa above PR4610",
+      "status": "not-attempted",
+      "evidence": "No new claimed/tested/uniform saving, compiler reserve, or outer admission."
+    }
+  ],
+  "obligations": [
+    {
+      "item": "all arbitrary inputs and helper ownership",
+      "status": "checked-scoped",
+      "evidence": "RESULT all640 columns, exacthelperreturns, one target per credited input, retained couplings."
+    },
+    {
+      "item": "actual new chart requests and raw-dual phase",
+      "status": "finite-checked",
+      "evidence": "Nested geometry census plus452 exactone-column source-compiler lowerings. Allf source semantics remain inherited."
+    },
+    {
+      "item": "scalar finite work",
+      "status": "counted",
+      "evidence": "113 terminal ADDs;758 joint boundary gates; emitted dyadic tariffs and l2 envelopes. No scalar/routing work converted into free ranks."
+    },
+    {
+      "item": "fixed tape movement, wrappers, storage, precision",
+      "status": "separate-open",
+      "evidence": "Actual primitive/owner word and452 wrapper upper count retained; no unproved O(V) route shortcut or rounded execution theorem."
+    },
+    {
+      "item": "ordinary center job and native matching",
+      "status": "optimistic-only",
+      "evidence": "Independent inverse8 job; ordinary positive-job normalization granted free only for a losing upper bound."
+    },
+    {
+      "item": "complete controller/source-target census",
+      "status": "open",
+      "evidence": "Single local partial terminal is not an independently diagonalized all-main-job caller. Pair/joint calculations are declared-policy accounting."
+    },
+    {
+      "item": "PR4610 compiler capacity and charged tails",
+      "status": "blocked-by-profitability",
+      "evidence": "Newrank9 paid requests and center job types require new matching/tail/finitewidth derivations. Keep originalroute; no source applicability inferred."
+    },
+    {
+      "item": "54 strict predicates and4 functional guards",
+      "status": "no-new-admission",
+      "evidence": "Rebind all retained premises and add any new interface/stock/precision guards before later use. Formula count alone is not completeness."
+    },
+    {
+      "item": "PR4483/4544 cancellation lemma",
+      "status": "not-used",
+      "evidence": "Would need binding of actual boundary masses, source/exit quantities and rootdebit to proposedcompiler."
+    },
+    {
+      "item": "BIT4760 compatibility",
+      "status": "later-only",
+      "evidence": "Not a dependency or budget source of this local study."
+    }
+  ]
+}

research/center-chain-terminal-feasibility/PROOF.md

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+# A charged terminal decoder for the heterogeneous centre offer
+
+Chris R / Clear1nsight, with substantial OpenAI Codex assistance. Apache-2.0.
+
+The result is a scoped negative construction. The emitted decoder obeys the
+retained Gaussian/common-chart primitive model and preserves arbitrary inputs.
+Its extra recursive bill exceeds its most favourable new target reward at every
+candidate saving C in [0,1]. This is not an impossibility theorem for affordable
+centre consumers, a full physical multiplier, or a new kappa certificate.
+
+## Actual offer and the changed protocol
+
+Use the exact #4698 normalized #4694 interface on 640 independent inputs. Write
+its scalar-coordinate operator as
+
+    M = [[A,B],[R,I]],       A(X,Y,h) = (X,Y+X,h).
+
+Here x=(X,Y,h) has 632 coordinates and z has eight. Each row of R is its
+56-term X observation plus one distinct pivot helper. B has 56 nonzero Y
+entries per column. R A = R and R B = 0. Physical operators have the form
+
+    E_out M E_in^-1.
+
+E_in and E_out are the actual direct sums of the source's canonical Gaussian
+frames. They are not omitted. Each pivot helper enters at the empty chart and
+exits at rank nine. X exits at rank nine; Y exits at its actual rank-eight chart;
+centres enter at their actual rank-eight charts and exit at the empty chart.
+The selected source is not homogeneous transverse progress on three fields.
+
+For a selected set S of centres, leave the complete source native word and the
+full joint 379-gate header and 379-gate footer intact. After that word:
+
+1. Apply the inverse actual output frame to every X and Y field used by the
+   selected rows/columns and every selected pivot helper. Each is a paid
+   Gaussian request, with literal monomial lowering wrappers.
+2. At the now common empty chart, subtract each entire R_j observation from
+   the actual centre j, including its pivot-helper term and all 56 X terms.
+3. Subtract B_j times that now decontaminated centre from every affected Y.
+4. Reapply every stripped output frame. Restore every touched endpoint chart.
+
+These are in-place operations. No field is initialized, discarded or identified
+with another arbitrary input. The original 640-owner population is retained;
+this does not establish that eight promoted arbitrary owners are free in a
+complete recursive caller. There is no new owner beyond the source extension.
