diff --git a/experiments/universality/README.md b/experiments/universality/README.md
index bf668e889..c9d904919 100644
--- a/experiments/universality/README.md
+++ b/experiments/universality/README.md
@@ -113,12 +113,41 @@ the exact source and the remaining coalescing-contact factorial scope.
 Independent copies do not establish a same-field spatial expanding-domain
 process or retire the existing regional additional-witness lane.
 
+The [spectral Gaussian admission theorem](spectral_gaussian_admission/PROOF.md)
+extends the actual H0 intensity and iid-copy process to stationary Gaussian
+spectra with a positive K5 block and fourth-order square-root summability.
+It independently proves the C4 interfaces, permits arbitrary outside support
+holes and supplies optional zero-mode removal and general finite carriers.
+A dyadic argument admits polynomial tails beyond that summability test.
+The coefficient remains the actual jet functional of each law; explicit
+spectral degeneracy examples exclude automatic family uniformity. Its
+[review](spectral_gaussian_admission/REVIEW.md) binds the original and current
+source verdicts and finite-regularity classical imports. This admits no
+covariance-matched non-Gaussian sampler or numerical approximation by itself.
+
+The [higher-homology fold theorem](higher_homology_fold/PROOF.md) now proves,
+under its stated same-field C4 continuation and complete critical-list
+hypotheses, that an isolated regular cubic fold creates an actual ordinary
+Hp interval with p=d-j-1, where j is the transverse Hessian index. Exact transverse
+splitting gives one adapted-flow connection; a fixed-annulus height cost
+excludes other connecting orbits, making the relative handle incidence a
+unit. Aligned regular-level isotopy transfers smooth handle incidence to
+the actual C4 filtration. Under the admitted SG contracts, every degree
+p=0,...,d-1 has lifetime density asymptotic to c_(q,p)*ell^(-1/3)
+as ell tends to zero, with law-specific inertia-cone coefficient and symmetry
+c_(q,p)=c_(q,d-1-p). The all-degree weighted
+cluster estimate yields a degree-marked iid-copy Poisson limit. Its
+[review](higher_homology_fold/REVIEW.md) binds the exact topological proof,
+classical imports and source exposure. This does not admit a numerical
+Hp algorithm, arbitrary coefficient laws/geometries or spatial process.
+
 These conventional analytic results discharge selected-bar counting and the
-leading near/far intensity composition for the stated ideal finite laws and the
-specific exact periodized Gaussian law. Further
+leading near/far intensity composition for the stated ideal finite laws and
+admitted stationary Gaussian spectral/carrier laws. Further
 conditional ranks, literal critical formal alignment, certified sampler and
-numerical/count coupling, genuinely blind law-by-law admission, higher homology
-and spatial expanding-domain process convergence remain separate obligations. The sampler
+numerical/count coupling, genuinely blind law-by-law admission, further field laws
+and geometries, and spatial expanding-domain process convergence remain separate
+obligations. The sampler
 and catalogue admissibility labels retain their original meaning.
 
 `models.py` defines the implemented finite-cutoff normalized sampler. Only
@@ -190,6 +219,6 @@ python3 -B -S experiments/universality/verify_published.py experiments/universal
 No held-out seeds have been consumed by this suite. Before a blind experiment,
 supply each model's actual law admission, coefficient/remainder/window,
 implementation coupling, numerical/count transfer, field-level precision and
-independent holdout custody described in the protocol. Further field laws,
-higher homology and geometry changes need their own actual analytic adapters
-and computational contracts; spatial expanding-domain process limits remain separate.
+independent holdout custody described in the protocol. Further field laws and geometry changes need their own actual analytic adapters;
+all admitted degrees still need their actual computational/count contracts.
+Spatial expanding-domain process limits remain separate.
diff --git a/experiments/universality/higher_homology_fold/PROOF.md b/experiments/universality/higher_homology_fold/PROOF.md
new file mode 100644
index 000000000..8e18a7900
--- /dev/null
+++ b/experiments/universality/higher_homology_fold/PROOF.md
@@ -0,0 +1,911 @@
+# Higher homology: attaching obstructions and isolated-fold unit incidence
+
+Incorporated on 8 October 2026 by OpenAI/Codex root, task
+01a11cbf-068d-7102-861e-75814e715c98, under GitHub307 claim6070584604
+and amendments6071035594/6071181708. Mathematical author:
+OpenAI/Codex universality_review. This source preserves Sections1-12
+of the frozen 44,258-byte/884-line preparation at SHA256
+`de1cad971e113f097a8e29ba82fb295d93b23f57aeda5b265e6f3c88cef47f2e`.
+That actual preparation has two separate fresh FULL nonauthor mathematical
+PASS verdicts. This changed introduction, source-binding section and README
+are a new construction, pending their own incorporated-source review.
+The initial worktree base is native spectral draft330 head
+`0c8ad2e7acbbf288b6ef2eeb4f96b6824ba8abaa`; no main landing is implied.
+All performers are source/route-exposed OpenAI/Codex AI agents;
+organizational independence0. No compiler, numerical, blind or scientific
+promotion is supplied by these conventional analytic results.
+
+Adjacent handle indices and a small height gap alone permit global
+attaching alternatives. An isolated regular cubic fold, with the
+same-field continuation and complete critical-list control stated below,
+supplies a unit incidence. That forces an actual short ordinary interval
+in the appropriate degree over every fixed coefficient field. Mixed
+transverse inertia changes the degree under these hypotheses.
+
+## 1. Conventions and the exact deterministic claim
+
+Let X be a closed oriented smooth d-manifold, d>=2, with a fixed
+background Riemannian metric. The probabilistic specialization below
+is the flat torus R^d/(L Z^d). Fix any coefficient field K. All
+homology and ordinary persistence use K. Essential intervals are
+excluded from the finite-lifetime population.
+
+For a field F, its decreasing superlevel filtration {F>=h} is the
+increasing sublevel filtration of g=-F at time tau=-h. An ordinary
+finite Hp interval born at field height b and dying at s<b has lifetime
+ell=b-s>0. The birth and death critical points are ordered in that
+filtration, regardless of whether either is a local extremum of F.
+
+Consider actual C4 fields F_r, r down to zero, converging to F_0
+globally in C4. In fixed physical coordinates (s,t) near x, where
+t is in R^(d-1), require:
+
+* x is the only degenerate critical point of F_0;
+* F_0(x)=b, grad F_0(x)=0, F_0,ss(x)=0,
+  F_0,st(x)=0 and F_0,sss(x)=12k, with k>0;
+* A_0=F_0,tt(x) is nonsingular, with j negative eigenvalues;
+* every other critical point of F_0 is Morse, and their critical
+  values are distinct and different from b;
+* F_r has the literal two pinned critical points
+  M_r=(-r/2,0), S_r=(r/2,0), with
+
+      F_r(M_r)=b, F_r(S_r)=b-k r^3.                     (1)
+
+These are assumptions on a same physical field continuation, not
+on unrelated arbitrary jets. In the Gaussian application they are
+proved on each fixed contact-law fiber almost surely by SG below.
+They imply the other critical set is finite: the regular contact
+is isolated, and every other root is isolated on a compact complement.
+
+**Deterministic unit-fold theorem.** Under these assumptions, for
+all sufficiently small r, F_r is globally Morse with distinct
+critical values and its actual ordinary finite persistence contains
+the interval
+
+      Hp: birth M_r, death S_r, lifetime k r^3,
+      p=d-j-1.                                         (2)
+
+Both endpoints have their unique birth/death roles in this interval.
+No other critical value lies between them. No H0 elder enclosure,
+negative-definite transverse block, preselected global attaching map,
+or simultaneous event for all contact targets is assumed.
+
+The threshold depends on this realization and these fixed marks.
+There is no uniform physical chart radius or deterministic convergence
+rate. This assertion concerns the actual filtration of F_r; an
+adapted gradientlike field is only a tool to calculate its handle
+incidence. It does not replace F_r by a different persistence law.
+
+## 2. Transverse inertia and the persistence degree
+
+At the two pins the axial Schur complement tends, after division
+by r, to -6k at M_r and +6k at S_r. The transverse blocks tend to
+A_0. Thus, for small r,
+
+      index_F(M_r)=j+1, index_F(S_r)=j.                 (3)
+
+Here index is the number of negative Hessian eigenvalues. Since
+Hess g=-Hess F, their sublevel handle degrees for g are
+
+      degree(M_r)=d-j-1=p,
+      degree(S_r)=d-j=p+1.                              (4)
+
+The earlier cell is M_r, at tau_a=-b; the later cell is S_r, at
+tau_b=-b+k r^3. In particular a negative-definite transverse A_0
+(j=d-1) gives p=0, but an index-j block with j<d-1 gives a higher
+homology interval. The saddle type must be the full inertia in (3).
+A determinant-sign predicate from d=2 is not a valid general-d
+replacement.
+
+One can obtain the signs in (3) directly from the reduced ridge
+constructed in Section 4; that proof also establishes the exact
+number of local roots. No conclusion about an actual interval follows
+from (3)-(4) alone. The unit attaching incidence is the decisive step.
+
+## 3. Finite-chain lemma, and what zero incidence really permits
+
+First consider a finite filtered chain complex over K. Its old
+subcomplex C^- is followed by one generator a of degree p and then
+one generator b' of degree p+1, with no intervening generator. Suppose
+
+      partial b'=u a+z, u!=0, z in C^-_p.              (5)
+
+The chain identity gives partial a=-u^(-1) partial z.
+Hence a+u^(-1)z is a cycle. It cannot have been a boundary before
+the later cell enters: any earlier boundary belongs to C^-, while
+this cycle has coefficient one on a. It gives a new p-dimensional
+homology direction. The next boundary kills that direction.
+
+Equivalently, reduce the boundary columns in filtration order.
+All earlier columns have zero a-row. The b'-column retains its
+nonzero a coefficient after reduction against them; a is its latest
+nonzero row. Thus its persistence pivot is a. This creates the exact
+finite interval [tau_a,tau_b) in degree p. The earlier cell cannot
+instead kill an old H_(p-1) class under (5). The conclusion is
+independent of old attaching maps, old cycles and characteristic:
+a signed incidence +/-1 is a unit in every field.
+
+For later use, there is a version requiring no global choice of
+a cellular boundary matrix. Suppose X^- is the sublevel set before
+the first handle, X^a after it and X^+ after the second, and
+
+      H_*(X^a,X^-)=K in degree p,
+      H_*(X^+,X^a)=K in degree p+1.                    (6)
+
+If the triple connecting homomorphism
+
+      delta:H_(p+1)(X^+,X^a)->H_p(X^a,X^-)
+
+is a unit, the triple exact sequence gives H_*(X^+,X^-)=0.
+The inclusion X^- -> X^+ is consequently a homology isomorphism.
+The pair sequence for X^a shows that H_p(X^-) injects into
+H_p(X^a) with quotient K: the next connecting map is zero because
+its composition with the surjective delta is zero. The map to
+H_p(X^+) kills one direction complementary to the old image and
+is an isomorphism on that old image. Therefore the newly born
+interval dies at the next handle. This also proves (2) from a
+relative unit incidence, without rearranging critical values by
+handle dimension or assuming a global Morse-Smale complex.
+
+There are real attaching alternatives when (5) is absent:
+
+* For p>=1, take an old p-cycle c, a new p-cycle a at time t_1,
+  a (p+1)-cell b' at t_2 just above t_1 with partial b'=c,
+  and a later (p+1)-cell e with partial e=a. The two Hp bars
+  are [t_0,t_2) and [t_1,t_3), not [t_1,t_2). This is realized
+  by a filtered CW complex: begin with a point and S^p, wedge
+  another S^p, and attach the two filling disks in the stated order.
+* For p=0, begin with old vertices c_0,c_1, add the new vertex a,
+  then an edge joining c_0,c_1, then a later edge joining a,c_0.
+  The middle edge's boundary has no a coefficient. The old
+  c_1 component dies first; the new a component has a long bar.
+  The oldest c_0 component is essential. This is an actual filtered
+  graph and respects the ordinary H0 augmentation constraint.
+* An earlier p-cell can instead kill an old H_(p-1) class and a
+  later (p+1)-cell create H_(p+1): set partial a=c_old and
+  partial b'=0. For p>=2 this is realized by filling an old
+  S^(p-1) and then wedging S^(p+1). For p=1 use an edge joining
+  two old components, then wedge S^2. Again the a coefficient
+  in partial b' is zero.
+
+These are rigorous filtered-chain/CW diagnostics. They are NOT
+realizations of the isolated generic fold in Section 1. Calling
+their arbitrary nearby cell times a geometric birth-death fold
+would erase exactly the unit-incidence obligation.
+
+Even a nonzero a coefficient is insufficient for *that exact pair*
+if a younger p-cell c enters between a and b': with
+partial b'=a+c, the reduction pivot is c. An isolated limiting
+contact whose other critical heights avoid b excludes this extra
+cell. Both unit incidence and no intervening critical event matter.
+
+## 4. Exact transverse splitting at finite regularity
+
+We prove the geometric input without assuming a smooth unfolding
+parameter lambda=r^2. The C4 continuation alone suffices.
+
+Shrink one fixed physical cylinder about x. Invertibility of A_0
+and the implicit function theorem give unique transverse critical
+ridges t=h_r(s), with h_r converging to h_0 in C3. At the pins
+h_r(+-r/2)=0. Put phi_r(s)=F_r(s,h_r(s)). Since
+
+      phi_r'(s)=F_r,s(s,h_r(s)),
+
+phi_r is C4, although the ridge need only be C3. At contact,
+h_0'(0)=-A_0^(-1)F_0,ts(0)=0. Differentiating the displayed
+identity gives phi_0'(0)=phi_0''(0)=0 and phi_0'''(0)=12k.
+The term containing h_0'' is multiplied by F_0,st=0.
+On a smaller fixed cylinder phi_r''' stays positive for small r.
