diff --git a/.github/workflows/remote-inverse-separation.yml b/.github/workflows/remote-inverse-separation.yml
new file mode 100644
index 000000000..b6131b954
--- /dev/null
+++ b/.github/workflows/remote-inverse-separation.yml
@@ -0,0 +1,33 @@
+name: Remote inverse-separation finite controls
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/remote_inverse_separation_20260928/**'
+      - '.github/workflows/remote-inverse-separation.yml'
+  push:
+    branches: [main]
+    paths:
+      - 'frontiers/remote_inverse_separation_20260928/**'
+      - '.github/workflows/remote-inverse-separation.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  exact-inverse-separation:
+    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 packet and both-mode intended negatives
+        run: |
+          set -euo pipefail
+          python -B -S frontiers/remote_inverse_separation_20260928/verify.py --output "${RUNNER_TEMP}/inverse-separation-${GITHUB_RUN_ID}-${GITHUB_RUN_ATTEMPT}"
+          git diff --exit-code
diff --git a/frontiers/remote_inverse_separation_20260928/PROOF.md b/frontiers/remote_inverse_separation_20260928/PROOF.md
new file mode 100644
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@@ -0,0 +1,351 @@
+# Inverse separation of remote critical-point pairs: the missing moment boundary
+
+Object: OA-REMOTE-INVERSE-SEPARATION-20260928-v1.
+Author: OpenAI / ChatGPT. Disposition: AUTHOR-SIDE CANDIDATE; nonauthor analytic review required.
+Scientific effect NONE. No existing source, review verdict or status register is changed.
+
+## 1. Objects, conventions, and conclusions
+
+Use the exact periodized field and full pair-pinned law of RM and RC in
+SOURCE_MAP.json. Fix d>=2, torus side L, remote exclusion rho>0, compact real birth
+marks b and compact positive gap marks k with k_->0. The endpoints are
+M=-(r/2)u, S=(r/2)u, pinned to heights b,b-k r^3 with zero gradients. Q_r is the
+original Gaussian conditional law. Keep W_r=F_d(H_M)F_(d-1)(H_S), Z_r=E_Qr W_r,
+and Q_r^W=(W_r/Z_r)Q_r exactly. Here F_j is the absolute determinant on index j
+(counting negative eigenvalues), zero at singular matrices. No additional
+normalizer or adjacency event is inserted.
+
+Let E be a FIXED deterministic positive-volume Borel subset of D_rho. Count all
+indices of critical points in E with heights in I_r=(b-k r^3,b). All pairs below
+are ordered and distinct. Let
+
+    M_p(r)=E_Qr^W sum_{x!=x' counted in E} dist(x,x')^p, p real.       (I1)
+
+The expectation is allowed to be infinite even though the random field has
+finitely many Morse critical points almost surely. Distances are physical torus
+distances. At p=0, M_0 is the second factorial moment. We always distinguish a
+normalized EXPECTED PAIR MEASURE from sampling a pair after conditioning a
+realized field on having two or more points.
+
+RP proves M_0(r)~r^5 A_0 and its blown-up pair-density limit. The separate RD
+manuscript addresses p>=0 only. This note fills the negative-moment boundary
+without importing RD's p=1 transition or its large-separation tail claims.
+
+**Theorem I (extended microscopic moments).** For every fixed -2<p<1,
+
+    M_p(r)=r^(5+p) A_p+o(r^(5+p)),                              (I2)
+
+where A_p is the positive finite contact integral in Section 2. Convergence is
+uniform on the fixed compact mark/frame sets for this fixed E. There is no rate
+for the little-o and no uniformity over arbitrary oscillating sets E_r.
+
+**Theorem II (fixed-r collision and sharp inverse threshold).** For every fixed
+sufficiently small r>0 there is J_r in (0,infinity) such that the expected pair
+separation measure, in a neighborhood of distance zero, has a density j_r with
+
+    j_r(delta)=delta[J_r+o(1)], delta down to zero.                (I3)
+
+Consequently, for this fixed r, M_p(r) is finite exactly when p>-2. In particular,
+
+    E_Qr^W sum dist(x,x')^(-q) < infinity iff q<2, q>=0.           (I4)
+
+At any fixed small delta_0>0, let I_q(r;epsilon,delta_0) restrict that inverse
+sum to epsilon<dist<=delta_0. As epsilon down to zero, at FIXED r,
+
+    q=2: I_2 ~ J_r log(delta_0/epsilon);
+    q>2: I_q ~ [J_r/(q-2)] epsilon^(2-q).                        (I5)
+
+For p>=1, Theorem II asserts finiteness, not a new r-asymptotic. For unordered
+pairs, divide j_r, J_r and all unnormalized pair moments by two.
