Sep 28, 2026 · Sep 28, 2026
@@ -0,0 +1,30 @@
+name: Remote within-window height-decoupling controls
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/remote_height_decoupling_20260928/**'
+      - '.github/workflows/remote-height-decoupling.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  exact-height:
+    runs-on: ubuntu-latest
+    timeout-minutes: 10
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Verify exact sources and both-mode semantic controls
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          set -euo pipefail
+          python -B -S frontiers/remote_height_decoupling_20260928/verify.py
+          git diff --exit-code
@@ -0,0 +1,284 @@
+# Faster within-window height decoupling at fixed remote separation
+
+Object: OA-REMOTE-HEIGHT-DECOUPLING-20260928-v1.
+Author: OpenAI / ChatGPT, foreground continuation, 28 September 2026.
+Disposition: AUTHOR-SIDE PROOF CANDIDATE; source-bound nonauthor review required.
+Scientific effect NONE. No parent source, verdict, graph or scientific status changes.
+
+## 1. Precise improvement and inputs
+
+Use the exact fixed-remote setting of RM and RC in SOURCE_MAP.json: the normalized
+periodic Gaussian field on the fixed d-dimensional torus, d>=2, fixed rho>0,
+compact birth marks b and gap marks 0<k_-<=k<=k_+, all orthonormal endpoint frames,
+original endpoint pins M=-(r/2)u and S=(r/2)u with heights b and b-k r^3 and zero
+gradients, Gaussian law Q_r and the SAME full endpoint determinant tilt W_r/Z_r.
+Here W_r=F_d(H_M)F_(d-1)(H_S). Put l=k r^3 and I_r=(b-l,b).
+
+Let E=E_r be any deterministic positive-volume Borel subset of the SAME fixed
+remote domain D_rho. Count all Hessian indices of critical points in E with
+heights in I_r; N_r is the count. Let theta=(b-f(x))/l be the normalized deficit,
+and put Y=(x,index). Thus theta is in (0,1). Adding marks does not change N_r.
+
+The prior HEIGHT_MARKS candidate establishes O(r) total variation (TV) to the
+LIMITING contact location/index profile times Uniform(0,1), by comparing the
+full mean density with its contact limit. This note proves a stronger but
+DIFFERENT target: comparison to the point's OWN finite-r location/index law.
+No change is made to that prior note or to the O(r) rate for the contact profile.
+
+Probability TV is sup_A |P(A)-Q(A)|, half the full variation norm. Write lambda
+for Uniform(0,1), mu_r for the expected marked point measure, m_r=E N_r, and
+q_r=E[N_r(N_r-1)]. Denote the normalized mean location/index marginal by pi_r.
+Let S_r be the joint (Y,theta) law conditioned on N_r=1, and pi_{r,1} its actual
+location/index marginal.
+
+**Theorem H.** Uniformly on the above fixed parameter sets and deterministic E_r,
+for all sufficiently small r:
+
+    dTV(mu_r/m_r, pi_r tensor lambda) <= C r^3;                       (H1)
+    dTV(S_r, pi_{r,1} tensor lambda) <= C r^2;                        (H2)
+    dTV(Law(theta | N_r=1), lambda) <= C r^2.                         (H3)
+
+Consequently, for the height H=f(x) of that unique point,
+
+    E[H | N_r=1] = b - k r^3/2 + O(r^5),
+    Var(H | N_r=1) = k^2 r^6/12 + O(r^8).                            (H4)
+
+A complete-configuration version, conditioned on N_r>=1, is within O(r^2) of a
+single point with location/index law pi_{r,1} and independent uniform theta.
+A measurable point-selection rule on nonempty configurations which returns the
+unique point when N_r=1 has the same O(r^2) height-uniformity and own-marginal
+independence conclusions. This is not an assertion that a finite-r location and
+index are mutually independent.
+
+The full joint law compared to the LIMITING contact profile pi_E tensor lambda
+remains O(r), as before. The stronger height/decoupling estimates do not supply
+a faster location-profile convergence rate or a height law for persistence
+partners. No numerical C or cutoff, global/no-rho statement, or optimal Gaussian
+rate is claimed.
+
+### Exact imported premises
+
+RM Sections 2-5 supply the uniform positive covariance sandwich for the transformed
+pins U_r plus Y_x=(grad f(x),f(x)), all fixed C^q Gaussian regression moments,
+the filtered-determinant estimate (8), Z_r/r^2 bounded above and away from zero,
+and the full pre-height-integrated density
+
+    R_{r,j}(x,h)=p_{Y_x|U_r=v_r}(0,h)
+       E[W_r F_j(H_x)|U_r=v_r,Y_x=(0,h)]/Z_r.                       (H5)
+
+Its pointwise contact comparison and positive contact kernels give
+
+    c r^3 |E| <= m_r <= C r^3 |E|.                                 (H6)
+
+RC Corollary D supplies q_r<=C r^5 |E|. These premises concern the same fixed
+remote law and all indices. They are not imported from a global cap event or
+from a probability bound in the wrong direction. The new work below is a
+same-r comparison in the actual height h, retaining the homogeneous endpoint
+constraints so that division by Z_r does not lose r^2.
