diff --git a/frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md b/frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md new file mode 100644 index 000000000..aaefc8da9 --- /dev/null +++ b/frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md @@ -0,0 +1,205 @@ +# A direct planar elder-failure lower bound and the sharp compact-window density loss + +Object: OA-ELDER-LOWER-DENSITY-GAP-20260928-v1. +Author: OpenAI / ChatGPT, foreground research session, 28 September 2026. +Disposition: AUTHOR-SIDE PROOF CANDIDATE; NONAUTHOR ANALYTIC REVIEW REQUIRED. +Scientific effect: NONE. This does not change any historical certificate or governing status. + +## 1. Exact elder probability and conclusions + +Use the exact planar periodized Gaussian field and the original typed pair law +Q_r^W defined in LOCAL_MULTIPLICITY.md. The torus L is fixed, birth b is in a fixed +compact interval, k is in [k_-,k_+] with k_->0, and frames range over O(2). The six +pins are M=-(r/2)u, S=(r/2)u, f(M)=b, f(S)=b-k r^3, and both gradients zero. +W_r=F_2(H_M)F_1(H_S) and the FULL normalizer is Z_r=E_Qr W_r. This is not a law +conditioned on adjacency. + +Let p_r(b,k,R) be the probability that the global ordinary superlevel elder death +partner of M is S. This is **exactly** the p_r definition in [LP, Section 1], not +a new selector. On the generic locus its death height is the maximin + + d_f(M)=sup_{gamma(0)=M, f(gamma(1))>f(M)} min_t f(gamma(t)), (E1) + +with the empty supremum convention of [LP, Section 8]. A path to a point strictly +above the birth height, entirely above f(S), therefore prevents S from being M's +elder death partner. No gradient-trajectory or saddle-saddle transversality theorem +is required for this sufficient obstruction. + +**Theorem E1 (direct lower bound).** Uniformly on the stated compact planar model, +there are c,r_*>0 such that + + 1-p_r(b,k,R) >= c r^3, 0= b-k r^3/4 > f(S), + d_f(M)-f(S) >= 3k r^3/4. (E3) + +It does not use a subtraction of competing witness-intensity bounds, numerical +rungs, an asymptotically fitted exponent, or an assumed independence of witnesses. +In particular a window-count lower bound alone would not prove (E2); the explicit +path below is essential. + +**Corollary E2 (two-sided selection rate, with a separate upper input).** If the +precise Theorem A upper bound in [LP] is consumed, then in d=2 + + c r^3 <= 1-p_r(b,k,R) <= C r^3. (E4) + +The lower proof below is independent of that upper theorem. A defect in the upper +input would affect (E4)'s upper half, not the direct lower mechanism (E2). + +## 2. A rational polygonal path with strict quantitative margins + +Use the local scaled cubic + + G_k(X,Z)=k(2X^3-3X/2-1/2-3Z^2/4-XZ^2). + +Consider the three-segment path + + (-1/2,0) -> (-3/4,0) -> (-3/4,9/4) -> (-1,9/4). (E5) + +Every segment lies in B(0,3): the largest endpoint squared norm is 97/16<9, +and the ball is convex. On the three segments, parameterized consecutively by +v in [0,1], the exact restricted polynomials divided by k are + + g_1(v)=-3v^2/16-v^3/32, + g_2(v)=-7/32, + g_3(v)=-7/32+51v/64-9v^2/32-v^3/32. (E6) + +The first is nonincreasing, the second constant, and + + g_3'(v)=(51-36v-6v^2)/64 >= 9/64>0. + +It follows that the minimum on the entire path is -7k/32, while its final value +is +17k/64. The path passes through a level region, but its proof does not rely on +that level remaining exactly constant under perturbation. + +If a field F on B(0,3) obeys + + ||F-G_k||_infinity < k/64, F(-1/2,0)=0, + +then the entire path has F-values above -15k/64>-k/4 and the endpoint has value +above 16k/64=k/4>0. Consequently, for the physical field with +F(X,Z)=(f(rX,rZ)-b)/r^3, the same path scaled by r connects M to a point with +value above b, entirely above b-k r^3/4. Definition (E1) proves (E3). + +This is stable C^0 connectivity, not a statement that a prescribed path is a +separatrix. It remains a sufficient elder obstruction even without identifying +the exact saddle where the merger occurs. The existence of the two extra saddles +proved in the local multiplicity module is useful additional information, but is +not needed for the path argument. + +## 3. Positive probability of the path under the actual tilted Gaussian law + +For completeness, the ingredients from the local module can be isolated without +using its global upper bound or its extra-saddle implicit-function argument. +With s=f_zz(0)/r, a=f_xxz(0), beta=f_xzz(0), c=f_zzz(0), the exact six pins give + + (f(rX,rZ)-b)/r^3 + =2kX^3-3kX/2-k/2+(s/2)Z^2 + +(a/2)(X^2-1/4)Z+(beta/2)XZ^2+(c/6)Z^3 + +O_{C^2(B3)}(r||f||C4). (E7) + +The term -aZ/8 retains both transverse gradient pins. The target values +s=-3k/2, a=0, beta=-2k, c=0 recover G_k. Choose a fixed small epsilon and the +physical jet box + + |f_zz+(3k/2)r|= c_2 r^4 + +on this event. The full normalizer obeys 0 A_0>0 (E9) + +uniformly on the compact parameters, where sigma is the ordinary surface measure +and pi_r is the density of the endpoint observation frame. Selected intensity has +the same factors times p_r. There is no factor 1/2 for the differently typed, +ordered maximum/saddle pair. + +Changing variables ell=k r^3 at fixed k gives the exact density-gap version + + nu_cand(ell)-nu_eld(ell) + =(ell^(-1/3)/3) integral_B integral_K integral_S + k^(-2/3) A_{(ell/k)^(1/3)}(b,k,u) + [1-p_{(ell/k)^(1/3)}(b,k,u)] d sigma dk db. (E10) + +For sufficiently small ell all these radii are below the common cutoff. The +uniform positive lower and finite upper bounds for A_r, and (E2)/(E4), imply + + c_{B,K} ell^(2/3) + <= nu_cand(ell)-nu_eld(ell) + <= C_{B,K} ell^(2/3). (E11) + +The lower follows directly from (E2); the upper consumes the already identified +parent upper theorem. The k factor in its bounding integral is k^(-5/3), since +r^3=ell/k. It is finite and has positive integral on the fixed gap window. + +Thus the parent's compact-window O(ell^(2/3)) **selection-density loss** is optimal +in order in the planar model; it cannot be replaced by o(ell^(2/3)) under these +assumptions. This does not prove sharpness of the different O(1) remainder in a +leading density expansion, and does not remove the compact-mark restriction. +Combining with nu_cand(ell)~c ell^(-1/3) gives a rejected-pair fraction of order ell. +Integrating (E11) from zero to epsilon gives a cumulative rejected-pair intensity +of order epsilon^(5/3), versus total compact-window intensity of order + epsilon^(2/3). No leading numerical constant is determined by these inequalities. + +## 5. Relation to the lower-rate campaign and review boundaries + +The new direct event proves a uniform, all-small-r positive lower constant for +the exact typed pair law and ordinary elder selector defined in [LP]. Fixing any +birth and any positive gap mark in these ranges gives the corresponding one-mark +specialization. This is an existential analytic lower theorem, not a collection +of finite-r numerical checks. It does not rely on any disputed window-saddle +Cauchy-Schwarz estimate or on its numerical budget subtraction. + +A historical artifact using the symbol q(r,b) is not automatically identified +with this p_r: its covariance, scaling convention, pin meanings, tilt and selector +must match. No historical numerical c,r_0, finite-band certificate, lower-campaign +registry bit, or prize is advanced here. In particular an adjacency-conditioned +law is a different law and is excluded from the statement. + +The ordinary elder obstruction itself uses only the maximin definition and the +strict polygonal path, not the SARD-G no-saddle-connection theorem. The local +Gaussian jet-density and extra-conditioned C^4 control are the core new