+
+**Theorem III (small scaled separation and compatible limits).** Under the
+normalized expected ordered-pair measure, S_r=dist/r converges in probability
+TV to a law S on (0,infinity). With J_0 defined in Section 5,
+
+    J_r/r^3 -> J_0>0,
+    g_S(s) ~ (J_0/A_0)s as s down to zero,
+    P(S<=epsilon) ~ [J_0/(2A_0)]epsilon^2.                      (I6)
+
+For 0<q<2,
+
+    E[S_r^(-q)] -> E[S^(-q)]=A_(-q)/A_0<infinity,               (I7)
+
+whereas both expectations are infinite for q>=2. The assertions in (I3),(I5)
+are fixed-r limits, and the first assertion of (I6) is a subsequent r limit.
+No arbitrary joint cutoff epsilon(r) follows just from these iterated limits.
+
+These are local-separation facts within the SAME FIXED REMOTE region. They do
+not extend rho to zero, change d or L with r, permit random field-selected E,
+identify persistence partners, or give an event-conditioned pair law. They also
+do not imply that same-index pairs have the same inverse threshold; the all-index
+sum is essential for the positive leading diagonal coefficient used here.
+
+## 2. Negative moments of the blown-up pair measure
+
+Use RP's contact law Q_0=Law(f|U_0=v_0), with endpoint contact weight w_0 and
+normalizer z_0=E_Q0 w_0>0. At remote x and unit direction e, set
+
+    Y=(grad f(x),H_x e,f(x)), T=D^3f(x)[e,e,e],
+    A=H_x restricted to e-perp.
+
+All correlations of these variables with the endpoint contact Hessian remain
+inside the same conditional expectation. Ordinary, not probability-normalized,
+sphere area is denoted de. Define, for -2<p<1,
+
+    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)].         (I8)
+
+There is no inverse T singularity in this range: (4-p)/3>1. The contact covariance
+floor, moment bounds, and full conditional support from RP make A_p finite,
+continuous, and positive. At p=0 this is exactly the all-index coefficient A_0.
+The formula does not assert a pure k power law: Q_0 and the integrand also depend
+on k through the endpoint target f_xxx(0)=12k.
+
+RP's exact coordinates x'=x+delta e, delta=r s, y=b-r^3 z and
+
+    t=(y'-y)/delta^3
+
+give the window region
+
+    0<z<k, (z-k)/s^3<t<z/s^3.                                (I9)
+
+After extracting r^5, the nonnegative pair-density integrand is dominated by
+C s(1+|t|)^(-m) times this indicator, with m sufficiently large. Integrating z,t
+gives C s min(1,s^-3). To insert dist^p, multiply by r^p s^p. The new envelope is
+
+    C s^(p+1) for s<=1; C s^(p-2) for s>=1.                   (I10)
+
+It is integrable precisely for -2<p<1. Thus RP's coupled determinant convergence
+and dominated-convergence proof applies, including unbounded weights at s=0.
+The fixed-Borel spatial indicators 1_E(x)1_E(x+rse) are handled as in RP: truncate
+s,t to compact ranges, use L1 continuity of torus translations, then remove the
+tails using (I10). One cannot infer this step from pointwise convergence at every
+point of an arbitrary Borel boundary.
+
+Integrating z gives the overlap (k-s^3|t|)_+. For t!=0 and p>-2, elementary
+integration up to s=(k/|t|)^(1/3) gives
+
+    integral_0^infinity s^(p+1)(k-s^3|t|)_+ ds
+      = 3 k^((p+5)/3)|t|^(-(p+2)/3)/[(p+2)(p+5)].             (I11)
+
+RP's limiting determinant product is 36t^2(det A)^2. Integrate the density of
+t=-T/12 and use
+
+    108*12^(-(4-p)/3)=(3/4)*12^((p+2)/3).
+
+This yields exactly (I8). The exceptional t=0 is a null target; Tonelli with
+|t|>eta and then eta down to zero avoids assigning a value to an undefined
+zero-times-infinity expression. For delta>=eta_0>0, RC's separated bound is
+O(r^6); the distance weight is bounded there even for negative p. After division
+by r^(5+p) it is O(r^(1-p)) and vanishes. This proves (I2). The endpoint p=-2
+cannot be decided from failure of this upper envelope alone; the next section
+provides the positive lower asymptotic that resolves it.
+
+## 3. A different collision frame when r is held fixed
+
+Fix 0<r<=r_1, with r_1 small enough that the pins lie strictly inside the excluded
+remote ball. For 0<delta<eta_0<rho/2 use the GRADIENT-ONLY pair frame
+
+    G_delta=(grad f(x),[grad f(x+delta e)-grad f(x)]/delta),
+    G_0=(grad f(x),H_x e).                                    (I12)
+
+Its transformation from the original pair gradients has absolute determinant
+Delta=delta^(-d). There is NO delta^(-3) height-coordinate Jacobian here: the
+height window is retained as a field mark and no divided height observation is
+included. Reusing the joint value-gradient frame's delta^(-d-3) would be wrong.
