LM006: conditional sign-band integration and cubic saddle-sign bound

reviews/lm006_cubic_sign_band_20261006/NOTE.md

@@ -0,0 +1,314 @@
+# LM006: conditional sign-band integration and a cubic saddle-sign bound
+
+Dylan Roy — delegated AI mathematical work. Actual author OpenAI / GPT-6
+Astra Pro, session `round18-cubic-sign-band-audit-20261006`.
+Scientific effect NONE. This is an additive analytic companion, not a Lean
+formalization, a replacement of a historical proof, or program acceptance.
+
+## 1. Scope, exact predecessors, and what is additional
+
+Use the six adjusted endpoint coordinates from #301 and the exact typed
+weight from #315, with the conventions of #318/#325:
+
+    v=(a_minus,a_plus,b_minus,b_plus,c_minus,c_plus),
+    T_r(v)=[(a_minus)_- (c_minus)_- - r b_minus^2]_+
+           [r b_plus^2-a_plus c_plus]_+,       0<=r<=1.       (1)
+
+Here x_+=max(x,0), x_-=max(-x,0). For physical r>0, T_r=W_r/r^2.
+Let E={c_plus>=0}; this is the SADDLE transverse sign, not the sign already
+forced at the maximum. Write z_r=E_Qr T_r and nu_r(E)=E_Qr[T_r 1_E]/z_r.
+Qr is the original Gaussian pin law; it has no additional good-event
+conditioning. No pin density or coordinate Jacobian is inserted into z_r.
+
+The #325 moment-only estimate is of order r e+e^4, hence O(r^2) when a
+valid coupling error satisfies e=O(r). We add a CONDITIONAL density bound
+for the scalar c_plus after all other endpoint coordinates are specified.
+Integrating the actual sign interval yields order r e^2+e^5 and hence O(r^3).
+Marginal density, full density at each positive r, and bounded covariance
+alone do not imply the uniform conditional-density premise.
+
+Prior-work credit is essential: P, `UNIFORM_MATRIX_CAP_AND_LIFETIME.md`,
+sections 6–7, already uses two soft determinant factors and a boundary-layer
+width to produce the cubic power, including its separate scalar case. This
+note is not the first cubic soft-layer argument or a new generic Gaussian
+regression method. It supplies the explicit one-dimensional endpoint-band
+formula, a sufficient 11-observation confluent Gram proof using a finite tenth
+spectral moment, and the source-matched consequence for this particular sign
+event. P's global selection and lifetime theorems are NOT premises here.
+Neither #329's weaker-regularity rate nor #333's application is rewritten.
+
+## 2. Exact deterministic band and its Lebesgue integral
+
+Put t=c_plus, g=c_plus-c_minus, so c_minus=t-g. Hold
+h=(a_minus,a_plus,b_minus,b_plus,g) fixed. On E and positive typed weight,
+necessarily a_minus<0 and 0<=t<g. In particular the weight vanishes on E
+when g<=0. For all real h and r in [0,1],
+
+    T_r(h,t) 1_{t>=0}
+      <= |a_minus| (g_+-t)(r b_plus^2+(a_plus)_- t)
+                    1_{0<=t<=g_+}.                        (2)
+
+Indeed the first positive part in (1) is at most
+|a_minus|(g-t) on its support; the second is at most
+r b_plus^2+(a_plus)_-t for t>=0. At either type boundary the determinant
+weight is zero, so no boundary mass is removed in (2).
+
+Integrating the nonnegative quadratic polynomial gives
+
+    int_0^infty T_r(h,t) dt
+      <= |a_minus| [r b_plus^2 g_+^2/2
+                          +(a_plus)_- g_+^3/6].            (3)
+
+The two coefficients are individually sharp for this envelope: with
+b_minus=0 and a_plus=0 the first integral is exactly the first term; with
+b_minus=b_plus=0 and a_plus<0 it is exactly the second term. These witnesses
+work at positive r as well. This does not assert simultaneous sharpness of
+all later Gaussian constants or sharpness for a particular field.
+
+## 3. Conditional density is the load-bearing new hypothesis
+
+Let X be any measurable R6-valued random vector and
+H=(X_a_minus,X_a_plus,X_b_minus,X_b_plus,G), G=X_c_plus-X_c_minus.