+
+## 2. Height regression changes along a smooth homogeneous-pin direction
+
+Fix r,x and all endpoint marks/frames. Order V_r=(U_r,grad f(x),f(x)), and let e_h
+be the target coordinate vector for its last entry. With one unconditioned field
+F, use RM's Gaussian regression realization for each height target h:
+
+    F_h(z)=F(z)+Cov(F(z),V_r) Sigma_r^{-1}(a_h-V_r),
+    a_h=(v_r,0,h),   Sigma_r=Cov(V_r).
+
+For h,h' in the closed between-pin interval this gives the exact pathwise identity
+
+    F_h-F_{h'}=(h-h') g_{r,x},
+    g_{r,x}(z)=Cov(F(z),V_r) Sigma_r^{-1} e_h.                       (H7)
+
+All fixed derivatives of the cross-covariance in z are uniformly bounded, by
+Cauchy-Schwarz and the smooth field's uniform derivative variances. The inverse
+covariance is uniformly bounded by RM Section 2. Thus for each fixed q,
+
+    ||g_{r,x}||_{C^q} <= C_q.                                      (H8)
+
+Applying the observation vector V_r to g_{r,x} gives Sigma_r Sigma_r^{-1}e_h=e_h.
+Therefore U_r g=0, grad g(x)=0 and g(x)=1. Since the transformed U_r is invertible
+as a re-expression of the ORIGINAL endpoint observations for r>0,
+
+    g(M)=g(S)=0,   grad g(M)=grad g(S)=0.                           (H9)
+
+This step uses the same exact pin law, not contact pins substituted at finite r.
+The direction is deterministic; the conditioned Gaussian residual is shared.
+All field supremum moments are uniform over h,h' by RM's extra-height regression
+argument. No independence of endpoint and witness Hessians is asserted.
+
+## 3. The vanishing endpoint columns survive the height comparison
+
+For any smooth g with both gradient pins zero, the average of D^2g u over the
+endpoint segment vanishes. At either endpoint i,
+
+    ||D^2g(i)u||
+      <= (1/r) integral_0^r t dt * ||D^3g||_infinity
+      = (r/2)||D^3g||_infinity.                                   (H10)
+
+For M, subtract D^2g(M+tu)u from D^2g(M)u inside its zero-average integral;
+for S reverse the segment. This is a vector integral identity, not an unjustified
+simultaneous vector Rolle point.
+
+For each conditioned F_h define its endpoint blocks as in RM Section 4:
+
+    H_i(h)=[[r alpha_i(h), r beta_i(h)^T],
+            [r beta_i(h), A_i(h)]],
+    K_i(h)=[[alpha_i(h), sqrt(r) beta_i(h)^T],
+            [sqrt(r) beta_i(h), A_i(h)]].
+
+Equations (H7)-(H10) give, uniformly in small r,
+
+    |alpha_i(h)-alpha_i(h')|+||beta_i(h)-beta_i(h')||
+                              <= C|h-h'|,
+    ||A_i(h)-A_i(h')|| <= C|h-h'|,
+    ||K_i(h)-K_i(h')||+||H_x(h)-H_x(h')|| <= C|h-h'|.               (H11)
+
+The normalizations alpha=f_uu/r and beta=grad_perp f_u/r therefore do NOT cause
+an r^-1 loss. Every matrix in (H11) has moments of all fixed orders uniformly in
+h, by the gradient-pin bounds and the extra-conditioned Gaussian moments.
+
+Positive congruence gives exactly, including at singular Hessians,
+
+    W_r(F_h)/r^2 = F_d(K_M(h)) F_(d-1)(K_S(h)).                    (H12)
+
+## 4. Uniform Lipschitz continuity of the full height density
+
+The deterministic estimate in RM (8) is
+
+    |F_j(A)-F_j(B)|
+       <= d max(||A||,||B||)^(d-1)||A-B||.                        (H13)
+
+It concerns the determinant TIMES the index indicator, not the indicator alone.
+Across index changes the matrix segment reaches determinant zero, so this bound
+does not require eigenvalues bounded away from zero.
+
+Apply (H13), (H11), and the product difference identity to the three factors in
+
+    T_h=F_d(K_M(h))F_(d-1)(K_S(h))F_j(H_x(h)).
