From 64c0a203b94105f0fccbe6f8116a94e13431f84e Mon Sep 17 00:00:00 2001 From: Dylan Roy <249851642+d6g8k5htny-coder@users.noreply.github.com> Date: Mon, 5 Oct 2026 22:24:05 -0700 Subject: [PATCH] LM006: positive actual-field leading coefficient for the cubic sign layer MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Dylan Roy — delegated AI mathematical work. Actual author OpenAI / GPT-6 Astra Pro, lm006-sign-asymptotic-20261006-r12, main229/6009740815. Add four isolated source-bound files only. Exact product line-process split, finite-radius scalar disintegration and positive C r^3 + o(r^3) asymptotic for the fixed saddle-transverse sign event. Twelve exact finite diagnostics pass in both Python modes; four source variants rejected. Nonauthor mathematical review and current-candidate checks remain required. No elder/Palm/global conclusion or scientific-register change. Scientific effect NONE. --- .../NOTE.md | 333 ++++++++++++++++++ .../SOURCES.json | 60 ++++ .../sign_coefficient.py | 171 +++++++++ .../test_sign_coefficient.py | 122 +++++++ 4 files changed, 686 insertions(+) create mode 100644 reviews/lm006_wrong_sign_asymptotic_20261006/NOTE.md create mode 100644 reviews/lm006_wrong_sign_asymptotic_20261006/SOURCES.json create mode 100644 reviews/lm006_wrong_sign_asymptotic_20261006/sign_coefficient.py create mode 100644 reviews/lm006_wrong_sign_asymptotic_20261006/test_sign_coefficient.py diff --git a/reviews/lm006_wrong_sign_asymptotic_20261006/NOTE.md b/reviews/lm006_wrong_sign_asymptotic_20261006/NOTE.md new file mode 100644 index 000000000..ea914c1b8 --- /dev/null +++ b/reviews/lm006_wrong_sign_asymptotic_20261006/NOTE.md @@ -0,0 +1,333 @@ +# 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 00 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=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<=tp_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=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 F: + """Integral of (d-u-t)_+ (u-t)_+ over 0 <= t <= d_+.""" + d, u = F(d), F(u) + if u < 0: + raise ValueError('u is a square and must be nonnegative') + if u == 0 or d <= u: + return F(0) + lo, hi = min(u, d-u), max(u, d-u) + return lo**2 * (3*hi-lo) / 6 + + +def layer_weight(am: Rational, ap: Rational, bm: Rational, bp: Rational, + d: Rational, t: Rational) -> F: + """Exact two-factor rescaled sign layer, with its t >= 0 event.""" + am, ap, bm, bp, d, t = map(F, (am, ap, bm, bp, d, t)) + if t < 0: + return F(0) + return max(max(-am, 0)*max(d-t, 0)-bm**2, 0)*max(bp**2-ap*t, 0) + + +def physical_weight(r: Rational, am: Rational, ap: Rational, bm: Rational, + bp: Rational, cm: Rational, cp: Rational) -> F: + """Original determinant/type weight for the two physical Hessians.""" + r, am, ap, bm, bp, cm, cp = map(F, (r, am, ap, bm, bp, cm, cp)) + if r <= 0: + raise ValueError('physical radius must be positive') + det_m, det_s = r*am*cm-r*r*bm*bm, r*ap*cp-r*r*bp*bp + if r*am < 0 and det_m > 0 and det_s < 0: + return -det_m*det_s + return F(0) + + +def transverse_means(m2: Rational, b: Rational, r: Rational) -> tuple[F, F]: + m2, b, r = map(F, (m2, b, r)) + return -m2*(b-r**3/12), m2*r**2/6 + + +def transpose(a: Sequence[Sequence[Rational]]) -> list[list[F]]: + return [list(map(F, row)) for row in zip(*a)] + + +def matmul(a: Sequence[Sequence[Rational]], b: Sequence[Sequence[Rational]]) -> list[list[F]]: + if not a or not b or any(len(row) != len(b) for row in a): + raise ValueError('incompatible matrix dimensions') + if any(len(row) != len(b[0]) for row in b): + raise ValueError('ragged matrix') + return [[sum((F(x)*F(y) for x,y in zip(row,col)), F(0)) + for col in zip(*b)] for row in a] + + +def inverse(a: Sequence[Sequence[Rational]]) -> list[list[F]]: + n = len(a) + if n == 0 or any(len(row) != n for row in a): + raise ValueError('nonempty square matrix required') + rows = [list(map(F,row))+[F(i==j) for j in range(n)] for i,row in enumerate(a)] + for j in range(n): + pivot = next((i for i in range(j,n) if rows[i][j]), None) + if pivot is None: + raise ValueError('singular matrix') + rows[j], rows[pivot] = rows[pivot], rows[j] + scale = rows[j][j] + rows[j] = [x/scale for x in rows[j]] + for i in range(n): + if i != j: + factor = rows[i][j] + rows[i] = [x-factor*y for x,y in zip(rows[i],rows[j])] + return [row[n:] for row in rows] + + +def diagnostic_spectra() -> list[dict[int,F]]: + """Three finite even probability spectra; NONE is the exact torus spectrum.""" + return [{0:F(1,3),-1:F(1,6),1:F(1,6),-2:F(1,6),2:F(1,6)}, + {i:F(1,7) for i in range(-3,4)}, + {0:F(1,2),-1:F(1,8),1:F(1,8),-3:F(1,8),3:F(1,8)}] + + +def moments(weights: Mapping[int,Rational], order: int) -> list[F]: + w = {k:F(v) for k,v in weights.items()} + if sum(w.values(),F(0)) != 1 or any(v <= 0 for v in w.values()): + raise ValueError('positive probability masses required') + if any(w.get(-k,F(0)) != v for k,v in w.items()): + raise ValueError('even spectrum required') + return [sum((v*F(k)**j for k,v in w.items()),F(0)) for j in range(order+1)] + + +def derivative_cov(alpha: tuple[int,int], beta: tuple[int,int], m: Sequence[F]) -> F: + powers = (alpha[0]+beta[0], alpha[1]+beta[1]) + if any(x % 2 for x in powers): + return F(0) + sign = -1 if (sum(beta)+sum(powers)//2) % 2 else 1 + return sign*m[powers[0]]*m[powers[1]] + + +PINS = ((0,0),(1,0),(0,1),(2,0),(1,1),(3,0)) +TARGETS = ((0,2),(2,1),(1,2)) # q, twice B, D + + +def contact_regression(m: Sequence[F], b: Rational) -> tuple[list[F],list[list[F]]]: + """Compute from the full finite product-jet covariance, not from the answer.""" + gram = [[derivative_cov(x,y,m) for y in PINS] for x in PINS] + cross = [[derivative_cov(x,y,m) for y in PINS] for x in TARGETS] + target = [[derivative_cov(x,y,m) for y in TARGETS] for x in TARGETS] + regression = matmul(cross,inverse(gram)) + mean = [row[0] for row in matmul(regression,[[F(b)],[0],[0],[0],[0],[2]])] + removed = matmul(regression,transpose(cross)) + return mean, [[x-y for x,y in zip(row,loss)] for row,loss in zip(target,removed)] + + +def contact_laws(m2: Rational, m4: Rational, b: Rational) -> dict[str,F]: + m2,m4,b = map(F,(m2,m4,b)) + delta = m4-m2*m2 + if m2 <= 0 or delta <= 0: + raise ValueError('positive m2 and spectral-square variance required') + return {'mean_q':-m2*b,'variance_q':delta, + 'variance_B':m2*delta/4,'variance_D':m2*delta} + + +def residual_cross(joint: Mapping[tuple[int,int],Rational]) -> F: + """Cov(Q + E[Y^2] F, F_xx) at contact, without a product assumption.""" + if sum(map(F,joint.values()),F(0)) != 1: + raise ValueError('probability weights required') + x2 = sum((F(p)*x*x for (x,y),p in joint.items()),F(0)) + y2 = sum((F(p)*y*y for (x,y),p in joint.items()),F(0)) + x2y2 = sum((F(p)*x*x*y*y for (x,y),p in joint.items()),F(0)) + return x2y2-x2*y2 + + +def power_ledger() -> dict[str,int]: + return {'transverse_factors':2,'density_variable':1,'normalized_numerator':3, + 'physical_numerator':5,'physical_normalizer':2,'probability':3} + + +def diagnostics() -> dict: + rows = [] + for weights in diagnostic_spectra(): + m = moments(weights,10) + mean,cov = contact_regression(m,F(6,5)) + laws = contact_laws(m[2],m[4],F(6,5)) + expected = [[laws['variance_q'],0,0], + [0,4*laws['variance_B'],0],[0,0,laws['variance_D']]] + if mean != [laws['mean_q'],0,0] or cov != expected: + raise AssertionError('contact regression does not match the stated laws') + rows.append({'m2':str(m[2]),'m4':str(m[4]), + 'laws':{k:str(v) for k,v in laws.items()}}) + return {'object':'LM006-WRONG-SIGN-ASYMPTOTIC-20261006', + 'scientific_effect':'NONE','mathematical_acceptance':False, + 'scope':'finite rational identities only; no infinite-field coefficient evaluated', + 'spectral_models':rows,'positive_box_integral_floor':'5/1024', + 'band_examples':{'J(2,1)':str(band_integral(2,1)), + 'J(3,1)':str(band_integral(3,1))}, + 'power_ledger':power_ledger()} + + +if __name__=='__main__': + print(json.dumps(diagnostics(),sort_keys=True,indent=2)) diff --git a/reviews/lm006_wrong_sign_asymptotic_20261006/test_sign_coefficient.py b/reviews/lm006_wrong_sign_asymptotic_20261006/test_sign_coefficient.py new file mode 100644 index 000000000..b614f6b38 --- /dev/null +++ b/reviews/lm006_wrong_sign_asymptotic_20261006/test_sign_coefficient.py @@ -0,0 +1,122 @@ +"""Exact finite diagnostics, not a proof of the infinite-field limit.""" +from fractions import Fraction as F +import itertools +import unittest + +try: + import sign_coefficient as sc +except ModuleNotFoundError: + sc = None + + +def integrate_clipped_lines(d, u): + """Independent piecewise integration; split at both actual zeros.""" + end = max(d, F(0)) + knots = sorted({F(0), end, *[x for x in (d-u, u) if 0 < x < end]}) + total = F(0) + for lo, hi in zip(knots, knots[1:]): + mid = (lo+hi)/2 + if d-u-mid > 0 and u-mid > 0: + total += u*(d-u)*(hi-lo)-d*(hi**2-lo**2)/2+(hi**3-lo**3)/3 + return total + + +class CoefficientTests(unittest.TestCase): + def setUp(self): + self.assertIsNotNone(sc, 'implementation has not been supplied') + + def test_integral_against_piecewise_reference(self): + for d in map(F, range(-3, 8)): + for u in (F(i, 4) for i in range(21)): + with self.subTest(d=d, u=u): + self.assertEqual(sc.band_integral(d, u), integrate_clipped_lines(d, u)) + + def test_exact_examples_and_boundaries(self): + examples = [(2, 1, F(1,3)), (3,1,F(5,6)), (1,1,0), (0,1,0), + (-1,1,0), (1,0,0), (F(3,2),F(1,2),F(5,48))] + for d,u,value in examples: + self.assertEqual(sc.band_integral(F(d),F(u)), F(value)) + with self.assertRaises(ValueError): + sc.band_integral(F(1),F(-1)) + + def test_symmetry_and_homogeneity(self): + for