Research: bound additive idle-axis repair costs

research/idle-additive-repair-budget/PROOF.md

@@ -0,0 +1,184 @@
+# An additive repair cannot pay for one idle-group consolidation
+
+This is a scoped negative result for a concrete changed interface: retain the
+original local and hosted calls, reassign the existing idle axes, and repair the
+changed physical operator using additional recursive phase children. The
+correction is charged **beyond the fixed original residual rank mass**. It is
+not a proof against changing or absorbing work into the original calls.
+
+## 1. The exact comparison model
+
+Let `0 <= a <= 1/400`, `p=1-a`. The raw cost of a rank-`r` recursive child is
+`r^p`, with `0^p=0`. Normalized cost is `(r/73)^p`. The same positive stock
+division and all other invoice terms are held fixed on both sides.
+
+One original replica/port group has at most twenty residual slots of integer
+ranks `0 <= r_i <= 51`. Let `M=sum r_i` and `A=sum r_i^p`. The candidate
+redistributes exactly this same residual mass into ranks at most 51. To give it
+every advantage, permit even nonnegative real counts and real ranks: write
+`A'=sum_j w_j s_j^p`, with `0 <= s_j <= 51`, `w_j >= 0`, and
+`sum_j w_j s_j=M`. Any actual reassignment is included in this relaxation.
+
+In addition, charge at least one actual repair child, of integer rank `q>=1`
+and full multiplicity one. With several repairs let their raw cost be
+`C=sum_l q_l^p`. We grant the repair exact arbitrary-input correctness and no
+other overhead; in particular this favorable screen gives it free movement,
+adapters and restoration. It does not construct such a free implementation.
+
+The original local and hosted profile, stock, widths and normalization remain
+fixed. Enlarging banks, changing original children, crediting repair rank against
+residual mass, or sharing one repair across several groups requires a different
+invoice. Nonnegative extra costs under the fixed invoice only worsen the result.
+
+## 2. An all-parameter allowance bound
+
+For every positive `s <= 51`, `s^p=s*s^(-a) >= s*51^(-a)`. Hence
+
+```
+A' >= M*51^(-a),
+A-A' <= B(a) := sum_i (r_i^p-r_i*51^(-a)).
+```
+
+Rank zero contributes zero. For integer `1 <= r <= 51`, put
+`t=a*ln(51/r) >= 0`. Since `1-exp(-t) <= t` and `r^p <= r`,
+
+```
+r^p-r*51^(-a)
+ = r^p*(1-exp(-t))
+ <= a*r^p*ln(51/r)
+ <= a*r*ln(51/r)
+ <= a*51/e.
+```
+
+The last inequality follows by differentiating `x*ln(51/x)`: its maximum over
+positive `x` is `51/e`, attained at `x=51/e`. The exponential series gives
+`e > 1+1+1/2+1/6 = 8/3`. Therefore, for `0<a<=1/400`,
+
+```
+0 <= B(a) <= 20*a*51/e <= 51/(20*e) < 153/160.
+```
+
+Each actual positive-rank repair child costs at least one raw unit, so
+
+```
+(A'+C)-A >= 1-B(a) > 7/160 > 0.
+```
+
+At `a=0`, both residual profiles cost exactly `M`; the repair costs at least
+one, which also proves the strict stated margin. The normalized increase is
+greater than `(7/160)*73^(-p)`, followed by any unchanged stock division.
+Thus this additive repair cannot improve the fixed-stock recursive moment
+inequality anywhere in the stated interval. This does not claim a new root,
+a new kappa, or rejection of another supplier interface.
+
+This is a written proof for every real `a` in the interval and every profile
+in the declared model. The finite rational endpoint calculation is a control,
+not the proof of the whole interval. No formal proof kernel has checked it.
+
+## 3. Binding the source geometry and the proposed repair
+
+The source base is `463113d675af7410ac91d53a4fed51ec6494eb9a`.
+The unchanged DATA-PHASES recipes contain twenty idle positions at every one
+of 112 ports, with full ranks only `0,8,16,32,48`. A valid hosted prefix removes
+axes from these bases, so its residuals have at most twenty slots and ranks at
+most 48. The retained halving guard permits new residual rank at most 51:
+`2*51^2=5202 < 73^2=5329 <= 2*52^2=5408`.
+
+The verifier authenticates all 2,717,280 decoded T740 fill rows through the
+retained PR3148 payload hash, then reconstructs actual residual inventories for
+owner replica zero at all 112 ports. Its actual endpoint allowance at `a=1/400`
+is below the more permissive analytic bound. The generic twenty-slot theorem
+also applies to any other valid residual partition meeting its premises; this
+package does not independently rerun hosting/native/owner admission for all
+other replicas or certify those partitions by a new full controller replay.