+
+The function phi_r' is therefore strictly convex. Its pinned roots
+are s=-r/2 and s=r/2; they are its only roots and phi_r'<0
+between them. Its derivatives at the left/right root are respectively
+negative/positive. Every local full critical point lies on the ridge,
+so these are exactly the two local critical points. For F_0 the
+expansion phi_0'(s)=6k s^2+o(s^2) proves contact is isolated.
+
+Now write, exactly rather than as a Taylor truncation,
+
+      F_r(s,h_r(s)+t)=phi_r(s)+t^T B_r(s,t)t,
+      B_r(s,t)=integral_0^1 (1-v) F_r,tt(s,h_r(s)+v t)dv.  (7)
+
+The symmetric B_r is C2 and converges in C2 on the fixed cylinder.
+At (0,0) its limit is A_0/2. Diagonalize A_0 once. On a sufficiently
+small matrix neighborhood, signed LDL factorization followed by
+positive square roots of the absolute pivots gives a smooth matrix
+map S(B), with
+
+      B=S(B)^T J S(B),
+      J=diag(-I_j,+I_(d-1-j)).                          (8)
+
+The pivots are bounded away from zero on this neighborhood. Set
+w=S(B_r(s,t))t. Its derivative in t at zero is invertible. The
+inverse function theorem, after shrinking once more, supplies
+uniformly invertible C2 coordinates (s,w), converging in C2 with r,
+in which
+
+      F_r=phi_r(s)-|w_-|^2+|w_+|^2.                   (9)
+
+This is an exact transverse splitting. No concavity is asserted
+when w_+ is nonempty. Uniform invertibility here is for this fixed
+realization's nonzero A_0; it is not an inverse-curvature moment
+or a common radius over the Gaussian law.
+
+Outside this cylinder, use implicit continuations of the finitely
+many old Morse critical points. On their compact complement and
+outside a smaller contact neighborhood, grad F_0 has a positive
+floor. Global C4 convergence therefore excludes any additional
+root. The old critical values converge to their distinct heights,
+all separated from b. Together with (1) this proves F_r is globally
+Morse/distinct eventually and that no critical value lies between
+F_r(S_r) and F_r(M_r).
+
+## 5. A unique local connection and a uniform no-escape argument
+
+In the splitting coordinates define the ascent vector field for F_r
+
+      dot s=phi_r'(s), dot w_-=-2w_-, dot w_+=2w_+.      (10)
+
+It is descending for g_r=-F_r. Along the axial segment between
+the two roots, s decreases from S_r to M_r. The scalar autonomous
+equation has exactly one such trajectory up to time translation:
+separation of variables determines it, and the simple endpoint
+zeros put the limits at infinite times.
+
+Any trajectory confined to this chart and converging to S_r as
+time -> -infinity and M_r as time -> +infinity must have w=0.
+A nonzero w_- explodes at the first limit; a nonzero w_+ explodes
+at the second. Thus the connecting trajectory is unique locally.
+Along it the unstable tangent of S_r is spanned by the axial
+direction and the w_+ directions; the stable tangent of M_r is
+spanned by the axial direction and the w_- directions. They sum
+to the whole tangent space and intersect in the orbit direction.
+The connection is transverse. On a regular level between the
+critical heights, the corresponding attaching/belt spheres meet
+transversely in one point.
+
+Pull (10) back to physical coordinates. The uniformly invertible
+coordinate changes give constants c,C>0, independent of small r,
+such that the resulting X_r satisfies
+
+      dF_r(X_r)>=c |grad F_r|^2,
+      |X_r|<=C |grad F_r|.                              (11)
+
+Choose fixed concentric physical balls B_R and B_(2R) inside a
+common splitting neighborhood, with both pins in B_(R/2). Make
+X_r equal to this product field on B_(2R), and blend with a
+background ascent gradient outside, using a nonnegative partition.
+The estimates (11) survive the blend; the resulting field is C1,
+is zero precisely at the critical points and is strictly ascending
+elsewhere. On the compact annulus B_(2R) minus B_R, F_0 has no
+critical point. For all small r there is a common gamma>0 with
+|grad F_r|>=gamma there.
+
+For any portion of a trajectory traversing that annulus, parameterize
+by background spatial arclength l. Equations (11) yield
+
+      dF_r/dl=dF_r(X_r)/|X_r| >= (c/C)gamma.
+
+Traversing from radius R to 2R costs at least
+
+      Delta_*=(c/C)gamma R>0                            (12)
+
+in F height. A trajectory connecting S_r to M_r has total height
+increase k r^3. Once k r^3<Delta_*, it cannot exit B_(2R).
+It is therefore entirely in the product region and is the unique
+transverse trajectory in (10). This excludes additional global
+connecting paths that could otherwise cancel the local signed
+incidence. It is a gradient-trajectory energy argument, not an
+assertion that all superlevel paths are trapped in a concave cylinder.
+
+The adapted field can depend on the realization and r. Handle
+incidence and persistence are invariants of the original function,
+so proving uniqueness for this one admissible field suffices.
+No uniqueness claim for every physical metric is needed.
+
+## 6. The classical unit-incidence import and the C4 bridge
+
+The classical smooth handle fact used here is specific: if two
+consecutive critical points have sublevel degrees p and p+1,
+the coefficient of the first handle in the later relative boundary
+is the signed intersection number of the later attaching sphere
+with the earlier belt sphere on a separating regular level.
+One transverse point gives +/-1. In Milnor's primary account this
+is the relative boundary formula of Corollary 7.3. The unique-orbit
+geometric cancellation criterion is separately Theorem 5.4; it has
+no middle-dimension or simply-connected restriction. Its stronger
+algebraic cancellation variants must not be substituted for the
+unique-point criterion.
+
+We do not silently apply smooth handle theory to a C4 field. Here
+is the finite-regularity bridge. For a fixed sufficiently small r,
+F_r is Morse/distinct on the compact X. A sufficiently close smooth
+approximation in C4 has exactly the same critical branches and
+indices; the segment joining it to F_r stays in a C2 neighborhood
+with no new critical point or critical-value crossing. The finitely
+many values vary C1 along this segment. A smooth-in-height increasing
+reparametrization aligns them to the original values: use disjoint
+height bumps, equal to one near the moving values, whose amplitudes
+are their small value shifts. Taking the approximation close enough
+keeps the derivative positive.
+
+Let g_v be the resulting aligned path of negative fields, v in [0,1].
+It is C4 in space, C1 in v; g_0=-F_r, and g_1 is smooth in space.
+Its critical values and order are constant. Select one regular
+level tau_i in each gap, including levels below/above all critical
+values. On small disjoint collars of these finitely many levels the
+gradient has a uniform positive floor along the path. Set
+
+      Y_v=-sum_i chi_i(g_v) (partial_v g_v)
+                                grad g_v/|grad g_v|^2,  (13)
+
+where each chi_i is one near tau_i and supported in its collar.
+The ordinary time-dependent flow is an ambient diffeomorphism
+and carries every {g_0<=tau_i} onto {g_v<=tau_i}, simultaneously.
+Indeed the derivative of g_v along the flow is zero on the relevant
+boundary; that hypersurface is invariant in the extended (v,x)
+space, so no trajectory crosses it. The same ambient maps respect
+all the inclusions between these finitely many sublevel sets.
+Thus their finite persistence modules are isomorphic. They determine
+the whole barcode because each between-critical band is carried by
+the ordinary critical-free level flow. Aligned values retain the
+actual endpoints, not only their ordering.
+
+The final smooth approximation may be made arbitrarily close in
+C4 to F_r after alignment. This requires choosing its accuracy
+relative to this r's shrinking critical-value gap; no uniform
+smoothing tolerance is asserted. Choose the final C4 error also
+tending to zero as r tends to zero. Its ridge splitting, positive
+third derivative, complete critical list and annulus gradient floor
+then satisfy Sections 4-5. The two roots can move slightly from
+the literal pins, but remain the same continued pair, and the
+aligned gap is still k r^3. The product field is now smooth.
+It supplies exactly one transverse attaching/belt intersection.
+Apply the smooth relative unit-incidence formula, then the relative
+lemma (6), and transfer the interval through (13) to F_r.
+This proves the deterministic theorem (2).
+
+For each fixed r the product flow is a conventional pseudogradient:
+it strictly decreases g and Xg has a nondegenerate local maximum
+at each nondegenerate root. If a chosen smooth handle convention
+requires the usual linear field in Morse coordinates at a root,
+one may use the one-dimensional Morse coordinate for phi there
+and adjust its positive axial speed locally. The transverse coordinate
+planes and the unique connection are unchanged; the annulus field
+is unchanged. This removes a convention issue without assuming a
+global Morse-Smale perturbation or changing a critical value.
+
+The bridge uses ordinary smooth approximation, implicit function
+continuation and a C1 time-dependent flow on critical-free collars.
+It supplies the required finite filtered homology equivalence;
+it does not assert an all-orders smooth family in r or lambda.
+Antony's primary unique-trajectory theorem assumes a smooth
+one-parameter birth-death unfolding and smooth metric family.
+Its main theorem corroborates the genuine smooth-fold picture,
+but its mixed unfolding derivative is not an extra premise silently
+imposed on this C4 regression continuation.
+
+## 7. Borel selectors and unique finite-bar counting in degree p
+
+On the open C4 Morse/distinct locus, each field has finitely many
+critical points. At a nondegenerate critical point, crossing its
+g-level attaches one handle, of degree equal to index_g. Its
+relative homology is K in that degree. The long exact sequence
+shows that the event either births a class in that degree or kills
+a class one degree below, with rank one; no other degree changes.
+These statements for C4 follow from Section 6's smooth approximation
+and regular-level equivalence. In particular every finite Hp interval
+has exactly one degree-p birth critical point and one degree-(p+1)
+death critical point. Each critical point has at most one endpoint
+role. Essential classes have a birth and no finite death.
+
+For clarity, finite persistence does not need an unspecified global
+chain basis: sample the sublevel sets at one regular value in each
+critical gap. The finite-dimensional homology vector spaces and
+their inclusion maps form a finite persistence module over K.
+Successive basis elimination gives its interval decomposition.
+Because the values are distinct and each handle changes one rank,
+an interval's endpoints identify unique critical points. The same
+argument is equivalently ordinary filtered boundary reduction when
+a compatible handle-cell model is used.
+
+Define sigma_p(F,M,S)=1 if F is Morse/distinct and that ordered
+pair is precisely one of its finite Hp intervals; set it to zero
+otherwise. It includes the appropriate inertia condition. This is
+a Borel mark of the full C4 field and ordered pair, as follows.
+
+In a small C2 neighborhood of one good field, all critical branches
+and their order persist by the implicit function theorem and a
+compact gradient floor. Apply the aligned-level isotopy (13) along
+any sufficiently short segment in this neighborhood. Its sampled
+persistence module is constant. Hence the pairing of the labeled
+critical branches is locally constant. On a countable chart cover
+of the separable C4 good locus, each finite branch graph is continuous
+and its selected endpoint relation is fixed. Taking their countable
+union gives the Borel graph of all selected pairs. The complement
+of the open good locus is Borel and receives zero. This proves the
+selector, not merely an unproved assertion that a root count is
+measurable. It allows globally different attaching maps in different
+Morse chambers; a critical-value crossing may change their pairing.
+
+The unique finite counting assertion is consequently
+
+      N_(p,t)(F)=sum_(M!=S, grad F(M)=grad F(S)=0)
+          sigma_p(F,M,S) 1{0<F(M)-F(S)<=t}.             (14)
+
+It is jointly Borel in field and t on good charts, extended by zero
+on bad fields. It excludes all essential Hp intervals. On the
+connected torus finite intervals exist only in p=0,...,d-1; no
+degree-(d+1) handle is available to kill a top-dimensional birth.
+We will make no claim for extended, relative, zigzag or torsion-valued
+persistence, whose endpoint rules are different.
+
+## 8. Probabilistic interfaces: what transfers and what is newly proved
+
+For the rest let the actual law be SG's fixed centered stationary
+Gaussian field on X=R^d/(L Z^d), d>=2. Its symmetric spectral
+weights q_n>=0 sum to one. A simple sufficient verifier is full
+positivity on |n|infinity<=5 and
+
+      sum_n sqrt(q_n)(1+|2pi n/L|)^4<infinity.          (15)
+
+More generally SG's explicitly proved finite carrier ranks plus
+its actual C4 version with E||F||C4^(2d)<infinity suffice. They are
+the joint value/gradient rank at every distinct pair, the named
+contact rows plus all independent entries of A, and the contact
+rows plus one/two distinct off-contact first jets. SG proves the
+residual ranks on the conditioned contact kernel, including the
+Bayes-weighted tail. No arbitrary-jet interpolation, four-site K7
+rank, empirical covariance or covariance-only non-Gaussian premise
+is substituted. All constants are for this fixed law, d and L.
+
+For p in {0,...,d-1}, set j=d-p-1 and use the typed determinant
+weight
+
+      W_(p,r)=|det H_M det H_S|
+           1{index H_M=j+1, index H_S=j},
+      V_(p,r)=W_(p,r)/r^2.                              (16)
+
+The selector sigma_p and this weight supply the following five
+interfaces. Their names describe mathematical premises, not a new
+registry or implementation.
+
+| Interface | Exact source of its proof here |
+| --- | --- |
+| COUNT_p | Section 7's Borel unique finite interval pairs, combined with SG's all-Borel two-site finite-incidence formula and two-height disintegration. |
+| CONTACT_p | SG's literal observation rows, canonical full-field regression and determinant identity, with the inertia cone in (17) below. |
+| SELECT_p | The new deterministic unit-fold theorem on the actual contact-good coupling; Sections 9-10 prove the weighted convergence. |
+| ENVELOPE_p | W_(p,r) is bounded by the untyped determinant product, so SG's degree-2d norm and Gaussian target envelope apply. |
+| FAR_p | SG's original two-site observation covariance floor and conditional determinant moments, with sigma_p<=1. |
+
+COUNT does not inherit an H0 elder selector. SELECT does not inherit
+H0 concavity when A is mixed. Their missing higher-degree mechanisms
+are proved respectively by finite persistence/regular-level isotopy
+and by unit incidence. The other three analytic interfaces retain
+the exact same law and rows.