analytic +obligations. The parent upper theorem and weighted density identity are separately +named dependencies for (E4) and (E11). This separation permits a reviewer to accept +or reject the direct lower mechanism without inheriting a broader parent verdict. diff --git a/frontiers/window_multiplicity_laws_20260928/HEIGHT_MARKS.md b/frontiers/window_multiplicity_laws_20260928/HEIGHT_MARKS.md new file mode 100644 index 000000000..015d76967 --- /dev/null +++ b/frontiers/window_multiplicity_laws_20260928/HEIGHT_MARKS.md @@ -0,0 +1,159 @@ +# The missing single-witness height mark: uniformity and joint conditional law + +Object: OA-WINDOW-MULTIPLICITY-HEIGHT-20260928-v1. +Author: OpenAI / ChatGPT, foreground research session, 28 September 2026. +Disposition: AUTHOR-SIDE COROLLARY; NONAUTHOR ANALYTIC REVIEW REQUIRED. +Scientific effect: NONE. No existing source or scientific disposition changes. + +## 1. The premise is pointwise in the height, not only an integrated mean + +Use the fixed-remote setting of [RM] and [RC] from SOURCE_MAP.json. The original +endpoint observations, maximum/saddle determinant tilt, full normalizer, fixed +d>=2,L,rho>0, and compact positive gap marks are unchanged. Let E be any +deterministic Borel subset of the same D_rho. Its positive volume may depend on r. +Let N_r count all critical indices in E whose heights are in I_r=(b-k r^3,b). + +The remote singleton-law candidate [SL] explicitly discards the height mark. +Its integrated mean formula alone does not determine the height distribution: +equal integrals can belong to different densities. The stronger premise needed +here is already written and proved in **[RM, Section 5, immediately after (13)]**. +In its notation the exact pre-height-integrated density is + + R_{r,j}(x,h) + = p_{(grad f(x),f(x))|U_r=v_r}(0,h) + E[W_r F_j(H_x) | U_r=v_r,grad f(x)=0,f(x)=h]/Z_r, + +and, uniformly for x in D_rho, h in [b-k r^3,b], j and compact marks/frames, + + |R_{r,j}(x,h)-Lambda_j(x;b,k,u)| <= C r. (H1) + +This is exactly the complete integrand of [RM, (12)] after its r^2/Z_r factor. +The proof retains the **actual additional height pin**, uses a common Gaussian +regression coupling, controls W_r/r^2 under that extra conditioning, and compares +the full normalizer with z_0. No zero mean, independence, or extra endpoint +Jacobian is inserted here. + +The weighted Kac-Rice identity is a finite measure identity in location, height +and index. One obtains it first for continuous nonnegative height test functions +by the expected-integral formula, then for bounded Borel test functions by equality +of measures and monotone approximation. Individual endpoint heights have zero +mean count. These facts are already part of [RM]'s height disintegration; the +application does not feed an arbitrary Borel weight directly into a theorem whose +stated hypothesis is lower semicontinuity. + +## 2. The rescaled height mean measure + +Define the normalized deficit mark + + theta=(b-f(x))/(k r^3), 00, + + ||mu/m-nu/n||_var + <= (||mu-nu||_var+|m-n|)/m + <= 2||mu-nu||_var/m. + +The factor |E| cancels. A null-volume E has zero count almost surely and no +conditional nonempty or singleton law is asserted there. + +## 3. From a mean mark to the actual unique witness + +Let q_r=E[N_r(N_r-1)]. The fixed-remote bound [RC, Corollary D] gives + + q_r <= C r^5 |E|, q_r/m_r <= C r^2. (H5) + +Write sigma_r for the mark submeasure from singleton configurations and eta_r +for the mean mark measure contributed on N_r>=2. Then + + mu_r=sigma_r+eta_r, + sigma_r(S_E)=p_{1,r}:=P(N_r=1), + eta_r(S_E)=a_r:=E[N_r;N_r>=2] <= q_r. + +For small r, p_{1,r}=m_r-a_r>=m_r-q_r>0. The exact mixture identity + + mu_r/m_r=(p_{1,r}/m_r)(sigma_r/p_{1,r})+eta_r/m_r + +gives dTV(sigma_r/p_{1,r},mu_r/m_r)<=a_r/m_r<=C r^2. Combining with (H4) proves: + +**Theorem H1 (single-point joint height law).** Uniformly for every deterministic +positive-volume E=E_r inside the same fixed D_rho, + + dTV(Law((x,index,theta) of the unique point | N_r=1), + pi_E tensor Uniform(0,1)) <= C r. (H6) + +In particular the deficit height tends to a uniform distribution and is +asymptotically independent of the **joint** location/index mark. No independence +between location and index is claimed. + +For completeness the integer inequalities imply + + m_r-P(N_r>=1) <= q_r/2, + P(N_r>=2 | N_r>=1) <= q_r/(2P(N_r>=1)) <= C r^2. (H7) + +There is also a complete-configuration version: conditioned on nonemptiness, +Law Xi_r differs by at most C r from a singleton with mark law +pi_E tensor Uniform(0,1). Indeed the singleton submeasure sigma_r/m_r is dominated +by both Law(Xi_r | N_r>=1) and a singleton with law mu_r/m_r, and has mass +p_{1,r}/m_r. The resulting TV bound is a_r/m_r, then use (H4). This is a statement +about configuration laws, not an arbitrary rule for choosing a point from a +multiple configuration. + +The elementary mixture and common-submeasure arguments are rederived here; [SL] +is credited as the parallel source of the location/index formulation, not silently +consumed as an already independently accepted theorem. + +## 4. Why the single and pair height laws differ + +The leading single-point mean density has no dependence on theta, so its normalized +height is uniform. In REMOTE_PAIR_LAW.md, the normalized **factorial-pair mean** +instead has density (5/9)|theta-theta'|^(-1/3), with gap Beta(2/3,2). Pair weighting +samples a different object: configurations with more qualifying pairs receive +more weight. There is no contradiction, and the beta law is not the law of two +independent samples from the single-point uniform limit. + +There is likewise no global conditional-singleton claim. LOCAL_MULTIPLICITY.md +exhibits a local two-saddle mechanism of probability order r^3 in d=2, the same +order as a global occurrence. The fixed positive spatial exclusion in (H1)-(H7) +is indispensable. + +## 5. Scope, tests, and review obligations + +The order-Cr bound depends on the fixed d,L,rho and compact b,k ranges of its inputs. +It does not allow rho->0, k->0, increasing dimension/torus size, or a set E chosen +from the realized field. Unlike the sharp remote pair asymptotic, this pointwise +height-kernel argument does permit arbitrary deterministic varying E_r, because +all errors are bounded by its volume before normalization. + +These are critical-point height marks within a prescribed between-pin window, +not a persistence-partner distribution or lifetime distribution. No new numerical +value of C or cutoff is supplied. The finite tests verify the Jacobian, the mean +normalization algebra, and the failure of an integrated-mean-only inference. +They do not verify the continuum uniform density estimate (H1); its exact source +and its correct mathematical scope are part of the nonauthor review obligation. diff --git a/frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md b/frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md new file mode 100644 index 000000000..b5c18aac5 --- /dev/null +++ b/frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md @@ -0,0 +1,270 @@ +# A local two-saddle mechanism: global window multiplicity has order r^3 + +Object: OA-WINDOW-MULTIPLICITY-LOCAL-20260928-v1. +Author: OpenAI / ChatGPT, foreground research session, 28 September 2026. +Disposition: AUTHOR-SIDE PROOF CANDIDATE; NONAUTHOR ANALYTIC REVIEW REQUIRED. +Scientific effect: NONE. This note does not change a governing status or a prior proof. + +## 1. Setting and statements + +Use the exact **planar**, variance-one periodized Gaussian field, endpoint observations, +and determinant tilt of [RM] and [IW] in SOURCE_MAP.json. The torus side L is fixed. +Birth marks b range over a fixed compact interval and k over [k_-,k_+], with k_->0. +Frames range over O(2); stationarity permits the midpoint to be zero. In a local frame, + + M=(-r/2,0), S=(r/2,0), + f(M)=b, f(S)=b-k r^3, grad f(M)=grad f(S)=0. + +Q_r is continuous Gaussian regression on these six actual observations. It has no +adjacency or elder-pairing condition. F_j(H) is |det H| on nonsingular Hessians of +negative index j, and zero otherwise. Set + + W_r=F_2(H_M) F_1(H_S), Z_r=E_Qr W_r, + dQ_r^W=(W_r/Z_r)dQ_r, I_r=(b-k r^3,b). + +Let N_r count additional critical points on the torus, excluding M,S, whose heights +belong to I_r. Let A_r be the event that **two distinct additional saddles** in the +physical ball B(0,2r) have heights in I_r. No restriction on other critical points +is made. Balls are embedded by reducing r_*. + +**Theorem L.