+
+Hölder and the uniform moments give
+
+    |E T_h-E T_{h'}| <= E|T_h-T_{h'}| <= C|h-h'|,
+    sup_h E|T_h| <= C.                                           (H14)
+
+The conditional Gaussian density p_{Y_x|U_r=v_r}(0,h) and its first h derivative
+are uniformly bounded on this compact target/covariance domain. Its difference
+is at most C|h-h'|. Multiply (H14) by this density and by r^2/Z_r, which is bounded
+and does NOT depend on the additional witness height h. Equation (H5) yields
+
+    |R_{r,j}(x,h)-R_{r,j}(x,h')| <= L |h-h'|                      (H15)
+
+with L uniform over remote x, compact endpoint marks/frames and small r.
+This is stronger for within-window variation than comparing each height to the
+contact kernel with an O(r) error. No derivative of a discontinuous index
+indicator or differentiation under an unbounded expectation is being used.
+
+## 5. Exact finite-r mean decoupling and its third-order rate
+
+The finite measure form of weighted Kac-Rice in RM gives, after h=b-l theta,
+
+    mu_r(dx,{j},dtheta)=l R_{r,j}(x,b-l theta) dx dtheta.            (H16)
+
+Write Rbar_{r,j}(x)=integral_0^1 R_{r,j}(x,b-l t)dt. The location/index marginal
+of mu_r has density l Rbar. By (H15),
+
+    integral_0^1 |R(x,b-l theta)-Rbar(x)| dtheta
+       <= L l integral_0^1 integral_0^1 |theta-t| dt dtheta
+       = L l/3.
+
+Sum over the d+1 indices and integrate over E. The FULL variation estimate is
+
+    ||mu_r - mu_r^Y tensor lambda||_var
+       <= (d+1)L l^2 |E|/3 <= C r^6 |E|.                          (H17)
+
+Both measures have the SAME mass m_r. Divide by 2m_r and use (H6), obtaining (H1).
+Unlike a comparison with the limiting contact profile, there is no normalization
+error between two different total masses. The volume factor cancels even when
+E_r varies and tends to zero in volume. A zero-volume E has no conditional law
+and is explicitly excluded.
+
+## 6. Conditional singleton law and physical-height moments
+
+Let sigma be the expected marked measure restricted to N_r=1, of mass p_1.
+Let eta be the mean marked measure on N_r>=2, of mass a=E[N_r;N_r>=2]. Then
+
+    mu_r=sigma+eta,    m_r=p_1+a,    a<=q_r.                       (H18)
+
+For small r, p_1>=m_r-q_r>0. The mixture identity implies
+
+    dTV(S_r, mu_r/m_r) <= a/m_r,   S_r=sigma/p_1.                 (H19)
+
+Marginal contraction gives dTV(pi_{r,1},pi_r)<=a/m_r. Tensoring with a fixed
+probability measure preserves TV. The triangle inequality and (H1) therefore give
+
+    dTV(S_r,pi_{r,1} tensor lambda) <= 2a/m_r+C r^3 <= C r^2.
+
+For the height marginal alone, compare first with mu_r/m_r and then with pi_r
+tensor lambda. Its bound is a/m_r+C r^3, again O(r^2). This proves (H2)-(H3).
+The factor two in the joint own-marginal comparison is kept, not silently removed.
+
+Since 0<=theta<=1, TV controls its first two moments:
+E theta=1/2+O(r^2), E theta^2=1/3+O(r^2), Var theta=1/12+O(r^2).
+Using H=b-l theta and l=k r^3 proves (H4), with uniform compact-k constants.
+
+Bonferroni and the factorial bound give
+
+    P(N_r>=1)>=m_r-q_r/2,
+    epsilon_r:=P(N_r>=2 | N_r>=1)<=q_r/(2m_r-q_r)=O(r^2).          (H20)
+
+The nonempty configuration law is a mixture of its singleton part and a
+multiple-configuration part with mass epsilon_r. Its TV distance from the
+singleton law S_r, embedded into configurations, is exactly epsilon_r. Combining
+with (H2) proves the complete-configuration assertion.
+
+Any measurable rule selecting one actual point from a nonempty configuration
+agrees with S_r on N_r=1, hence its selected-mark law differs from S_r by at most
+epsilon_r. Applying marginal contraction once more proves O(r^2) decoupling from
+its OWN selected location/index marginal, and O(r^2) height uniformity. A rule
+may be randomized via a probability kernel; the same mixture argument applies.
+
+## 7. Why this does not improve the contact-profile rate
+
+If pi_E is the limiting contact spatial/index profile, RM gives only
+TV(pi_r,pi_E)=O(r). Therefore
+
+    TV(S_r,pi_E tensor lambda) <= O(r^2)+TV(pi_{r,1},pi_E)=O(r).
