Three-colour native n+1 controller: bounded six/eight-bank feasibility

[Clear1nsight]

Status: FINDING / RELEASED. The exact declared no-IDLE colour-word policy is stopped. Priority revision7. No physical endpoint or new kappa is established.

All2187 six-bank and19683 eight-bank assignments were checked in the exact C4^4 group algebra, with zero diagonal endpoints of any degree. Runtime was5.770seconds locally. A code-disjoint independent certificate proves the entire declared domain at once: specialize the valid physical character Q=1,D_A=D_B=D_C=i. Every colour choice then gives the same operator. Dense integer real2n-by2n multiplication produces off-diagonal owner entry(0,1)=2i in BOTH frozen cases, with14/26 nonzero off-diagonal blocks. This character exists at every positive column width: choose central eigenweight0 and weight1 in each disjoint8f transverse block. Therefore neither frozen no-IDLE word can be diagonal for any colour assignment at any positive width. Recolouring or adding colours cannot repair this unchanged word's common-i sector.

Controls recover the existing source endpoint at D=1 and reject wrong backward sign, wrong Q inverse and an omitted scalar bridge. The native local binding was separately checked at all112 ports, three colours and two orientations (672 cases), including orientation-specific rawD0 translation corrections and exact phase. These checks validate the local offers; they do not alter the failed full endpoint. Artifacts are currently local research outputs, not a new public standalone package. This small stopping certificate is retained here as a coordination finding rather than a separate negative PR. No further assignment/seed/variant expansion was run.

Reopen only with a changed full word or legitimately priced IDLE/restoration work and a favourable complete invoice. Retaining the source IDLEs and pricing their actual Gaussian ranks is a different candidate; it first needs the source-bound diagonal target and full obligation census. Existing central n+1 identities and numerical margins cannot replace those gates.

A separate read/derive gate also closes the immediate retained-IDLE variant, before any new colour search or prospective claim. Actual source output pairs have odd prefix lengths3,5,7[,9], so under three colours their Gaussian ranks are9 or25. Each earlier output pair has an associated even-prefix IDLE of rank0 or16; a rank25 output requires that rank16 bill. Every occurrence acts on224 actual main fields, not one algebraic representative. Concavity gives 25^t-16^t<=9^t, t=1-C. Even granting ALL middle rank8 IDLEs free, the complete optimistic upper margin is

V_p=224*((p-1)*9^t+25^t)+(2p+1)*(H(t)-P(t)).

At C=.0030217758 this is[-3526.545252253298176,-3526.545252253298175] forp3 and[-5260.359842259440112,-5260.359842259440111] forp4. Its rank masses at C=0 are already-3360/-5056. The signed coefficients are negative only at ranks<=5 and positive only above5, so V_p/5^t increases witht and its negative value at1 proves negativity for everyC in[0,1]. Charging the mandatory middle IDLEs makes the bound still lower. The independent source/accounting review agrees. This is a fixed source-word/target-inventory stop, not a universal exclusion of changed multi-bank controllers. No additional assignment scan was performed.

Original prospective assignment (retained for provenance):

Determine whether the existing n+1 central controller can become an affordable native caller by reusing three transverse eight-axis blocks, with the source IDLE calls omitted. The primary promising invoice has eight independent112-field main banks, nine FULL native invocations and nine disjoint408-helper inventories. The proposed target is672H17+224H25+9H against9P, ambient25=max productive25. Its exact optimistic margin at C=.00339 is[26.543738781504416,26.543738781504417], before finite work, precision, compiler reserve or outer funding. At the developed comparison C=.0030217758 it is[58.275742103391407,58.275742103391408]. These are conditional target rewards, not an available budget or kappa result. Retained ordinaryBIT admission.0034 still constrains any outer claim.

The central n+1 word is PREDECESSOR WORK, not a new synthesis claim: source#4610 at20bb93cd990d2ff683e4d0fc56b87a42cd5d3296, research/kappa-rebuild/bit/COMPLEX-CONTROLLER-FACTOR.json, contains p3keep15 andp4keep63. Specializing every D_s=1 makes all source IDLEs identity and gives seven/nine F/B letters with integral scalar boundaries/bridges and diagonal three/four Q^-1, three/four Q. The independent exact Laurent-Q replay passes. The missing implication is a literal native transverse word, not the central identity.

