Conditional κ = 0.00328637739144124222748802 (+1.626e-7 over #6398): a third group-1091 anchor

research/mixed-projector-query-root/PROOF.md

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+# A mixed projector root for independent Gaussian outputs
+
+This construction changes the caller. It keeps the complete PR6349 native
+source. Four binary query suppliers, with 7, 2, 2 and 7 native leaves, enter
+a six-field root. The root has three thin queries and two individual
+whole idle jobs. It gives six independent Gaussian outputs. The child
+suppliers give another 28 independent main outputs.
+
+The conditional claim is kappa 0.00328467. The five inherited framework
+assumptions remain. This is not an unconditional multiplication theorem,
+a materialized uniform word or an independent scientific review.
+
+## 1. An ordinary normalization of the native query
+
+Write q = Q^-1 for the actual common central Gaussian. The native checker
+verifies Q^2 = Q^-2 = X with absolute phase +1. X is an ordinary translation
+and X^2 = I. These identities hold under the retained tensor implementation.
+Define
+
+```text
+lambda = (1+i)I/2 + (1-i)X/2,
+lambda^-1 = (1-i)I/2 + (1+i)X/2,
+S = lambda q.
+```
+
+Then lambda^2 = X, S^2 = I, and lambda^-1 S = q. All these operators commute
+with the fresh stage products D. The normalization is applied to the complete
+field, using its global translation X. Its scalar coefficients stay outside
+tensor powers. In general S is not the tensor power of the one-column S.
+
+The translation has disjoint two-address orbits. On one orbit lambda has
+matrix [[a,b],[b,a]], where a=(1+i)/2 and b=(1-i)/2. Four ordinary complex
+scalar operations implement it in place:
+
+```text
+x <- x - i*y
+x <- a*x
+y <- (1+i)*y
+y <- y - i*x.
+```
+
+Conjugate each coefficient to implement the inverse. No owned field, paid
+Gaussian job or productive input is added. The pair map is unitary; each
+displayed operation has growth at most two in the maximum norm. There are
+four operations per address pair. A loop over the existing translation
+pairs has linear data work. The ordinary-work bill below includes these
+operations and translations.
+
+A child supplies D U(q), where U(q)=I+q E01, on two arbitrary fields.
+Conjugation by diag(lambda,1), followed by U(-1), gives
+D U(S-1). A thin query has the same conversion and retains its paid H1
+charge. Thus the central-involution root below has an actual native
+implementation. Its output S is converted back to q by lambda^-1.
+
+## 2. The six-field root
+
+Use the shortest-query normal form from
+`research/five-call-projector-rigidity/PROOF.md`. For i=0,...,5 set u_i=e_i.
+Set u_6=(-1,1,-1,1,-1,1), and (v_i)_j=sign(i-j). Let R_i=u_i v_i.
+Each R_i is square-zero and has rank one. Direct exact multiplication gives
+
+```text
+(I-2R_6) ... (I-2R_0) = -I_6.
+```
+
+`ROOT.json` specifies an invertible rational basis F, four stage projectors
+P_i, and their native two-field bases C_i. Full sites are 0, 1, 5 and 6.
+Sites 2, 3 and 4 are thin. In F coordinates the four projector masks are
+
+| Site | Output coordinates in the stage | Rank | Native leaves | Extra whole idles |
+| --- | --- | ---: | ---: | --- |
+| 0 | 0,1 | 2 | 7 | none |
+| 1 | 0,1,5 | 3 | 2 | one H16 |
+| 5 | 0,2,4 | 3 | 2 | one H16 |
+| 6 | 0,2 | 2 | 7 | none |
+
+The exact checker verifies P_i^2=P_i, P_i R_i=R_i P_i=R_i and pairwise
+commutation of the four P_i. It also verifies that P_i commutes with the
+query prefix (I-2R_i)...(I-2R_0). These conditions permit each independent
+stage factor to move past its own query prefix. The two central characters
+then give the same diagonal stage product, with scalar factor +1 or -1.
+The checker also multiplies every independent stage coefficient directly.
+
+At a full site, C_i takes R_i to E01 and P_i to diag(I_r,0). Apply the child
+supplier to the first two coordinates. Apply D_i separately to each of the
+remaining r-2 coordinates. These are exactly the charged whole idle jobs.
+All basis changes are emitted as rational ADD, SCALE and SWAP operations.
+At a thin site, the corresponding C_i converts one paid query to R_i.
+
+Apply F at entry and F^-1 at exit. Apply lambda^-1 to all six root outputs.
+Their independent stage supports are
+
+```text
+ABCD, AB, CD, empty, C, B.
+```
+
+For child sizes 7,2,2,7, the six Gaussian ranks are 145,73,73,1,17,17.