+
+## 9. Contact cone, same-field genericity and weighted selection
+
+SG's literal contact observations, in row order, are
+
+      U_r=((F(M)+F(S))/2,(F(S)-F(M))/r,
+          (F_u(S)-F_u(M))/r,
+          (6/r^2)[F_u(M)+F_u(S)-2(F(S)-F(M))/r],
+          ((F_vi(M)+F_vi(S))/2,(F_vi(S)-F_vi(M))/r)_i),
+      v_r=(b-k r^3/2,-k r^2,0,12k,(0,0)_i), k>0.
+
+They have 2d+2 components. Their contact limit pins F=b,
+grad F=0, the axial Hessian column V=0 and F_uuu=12k.
+The Gaussian contact density pi_r(v_r) converges to pi_0(v_0),
+and its covariance has a common floor in all frames at small r.
+The original two-site height/gradient density is exactly
+12r^(-(d+3)) pi_r(v_r).
+
+On the canonical full-field conditional coupling Q_r, SG proves
+F_r -> F_0 globally C4 and a bound
+||F_r||C4<=C||F||C4+C(|b|+k). Under Q_0 all other critical points
+are Morse, have distinct heights, and avoid b almost surely,
+for each fixed contact target/frame. The contact transverse block
+A_0 has a full Gaussian density on symmetric matrices. It is
+nonsingular almost surely. These are actual conditional statements,
+not a disintegration of unconditional genericity onto every null
+target or an uncountable simultaneous event.
+
+For the endpoint Hessians the exact block identity is
+
+      H_i=[r alpha_i, r beta_i^T; r beta_i,A_i],
+      det H_i/r=alpha_i det A_i-r beta_i^T adj(A_i)beta_i.
+
+Its validity for singular A_i follows because both sides are
+polynomials. Here alpha_M -> -6k, alpha_S -> 6k, beta_i is bounded
+and A_i -> A_0. The congruence diag(r^(-1/2),I) has limit
+diag(-6k,A_0) at M and diag(6k,A_0) at S. Thus if A_0 is nonsingular
+its inertia gives exactly (3) on its index-j cone. A different
+inertia eventually fails (16); at a singular limit the determinant
+product tends to zero irrespective of indicator oscillations.
+Consequently
+
+      V_(p,r) -> V_(p,0)
+         =36k^2(det A_0)^2 1{index A_0=j},
+      V_(p,r)<=C(1+||F_r||C3)^(2d).                    (17)
+
+Each index cone is open and nonempty. Full conditional Gaussian
+matrix support makes
+
+      z_(p,0)(b,k,u)=E_Q0 V_(p,0)
+
+strictly positive and finite at every fixed b,k>0,u. Domination
+gives E_Qr V_(p,r) -> z_(p,0). On V_(p,0)>0, the contact-good
+realization meets Section 1, so its actual pinned pair satisfies
+sigma_p(F_r,M_r,S_r)=1 for all sufficiently small r. On
+V_(p,0)=0 the product V_(p,r)(1-sigma_p) tends to zero without
+requiring any finite-r genericity on that weight-zero support.
+The same norm moment yields
+
+      E_Qr[V_(p,r)(1-sigma_p)]->0,
+      E_Qr[V_(p,r) sigma_p]->z_(p,0).                  (18)
+
+No normalized weighted law is needed outside a compact small-r
+band where its normalizer is positive. The unnormalized formulation
+(18) has no global inverse-normalizer or inverse-curvature premise.
+
+SG's target coercivity and Gaussian regression bounds give one
+all-mark envelope, also for (16):
+
+      B_(p,r)=12 pi_r(v_r) E_Qr[V_(p,r) sigma_p]
+        <=H(b,k)=C(1+|b|+k)^(2d) exp[-c(b^2+k^2)],
+      integral_Rx(0,infinity) k^(-2/3)H db dk<infinity.  (19)
+
+The constants can be enlarged over the finite set of p. They
+are not common over arbitrary spectral laws or dimensions.
+
+## 10. Actual Hp lifetime density and its physical coefficient
+
+Apply SG's all-Borel area identity with mark sigma_p and the
+inertia predicate in (16). A single absolute determinant product
+enters. Two-height disintegration gives the finite Borel density
+representative
+
+      nu_p(ell)=L^(-d) integral_(M!=S) integral_R
+        p_O(b,0,b-ell,0)
+        E_Q[|det H_M det H_S| sigma_p] db dM dS.         (20)
+
+The equality is an expected finite-interval counting identity,
+not a claim that all critical pairs are bars. The Gaussian kernel
+is the actual full-field regression version at the stated target;
+the selector is zero on its bad-field subset. There is no inherited
+continuity assertion for this discontinuous global mark.
+
+A common spatial translation carries all sublevel sets and their
+inclusions homeomorphically, with the same critical labels and heights.
+Thus sigma_p is translation invariant. Stationarity then removes the
+midpoint x from the conditional kernel; its integral is exactly L^d.
+This justifies cancellation of the per-volume factor in the near
+formulas below. Orthogonal transverse frame changes preserve the
+Hessian inertia, determinant square and observation-density product;
+local Borel frame charts suffice.
+
+For distances at least r_0, SG's original two-site covariance
+floor and conditional determinant moments give
+
+      0<=nu_p,far(ell)<=C, 0<ell<=1.                    (21)
+
+It also gives a finite bound on any compact positive-lifetime band.
+In the unique short-distance band, use midpoint x and displacement
+r u, with u oriented from birth point M to death point S. The
+exact flat Jacobian factors are
+
+      observations: 12 r^(-(d+3)),
+      spatial polar: r^(d-1) dr dx d sigma(u),
+      two heights: r^3 db dk,
+      determinants: r^2 E_Qr[V_(p,r) sigma_p].
+
+Their product is r B_(p,r) dr db dk dx d sigma(u).
+This radial order, rather than the cubic gap alone, fixes the
+exponent. Set ell=k r^3 in the specific density (20):
+
+      ell^(1/3) nu_p,near(ell)
+        =integral 1{k>=ell/r_0^3}
+          B_(p,(ell/k)^(1/3))(b,k,u)/(3k^(2/3))
+                                             db dk d sigma(u).
+
+Equations (18)-(19) give the full all-mark dominated limit.
+Equation (21) removes the far term, so
+
+      nu_p(ell) ~ c_(q,p) ell^(-1/3),
+      E N_(p,t)/L^d ~ (3/2)c_(q,p) t^(2/3),
+      c_(q,p)=4 integral k^(-2/3) pi_0(v_0)
+                             z_(p,0) db dk d sigma(u).  (22)
+
+The coefficient is finite and strictly positive for every
+p=0,...,d-1 in this admitted Gaussian spectral class. The density
+statement is pointwise for the explicitly specified finite Borel
+representative (20). Arbitrary null-set changes to a Radon-Nikodym
+version need not retain a pointwise asymptotic. Density continuity
+is not asserted. The cumulative statement follows by integration,
+not by differentiating a cumulative asymptotic. The missed near
+candidate population is only o(ell^(-1/3)); no O(1) near remainder
+is proved.
+
+Here is the exact law-specific Gaussian coefficient. Let
+G=grad F, V_u=(F_uu,F_uv_1,...,F_uv_(d-1)),
+A_u=Hess F restricted to u-perp, and
+tau_u^2=Var(F_uuu|G=0)>0. Define
+
+      D_(u,j)=E[(det A_u)^2 1{index A_u=j}|V_u=0].       (23)
+
+For the fixed spectral law, all these are its actual covariances
+and conditional Gaussian matrix law. Stationary parity makes
+(G,F_uuu) independent of (F,V_u,A_u). Integrating the birth b
+in (22) removes conditioning on F=b but retains V_u=0 and the
+correlations of V_u with A_u. It gives
+
+      pi_0(v_0) z_(p,0), integrated in b
+         =36k^2 p_G(0) p_Vu(0) phi_tau_u(12k) D_(u,j).
+
+The remaining k factor in (22) is
+
+      144 integral_0^infinity k^(4/3) phi_tau(12k)dk
+        =12^(-1/3) integral_0^infinity z^(4/3) phi_tau(z)dz
+        =Gamma(7/6) tau^(4/3)/(24^(1/3)sqrt(pi)).
+
+Thus the physical coefficient is
+
+      c_(q,p)=Gamma(7/6)/(24^(1/3)sqrt(pi))
+          *integral_S^(d-1) p_G(0) p_Vu(0)
+                    tau_u^(4/3) D_(u,d-p-1) d sigma(u).  (24)
+
+There is no factor one half for ordered birth-to-death directions.
+The numeral 24 is a change-of-variables constant, not the torus
+side L. The coefficient depends on q,d,L, and its transverse
+cone; covariance-independent amplitudes or infinite-cutoff uniformity
+are not inferred. Its leading value is independent of the chosen
+coefficient field K because the local incidence is +/-1. Global
+bar pairings and lower-order populations can depend on K.
+
+Conditioned only on V_u=0, A_u is centered. Its symmetry under
+A_u -> -A_u gives D_(u,j)=D_(u,d-1-j); hence
+
+      c_(q,p)=c_(q,d-1-p).                              (25)
+
+This is a Gaussian cone identity, not a claim that the actual
+entire barcode in those two degrees is identical. Summing over
+p replaces the cone in (23) by the full second determinant moment,
+since singular matrices have probability zero. In d=2 the two
+leading coefficients c_(q,0),c_(q,1) are equal; no finite H2
+lifetime population is created by these ordinary filtrations.
+
+## 11. A degree-marked iid-copy corollary, with its actual cluster input
+
+The preceding result admits the same narrow process regime as SG;
+this section explains the additional global step rather than inferring
+Poisson from the first moment. Define
+
+      N_t^all=sum_(p=0)^(d-1) N_(p,t)
+
+on the good locus and zero otherwise. The Borel construction in
+Section 7 applies simultaneously to this finite set of degrees.
+A single field has a finite positive lifetime list, so its process
+on bounded scaled-lifetime windows tends to empty almost surely;
+the mean also tends to zero at order t^(2/3).
+
+At a fixed contact-good realization with V_(p,0)>0, the unit-fold
+theorem supplies the actual Hp bar and consumes both new critical
+points' unique roles. Every other finite bar, in every degree,
+uses two of the finitely many old critical branches. Their limiting
+heights are distinct, so the minimum of all their nonzero possible
+endpoint gaps is positive. This is true whether or not their old
+pairing changes with r. There are no extra critical branches near
+b. At r=(ta/k)^(1/3), with fixed 0<a<=T, exactly the pinned bar
+has lifetime at most Tt eventually. Hence N_(Tt)^all=1 there.
+When V_(p,0)=0, the normalized determinant weight tends to zero
+regardless of the number or pairing of finite-r bars.
+
+Insert the bounded Borel mark 1{N_(Tt)^all>=2} into (20), and
+sum over p. Pointwise the weighted contact integrand tends to
+zero by the preceding endpoint exhaustion. Its bound is a finite
+multiple of
+
+      a^(-1/3) k^(-2/3) H(b,k), 0<a<=T,
+
+integrable over the complete marks, including a down to zero.
+Far anchored counts are O(t) by (21). Therefore
+
+      E[N_(Tt)^all 1{N_(Tt)^all>=2}]=o(t^(2/3)).         (26)
+
+This does not bound the second factorial moment. It deliberately
+inserts a bounded global mark into a once-counted endpoint sum,
+not an additional unbounded witness count or a four-site rank claim.
+
+Give each finite bar marks (p,a,b,k,M,u), with a=ell/t,
+k=ell/dist(M,S)^3 and ordered u from its birth point M to its
+death point S. In the near band these are unique physical
+displacements. A fixed Borel shortest-displacement convention
+outside that band is harmless because its contribution is O(t).
+The limiting one-bar mean measure is
+
+      d eta_(q,p)=4 a^(-1/3) k^(-2/3) pi_0(v_0)
+               z_(p,0)(b,k,u) da db dk dx d sigma(u).    (27)
+
+Its mass on a<=T is (3/2)L^d c_(q,p) T^(2/3); the lifetime
+marginal is L^d c_(q,p) a^(-1/3) da. Midpoint x and the actual
+birth position M=x-r u/2 have the same limiting mark. In higher
+degrees M is the birth critical point, not generally a maximum.
+
+For n independent whole copies of this one fixed actual law,
+choose t_n down to zero with n t_n^(2/3)->lambda in (0,infinity).
+For any compact-supported nonnegative process test f, the error
+between its single-copy expected sum of 1-exp(-f) and its
+single-copy expected 1-exp(-sum f) is bounded by the left side
+of (26). Independence and the elementary Laplace product give
+the Poisson random measure with intensity
+
+      lambda sum_(p=0)^(d-1) delta_p tensor eta_(q,p).    (28)
+
+The same all-mark bound proves convergence on finite windows a<=T.
+The resulting limiting degree-mark populations are independent
+Poisson restrictions of this marked measure; the finite-field
+degree populations are not asserted independent.
+This is an independent-whole-copy rare-event limit. It is not a
+spatial expanding-domain limit, a within-one-field Poisson law,
+a coalescing-contact factorial theorem or the separately owned
+regional additional-witness estimate. Those regimes do not follow
+from (26). No K7 factorial side result is inherited from K5 positivity.
+
+## 12. Failure boundaries and what the result does not universalize
+
+The reusable higher-degree mechanism is now explicit:
+
+1. Nonzero cubic axial derivative and nonsingular transverse block
+   give the adjacent degrees and exact local product connection.
+2. Same-field C4 convergence and off-contact genericity give the
+   complete critical list, old height gaps and annulus gradient floor.
+3. The small height gap plus that floor excludes other connecting
+   trajectories; the relative coefficient is a unit.
+4. No intervening critical value lets that unit create and kill
+   the new persistence direction at those exact two endpoints.
+5. The Borel once-count mark plus the actual probabilistic contact,
+   all-mark envelope and far interfaces give the lifetime density.
