diff --git a/experiments/universality/README.md b/experiments/universality/README.md
index 9c95539f8..b3ecfd707 100644
--- a/experiments/universality/README.md
+++ b/experiments/universality/README.md
@@ -178,6 +178,24 @@ This restricted parameter path is C1 but not C2 at zero and its two-site
 observation law is singular; it supplies no generic two-parameter cusp,
 Gaussian jet-law admission, numerical coupling or spatial process.
 
+The [smooth two-control cusp population](two_parameter_cusp_population/PROOF.md)
+extends the explicit surgery to a full smooth control rectangle through the
+quartic contact. Uniform root and height guards give the complete critical
+list and actual global H0 merger. Almost every field has zero or one finite
+H0 bar, and the total probability of a bar is at most 1/5 for this rectangle.
+Its leading whole-field lifetime density is asymptotic to
+C_edge*ell^(-1/3), with cumulative count asymptotic to
+(3/2)*C_edge*t^(2/3); these short bars approach the ordinary fold edges.
+Restricted away from those edges, the quartic-tip density is exactly
+C_tip*ell^(1/4), and the count exactly (4/5)*C_tip*t^(5/4), below the
+stated cutoff. Contact sampling weight and stratum therefore both matter.
+The independent-whole-copy PRM limit retains latent control, sign and
+location marks; physical birth and cubic marks keep their shared-control
+correlation. Its [review](two_parameter_cusp_population/REVIEW.md) preserves
+original-source reviews and completed incorporated-source review. This is one
+specified stationary law, with no generic full A3, Gaussian contact, Hk,
+same-field spatial, numerical, formal or blind admission.
+
 These conventional analytic results discharge selected-bar counting and the
 leading near/far intensity composition for the stated ideal finite laws and
 admitted stationary Gaussian spectral/carrier laws. Further
diff --git a/experiments/universality/two_parameter_cusp_population/PROOF.md b/experiments/universality/two_parameter_cusp_population/PROOF.md
new file mode 100644
index 000000000..d86cc513b
--- /dev/null
+++ b/experiments/universality/two_parameter_cusp_population/PROOF.md
@@ -0,0 +1,782 @@
+# Smooth two-control cusp population: actual H0 bars and fold-edge dominance
+
+Repository incorporation by OpenAI/Codex `final_contract_review`, the bounded
+wrapper constructor, under integration owner `root`, task
+`01a11cbf-068d-7102-861e-75814e715c98`. Mathematical author:
+`universality_review`. Native claim `6071732615` authorizes these three paths
+within its eleven-path scope through 02:20 UTC on 9 October 2026. The initial
+detached base is published quartic PR336 head
+`3ab12c6542c88554ad0d42e0e19aff2461dc3c0d`; no main integration is implied.
+
+Sections 1-10 preserve the frozen external preparation exactly: 32,809 UTF-8
+bytes / 696 lines, SHA256
+`001c4f609e167fcf139b5f46611bacdbf7203e4191060953d34485f78e2fc7c6`.
+Its 28,073-byte / 620-line mathematical core has SHA256
+`d4c2f0bc923d95a8a1905f4745dd4c2cad6025c1f1dcda99441e9f5285f5a213`.
+Both `final_contract_review` and `benchmark_formal_audit` completed separate
+fresh FULL original-source mathematical PASS reviews, with no must-fix.
+Root fully read the actual original 696 lines. Their exact reading ranges
+and the current context bindings are recorded in Section 11.
+
+The constructor first reviewed that original source and now authors this
+repository wrapper, review custody and README addition. This is an integration
+contribution, not an independent review of the constructor's own wrapper.
+A fresh full mathematical review of the incorporated source, root's current
+full read and new metadata review are **PENDING** at this construction snapshot.
+Original verdicts do not certify the new wrapper bytes. The
+[review record](REVIEW.md) preserves these separate source-bound states.
+
+The result concerns one specified stationary non-Gaussian field law. Its
+smooth two-control rectangle yields an explicit globally selected H0 bar,
+with fold-edge leading density and a lower-order restricted-tip contribution.
+All performers are source/route-exposed members of the same OpenAI/Codex team;
+organizational independence is zero. No general A3, Gaussian, Hk, numerical,
+spatial, formal, blind or scientific-status admission is implied.
+
+## 1. The fixed law and the exact assertions
+
+Fix an integer d>=2, L>0, X=(R/LZ)^d, spatial volume V=L^d, and
+omega=2 pi/L. Persistent homology is ordinary superlevel H0 over a fixed
+field K, with the elder rule and the essential global-maximum class excluded.
+Use the explicit smooth seed and quartic surgery of Section 2. The family is
+
+    F_(u,v)=F_0+chi(s,z_perp)[u s^2/2+v s],              (1)
+
+on the full control rectangle
+
+    -u_0<u<u_0,      -v_0<v<v_0,
+    v_0>=eta_c u_0^(3/2),      eta_c=2/(3 sqrt(3)).       (2)
+
+Section 2 supplies compatible explicit smallness conditions, including
+root containment, annular exclusion and a global high-level barrier.
+All constants and rectangle sides are fixed before taking any lifetime limit.
+Choose (u,v) uniformly in this full rectangle, with density
+
+    c_0=1/(4u_0 v_0),                                   (3)
+
+and independently choose Haar-uniform Y in X. The random field is
+
+    G(x)=F_(u,v)(x-Y).
+
+We leave its scale unnormalized in the main theorem. This law is stationary,
+has smooth sample fields and uniform bounds on every spatial C^q norm, and
+is almost surely Morse with distinct critical values. The density (3) is
+positive and smooth in the interior, including at the control origin;
+its extension by zero across the rectangle boundary is not claimed smooth.
+
+The only controls producing a finite H0 bar are the open cusp wedge
+
+    W={0<u<u_0, |v|<eta_c u^(3/2)}.                      (4)
+
+At good controls in W there is exactly one finite H0 interval, with
+
+    eta=|v|/u^(3/2),
+    ell=u^2 kappa(eta),        0<eta<eta_c,              (5)
+
+where kappa decreases strictly from 1/4 to zero. Controls with v=0 in W
+have tied maxima and form a null exception. Outside the wedge, away from
+the discriminant, the field has one maximum and no finite H0 intervals.
+The discriminant and tied-value locus have two-dimensional control measure
+zero, so their arbitrary zero-count convention does not change the law.