+No root input is owned or initialized. The scalar basis maps are invertible.
+
+## 3. Binary children and full endpoint
+
+PR6348 supplies the unchanged four-field binary query extension. Each node
+combines two query suppliers, one paid thin query and two whole idle jobs.
+It returns another query supplier and two independent Gaussian outputs.
+Its separate Laurent checker is replayed here under its source hash.
+
+The four child trees have 18 native leaves and 14 binary nodes. Their node
+size counts are {2:8,3:2,4:2,7:2}. The root adds three thin queries.
+The complete caller has 34 main fields per port, 18 distinct native stages,
+17 thin queries, 14 paired whole idles and two individual whole idles.
+Its 570 literal instructions include all basis maps and 20 normalizations.
+
+The full replay uses exact coefficients in Q(i)[Q,D_1,...,D_18]/(Q^4-1).
+It checks all 1,156 input/output entries. Stage variables remain distinct.
+It checks the complete Gaussian endpoint, not only the S=+1 and S=-1 cases.
+Missing native calls, thin queries, paired idles, individual idles,
+normalizations and central translations are rejected.
+
+Each native call receives 562 independent helper inputs. These are separate
+from all main fields and from every other call's helpers. There are
+18*562+34*104=13,652 productive fields. Calls execute in sequence. The 29
+owned labels, 32 native lifetimes and peak 13 fields remain as in PR6349.
+Thin-query storage is reused between calls. Normalization adds no storage.
+
+The physical checker verifies all 7,488 native contacts, their absolute
+phases, every output Gaussian and every paired or individual idle. The
+fresh stage axes are orthonormal and fix the central port. The source word
+receives its complete arbitrary-input, dense-transpose and native replay.
+
+## 4. Complete price and outer transfer
+
+The complete invoice has 66,988 paid jobs and 13,652 productive jobs.
+Paid rank mass is 185,340; productive rank mass is 186,100. The surplus is
+760. The maximum target rank is 145 and maximum paid rank is 56. The
+emitted-event count agrees with a separate fixed-tree and root count.
+
+The strict raw crossing is in (0.00328468791927,0.00328468791928), between
+adjacent points on the 10^-14 grid. `RESULT.json` records both rational
+moment engines, completion, compiler and outer ledger. The claim is
+0.00328467; the tested
+saving is 0.003284675. The completion uses power 1024, cutoff 12, floor 13,
+offset 1/7, reserve 10^-12 and decay 31/32. The logarithm power is 1100.
+All local native, route and scalar costs remain charged. The new caller
+uses a conservative 32 real scalar units per field per instruction,
+including the normalization. Its complete ordinary bound is below 10^9.
+The retained admitted BIT saving is .0034. All 54 strict and four functional
+outer predicates must pass.
+
+The comparison is PR6353's published conditional .00315057. The present
+source is PR6349; it does not consume PR6353's output-contracted word.
+The source-relative gain is from the new caller, not new digits of an old
+crossing. PR6353 and PR6350 retain their separate construction credit.
+
+The basis search checked 241,886,080 bounded bases modulo 65521. It found
+695 support patterns. Floating searches selected the rank 2,3,3,2 pattern.
+The exact rational checker then verified the chosen basis and every stage
+coefficient. This finite search does not prove a global basis optimum.
+The new child-size search selected 7,2,2,7. Section 5 gives an exact ceiling
+for all binary children of this fixed root.
+
+## 5. A ceiling for every binary child size of this root
+
+This statement keeps the root and the PR6349 native source fixed. It permits
+all positive integer child sizes and all binary tree shapes. Let H(r)=r^t,
+t=1-kappa, and let L be the native paid-minus-helper moment per port.
+Let C(n) be the minimum binary-node cost for n leaves. Then
+
+```text
+C(1)=0,
+C(n)=min_{1 <= a < n} (C(a)+C(n-a)+2H(8n)+1-2H(8n+1)).
+```
+
+For a pair with total n, the best root contribution is
+
+```text
+G(n)=max_{1 <= b < n}
+     (H(8n+1)-n L-C(n-b)-C(b)+H(8b+1)-H(8b)).
+```
+
+For four positive children, with pair totals x and y, the complete margin
+per port is H(8(x+y)+1)+G(x)+G(y)-2. The two pairs contain all root outputs,
+three thin queries, both individual idles and every binary-node job.
+
+At kappa .00328469, `ceiling.py` uses rational outward intervals. It checks
+all 4,851 binary splits below 100 leaves and all 4,656 root pair choices with
+four through 99 leaves. Every margin upper bound is strictly negative.
+
+For larger totals f, concavity gives C(n) >= -(n-1) and
+H(8b+1)-H(8b) <= H(9)-H(8). Jensen's inequality gives the upper bound
+
+```text
+B(f)=H(8f+1)+2H(4f+1)-f(L-1)+2(H(9)-H(8))-6.