+Assume the conditional law of X_c_plus given H has a Lebesgue density
+f(t|H)<=L almost everywhere for a fixed finite L. All expectations in this
+section are under the SAME law of X. Disintegration and Tonelli, followed
+by (3), give
+
+ N_r:=E[T_r(X)1_E]
+   <= L { (r/2) E[|X_a_minus| X_b_plus^2 G_+^2]
+          +(1/6) E[|X_a_minus| (X_a_plus)_- G_+^3] }.       (4)
+
+This holds as an extended nonnegative inequality; the following moment
+assumptions make its right-hand side finite. No independence is assumed.
+
+One cannot substitute a bound on the marginal density of X_c_plus. For a
+rational counterexample, take t uniform on [0,1], g=3t/2, a_minus=a_plus=-1,
+and b_minus=b_plus=0. Then T=t^2/2, so N=1/6. Inserting the marginal density
+cap L=1 into the proposed right-hand side gives E[g^3]/6=9/64<1/6.
+H determines t; its conditional law has no density. This example is not a
+Gaussian field. Even for Gaussian variables, a nonsingular marginal can
+become deterministic after conditioning on a correlated coordinate.
+
+## 4. Explicit Gaussian-error consequence
+
+Define U=X_a_plus-1, u2=E U^2, g2=E G^2,
+K2=E X_a_minus^2, K6=E[X_a_minus^2 X_b_plus^4]. Assume K2,K6 finite and
+U,G individually Gaussian, possibly noncentered, degenerate and dependent.
+The elementary inequality (x)_- <= (x-1)^2/4 gives
+
+ N_r <= L [ (sqrt(3 K6)/2) r g2
+       +(105*10395)^(1/4) sqrt(K2) u2 g2^(3/2)/24 ].        (5)
+
+Details of the moment step, without independence:
+
+ E[|X_a_minus| X_b_plus^2 G_+^2]
+      <= sqrt(K6) sqrt(E G^4) <= sqrt(3 K6) g2,
+ E[|X_a_minus| U^2 |G|^3]
+      <= sqrt(K2) (E U^4 G^6)^(1/2)
+      <= sqrt(K2) (E U^8 E G^12)^(1/4)
+      <= (105*10395)^(1/4) sqrt(K2) u2 g2^(3/2).
+
+For any real Gaussian Z=m+sZ0, its binomial even-moment expansion implies
+E Z^(2k) <= (2k-1)!! (E Z^2)^k. Coefficientwise, the ratio of the j-th
+variance coefficient to the corresponding upper coefficient is
+1/(2(k-j)-1)!!<=1, with (-1)!!=1. The needed constants are 3,105,10395
+for orders 4,8,12. Thus (5) also covers zero variances and nonzero means.
+
+Now assume a specified coupling to Y with Y_a_plus=1 and Y_c_plus=Y_c_minus
+almost surely, and e^2=E||X-Y||^2. Then u2<=e^2 and g2<=2e^2, so
+
+ N_r <= L [sqrt(3 K6) r e^2
+             +(105*10395)^(1/4) sqrt(2 K2) e^5/12].        (6)
+
+If 0<z_*<=z_r and e<=K_e r, and K2<=C2, K6<=C6 on a fixed interval,
+
+ nu_r(E) <= min(1, (L/z_*) [sqrt(3 C6) K_e^2 r^3
+             +(105*10395)^(1/4) sqrt(2 C2) K_e^5 r^5/12]). (7)
+
+In particular the probability is O(r^3). The division is by the normalized
+normalizer z_r=Z_r/r^2, not the physical Z_r. The estimate bounds the actual
+typed numerator directly; no L1 weight error is silently set to zero.
+Neither (6) nor (7) is valid from a mere second-moment error without the
+stated Gaussian higher-moment and conditional-density inputs.
+
+## 5. A uniform conditional-density bound from a confluent 11-jet frame
+
+Work in the original centered Gaussian Hilbert space, before imposing pins.
+Set F(x)=f(x,0), H_y(x)=f_y(x,0), Q(x)=f_yy(x,0). Assume on a common open
+interval around zero that F is C5, H_y is C3, Q is C1 in mean square and
+that the indicated derivatives are actual jointly Gaussian derivatives.
+Let
+
+ J11=(F(0),F'(0),...,F^(5)(0),
+                H_y(0),...,H_y'''(0),Q'(0)).
+
+Assume the covariance of the TWELVE-component vector (J11,Q(0)) is positive
+definite. This is stronger than #301's six-pin Gram hypothesis and will be
+verified for a stated spectral class in section 6. Define
+
+    sigma0^2=Var(Q(0) | J11)>0.