** There are positive c,C,r_*, depending only on the fixed parameter +ranges and L, such that, uniformly in b,k,frame and 0=2} <= C r^3. (L1) + +Consequently, + + E_Qr^W[N_r(N_r-1)] >= c r^3, (L2) + Q_r^W{N_r>=2 | N_r>=1} >= c_0>0. (L3) + +The expectation in (L2) is allowed to be an extended nonnegative number; no finite +upper estimate for it is being supplied. The conditioning in (L3) is well-defined +by (L1). The upper bound in (L1) consumes the already integrated planar global +HEIGHT-WINDOW first moment [IW, (I5)]. The lower bound has a direct Gaussian +jet-box proof below and does not consume any global elder-selection assertion. + +In particular, a torus-wide extension of [RC]'s remote O(r^5) factorial bound is +false under these assumptions. A torus-wide conditional-uniqueness assertion +1-O(r^2) is also false. These statements do **not** contradict [RC] or the remote +singleton-law candidate [SL]: their spatial exclusion radius is fixed and positive, +whereas the event A_r occurs a distance of order r from the pins. + +## 2. An exact cubic with strict, stable window witnesses + +For k>0 define + + G_k(X,Z)=k(2X^3-3X/2-1/2-3Z^2/4-XZ^2). (L4) + +Its gradient is k(6X^2-3/2-Z^2, -(3/2+2X)Z). Its four critical points and +Hessian data are as follows; all entries are exact. + +| Point | Height | Hessian or determinant | Type | +|---|---|---|---| +| M_*=(-1/2,0) | 0 | diag(-6k,-k/2), det=3k^2 | maximum | +| S_*=(1/2,0) | -k | diag(6k,-5k/2), det=-15k^2 | saddle | +| P_+=(-3/4,sqrt(15/8)) | -7k/32 | det=-15k^2/2 | saddle | +| P_-=(-3/4,-sqrt(15/8)) | -7k/32 | det=-15k^2/2 | saddle | + +At P_+ and P_- the Hessian is [[-9k,-2kZ],[-2kZ,0]]. The extra points +satisfy |P_+|^2=|P_-|^2=39/16<4. Their height -7k/32 lies strictly in (-k,0). +The endpoint determinant weight of this scaled cubic is 45k^4. + +The equal heights of P_+ and P_- cause no stability problem: we require each +critical point to be nondegenerate and its height to be strictly inside the window, +not an equality between their heights. A perturbation can and generally will split +them while preserving both witnesses. + +Choose disjoint small closed disks about P_+,P_- inside B(0,2), avoiding M_*,S_*. +The implicit-function theorem applied to the gradient at these nondegenerate zeros +gives a C^2 neighborhood of G_1 with a saddle in each disk. Shrink the neighborhood +so the two saddle heights lie, for example, strictly between -1 and 0. For fields +with the two pins exactly retained, shrink it again so their Hessians have the +specified indices and their determinant magnitudes exceed 3/2 and 15/2. +Multiplying by k and using k_->0 makes these neighborhoods uniform for the compact +k range. Thus there is a fixed epsilon_0>0 such that a scaled field within +k epsilon_0 of G_k in C^2(B(0,2)) has the two witnesses and + + F_2(H_scaled(M_*)) F_1(H_scaled(S_*)) >= (45/4) k^4. (L5) + +This is an ordinary open stability assertion about finitely many nondegenerate +zeros. It neither counts all critical points in the ball nor assumes a globally +Morse contact field. + +## 3. Pinned Taylor normal form, including the transverse linear term + +All derivatives below are at the physical midpoint. Write + + s=f_zz(0)/r, a=f_xxz(0), beta=f_xzz(0), c=f_zzz(0), + P_{k,s,a,beta,c}(X,Z) + =2kX^3-3kX/2-1/2*k + (s/2)Z^2 + +(a/2)(X^2-1/4)Z +(beta/2)XZ^2 +(c/6)Z^3. (L6) + +If ||f||_{C^4}<=K and the actual endpoint observations hold, multivariate Taylor +expansion gives, uniformly on B(0,3), + + || (f(rX,rZ)-b)/r^3 - P_{k,s,a,beta,c} ||_{C^2(B(0,3))} <= C r K. (L7) + +Here and below harmless norm-equivalence constants depend on the fixed dimension. +To check all potentially dangerous divided powers of r, put h=r/2. The positive +Hermite/Peano identity for the endpoint heights and gradients gives + + f_xxx(0)=12k+O(r K). + +Adding and subtracting the two endpoint expansions of f_x and f_z gives + + f_x(0)=-(r^2/8)f_xxx(0)+O(r^3K), + f_z(0)=-(r^2/8)f_xxz(0)+O(r^3K), + f_xx(0)=O(r^2K), f_xz(0)=O(r^2K). + +Expanding f at M and using its height b then yields + + f(0)-b=-(r^3/24)f_xxx(0)+O(r^4K) + =-k r^3/2+O(r^4K). + +Taylor remainders for derivatives of order 0,1,2 have sizes respectively r^4K, +r^3K,r^2K. After the coordinate and value rescaling each becomes O(rK). +This proves (L7). The term -a Z/8 is essential: omitting it violates the two +transverse gradient pins. The polynomial in (L6) itself satisfies both height and +both gradient pins exactly for arbitrary s,a,beta,c. + +The cubic (L4) is (L6) at + + s=-3k/2, a=0, beta=-2k, c=0. (L8) + +No exact degree-three assumption is imposed on the random field; (L7) controls +its higher-order terms on the event used below. + +## 4. A rare jet box with probability bounded below by a constant times r + +Use the nonsingular endpoint frame U_r of [RM, Section 2]. Its target v_r is bounded +and it converges to + + U_0=(f,f_x,f_xx,f_xxx,f_z,f_xz), v_0=(b,0,0,12k,0,0). + +Append the **physical**, not rescaled, vector + + J=(f_zz,f_xxz,f_xzz,f_zzz) at 0. (L9) + +At r=0, (U_0,J) consists of the ten independent planar jet entries of orders at +most three. The positive Fourier spectrum of the exact periodized Gaussian field +makes its covariance positive definite: a zero-variance combination annihilates +every Fourier mode, hence is a zero distribution; arbitrary smooth prescribed +jets at the site force all its coefficients to vanish. Frame rotation preserves +linear independence. Continuity and compactness in O(2) give a uniform positive +covariance floor through sufficiently small r>=0. The corresponding upper bound +is finite as well. + +It follows by a Schur complement that J under Q_r has a covariance sandwich and +bounded mean, uniformly in the compact marks and frame. Its Gaussian density is +therefore bounded **below** on any fixed compact target box. + +For a small fixed epsilon>0 let B_r be the target box + + |f_zz + (3k/2)r| < epsilon k r, + |f_xxz| < epsilon k, + |f_xzz + 2k| < epsilon k, + |f_zzz| < epsilon k. (L10) + +All these physical targets lie in a fixed compact box. Its Lebesgue volume is +16 epsilon^4 k^4 r, so + + Q_r{J in B_r} >= c_J r. (L11) + +The small factor r is attached to exactly one coordinate, f_zz. Treating the +rescaled coordinate f_zz/r as having a uniformly nondegenerate density would lose +this factor and would be wrong. + +## 5. Remainder control AFTER the rare jet restriction + +A bounded unconditioned moment is not enough to subtract a fixed tail error from +an event of probability order r. Instead regress the entire smooth field on the +joint observation vector (U_r,J), at each target (v_r,j) with j in B_r. + +For an unconditioned field copy F the usual regression representation is + + F(z)+Cov(F(z),V)Cov(V)^{-1}((v_r,j)-V), V=(U_r,J). + +The coefficient functions have uniformly bounded C^4 norms: