Math candidates: direct elder lower bound, sharp density loss, local clusters and remote pair laws

frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md

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+# 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<r<=r_*.                      (E2)
+
+The proof supplies a more specific subevent of this probability on which
+
+    d_f(M) >= 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|<epsilon k r,
+    |f_xxz|<epsilon k, |f_xzz+2k|<epsilon k, |f_zzz|<epsilon k.
+
+The endpoint contact observations plus these four physical jets are exactly the
+ten independent planar jet coordinates of degree at most three. Positive Fourier
+weights, continuity of the observation frame, and compactness give a uniformly
+positive conditional density on these bounded targets. The box volume is
+16epsilon^4 k^4 r, so its Q_r probability is at least c_1 r.
+
+Condition on those **physical** jet values as well as the six pins before bounding
+the C^4 norm. The joint covariance remains uniformly invertible and targets are
+bounded, so full-field Gaussian regression yields a uniform conditional C^4 moment.
+Choose a fixed finite K so the conditional probability of ||f||C4<=K is at least
+1/2 at every jet target in the box. Integrating over the box leaves probability
+at least (c_1/2)r. One may not replace this step by subtracting a fixed unconditional
+tail probability from a rare event of size r.
+
+Choose epsilon and then r_* small enough that (E7) is within k/64 of G_k in C^0
+on B3 and sufficiently close in C^2 at the two pinned endpoints. Their scaled
+Hessians at the cubic are diag(-6k,-k/2) and diag(6k,-5k/2), respectively. Their
+indices persist and their determinant magnitudes retain positive lower bounds.
+Since each physical Hessian is r times its scaled Hessian,
+
+    W_r >= c_2 r^4
+
+on this event. The full normalizer obeys 0<Z_r<=C_Z r^2 by the gradient-pin
+averages and uniform Q_r C^3 moments (or [RM, (11)]). Thus its tilted probability
+is at least
+
+    (c_1 c_2/(2C_Z)) r*r^4/r^2 = c r^3.                  (E8)
+
+Every field in this event satisfies the path obstruction (E3). This proves (E2).
+The sufficient event also has the two local window saddles when the stronger
+C^2 stability neighborhood from LOCAL_MULTIPLICITY.md is used. All constants are
+existential but independent of r,b,k,frame on the stipulated compact ranges.
+
+## 4. Sharp loss between candidate and elder lifetime densities
+
+This section additionally consumes the source-bound weighted Kac-Rice and radial
+ledger of [LP, Sections 9-11]. Let B and K have positive lengths, and let
+nu_cand(ell), nu_eld(ell) be exactly its **compact birth/gap-window** candidate and
+ordinary elder densities, per unit midpoint volume. The candidate intensity is
+
+    r A_r(b,k,u) dr db dk d sigma(u),
+    A_r=12 pi_r(v_r)(Z_r/r^2) -> 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.

frontiers/window_multiplicity_laws_20260928/HEIGHT_MARKS.md

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+# 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), 0<theta<1.
+
+Let Xi_r put one atom (x,j,theta) at every counted critical point, on
+S_E=E x {0,...,d} x (0,1). Its total count is still N_r; adding marks has not
+created extra points. Changing variables h=b-k r^3 theta gives the exact mean
+measure
+
+    mu_r(dx,{j},dtheta)
+       =k r^3 R_{r,j}(x,b-k r^3 theta) dx dtheta.           (H2)
+
+The absolute Jacobian is k r^3, not r, r^2, or a pin-frame Jacobian. Equation (H1)
+therefore implies the uniform density statement
+
+    | dmu_r/(k r^3 dx dtheta) - Lambda_j(x) | <= C r.       (H3)
+
+Put
+
+    A_E=sum_j integral_E Lambda_j(x) dx,
+    pi_E(dx,{j})=Lambda_j(x)dx/A_E,
+    mu_{0,r}=k r^3 Lambda_j(x)dx dtheta.
+
+On the fixed compact remote parameter set Lambda_j are continuous, positive and
+bounded. Thus for positive-volume E, uniformly over these E,
+
+    c r^3 |E| <= m_r:=E N_r <= C r^3 |E|,
+    ||mu_r-mu_{0,r}||_var <= C r^4 |E|,
+    dTV(mu_r/m_r, pi_E tensor Uniform(0,1)) <= C r.         (H4)
+
+Probability TV is one half of the full variation norm. To see the normalization
+bound directly, for positive measures mu,nu of masses m,n>0,
+
+    ||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.