+
+Let q(h)=|d eta/d kappa| at eta=kappa^(-1)(h), 0<h<1/4.
+The whole-torus expected finite-H0 lifetime measure has the explicit density
+
+    nu(t)=2c_0 integral_(2 sqrt(t))^u_0
+                         u^(-1/2) q(t/u^2) du,
+                         0<t<u_0^2/4,                  (6)
+
+and is zero for t>=u_0^2/4. This is derived by a direct pushforward of
+control measure, not by differentiating a cumulative asymptotic. Write
+
+    K_c=4/3^(5/4),
+    Q_c=(2/3) K_c^(-2/3),
+    C_edge=(8c_0/7) K_c^(-2/3) u_0^(7/6).              (7)
+
+Then
+
+    nu(t) ~ C_edge t^(-1/3),
+    E N_t ~ (3/2) C_edge t^(2/3),        t down to zero. (8)
+
+These are whole-field coefficients. Per unit spatial volume divide by V.
+For any fixed 0<epsilon<eta_c, the population restricted to
+eta<=eta_c-epsilon has, for all sufficiently small t, the exact density
+
+    nu_tip,epsilon(t)=C_tip,epsilon t^(1/4),
+    C_tip,epsilon=c_0 integral_0^(eta_c-epsilon)
+                                      kappa(eta)^(-5/4) d eta. (9)
+
+Its cumulative contribution is exactly (4/5)C_tip,epsilon t^(5/4)
+there. The uniform small-u bound in Section 5 makes fold-edge dominance
+and the absence of an atom at the A3 tip precise.
+
+For n independent whole-field copies, if t_n->0 and n t_n^(2/3)->lambda
+in (0,infinity), their actual short-H0 points with lifetime mark ell/t_n
+and birth position x converge to a Poisson random measure with intensity
+
+    lambda C_edge a^(-1/3) da dx/V.                     (10)
+
+The full latent-control, sign, physical birth and cubic marks are specified
+in Sections 7--8. This uses the exact single-field bound N_t in {0,1}.
+It is not a same-field spatial or expanding-domain process claim.
+
+## 2. Actual global construction and uniform rectangle bounds
+
+Here we reproduce the needed global surgery, rather than presuming a local
+normal form has a harmless completion. Put w_i=3^(-i) and
+
+    h(y)=sum_i w_i cos(omega y_i),       b=sum_i w_i.
+
+Its 2^d critical points have y_i in {0,L/2}, diagonal nonzero Hessians,
+and distinct values. In comparing two sign patterns, their first differing
+weight dominates the sum of all later weights. The origin is the unique
+maximum, and all other critical values are <=b-Delta, Delta=2w_d.
+
+In |y_i|<L/4 use z_i=sqrt(2w_i)sin(omega y_i/2), s=z_1, and
+z_perp=(z_2,...,z_d). These smooth coordinates satisfy exactly
+
+    h=b-s^2-|z_perp|^2.
+
+Choose R^2<min(w_d,1). The superlevel {h>b-R^2} is precisely the
+chart ball D_R. Indeed each coordinate's nonnegative sine-square loss
+is then <R^2<w_i, forcing |y_i|<L/4 in torus representatives.
+Outside this ball h<=b-R^2, and no other old critical point lies in it.
+Let r_1=R/4. Take a smooth nondecreasing c:[0,infinity)->[0,1] with
+
+    c(r)=r/4 for r<=r_1^2,
+    c(r)=1 for r>=R^2/2,
+    c(r)>0 for r>0,       c'(r)>=0.
+
+For example integrate a nonnegative smooth derivative equal to 1/4
+on [0,r_1^2], tapering to zero before R^2/2, and add an interior bump
+to make its total integral exactly one. This is possible because R^2<1.
+Define on D_R, retaining h outside,
+
+    F_0=b-s^2 c(s^2+|z_perp|^2)-|z_perp|^2.             (11)
+
+It equals h on a full boundary collar, so is globally C-infinity.
+Its derivatives are
+
+    (F_0)_s=-2s[c(r)+s^2 c'(r)],
+    (F_0)_(z_i)=-2z_i[1+s^2 c'(r)],     r=s^2+|z_perp|^2.
+
+They vanish in D_R only at the origin. All old critical points outside
+are unchanged. In |z|<=r_1 the exact formula is
+
+    F_0=b-s^4/4-(1+s^2/4)|z_perp|^2.                   (12)
+
+Let chi be a fixed radial smooth cutoff, equal to one for |z|<=r_1/2,
+zero for |z|>=r_1, and between zero and one elsewhere. Its zero extension
+is smooth on X. Formula (1) therefore defines a global field jointly
+C-infinity in x,u,v, including at u=v=0. In its exact core,
+
+    F_(u,v)=b+P_(u,v)(s)-(1+s^2/4)|z_perp|^2,
+    P_(u,v)(s)=-s^4/4+u s^2/2+v s.                    (13)
+
+For explicit uniform guards, set
+
+    a_*=r_1/4,        delta=r_1^4/64,
+    g_*=min_(r_1/2<=|z|<=r_1)|grad_z F_0|>0,
+    B_j=||chi s^2/2||_(C^j),    D_j=||chi s||_(C^j), j=0,1,
+
+where for j=1 the gradient norm alone suffices. Fix a constant
+c_v>=eta_c, set v_0=c_v u_0^(3/2), and choose u_0>0 small enough that
+
+    u_0<a_*^2/4,          v_0<a_*^3/4,
+    B_1 u_0+D_1 v_0<g_*/2,
+    B_0 u_0+D_0 v_0<delta/4.                           (14)
+
+These strict inequalities are compatible: their left sides tend to zero
+as u_0 tends to zero. They also permit any larger v_0 satisfying (2)
+and the same upper bounds. Coverage of the wedge is not being confused
+with the smallness guards.
+
+Every real root of -s^3+u s+v=0 in the rectangle has |s|<a_*.
+For if |s|>=a_*, its root equation and (14) would give
+
+    |s|^3<=u_0|s|+v_0<|s|^3/4+|s|^3/4,
+
+a contradiction. Thus all axial roots are strictly inside the region
+where chi=1. On the transition annulus the perturbation gradient is
+<g_*/2, so no new critical points occur. Elsewhere in D_R the unperturbed
+gradient excludes roots. Old critical points outside are unchanged.