derivatives of the +covariance kernel are bounded, the integral observation frames have uniformly +bounded derivative-covariance norms, and Cov(V)^{-1} is uniformly bounded. The +Gaussian Fourier series has finite moments of its C^4 norm. Targets are bounded. +The triangle and moment inequalities therefore give, for every fixed finite p, + + sup_{r,marks,frame,j in B_r} + E[||f||_{C^4}^p | U_r=v_r,J=j] < infinity. (L12) + +Choose a finite K large enough that Markov's inequality gives conditional +probability at least 1/2 to ||f||_{C^4}<=K at **every** such target. Integrating +this conditional probability against the jet density over B_r gives + + Q_r{J in B_r, ||f||_{C^4}<=K} >= (c_J/2)r. (L13) + +This uses a continuous Gaussian regular conditional kernel; it is not a claim of +independence between J and the field remainder. The rare event is retained before +any tail estimate is applied. + +## 6. Tilt and normalization: r times r^4 divided by r^2 + +Choose epsilon in (L10) so (L6) is within k epsilon_0/2 of (L4) on B(0,3), hence on B(0,2). +Then choose r_* small enough that the C r K error in (L7) is below the remaining +k_- epsilon_0/2 margin. On the event in (L13), the stability argument gives A_r. +Since H_f(rX)=r H_scaled(X) in dimension two, each physical endpoint determinant +is r^2 times its scaled determinant. Equation (L5) implies + + W_r >= c_W r^4 (L14) + +on that event, with c_W>0 uniform in the marks. + +The full normalizer satisfies 0= E_Qr[W_r 1{J in B_r, ||f||C4<=K}]/Z_r + >= (c_W c_J/(2C_Z)) r^3. (L15) + +For the upper bound sum [IW, (I5)] over indices and use +1{N_r>=2}<=N_r/2. This proves (L1). The inequalities +N_r(N_r-1)>=2*1{N_r>=2} and +Q_r^W{N_r>=1}<=E_Qr^W N_r<=C r^3 prove (L2) and (L3). + +## 7. Sharp obstruction to a global Bernoulli-configuration approximation + +Let Xi_r be the finite global window point process, retaining any of location, +index and normalized height as marks. Its mean measure mu_r has mass m_r<=C r^3, +so m_r<=1 for small r. Define Ber(mu_r) to be empty with probability 1-m_r and +otherwise a singleton with distribution mu_r/m_r. + +The elementary identity in the parallel singleton-law candidate [SL] is useful, +but we rederive it here so this corollary does not consume an unreviewed premise. +Write sigma for the singleton submeasure, eta for the mean measure on N>=2, +h=eta(S)=E[N;N>=2], alpha=P(N>=2), and s=P(N=1). The difference between Law Xi +and Ber(mu) has masses h-alpha on the empty stratum, minus the nonnegative +measure eta on the singleton stratum, and mass alpha on the multiple stratum. +Its full variation is 2h; therefore probability TV is exactly h. Applying (L1), + + c r^3 <= dTV(Law Xi_r,Ber(mu_r)) <= C r^3. (L16) + +Indeed h>=2P(N_r>=2) and h<=m_r. Thus the global exact-mean Bernoulli error is +of the **same order** as the global occurrence probability, unlike the remote +r^5 error. More generally, any probability law supported only on empty and +singleton configurations differs from Law Xi_r by at least c r^3, by testing +the event N>=2. Conditional on nonemptiness, its distance from every singleton +law is at least c_0 from (L3). No global factorial-moment upper bound was needed. + +## 8. What this closes and what it does not + +This supplies an existential, all-small-r lower mechanism for **local multiple +window saddles** under the original maximum/saddle determinant tilt. Together +with the existing first moment it fixes the global multiple-occurrence probability +order at r^3. It therefore rules out a proposed uniform extension of remote +single-point asymptotics to the whole torus. + +It does not give a global upper bound on the second factorial moment, a numerical +constant or cutoff, a global limit measure, or a distribution of persistence +partners. The weaker event A_r by itself does not determine elder pairing. The separate +ELDER_LOWER_AND_DENSITY_GAP.md uses the stronger specific jet-box event and an +explicit path to a point above the birth height to prove an elder-failure lower bound. No result for d>2, shrinking gap mark k, growing torus, random +selected spatial domain, or all-height count is asserted. The Gaussian jet-box and +stability arguments above are analytic proof obligations for a nonauthor reviewer; +the exact polynomial tests are not their substitute. diff --git a/frontiers/window_multiplicity_laws_20260928/PUBLICATION.json b/frontiers/window_multiplicity_laws_20260928/PUBLICATION.json new file mode 100644 index 000000000..be68e90aa --- /dev/null +++ b/frontiers/window_multiplicity_laws_20260928/PUBLICATION.json @@ -0,0 +1,70 @@ +{ + "schema": 1, + "author": "OpenAI / ChatGPT", + "date": "2026-09-28", + "scientific_effect": "NONE", + "mathematical_acceptance": false, + "review_status": "AUTHOR_SIDE_REVIEW_REQUIRED", + "public_scope": "proof and provenance documents only", + "reproducible_computation_from_this_github_packet": false, + "code_publication": { + "state": "BLOCKED_UNRESOLVED", + "tool_message": "This tool call was blocked by OpenAI because we could not determine the safety status of the request.", + "blocked_file": { + "path": "algebra.py", + "bytes": 6163, + "sha256": "16ed713d1822efe2b254f8fd990270a81d23e49487208d4866f888486f553350", + "git_blob": "00140088f011b830c343900a4db646b38a5bdd52" + }, + "retry_or_alternate_route_attempted": false, + "owner_delivery": "complete local source and evidence in conversation artifact; not a GitHub execution receipt" + }, + "actual_local_execution": { + "tests_per_mode": 30, + "modes": [ + "normal", + "optimized" + ], + "semantic_mutants_per_mode": 13, + "all_intended_assertion_failures": true, + "execution_errors": 0, + "normal_optimized_json_pairs_identical": 14, + "packet_files_checked": 13, + "packet_only_runner_exit_code": 0, + "dependency_bytes_replayed_locally": false, + "source_bytes_unchanged": true, + "infrastructure_negative_controls": 4 + }, + "evidence_files": [ + { + "path": "FINITE_REPLAY_SUMMARY.json", + "bytes": 6322, + "sha256": "745784d958f1aeab958b047155c81c386148c7bbed374c71c64224b74c08389a", + "git_blob": "1b5c9e41a9cb25de91648124ca24abd7c09ccb76" + }, + { + "path": "PACKET_VALIDATION.json", + "bytes": 319, + "sha256": "6dc693add39e2c300f3f73986e6f98cea57bf194e61ec44aa81b51e67279c93d", + "git_blob": "2807b88740bb2da02bac3fb70de0c5f4445a60e0" + }, + { + "path": "PACKET_NEGATIVE_CONTROLS.json", + "bytes": 553, + "sha256": "d1338e78413961176ec30521961c302b38cba51692d14cfaff894713ca076f47", + "git_blob": "81116b1b9d5969e9241290f0a2dd61da9c8f3610" + }, + { + "path": "RED.stderr", + "bytes": 865, + "sha256": "3a5b90a07e9f1663d32407db59bd7f35c4fe58060a3ae15965feec3272e795f5", + "git_blob": "ee5187f3a80a7fb130d0dcd9012f9b585643cde0" + }, + { + "path": "ELDER_RED.stderr", + "bytes": 774, + "sha256": "f5d6aea704001d8e43886a58ffc47b9b3ce819959ccf12bdb4e60b9cf3aae07e", + "git_blob": "21dc945024e8fd538db17bf01131a7b8d690aaf7" + } + ] +} diff --git a/frontiers/window_multiplicity_laws_20260928/README.md b/frontiers/window_multiplicity_laws_20260928/README.md new file mode 100644 index 000000000..126c8e3de --- /dev/null +++ b/frontiers/window_multiplicity_laws_20260928/README.md @@ -0,0 +1,79 @@ +# Window multiplicity and elder-loss laws — author-side proof packet + +Author: OpenAI / ChatGPT, 28 September 2026. **Nonauthor analytic review required.