Ordinary BIT saving 0.0034597 (+0.67939%) via compact partial helpers

research/bit-compact-productive-controller/PRICE-PROPOSAL.json

@@ -0,0 +1,99 @@
+{
+  "invoice": {
+    "targets": {
+      "4": 72,
+      "8": 7776,
+      "15": 444,
+      "22": 444,
+      "29": 444,
+      "36": 444,
+      "43": 444,
+      "50": 444,
+      "57": 444,
+      "64": 222,
+      "127": 222
+    },
+    "paid": {
+      "1": 25407,
+      "2": 24156,
+      "3": 1404,
+      "4": 702,
+      "5": 1836,
+      "6": 180,
+      "7": 324,
+      "14": 444,
+      "21": 444,
+      "28": 444,
+      "35": 444,
+      "42": 444,
+      "49": 444,
+      "56": 444,
+      "63": 222
+    },
+    "rank_surplus": 753,
+    "thick_calls": 18,
+    "thin_queries": 1665,
+    "original_fields": 11400,
+    "main_original_fields": 3552,
+    "helper_original_fields": 7848
+  },
+  "certificate": {
+    "params": {
+      "tested": "69509/20000000",
+      "uniform": "43443/12500000",
+      "cover_gap": "9765881/2560000000000",
+      "atom": "173/50000",
+      "admitted": "34597/10000000",
+      "old_backend": "384599/10000000000",
+      "alpha_gap": "1/4294967296",
+      "prime_coefficient": 67108864
+    },
+    "after_cover": "8887360519/2560000000000",
+    "ordinary_exact": "17713901828029/5120000000000000",
+    "admitted": "34597/10000000",
+    "tail": {
+      "tested_saving": "69509/20000000",
+      "uniform_saving": "43443/12500000",
+      "alpha": {
+        "lower": "0.999999991905059684301510340131",
+        "upper": "0.999999991905059684301510340132"
+      },
+      "gamma": {
+        "lower": "0.958113118679772899937220732700",
+        "upper": "0.958113118679772899937220732701"
+      },
+      "delta": {
+        "lower": "0.948389236578657043633140116103",
+        "upper": "0.948389236578657043633140116104"
+      },
+      "tensor_power": 1024,
+      "cutoff_base": 17,
+      "target_floor_base": 18,
+      "tail_exponent": "1/5",
+      "normalized_surplus_lower": "1069547519/4503600701112320",
+      "compiler_H0": 377227157504,
+      "uniform_H1": 262206447682297857,
+      "uniform_N": "2^262206447682297857",
+      "winning_word_materialized": false
+    },
+    "comparators": {
+      "current_admitted": "17/5000",
+      "current_exact_ordinary": "8590884599/2500000000000",
+      "PR4697_fixed_C0_prime_saturated_sup": "8845274375/2568746817031"
+    },
+    "relative_gains": {
+      "current_admitted": "597/34000",
+      "current_exact_ordinary": "58365401/8590884599",
+      "PR4697_fixed_C0_prime_saturated_sup": "418189878821507/88452743750000000"
+    },
+    "margins": {
+      "adapter": "1298171971/5120000000000000",
+      "ordinary_relaxation": "237828029/5120000000000000",
+      "stock": "5084570898171971/5120000000000000",
+      "prime_allowance": "17179869184/9765881"
+    },
+    "WORD_changed": true,
+    "new_whole_kappa": null,
+    "scope": "Conditional ordinary BIT supplier only; main selected/complex result unchanged."
+  }
+}

research/bit-compact-productive-controller/PROOF.md

@@ -0,0 +1,218 @@
+# Compact partial-helper BIT supplier
+
+This is a conditional ordinary BIT improvement. All analytic, exact-recovery,
+physical-tape, all-size native/compiler and fixed-weight assumptions of the
+selected construction are retained. The selected multiplication exponent is
+unchanged. This package supplies a changed finite native unit, its complete
+paid mixed-target bill, and fresh strictification and ordinary parameters.
+
+## 1. The local interface
+
+The input is the already merged compact436 word from #4326 and its retained
+predecessor lineage. No new local rewrite is claimed. Its 658 original ARRAYs
+are 111 main SOURCEs, 111 main receivers and 436 arbitrary independent helpers.
+All helpers start in frame zero. The nine owned COPY lifetimes use one temporary
+slot, charged at rank seven, and return it safely; they add no productive input.
+
+Every MOVE changes an actual nested pair of rational QQ frames, with a proper
+paid rank. Every signed VALUE addition uses a common actual frame. The complete
+Boolean linear map on **all 658 original symbols** has own-input rows on the
+SOURCEs and helpers and the required SOURCE addition on the main receivers.
