LM006: positive actual-field leading coefficient for the cubic saddle-sign layer

reviews/lm006_wrong_sign_asymptotic_20261006/NOTE.md

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+# LM006: a positive leading coefficient for the actual saddle-transverse sign layer
+
+Dylan Roy — delegated AI mathematical work. Actual author OpenAI / GPT-6
+Astra Pro, `lm006-sign-asymptotic-20261006-r12`, claim main#229/6009740815.
+Scientific effect NONE; organizational-independence credit 0. This is a new
+additive author-side analytic proposal requiring its own nonauthor review.
+It does not revise historical LM006 proof/review bytes or a scientific register.
+
+## 1. Exact object, result, and prior-work boundary
+
+Use P's normalized periodized Gaussian field in dimension two, side L=24,
+with its original coordinate axes. Fix b=6/5, h=r/2, M=(-h,0), S=(h,0), and
+condition by canonical Gaussian regression on the SIX prescribed observations
+
+    f(M)=b, f(S)=b-r^3/6, grad f(M)=grad f(S)=0.
+
+Write Q_r for this original pin law. No good event is additionally conditioned
+on. At positive r the endpoint variables of E are exactly
+
+    (a_-,a_+,b_-,b_+,c_-,c_+)
+       =(f_xx(M)/r,f_xx(S)/r,f_xy(M)/r,f_xy(S)/r,f_yy(M),f_yy(S)).
+
+The original physical determinant/type weight and full normalizer are
+
+    W_r=|det H_M det H_S| 1{H_M<0,det H_S<0}, Z_r=E_Qr W_r,
+    T_r=W_r/r^2, z_r=Z_r/r^2, d nu_r=(T_r/z_r) d Q_r.       (1)
+
+N gives the exact continuous formula, including every singular boundary,
+
+    T_r=[(a_-)_- (c_-)_- - r b_-^2]_+
+        [r b_+^2-a_+ c_+]_+.                              (2)
+
+Here x_+=max(x,0), x_-=max(-x,0). The event under investigation is ONLY
+
+    E_r={f_yy(S)>=0}={c_+>=0},
+
+not the maximum's sign, elder-partner failure, adjacency, capture or persistence.
+All limits below mean r decreases to zero through positive radii; they stay
+inside 0<r<=r_*<min(1,24) with r_* sufficiently small. No numerical r_* is given.
+
+Define the one-dimensional spectral probability and raw frequency moments
+
+    rho_n=exp(-2 pi^2 n^2/L^2)/sum_j exp(-2 pi^2 j^2/L^2),
+    omega_n=2 pi n/L, m_j=sum_n rho_n omega_n^j,
+    delta=m_4-m_2^2, s^2=m_2 delta.                       (3)
+
+These are the EXACT torus moments, not planar substitutions. Full positive
+support implies m_2>0 and delta=Var(omega_N^2)>0. All moments are finite.
+Let Q, B, D be mutually independent real Gaussians with
+
+    Q ~ N(-m_2 b,delta), B ~ N(0,s^2/4), D ~ N(0,s^2).
+
+The variable B is one HALF of the conditioned f_xxy(0), not that derivative
+itself. Define
+
+    J(d,u)=int_0^{d_+} (d-u-t)_+ (u-t)_+ dt, u>=0,
+    Gamma=E J(D,B^2), z_0=E (Q_-)^2,
+    p_0(0)=exp(-(m_2 b)^2/(2 delta))/sqrt(2 pi delta).
+
+**Theorem (new actual-field asymptotic).** For exactly the object above,
+
+    nu_r(E_r)=C_{L,b} r^3+o(r^3),
+    C_{L,b}=p_0(0) Gamma/z_0,       0<C_{L,b}<infinity.     (4)
+
+In particular this sign probability is Theta(r^3), with a strictly positive,
+explicit Gaussian-integral coefficient. The coefficient is NOT numerically
+evaluated or certified by this packet. If Phi and phi are standard normal
+CDF/density and a=m_2 b/sqrt(delta), the optional denominator identity is
+
+    z_0=((m_2 b)^2+delta) Phi(a)+m_2 b sqrt(delta) phi(a).   (5)
+
+E and N already supply endpoint convergence and the full-normalizer limit.