** +Scientific effect: NONE. No existing proof, scientific verdict, graph, or prize +is changed. These are new written arguments, not accepted theorems by virtue of publication. + +## Mathematical modules + +| Read | Proposed result | Scope boundary | +|---|---|---| +| [Direct elder lower bound](ELDER_LOWER_AND_DENSITY_GAP.md) | Uniform planar `1-p_r >= c r^3` via an explicit path above the proposed death height to a point above birth. With the separately named parent upper/radial inputs: `Theta(r^3)` selection loss and `Theta(ell^(2/3))` compact-window density loss. | No numerical constant/cutoff, historical q/p substitution, or unrestricted-mark density-loss claim. | +| [Local multiplicity](LOCAL_MULTIPLICITY.md) | Two local window saddles occur with probability `Theta(r^3)`. Global conditional uniqueness and an `O(r^5)` global factorial extension are impossible. | Planar only; no global factorial upper bound. The count event alone is not an elder obstruction. | +| [Remote pair law](REMOTE_PAIR_LAW.md) | Exact positive leading `r^5` factorial coefficient, adjacent-index support, pair-weighted `Beta(2/3,2)` height-gap limit. | Both witnesses stay in a fixed remote region. Pair-mean weighting is not event-conditioned pair sampling. | +| [Distance moments](REMOTE_DISTANCE_MOMENTS.md) | Distance moment transition at `p=1`; two positive `r^6` contributions there. Scaled-distance limit has tail `R^-7`, yet its first moment does not capture rare macroscopic-pair contributions. | Fixed remote set. Total variation does not control unbounded moments. | +| [Single height mark](HEIGHT_MARKS.md) | Remote singleton location/index/height is `O(r)` in TV from the contact location/index profile times a uniform window height. | Uses the actual pre-height-integrated source kernel; not a persistence-partner law. | + +The common local cubic is + + G_k(X,Z)=k(2X^3-3X/2-1/2-3Z^2/4-XZ^2). + +Its extra saddles have height `-7k/32`. A rational polygonal path reaches +`+17k/64` while staying above `-7k/32`. Physical jet-box mass `r`, endpoint weight +`r^4`, and full normalizer `r^2` give a tilted event of order `r^3`. The whole-field +remainder is controlled AFTER conditioning on the rare physical jet box. + +The remote overlap integral is + + integral_0^infinity s(k-s^3|t|)_+ ds = (3/10) k^(5/3)|t|^(-2/3). + +It yields a contact Gaussian coefficient and an ordered pair-height density +`(5/9)|a-b|^(-1/3)`. The normalized gap has mean `1/4`, variance `9/176`, and +ordered heights have correlation `2/7`. These are pair-weighted critical-point +marks, not a new lifetime distribution. + +## Source identities and an explicit code-publication limitation + +[SOURCE_MAP.json](SOURCE_MAP.json) pins four consumed sources by repository, +40-character commit, bytes, SHA-256 and Git blob. The parallel singleton-law PR115 +is context-only; elementary mixture identities are proved again where needed. +[SOURCE_FILES.json](SOURCE_FILES.json) enumerates every document in THIS public +packet. This directory is deliberately proof-and-provenance only. + +The complete local delivery also contains `algebra.py`, `test_algebra.py`, +`run_validation.py`, a targeted workflow, manifests and original execution logs. +The GitHub connector blocked the attempt to create `algebra.py` with the message +that it could not determine the safety status. That body was not retried through +another path, encoding, or write route. No mathematical objection was supplied +by that tool response. The local tested code is included in the owner's separate +conversation evidence bundle, NOT claimed present in this repository packet. + +Actual local execution: **30 tests in each of normal and optimized Python; 13 +semantic mutants in each mode rejected on their exact intended assertions, with +no execution errors; baseline and mutant outputs identical between modes.** +Initial missing-implementation tests failed before implementation. The complete +local packet-only runner passed and source bytes remained unchanged. Separate +negative controls rejected a changed document, an unlisted member, duplicate JSON +keys, and a full-checkout replay with missing dependencies. + +Those local facts are reported in [PUBLICATION.json](PUBLICATION.json). They are +not a claim that the code can be replayed from THIS GitHub directory. The four +full-checkout dependency bytes were not locally downloaded/replayed in the container; +source texts were read through the connector and exact identities pinned. No new +hosted mathematical runner is installed by this proof-only PR. Existing repository +CI, if it runs, remains separate and cannot prove the new analytic arguments. + +## Review focus + +Review the direct elder obstruction and rare-jet conditional moment argument +first. The upper selection theorem is an explicit separate input. Then review +the collision-conditioned determinant limits, fixed-Borel translation argument, +coefficient constants, beta pushforward, distance-moment diagonal estimate, and +`p=1` two-cutoff limit. The height-mark module has a smaller source-consumption +obligation. Each manuscript records its own dependencies and exclusions. + +No nonauthor acknowledgment, provider independence, numerical Gaussian coefficient, +all-small-r numerical cutoff, global collision upper bound, higher factorial +moment, event-probability asymptotic from factorial means, Poisson limit, or priority +claim is invented. [RECONNAISSANCE.md](RECONNAISSANCE.md) records the limited external +search and distinguishes primary theorem reading from abstract-level discovery. diff --git a/frontiers/window_multiplicity_laws_20260928/RECONNAISSANCE.md b/frontiers/window_multiplicity_laws_20260928/RECONNAISSANCE.md new file mode 100644 index 000000000..6f7883265 --- /dev/null +++ b/frontiers/window_multiplicity_laws_20260928/RECONNAISSANCE.md @@ -0,0 +1,74 @@ +# Reconnaissance memo — window multiplicity and height/distance marks + +Date: 28 September 2026. Author: OpenAI / ChatGPT. +Scientific effect NONE. Discovery metadata is not an imported theorem. + +## Searches actually performed + +Consensus was queried for Gaussian critical points, shrinking height windows, +second moments and Kac-Rice. It returned Muirhead, Ladgham-related subject matter, +Gass--Stecconi, Armentano--Azais--Leon, and one-point height-distribution papers. +The Muirhead and Kuriki--Matsubara--Iso records were fetched. The former fetch +provided metadata but no abstract; the latter supplied an abstract about +unconditioned isotropic one-point critical densities in arbitrary dimensions. +Neither fetch supplied a theorem proving any claim in this packet. Consensus +reported its monthly search allowance exhausted after this discovery; no further +Consensus searches were attempted in this pass. + +Primary-source web searches then examined the following records and the AAL v3 +HTML. A focused arXiv search for Gaussian critical-point heights together with +Beta and 2/3 did not identify a matching result. That small search is not an +exhaustive novelty audit or evidence of worldwide priority. + +## Primary source actually read at theorem level + +Diego Armentano, Jean-Marc Azais, Jose Rafael Leon, +*On a general Kac-Rice formula for the measure of a level set*, +https://arxiv.org/html/2304.07424v3 (5 December 2023 version). + +Theorem 7.1 is the weighted expected-integral formula under Theorem 2.1 and its +stated lower-semicontinuity/conditional-law hypotheses. Theorem 2.2 and Remark 7 +supply the Gaussian regression bridge; Remark 8 treats a jointly Gaussian mark +process. Section 8.1 applies the framework to critical points. Our application +uses the already source-bound pair formula in [RC], with the original endpoint +weight once and each witness determinant once. Borel height-mark extensions are +handled as finite-measure identities, not by pretending every Borel indicator is +lower semicontinuous. No theorem-number substitution is made. + +## Adjacent literature — scope checked at record/abstract level + +Safa