+
+The core transverse equations force z_perp=0. Hence the complete new
+critical list is exactly the real roots of that axial cubic, together
+with the old 2^d-1 critical points. Its core Hessian is diagonal,
+
+    diag(u-3s^2, -2(1+s^2/4) I_(d-1)).                 (15)
+
+Coordinate changes preserve its inertia at critical points.
+Every new critical value is within delta/4 of b: using |s|<a_* and
+a_*^4=delta/4 gives
+
+    |P_(u,v)(s)|<=a_*^4/4+u_0 a_*^2/2+v_0 a_*
+                   <5delta/32<delta/4.                (16)
+
+Also delta=R^4/16384<Delta. Thus these new values remain uniformly
+separated from every old one by more than 3Delta/4. The old values are
+fixed and mutually distinct, so all their distinct differences have a
+positive minimum. This excludes hidden old-level collisions in the
+entire rectangle.
+
+Finally, outside the core |z|<r_1/2, monotonicity and c<=1 imply
+
+    b-F_0>= (r_1^2/16)|z|^2>=delta
+
+in the chart, while the exterior chart loss is at least R^2>delta.
+The last inequality of (14) gives the global barrier
+
+    F_(u,v)<=b-3delta/4 outside the core,
+    all new critical heights >b-delta/4.               (17)
+
+These are uniform bounds over the full rectangle, not only over the wedge.
+
+## 3. Complete root classification and actual global H0 selection
+
+The cubic -s^3+u s+v has three simple real roots exactly when
+
+    u>0,       27v^2<4u^3.
+
+This follows from evaluating the cubic at its two turning points
+s=+-sqrt(u/3). For u<0 it is strictly decreasing and has one real root.
+For u=0,v!=0 it also has one simple real root. For u>0 outside the
+closed three-root wedge it has one root beyond the turning points, at
+which u-3s^2<0. Thus outside W and off the discriminant there is one
+new maximum; with the unchanged old list, it is the only maximum on X.
+The field is Morse and has distinct critical values by (16). It has
+no finite H0 intervals: a connected closed torus with just one maximum
+has only the essential H0 birth, and no second component birth.
+
+Inside W scale s=sqrt(u)z. For epsilon=sign(v) and eta=|v|/u^(3/2),
+the axial profile is u^2 Phi_(epsilon eta), where
+
+    Phi_eta(z)=-z^4/4+z^2/2+eta z.
+
+For positive eta denote its three roots of Phi'=0 by z_-<z_0<z_+.
+They lie respectively in (-1,-1/sqrt(3)), (-1/sqrt(3),0), and (1,infinity).
+The outer roots give two maxima, the middle root gives the saddle with
+one positive and d-1 negative physical Hessian eigenvalues. The left
+maximum is lower. To verify the ordering rather than prescribe it,
+Phi(z_-)>Phi(z_0) follows by integrating its negative derivative between
+these two roots. The right-minus-left peak value is zero at eta=0 and
+has derivative z_+-z_->0, proving strict positivity at eta>0.
+For negative v, reflect s to -s: the right maximum is now younger.
+
+All critical heights near this pair satisfy (17). Therefore every
+superlevel set between its saddle and its lower maximum is entirely
+inside the exact core. At a height q there its axial section is
+{P_(u,v)(s)>=q-b} and its transverse fibers are the balls
+
+    |z_perp|^2 <= [b+P_(u,v)(s)-q]/(1+s^2/4).          (18)
+
+For levels strictly between the saddle and the lower maximum there
+are two axial intervals. Contracting each transverse fiber to its
+center increases F and stays in the core, so the full global superlevel
+has exactly two components, one containing each maximum. At the saddle
+level the intervals first touch at its axial coordinate, with a
+single-point transverse fiber; just below, the superlevel is connected.
+The global barrier rules out an earlier external bypass. This proves
+an actual component merger in the whole torus field.
+
+The higher maximum is older, and its component survives. The lower
+maximum is paired with the middle saddle, with lifetime (5). There are
+exactly two global maxima and therefore exactly one finite H0 interval;
+the higher maximum gives the essential class. This completes the
+actual selection proof in both regions of the control rectangle.
+
+On the discriminant v=+-eta_c u^(3/2), u>0, the cubic has a double
+root and another simple root. At u=v=0 it has a triple root. These
+fields are not Morse at the contact. At v=0,u>0 the two maxima are
+nondegenerate but have equal values. These graphs and the line v=0
+are Lebesgue-null in the control plane by Fubini. We set counts to
+zero there for the probabilistic formulas, without asserting a
+distinct-values elder convention on that exceptional set.
+
+## 4. The gap function and a derived fold-edge inverse density
+
+For 0<=eta<eta_c define by endpoint continuation
+
+    kappa(eta)=Phi_eta(z_-(eta))-Phi_eta(z_0(eta)).
+
+It is smooth before eta_c and
+
+    kappa(0)=1/4,
+    kappa'(eta)=z_-(eta)-z_0(eta)<0,
+    kappa(eta) -> 0 as eta increases to eta_c.           (19)
+
+The derivative formula uses that Phi' vanishes at each critical root;
+there is no endpoint differentiation error. At eta_c the two roots
+coalesce at z_c=-1/sqrt(3). Thus kappa maps (0,eta_c) bijectively to
+(0,1/4), and its exact inverse derivative is
+
+    q(h)=1/[z_0(kappa^(-1)(h))-z_-(kappa^(-1)(h))].     (20)
+
+It is positive, smooth for 0<h<1/4, and tends to one as h increases
+to 1/4. We use this extension at the upper endpoint if needed.
+
+We now derive the singular endpoint constant. Set delta_eta=eta_c-eta
+and write z=z_c+H. The cubic is exactly
+
+    Phi_eta'(z_c+H)=sqrt(3) H^2-H^3-delta_eta.           (21)
+
+Let rho=sqrt(delta_eta), H=rho w. The two nearby roots solve
+sqrt(3)w^2-rho w^3-1=0. At rho=0 they are +-a_c, a_c=3^(-1/4),
+and are simple in w. The implicit function theorem gives
+
+    w_-(rho)=-a_c+O(rho),       w_0(rho)=a_c+O(rho).
+
+Their order identifies them with the lower-maximum and saddle roots.