+C (#341) already gives the cubic upper bound and an ABSTRACT Gaussian example
+attaining cubic order. P sections 6–7 already contain the soft determinant /
+boundary-layer mechanism. The new contribution is the product-covariance
+split, exact disintegration of THIS field's sign layer, and the strictly
+positive leading coefficient for this fixed physical object. It is not a new
+generic Gaussian regression method or a first cubic-power argument. This note
+does not use C's conditional-density lemma or global claims from P.
+
+## 2. Exact independent line-process decomposition
+
+P's Fourier weights factor along the original axes. Thus its covariance is
+
+    K_L(x,y)=k(x) k(y), k(x)=sum_n rho_n exp(i omega_n x),
+    k(0)=1, k''(0)=-m_2, k''''(0)=m_4.                   (6)
+
+Set F(x)=f(x,0), H(x)=f_y(x,0), Q(x)=f_yy(x,0), before conditioning. Smooth
+mean-square differentiation of (6), justified by summable spectral moments,
+gives for every x,y on the line
+
+    Cov(F(x),F(y))=k(x-y), Cov(H(x),H(y))=m_2 k(x-y),
+    Cov(Q(x),Q(y))=m_4 k(x-y),
+    Cov(Q(x),F(y))=-m_2 k(x-y),
+    Cov(F(x),H(y))=Cov(Q(x),H(y))=0.                     (7)
+
+Consequently the centered Gaussian process
+
+    R(x)=Q(x)+m_2 F(x)
+
+has covariance delta k(x-y) and is independent of the JOINT line processes
+(F,H). Every finite family factors by its zero cross-covariance; this also
+applies to mean-square derivatives. F and H are themselves independent.
+Canonical conditioning on the four axial F observations and the two H values
+therefore leaves R unchanged and independent of their conditioned lines.
+This is an exact Gaussian-regression statement at every prescribed pin value,
+not conditioning a zero-probability event by an informal ratio.
+
+Under the specified pins we have exactly
+
+    c_-=-m_2 b+R(-h), c_+=-m_2(b-r^3/6)+R(h).
+
+Let A_r=(a_-,a_+,b_-,b_+), and put
+
+    U_r=(c_-+c_+)/2=-m_2(b-r^3/12)+(R(-h)+R(h))/2,
+    D_r=(c_+-c_-)/r=m_2 r^2/6+(R(h)-R(-h))/r.            (8)
+
+Stationarity and evenness of k give zero covariance of the R average and
+difference. They are Gaussian, so U_r and D_r are independent. Both are also
+independent of A_r: the four Hessian coordinates use only F and H. Thus
+
+    U_r independent of (A_r,D_r), D_r independent of A_r,
+    U_r~N(mu_r,v_r), mu_r=-m_2(b-r^3/12),
+    v_r=delta (1+k(r))/2,
+    D_r~N(m_2 r^2/6, 2 delta (1-k(r))/r^2).              (9)
+
+For sufficiently small positive r, v_r>=delta/2. These identities hold BEFORE
+applying the determinant tilt. No corresponding independence under nu_r is
+asserted. In particular one cannot factor expectations under nu_r as though
+the tilt preserved the Gaussian process independence.
+
+## 3. The three limiting Gaussian coordinates and the normalizer
+
+The spectral regularity and six-jet positivity required by E are verified by
+P section 2. E's original adjusted linear functionals, after imposing the same
+pins, are precisely the variables used here. Its limiting conditional law is
+
+    A_r -> (-1,1,-A/2,A/2),
+    A=Law(H''(0) | H(0)=H'(0)=0).                       (10)
+
+The axial limiting cubic pin is f_xxx(0)=2, fixed by the actual r^3/6 gap.
+The H line is independent of the F line, so no additional axial pin changes A.
+The H covariance in (7) gives
+
+    Var H(0)=m_2, Var H'(0)=m_2^2,
+    Var H''(0)=m_2 m_4,
+    Cov(H''(0),H(0))=-m_2^2,
+    Cov(H''(0),H'(0))=Cov(H(0),H'(0))=0.