+
+Crucially, the source coverage tool defines `idle_mass=total_mass-local_mass`.
+The theorem above is **not** an allowance to insert a repair into that coverage
+law and also reduce `M`. Here the repaired proposal is the literal one: assign
+each original residual axis once, then insert correction operators in addition.
+Its reassigned residual mass is still exactly `M`. All correction children are
+extra to that mass. A controller that instead eliminates other existing paid
+work has changed this hypothesis and must be priced separately.
+
+## 4. A physical operator witness and the sharp insertion count
+
+Use PR3148 owner replica zero, port zero. Move axis `v=16` from child 10 to
+child 12, as in a component of the pinned PR3184 star. The donor and receiver
+residual ranks change `48,48 -> 47,49`. The axis is in the donor residual and
+is orthogonal to the receiver's full basis. The total rank mass and group axis
+multiset are unchanged.
+
+On the translation character `x=16` in `F2^73`, the canonical-cover baseline
+eight-stream operator is `i I_8`. With zero-based indices the reassigned operator differs
+from it at exactly
+
+```
+Delta[1,1] = -1-i,
+Delta[3,1] = -1-i,
+Delta[7,7] =  1-i.
+```
+
+The submatrix on rows and columns `{1,7}` has determinant `-2`, and the only
+nonzero columns are 1 and 7. Thus its rank over `Q(i)` is exactly two.
+These are operator identities on arbitrary inputs, including dirty streams;
+no zero-state promise is used. The global phase is retained in `i I_8`.
+
+An inserted single-stream phase child of any axis rank has, at a fixed
+character, an eight-stream action `D=I+(z-1)e_s e_s^T`. If the retained prefix
+and suffix are `A,B`, adding it changes the operator by
+`B(D-I)A`, of rank at most one. At any cut this cannot equal the required
+rank-two correction. Therefore at least two such insertions are necessary at
+caller-level cuts in this exact retained-interaction example. It does not
+cover edits inside an opaque recursive child's helper-coupled implementation.
+The claim concerns completed
+Fourier-diagonal single-stream phase insertions; it does not treat new
+multi-stream interactions or an unpriced bundled decoder as one such child.
+
+Two rank-one calls suffice algebraically: insert `C_v` next to the donor's
+reduced residual phase and `C_v^-1` next to the receiver's enlarged phase.
+Here `C_v` has eigenvalue `i^(x dot v)` at character `x`, so `C_v^4=I` exactly.
+The donor phase is restored and the receiver's extra factor is cancelled.
+This restores each affected original factor and hence the entire operator
+for all `2^73` characters, including global phase. It simply repays and
+reverts the transfer; it does not achieve a cheaper supplier.
+
+The identity `C_v^2=X_v` also gives `C_v^-1=C_v X_v`. Thus the inverse is one
+rank-one selected child plus the explicit rank-zero translation `X_v`, whose
+movement is granted free in the favorable comparison of section 1. The
+interpreter represents the inverse by three forward phase exponents to test
+the same identity. The invoice charges one rank-one child and grants the
+translation free, rather than charging three forward calls. No repeated-axis
+list in this test is accepted as an orthonormal
+residual basis. For the algebraic repair, the generous budget theorem already
+loses with one added child, before charging its necessary second insertion.
+
+## 5. Controls and precise reopening conditions
+
+The source-bound replay rejects a wrong inverse, omission of either repair,
+and acceptance based only on the unchanged axis multiset. A positive
+single-insertion control has difference rank one. The retained interpreter is
+reused; a separate specialist implementation independently checked the exact
+operator witness. These are AI-assisted checks, not independent human review.
+
+The independent integer-root routine encloses every `r^(399/400)`, `r=0..51`,
+using exact integer-power bisection. It certifies the endpoint single-slot
+maximizer `r=19` and four **positive relaxed-invoice** controls outside the
+model: 22 original slots, half an actual repair call, one repair shared across
+two groups, and repair rank deducted from original residual mass. These
+relaxations are not claimed to realize correct operators. They demonstrate
+why those hypotheses cannot be dropped from this budget argument.
+
+The bounded investigation stops here. A constructive route can reopen it by
+absorbing corrections into existing calls, replacing original interactions,
+providing an actual shared repair across groups, changing the coverage/rank
+contract, or studying another saving interval. Each needs a new complete
+invoice and arbitrary-input restoration proof. This result supplies none of
+those interfaces and establishes no general barrier to a larger kappa.
+
+Credit: the source construction and physical-stream objection remain with
+Rohan Arun, Jeremy Diviney, dcharb78, Marius-Paul Dumitrean, Jeremy Kothe,
+Gonzalo-Mier, Gareth Davidson, Nick Petalas, Tekalign and the retained
+word/native/geometry lineage. This scoped repair allowance and its exact
+application were prepared by Ross Richardson with substantial OpenAI Codex
+assistance and independent mathematical-specialist implementations.