+The exact difference of potential values is the integral of minus
+(21) between these roots. It gives
+
+    kappa(eta)=delta_eta^(3/2)
+       integral_(w_-(rho))^(w_0(rho))
+                [1-sqrt(3)w^2+rho w^3] dw
+       =K_c delta_eta^(3/2)[1+O(sqrt(delta_eta))],
+    K_c=integral_(-a_c)^(a_c)(1-sqrt(3)w^2)dw
+          =(4/3)a_c=4/3^(5/4).                         (22)
+
+Also z_0-z_-=2a_c sqrt(delta_eta)[1+O(sqrt(delta_eta))].
+Substituting in the exact formula (20), rather than differentiating
+the asymptotic (22), proves
+
+    lim_(h down to zero) h^(1/3)q(h)
+       =K_c^(1/3)/(2a_c)=(2/3)K_c^(-2/3)=Q_c.          (23)
+
+In particular h^(1/3)q(h) extends continuously to [0,1/4] and is
+bounded there. For some finite A_q depending only on this fixed
+profile,
+
+    0<q(h)<=A_q h^(-1/3),       0<h<=1/4.              (24)
+
+This global normalized-gap bound controls the entire population;
+an endpoint expansion alone would not suffice for the later domination.
+
+## 5. Direct density, global domination and the leading coefficient
+
+Inside W use u>0, epsilon=+-1 and v=epsilon u^(3/2)eta. Its control
+measure is exactly c_0 u^(3/2)du d eta for each sign. For any nonnegative
+Borel lifetime test f, actual selected counting and Tonelli give
+
+    E sum_finiteH0 f(ell)
+      =2c_0 integral_0^u_0 integral_0^eta_c
+                        u^(3/2) f(u^2 kappa(eta)) d eta du. (25)
+
+The null tie and discriminant controls contribute nothing. For fixed u,
+the ordinary one-variable substitution h=kappa(eta), then ell=u^2 h,
+turns (25) into integral f(ell)nu(ell)d ell, with exactly (6).
+Thus nu is an actual density representative of the expected selected
+bar measure. For each t>0 it is finite. It is continuous on positive
+t: the q argument lies in a compact subinterval of (0,1/4] locally
+in t, the moving lower endpoint is regular, and the integration interval
+shrinks to zero as t approaches u_0^2/4. No continuity of an arbitrary
+global selected Kac--Rice kernel is being imported.
+
+The total mass checks the rectangle normalization:
+
+    P(one finite H0 bar)=E total finiteH0
+      =2c_0 eta_c integral_0^u_0 u^(3/2)du
+      =(4c_0 eta_c/5)u_0^(5/2)
+      =eta_c u_0^(3/2)/(5v_0)<=1/5.                    (26)
+
+With the simplest choice v_0=eta_c u_0^(3/2), it is exactly 1/5.
+The remaining controls have no finite H0 bar. Nu is therefore not a
+probability density of total mass one; treating it as one would lose
+the full symmetric rectangle's population factor.
+
+Extend the integrand in (6) by zero for u<=2sqrt(t). Then
+
+    t^(1/3)nu(t)=2c_0 integral_0^u_0
+       1{t/u^2<1/4} u^(1/6)
+                    [(t/u^2)^(1/3)q(t/u^2)] du.        (27)
+
+For each fixed u>0 its bracket tends to Q_c. Bound (24) supplies the
+majorant A_q u^(1/6), which is integrable both at u=0 and at u=u_0.
+Dominated convergence therefore gives
+
+    lim t^(1/3)nu(t)=2c_0 Q_c (6/7)u_0^(7/6)=C_edge.
+
+This proves the first part of (8), with the constant (7). There is a
+uniform positive-lifetime bound nu(t)<=C_bound t^(-1/3), from (27).
+Integrating the already established density, or applying dominated
+convergence after ell=t a, gives its cumulative asymptotic in (8).
+We have not differentiated an asymptotic count to infer a density.
+
+The small-u control is quantitative. The contribution from u<=r to
+t^(1/3)nu(t) is at most
+
+    (12c_0 A_q/7) r^(7/6),       0<r<=u_0,             (28)
+
+uniformly in t. Its limiting coefficient is exactly
+(12c_0 Q_c/7)r^(7/6). Hence taking r down to zero removes the A3-tip
+tail uniformly. For every fixed positive u the asymptotic samples
+approach one of its two ordinary fold edges. No mass concentrating at
+u=0 is hidden in the all-u limit.
+
+## 6. The genuinely quartic-tip contribution away from the fold edges
+
+Fix 0<epsilon<eta_c and restrict (25) to 0<=eta<=eta_c-epsilon.
+On that interval kappa is bounded below by
+kappa_min=kappa(eta_c-epsilon)>0. Change variables u=sqrt(ell/kappa(eta))
+for fixed eta. The factor two for the signs cancels the derivative's
+one-half, and the selected measure has density
+
+    c_0 ell^(1/4) integral_0^(eta_c-epsilon)
+          kappa(eta)^(-5/4)
+                       1{ell<u_0^2 kappa(eta)} d eta. (29)
+
+For 0<ell<u_0^2 kappa_min the indicator is identically one, proving
+the exact formula (9). Its coefficient is finite and positive, and
+its cumulative is exactly (4/5)C_tip,epsilon t^(5/4) below that cutoff.
+It is lower order than t^(2/3), and its density is lower order than
+t^(-1/3). The controls producing these small lifetimes necessarily
+have u=O(sqrt(t)) and v=O(t^(3/4)), so they approach the actual quartic
+tip while staying away from the two fold-edge directions.
+
+The area element u^(3/2)du d eta is the reason this tip population
+differs from a flat one-parameter restricted path. A fixed-eta path
+sampled with flat u density gave a lifetime density ell^(-1/2) in the
+earlier construction. Integrating a full smooth two-dimensional control
+density through its shrinking cusp width instead gives the tip power
+ell^(1/4). Neither calculation assigns one sampling-independent power
+to the A3 label.
+
+## 7. The actual limiting marks and their physical coordinate conversion
+
+Retain epsilon=sign(v), the positive control u, and actual birth position
+x. On the cusp these are Borel. The birth location is Haar-uniform and
+independent of every control: conditional on controls it is Y plus a
+deterministic critical-point location. The direction from the younger
+maximum to its saddle is exactly epsilon e_1. All roots lie in a fixed
+small coordinate chart, so their physical displacement is the unique
+shortest one. For epsilon=+1, their chart positions are sqrt(u)z_- and
+sqrt(u)z_0; for epsilon=-1 they are their reflected positions.
