Preserve reviewed smoothing, A4 sector and finite-L coefficient proofs

frontiers/gaussian_a4_elder_sector_20261017/EVERYWHERE_MARK_EXTENSION.md

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+# Exact observable extension at the full-jet Rice join
+
+This records an interpretation-sensitive but valid join in §§2,3,7,8 of the
+frozen A4 proposal, SHA256 abed6f94da080a33458b78f1e60ae92ad90083323af41d241de05939d4377ff0.
+It is reviewer reconstruction, not an author-source edit or a new hypothesis.
+
+Let S(x,f) be the local sector tests in §2, including criticality, the simple
+negative Hessian with the specified hard eigenvalue, the unique Q5>0 frame,
+the compact jet boxes, 0<r<r0 and the actual residual C7 cutoff. Do not include
+a global Morse/distinct-height indicator in S. These tests make sense and are
+Borel on all smooth fields; extend S by zero where their local denominators
+or simple eigenframe are unavailable.
+
+Define D_f(x) on all smooth fields as in §7: the supremum over rational c<f(x)
+at which x connects in {f>c} to a point with value greater than f(x), with empty
+supremum -infinity. Countable geodesic-ball chains and dense endpoint tests
+make this Borel. For Borel B subset (0,infinity), use the everywhere Borel mark
+
+    w_B(x,f)=S(x,f) 1{D_f(x) is finite}
+                    1{f(x)-D_f(x) in B}.
+
+The original field F is Morse with distinct critical values almost surely.
+On this event the sum of w_B over its critical points is exactly the actual
+sector-bar count N_A4(B): classical elder H0 identifies D with the death,
+and the deterministic local enclosure proves that every marked birth has
+an older point and hence is not essential. The count defined to be zero on
+the original null complement therefore has the same expectation as this
+extended critical-root sum. This is an equality of observables almost surely
+under the ORIGINAL field law, not a claim of pointwise equality on every
+conditional fiber.
+
+Apply the one-point all-Borel Rice identity to w_B. Its canonical full-jet
+kernel is f_z=omega+m(z). Since Hm(z)=0 and H omega=omega, its sector residual
+cutoff is exactly 1{||omega||C7<=R}; the actual mark has not been replaced by an
+independent auxiliary draw. At every target z_r in the sector, for EVERY
+cutoff residual, §7 proves D_(f_zr)(x)=b-Delta(r,omega) by the boundary barrier
+and outgoing ridge path. It follows directly that the conditional mark equals
+
+    1{||omega||C7<=R} 1{Delta(r,omega) in B}
+
+when the other displayed jet-sector tests are met. The coordinate changes then
+give (25). No conditional global Morse indicator has been deleted. One was
+never present in the Borel extension to which Rice was applied.
+
+At the limiting target z0, the contact is forced degenerate; no global Morse
+statement about that law is needed either. The coefficient limit concerns the
+explicit finite jet density, deterministic local lifetime and fixed residual
+cutoff, not an attempted exceptional-target extension of a genericity theorem.
+
+The source explicitly supplies the everywhere intrinsic threshold (§7), the
+original almost-sure actual-bar identification, and the all-Borel identity
+(§3), and says that its conditional trapping/density argument does not assume
+global Morse behavior. Thus this is a legitimate conventional inference from
+its written constructions. A sentence spelling out w_B before (25) would aid
+readers, but no required mathematical amendment or added hypothesis is needed.
+The reviewed interpretation is precisely this one. Inserting a global Morse
+indicator in the kernel and discarding it by unconditional genericity would
+be invalid and is not accepted.

frontiers/gaussian_a4_elder_sector_20261017/README.md

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+# Actual Gaussian compact A4 elder sector
+
+This is a source-author publication proposal carrying the full unchanged [proof](PROOF.md) and [actual integrated nonauthor review](REVIEW.md). Historical proposal/PENDING wording stays unchanged beside the actual later scoped acceptance. This index is not a new proof, review, source adoption, owner decision or scientific status register.
+
+One positive fixed compact regular marked Gaussian A4 ordinaryH0 elder sector on a fixed oriented flat two-torus; density C_A4 ell^(4/5)+O(ell), cumulative~(5/9)C_A4 t^(9/5), as ell,t decrease to0. These are the stated small-argument results and must not be treated as exact finite-scale power laws.
+
+Not whole-bar leading law/next whole-density correction; no negligibility of omitted discriminants, Q5=0/second-soft/large-jet/residual sectors, other dimensions/curvature/Hk/PPP, numerical or science. The full original proof and review control the exact hypotheses and limitations. All remaining original numerical/scientific obligations retain their evidence-specific dispositions; this packet does not close all five groups or the113retained failures.
+
+Exact GAD literal kernels/all-Borel one-point incidence plus shared finite-incidence supplement; named classical local/global Morse and differential facts; coauthor source is authorship, mark extension is reviewed reconstruction.
+
+The exact [everywhere Borel mark extension](EVERYWHERE_MARK_EXTENSION.md) is accepted reviewer reconstruction. The [ridge/frame derivation](coauthor/DERIVATION.md) is exposed COAUTHOR source, not nonauthor review or independent standalone theorem acceptance.
+
+The shared [source/dependency/exclusion index](../nongaussian_smoothing_admission_20261017/FOLLOWON_SOURCE_INDEX.json) records exact original prose identities, canonical source/review publications, immutable existing Math475/GAD and Main331/HF references, neutral historical-seal hashes and scope boundaries. Already landed GAD proofs/reviews and existing HF sources are referenced, not duplicated or replaced. Original sources and old seals were not edited.
+
+No Lean source, package, manifest, workflow, runtime, compiler target or execution receipt is registered here. Full books/PDFs/PNGs, private filesystem/fileID metadata, programs, failed attempts and recovery archives remain local; this is not a public ZIP, hermetic proof closure or portable execution package. New routing metadata has source-author provenance. The completed author/review sources here are source-exposed OpenAI, organizational/provider-distinct/blind/human credit0. Supplementary Claude input on the separate finite-L binding (moment/He/normalization only) is PENDING at Math193/6243336796; it supplies no completed review, adoption or independence credit. Existing C261 workflow/source history at this base is separate; this candidate changes none of it. Scientific effect NONE.