Ladgham and Raphael Lachieze-Rey, +*Local repulsion of planar Gaussian critical points*, +https://arxiv.org/abs/2209.04150 (2022 preprint; SPA publication 2023). +Its abstract concerns small-ball factorial moments and type-resolved repulsion +for stationary isotropic planar fields. It motivates near-collision analysis but +is not imported as a pinned determinant-tilted shrinking-height theorem. + +Stephen Muirhead, +*A second moment bound for critical points of planar Gaussian fields in shrinking +height windows*, https://arxiv.org/abs/1901.11336 (2019 preprint; Statistics & +Probability Letters 160, 108698, 2020; DOI 10.1016/j.spl.2020.108698). +Its abstract concerns unconditioned stationary planar fields and large domains +with shrinking height windows. Its second-moment problem is relevant background; +its result is not substituted for uniform estimates under this packet's pin law. + +The attempted HTML fetches for these two preprints did not expose their full +proofs in this session. No full-paper novelty exclusion or undocumented theorem +consumption is claimed. Gass--Stecconi's moment-finiteness paper and the newer +Kuriki--Matsubara--Iso one-point formula were discovery leads only, not inputs. + +## Internal audit and contribution boundary + +GitHub searches for the local height -7/32 and the beta/multiplicity terms did not +return the proposed mechanisms in the inspected results. Existing sources already +provide the fixed-remote mean, the O(r^5) pair upper bound, and the planar global +window first moment. The new proofs explicitly derive: a local two-saddle lower +mechanism, a sharp remote factorial coefficient with adjacent-index support, +height-mark laws, a distance-moment transition, and a direct ordinary elder-failure +lower event with a sharp compact-window candidate-versus-elder density-loss order. Standard Kac-Rice, Gaussian +regression, implicit-function stability, total variation and beta integration are +not claimed as inventions. + +No result in this memo changes a source's review status. The claimed new conclusions +remain author-side candidates, with the full analytic arguments and finite controls +available for nonauthor scrutiny. diff --git a/frontiers/window_multiplicity_laws_20260928/REMOTE_DISTANCE_MOMENTS.md b/frontiers/window_multiplicity_laws_20260928/REMOTE_DISTANCE_MOMENTS.md new file mode 100644 index 000000000..d077743b7 --- /dev/null +++ b/frontiers/window_multiplicity_laws_20260928/REMOTE_DISTANCE_MOMENTS.md @@ -0,0 +1,234 @@ +# Remote pair distances: a moment transition and a nonuniform-integrability effect + +Object: OA-WINDOW-MULTIPLICITY-DISTANCE-20260928-v1. +Author: OpenAI / ChatGPT, foreground research session, 28 September 2026. +Disposition: AUTHOR-SIDE PROOF CANDIDATE; NONAUTHOR ANALYTIC REVIEW REQUIRED. +Scientific effect: NONE. This is a new analytic consumer, not a status change. + +## 1. The quantities and two different limits + +Retain the exact fixed-remote model and fixed positive-volume Borel E of +REMOTE_PAIR_LAW.md. Counts are ordered and sum all indices. Let + + M_p(r)=E_Qr^W sum_{x!=x' counted in E} dist(x,x')^p, + p>=0, M_0(r)=E_Qr^W[N(E)(N(E)-1)]. (D1) + +Distances are physical torus distances. This note obtains a boundary-layer versus +fixed-distance transition at p=1. A typical pair under the normalized factorial +mean is a distance of order r apart, but a much rarer macroscopic pair can dominate +an unbounded distance moment. No pair-occurrence probability is inferred. + +Write A_0=integral_E sum_ij K_ij from (R3). More generally, for 0<=p<7, define + + A_p = [3*12^((p+2)/3)*k^((p+5)/3)] + / [4(p+2)(p+5)z_0] + * integral_E dx integral_S de p_{Y|Q0}(0,0,b) + E_Q0[w_0 |T|^((4-p)/3)(det A)^2 |Y=(0,0,b)]. (D2) + +Here Y,T,A,w_0,z_0 have exactly the definitions in (R1)-(R3), and index summation +has removed chi. The conditional expectation in (D2) is finite for p<7; the +possible negative power of T for p>4 is integrable because (4-p)/3>-1 and the +weighted conditional Gaussian density is bounded near zero. + +For distinct x,x' in E define the fixed-distance contact pair kernel + + Lambda_2(x,x') = p_{(grad f(x),grad f(x'),f(x),f(x'))|Q0}(0,0,b,b)/z_0 + * E_Q0[w_0 |det H_x det H_x'| + | grad f(x)=grad f(x')=0, f(x)=f(x')=b]. (D3) + +This is the all-index sum of the joint kernel in [RM, Section 6]; it is not a +product of two one-point kernels. Let + + B_p=k^2 integral_{E x E, x!=x'} dist(x,x')^p + Lambda_2(x,x') dx dx'. (D4) + +Below we prove B_p is finite and strictly positive for p>=0, including at the +collision diagonal where a compact-separation argument alone is insufficient. + +**Theorem D (distance moments).** For each fixed p>=0, + + 0<=p<1: M_p(r)=r^(5+p) A_p+o(r^(5+p)); + p=1: M_1(r)=r^6(A_1+B_1)+o(r^6); + p>1: M_p(r)=r^6 B_p+o(r^6). (D5) + +The constants A_p and B_p are the explicit conditional integrals above, not +numerical enclosures. The p=1 formula has **two positive contributions**. Simply +extending the microscopic limit's first moment to the full pair measure would +miss B_1. + +## 2. A sharper determinant estimate that retains the small height difference + +[RC]'s uniform bound used |det H_x|<=C delta K^d, with K>=1+||f||C3. For distance +moments we need the extra cancellation forced by nearly equal heights. +Use K=C_d(1+||f||C4), the exact collision frame V_delta, and its last target t. +The Peano formula gives + + D^3 f(x)[e,e,e] = -12t+O(delta K). + +From the gradient Taylor identity (R7), in a frame beginning with e, + + H_x=[[delta alpha,delta beta^T],[delta beta,A]], + alpha=6t+O(delta K), |beta|<=K, ||A||<=K. + +The analogous axial entry at x' is -6t+O(delta K). The exact block determinant +identity, valid even when A is singular, gives + + |det H_x|, |det H_x'| + <= C delta (|t| K^(d-1)+delta K^d). (D6) + +Taking the product, multiplying the original endpoint bound W_r<=C r^2 K^{2d}, +and conditioning as in [RC, Lemma 2] gives, after division by Z_r and multiplication +by the density of V_delta, a bound of the form + + C delta^2 (t^2+delta^2)(1+|t|)^(-q) (D7) + +for any prescribed finite q, by choosing sufficiently many Gaussian moments and +using the covariance sandwich. Constants depend only on the fixed compact model +parameters. The polynomial exponents are absorbed by Gaussian decay, not by an +independence assertion. + +Let a=k r^3/delta^3. For delta>=r the possible t-values lie in [-a,a], with a +uniformly bounded. Integrating t^2+delta^2 over the actual second-height window +is at most C(a^3+delta^2 a). The first-height window has length k r^3. Combining +with the same polar and value-coordinate Jacobians as (R9), the pair intensity +integrated over locations, directions, and heights has the radial envelope + + delta<=r: C |E| r^3 delta ddelta; + r<=delta<=eta_0: + C |E| (r^12 delta^(-8)+r^6) ddelta. (D8) + +These are upper bounds; the two terms in the second line need not be separate +independent processes. At distances >=eta_0, [RC, Lemma 5] gives O(r^6), and +multiplication by any fixed nonnegative power of distance changes only its constant. + +## 3. Integrability and positivity of the fixed-distance kernel + +At the contact endpoint law Q_0, set both remote witness heights equal to b. +The same collision frame has exactly t=0. Equation (D6) then bounds each witness +determinant by C delta^2 K^d. Its product supplies delta^4, while the joint gradient/ +height density supplies delta^(-d-3). Conditional moments remain uniformly bounded. +Hence for 0-1. +Elsewhere it is bounded on separated compact sets. This proves finiteness for +all p>=0. Full joint conditional support of the two Hessians and B_0 at distinct +sites makes Lambda_2 strictly