+
+At the positive edge, the coalescing chart point is
+s_c=-sqrt(u)/sqrt(3). The reflected edge has +sqrt(u)/sqrt(3).
+The inverse physical chart is
+
+    y_1(s)=(2/omega)arcsin(s/sqrt(2w_1)),
+    g(u)=y_1'(s_c)
+        =2/[omega sqrt(2w_1)] [1-u/(6w_1)]^(-1/2)>0.    (30)
+
+The smallness conditions ensure the displayed denominator is positive.
+Since g is even in the coalescing sign, it applies to both edges.
+For fixed u>0, as eta increases to eta_c, the physical pair distance r is
+
+    r=2g(u)sqrt(u)a_c sqrt(delta_eta)
+                                      [1+O(sqrt(delta_eta))].
+
+Combining this with ell=u^2 kappa(eta) and (22) gives the actual
+physical cubic mark limit
+
+    k_phys=ell/r^3 -> k(u)
+          =sqrt(u)/(2sqrt(3)g(u)^3)>0.                 (31)
+
+Equivalently the oriented axial third derivative at the coalescing point
+is 2sqrt(3)sqrt(u)/g(u)^3=12k(u): the axial first and second derivatives
+vanish there, so a coordinate change contributes no lower-jet terms.
+This checks the physical normalization, not just a normalized chart ratio.
+
+The physical birth height is b+u^2 Phi_eta(z_-), with reflection at the
+negative edge. At eta=eta_c, Phi_eta(z_c)=-1/12, giving
+
+    B -> B_c(u)=b-u^2/12.                               (32)
+
+The older maximum at the edge has value b+2u^2/3, so it remains strictly
+older for each fixed u>0. This gap need not be uniform as u tends to
+zero; uniformity is instead supplied by the integrable tail (28).
+
+For each sign the joint lifetime/control/spatial density before scaling is
+
+    c_0 1{0<ell<u^2/4} u^(-1/2)q(ell/u^2)
+                                  d ell du dx/V.      (33)
+
+Therefore t^(-2/3) times the expected point measure with ell/t=a has
+the weak limit
+
+    eta_edge(da,du,d epsilon,dx)
+      =c_0 Q_c a^(-1/3) da u^(1/6)du dx/V
+                  [delta_(+1)+delta_(-1)](d epsilon),
+                  a>0, 0<u<u_0.                       (34)
+
+This follows directly by substituting ell=t a in (33), using (23),
+and dominating by c_0 A_q a^(-1/3)u^(1/6). The majorant is integrable
+on 0<a<=T, 0<u<u_0, including both small-lifetime and small-u boundaries.
+Bounded continuous tests of the additional actual marks converge to the
+pushforward of (34) under B=B_c(u), k_phys=k(u), direction=epsilon e_1.
+Continuity for fixed positive u,a and this same majorant prove the
+statement; no unjustified uniform positive-u floor is required.
+
+After normalizing the latent-u part of (34), its density is
+
+    (7/6)u_0^(-7/6)u^(1/6)du,
+
+and the two signs are equiprobable. Lifetime, latent u, sign and spatial
+position factor in this limiting intensity. Birth and cubic marks are
+deterministic functions of u and retain that correlation. There is no
+atom at u=0 or k_phys=0. In fact k(u) is asymptotic to a positive constant
+times sqrt(u) at zero, so the mass below a small cubic mark has order
+k_phys^(7/3). Summing the two signs and integrating u in (34) gives
+C_edge a^(-1/3)da dx/V, confirming (10)'s coefficient.
+
+## 8. Independent-copy PRM from exact single-field multiplicity
+
+For almost every control the field has zero or one finite H0 interval.
+Define its marked point measure on good controls by the explicit selected
+pair above, and zero on the null bad controls. This is Borel: the root
+branches are smooth on each sign/wedge component, their values and
+positions are continuous, and the control-region indicators are Borel.
+Outside the wedge the measure is empty. In particular for every t,
+
+    N_t in {0,1},
+    E[N_t;N_t>=2]=0.                                   (35)
+
+For a single field, its point process on lifetime window (0,Tt] is
+eventually empty almost surely as t decreases to zero. If a sample has
+a finite bar, its lifetime is fixed and positive; otherwise its process
+is always empty. This also follows in probability from (8).
+
+Let f>=0 be a bounded continuous test of the scaled lifetime and the
+retained marks, supported in 0<a<=T. Because there is at most one point,
+the exact one-copy Laplace deficit equals its one-point integral:
+
+    D_t(f)=E[1-exp(-sum f)]
+           =E sum [1-exp(-f)].                          (36)
+
+The measure convergence in (34), with its integrable domination, gives
+
+    t^(-2/3)D_t(f) -> integral [1-exp(-f)]d eta_edge,    (37)
+
+or the corresponding physical-mark pushforward if those marks are used.
+For n independent copies the Laplace functional is (1-D_(t_n)(f))^n.
+Under t_n->0 and n t_n^(2/3)->lambda>0 it tends to
+
+    exp(-lambda integral [1-exp(-f)]d eta_edge),        (38)
+
+the PRM Laplace functional. For lifetime and birth position alone this
+is exactly (10). Its mean on (0,T]xX is
+lambda(3/2)C_edge T^(2/3). The location measure is normalized Haar
+dx/V; C_edge was already a whole-field coefficient, so multiplying it
+by a further V would be incorrect.
+
+This is a genuine rare-event process theorem for the actual constructed
+law, and uses more than a first-moment exponent: the complete field
+construction proves the multiplicity statement (35). No same-field
+Poisson, factorial, spatial-growth, coalescing-contact or expanding-domain
+result is asserted. There is no numerical field draw or independent seed
+in this argument.
+
+## 9. Smooth local controls, density deficits and optional normalization
+
+The core coordinate change w=z_perp sqrt(1+s^2/4), independent of the
+controls, makes the whole local family exactly
+
+    b-s^4/4+u s^2/2+v s-|w|^2.                         (39)
+
+Thus its two controls are the actual canonical quadratic and linear
+controls of the quartic potential, smooth on an open rectangle through
+the origin. At the origin the germ's derivative ideal is generated by
+s^3,w_1,...,w_(d-1). Taylor's formula identifies its local Jacobian
+quotient with the span of 1,s,s^2; after omitting the additive height
+constant, the derivatives with respect to v and u supply its two
+independent generators s and s^2/2. This verifies the essential two
+local directions directly. We invoke no unseen classification or
+versal-deformation theorem to assert equivalence of arbitrary smooth
+families to (39). The exact displayed canonical unfolding suffices.