+```
+
+The checker proves B(100)<0 and B'(100)<0 with rational intervals. Since
+t is strictly between zero and one, B' decreases. Thus B(f)<0 for every
+real f>=100. No larger binary tree is omitted.
+
+The native deficit L has strictly positive derivative throughout
+0<=kappa<=.0034. The checker proves this with an interval lower bound above
+42. Each binary-node cost increases with kappa, while each positive H term
+and each increment H(r+1)-H(r) decreases. Thus every root margin decreases
+throughout the retained BIT admission range. Failure at .00328469 also
+excludes every larger saving in that range.
+
+This fixed root cannot reach .00328469 with the retained native source.
+The claim .00328467 is within .00000002 of that ceiling. This is a scoped
+ceiling, not a bound for other roots, source words or query interfaces.
+
+## Scope and open next questions
+
+The inherited assumptions are native_all_columns, owned_erasure, compiler,
+precision_tape and analytic_recovery. Finite operator and compiler checks
+do not discharge their full quantified statements. The large uniform word
+and rounded multiplier are not materialized. No global optimality or
+practical speedup is claimed. The selected frontier stays unchanged.
+
+An independent reviewer should first check the global-field normalization,
+the two charged rank-three projectors, and input independence.
+Then replay the exact source and complete invoice. A later improvement can
+change this root or combine it with a verified source, but must include
+every changed productive debit, ordinary operation and compiler cost.

research/mixed-projector-query-root/README.md

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+# Conditional kappa 0.00328467 from a mixed projector root
+
+This caller gives conditional **kappa 0.00328467**, an increase of
+**0.00013410 (about 4.26 percent)** over PR6353's .00315057. It keeps the
+PR6349 native source and combines four PR6348 binary query suppliers in a
+new six-field root. The selected frontier stays unchanged.
+
+The root permits paid stage projectors of ranks two and three. An
+ordinary translation-based normalization supplies its central involution
+and restores Gaussian outputs. The literal caller includes every
+normalization, basis change, native call, thin query and complementary idle.
+It has 18 native calls, 34 main fields per port and 66,988 paid jobs.
+
+An exact check bounds all binary child sizes for this fixed root below
+.00328469. The new claim is within .00000002 of that scoped ceiling. It is
+.1485 percent above the earlier local .00327980 root.
+
+Start with [PROOF.md](PROOF.md), [ROOT.json](ROOT.json), the literal
+[CONTROLLER.json](CONTROLLER.json) and [RESULT.json](RESULT.json).
+[VALIDATION.md](VALIDATION.md) states which checks actually ran.
+
+```sh
+python3.12 -B scripts/research_package.py --package research/mixed-projector-query-root
+python3.12 -I -B research/mixed-projector-query-root/verify.py
+python3.12 -I -O -B research/mixed-projector-query-root/verify.py
+```
+
+Expected marker: `PASS_MIXED_PROJECTOR_QUERY_ROOT_KAPPA_00328467`.
+The registered verifier uses only the Python standard library.
+
+Five inherited framework assumptions remain. The uniform word and rounded
+multiplier are not materialized. Author replay and hosted CI do not replace
+independent scientific review or formal-kernel acceptance. No global
+optimum or practical runtime improvement is claimed.
+
+The [final source composition](../mixed-root-contact-composition/README.md)
+gives 0.00328621476572836015146308 after the PR6353, PR6372 and PR6375 source changes. This package retains
+the separate root construction and its original source for review. The
+[headroom proof](../mixed-projector-root-headroom/PROOF.md) now classifies
+the full minimum-idle family. Historical validation receipts retain the
+publication status at the time they were made.
+
+## Open next questions
+
+- Independently check the normalization on the complete field, the four
+  projectors, all arbitrary inputs and the complete paid bill.
+- Test a different root or a verified improved source only with a new full
+  invoice, native binding, compiler and outer replay.
+
+[TASK.json](TASK.json) gives concrete first checks and remaining obligations.
+
+## Attribution and coordination
+
+New root, normalization, complete caller and verification: Ali Romman with
+substantial OpenAI Codex assistance; no subagents. The query normal form,
+PR6348 binary module and PR6349 source retain their separate credit.
+Cinna Davis supplies the native query and thin-query interface.
+DreamingOfClouds supplies PR6314's source compression with Claude Opus 5.5
+assistance. [NOTICE](NOTICE) retains the full native, helper, compiler, BIT
+and outer lineage. Apache-2.0.
+
+[Canonical investigation](https://github.com/CrocSwap/integer-mult-bounds/discussions/5600#discussioncomment-19209322).
+The source and owned-word searches in adjacent Discussions were excluded.