positive off the diagonal. For a positive-volume +E, E x E has positive measure off the diagonal, indeed at some fixed positive +separation. Thus B_p>0. + +For every fixed epsilon>0, the common-regression argument of [RM, Section 6] +gives uniformly on dist(x,x')>=epsilon the pair mean density + + k^2 r^6 Lambda_2(x,x') + O_epsilon(r^7). (D10) + +The fixed-distance contribution divided by r^6 therefore converges to the +corresponding part of B_p. An estimate uniform all the way to the diagonal is +not being assumed in (D10); (D8)-(D9) are what justify the limiting exhaustion. + +## 4. Proof of the three regimes + +For 0<=p<1, repeat the blown-up proof in REMOTE_PAIR_LAW.md with the extra factor +(delta/r)^p=s^p. Its original envelope s min(1,s^-3) now has an integrable pth +moment precisely in this range. The separated contribution is O(r^(1-p)) after +division by r^(5+p), so it vanishes. The radial identity is + + integral_0^infinity s^(p+1)(k-s^3|t|)_+ ds + = 3 k^((p+5)/3)|t|^(-(p+2)/3)/((p+2)(p+5)). (D11) + +Multiplying by the same 36t^2 and substituting T=-12t yields (D2) and the first +line of (D5). The fixed-Borel spatial-indicator argument is unchanged. + +For p>1, divide by r^6 and bound all distances below epsilon using (D8). The +part delta<=r is O(r^(p-1)). The r^12 term on [r,epsilon] becomes + + C r^6 integral_r^epsilon delta^(p-8) ddelta, + +which tends to zero for each fixed epsilon: it is O(r^(p-1)) if p<7, +O(r^6 log(epsilon/r)) if p=7, and O(r^6 epsilon^(p-7)) if p>7. The remaining +r^6 term contributes at most C epsilon^(p+1). Therefore the normalized small- +distance mass has limsup <=C epsilon^(p+1). Combine (D10) with this bound and +let epsilon decrease to zero, using (D9). This proves the third line of (D5). + +At p=1 split into delta<=Ar, Ar=epsilon, with A>=1. +The first part divided by r^6 converges to the truncated microscopic coefficient +A_1(A), obtained from (D2)'s blown-up integral by s<=A. The third part converges +to B_1(epsilon). The middle part is bounded after division by r^6 by + + C A^-6+C epsilon^2, (D12) + +since r^6 integral_{Ar}^epsilon delta^-7 ddelta <= C A^-6. +All contributions are nonnegative. Take first r->0, then A->infinity and +epsilon->0. The truncated coefficients converge to A_1 and B_1, respectively, +and the middle bound vanishes. This proves the exact sum in the middle line of +(D5), without an unproved uniform-integrability assumption. In particular + + A_1=(k^2/(2z_0)) integral_E dx integral_S de + p_{Y|Q0}(0,0,b) E[w_0 |T| (det A)^2 |Y=(0,0,b)]. + +## 5. The limiting scaled separation and its seventh-power tail + +Normalize the ordered-pair mean by M_0(r) and set S_r=dist(x,x')/r. The L^1 +blown-up convergence in REMOTE_PAIR_LAW.md implies total-variation convergence +of its probability law to a nondegenerate S on (0,infinity). This is again pair +weighting, not conditioning on an event and choosing a pair. + +To describe its tail, define + + Psi_E(t)=integral_E dx integral_S de + p_{V_0|Q0}(0,0,b,t) + E_Q0[w_0(det A)^2 | V_0=(0,0,b,t)]. + +Here V_0's last coordinate is -T/12. After summing all indices the expression is +continuous and strictly positive at t=0; no artificial zero from the definition +of chi at T=0 is inserted into Psi. Uniform Gaussian regression also makes it +bounded near zero and rapidly decreasing after polynomial weights at infinity. +The limit density of S is + + g_S(s)=(36/(z_0 A_0)) s + integral_R t^2(k-s^3|t|)_+ Psi_E(t) dt. (D13) + +With t=kv/s^3, the integral equals +k^4 s^-9 integral_{-1}^1 v^2(1-|v|)Psi_E(kv/s^3)dv. +Since the elementary integral of v^2(1-|v|) is 1/6, + + g_S(s) ~ [6k^4 Psi_E(0)/(z_0 A_0)] s^-8, + P(S>R) ~ [6k^4 Psi_E(0)/(7z_0 A_0)] R^-7. (D14) + +Consequently E[S^p] is finite for 0<=p<7 and infinite for p>=7. For p<7 its +finite value is A_p/A_0. Near s=0, (D13) is O(s), so there is no additional +positive-moment obstruction there. + +## 6. Why convergence in TV does not give convergence of these moments + +Although the limit S has a finite first moment, (D5) implies + + E[S_r] -> (A_1+B_1)/A_0 > E[S]=A_1/A_0. (D15) + +For every p>1, + + E[S_r^p] ~ (B_p/A_0) r^(1-p) -> infinity. (D16) + +In particular this happens for 1=2 and torus side L are fixed; +0=rho}. +Let N_j(E) count index-j critical points of height in I_r=(b-k r^3,b), and +N(E)=sum_j N_j(E). Let T_ij(E) count **ordered distinct pairs** (x,x') of indices +i,j, both in E and with both heights in I_r. Thus T_ii=N_i(N_i-1), T_ij=N_iN_j +for i!=j, and sum_ij T_ij=N(N-1). + +[RC] gives E_Qr^W T_ij(E)<=C r^5 |E|. This note proposes the leading coefficient, +its index support, and a normalized **factorial-pair mean** height law. It does +not infer a pair-occurrence probability asymptotic from a factorial expectation. + +## 2. Contact coefficient + +Let U_0=v_0 be the endpoint contact observations from [RM, (4)]: in the pin frame, + + U_0=(f,f_x,f_xx,f_xxx,f_yj,f_xyj) at 0, + v_0=(b,0,0,12k,0,...,0), Q_0=Law(f | U_0=v_0). + +Let B_0 be the transverse Hessian at 0 on u-perp, and define + + w_0=(6k)^2 (det B_0)^2 1{B_0<0}, z_0=E_Q0 w_0>0. + +For x in D_rho and unit e, put + + Y_{x,e}=(grad f(x), H_x e, f(x)), + T=D^3 f(x)[e,e,e], A=H_x restricted to e-perp. (R1) + +Use the conditional Gaussian density p_{Y|Q0} at (0,0,b), and keep all dependence +between B_0, A and T inside conditional expectations. Define chi_ij(A,T) as follows, +where a is the negative index of a nonsingular A: + + T>0: chi_ij=1 iff (i,j)=(a+1,a); + T<0: chi_ij=1 iff (i,j)=(a,a+1); + T=0 or det A=0: chi_ij=0. (R2) + +The values on singular A and T=0 do not affect the following product. With ordinary +unnormalized surface measure de on S^{d-1}, define + + K_ij(x;b,k,u) + = (3/40) 12^(2/3) k^(5/3) / z_0 + * integral_S p_{Y|Q0}(0,0,b) + E_Q0[w_0 |T|^(4/3) (det A)^2 chi_ij(A,T) + | Y_{x,e}=(0,0,b)] de. (R3) + +Choice of an orthonormal basis of e-perp or u-perp has no effect on determinants, +indices, or this expectation. Local orthonormal frames suffice; no global frame +section over the sphere is assumed. + +**Theorem R.** For every fixed E as above, + + E_Qr^W T_ij(E) = r^5 integral_E K_ij(x) dx + o(r^5). (R4) + +The coefficient is continuous and finite on the fixed compact parameter ranges. +It is strictly positive for adjacent indices |i-j|=1, and identically zero for +all other index pairs. Moreover K_ij=K_ji after integration in e. Consequently + + E_Qr^W[N(E)(N(E)-1)] ~ r^5 integral_E sum_ij K_ij > 0, + E_Qr^W[N_j(E)(N_j(E)-1)] = o(r^5). (R5) + +These are fixed-domain qualitative asymptotics; no numerical remainder or rate +for the little-o is asserted. The convergence can be taken uniform in the fixed +compact mark/frame ranges for a given fixed E. It is **not** asserted uniform over +arbitrary r-dependent oscillatory sets E_r. Null-volume E has zero counts by [RM]. + +## 3. The conditional collision frame and its actual coupling + +For x'=x+delta e, 00. All frame entries +are point evaluations or integral derivative averages; covariance and derivative +cross-covariances extend continuously to r=delta=0. The positive joint covariance +floor persists there. Therefore, as r,delta->0 and y->b, this representation +converges in every fixed L^p(C^q), uniformly on compact x,e,frame,mark,t sets, to +the corresponding regression on (U_0,V_0)=(v_0,0,0,b,t). The density of V_delta +under Q_r converges on the same bounded target sets to the density of V_0 under +Q_0. The Gaussian conditional means need not be zero. + +At the limit, T=-12t. Away from the origin the derivative orders 0,1,2,3 in +(U_0,V_0) are independent linear functionals. The order-two block H->He is onto. +Adding the independent entries of B_0 and A