+
+This is the eleven-observation residual variance, not the variance of Q_L
+conditioned only on the six contact pins used to define z0 below.
+
+For h=r/2>0 consider the 11 observations
+
+ O_r=(F(-h),F'(-h),F''(-h), F(h),F'(h),F''(h),
+      H_y(-h),H_y'(-h),H_y(h),H_y'(h), Q(h)-Q(-h)).        (8)
+
+These are exactly the six raw pins together with the four axial/mixed
+Hessian entries and the transverse difference, up to permutation and
+nonzero scaling/subtraction of known pin rows. They do NOT include Q(h)
+as an extra conditioned observation. Conditioning first on the six pins
+and then on H in section 3 is equivalent to conditioning on (8).
+
+Here is an explicit invertible transformation removing the coalescing
+singularity. For m=3 or 2 define the fixed 2m by 2m real matrix
+
+ V_m[(s,ell),j] = (d^ell/dt^ell t^j)|_{t=s},
+ s=-1,+1; ell=0,...,m-1; j=0,...,2m-1.                    (9)
+
+It is invertible: a polynomial of degree <2m with all m derivatives
+vanishing at both nodes would be divisible by (t-1)^m(t+1)^m, so is zero.
+Use V_3 on the six F rows in (8), multiplying the ell-th observed derivative
+by h^ell first. If c_j is the resulting coefficient after V_3^-1, set
+R_j=j! h^-j c_j. Hilbert-space Taylor expansion gives, for ell<=2,
+
+ h^ell F^(ell)(s h)
+    = sum_{j=ell}^5 [F^(j)(0) h^j/j!] V_3[(s,ell),j]+o(h^5).
+
+Thus R_j=F^(j)(0)+o(h^(5-j)) and all six rows converge in L2.
+Apply the identical construction with m=2 to H_y and its first derivative;
+the four rows converge to H_y^(j)(0), j=0,...,3. Divide the last row in (8)
+by 2h to obtain convergence to Q'(0). The combined change of rows is
+invertible for every h>0. Call the transformed vector R_r; then
+
+    (R_r,Q(h)) -> (J11,Q(0)) in L2.                       (10)
+
+The 12 by 12 covariance matrices converge. Since the limiting one is
+positive definite, its 11 by 11 observation block is invertible for small r,
+and the scalar Schur complement is continuous and tends to sigma0^2.
+Consequently
+
+    Var(Q(h) | O_r) >= sigma0^2/2                        (11)
+
+on a sufficiently small fixed interval. In a jointly Gaussian family this
+conditional variance does not depend on the prescribed values. The canonical
+conditional density of Q(h) is therefore bounded by
+
+    L=(pi sigma0^2)^(-1/2).                              (12)
+
+The actual successive pin/endpoint conditionings in section 3 have precisely
+this variance: their raw observations are related to O_r by the invertible
+maps just described. Their means may depend on all prescribed values, but
+a Gaussian density's supremum does not depend on its mean. The raw or scaled
+six-endpoint covariance is not assumed uniformly invertible; several of its
+eigenvalues may tend to zero. Equation (11) is a scalar residual statement
+proved through the DIFFERENT rescaled observation frame R_r.
+
+## 6. Spectral-class and source-matched fixed-field corollaries
+
+Suppose the actual stationary covariance is
+
+ C(z)=sum_{k in Z2} p_k exp(i nu k.z),
+ p_k>0, p_-k=p_k, nu>0, sum_k p_k(1+|nu k|^10)<infinity.   (13)
+
+The tenth moment gives the mean-square derivative regularity in section 5
+by dominated convergence in the weighted spectral L2 space. The 12 derivative
+symbols, after nonzero scale/phase factors, are the distinct monomials
+
+    1,x,x^2,x^3,x^4,x^5, y,xy,x^2y,x^3y, xy^2,y^2.
+
+A zero-variance linear combination would vanish at every integer-frequency
+pair because every p_k is positive. Fixing each integer y and then each
+integer x shows that such a polynomial must be identically zero. Hence the
+12-jet covariance is positive definite. This verifies the extra premise,
+not merely the six-pin Gram. Removing the Q(0) mode in a finite model can
+leave a positive residual variance at every r>0 but make it vanish at contact;
+the checker retains that countercontrol.