d in (F(1,2),F(1),F(2),F(5)): + for ratio in (F(0),F(1,5),F(1,2),F(4,5),F(1)): + u=d*ratio + self.assertEqual(sc.band_integral(d,u),sc.band_integral(d,d-u)) + for c in (F(1,3),F(2),F(7,2)): + self.assertEqual(sc.band_integral(c*d,c*u),c**3*sc.band_integral(d,u)) + + def test_positive_box_witness(self): + for b,d,t in itertools.product((F(1,2),F(5,8),F(3,4)), + (F(1),F(9,8),F(5,4)), + (F(0),F(1,16),F(1,8))): + self.assertGreaterEqual(d-b*b-t,F(5,16)) + self.assertGreaterEqual(b*b-t,F(1,8)) + self.assertGreaterEqual(sc.band_integral(d,b*b),F(5,1024)) + + def test_full_physical_scaling(self): + values=(F(-2),F(-1),F(0),F(1),F(2)) + for r in (F(1,2),F(1,5)): + for am,ap,bm,bp,d,t in itertools.product(values,values,(F(0),F(1)), + (F(0),F(1)),values, + (F(-1,2),F(0),F(1,2),F(1))): + w=sc.physical_weight(r,am,ap,bm,bp,r*(t-d),r*t) + actual = w/r**4 if t>=0 else F(0) + self.assertEqual(actual,sc.layer_weight(am,ap,bm,bp,d,t)) + + def test_layer_support(self): + for d in (F(-2),F(0),F(1),F(3)): + for t in (F(-1),F(0),F(1),F(2),F(4)): + if t<0 or t>max(d,F(0)): + self.assertEqual(sc.layer_weight(-1,1,-1,1,d,t),0) + + def test_endpoint_average_difference(self): + delta=F(13,7) + for correlation,r in itertools.product((F(-1,2),F(0),F(3,4),F(1)), + (F(1,3),F(2,5))): + covariance=[[delta,delta*correlation],[delta*correlation,delta]] + transform=[[F(1,2),F(1,2)],[-1/r,1/r]] + out=sc.matmul(sc.matmul(transform,covariance),sc.transpose(transform)) + self.assertEqual(out,[[delta*(1+correlation)/2,F(0)], + [F(0),2*delta*(1-correlation)/r**2]]) + + def test_exact_pin_drift(self): + for r,m2,b in itertools.product((F(1,2),F(1,5)),(F(1),F(5,3)),(F(6,5),F(-2))): + mean_u,mean_d=sc.transverse_means(m2,b,r) + self.assertEqual(mean_u-r*mean_d/2,-m2*b) + self.assertEqual(mean_u+r*mean_d/2,-m2*(b-r**3/6)) + + def test_actual_contact_regression_in_finite_product_models(self): + for weights in sc.diagnostic_spectra(): + moments=sc.moments(weights,10) + for b in (F(6,5),F(-1),F(0)): + mean,covariance=sc.contact_regression(moments,b) + m2,m4=moments[2],moments[4];delta=m4-m2*m2 + self.assertGreater(delta,0) + self.assertEqual(mean,[-m2*b,F(0),F(0)]) + self.assertEqual(covariance,[[delta,0,0],[0,m2*delta,0],[0,0,m2*delta]]) + + def test_mixed_derivative_factor_and_width_laws(self): + for weights in sc.diagnostic_spectra(): + m=sc.moments(weights,10);delta=m[4]-m[2]**2 + laws=sc.contact_laws(m[2],m[4],F(6,5)) + self.assertEqual(laws['variance_q'],delta) + self.assertEqual(laws['variance_B'],m[2]*delta/4) + self.assertEqual(laws['variance_D'],4*laws['variance_B']) + + def test_nonproduct_obstruction(self): + joint={(sx*a,sy*a):F(1,8) for a in (1,2) for sx in (-1,1) for sy in (-1,1)} + self.assertEqual(sc.residual_cross(joint),F(9,4)) + weights=sc.diagnostic_spectra()[0] + product={(x,y):p*q for x,p in weights.items() for y,q in weights.items()} + self.assertEqual(sc.residual_cross(product),0) + + def test_normalizer_power_ledger(self): + self.assertEqual(sc.power_ledger(),{'transverse_factors':2,'density_variable':1, + 'normalized_numerator':3,'physical_numerator':5,'physical_normalizer':2, + 'probability':3}) + + +if __name__=='__main__': + unittest.main(verbosity=2)