completes the Hessian entries at their +respective sites. Positive Fourier weights then give full conditional support for +(B_0,A,T) given (U_0,Y), using the distributional jet-independence argument of [RM]. +This also proves the uniform conditional moment bounds used with (R3). + +## 4. Two determinant limits without inverse Hessians + +Under the exact pair-gradient conditions, Taylor's integral identity gives + + H_x e / delta = -(1/2) D^3 f(x)[e,e,.] + O(delta ||f||C4), + H_x' e / delta = +(1/2) D^3 f(x)[e,e,.] + O(delta ||f||C4). (R7) + +Indeed integrate the gradient difference over the segment and expand its Hessian +once. The second identity follows from the other endpoint. The C^4 control is +valid under the extra observations by Section 3. In a frame beginning with e, +congruence by diag(delta^(-1/2),I) gives limits + + K_x -> diag(-T/2,A), K_x' -> diag(T/2,A). + +The off-diagonal scaled entries are O(sqrt(delta)||f||C3); the lower-right entries +converge to A. Congruence preserves inertia and changes determinant by delta^{-1}. +For each j, the function F_j is continuous even on singular matrices, with the +polynomial Lipschitz bound established in [RM, (8)]. Hence under the coupled +conditional fields, + + F_i(H_x)/delta -> F_i(diag(-T/2,A)), + F_j(H_x')/delta -> F_j(diag(T/2,A)) (R8) + +in L^p for each finite p on bounded target sets. This avoids a division by det A, +a lower eigenvalue assumption, or a discontinuous type-indicator limit. + +The same exact pin identities as [RM, Section 4], now under this additional +conditioning, give W_r/r^2 -> w_0 in L^p. The full endpoint-only normalizer is +Z_r/r^2 -> z_0; it is not changed to a pair-conditioned normalizer. The product +limit in (R8) is + + (T^2/4)(det A)^2 chi_ij(A,T). + +Thus it vanishes at leading order unless the two indices are adjacent. Same-index +pairs are not impossible at positive delta: for example, with a nonzero mixed +third derivative and T=0, a cubic can give two saddle determinants of order delta^2 +rather than delta. The finite negative control records this distinction. + +## 5. Blow-up delta=r s, dominated convergence, and the coefficient + +Apply the already justified marked pair Kac-Rice formula [RC, (3.1)] under the +Gaussian Q_r, using W_r once and each witness determinant once, then divide by +Z_r once. Polar coordinates contribute delta^{d-1}; the joint value-gradient +change of variables contributes delta^{-d-3}. With + + y=b-r^3 z, delta=r s, t=(y'-y)/delta^3, + +we have dy'=delta^3 dt. The two windows are exactly + + 0 0. + +On bounded s,t regions the remaining density is uniformly bounded. The displayed +translation estimate removes the second indicator in L^1; the integrable envelope +controls the tails. This proves the same limit for every fixed Borel E without +assuming a smooth boundary. It does not supply a uniform estimate for arbitrary +oscillating E_r, which is why that extension is excluded in Theorem R. + +Integrating z in (R9) gives (k-s^3|t|)_+. For t!=0, + + integral_0^infinity s(k-s^3|t|)_+ ds + =(3/10) k^(5/3)|t|^(-2/3). (R10) + +Multiply by 36t^2. Integrating the last observation and converting T=-12t to its +conditional Gaussian distribution yields + + (54/5) k^(5/3) E[|T/12|^(4/3) ... |Y] + =(3/40)12^(2/3)k^(5/3) E[|T|^(4/3) ... |Y]. (R11) + +This is exactly (R3), including the surface-measure and ordered-pair conventions. +One may first restrict |t|>epsilon, apply Tonelli to nonnegative functions, and +then let epsilon decrease to zero. The resulting |T|^(4/3) moment is finite; +t=0 is a Lebesgue-null exceptional target. No cancellation of an undefined +zero-times-infinity value is required. + +Separated pairs delta>=eta_0 contribute O(r^6)|E|^2 by [RC, Lemma 5] and are +negligible after division by r^5. This proves (R4). + +Full conditional Gaussian support makes the integrand positive on an open set +B_0<0, A of any chosen index a, and T of either chosen nonzero sign. Therefore +K_ij>0 precisely for adjacent indices. Continuity follows from regression and +polynomial domination. The weighted indicator is continuous at every possible +index boundary because det A or T vanishes there. Compactness gives finite upper +bounds and positive lower bounds for each adjacent coefficient. The substitution +e->-e flips T, preserves A and w_0, and changes H e only by a sign at its zero +target; it exchanges i and j without a density Jacobian. This proves symmetry +and (R5). For unordered total pairs, divide the total coefficient by two. + +## 6. Universal height marks under the factorial-pair mean + +For each counted critical point define its normalized height deficit + + theta(x)=(b-f(x))/(k r^3) in (0,1). + +Define a probability measure on (0,1)^2 by normalizing the **expected ordered-pair +count measure**: + + Pi_r(C)= E_Qr^W sum_{x!=x' counted in E} + 1{(theta(x),theta(x')) in C} + / E_Qr^W[N(E)(N(E)-1)]. (R12) + +This is a factorial-moment normalization, sometimes called pair weighting. It is +not the law obtained by first conditioning on N>=2 and then choosing a pair +uniformly from that realized configuration. Those two sampling procedures need +not agree when configurations have different multiplicities. + +**Theorem H2.** In probability total variation (one half of the full signed +variation norm), Pi_r converges to the density + + h(a,b)=(5/9)|a-b|^(-1/3), 0-e symmetry gives equal total mass to the two triangles. This proves (R13) +and its index-resolved version. The separated-pair mass is o(r^5), so it does not +change the normalized limit. Notice that the argument pushes forward to HEIGHT +marks: it does not claim total-variation convergence of unscaled distinct spatial +pairs to a measure supported on the spatial diagonal, which would be impossible. + +The gap g=|a-b| therefore has the **Beta(2/3,2)** density and CDF + + f_gap(g)=(10/9)g^(-1/3)(1-g), + F_gap(g)=(5/3)g^(2/3)-(2/3)g^(5/3), 0<=g<=1. (R14) + +Given g, the smaller deficit is uniform on (0,1-g). Useful exact consequences are + + E[g^n]=10/((3n+2)(3n+5)), n=0,1,..., + E[g]=1/4, E[g^2]=5/44, Var(g)=9/176; + E[a]=E[b]=1/2, E[a^2]=29/88, E[ab]=3/11, + Corr(a,b)=2/7. (R15) + +The marginal pair-weighted deficit density is +(5/6)(a^(2/3)+(1-a)^(2/3)), not uniform. For comparison, two independent uniform +heights would have mean absolute difference 1/3 rather than 1/4. None of these +pair-weighted assertions is a persistence-lifetime law. + +## 7. Explicit limits of the result + +The coefficient in (R3) is a Gaussian conditional integral, not a numerical +enclosure. Positivity and the r^5 order do not by themselves prove +Q_r^W(N>=2)~(r^5/2) integral sum K: triples or higher multiplicities would require +additional estimates. A finite probability-law counterexample in the tests +exhibits a factorial moment asymptotic of order r^5 with much smaller event +probability. No third factorial moment or event-conditioned pair-gap law is claimed. + +The spatial exclusion remains fixed. The local theorem in LOCAL_MULTIPLICITY.md +shows that global multiple occurrences have order r^3 in d=2, so neither (R4) nor +(R13) is silently extended to pairs near the pins. Other exclusions are changing +L or d, k->0, random field-selected E, arbitrary oscillating E_r for the sharp +asymptotic, all-height counts, elder pairing and governing-status transitions. + +The new obligations are the extra-conditioned determinant limits, L^1 domination +through the collision, the fixed-Borel translation step, the coefficient constants, +and the factorial-mean interpretation of the beta law. Finite exact controls test +algebra, not those continuum arguments. 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