+New material is Apache-2.0; retained sources are unchanged.

research/idle-additive-repair-budget/README.md

@@ -0,0 +1,86 @@
+# Added phase children cannot pay for one idle-group consolidation
+
+**Scoped negative result; no new kappa or supplier admission.** Retaining the
+original local and hosted calls, one replica/port group's idle-axis repacking
+cannot pay even one additional positive-rank repair child under the declared
+fixed-mass invoice. This tests a constructive repair route left outside the
+fixed-schedule transfer obstructions, rather than rerunning their census.
+
+For `0 <= a <= 1/400`, raw recursive child cost is `r^(1-a)`. With at most
+twenty original residual slots and new residual ranks at most 51, even ideal
+fractional rank-51 packing saves **less than `153/160` raw units**. One actual
+additional child costs at least one. The resulting raw moment therefore
+increases by **more than `7/160`**, before extra overhead. [PROOF.md](PROOF.md)
+gives the all-parameter argument, endpoint handling and normalization.
+
+The essential hypothesis is that repairs are paid **in addition to the fixed
+original residual rank mass**. The source coverage formula can instead deduct
+new local rank from idle mass; that alternative invoice is outside this result.
+Changing original children, stock, normalization, interactions or rank cap, or
+sharing one repair across groups, also requires a new comparison. Exact positive
+relaxed-invoice controls show why those distinctions matter.
+
+A concrete PR3184 component on PR3148 owner replica zero / port zero moves
+axis 16 from child 10 to child 12. It changes the exact eight-stream operator
+by rank two over `Q(i)` at character 16; a `2x2` minor has determinant `-2`.
+One inserted single-stream phase child changes it by rank at most one at any
+retained caller-level cut. Two signed rank-one phase children restore the
+original factors, including global phase, for every character; they simply
+repay the transfer. The inverse uses `C_v^-1=C_v X_v`, with translation movement
+granted free by the optimistic screen. This is an arbitrary-input operator statement, not a
+zero-helper experiment. Omitted repairs and a wrong inverse fail.
+
+## Sources and coverage
+
+Base: `463113d675af7410ac91d53a4fed51ec6494eb9a`, priority revision 6.
+[SOURCES.json](SOURCES.json) pins nineteen live inputs, plus catalogue and
+selected-record provenance. The verifier authenticates all 2,717,280 decoded
+T740 fill rows and reconstructs owner-replica-zero inventories at all 112
+ports. Every source recipe has twenty idle slots and full ranks in
+`{0,8,16,32,48}`. The inherited strict halving cap is 51.
+
+The retained exact eight-stream interpreter and rational moment enclosures are
+reused unchanged. An independent mathematical-specialist implementation checked
+the operator witness, and a separate cost reviewer rederived the continuous
+bound. The latter's independent integer-power method is included in
+[allowance.py](allowance.py), covering the endpoint `a=1/400`, integer ranks
+0..51 and four excluded relaxed invoices. It does not replace the written
+proof on the whole interval. There is no formal-kernel or independent human
+review, new full native/controller replay, supplier admission or outer theorem.
+
+## Reproduce
+
+From the repository root:
+
+```sh
+python3.12 scripts/research_package.py --package research/idle-additive-repair-budget
+```
+
+Expected: `PASS_IDLE_ADDITIVE_REPAIR_BUDGET` and
+`PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=negative-result`.
+The registered command verifies the committed [RESULT.json](RESULT.json).
+For a fresh deterministic receipt use `verify.py --output /absolute/new.json`;
+the output must be new and outside this package. It does not modify sources.
+
+## Open next questions
+
+The bounded repair test is complete. [TASK.json](TASK.json) records two precise
+reopening routes: absorb a correction into original children while proving
+what existing cost disappears, or construct a shared correction across groups
+and account for its actual joint operator, stock, routing and restoration.
+Neither route is claimed to work or reserved by this package.
+
+## Credit and license
+
+The source construction, proposals and physical-stream objections remain due
+to Rohan Arun (#3184/#3190), Gonzalo-Mier (#1252/#3196/#3199), Gareth Davidson
+(#1387), Nick Petalas (#3187), Tekalign (#3204), Jeremy Diviney (#3148),
+dcharb78, Marius-Paul Dumitrean, Jeremy Kothe and all retained word, native,
+geometry and compiler predecessors. The selected record is unchanged;
+archival of a predecessor is not scientific admission.
+
+This package adds the explicit additive-repair allowance, its source-bound
+application, exact insertion witness and scope-sensitive invoice controls.
+Prepared by Ross Richardson with substantial OpenAI Codex assistance, including
+independent mathematical-specialist checks. New material is Apache-2.0.
+Frozen predecessors and their notices remain unchanged.