+
+Its conditional mean is zero, and its conditional variance is
+
+    m_2 m_4-(-m_2^2)^2/m_2=m_2 delta.                   (11)
+
+Thus B=A/2 has variance s^2/4. From k(r)=1-m_2 r^2/2+o(r^2), (9) gives
+D_r -> D~N(0,s^2). The independence in (9) persists in the limit, yielding
+
+    (A_r,D_r) -> (-1,1,-B,B,D), B independent of D.       (12)
+
+Also U_r -> Q~N(-m_2 b,delta), independent of B,D. The raw six-pin contact law
+of f_yy(0) is therefore this Q. This re-identifies the actual scalar coordinate
+through (7), not a numerical replacement of its law by N(-b,2).
+
+E gives bounded means/covariances of A_r. Formula (9) bounds the mean/variance
+of D_r. Hence every fixed moment of ||(A_r,D_r)|| is uniformly finite for small
+r, by its finite-dimensional Gaussian law, including singular covariance.
+There is no uniform inverse for the entire six-endpoint covariance: it becomes
+singular at contact. No rate for a covariance square root is assumed.
+
+We consume the already established E/N normalizer interface, not a repeat of
+its determinant/type/uniform-integrability proof:
+
+    z_r=Z_r/r^2 -> z_0=E (Q_-)^2>0.                     (13)
+
+The strict positivity follows from the nondegenerate Gaussian Q; finiteness
+follows from its second moment. Formula (5), if used, follows by integrating
+(m+sigma x)^2 on x<-m/sigma with m=-m_2 b and sigma^2=delta. This normalizer
+is still the FULL normalizer in (1), not conditioned on E_r.
+
+## 4. Exact finite-radius disintegration of the sign event
+
+Let p_r be the density of U_r in (9). Conditional on (A_r,D_r)=(a,d), change
+only the scalar integration variable by c_+=r t. Then
+
+    c_-=r(t-d), U_r=r(t-d/2), dU_r=r dt.
+
+On E_r, a positive maximum determinant forces c_-<0, hence 0<=t<d. When
+d<=0 the entire tilted sign weight vanishes. By (2), for 0<=t<=d_+,
+
+    T_r=r^2 Psi(a,d,t),
+    Psi(a,d,t)=[(a_-)_- (d-t)-b_-^2]_+ [b_+^2-a_+ t]_+.
+
+Therefore, EXACTLY for each sufficiently small positive r,
+
+    N_r:=E_Qr[T_r 1_{E_r}]
+      =r^3 E int_0^{(D_r)_+} Psi(A_r,D_r,t)
+                              p_r(r(t-D_r/2)) dt.       (14)
+
+Tonelli applies to nonnegative integrands. The r^3 consists of TWO factors r
+from the normalized determinant factors and ONE factor r from scalar density
+integration. It is not the determinant of the full six-variable scaling map,
+and there is no additional pin density or adjacency factor. Type-boundary
+points carry zero weight and do not affect (14).
+
+The two laws in this expectation are the UNTILTED Gaussian laws specified in
+(9). Formula (14) does not replace finite-r density by a limiting density
+before integration or assume a uniform density for all endpoint coordinates.
+
+## 5. Rigorous limit of the rescaled numerator
+
+For small r, sup_x p_r(x)<=1/sqrt(pi delta), and p_r(x)->p_0(0) as (r,x)->(0,0).
+This follows directly from the continuous mean and positive limiting variance
+in (9). On each compact set of (a,d), the integrand in (14), after integration
+on [0,d_+], converges uniformly to
+
+    p_0(0) int_0^{d_+} Psi(a,d,t) dt.
+
+Positive part is continuous, and the integration interval has bounded length
+on such a compact set. No assertion of uniform convergence over all d is used.
+
+For all real a,d and 0<=t<=d_+, the elementary envelope is
+
+    0<=Psi(a,d,t)
+      <=|a_-|(d_+-t)(b_+^2+|a_+|t).