+
+The full random law still does not have the Gaussian two-site observation
+contract. At fixed distinct sites p,q, their two values and two gradients
+form a vector in R^(2d+2), a smooth function of the d translation
+parameters and the two controls. Since d+2<2d+2 for d>=2, its image is
+Lebesgue-null. Explicitly, cover the parameter space by countably many
+compact coordinate charts, use their bounded derivatives to make the
+maps Lipschitz, then cover each d+2 dimensional chart by epsilon cubes.
+Their images have total (2d+2)-volume O(epsilon^d), tending to zero.
+The observation law is therefore singular. This does not imply that
+its covariance matrix is singular; nonlinear distributions can have
+full covariance on a lower-dimensional support.
+
+Stationarity follows exactly from Haar translation Y, independently of
+controls. Uniform spatial C^q bounds follow from the fixed smooth
+surgery/cutoff and bounded u,v, for each q, so all positive norm moments
+are finite. The scalar G(0) is bounded and has positive variance:
+conditional on each control it is a nonconstant smooth function at a
+uniform point. It is therefore not a nonconstant Gaussian variable.
+We neither supply independent Gaussian coefficients nor claim any
+covariance match, P5 spectral carrier, arbitrary-jet rank, full contact
+density, or Gaussian theorem admission.
+
+If deterministic centering and variance normalization are desired, put
+mu=E G(0), sigma^2=Var G(0)>0 and Z=(G-mu)/sigma. Its pairing and
+positions are unchanged, and every lifetime is divided by sigma.
+Consequently
+
+    nu_Z(t)=sigma nu_G(sigma t),
+    C_edge,Z=sigma^(2/3)C_edge,G,
+    C_tip,epsilon,Z=sigma^(5/4)C_tip,epsilon,G.           (40)
+
+The limiting physical birth height becomes (B_c(u)-mu)/sigma and the
+cubic mark becomes k(u)/sigma. These are fixed-law scale transformations,
+not comparisons of covariances between different populations.
+
+## 10. What the actual population establishes about taxonomy
+
+The field at u=v=0 has the explicit A3 function-germ normal form
+b-s^4/4-|w|^2. The family now samples both of its essential local
+coefficients with a positive interior density, rather than following
+one prescribed C1-not-C2 path. Almost every realization is Morse,
+and the actual bar has been selected globally, not assumed.
+
+Nevertheless the dominant short bars in the full population approach
+ordinary cubic folds at fixed positive u. Their expected density has
+power -1/3 and a coefficient determined by the control population,
+while the away-edge quartic-tip contribution has density power +1/4
+and cumulative power 5/4. The small-u domination (28) is what permits
+this conclusion without silently replacing the whole cusp population
+by its fixed-u edges.
+
+Together with the restricted-path example, this falsifies two shortcuts:
+an A3 label alone does not assign a sampling-independent density power,
+and a population containing an A3 contact need not have its leading
+short-bar statistics governed by the tip. One must identify the actual
+sampling measure, the singular strata it weights, the physical height-gap
+law and the global selector. The present result supplies these inputs
+for this particular family; it is not a theorem about every A3 ensemble.
+
+The -1/3 exponent in (8) is compatible with the ordinary cubic edge
+relation (31), despite the law's singular full observation distribution.
+This does not extend the Gaussian counting theorem by covariance alone.
+We counted by explicit control pushforward, and the coefficient (7) is
+not inferred from the Gaussian jet functional. No Hk, scientific uptake,
+formal implementation, sampler coupling or full singularity taxonomy is
+closed by the present preparation.
+
+## 11. Operational context, source review and constructor boundary
+
+The following current repository sources were freshly hashed in this detached
+construction checkout. The quartic seed and matched surgery are reused narrowly;
+the full rectangle guards, root classification, selected counting and population
+limits are explicitly rederived in Sections 1-10. The earlier one-parameter
+probability law is not transferred. ST is conditional-taxonomy context only.
+
+| Context | Current repository source | UTF-8 bytes / lines | SHA256 |
+| --- | --- | --- | --- |
+| Quartic construction | [stationary quartic field](../quartic_cusp_field/PROOF.md) | 35931 / 751 | `33ba8362069346058ba33bbd8c1bd1feed62b749a96b6cbbc8836ddcbc6e4f2e` |
+| ST taxonomy | [regular-fold structural theorem](../fold_structural_universality/PROOF.md) | 43927 / 804 | `3d952a906553d578876456539a8f0dcd64aedd45099e8394cbe9db99866ec531` |
+
+The mathematical author consumed the original quartic preparation, 32,334 bytes /
+695 lines, SHA256
+`24e6f4149c0a2df2f8df1dbfa8d66da3e01a7d12bee4b803bac093cfea36bf3f`.
+The author had a prior whole saved-byte read of that source and additionally
+reread its actual lines 72-353 for this two-control task. The original/current
+quartic Sections 1-11 match exactly: 26,625 bytes / 602 lines, SHA256
+`9f1426a21b40b524ad4b25c81368489b0aa7a0c29f79659eb556bb4d1579f6d0`.
+The constructor checked the actual core mapping here. It does not create a new
+whole-751-line author read. The author's original ST whole read and earlier
+bounded lines 691-737 remain separately exposed in the historical ledger.
+No Gaussian, higher-homology or amplitude theorem premise is consumed here.
+
+The original two-control preparation is
+`two-parameter-cusp-population-preparation.md`, 32,809 bytes / 696 lines,
+SHA256 `001c4f609e167fcf139b5f46611bacdbf7203e4191060953d34485f78e2fc7c6`.
+`final_contract_review` freshly read all actual lines in untruncated ranges
+1-240, 241-480 and 481-696: mathematical PASS, no must-fix.
+`benchmark_formal_audit` separately freshly read all actual lines in untruncated
+ranges 1-232, 233-464 and 465-696: mathematical PASS, no must-fix.
+Root fully read the actual original in untruncated ranges 1-230, 231-464 and
+465-696. These original-source performances do not review the new incorporated
+wrapper or become provider-independent acceptance.