+
+Removing a premise has an identifiable obstruction, rather than
+an automatic replacement law. Arbitrary adjacent cells with zero
+incidence can exchange long bars as Section 3 shows. A second
+critical level entering the shrinking band can change the exact
+pivot even with a nonzero incidence. A singular transverse block
+is outside the regular-fold theorem. Uncontrolled global continuation
+can create extra endpoints or destroy the fixed-annulus floor.
+Other singularity orders/contact densities can change the radial
+weight and exponent. A non-Gaussian law needs its own actual
+contact density, conditional selection and envelope; its covariance
+does not supply them.
+
+There is therefore no H0 elder-rule generalization by terminology:
+in p>0 the algebraic pivot and unit relative boundary replace elder
+component enclosure. No arbitrary higher-homology fold, arbitrary
+manifold, non-Gaussian class, numerical sample, formal theorem or
+coefficient interval is admitted by the word 'generic'. The stated
+Gaussian spectral class is admitted by its named ranks, moments,
+conditional genericity and the new topological argument above.
+
+The qualitative unit-fold theorem works on the deterministic
+oriented closed manifold in Section 1, but the density/process
+specialization here is only the exact flat-torus Gaussian law.
+A different geometry still needs its actual all-Borel count,
+contact observations, volume Jacobian, envelope and far contracts.
+No new scientific promotion or model-status change is made.
+
+## 13. Exact operational sources, review extents and classical imports
+
+The following incorporated operational bindings were freshly hashed by
+root. SG's mathematical Sections1-11 are byte-identical to its original
+825-line preparation; the original source consumed by the Hk author and
+the new incorporated SG wrapper remain separately identified.
+
+| Symbol | Incorporated proof | Bytes / lines | SHA256 |
+| --- | --- | --- | --- |
+| SG | [spectral Gaussian admission](../spectral_gaussian_admission/PROOF.md) | 42478 / 826 | 6c0f34cb318d337b3b8c2c306f3b965f5e5a9d2fe38d5e001b8044de74b7f225 |
+| ST | [regular-fold structural theorem](../fold_structural_universality/PROOF.md) | 43927 / 804 | 3d952a906553d578876456539a8f0dcd64aedd45099e8394cbe9db99866ec531 |
+| IG | [periodized Gaussian H0](../periodized_gaussian_h0/PROOF.md) | 34629 / 637 | c8bd489ca3fbb3760f7b2d53e7385131ac5336c51ba983e00e4d540fe2f00aaa |
+| PP | [iid short-bar process](../iid_short_bar_process/PROOF.md) | 30935 / 618 | 320bc988c468bea4938bb777a2d05952c6ab8e0a384f33725b50de847aeec1eb |
+| ET | [finite elder transfer](../finite_elder_transfer/PROOF.md) | 23774 / 458 | c9e4fb09900888cd2b1bd96b49f85072b51c822b2503fe9f3d8891049bf66201 |
+
+The Hk mathematical author consumed SG's original42038B/825-line source,
+SHA25643e8180947b559e53f7afcf95ba61136f3fffcbe05a19e7a5953018fda27e294:
+a full prior saved-file read and fresh lines1-645/740-825 during the Hk
+construction. Optional diagnostics646-739 are not a new Hk premise.
+The current SG source has the same36656B Sections1-11 core, SHA256
+4cad43d3f80a8e47da886762d402c4981fe6834e278579b36673055539d6cf32;
+root checked actual complete original/current core and incorporation diff.
+No new whole current826-line SG read is attributed to the Hk author.
+ST/IG/PP received that author's full actual reads in the immediately
+preceding spectral derivation, including the separately filled truncated
+ST middle; their identities were freshly rechecked during Hk preparation.
+ET's incorporated bytes were rehashed; the author's full original ET read
+was prior, with no fresh full incorporated ET wrapper read claimed.
+
+The proof consumes SG Sections2-5 for actual C4 law/moments, named finite
+ranks and all-Borel incidence; Section6 for every fixed contact posterior's
+genericity and complete old critical list; Section7 for the radial and
+coefficient ledger and far bound; Section8 for the independently explained
+bounded-cluster/Laplace mechanism. SG's H0 selector is supplied here by
+Sections3-7's new relative unit-incidence/persistence argument. ST supplies
+the observation/determinant ledger, IG the preceding physical-law route,
+and PP the process-regime distinction. No K7/factorial theorem is consumed.
+
+Fresh FULL original-source mathematical reviews are separate:
+final_contract_review read all884 actual lines in four untruncated ranges,
+verified the exactde1cad identity, and found no must-fix; benchmark_formal_audit
+independently read all884 actual lines and found no must-fix. Both freshly
+rehashed the five original declared inputs. Final's prior separate full
+actual input audits include SG original/current; this pass does not claim
+five extra dependency reads. Benchmark's prior full input audits and ET
+authorship remain separately exposed; this pass does not claim five new
+whole dependency rereads. Root read original actual1-310/311-620/621-884
+untruncated. Changed incorporated bytes/readme await their own full review.
+
+Final's new Hk primary exposure was bounded parsed Milnor Section5 setup
+and Theorem5.4, Section7 setup/Corollary7.3, bounded Laudenbach convention
+and theorem clauses, and bounded Antony introduction/main theorem/normal
+form. Benchmark's new Hk exposure was bounded parsed Milnor Section5 and
+Section7 setup/Lemma7.2/Corollary7.3 and Laudenbach convention/extra-orbit
+statement; no new Antony reading is credited to that reviewer. Root's new
+bounded Milnor find results supplied actual Corollary7.3 and Theorem5.4
+passages, without full-book or raw-PDF credit. Neither root nor reviewers
+inherit the author's complete six-page Laudenbach read described below.
+
+Original author supplied C/window/S/ST/IG/PP/SG. Benchmark authored CG/ET
+and catalogue code. Final authored the separate fold-chart route and
+provided earlier carrier witnesses. Root supplied bounded prefile unit
+incidence, IGtheta and PP endpoint/Borel challenges. These are substantial
+same-provider source and route exposures, with organizational independence0.
+The [review record](REVIEW.md) retains exact original/current verdicts.
+
+The following author-side primary-exposure account and import ledger is
+preserved from the original preparation. Its final request for exact-file
+review is historical; the two new original-source verdicts are recorded
+above and separately from pending incorporated-source review.
+
+Fresh primary readings for the new topological step are explicit:
+
+* [Milnor, Lectures on the h-Cobordism Theorem](https://webhomes.maths.ed.ac.uk/~v1ranick/surgery/hcobord.pdf),
+  1965 primary book scan: bounded parsed-PDF readings of the
+  Section 5 setup/first cancellation theorem (printed pages 45-48,
+  PDF pages 49-52), and Section 7's relative boundary formula and
+  handle homology construction (printed pages 85-90, PDF pages
+  89-94). Corollary 7.3 is the incidence import actually used.
+  Other cancellation hypotheses elsewhere in that book are not
+  imported. A bounded earlier opening also included its equal-N
+  Morse perturbation lemma, but the new topology proof does not
+  depend on a new high-dimensional Sard import.
+* [Laudenbach, A proof of Morse's theorem about cancellation](https://arxiv.org/pdf/1307.2545),
+  arXiv:1307.2545v1: the complete parsed six-page/291-line primary
+  text was read. Its unique transverse orbit criterion and
+  pseudogradient convention corroborate the geometric mechanism.
+  Its additional global orbit condition is not assumed implicitly
+  as a replacement for the direct relative incidence calculation.
+* [Antony, Gradient Flow Line Near Birth-Death Critical Points](https://arxiv.org/pdf/1706.07746),
+  arXiv:1706.07746v3: bounded actual parsed reading of the main
+  theorem, normal-form discussion and introductory outline only,
+  not the full 39-page proof. Its smooth unfolding and metric
+  assumptions are stated in Section 6 and are not claimed satisfied
+  merely by the C4 contact coupling.
+
+These were primary web text reads, not downloaded PDF byte-custody
+verifications. No raw-PDF SHA or full-book/full-Antony read is
+invented. No failed primary web operation was retried. The new construction
+itself proves the adapted-flow no-escape step and the C4-to-smooth
+filtration equivalence. Classical imports remaining are smooth
+Morse handle attachment and its relative incidence formula,
+ordinary smooth approximation/implicit continuation/ODE flow,
+finite persistence interval decomposition, and SG's explicitly
+declared locally Lipschitz area formula/Gaussian regression.
+All integrability and limiting uses are supported by the actual
+SG moments, target coercivity and domains in Sections 8-11.
+
+The strongest proved preparation claim is the unit-fold mechanism,
+and, under the actual SG spectral assumptions, the resulting
+ordinary Hp density and independent-copy degree-marked process.
+An exact-file nonauthor attack must still assess these new
+topological imports/bridges and composition. No review, compiler,
+organizational independence or scientific credit is manufactured
+by the author-side derivation or saved-byte readback.
diff --git a/experiments/universality/higher_homology_fold/REVIEW.md b/experiments/universality/higher_homology_fold/REVIEW.md
new file mode 100644
index 000000000..c5910fc14
--- /dev/null
+++ b/experiments/universality/higher_homology_fold/REVIEW.md
@@ -0,0 +1,143 @@
+# Higher-homology fold: source-bound review record
+
+Construction snapshot,8October2026. Mathematical author OpenAI/Codex
+universality_review; wrapper/integrator root task
+01a11cbf-068d-7102-861e-75814e715c98, claim6070584604 with
+amendments6071035594/6071181708, lease00:30UTC9October.
+
+Current [proof](PROOF.md): 46174 UTF-8 bytes/911 lines,
+SHA25687f43e6ed94c5c63c385ecf177f68212afa5db54b24062fa008f081f47a78999.
+Preserved Sections1-12: 37220B,
+SHA256c770d4f7a28bff3d9f59a5baa004f052fb6db2c9e27a11799db0e7420d124471.
+Original44258B/884-line source SHA256
+de1cad971e113f097a8e29ba82fb295d93b23f57aeda5b265e6f3c88cef47f2e
+has two distinct fresh FULL saved-file mathematical PASS verdicts,
+final_contract_review and benchmark_formal_audit, no must-fix.
+Root fully read the actual original884 lines. Current full root read,
+current whole-source mathematical review and new README metadata review
+are PENDING at this construction snapshot. Original verdicts do not
+certify these new incorporated bindings.
+
+Actual isolated-fold unit relative incidence supplies the finite Hp bar
+in degree p=d-j-1. SG named ranks, C4 moments, canonical contact and
+all-mark/far bounds then supply actual degree-specific density,
+law-specific inertia-cone coefficient and iid whole-copy marked process.
+The all-degree weighted cluster estimate is proved; no second factorial
+or same-field spatial claim is inherited. Ordinary homology over every
+fixed field of coefficients; no torsion/integral barcode claim.
+
+All agents are OpenAI/Codex AI performers, prior source/route exposed,
+organizational independence0. Full and bounded dependency/primary reading
+extents are separately disclosed in PROOF Section13. Critical Lean,
+computational/count coupling, numerical enclosures, blind50 custody,
+spatial/expanding-domain process and scientific promotion remain open.
+Compiler/tests/sampler/seeds:NOT_RUN. Full landmark goal ACTIVE.
+## Completed incorporated-source review and companion amendment
+
+The construction snapshot above is preserved verbatim: 1,908 bytes /
+34 lines, SHA256 `e1c109981a613c3e96283d8fe8f06200d344dce5f46aa0c3991c30a59d362769`.
+Its PENDING states describe that historical construction.
+
+**final_contract_review: fresh FULL current-proof mathematical PASS.**
+The reviewer read all 911 actual saved lines in four untruncated ranges,
+verified 46,174 bytes / SHA256
+`87f43e6ed94c5c63c385ecf177f68212afa5db54b24062fa008f081f47a78999`,
+read the complete original884-to-current911 diff and independently matched
+the 37,220-byte Sections1-12 core. No mathematical must-fix emerged.
+This is a new verdict, separate from the original884-line proof reviews.
+
+The fresh attack covered C4 splitting, the full critical list, the unique
+product-flow connection, fixed-annulus height exclusion, relative unit
+incidence, aligned smoothing and regular-level filtration equivalence,
+Borel once-counting, inertia-cone constants and the all-degree cluster
+estimate. Five current operational dependency hashes, including SG826,
+match the table. Original/current SG Sections1-11 also match. These are
+fresh hashes and core comparisons, with prior whole dependency audits
+separately exposed; no five new whole dependency rereads are claimed.
+No new primary-source fetch occurred in this incorporation pass.
+
+The reviewer fully read the construction REVIEW34/e1c109 and
+README222/9eaf2a, its complete predecessor diff, and checked all33 local
+links. Benchmark independently performed a full construction REVIEW34 /
+README222 metadata read, complete source and README diffs, both preserved
+core comparisons, five dependency hash checks and all33 links; this was a
+bounded metadata audit, not a new whole911-line mathematical review.
+
+**Both companion audits: README wording AMEND.** The 14,761-byte /
+222-line README at SHA256
+`9eaf2a0c036fba117fb1ea894b03a889fc92f56fd4542a522375f6c113fd4095`
+said each degree "has density c_(q,p)*ell^(-1/3)" without an asymptotic
+qualifier. Read literally, this overstated the actual small-lifetime
+asymptotic as an exact global density. The proof itself correctly states
+(22)'s asymptotic for its canonical Borel representative.
+
+Root preserved the actual construction README in an ordinary external
+snapshot and corrected only that sentence to say lifetime density is
+asymptotic to the stated expression as ell tends to zero. The corrected
+README is 14,805 bytes / 223 lines, SHA256
+`303a83b01c05023d2cbf9728de4e40851c63da3afb8056835141c74786df8311`.
+The proof remains unchanged. The corrected README and this appended
+review record await their own exact-source metadata audit at this stage;
+the earlier AMEND is retained, not overwritten as a PASS.
+
+Root completed the actual current911-line read in untruncated1-455 /
+456-911 ranges, the complete actual incorporation diff, and full actual
+construction REVIEW34 / README222 and its diff. The first combined tool
+output truncated its diff/metadata portions; those portions were repeated
+and fully read in a separate untruncated result. Five current inputs were
+freshly rehashed. No truncated portion is credited as a complete read.