+
+## Exact all-input endpoint
+
+After the source, x'=Ax+Bz and z'=Rx+z. R reads X and helpers only, which the
+scalar source leaves unchanged. Thus the centre subtraction yields z_j exactly.
+The Y subtraction then removes precisely B_j z_j. For every subset S,
+
+    x'' = A x + sum_(j not in S) B_j z_j,
+    z''_j = z_j                  (j in S),
+    z''_j = R_j x + z_j          (j not in S).
+
+The implemented scalar terminal is the product of two invertible shears. Its
+block matrix has upper-left I+B E_S R, upper-right -B E_S, lower-left -E_S R
+and lower-right I. The physical endpoint is E_out D_S M E_in^-1.
+
+For S={0}, roles 224 and 632 are the pivot and promoted centre. All 640 primal
+columns are freshly checked, including every remaining coupled centre and every
+centre-to-Y column. The output of role 632 is the independent inverse Gaussian
+of its original arbitrary input, with rank eight. The pivot retains its original
+rank-nine helper job. It is reclassified from autonomous helper to consumer
+pivot return and is counted exactly once. The other 407 helper jobs are unchanged.
+The seven other centres and the main provider contract remain coupled; they
+receive no new independent-job credit in the incremental screen.
+
+The verifier also emits the actual gatewise raw inverse transpose. Scalar ADDs
+are transposed and negated; a Gaussian unitary contributes its complex conjugate,
+not the same forward Gaussian. The inherited source convention dualizes the
+chart labels. A separate sparse rational/free-word evaluator checks its full
+640-row result against the transpose of an independently checked two-sided
+inverse. Forward-tail and forward-Gaussian substitutions reject.
+
+The all-width statement is conditional on the retained source native and chart
+contracts. Each address-operator block and its actual inverse are tensor-lifted;
+no tensor of a field-level sum is substituted for this chronology. New nested
+requests are lowered at one column with the source's exact Clifford compiler,
+including their absolute phase and raw-dual conjugates. The original native
+phase suite, tape execution and rounded recursion are not freshly replayed here.
+
+## Physical accounting unit and target ownership
+
+One normalized invocation costs the entire 379+379 joint boundary word. A chain
+is not assigned one eighth of that cost. Two separate normalized invocations
+cost 1516 boundary gates unless a later actual word proves sharing. This study
+uses one invocation and does not claim such a cancellation.
+
+The native extension scalar tally is 158136 real scalar units. The full joint
+normalization adds 6412 units and the chosen terminal adds 1128 units, for a
+total of 165676 units under the retained scalar tariff. These are finite
+whole-field scalar charges, not recursive ranks or a complete tape-time bill.
+
+The target census has an input identity, output type and sector for every
+credited job. The separate 640-row inventory records every arbitrary input,
+initial/final chart and digest of its actual chronological contact history.
+A selected pivot's rank-nine job remains one inherited job. Only its centre's
+independent inverse rank-eight job can be an additional reward. Positive ordinary
+credit is granted optimistically: its required positive-job normalization has
+not been admitted or priced as a free physical operation.
+
+## A separate zero-chart-change obstruction
+
+There is also a small optimistic rejection that does not depend on the chosen
+strip schedule. Permit arbitrary invertible scalar postprocessing within each
+actual terminal chart, with no paid chart transport or new cross-chart operation.
+The empty output chart contains precisely the eight centres. Their coefficient
+block on the eight independent pivot-helper inputs is I8. For an invertible
+centre-bank postprocessor L that block becomes L. No row of an invertible L is
+zero, so no centre-bank output becomes independent of all pivot inputs. The
+other terminal chart blocks cannot enter this bank under the declared alphabet.
+Thus zero-chart-change postprocessing cannot deliver any pure independent centre
+job. Allowing operator-valued mixing restricted to the centre bank itself also
+cannot annihilate its identity helper block without losing invertibility.
+
+This is a rank/ownership obstruction to that particular free-decoder model.
+Cross-chart reads, paid Gaussian jobs, a changed producer, joint bank operations
+and different output locations remain open. It supplies neither a rank-nine
+universal query floor nor an exclusion of affordable heterogeneous embeddings.
+The authenticated full source replay already checks the exact identity helper
+block used by this argument.
+
+## Declared complete-cost screen
+
+The domain is exactly the strip-all-touched-fields, subtract-complete-observation,
+subtract-contamination, restore-all-stripped-fields policy above. It keeps the
+source word, helper normalization and all untargeted output contracts. It allows
+all eight singleton choices, all 28 distinct pairs, and a separate jointly
+amortized eight-centre accounting case. It does not minimize over other decoder
+algorithms, gather orders, intermediate shared chart routes or source rewrites.
+
+Set t=1-C and let s=|S|. Let n_X and n_Y be the distinct X and Y support counts.