+
+Both nonauthor original-source reviewers freshly reproduced the quartic695
+and ST804 hashes, with bounded rereads of quartic lines 72-353 and ST lines
+691-737. Their preceding separate whole-source exposure is retained; two
+additional whole dependency audits are not claimed. The constructor now checks
+the preserved two-control mathematical core, the historical-ledger body,
+current context identities, baseline README preservation and local links.
+These are bounded integration checks, not a new full incorporated-source
+mathematical verdict. Current source and metadata review remain PENDING until
+separately completed against these new actual bytes.
+
+Source/route exposure is substantial. The mathematical author wrote the earlier
+quartic, spectral, selected-bar, process, higher-homology and amplitude sources.
+Root supplied the prefile full-rectangle, edge/tip and physical-mark questions.
+Benchmark supplied the separate prefile warning that wedge coverage does not
+supply the full global guards. Final_contract_review contributed a separate
+fold-chart route, prior carrier/amplitude attacks and the amplitude wrapper;
+for this source it is the original696 reviewer and subsequent wrapper constructor.
+All performers are OpenAI/Codex AI agents. Organizational independence remains zero.
+
+No new primary-source fetch is credited in this incorporation. The author's
+prior bounded AGV nomenclature preview is recorded below; no full classification,
+versal-deformation theorem or new full primary-book read is asserted. The
+local canonical family and quotient statement are explicitly derived in Section 9.
+Earlier failed preview opens by root and final remain without primary-content
+credit and were not retried as part of this construction.
+
+Section 12 preserves the original author's ledger body verbatim, including its
+historical request for an exact-file review and original artifact paths. Those
+statements describe preparation chronology. The completed original696 reviews
+above and pending incorporated-source reviews are separate. Publication, PR
+attachment, hosted checks, merge and ordinary receipt bindings will be recorded
+only when obtained; this source records none of those new integration facts.
+No compiler, sampler, seed, installation, scientific-status change or external
+post belongs to this construction.
+
+## 12. Preserved original author bindings, reading extents and exposure
+
+The only reused actual-field input is
+`/Users/dylanroy/Documents/Codex/2026-10-08/turn-the-below-into-a-reality/quartic-cusp-field-preparation.md`,
+32,334 UTF-8 bytes / 695 lines, SHA256
+`24e6f4149c0a2df2f8df1dbfa8d66da3e01a7d12bee4b803bac093cfea36bf3f`.
+The author previously reread all actual saved bytes before freezing it.
+For this task its hash was freshly reproduced and actual lines 72--353
+were freshly read, covering the seed, matching surgery, restricted
+roots, complete critical list and enclosing barrier. We reuse that
+construction narrowly and explicitly rederive the full-rectangle
+guards, root classification and selection; its one-parameter probability
+law is not transferred to the new two-dimensional population. No
+completed independent exact-source verdict on this new file is inferred
+from a review of the preceding source.
+
+The structural source's conditional-order taxonomy is context, not a
+counting theorem imported into this singular law:
+`fold-universality-worktree/experiments/universality/fold_structural_universality/PROOF.md`,
+SHA256 `3d952a906553d578876456539a8f0dcd64aedd45099e8394cbe9db99866ec531`.
+It had a prior full actual-source read; its Section 10, lines 691--737,
+was freshly read during the preceding quartic task. It separates height
+order, observation density, radial measure and selected weight rather
+than assigning an unproved higher-order class law.
+
+Prior SG, higher-homology, selected-bar, iid-process and amplitude source
+authorship supplied substantial route exposure. Their mathematical
+conclusions are not needed for (6)--(10): the actual dependencies are
+the explicit smooth cutoff construction, root analysis, elementary
+connected-component topology, Haar translation, ordinary substitutions,
+Tonelli, the implicit function theorem, dominated convergence and the
+iid Laplace product. In particular no arbitrary conditioned-Gaussian
+genericity or independence premise is transferred.
+
+Root's prefile contributions included the proposed full rectangle,
+coverage condition, fold-edge expansion and constants, tip pushforward,
+latent-u weighting, birth-height and physical-cubic-mark targets. The
+author checked them by the calculations (19)--(40). The communicated
+benchmark prefile attack highlighted that wedge coverage alone does not
+supply the global smallness guards; (14)--(17) supply those independently.
+Any later exact-file verdict must be separately bound to the final saved
+bytes, not relabel this communicated route discussion as a whole-file read.
+
+For A_k nomenclature only, the author had actually read the official AGV
+publisher pages and the two-page primary *Lists of singularities* preview,
+printed pages 242--243, during the preceding task. It identifies the A_k
+function-germ series. No whole book, real-classification theorem or
+versal criterion was read or consumed here. [Actual primary preview](https://page-one.springer.com/pdf/preview/10.1007/978-1-4612-5154-5_15).
+Formula (39) and its elementary quotient calculation provide the exact
+local statement required by this proof, without an unseen primary import.
+
+No repository, frozen predecessor, GitHub artifact, model registry,
+formal source or status was changed. No compiler, sampler, seed,
+installation, empirical draw or external post was used. This is a
+source-exposed same-team analytic preparation, organizational credit zero,
+awaiting a fresh nonauthor full actual-byte review.
diff --git a/experiments/universality/two_parameter_cusp_population/REVIEW.md b/experiments/universality/two_parameter_cusp_population/REVIEW.md
new file mode 100644
index 000000000..424674a4f
--- /dev/null
+++ b/experiments/universality/two_parameter_cusp_population/REVIEW.md
@@ -0,0 +1,116 @@
+# Two-control cusp population: construction and source-bound review
+
+Construction snapshot. Mathematical author: OpenAI/Codex universality_review.
+Wrapper constructor: final_contract_review; integration owner: root, task
+01a11cbf-068d-7102-861e-75814e715c98. Native claim6071732615 has eleven-path
+scope through02:20UTC9October2026. Initial detached base is published quartic
+PR336 head3ab12c6542c88554ad0d42e0e19aff2461dc3c0d. No main integration implied.
+
+Current [proof](PROOF.md): 38350 UTF-8 bytes/782 lines,
+SHA256 `9e8caa52b6448e23e465e646ca98d6e42e0cd71442ad4118b59145ee849e1326`.
+Sections1-10 preserve exactly 28073B/620lines,
+SHA256 `d4c2f0bc923d95a8a1905f4745dd4c2cad6025c1f1dcda99441e9f5285f5a213`.
+The original author-ledger body remains verbatim,3421B/56lines,
+SHA256 `18ae44d25c6bdcaf3991cf6dcd64566528bc5285c79d96076ddf41c146d70c51`.