+
+Consequently the integral including p_r is bounded by
+
+    (pi delta)^(-1/2) |a_-|
+       [b_+^2 d_+^2/2+|a_+| d_+^3/6].                   (15)
+
+This is a fixed polynomial-growth bound of degree five. The uniform Gaussian
+moments from section 3 imply a bounded SECOND moment of that envelope.
+They also make its contribution outside any growing Euclidean ball uniformly
+tend to zero (for example combine Cauchy–Schwarz with a uniform second-moment
+tail bound on (A_r,D_r)). Truncate with a continuous compact cutoff, apply
+(12) and compact uniform convergence there, and then let the cutoff radius
+increase. This proves convergence of expectations in (14), not merely
+pointwise convergence or convergence in distribution:
+
+    N_r/r^3 -> p_0(0) E int_0^{D_+}
+                         (D-B^2-t)_+ (B^2-t)_+ dt
+             =p_0(0) Gamma.                             (16)
+
+Thus the actual physical numerator has the DIFFERENT power
+
+    E_Qr[W_r 1_{E_r}]=r^2 N_r
+       =p_0(0) Gamma r^5+o(r^5).                        (17)
+
+Dividing (16) by the positive limit (13) proves (4). The proof does not require
+an O(r) error for every other endpoint coordinate: its first-order random
+transverse DIFFERENCE is explicit, and convergence plus bounded Gaussian
+moments of A_r suffices for the leading coefficient.
+
+## 6. Closed elementary inner integral, finiteness and strict positivity
+
+For u>=0, J(d,u)=0 if u=0 or d<=u. If d>u>0, set
+
+    ell=min(u,d-u), M=max(u,d-u).
+
+The interval where both factors are positive is precisely 0<t<ell. Integrating
+the quadratic, without dropping either threshold, gives
+
+    J(d,u)=u(d-u)ell-d ell^2/2+ell^3/3
+          =ell^2(3M-ell)/6.                             (18)
+
+For example J(2,1)=1/3 and J(3,1)=5/6. Formula (18) is symmetric in u and
+d-u where both are nonnegative. It is homogeneous of degree three in (d,u),
+not in (d,B). It must not be extended as an unclipped polynomial to d<=u.
+An optional completely explicit coefficient representation is
+
+    Gamma=int_R int_R J(x,beta^2)
+             [exp(-x^2/(2s^2))/sqrt(2 pi s^2)]
+             [2 exp(-2 beta^2/s^2)/sqrt(2 pi s^2)] dx d beta. (19)
+
+The factor 2 in the B density is required by Var(B)=s^2/4.
+The degree-five envelope (15) at a=(-1,1,-B,B) proves finiteness.
+
+For strict positivity use a fixed positive-probability rectangle, not a sampled
+positive value. When
+
+    1/2<=B<=3/4, 1<=D<=5/4, 0<=t<=1/8,
+
+one has D-B^2-t>=5/16 and B^2-t>=1/8. Thus
+
+    J(D,B^2)>=5/1024
+
+throughout that rectangle, and independence gives
+
+    Gamma >= (5/1024) P(1/2<=B<=3/4) P(1<=D<=5/4)>0.      (20)
+
+Each Gaussian assigns positive mass to its interval since s^2>0. Together with
+p_0(0)>0 and 0<z_0<infinity, this proves the strict coefficient assertion.
+Equation (20) is symbolic and is NOT an outward numerical torus certificate.
+
+## 7. Tests, assumptions that cannot be discarded, and release boundary
+
+`sign_coefficient.py` and `test_sign_coefficient.py` use exact rational arithmetic
+and the standard library. Their twelve methods check the clipped integral
+against separate root-split polynomial integration, thresholds, homogeneity,
+the positive rectangle, exact physical determinant/type scaling, average /
+difference covariance, pin-target drift, and complete contact regression for
+three FINITE PRODUCT spectra. That regression is computed from the six-by-six
+pin Gram and the three-target cross-covariance, rather than inserted from the
+answer. It recovers diag(delta,m_2 delta,m_2 delta) for (Q,2B,D).
+
+Those finite spectra are diagnostic models, not the exact infinite torus.