+The SOURCE entry/exit frames and receiver entry/exit frames are exactly those
+of the PR589 primitive used by the selected controller. The signed programs
+also compose with their actual inverses in both orders.
+
+The physical effect includes the endpoint frames. For a helper beginning at
+zero, own-input VALUE row and final frame U imply the actual job D_H on that
+arbitrary input, with
+
+    H = B^T (B B^T)^(-1) B.
+
+432 helpers have H=I8. Roles 462,463,466,468 end at frames506,506,508,509,
+respectively, and have actual rational rank-four idempotents H. They retain
+their own arbitrary inputs without cross terms. Their output credit is four,
+not eight. Their omitted full-rank completion would have projector I-H.
+
+The new local paid inventory is
+
+    {1:1319, 2:1342, 3:78, 4:39, 5:102, 6:10, 7:18}.
+
+It includes all nine rank-seven COPY charges. The 19,071 unit additions and
+all native source/frame conditions remain part of the unit. The fallback
+`local_word.py --completed` pays four additional rank-four calls; it is not
+the submitted partial-helper interface.
+
+## 2. Actual chronological p16 caller
+
+Retain the query-thinned p16,a7 scalar word, both physical boundaries, all15
+query-only stages, full idles and main port charts. At each of its18 full
+invocations, substitute the new local word. It accepts arbitrary current
+SOURCE/receiver data, so chronological composition permits correlations in
+those values. Each invocation has its own436 fresh arbitrary helper owners.
+The input count is
+
+    32*111 + 18*436 = 11,400.
+
+`global_word.py` emits446,400 chronological records. The separate interpreter
+checks all11,400 decoded RAW columns, current frame states, injective helper
+ownership, actual SOURCE versions and all1,827 owned lifetimes. The all-function
+argument combines those checks with the exact main group-polynomial identity,
+the local common-frame invariant and111 rational port-chart identities.
+The polynomial identity is checked coefficient by coefficient, not at a few
+character evaluations. The old LOCAL only changes the main receiver/shear
+interface; its replacement has the identical interface.
+
+The active-stage injection uses the old eight-space plus17 distinct fresh
+seven-spaces, within127 head coordinates. Thin stages allocate no helper bank.
+Each main coefficient has its stated rank, so the complete productive inventory
+is
+
+    T={4:72,8:7776,15:444,22:444,29:444,36:444,
+       43:444,50:444,57:444,64:222,127:222}.
+
+Every actual recursive call remains in
+
+    P={1:25407,2:24156,3:1404,4:702,5:1836,6:180,7:324,
+       14:444,21:444,28:444,35:444,42:444,49:444,56:444,63:222}.
+
+There are57,339 queries. The rank-one surplus is still753. This includes every
+copied thin query and whole idle. All11,400 originals of the new supplier are
+checked. Its54 fewer helper originals also lose their old output credit;
+they are not omitted columns of an unchanged11,454-input program.
+`audit_stream.py` independently reads the emitted records and
+rejects duplicate owners, fake rank-eight credit, missing CLEAR, a wrong nested
+residual and a SOURCE modification during a capture.
+
+## 3. Whole-job normalization and all column multiplicities
+
+Let Q4=diag(I4,0) on the local eight-space. Each special helper has explicit
+image and kernel bases. With S=[B^T|A^T], exact arithmetic checks
+
+    S Q4 S^(-1)=H,     S^(-1) H S=Q4.
+
+The reversed-tail16-dimensional covariance is also checked. In physical
+paired coordinates the matching matrix is diag(S,C8 S C8); its inverse
+conjugates the actual D_H into D_Q4. Tensoring by I_f proves the identity for
+every positive column multiplicity, using C_(8f)=C8 tensor Cf.
+
+Use the source delayed **whole-GL cover**, introduced after the finite word
+has been compiled. For a raw H output the physical input fiber route is
+g -> g S^(-1), with inverse return g -> g S. Both routes are paid. There are
+72 such output jobs and144 additional routes. This protocol uses fresh
+independent complete-cover rows and one common ancestor-only external weight;
+it does not require a free dense operation on an uncovered ARRAY. The main
+port routes and all SOURCE INDEX/lifetime conditions remain explicit. Only
+final rank-one leaves use the inherited weighted normalization.
+
+Together with7,548 inherited nonidentity main port routes, the raw route
+census is7,692. The finite full-group cover is algorithm-defined; it is not
+materialized by these tests. Its legality retains the selected source's
+whole-job and native implementation hypotheses.