+Only its heading is renamed; new operational Section11 is separate.
+
+Original frozen preparation:32809B/696lines,
+SHA256 `001c4f609e167fcf139b5f46611bacdbf7203e4191060953d34485f78e2fc7c6`.
+Two separate fresh FULL original-source mathematical PASS verdicts, no must-fix:
+final_contract_review actual1-240/241-480/481-696, and benchmark_formal_audit
+actual1-232/233-464/465-696. Root fully read actual1-230/231-464/465-696.
+Both reviewers freshly hashed quartic695/ST804 and bounded-reread quartic72-353
+and ST691-737; prior whole-source exposure is separate, not two new whole
+input reviews. The constructor then authored this wrapper, companion and README.
+It is not an independent reviewer of its own wrapper. New incorporated full
+mathematical review, root current full read and new companion/README metadata
+review are **PENDING** at this construction snapshot. Original verdicts do not
+certify these new source bytes.
+
+The exact full rectangle has a jointly smooth field family. Root guards,
+annular exclusion, old-height separation and the global barrier exhaust its
+critical list and prove the actual finite H0 elder merger inside the wedge;
+outside there is no finite H0 bar. Discriminant/tie controls are null.
+The whole-field density/count have leading fold-edge powers -1/3 and2/3,
+while the population restricted away from the edges has exact tip density
+C_tip*t^(1/4) and count(4/5)*C_tip*t^(5/4) below its stated cutoff.
+The density has total mass at most1/5, not one. Integrable all-control
+bounds justify the edge limit, latent marks and independent-whole-copy PRM;
+t_n->0 is explicit. Physical birth/cubic marks retain their correlation.
+
+Current quartic751/33ba8362 and ST804/3d952a90 are linked and fully identified
+in proof Section11. The author consumed original quartic695/24e6f414, with
+prior whole read and actual bounded72-353 reread. Its original/current
+Sections1-11 match26625B/602/9f1426a2; no new whole751 author read is inferred.
+The earlier restricted-path probability law is not transferred. ST is context
+only. No Gaussian, higher-homology or amplitude theorem premise is consumed.
+
+The [README](../README.md) retains all261 baseline lines byte-for-byte and
+adds the actual population, edge/tip distinction, count probability and marked
+iid-copy limit. Constructor source/diff/core/ledger/hash/link checks are bounded
+integration evidence; fresh nonauthor source and metadata verdicts are pending.
+
+All performers are source/route-exposed OpenAI/Codex AI agents, org0. The
+constructor is a prior fold-chart/amplitude contributor and original696 reviewer;
+root supplied prefile questions and benchmark the separate global-guards attack.
+Author's prior AGV exposure is bounded nomenclature only, without classification
+or versality credit. No new primary fetch/full book, compiler, tests, sampler,
+seeds, empirical experiment or scientific promotion is credited. No generic
+full A3, Gaussian-law, Hk, numerical, spatial or formal/blind admission.
+Publication, attachment, hosted checks, merge and receipts will be recorded
+separately only when obtained. Full landmark goal ACTIVE.
+
+## Completed incorporated-source review: 9 October 2026 UTC
+
+This append records performances completed after the construction snapshot
+above. That snapshot remains verbatim in the ordinary external
+two-control-review-construction.md:3962B/61lines,
+SHA256 cd9c6940e9f1de644e86a5aca4285c6bb0787edb1112ec7943413259cff52cd2.
+The original construction README is preserved as
+two-control-readme-construction.md:18600B/279lines,
+SHA256 96386db9e173c11d397ed30a2be08bf24078e845f870f288a8060fa4d2ad8dc1.
+The proof construction chronology in the header/Section11 is historical;
+the following current-source verdict supersedes its pending review state.
+
+benchmark_formal_audit freshly read the complete actual incorporated proof,
+38350B/782lines/SHA256
+9e8caa52b6448e23e465e646ca98d6e42e0cd71442ad4118b59145ee849e1326,
+in untruncated ranges1-260,261-520,521-782, and the complete696-to-782 diff.
+Mathematical PASS, no must-fix. This is a new verdict on these actual source
+bytes, separate from the two original696 verdicts. The proof exhausts the
+global critical list and younger-component merger across the rectangle,
+including the root/height/annular guards and null discriminant/tie sets.
+The lifetime pushforward, integrable edge majorant and away-edge tip cutoff,
+total bar mass, latent/physical marks and iid-copy scaling have their
+declared scope. No new general singularity or field-law admission follows.
+
+Benchmark also fully read the actual construction REVIEW61 and README279:
+metadata PASS. universality_review separately fully read both actual documents
+and the complete696-to-782 proof diff: metadata PASS, no must-fix.
+The latter is the mathematical author's metadata audit, not a new nonauthor
+whole782 mathematical verdict. Both freshly reproduced current quartic751
+and ST804 identities and the exact original/current quartic core mapping.
+All37 genuine local links across the three files resolve. External AGV
+nomenclature was not newly fetched; no classification theorem is imported.
+
+Root fully read the current proof in actual untruncated1-260,261-520,521-782
+ranges, the whole61-line companion and279-line README, and the complete
+original/current diff. Root reproduced the unchanged28073B/620line core,
+the two context hashes and26625B quartic core mapping. The preserved author
+ledger's3421B/56line identity includes its heading separator newline;
+excluding only that delimiter gives3420B/55lines/SHA256
+6b3656801994fa71372e1d0200c4197503b8b28b03603f2d8df2f74108cf46a3.
+The original/current body is byte-identical under either consistent boundary.
+Removing the1213B/18line README insertion restores the prior261-line bytes.
+Fresh read-only navigation reports29pages/167local links/6fragments/0problems;
+2380 external links were not crawled. The initial wrong navigation-script
+path failed before execution; discovery found tools/navigation_check.py,
+whose successful result is the evidence recorded here.
+
+Root authored this append and the README's completed-review phrase after those
+performances. A fresh bounded metadata audit of these final edits remains
+separate work; the prior mathematical verdict is on the unchanged782 proof.
+All performers remain source/route-exposed OpenAI/Codex AI agents, org0.
+No human mathematical authorship is inferred from configured Git identity.
+No hosted checks, ready state, main merge, compiler, empirical experiment,
+sampler/count coupling, blind admission or scientific promotion is claimed.