+The exact emitted additional recursive cost, and an optimistic upper bound on
+new productive reward, are
+
+    P_under(C) = 2 n_Y 8^t + 2(n_X+s) 9^t,
+    R_bar(C)   = s 8^t.
+
+All scalar, routing, storage and precision bills are separate; setting their
+incremental cost to zero only makes this screen more favourable. The new centre
+normalization is also granted zero cost. The old main/helper reward cancels in
+the comparison, because those contracts are retained and no pivot is duplicated.
+
+    Delta M(C) <= R_bar(C) - P_under(C).
+
+The source-authenticated support census gives:
+
+| Selected centres | n_X | n_Y | Added paid requests | Optimistic new reward | Delta M upper bound |
+| --- | ---: | ---: | --- | --- | --- |
+| Each singleton | 56 | 56 | 112 H8 + 114 H9 | H8 | -111*8^t -114*9^t |
+| Each distinct pair | 88 | 88 | 176 H8 + 180 H9 | 2 H8 | -174*8^t -180*9^t |
+| All eight together | 112 | 112 | 224 H8 + 240 H9 | 8 H8 | -216*8^t -240*9^t |
+
+For every C in [0,1], both powers are at least one. These upper bounds are at
+most -225, -354 and -456 respectively. The sign is therefore certified on the
+entire declared interval, with no one-point extrapolation or unproved monotonicity.
+At C=0 the exact deficits are 1914, 3012 and 3888 rank units. These are costs of
+this emitted policy, not unavoidable ranks for all physical embeddings.
+
+For S={0}, the complete local paid profile is
+
+    {1:2201, 2:412, 3:510, 4:127, 5:40, 8:120, 9:114}.
+
+The unchanged source profile is
+
+    {1:2201, 2:412, 3:510, 4:127, 5:40, 8:8}.
+
+The terminal has 113 dyadic scalar ADDs. RESULT.json retains the complete word,
+scalar tariffs, prefix norm envelopes, all 37 support/cost cases and ownership.
+The chosen 226 added Gaussian requests have at most 452 left/right monomial
+wrapper invocations. Their actual lowering recipes are retained. A generic
+finite-work ledger is the inherited source work plus 758 normalization gates,
+113 terminal ADDs, the 226 Gaussian requests and their emitted wrappers, along
+with the actual stream movement and recursive-call storage/precision charges.
+No O(V) route shortcut, zero-cost wrapper, or rounded precision theorem is assumed.
+The scalar l2 envelope is a triangle bound, not a machine-bit guard certificate.
+
+The strict negative recursive comparison stops expansion of this policy before
+its finite work needs a new outer or compiler admission campaign.
+
+## Homogeneous-template compatibility rejection
+
+The exact first pivot diagonal Gaussian has column support 512; its centre's
+inverse rank-eight Gaussian has support 256. The retained literal Clifford
+circuits reproduce both, including phase. Per-field monomial pre/post wrappers
+preserve support cardinality. Thus the unmodified assignment of these fields
+cannot have the same transverse diagonal multiplier required by the homogeneous
+three-field offer. This rejects direct identification with only those wrappers.
+It does not reject a paid embedding, a further joint basis, a different query Q,
+a heterogeneous consumer or another choice of shared recursive job type.
+The physical query inventory records each actual inverse canonical centre frame,
+its exact monomial square and its order-four check. It does not substitute that
+operator for both heterogeneous chain links or assume symmetry. In fact, all
+eight candidate centre inverse frames fail the literal symmetry test. For centre
+0, the exact matrix entries are Q[10,0]=1/16 and Q[0,10]=-1/16. This is a
+rejection of using that actual centre diagonal operator directly as the template
+Q, not a rejection of other Q choices or a paid heterogeneous embedding. A last-contact
+audit of the chosen 113 stripped fields finds a cross-field ADD at every final
+source contact. Commuting a strip through operations on unrelated fields alone
+therefore does not expose an adjacent source Gaussian inverse to cancel. Joint
+changes across those scalar contacts remain outside this terminal policy.
+The five failure controls in the retained group-ring consumer are rerun, without
+claiming its algebraic identity as a new contribution.
+
+## Compiler and ceiling boundaries
+
+The PR4610 compiler route is fixed. No #4483/#4544 implication is substituted for
+it. No profile is admitted by passing 54 outer predicates or four guards. This
+new local profile even introduces rank-nine paid requests and inverse centre jobs;
+a surviving complete caller would require a new source/target matching and tail
+obligation map before using the retained compiler. The failed local screen makes
+that later work unnecessary for this policy. See OBLIGATIONS.json.
+
+PR4700's monotone single-centre-call protocol and PR4696's fixed graph/subspace
+phase grammar are not invoked as universal bounds. The new terminal changes the
+centre return, adds descending/ascending boundary calls, and uses a new coupled
+consumer. The centre's original paid call is retained, never silently deleted.