+
+Use the same axial pins as #301: F(-h)=b, F(h)=b-r^3/6 and both gradient
+pins zero, with b=6/5. This is k=1/6 in P's convention. The #301 pin frame
+has d_r=(b-r^3/12,0,0,0,0,2); after pinning, the adjusted axial/mixed Hessian
+entries are precisely the physical entries divided by r. There is no switch
+of endpoint coordinates or height-gap coefficient in applying (8).
+
+Under (13), the stronger derivative assumptions of #329 hold, so its
+explicit common regression coupling gives e<=K_e r, bounded Gaussian means
+and covariances, and z_r>=z0/2>0 on a sufficiently small interval. The same
+marginals and coupling are used in (5)--(7). For example its moment bounds
+supply C2=M^2+6Lambda and C6=32(M^6+480Lambda^3). The limiting contact
+transverse Gaussian has positive variance by the 12-jet Gram, hence
+z0=E(Q_L)_-^2>0. Intersect that interval with the one in (11) and r<=1.
+Equation (7) proves the cubic sign bound for this spectral class.
+
+For P's actual periodized Gaussian field, specialize d=2, L=24 and nu=2pi/24.
+P sections 1--2 specify exactly
+
+ p_k=exp(-2pi^2|k|^2/24^2) / sum_j exp(-2pi^2|j|^2/24^2).
+
+These weights are strictly positive and have finite moments of every order;
+thus (13) holds. This is a check of the actual model and pins, not an import
+of P's global theorems. We obtain constants C<infinity and r_*>0 such that
+
+  nu_r{ f_yy(S)>=0 } <= C r^3,             0<r<=r_*<24.    (14)
+
+All constants are existence/symbolic constants, not numerical enclosures.
+This statement is fixed in dimension, period, birth, direction and gap mark.
+It is NOT a result about 1-p_r, the elder partner, extra critical points,
+capture, persistence lifetimes, all marks, or growing windows. It does not
+make the endpoint total-variation obstruction disappear. P's boundary-layer
+mechanism is explicitly credited; this is its independently checkable
+finite-jet/sign-coordinate refinement, not a new whole-program closure.
+
+## 7. Cubic order is attained in an abstract Gaussian family
+
+Let Q be standard normal and, for 0<r<=1, define
+
+    X_r=(-1,1,-1,1,Q-3r,Q),
+    Y=(-1,1,-1,1,Q,Q).
+
+Here H is deterministic, Q|H remains
+standard normal, e=3r and the relevant moments are bounded. The exact weight
+is (2r-Q)_+(r-Q)_+. Its normalizer tends to E Q_-^2=1/2 by domination.
+The numerator of E is
+
+    int_0^r (2r-t)(r-t) phi(t) dt
+       = r^3 int_0^1 (2-s)(1-s) phi(rs) ds.
+
+Continuity of phi and integral_0^1(2-s)(1-s)ds=5/6 yield
+
+    lim_{r->0} nu_r(E)/r^3=5 phi(0)/3>0.                 (15)
+
+Thus an exponent strictly larger than 3 cannot hold for the entire abstract
+Gaussian/density/coupling class. This rank-one Gaussian family is NOT claimed
+to be an actual six-pin periodic field, or an actual-field lower asymptotic.
+It establishes sharp order for the assumptions, not equality of any coarse
+constant in (7).
+
+## 8. Reproduction, controls, and boundaries
+
+`python -B -S test_cubic_band.py` and its `-O` version run 12 methods.
+`cubic_band.py` runs 2214 exact controls: 2160 band integrations, 40 Hermite
+monomial recoveries, four conditional variances, four sharp cubic integrals
+and six isolated witnesses. Its outputs are deterministic. The finite
+Gaussian model uses independent coefficients of 15 explicitly defined
+monomials. Exact Schur computation gives 1+h^4/4; deleting its constant Q
+mode gives h^4/4, so positive-radius density alone does not supply a uniform
+floor. It is not an evaluation of the infinite periodic covariance.
+
+M1--M6 must exit1 with the respective reasons BAND_R_COEFFICIENT,
+BAND_CUBIC_COEFFICIENT, HERMITE_DERIVATIVE_SCALE, GAUSSIAN_TWELFTH,
+CONDITIONAL_NOT_MARGINAL, MARGINAL_DENSITY_NOT_SUFFICIENT. Unknown arguments
+exit2. Numerical checks are not proofs of the continuum claims. No existing
+workflow automatically runs this new checker; no Lean target is added.