+
+Same-provider source/route exposure and organizational independence0
+remain explicit. Ordinary fixed-field homology, exact SG law/contracts,
+no density continuity or near O(1) remainder, no numerical Hp coupling,
+critical Lean or blind closure, no same-field spatial process and no
+scientific promotion retain their stated boundaries. Goal ACTIVE.
+
+## Retained-author documentary clarification, 18 October 2026
+
+This appended record is source-author preparation by Root task
+01a11cbf-068d-7102-861e-75814e715c98 with the OpenAI/Codex local
+SG/HF source-composition preparer. Every byte of the original 94-line review
+above is preserved, SHA256
+`4a24a4fe3e6bdd6e380ce1436f0b07eed587a03e01dc3ad6c59c59c4fbfba5fb`.
+The historical pending words and completed proof-review records retain their
+actual dates and scopes; this is not an independent review of the append.
+
+### D1: actual body-only correction
+
+[Claude S's actual review6139276556](https://github.com/d6g8k5htny-coder/main/pull/331#issuecomment-6139276556)
+reported PASS_MATHEMATICS_SCOPED for the deterministic unit-fold theorem and
+its SG specialization, together with documentary D1. Its
+[actual amendment route6139465647](https://github.com/d6g8k5htny-coder/main/pull/331#issuecomment-6139465647)
+permits correcting the unsupported completed-audit claim in the PR body to
+pending, while preserving the proof and historical review bytes.
+Root actually made that body-only correction; the saved API `updated_at`
+is `2026-10-18T00:32:32Z`. Root verified the full native replacement-body
+readback at `2026-10-18T00:33:31Z`; source head
+`89657c7c48289378949dc59e7fbe6dc1a55d9c62` stayed unchanged.
+[Actual correction delivery6243772927](https://github.com/d6g8k5htny-coder/main/pull/331#issuecomment-6243772927)
+and [finite author pickup6243752128](https://github.com/d6g8k5htny-coder/main/pull/330#issuecomment-6243752128)
+identify this action. The full old body remains in custody. No previous
+metadata audit is invented, and no mathematical verdict is enlarged.
+
+### Optional file-attribution note: SG generalizations and refusals
+
+The diagnostics-reading sentence in PROOF Section 13 concerns the source
+`spectral_gaussian_admission/PROOF.md` (SG), not section numbering in this
+higher-homology proof. The relevant SG portion includes optional
+generalizations and the opening of its refusals/scope discussion. This
+no-range attribution note follows the corrected
+[Claude S note6154541570](https://github.com/d6g8k5htny-coder/main/pull/331#issuecomment-6154541570)
+and [Muse Spark route read6154559578](https://github.com/d6g8k5htny-coder/main/pull/331#issuecomment-6154559578).
+It changes neither proof bytes nor the author's actual prior and bounded SG
+reading extent, and does not claim any additional reading or premise.
+The earlier wrong-file AMEND and its correction stay in their native history.
+
+A distinct final metadata/composition review must bind this actual frozen
+append and combined README externally; it is not self-certified here. The
+ordinary fixed-coefficient-field unit-fold theorem retains its complete
+same-field C4 continuation and critical-list hypotheses. Its probabilistic
+claim retains the exact fixed stationary Gaussian SG contracts and iid
+whole-copy regime. No numerical Hp algorithm, general manifold/nonGaussian
+probability adapter, same-field spatial process, literal critical Lean or
+scientific promotion is supplied by this source composition.
diff --git a/experiments/universality/spectral_gaussian_admission/PROOF.md b/experiments/universality/spectral_gaussian_admission/PROOF.md
new file mode 100644
index 000000000..72c896474
--- /dev/null
+++ b/experiments/universality/spectral_gaussian_admission/PROOF.md
@@ -0,0 +1,826 @@
+# Spectral Gaussian admission for regular-fold H0 lifetimes and iid-copy processes
+
+Incorporated mathematical source,8October2026. Original author OpenAI/Codex
+universality_review; repository wrapper and binding author root,
+task01a11cbf-068d-7102-861e-75814e715c98, exactclaim6070584604 amended6071035594.
+This records a fixed-law spectral/carrier admission theorem with named
+classical imports. Its actual C4 regularity is weaker than smooth samples;
+no existing smooth premise is silently declared met.
+
+Original42038B/825-line source SHA43e8180947b559e53f7afcf95ba61136f3fffcbe05a19e7a5953018fda27e294
+received two separate fresh full nonauthor mathematical PASS verdicts.
+The preserved mathematical Sections1-11 are unchanged. This repository
+source has its own identity and fresh review in REVIEW.md; original-file
+verdicts do not certify later wrapper or dependency bytes.
+All performers are AI agents with substantial same-provider source/route
+exposure, organizational independence0. No numerical draw,blind seed,
+compiler,formal closure or scientific promotion is inferred.
+
+## 1. Admission theorem and a concrete spectral verifier
+
+Fix an integer d>=2 and L>0. Let X=R^d/(L Z^d), omega=2pi/L, and
+let a declared real stationary Gaussian law have spectral weights
+
+    q_n=q_(-n)>=0, sum_(n in Z^d) q_n=1.                  (1)
+
+The field is ensemble centered. If q_0>0, it has its independent random
+constant mode; if q_0=0, that mode is absent. In either case its point
+variance is one. Its physical covariance is
+
+    K(z)=sum_n q_n exp(i omega n.z),                      (2)
+
+not a covariance inferred from a name, approximate grid or truncated
+experiment. The coefficient law is Gaussian; specifying this covariance
+alone for a non-Gaussian field would not specify the law used below.
+
+Here is a simple sufficient verifier:
+
+    (P5) q_n>0 for every |n|infinity<=5;
+    (A4) sum_n sqrt(q_n)(1+|omega n|)^4<infinity.          (3)
+
+This is full positivity on the fixed box K5, including n=0. An optional
+extension replaces it by (P5^0): positivity only for 0<|n|infinity<=5,
+with q_0 unrestricted; Section 3 proves the additional zero-mean witness
+step. The main verifier (P5) does not need that extension. Modes outside the box
+may have arbitrary holes or may all vanish. No positive lower bound on
+all frequencies, isotropy, analytic covariance or Gaussian-shaped
+spectral decay is required. For each fixed law there is a positive
+minimum among its finitely many required weights. A common minimum
+over a family is a further condition, not part of (3).
+
+**Spectral admission theorem.** Under (1) and (3), the actual field
+has a C4 version and supplies the finite-regularity versions of COUNT,
+CONTACT, SELECT, ENVELOPE and FAR proved below. The expected ordinary
+finite superlevel H0 lifetime measure per volume has a specified finite
+nonnegative Borel density representative satisfying
+
+    nu_bar(ell) ~ c_q ell^(-1/3),
+    E N_t/L^d ~ (3/2)c_q t^(2/3), ell,t down to zero,       (4)
+
+where 0<c_q<infinity is the actual law-specific jet functional in (27).
+The candidate-pair coefficient is the same. Bars are selected by the
+global ordinary elder rule; the essential class is excluded.
+
+For independent whole copies of this one fixed field law, retain the
+marks lifetime a=ell/t, birth b, physical gap k=ell/dist(M,S)^3,
+birth location M and ordered direction from M to its killing saddle S.
+If n t_n^(2/3)->lambda in (0,infinity), their superposition converges
+to the Poisson random measure with intensity lambda eta_q, where
+
+    d eta_q=4 a^(-1/3) k^(-2/3) pi_0(v_0) z_0(b,k,u)
+                         da db dk dx d sigma(u).          (5)
+
+After integrating birth/gap/direction, its whole-torus lifetime
+intensity is lambda L^d c_q a^(-1/3) da. A single fixed field's process
+instead tends to empty on bounded rescaled-lifetime windows.
+
+This is a C4 theorem, not a statement that (A4) makes samples C-infinity.
+The existing structural and physical-law sources literally assume
+smooth samples and prove all-Cq norm bounds. Sections 2-8 independently
+check the finite regularity actually needed and prove the C4 variant;
+they do not declare that an A4 field meets those stronger literal words.
+No selected-density continuity, O(1) full near remainder, quantitative
+cutoff, Hk theorem, spatial expanding-domain limit, coalescing-contact
+estimate or global second-factorial theorem follows.
+
+The theorem remains true under the weaker alternative (R4): the actual
+Gaussian law has a C4 version with E||F||C4^(2d)<infinity. This version
+must be supplied rather than asserted from finite pointwise moments.
+Section 10 gives a spectral polynomial-tail family for which (R4) is
+proved directly while (A4) fails. Neither (P5) nor (A4) is claimed
+necessary or a minimal characterization.
+
+## 2. Gaussian representation, C4 moments and the exact regularity ledger
+
+Choose one representative from each nonzero pair {n,-n}, omitting
+zero-weight terms. Independent standard real Gaussian coordinates give
+
+    F(z)=sqrt(q_0) C_0+sum_pairs sqrt(2q_n)
+              [C_n cos(omega n.z)+S_n sin(omega n.z)].     (6)
+
+The constant term is omitted when q_0=0. Every real Fourier polynomial
+with support in a positive-weight finite set is a coefficient direction
+of this actual law; its coordinate scaling by sqrt(q_n) is invertible.
+The block retains the original weights. It is never renormalized to
+variance one after removal of the tail.
+
+Under (A4), a positive weighted sum of |C_n|+|S_n| bounds ||F||C4.
+Its expectation is finite, hence it is finite almost surely. Minkowski
+gives its Lp norm at most a constant times the sum in (A4), for every
+finite p>=1. Thus the series and derivatives through order four
+converge absolutely uniformly, with
+
+    F in C4 almost surely,
+    E||F||C4^p<infinity for every finite p>0.              (7)
+
+All derivative jets used below are the actual derivatives of this
+version. In particular A4 also implies sum_n q_n(1+|omega n|)^8<infinity:
+write a_n=sqrt(q_n)(1+|omega n|)^4 and use sum a_n^2<=(sum a_n)^2.
+This explains why mixed derivative covariances needed in C4 regression
+exist, without asserting pathwise eighth derivatives. Gaussian
+uniqueness on a countable dense evaluation set identifies the Borel
+law on C4. A theoretical representation is not an empirical draw.
+
+Under (R4), the same argument below starts with its actual C4 version.
+Its finite positive Fourier block, obtained by the continuous Fourier
+coefficient maps, is independent of its C4 Gaussian tail. All uses of
+field-norm moments need at most order 2d; all uses of Bochner covariance
+need order two, which is included for d>=2. No Fernique theorem is
+needed to upgrade the stated moment premise.
+Here ||F||C4 is any fixed norm formed from supremum norms of its
+coordinate derivatives through order four on this flat torus.
+
+The finite-regularity imports and their uses are explicit:
+
+| Step | Regularity actually used |
+| --- | --- |
+| Gradient coefficient incidence | A C4 tail gives a C3 map between equal-dimensional finite spaces. |
+| Critical-value nullity of that projection | The equal-dimensional C1 case, also directly obtained from the area formula by zero Jacobian on its critical set. No scalar high-dimensional Sard theorem is invoked. |
+| All-Borel critical counts | Weighted Euclidean area formula on locally Lipschitz incidence maps, localized to compact charts. C3 is ample. |
+| Morse local form and H0 handles | Classical C2 Morse local form; the quadratic local model and critical-free level flow suffice for component attachment. |
+| Critical-free bands | grad F/|grad F|^2 is C3 away from roots, so its ordinary flow is defined on each compact band. |
+| Local cubic ridge | F_t is C3, its implicit ridge is C3, and phi'=F_s on the ridge makes phi C4. |
+| Conditional convergence and endpoint estimates | Global C4 convergence, physical C3 bounds and finite 2d-th norm moments. |
+
+The C2 Morse local-form theorem is used only on nondegenerate critical
+points. Together with the displayed critical-free flow, its quadratic
+coordinate form proves the same H0 handle statement used by the smooth
+source: maxima create components, and an index-(d-1) critical point
+merges two components or attaches a loop without changing H0. Higher
+handle attaching sets are connected and do not merge H0 components.
+This depends on the local homeomorphism/flow, not on infinite
+differentiability. The conventional C2 Morse lemma and the Lipschitz
+Euclidean area formula are named classical imports, not new theorems
+about arbitrary low-regularity fields. No weakened rule below uses C1
+samples or a nonexistent pathwise higher derivative.
+
+## 3. Specific finite rank witnesses, including the absent-zero-mode case
+
+Put
+
+    theta_a(z)=sum_j[1-cos(omega(z_j-a_j))].
+
+It is positive away from a, has a zero of order two at a, and has
+coordinate frequency degree one. Theta_a^2 vanishes through derivative
+order three and has coordinate degree two.
+
+At p the coordinates sin(omega(z_j-p_j))/omega have derivative I.
+Composing their degree-three Taylor polynomials realizes any prescribed
+third-order jet at p. Their support lies in K3={|n|infinity<=3}.
+This proves independence of the contact rows, including the independent
+transverse Hessian entries, at every point and orthonormal frame.
+
+For two distinct first-jet sites p,q, take B_p=theta_q^2 and the
+functions B_p and B_p sin(omega(z_j-p_j))/omega. They kill the other
+first jet; the value/gradient map at p is triangular with positive
+diagonal B_p(p). The symmetric construction gives full two-site
+value/gradient rank. Support is in K3.
+
+For a contact x and an offsite p use B_p=theta_x^2; this kills the
+entire third jet at x while freely varying the value and gradient at
+p. For distinct x,p,q use
+
+    B_p=theta_x^2 theta_q^2, B_q=theta_x^2 theta_p^2.
+
+Their products with the local sine coordinates lie in K5, kill the
+other offsite first jet and the full third contact jet, and give both
+offsite value/gradient jets freely. Combining them with the single-site
+contact directions gives the joint ranks. This is a proof for the
+named rows, not an inference of arbitrary multijet rank.
+
+If q_0=0, a witness's constant Fourier coefficient must be removed.