+A separate nonproduct finite spectral example has
+Cov(Q+E[Y^2]F,F_xx)=E[X^2Y^2]-E[X^2]E[Y^2]=9/4, not zero. Thus positivity of
+all relevant marginal variances alone cannot justify (7)'s independence.
+The actual product identity (6), along the ORIGINAL torus axes, is essential.
+No arbitrary orientation, nonproduct covariance, varying mark, growing period
+or dimension is included in (4).
+
+Before implementation the twelve-method suite failed with twelve assertions
+identifying the missing module. Final normal and optimized runs pass all twelve.
+Four isolated source corruptions (doubling (18), losing the B/2 scale, reversing
+the pin drift, losing the physical r^2 multiplier) make the complete tests fail
+by assertions in both modes. These are finite diagnostic controls, not a
+computer proof of (16), independent mathematical review, or a Lean execution.
+
+P supplies ONLY the model/pins/spectral positivity/regularity consumed here;
+E supplies the conditional endpoint interface; N supplies the exact physical
+weight/normalizer interface; C is credited prior upper-bound/sharpness work.
+All immutable paths, commits and roles are in SOURCES.json. No earlier proof,
+review, workflow, source manifest or scientific register is modified.
+
+A new source-bound nonauthor mathematical read is required on the product-line
+split, prescribed-pin means, independence before tilt, exact layer Jacobian,
+Gaussian limit and uniform-integrability argument, and strict positivity.
+General existing CI does not automatically execute this standalone checker or
+formalize this new theorem. After review, an eligible separate integrator must
+reconcile the actual current base and its own required checks. No author
+self-merge, scientific flag change, full-program closure or claim about elder
+failure/capture/persistence follows from this sign-coordinate asymptotic.

reviews/lm006_wrong_sign_asymptotic_20261006/SOURCES.json

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+{
+  "object": "LM006-WRONG-SIGN-ASYMPTOTIC-20261006",
+  "author": "OpenAI / GPT-6 Astra Pro",
+  "session": "lm006-sign-asymptotic-20261006-r12",
+  "claim": "main#229/6009740815",
+  "scientific_effect": "NONE",
+  "nonauthor_review": "PENDING",
+  "source_repository": "d6g8k5htny-coder/Math-",
+  "source_cut": "8d78be3aebf7320d8df544c3961d16f25f6ec6ab",
+  "files": [
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+    },
+    {
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+    }
+  ],
+  "sources": [
+    {
+      "tag": "P",
+      "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
+      "commit": "8d78be3aebf7320d8df544c3961d16f25f6ec6ab",
+      "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2",
+      "role": "Model and pins in section 1; positive product spectrum and regularity in section 2. No global theorem imported."
+    },
+    {
+      "tag": "E",
+      "path": "reviews/lm006_endpoint_extension_20261005/NOTE.md",
+      "commit": "8d78be3aebf7320d8df544c3961d16f25f6ec6ab",
+      "git_blob": "551aa563562bc88a19e33d146259c03aa2d318f6",
+      "role": "Sections 1-5: exact six-pin adjusted endpoint interface, conditional mean/covariance convergence and Gaussian moments."
+    },
+    {
+      "tag": "N",
+      "path": "reviews/lm006_normalizer_stability_20261005/NOTE.md",
+      "commit": "8d78be3aebf7320d8df544c3961d16f25f6ec6ab",
+      "git_blob": "5790aaa0332bd8e79b276be1577e4c5f3502eed6",
+      "role": "Sections 1-2 and 5: exact T_r=W_r/r^2, contact weight and full-normalizer limit."
+    },
+    {
+      "tag": "C",
+      "path": "reviews/lm006_cubic_sign_band_20261006/NOTE.md",
+      "commit": "771913a7dffdc1283e491166d0faa9ec53e1f06a",
+      "git_blob": "88dae06c5eefd102239f976513eb5e97c4ac7580",
+      "role": "Prior-work credit only: cubic upper bound and abstract sharpness. Fetched at its immutable PR source; not asserted present on this branch."
+    }
+  ]
+}