Freeze exactly ONE source case per size. First screen all3^7=2187 assignments for six banks/seven FULL calls (p3keep15); then screen all3^9=19683 for eight banks/nine calls (p4keep63). Each D_s is replaced by one of three positive commuting E8 Gaussian words D_A,D_B,D_C, all orthogonal to the same central Q axis. Omit every source IDLE, keep its actual scalar boundaries, bridges, F/B orientations and main call roles. This defines a changed chronological word; its endpoint is not presumed to survive. No variant expansion, random seeds, larger bank count or additional address alphabet is part of this claim.

Use the exact group algebra of C4^4 generated by Q,D_A,D_B,D_C. Squares are retained as the actual monomial translations; none is set to identity. A literal product lifts at every positive tensor width. This algebra is faithful for these commuting physical operators once f>=3, because all four eigenvalues of Q_f and of each D8_f occur and the disjoint blocks realize all256 characters. Thus an exact nonzero coefficient is a meaningful all-width obstruction for this declared word, while a positive candidate must have zero off-diagonals and each diagonal exactly one signed group word. Required main census is(n-2)H17+2H25: odd central exponent and precisely two/three odd transverse exponents. Sums of tensor words and one-column matrix coincidences cannot be accepted as Gaussian jobs.

Native local binding is a declared source premise and separately checked gate. The source17-axis reference has A_ref N_raw B_ref=D_B E(Q); embed untouched C8 spectators to25. For a q-fixing monomial route R_c sending reference B8 to colour c, use entry B_ref R_c^-1 and exit R_c A_ref. Routes for A/B/C are the source orthogonal A/B swap, identity, and B/C swap. They act only on main owners: all408 helper outputs remain their independent autonomous jobs, and source centres remain owned zero temporaries. RawD0's exact translation correction is retained; it is not incorrectly commuted through a phase. Every emitted normalizer, scalar bridge, monomial pass, centre initialization/cleanup, helper return and required storage/precision bill must be priced in a positive witness. No independent centre target or repeated helper reward is allowed.

The six-bank invoice448H17+224H25+7H versus7P is a smaller feasibility probe: its margin at C4610 is[19.211803988589503,19.211803988589504], raw crossing(.003306670,.003306671). The eight-bank raw crossing(.003698329,.003698330) lies beyond the retainedBIT cap. A numerical source-style completion screen passes at.00339 with ambient25, but no full caller, prefix bound or compiler/outer transfer is supplied. Using globally disjoint transverse blocks would instead give ambient57/73 with maximum target25, outside the unchanged source compiler premise; the three-colour endpoint must be proved rather than inferred from a graph colouring.

Stop this exact word family if the complete-domain endpoint screen has no qualifying target or if its legitimate complete reward-minus-paid bound cannot fund the candidate C interval. A positive result only authorizes actual native word/raw inverse-transpose emission, ownership-disjoint target verification and a complete finite/compiler obligation map. No kappa claim until the caller, compiler reserve, BIT/outer transfer and independent endpoint/next-grid validation are funded.

Predecessors/overlap: #4666 supplies the binary/full-controller restrictions and related source central word;#4694/#4698 supply the coupled-centre distinction;#4712 supplies mandatory dirty-helper/query failures;#4920 closes the two-FULL main-only endpoint in its two stated scopes;#4888 closes fixed-word centre-chart grammars. Source IDLE alignment is not silently free: its existing first bridge mixes D0D1 history with an untransformed bank. This experiment checks a genuinely changed complete word instead of removing that obligation by declaration. #4908 offers a useful cyclic-recovery module but does not establish these main endpoints. Latest PR titles through#4921 and targeted colour/FULL searches were checked; no same assigned finite colour word was found. Public/unpublished overlap coverage remains bounded.

Prepared for Chris R/Clear1nsight with substantial OpenAI Codex assistance. Separate gpt-6.1-sol/high agents handle exact word algebra and independent source/accounting review. Credit Cinna Davis/1337isnot1337, Cao Yifan/intqwq, zxv, Jacob Alan Sussman and the retained native, scalar, Clifford, precision, compiler and outer lineage. Apache-2.0. No independent human review.

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