+For each relevant jet configuration there is a nonnegative polynomial
+g with positive integral, zero at all prescribed jets:
+
+| Configuration | g, with the indicated jet vanishings |
+| --- | --- |
+| One contact through order three | theta_x^2 |
+| First jets at p,q | theta_p theta_q |
+| Contact at x and first jet at p | theta_x^2 theta_p |
+| Contact at x and first jets at p,q | theta_x^2 theta_p theta_q |
+
+The degrees are respectively 2,2,3,4. At finitely many distinct sites
+the factors are positive on a nonempty open set, so integral g>0.
+Replace each already constructed witness h by
+
+    h_tilde=h-(integral_X h/integral_X g) g.              (8)
+
+It has exactly the same specified jets and zero mean; its support
+stays in K5. Thus every required rank is witnessed using only nonzero
+positive K5 modes, proving the optional (P5^0) extension of the theorem.
+Removing q_0 still changes the law and birth-mark
+distribution; it is not declared to preserve the covariance or
+normalization of a field which previously had that mode.
+
+In particular the actual finite block and the full Gaussian law have
+positive covariance for precisely these vectors:
+
+    O_(p,q)=(F(p),grad F(p),F(q),grad F(q)), p!=q;
+    U_0=(F,F_u,F_uu,F_uuu,(F_vi,F_uvi)_i);
+    U_0 plus independent symmetric entries of A=Hess F|u-perp;
+    U_0 plus (F(p),grad F(p)), p!=x;
+    U_0 plus both first jets at p,q, x,p,q distinct.        (9)
+
+Indeed zero variance of a row combination would make it annihilate
+every positive Fourier direction, contradicting the witnesses. The
+enlarged Hessian list uses independent symmetric entries; it does not
+pretend all d^2 entries are independent. After U_0 is conditioned its
+coordinates are deterministic; the residual ranks are the positive
+Schur complements of these joint covariances.
+
+Continuity gives positive floors only on the declared compact separated
+domains. Contact floors are uniform in frames because O(d) is compact.
+No floor is inferred as additional sites approach contact or the pair
+diagonal. Nor is a K7 four-site rank result inferred from K5 support.
+The iid proof below needs the bounded one-contact global mark, not K7.
+
+## 4. COUNT by finite incidence and actual Gaussian fibers
+
+Split F=F_B+T into its actual K5 coefficient block and independent C4
+tail; omit the zero coordinate if absent. The finite block alone has
+all ranks (9), with its original positive weights. Given almost every
+tail realization, it is a deterministic C4 function added to a finite
+smooth Gaussian basis.
+
+At a point choose d coefficient columns whose gradient minor is
+invertible locally. Write the block coordinates as (a,eta). The zero
+gradient equation solves
+
+    a(p,eta)=-B(p)^(-1)[grad T(p)+C(p)eta].               (10)
+
+The coefficient projection (p,eta)->(a(p,eta),eta) is a C3 map between
+N-dimensional spaces. Its derivative is singular exactly when the
+physical Hessian at p is singular, after invertible minor factors.
+On compact chart pieces the map is Lipschitz. Area formula applied
+to its critical set, whose absolute Jacobian is zero, proves that
+its critical image has Lebesgue measure zero: every image point has
+at least one critical preimage. This is the needed equal-dimensional
+Sard conclusion, requiring only C1 here. The Gaussian block has a
+Lebesgue density. Countable chart/minor exhaustion and integration of
+the tail prove unconditional Morse genericity before any Rice count.
+Compactness then gives a finite critical set.
+
+For a pair p!=q use 2d invertible gradient coefficient columns. The
+spatial derivative is block diagonal (H_p,H_q). The coefficient
+projection Jacobian is |det H_p det H_q|/|det B(p,q)|, with the usual
+coordinate-volume factors. Weighted area formula, coefficient
+change of variables and nonnegative Tonelli give
+
+    E sum_(p!=q,grad F(p)=grad F(q)=0) w(p,q,F)
+      =integral_(p!=q) p_(grad F(p),grad F(q))(0)
+          E[|det H_p det H_q| w | both gradients=0] dp dq  (11)
+
+for every nonnegative Borel mark of the full C4 field and pair.
+Both sides initially may be infinite. Partition the countably many
+minor/chart pieces disjointly and exhaust relatively compact parameter
+domains. Local Lipschitz regularity suffices; no global coefficient
+Lipschitz assertion is made. Conditional determinant moments and
+covariance floors make unmarked integrals finite on compact annuli.
+
+The conditional version in (11) is fixed at every target. The
+finite coefficient fiber integral is an ordinary full-rank Gaussian
+disintegration. Integrating its Bayes-weighted tail gives, for each
+finite evaluation vector, the usual full Gaussian regression at that
+target. Equality on a countable dense evaluation set identifies the
+whole C4 kernel. Thus one does not extend a continuous-mark formula
+to Borel marks by assertion or select a favorable unspecified null
+fiber. All-Borel marks are allowed directly by area formula.
+
+The conditional two-height covariance at distinct sites is positive
+by O rank in (9). Apply (11) with F(p)=F(q) on each compact annulus:
+the conditional height density makes its contribution zero. Countable
+annulus exhaustion gives distinct critical values almost surely.
+Point incidence similarly gives finite expected critical count,
+using a uniform gradient covariance floor and Hessian moments.
+
+The intrinsic rational-superlevel connectivity selector of ST is
+Borel on C4 fields and zero outside the open Morse/distinct locus.
+The local-form/level-flow argument of Section 2 counts every finite
+H0 bar once as an ordered maximum/merging-saddle pair. Its unique
+essential global-maximum class is excluded. Apply (11) with the
+Hessian type indicator and selector; only one determinant product
+enters. Two-height disintegration yields the specific Borel kernel
+
+    K_sel(p,q,b,s)=p_O(b,0,s,0) E_Q[W_(p,q) sigma],
+    nu_bar(ell)=L^(-d) integral_(p!=q) integral_R
+                         K_sel(p,q,b,b-ell) db dp dq.      (12)
+
+This proves COUNT in the C4 setting, with the chosen actual versions.
+It does not use two-site full Hessian rank or a scalar Sard theorem
+on F:X^d->R.
+
+## 5. Actual contact regression and ENVELOPE
+
+At M=x-r u/2,S=x+r u/2 choose an orthonormal transverse frame v_i.
+Use the literal row order
+
+    U_r=((F(M)+F(S))/2,(F(S)-F(M))/r,
+         (F_u(S)-F_u(M))/r,
+         (6/r^2)[F_u(M)+F_u(S)-2(F(S)-F(M))/r],
+         ((F_vi(M)+F_vi(S))/2,(F_vi(S)-F_vi(M))/r)_i),
+    v_r=(b-k r^3/2,-k r^2,0,12k,(0,0)_i), k>0.           (13)
+
+The contact rows are exactly U_0 in (9), target
+v_0=(b,0,0,12k,(0,0)_i). In an axial C4 path psi(s), with a=r/2,
+the fourth row is the positive average
+
+    (3/(4a^3)) integral_(-a)^a (a^2-s^2) psi'''(s) ds.
+
+Integration by parts verifies this identity. Its kernel has mass one.
+All other rows are averages/divided differences of derivatives of
+order at most two. Thus, uniformly in x and frames,
+
+    |U_r F|<=C||F||C3,
+    |(U_r-U_0)F|<=Cr||F||C4.                             (14)
+
+Let Sigma_r=Cov(U_r), and let c_r=E[F U_r F] be a finite vector of
+C4-valued Bochner expectations. E||F||C4^2<infinity makes them exist.
+The estimates in (14) show
+
+    Sigma_r->Sigma_0 uniformly,
+    ||c_r-c_0||C4<=Cr E||F||C4^2.                         (15)
+
+This supplies global C4 cross-covariance convergence without pathwise
+C8 derivatives or an all-Cq claim. Stationarity removes x from the
+contact floor. Rank (9) and compactness of O(d) give Sigma_0>=lambda I,
+and then Sigma_r>=lambda I/2 for a common small r_0>0. Upper covariance
+bounds also hold. Density pi_r(v_r) therefore converges at every
+fixed b,k,u to pi_0(v_0).
+
+On one actual full-field probability space set
+
+    F_r=F+c_r Sigma_r^(-1)(v_r-U_r F).                    (16)
+
+This satisfies its pins exactly. Its centered residual is independent
+of U_r F by Gaussian orthogonality on a dense evaluation set. Thus it
+is the canonical full-field conditional law Q_r, including at null
+pin events. Equations (14)-(15) give F_r->F_0 globally in C4 and
+
+    ||F_r||C4<=C||F||C4+C(|b|+k), 0<r<=r_0.              (17)
+
+The constants depend on the declared fixed law, d,L, not on b,k,r.
+The same coupling is continuous along convergent compact parameter
+sequences. It is not a finite-truncation convergence argument.
+
+Under the zero endpoint gradients, the physical Hessians satisfy
+
+    H_i=[r alpha_i, r beta_i^T; r beta_i, A_i],
+    det H_i/r=alpha_i det A_i-r beta_i^T adj(A_i) beta_i.
+
+This polynomial identity holds also for singular A_i. Equation (14)
+and the endpoint zero average bound alpha_i,beta_i by C||F_r||C3.
+Global C4 convergence gives alpha_M->-6k, alpha_S->6k and A_i->A_0.
+The correct congruence diag(r^(-1/2),I) gives the maximum/index-(d-1)
+types when A_0<0. If A_0 is nonsingular but not negative definite,
+the maximum condition fails; if singular, the determinant kills the
+boundary. Consequently, with W_r the actual typed determinant product,
+
+    V_r=W_r/r^2 -> V_0=36k^2(det A_0)^2 1{A_0<0},
+    V_r<=C(1+||F_r||C3)^(2d).                            (18)
+
+The moment in (R4) suffices for domination. Enlarged contact rank
+makes the conditional Gaussian law of A_0 full-support on symmetric
+transverse matrices. Its b-conditioned mean may be nonzero, but
+every open negative-definite cone has positive probability. Thus
+z_0=E V_0 is positive and finite at every fixed b in R,k>0 and
+E_Qr V_r->z_0. No pin covariance itself remains positive after pinning.
+
+The target's first and fourth coordinates are b-k r^3/2 and 12k;
+for bounded r they give |v_r|^2>=c(b^2+k^2). Covariance floors and
+(17)-(18) therefore yield the all-mark unnormalized bound
+
+    12 pi_r(v_r) E_Qr V_r
+      <=H(b,k)=C(1+|b|+k)^(2d) exp[-c(b^2+k^2)],
+    integral_Rx(0,infinity) k^(-2/3) H db dk<infinity.      (19)
+
+This proves CONTACT and ENVELOPE. Continuous positivity of z_0 gives
+a floor on compact birth/gap/frame windows, k bounded away from zero.
+There is no all-mark inverse normalizer, inverse random curvature or
+unbounded-law-family uniformity.
+
+## 6. Contact posterior genericity and actual SELECT
+
+Fix x,frame,b in R,k>0 and condition on U_0=v_0. This conditional
+field has a forced degenerate contact at x; it is not globally Morse.
+Unconditional genericity cannot be substituted for its needed
+off-contact assertion.
+
+In the finite-block/tail split the block contact covariance Sigma_B
+is positive. The conditional tail has the exact posterior density
+
+    dmu_tail,v(t)/dmu_tail(t)
+      =p_(U_0 F_B)(v_0-U_0 t)/p_(U_0 F)(v_0).             (20)
+
+Its denominator is positive and the density integrates to one. The
+posterior is absolutely continuous with respect to the original C4
+tail, so it remains C4 almost surely. Given the tail, the block is a
+Gaussian on its affine contact fiber. Its free coefficient directions
+are the kernel of U_0; the centered free coordinates have a positive
+Lebesgue density independent of the affine mean.
+
+The residual ranks in (9) hold on that kernel. With the fixed posterior
+tail, apply the same C3 equal-dimensional point incidence on X minus
+{x}: countable compact exhaustion proves all other critical points
+Morse. The residual single value/gradient density makes the mark
+F(p)=b contribute zero; residual paired first-jet density makes
+F(p)=F(q) contribute zero when x,p,q are distinct. Use the all-Borel
+one- and two-point area formulas and separated compact exhaustions,
+then integrate (20). Thus, for each fixed contact target, all other
+critical heights are distinct and different from b almost surely.
+No event simultaneously covering uncountably many targets is asserted.
+
+The full-density conditional A_0 is nonsingular almost surely. With
+invertible A_0 the local C3 transverse ridge exists; its reduced
+function has phi_0'=phi_0''=0 and phi_0'''=12k at contact. Hence contact
+is isolated. The remaining compact critical set consists of isolated
+Morse points and is finite; their heights have positive finite gaps
+from each other and b.
+
+On V_0>0, A_0<0. The deterministic elder argument in ST Section 4
+needs only the C4 regularity checked in Section 2, so it applies to
+the actual coupling (16). To make its topological content explicit,
+take a fixed small cylinder in axial s and transverse t. Strict
+transverse concavity gives a unique ridge h_r(s), C3-converging to
+h_0. Its reduced phi_r has phi_r''' positive on the cylinder. The
+exact pins give its only critical roots at s=+-r/2, a maximum then
+an index-(d-1) saddle. A compact gradient floor and implicit old
+branches exclude any other critical point globally.
+
+The left cylinder cut at S has its left and transverse boundary
+below the death height; its right face is bounded above by the saddle
+height. For intermediate levels its maximum component is enclosed,
+connected along the concave fibers and ridge, and cannot connect
+outside. The positive ridge past S reaches a fixed point higher than
+the limiting birth b. At S the newborn component therefore merges
+with an older component; the elder rule gives precisely this actual
+finite pair. All old critical levels are separated from b, so the
+coupled field is globally Morse/distinct eventually as well.
+
+Thus sigma_r=1 eventually on V_0>0. On V_0=0, V_r(1-sigma_r)->0
+without any genericity claim at the corresponding finite-r fibers.