+
+There is no contradiction with the sharper own-marginal factorization. The old
+O(r) comparison can be dominated by spatial/index drift which is common to
+all heights and vanishes when that finite-r marginal is retained.
+
+Nor do the moment premises alone permit O(r^3) conditional height error. For a
+simple abstract marked process at two distinct locations A,B, choose m=r^3 and
+p_2=r^5/2. A two-point configuration occurs with probability p_2 and contains one
+point at each location with a common height uniform on (0,1/2). Choose the
+singleton submeasure to make the TOTAL mean mark measure exactly m times the
+uniform law on {A,B} times Uniform(0,1): its height density summed over locations
+is m-4p_2 on (0,1/2) and m on (1/2,1). These are nonnegative for r<=1/2 and have
+mass p_1=m-2p_2. Put the remaining probability 1-m+p_2 on the empty configuration.
+
+This process has an exactly height-flat mean and q=2p_2=r^5, but its singleton
+height TV from uniform is
+
+    p_2/(m-2p_2)=r^2/[2(1-r^2)].                                 (H21)
+
+It shows the O(r^2) multiplicity correction cannot be eliminated from just these
+mean/factorial premises. It is NOT an example of the conditioned Gaussian field
+and does not prove that the rate is optimal for that specific field.
+
+## 8. Review and verification boundaries
+
+The new load-bearing analytic step is (H15): the shared regression direction,
+homogeneous endpoint constraints, normalized Hessian columns, filtered
+polynomial determinant bound, and extra-conditioned moments. The finite tests
+check the TV convention, actual finite-r marginal, pair multiplicity weights,
+normalizations, the explicit homogeneous-pin polynomial, and the rate-barrier
+example. They do not establish a covariance floor, a Gaussian supremum bound,
+or Kac-Rice applicability.
+
+This note does not permit rho->0, changing d/L, unbounded birth marks or k->0,
+or field-selected random E. It does not imply global uniqueness, change a
+persistence-partner definition, or recover/retry the unrelated blocked code in
+Math-#116. Authorship/exposure is OpenAI, including the prior remote source;
+independent acceptance requires an actual source-bound nonauthor review.
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+# Within-window height decoupling — author-side candidate
+
+[PROOF.md](PROOF.md) proves the proposed improvement using the existing fixed-remote
+mean and factorial inputs: O(r^3) factorization of the normalized mean against its
+own finite-r location/index marginal, and O(r^2) height uniformity/independence for
+the actual unique witness. Joint convergence to the limiting contact profile stays
+O(r). Physical height mean and variance errors become O(r^5) and O(r^8).
+
+Scientific effect NONE. No parent source or scientific status is changed.
+This is a new derivation, not a replay of the absent Math-#116 algebra suite.
+
+Run `python -B -S verify.py` or `python -B -O -S verify.py` from this directory.
+The finite standard-library tests and semantic mutants do not certify the Gaussian
+argument. Source identities and comparison scope are in SOURCE_MAP.json.
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+# Reconnaissance and attribution — 28 September 2026
+
+This is a source-consumer derivation, not an independent Gaussian-field theory
+or a novelty/priority claim. The existing remote mean proof is read at its exact
+Git blob in SOURCE_MAP.json. Its same-r conditional Gaussian field construction,
+filtered determinant estimate and full normalizer are the load-bearing inputs.
+The same-r height-Lipschitz consequence and finite-r marginal target are derived
+explicitly in PROOF.md rather than inferred from an integrated mean.
+
+The neighboring primary-source Kac-Rice framework is Armentano, Azais and Leon,
+On a general Kac-Rice formula for the measure of a level set,
+arXiv:2304.07424v3. It is the underlying reference of the consumed remote source,
+not a new direct theorem import here. Gaussian regression, vector integral
+Taylor bounds, TV contraction and the singleton/mean mixture are classical.
+The earlier HEIGHT_MARKS and remote-singleton records are credited in the proof;
+neither is silently rewritten or treated as new independent review.
+
+The current Consensus request returned its monthly quota error. The continuation
+used direct primary-source records instead; no Consensus search hit or full-text
+read is fabricated. A future novelty audit would need substantially broader
+search than this dependency-focused pass. The finite tests do not prove the
+Gaussian covariance or expected-integral premises.
@@ -0,0 +1,11 @@
+{
+  "scientific_effect": "NONE",
+  "baseline": {
+    "errors": [],
+    "failures": [],
+    "passed": true,
+    "tests": 12
+  },
+  "scope": "Finite identities, normalization and counterexamples only; no Gaussian continuum verification.",
+  "nonauthor_acceptance": false
+}