+
+General linear Gaussian conditioning is prior methodology. The primary
+abstract record Majumdar--Majumdar, arXiv:1710.09285, was consulted as context,
+not imported as an unread proof of (10)--(14). The finite covariance/Schur and
+polynomial arguments above give the assumptions and derivation explicitly.
+A fresh nonauthor review must inspect the new conditional-density step,
+constants, exact pin correspondence, Gaussian moment use and exclusions.
+All historical proof bytes, review dispositions and scientific flags remain
+unchanged. In particular this does not repair the separate source-admission
+gate or resolve missing independent formal alignment/provenance.

reviews/lm006_cubic_sign_band_20261006/SOURCES.json

@@ -0,0 +1,74 @@
+{
+  "schema_version": 1,
+  "scientific_effect": "NONE",
+  "author": {
+    "provider": "OpenAI",
+    "model": "GPT-6 Astra Pro",
+    "session": "round18-cubic-sign-band-audit-20261006"
+  },
+  "files": {
+    "NOTE.md": {
+      "bytes": 15359,
+      "sha256": "21284fa5945752736483437a59e5c98477b9ded3e90c6107d1454bf5f3d8ce9c",
+      "git_blob": "88dae06c5eefd102239f976513eb5e97c4ac7580"
+    },
+    "cubic_band.py": {
+      "bytes": 7865,
+      "sha256": "932a9ec1f038dce87e27007c917efa40cc07702e77e789a055ccacf5aa1e67cb",
+      "git_blob": "ea3225adc01311f6bd50e80c9d629c3c04b42345"
+    },
+    "test_cubic_band.py": {
+      "bytes": 4983,
+      "sha256": "ab92fbd9d36a0342f0bd277e705b85286645c574afb6061b4e7814b30d66ff27",
+      "git_blob": "bd565daa715a3532f7b7d1e0e0a432ef2acef2f7"
+    }
+  },
+  "sources": [
+    {
+      "id": "P",
+      "repository": "d6g8k5htny-coder/Math-",
+      "commit": "b25cac0bcbb4032d624f14795257d9578c90d105",
+      "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
+      "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2",
+      "consumed": "sections1-2 model, pin convention, positive spectrum; sections6-7 prior-methodology credit only, no global theorem imported"
+    },
+    {
+      "id": "E",
+      "repository": "d6g8k5htny-coder/Math-",
+      "commit": "e31b6ad8356e0760c8a7fb6e816ff640c45176c9",
+      "path": "reviews/lm006_endpoint_extension_20261005/NOTE.md",
+      "git_blob": "551aa563562bc88a19e33d146259c03aa2d318f6",
+      "consumed": "sections2-3 exact pin frame and adjusted coordinates"
+    },
+    {
+      "id": "N",
+      "repository": "d6g8k5htny-coder/Math-",
+      "commit": "5e027afcae8c7bd42a6df8ae2025f82105b3e586",
+      "path": "reviews/lm006_normalizer_stability_20261005/NOTE.md",
+      "git_blob": "5790aaa0332bd8e79b276be1577e4c5f3502eed6",
+      "consumed": "exact normalized physical weight; new deterministic bound rederived"
+    },
+    {
+      "id": "R",
+      "repository": "d6g8k5htny-coder/Math-",
+      "commit": "5b22d6f8d3335e2d863702fb7bbd75b5a891178e",
+      "path": "reviews/lm006_quantified_regression_20261005/NOTE.md",
+      "git_blob": "a58948d251abdb34b334e1a1da83ec8982de29dc",
+      "sha256": "b68e077adbfdc551e6211cf8e61aa2d98719f1de4f653abcdbd5d90c613a11b1",
+      "consumed": "sections2-5 specified RMS coupling and uniform Gaussian moments/normalized lower floor; published immutable source, not a claim of merge"
+    },
+    {
+      "id": "S",
+      "repository": "d6g8k5htny-coder/Math-",
+      "commit": "1dfe801eb61a8335a2e42b886fca9b000fb3598c",
+      "path": "reviews/lm006_typed_sign_suppression_20261005/NOTE.md",
+      "git_blob": "9015971fa03f44ae99d410facc6f3d8c26cdc945",
+      "consumed": "prior r e+e^4 comparison only; not used to prove new band estimate"
+    }
+  ],
+  "external_context": {
+    "arxiv": "1710.09285",
+    "read": "abstract record only",
+    "role": "prior linear Gaussian conditioning; not proof premise"
+  }
+}