+The same norm domination gives
+
+    E_Qr[V_r(1-sigma_r)]->0,
+    E_Qr[V_r sigma_r]->z_0.                              (21)
+
+This is SELECT. A varying-parameter compactness argument gives
+qualitative compact-window convergence if desired, but the theorem
+uses the pointwise result and (19). Neither a common physical chart
+radius nor a probability rate is claimed.
+
+## 7. FAR, density composition and the coefficient
+
+For dist(p,q)>=r_0, the original O_(p,q) covariance has a positive
+floor by (9), continuity and compactness. Conditional regression at
+target (b,0,b-ell,0), 0<ell<=1, gives a C2-norm moment polynomial
+in |b| and a Gaussian observation density bounded by C exp(-c b^2).
+Integrating the determinant product over b and the compact pair domain
+gives
+
+    0<=nu_bar,far(ell)<=nu_cand,far(ell)<=C.               (22)
+
+Any fixed compact positive-lifetime band similarly has a finite
+local bound. This is a direct density bound for the representative
+(12), not differentiation of a cumulative estimate. It proves FAR.
+
+The literal linear observation transform in (13) has exact absolute
+determinant 12r^(-(d+3)). Its four height/axial rows contribute 12r^-4,
+and the d-1 transverse pairs contribute r^(-(d-1)). With spatial
+polar volume r^(d-1)dr, height Jacobian r^3 db dk and typed Hessian
+weight r^2 E[V_r sigma], the product is
+
+    r B_r dr db dk dx d sigma(u),
+    B_r=12 pi_r(v_r) E[V_r sigma].                       (23)
+
+The flat torus midpoint Jacobian is exactly one in the unique short
+band. The dimensional powers cancel to r; a cubic gap alone would
+not determine this cancellation. Substituting ell=k r^3 in the
+specific density (12) gives
+
+    ell^(1/3) nu_bar,near(ell)
+      =integral 1{k>=ell/r_0^3}
+           B_((ell/k)^(1/3))(b,k,u)/(3k^(2/3)) db dk d sigma.
+
+Equations (19),(21) give its all-mark dominated limit
+
+    c_q=4 integral_Rx(0,infinity)xS^(d-1)
+                  k^(-2/3) pi_0(v_0) z_0 db dk d sigma.   (24)
+
+It is finite and positive; the far term vanishes after multiplication
+by ell^(1/3). This proves (4) as a pointwise statement for the chosen
+canonical finite Borel representative. A different density modified
+on a null set need not retain that pointwise statement. Global
+selection's discontinuity supplies no density-continuity theorem.
+The cumulative factor 3/2 follows by integration.
+
+For explicit physical evaluation, write xi=omega n. The exact jet
+covariances include
+
+    M_G=sum q_n xi xi^T,
+    M_Vu=sum q_n (xi.u)^2 xi xi^T,
+    h_u=Cov(F_uuu,G)=-sum q_n (xi.u)^3 xi,
+    tau_u^2=sum q_n (xi.u)^6-h_u^T M_G^(-1)h_u.           (25)
+
+Hessian covariances use sum q_n xi_i xi_j xi_k xi_l, and
+Cov(F,H_ij)=-sum q_n xi_i xi_j. All these sums are actual and
+absolutely convergent. Stationarity and q_n=q_-n make the covariance
+even; the odd Gaussian block (G,F_uuu) is independent of the even
+block (F,V_u,A_u). No rotational or coordinate-product independence
+is inferred. Put
+
+    D_u=E[(det A_u)^2 1{A_u<0}|V_u=0].                   (26)
+
+The full V_u has d coordinates, not merely F_uu. Birth disintegration
+removes F=b while retaining V_u/A_u correlations. The b-conditioned
+A need not be centered; the A conditioned only on V_u=0 is centered.
+The half-Gaussian integral is
+2^(-1/3)Gamma(7/6)tau_u^(4/3)/sqrt(pi). Combining the exact 12 factor
+in (24) yields
+
+    c_q=Gamma(7/6)/(24^(1/3)sqrt(pi))
+         *integral_S p_G(0) p_Vu(0) tau_u^(4/3) D_u d sigma(u),
+    p_G(0)=(2pi)^(-d/2)(det M_G)^(-1/2),
+    p_Vu(0)=(2pi)^(-d/2)(det M_Vu)^(-1/2).               (27)
+
+Full enlarged contact rank makes D_u>0; frame compactness controls
+the integrals. For d=2 it is Var(A_u|V_u=0)/2. For higher d one
+keeps the negative-definite cone integral. Ordinary full-sphere area
+and ordered maximum-to-saddle direction introduce no factor 1/2.
+The numeral 24 is the Jacobian/gamma factor, not L. The exponent is
+shared in this admitted class, while c_q varies with the physical law.
+
+## 8. Marked iid-copy process: physical cluster avoidance, not factorial transfer
+
+On the open C4 Morse/distinct locus define N_t as the sum of its
+finite Borel elder pairs with gap <=t, and assign zero on its complement.
+Implicit critical branches on a countable parameter-chart cover make
+N_t jointly Borel in field and threshold. Use a Borel shortest torus
+displacement convention away from the near band; its far mark
+contribution vanishes under (22). The near ordered displacement is
+M=x-r u/2,S=x+r u/2. Local/Borel transverse frames suffice.
+Explicitly Xi_t is the sum of delta masses at (ell/t,b,k,M,u) over
+finite bars, on E=(0,infinity) x R x (0,infinity) x X x S^(d-1),
+and is zero on bad fields. Its restriction to a<=T has count N_(Tt).
+
+At ell=ta with 0<a<=T, substitution in (23) gives the rescaled mean
+integrand
+
+    a^(-1/3) 1{k>=ta/r_0^3}
+       B_((ta/k)^(1/3))(b,k,u)/(3k^(2/3))
+
+with birth location x-r u/2. Its majorant is a^(-1/3)H/(3k^(2/3)),
+integrable over the full window and all marks. Pointwise location
+convergence and (21) give the mean limit (5), including tightness of
+small-a and unbounded birth/gap tails. Far expected counts are O(t).
+In particular eta_q{a<=T}=(3/2)L^d c_q T^(2/3).
+
+For a fixed contact-law realization with V_0>0, Section 6 gives a
+complete list: only the pinned M,S collapse, and all old critical
+branches persist with distinct limiting heights. The pinned actual
+elder bar consumes their unique birth/merger roles. Every other bar
+has two old endpoints, whose finitely many possible limiting gaps
+have a positive minimum. Thus at r=(ta/k)^(1/3), N_(Tt)=1 eventually.
+The pairing among old endpoints need not stay fixed. On V_0=0 the
+weight V_r tends to zero regardless of the count.
+
+Insert the bounded global Borel mark 1{N_(Tt)>=2} in (11)-(12).
+At every fixed a,b,k,u,x its weighted contact expectation tends to
+zero. The preceding all-mark majorant applies without multiplying
+by an additional count. Dominated convergence and the O(t) far
+anchor bound prove the genuine physical estimate
+
+    E[N_(Tt)1{N_(Tt)>=2}]=o(t^(2/3)).                    (28)
+
+For a compact-supported nonnegative process test f, let h=1-exp(-f).
+The difference between E Xi_t h and E[1-exp(-Xi_t f)] is between
+zero and the left side of (28). Hence, for n independent whole
+copies with n t_n^(2/3)->lambda,
+
+    E exp[-sum_(j=1)^n Xi_(t_n)^(j) f]
+       ->exp[-lambda integral(1-exp(-f))d eta_q].        (29)
+
+Equivalently censor multi-point copies; their union probability is
+o(n t_n^(2/3))->0. The remaining independent Bernoulli singletons
+have convergent tight mark laws and binomial counts converging to
+Poisson. This proves the process theorem, including finite windows
+a<=T. It is independent whole copies, not within-one-field spatial
+independence. For one fixed field the finite positive lifetime list
+makes the local rescaled process eventually empty almost surely.
+
+Nothing in (28) bounds E[N_(Tt)(N_(Tt)-1)]: that would insert an
+unbounded multiplicity into the contact kernel. K5 does not license
+the auxiliary K7 four-site claims of PP. Coalescing-contact factorial
+and regional additional-witness problems therefore remain separate.
+PP's two-point-cluster comparison also still applies: first moment
+alone can give a compound Poisson limit, and (28) is the substantive
+extra ingredient used here.
+
+## 9. More general finite carriers and why K5 is not necessary
+
+The proof only needs a finite symmetric positive-weight carrier B
+whose real trigonometric span supplies the five exact joint ranks
+in (9), for every declared point/frame/distinct-site tuple. Under
+(R4), these **named carrier tests** replace (P5) throughout Sections
+4-8: select finite minors there, use that block's contact posterior,
+and retain the same full Gaussian regression and moments. A covariance
+floor still follows only on separated compact domains. This is a
+more general sufficient admission theorem, not an assertion of a
+minimal spectral criterion or a license to ignore the residual tests.
+
+There is a concrete example failing literal K5 positivity but meeting
+these tests in every d>=2. Start with any positive full K5 finite law
+H, and take an integer unimodular shear A with z_1 mapped to
+z_1+m z_2, m>5. Let F(z)=H(Az). It is again stationary Gaussian on
+the same torus and variance one, with positive support on A^T K5.
+The mode e_1 is absent: A^(-T)e_1=(1,-m,0,...,0) is outside K5.
+The finite C4 sum is automatic.
+
+All needed carrier tests nevertheless hold. Pull back the earlier
+finite witnesses through the torus diffeomorphism A. Full third jets
+at a contact transform by an invertible jet map; first jets at
+distinct sites transform invertibly too; full contact-jet vanishings
+are preserved. Thus the same joint/residual interpolation maps are
+surjective. The transformed support, rather than an imaginary full
+coordinate box, supplies the finite minors. The geometric constants
+depend on A, and A is not treated as an isometry. This proves that
+full coordinate K5 positivity is sufficient, not necessary.
+
+Even within the simple verifier, arbitrary outside holes are harmless.
+For example remove any chosen symmetric pair of modes outside K5
+from an admitted infinite A4 spectrum and renormalize the remaining
+weights. K5 remains positive and A4 remains finite, so the theorem
+applies to this changed physical law. The change may alter the jet
+coefficient and birth marks; it is not claimed to preserve them.
+
+## 10. A4 is not necessary: polynomial tails with a proved C4 moment version
+
+Let
+
+    q_n=Z^(-1)(1+|n|infinity)^(-beta),
+    d+8<beta<=2d+8.                                     (30)
+
+The normalizer is finite; symmetry, positivity and (P5) hold. On a
+dyadic frequency shell 2^j<=|n|infinity<2^(j+1), the A4 shell mass
+is comparable to 2^(j(d+4-beta/2)). Thus A4 diverges throughout the
+stated upper-bound range, including its endpoint. We now prove
+that the actual Gaussian Fourier law still meets (R4), without a
+claim from pointwise eighth moments alone.
+
+Let F_j be its finite Gaussian trigonometric shell. For each derivative
+alpha of order <=4, its value at any fixed point has standard
+deviation at most
+
+    M_j=C 2^(j(d+8-beta)/2).
+
+The elementary coefficient-L1 bound for the gradient of D^alpha F_j
+has Lp norm at most C_p 2^(j(d+5-beta/2)), for every finite p>=1.
+Use a torus grid of mesh delta_j=2^(-j(d/2+2)), with fixed-L
+constants. Its logarithmic cardinality is O(j+1). A union bound on
+Gaussian marginal tails, followed by tail integration, gives
+
+    ||max_grid |D^alpha F_j|||Lp
+       <=C_p sqrt(j+1) M_j.
+
+No independence of grid values is needed. Interpolation from the
+nearest grid point costs at most C delta_j||grad D^alpha F_j||infinity,
+whose Lp norm is at most C_p 2^(-j)M_j. There are finitely many
+derivatives through order four. Consequently,
+
+    ||F_j||Lp(C4)<=C_p sqrt(j+1)2^(j(d+8-beta)/2),
+    sum_j ||F_j||Lp(C4)<infinity.                        (31)
+
+The base finite block is handled separately. Minkowski proves all
+finite C4-norm moments for the block sum. In particular the sum of
+its C4 norms has finite expectation and is finite almost surely,
+so the shell series converges absolutely in the C4 Banach space.
+It has covariance (2) by summable q and is the actual Gaussian law.
+This proves (R4), hence the full density and iid-copy conclusions,
+despite the divergence of A4. It does not claim samples C-infinity.
+All constants in this calculation are for fixed d,L,beta.
+
+This alternative also shows why a finite positive carrier plus an
+actual C4/moment version is a more intrinsic admission condition than
+one particular Fourier summability test. No assertion is made that
+(30)'s range or C4 itself is necessary for a regular-fold theorem.
+
+## 11. Refusals, lack of law-family uniformity and structural meaning
+
+Smoothness or summability alone does not ensure admission. If q_0=1
+and all other weights vanish, the field is a random constant: all
+moments exist, but the rank, Morse and positive fold-weight conclusions
+fail. Spectra supported on a proper sublattice produce exact spatial
+copies of critical values, defeating distinct-value COUNT. Additive
+coordinate-only spectra can force mixed Hessian rows to vanish and
+collapse the contact density. These are refusal diagnostics for the
+present interfaces, not proved alternative exponent laws.
+
+Nor does variance normalization make the coefficient universal.
+Take any admitted normalized Gaussian H and an independent standard
+constant C. For a fixed 0<delta<1 put
+
+    F_delta=sqrt(1-delta^2) C+delta H.
+
+This is another normalized stationary Gaussian law with positive
+nonzero K5 modes and finite A4 when H has them. Its derivative
+covariance floors shrink as delta^2. Addition of its constant shifts
+all critical heights equally, while multiplication by delta scales
+every lifetime, so the exact density transformation gives
+
+    c_(F_delta)=delta^(-2/3)c_H.                         (32)
+
+It diverges as delta->0, although every fixed delta is admitted and
+even A4 can stay bounded. Thus uniform tail summability without a
+common finite-carrier floor does not supply law-family uniformity.
+No joint delta/lifetime assertion is imported from the fixed-law theorem.