+
+## 4. Strict price gain
+
+Write Phi(t)=sum_k T_k k^t - sum_r P_r r^t. One changed full invocation adds
+
+    Delta(t)=4 + 3^t + 9*4^t + 6^t + 7^t - 7*8^t.
+
+The positive terms split seven eights into
+
+    (1,7), (1,1,6), (1,3,4), (4,4), (4,4), (4,4), (4,4).
+
+Each group sums to eight. Strict subadditivity of x^t for0<t<1 gives
+Delta(t)>0, while Delta(1)=0. This compares the two verified circuits' prices.
+The complete p16 difference is18 Delta(t); main, query-only and idle terms
+are unchanged. Independent exact intervals put the new raw crossing in
+
+    (0.003475458064076428997514, 0.003475458064076428997515),
+
+against the old
+
+    (0.003464248972023961106948, 0.003464248972023961106949).
+
+The raw improvement is about0.323565%. It is a changed paid/target inventory.
+
+## 5. Charged completion, splitting and finite rank compiler
+
+Rank-four targets overlap the large idle ranks. Apply the source chronological
+tensor product and higher-moment theorem, retaining distinct physical owners,
+whole-job normalizers and every live owned bank. Complete every product target
+below18^1024 with a paid disjoint-axis job on that same owner, and split every
+original and added paid job at17^1024. New input credit is never created by
+completion.
+
+At C=0.00347545, t=1-C, u=1/5 and v=1024, use the complete bound
+
+    alpha^v + 2 gamma^v + 2 delta^v,
+    alpha=P_t/T_t,
+    gamma=P_(t+u)/(17^u T_t),
+    delta=11400*18/(17^C T_t).
+
+Two retained rational interval engines and a separately implemented positive-
+series engine verify alpha<1-2^-32, gamma<31/32 and delta<19/20.
+Both latter bounds raised to128 are below1/16, and
+
+    1/(1+1024*2^-32) + 4*2^-32 < 1.
+
+Thus both tails, including all target completion, fit. The new uniform saving
+before covering is C0=0.00347544. The source integer compiler inequalities,
+with17^64<18^63 and raw ambient127<256, give
+
+    H0=377227157504,
+    H1=262206447682297857,
+    N=2^H1.
+
+These are sufficient rank-compiler arithmetic bounds. They are not a claim
+that the predecessor's numerical precision, tape or stock thresholds apply
+unchanged to the new word.
+
+## 6. Cover, stock, finite constants and ordinary saving
+
+Set d=C0-C1=2^-18+10^-10, so
+
+    C1=8887360519/2560000000000.
+
+The same odd-prime rule q>2^26 N^5, avoiding the fixed denominators and rank
+witnesses, satisfies q>(256/d)N^5 and pays the full bad-class fallback. Fix
+all new finite word, route, height and owned-bank lists before varying the
+input chunk width e. The new raw cancellation-free integer prefix bound
+uses485 bits; the old21-bit bound is not inherited. The bound10^9 is a coarse
+raw-record allowance, not a final normalizer or machine-instruction count.
+After tensoring, completion, splitting and finite compilation, regenerate
+their fixed counts and heights by those source algorithms. In particular,
+tensor/normalizer multipliers do not remain bounded by10^9.
+
+These fixed constants multiply the source lower-order adapter, old-atom and
+ROW terms. The source proof permits this dependence on the fixed new
+(N,B,q,program). No route or normalizer is assigned zero cost. The borrowed
+reserve is the product W(w)^D S_old with
+
+    W(w)=B |GL_N(F_q)| q^(N^2(w-1)),
+    D >= floor(log_N ceil(e/w)).
+
+Its borrowed digits are O_(N,B,q)(w log e). Choose atom exponent
+rho=0.00346 and w=ceil(e^rho). With the retained old saving
+a_old=0.0000384599, the three root costs are
+
+    O(V[e^(1-rho)
+        +e^((1-rho)(1-C1)+rho(1-a_old))
+        +e^rho log e]).
+
+Consequently the exact ordinary saving is
+
+    a=(1-rho)C1+rho a_old
+     =17713901828029/5120000000000000.
+
+The exact positive gaps are
+
+    rho-a=1298171971/5120000000000000,
+    1-a-rho=5084570898171971/5120000000000000,
+    a-0.0034597=237828029/5120000000000000.
+
+They absorb the linear adapter and borrowed-stock logarithm for sufficiently
+large e with the newly generated finite constants. This admits the simpler
+ordinary BIT saving34597/10000000=0.0034597. The massive winning word and its
+full machine execution are not materialized. This is the same conditional
+native/compiler setting as the selected supplier, with the changed ABI and
+new finite constants stated explicitly. No larger complex or multiplication
+kappa follows from this BIT-only contribution.