+
+The admitted exponent is structural in a precise sense: the cubic
+height gap, regular contact density, spatial polar power and typed
+Hessian product have the orders in (23), giving r dr and then -1/3.
+The spectral tests verify those probabilistic and topological
+interfaces for this Gaussian family. They do not say the cubic gap
+alone fixes the exponent, or that covariance fixes a non-Gaussian
+contact density. A forced transverse null block, contact rank loss,
+different singularity order, nonintegrable marks or a comparable far
+contribution needs a different proof. No higher-order singularity
+taxonomy is asserted as an established field law here.
+
+For a finite collection of already admitted laws one can take finite
+minima/maxima of analytic constants, while retaining each c_q. For
+an arbitrary family approaching a support boundary, varying dimension,
+side length or geometry, the present individual theorem makes no
+uniform claim. Hk, spatial mixing/expanding volume, regional rejected
+candidate/additional-witness populations, all factorial moments,
+numerical approximation windows, blind fields and formal owner targets
+remain distinct consumers.
+
+## 12. Exact repository bindings, proof reuse and review scope
+
+Current operational sources are the unchanged reviewed files of the
+structural/physical/process increment. The initial incorporation base is
+native1966a74693f5cebfc63ace00b85d9efa0d07a3d7, reconciled to mainf08dfdf9.
+These mathematical dependencies have their own actual source identities:
+
+| Input | Current source | UTF-8 bytes / lines | SHA256 |
+| --- | --- | --- | --- |
+| ST | [Structural fold theorem](../fold_structural_universality/PROOF.md) | 43927 / 804 | 3d952a906553d578876456539a8f0dcd64aedd45099e8394cbe9db99866ec531 |
+| IG | [Exact periodized law](../periodized_gaussian_h0/PROOF.md) | 34629 / 637 | c8bd489ca3fbb3760f7b2d53e7385131ac5336c51ba983e00e4d540fe2f00aaa |
+| PP | [Iid marked process](../iid_short_bar_process/PROOF.md) | 30935 / 618 | 320bc988c468bea4938bb777a2d05952c6ab8e0a384f33725b50de847aeec1eb |
+
+The original author freshly read all three actual current sources in full,
+including separately filling its initially truncated structural middle.
+Those reads covered these exact unchanged dependencies. Its prior original
+preparation reads remain distinct; neither is a claimed full read of new
+wrapper bytes. Root independently fully read all three current sources,
+all825 original spectral lines and the complete actual incorporation diff.
+
+Reuse is narrow. STSections2-4 supply the Borel selector, literal row
+transform, determinant identity and elder ridge route; Sections5-7 supply
+the geometric ledger and corrected parity coefficient. Sections2-8 here
+independently establish their C4 variants. IGSections3-6 supply theta
+interpolation, finite-block posterior incidence and full-field regression;
+the image-sum law, all-mode positivity and physical moment numerals are
+not general-family premises. PPSections2-5 supply Borel counts, complete
+endpoint isolation and iid Laplace; Section8 supplies the first-moment
+insufficiency comparison. Its auxiliary K7/factorial result is not admitted
+from K5. The original task began while PP review was pending; root later
+reported the unchanged incorporated sources' full reviews completed.
+That historical report was not the spectral author's own review of those
+records or a verdict on later spectral wrapper bytes.
+
+The original spectral saved source received final_contract_review fresh
+FULL825-line mathematical PASS, and benchmark_formal_audit separately
+fresh FULL825-line mathematical PASS. Both are nonauthor AI reviews with
+substantial prior route exposure and credited prefile contributions.
+The former freshly rehashed ST/IG/PP with its immediately preceding full
+incorporated-source reads; the latter freshly rehashed them and indirect
+sources, reading ST750-804,IG585-637,PP545-618 custody sections with prior
+full original-source reads and earlier regularity excerpts. Those hashes
+and bounded dependency reads are not additional whole dependency audits.
+Earlier prefile checks remain bounded route attacks rather than full
+saved-file reviews. Original preparations remain frozen in ordinary
+coordinator custody, not unverified repository links.
+
+The mathematical Sections1-11 above retain their exact original bytes.
+The local review companion separately binds this current whole file,
+construction snapshots, completed verdicts and any subsequent amendments.
+No metadata pending snapshot is rewritten as a completed earlier review.
+
+Named classical imports remain weighted Lipschitz Euclidean area formula,
+finite-differentiability Morse local form and critical-free flow/H0 handle
+consequences, with Gaussian regression,Bochner expectation,Tonelli/Fubini,
+Minkowski and dominated convergence under the displayed domains/moments.
+Both saved-file reviewers retained a wording qualification: actual use is
+on C4 functions, for which finite-differentiability Morse coordinates
+suffice; neither freshly verified the stronger C2 boundary statement.
+Root's separate bounded primary read of [Conrad's Morse handout,Remark2.2](https://math.stanford.edu/~conrad/diffgeomPage/handouts/morselemma.pdf)
+confirms that Cp,p>=3 has Cp-2 coordinates, sufficient for this C4 theorem.
+It is a parsed statement read, not a full-book/raw-PDF custody claim.
+Root's separate Milnor h-cobordism open returned initial passages only;
+no complete-book review is attributed to it or to the other agents.
+
+The admitted law/carrier tests must actually be checked for each proposed
+Gaussian support. No covariance-only non-Gaussian admission, spectral
+family uniformity, Hk result, spatial expanding-domain limit, coalescing
+or global factorial closure, numerical window, sampler/count coupling,
+blind50 confirmation or literal Lean discharge is supplied. Scientific
+promotion:NONE. Full landmark goal ACTIVE.
diff --git a/experiments/universality/spectral_gaussian_admission/REVIEW.md b/experiments/universality/spectral_gaussian_admission/REVIEW.md
new file mode 100644
index 000000000..52104b1fd
--- /dev/null
+++ b/experiments/universality/spectral_gaussian_admission/REVIEW.md
@@ -0,0 +1,134 @@
+# Spectral Gaussian admission: source-bound review record
+
+Construction snapshot,8October2026. Original mathematical author
+OpenAI/Codex universality_review; wrapper/binding author and integrator root,
+task01a11cbf-068d-7102-861e-75814e715c98, claim6070584604/amend6071035594.
+
+Current [proof](PROOF.md): 42478 UTF-8 bytes/826 lines,
+SHA2566c0f34cb318d337b3b8c2c306f3b965f5e5a9d2fe38d5e001b8044de74b7f225. Preserved Sections1-11
+36656B/SHA2564cad43d3f80a8e47da886762d402c4981fe6834e278579b36673055539d6cf32.
+Original42038B/825-line source SHA43e8180947b559e53f7afcf95ba61136f3fffcbe05a19e7a5953018fda27e294
+has two separately bound fresh FULL mathematical PASS verdicts,
+final_contract_review and benchmark_formal_audit, no must-fix.
+Root fully read original/current and the real complete incorporation diff.
+Fresh full current repository-source review is PENDING at this construction
+snapshot; original verdicts do not certify this changed wrapper/readme.
+
+All performers are AI agents,OpenAI/Codex,source and route exposed.
+Original author supplied C/window/S/ST/IG/PP; benchmark authored CG/ET and
+implemented the catalogue; final_contract authored the separate fold-chart
+route and supplied prefile zero-mean witnesses; root supplied IGtheta and
+PPendpoint/Borel challenges and bounded primary finite-regularity check.
+Organizational independence0; no human/provider-distinct/blind acceptance.
+Fresh full saved-file reviews are distinct from bounded prefile attacks,
+fresh dependency hashes, prior full reads and custody-section excerpts.
+
+The scope is the actual fixed-law Gaussian C4 carrier, named positive
+finite ranks, law-specific coefficient and independent-whole-copy process.
+No inherited K7/global factorial/spatial/Hk/criticalLean/numerical/blind
+closure. Actual topology uses C4, with the reviewers' unverified stronger
+C2 boundary qualification retained in PROOF.md. New compiler/tests/sampler/
+seeds:NOT_RUN. Scientific promotion:NONE. Full landmark goal ACTIVE.
+## Completed current-source review, 8 October 2026
+
+The construction snapshot above is preserved verbatim: 1,981 bytes / 31
+lines, SHA256 `12d7db0c580b4e5df0288454d734f064dab9ca0631304f340f1693ef9d75bb9d`.
+Its PENDING status is historical. The following new verdict completes the
+review of the incorporated source identified above.
+
+**final_contract_review: fresh FULL current-source mathematical PASS.**
+The reviewer read all 826 actual saved PROOF lines in four untruncated
+ranges, verified 42,478 bytes / SHA256
+`6c0f34cb318d337b3b8c2c306f3b965f5e5a9d2fe38d5e001b8044de74b7f225`,
+read the complete original-to-incorporated diff, and independently verified
+that the 36,656-byte Sections 1-11 core is unchanged. No mathematical
+must-fix or wrapper contradiction was found. This verdict is separate from
+the review of the original 825-line preparation.
+
+The attack covered actual C4 regularity and moments, finite-block rank and
+area-formula incidence, canonical posterior conditioning, global contact
+selection and elder pairing, weighted singular-transverse limits, the
+all-mark envelope and far contribution, law-specific coefficient, iid
+singleton censoring, zero-mode removal, sheared finite carriers, polynomial
+spectral tails and the stated failure of family-uniform constants.
+
+The reviewer freshly rehashed the current ST / IG / PP dependencies as
+`3d952a906553d578876456539a8f0dcd64aedd45099e8394cbe9db99866ec531`,
+`c8bd489ca3fbb3760f7b2d53e7385131ac5336c51ba983e00e4d540fe2f00aaa`,
+and `320bc988c468bea4938bb777a2d05952c6ab8e0a384f33725b50de847aeec1eb`.
+Their separate complete incorporated-source reads immediately preceded
+these SG reviews; this pass does not claim three additional full reads.
+The actual construction REVIEW (31 lines) and README (207 lines) were also
+fully read. All 28 local link targets resolved. The reviewer performed no
+fresh primary-source fetch, hosted operation, compiler, sampler or test.
+
+**benchmark_formal_audit: fresh FULL construction-metadata PASS.** The
+reviewer read the 31-line construction REVIEW and all 207 README lines,
+read the complete README predecessor diff and the complete incorporation
+diff, independently checked the preserved mathematical core and actual
+dependency hashes, and checked all 28 local links. This is a metadata
+verdict, with current mathematics pending at that snapshot. The benchmark
+reviewer did not newly read all 826 current proof lines. Original SG full
+review and bounded current dependency custody rereads retain their
+separate extents and identities recorded in PROOF Section 12.
+
+These same-provider, source-exposed reviews retain organizational
+independence credit 0. The actual C4 theorem, stronger-C2 qualification,
+root-only bounded primary-source exposure, no critical-Lean closure, no
+numerical or blind admission, no same-field spatial or Hk result, and no
+scientific promotion remain as stated. This appended review record awaits
+its own fresh metadata audit; the completed proof verdict is already bound
+to the unchanged proof bytes. Full landmark goal remains ACTIVE.
+
+## Retained-author documentary clarification, 18 October 2026
+
+This appended record is source-author preparation by Root task
+01a11cbf-068d-7102-861e-75814e715c98 with the OpenAI/Codex local
+SG/HF source-composition preparer. It preserves every byte of the preceding
+81-line review, SHA256
+`50424a8e0476c6c951b6c8cee65b318294dd96deaab1190a9afd3cf2154eb8ce`.
+The construction and completed-proof review records above retain their
+original dates, performers and reading extents. This append supplies no new
+mathematical review, independent-review credit or scientific promotion.
+
+### D1: import naming, source-preserving Route A
+
+[Claude S's actual review6139276363](https://github.com/d6g8k5htny-coder/main/pull/330#issuecomment-6139276363)
+reported PASS_MATHEMATICS_SCOPED with documentary D1/D2. The following
+import-naming clarification adopts its author's
+[Route A6139465410](https://github.com/d6g8k5htny-coder/main/pull/330#issuecomment-6139465410),
+whose proposed documentary route received
+[Muse Spark's bounded route read6154499494](https://github.com/d6g8k5htny-coder/main/pull/330#issuecomment-6154499494).
+Those are actual attributed reviews of their respective sources and routes;
+a route PASS is not a review of this newly appended composition.
+
+Section 2 of PROOF.md names the Morse local-form import as the classical
+C2 Morse local form / conventional C2 Morse lemma. The import actually
+consumed, as Section 12 records, is the Morse local form for C^k functions,
+k >= 3, with C^(k-2) coordinates, applied at k = 4 to the C4 field. No
+C2-function version is used. The Section 2 wording remains unchanged to
+preserve the exact whole proof and its Sections 1-11 core, including the
+identities consumed by the higher-homology and later source packets.
+This is the already-reviewed import-naming distinction, not a new classical
+proof, a stronger low-regularity theorem or an added hypothesis.
+
+### D2: actual body correction and current audit boundary
+
+Root corrected only the unsupported historical completed-metadata-audit
+assertion in the original Main330 body to pending. The saved API
+`updated_at` is `2026-10-18T00:32:30Z`. Root verified the complete native
+replacement-body readback at `2026-10-18T00:33:31Z`; source head
+`88ff9223e57bf38c7fc33cb3559a88793a7b5087` stayed unchanged.
+[Actual correction delivery6243772787](https://github.com/d6g8k5htny-coder/main/pull/330#issuecomment-6243772787)
+binds this action under
+[finite source-author pickup6243752128](https://github.com/d6g8k5htny-coder/main/pull/330#issuecomment-6243752128).
+The previous body and the old source record are preserved. No earlier audit
+is inferred from a route review, newly invented or backdated.
+
+The original prefix's pending metadata-audit sentence describes its dated
+source state. A distinct final metadata/composition review must bind the
+actual frozen source preimage of this append and the combined README; none
+is self-certified here. The existing mathematical verdicts remain scoped
+to their exact proof bytes, fixed Gaussian law/carrier assumptions and named
+classical imports. Critical Lean, numerical/count coupling, same-field spatial
+limits, blind validation and scientific promotion remain unclaimed.