+
+For this fixed r, the joint vector (original endpoint pins,G_delta) has a positive
+covariance floor uniformly over remote x,e and small delta including delta=0.
+To justify contact rank, H->H e is onto in symmetric matrices, and gradient and
+Hessian evaluations at x are independent linear functionals modulo the original
+value/gradient evaluations at M,S. Positive spectral weights imply that a
+zero-variance linear combination annihilates every Fourier mode. As a finite
+sum of derivatives of delta distributions at the three distinct sites, it is
+then the zero distribution; localized smooth test functions with prescribed
+jets force each coefficient to vanish. Compactness gives the fixed-r floor.
+Local orthonormal charts suffice for the transverse matrix A; no global sphere
+frame or arbitrary-rotation invariance is assumed.
+
+On one unconditioned smooth Gaussian field, use the usual finite-dimensional
+regression formula for these observations with the exact original targets and
+G_delta=0. Integral derivative representations of the divided gradient row give
+covariance convergence as delta->0; the bounded inverse and rapid Fourier
+summability give convergence of the conditional fields in every fixed
+L^a(C^b), with uniform moments over remote x,e. Constants in this section may
+depend on this fixed r. The original pin constraints are exact throughout.
+
+The exact gradient-pair identity and C4 Taylor remainder yield
+
+    H_x e/delta -> -D^3 f(x)[e,e,.]/2,
+    H_(x+delta e)e/delta -> +D^3 f(x)[e,e,.]/2.                  (I13)
+
+Positive congruence by diag(delta^-1/2,I) gives the two limits diag(-T/2,A) and
+diag(T/2,A). Taking absolute determinants, which are continuous even on singular
+matrices and have polynomial Lipschitz bounds, gives the coupled L^a limit
+
+    |det H_x det H_(x+delta e)|/delta^2
+           -> (T^2/4)(det A)^2.                               (I14)
+
+The endpoint weight W_r is also a product of continuous index-filtered
+determinants; its change under this coupling converges with polynomial moment
+control. The original denominator Z_r>0 is independent of delta and remains
+unchanged. For an a priori dominating bound, the zero-gradient segment identity
+gives |det H_x det H_(x+delta e)|<=C delta^2(1+||f||_C3)^(2d).
+
+The product of height indicators tends to the SINGLE indicator
+1{f(x) in I_r}. To justify this despite its discontinuity, append f(x) to G_0.
+Its conditional variance is positive by the same distinct-jet rank argument,
+so the endpoints of I_r have zero conditional probability. Uniform convergence
+also follows by bounding the probability of being within eta of either endpoint
+using the uniformly bounded conditional height density, then letting eta->0.
+Moment bounds and Holder control products with the determinant weights. This
+argument keeps the unconditioned-in-height frame (I12); the shrinking width is
+fixed and positive during this delta limit.
+
+## 4. Positive fixed-r diagonal coefficient and inverse divergence
+
+Let K_r(x,x') be the all-index, height-window pair intensity under Q_r^W, so the
+marked Kac-Rice identity reads
+
+    E_Qr^W sum F(x,x') = integral F(x,x')K_r(x,x') dx dx'
+
+on distinct remote pairs for nonnegative Borel F. This is the Gaussian formula
+under Q_r weighted ONCE by W_r and divided ONCE by Z_r. Exhaustion away from the
+diagonal and monotone convergence allow infinite nonnegative integrals.
+The continuous-field mark extension is the same interface used in RM/RC.
+
+Equations (I12)-(I14) and the density change give, uniformly over remote x,e for
+fixed r,
+
+    delta^(d-2)K_r(x,x+delta e) -> c_r(x,e),
+    c_r(x,e) = p_(G0|Qr)(0)/(4Z_r)
+          * E_Qr[W_r T^2(det A)^2 1{f(x) in I_r}|G_0=0].       (I15)
+
+The coefficient is finite and continuous. It is strictly positive: conditional
+on G_0=0 and the original endpoint pins, the independent symmetric entries of
+the two endpoint Hessians and the remote transverse Hessian A, together with
+remote height and T, have full joint Gaussian support. This follows by adding
+the remaining independent derivative evaluations to the distinct-site jet list.
+The open event of a typed maximum/saddle at the pinned endpoints, remote height
+strictly inside I_r, nonsingular A, and T nonzero has positive probability. The
+integrand in (I15) is positive there. No independence between these blocks is
+needed.
+
+In polar coordinates the radial power is
+
+    (d-1) from dx' - d from the gradient density
+                         +2 from the determinants = 1.       (I16)
+
+For delta below the injectivity scale, define the radial density explicitly by
+
+    j_r(delta)=delta^(d-1) integral_E dx integral_S de
+                   1_E(x+delta e) K_r(x,x+delta e).
+
+The convergence in (I15) is uniform for fixed r. The bounded limit and
+sup_{|h|<=delta}||1_E(.+h)-1_E||_1->0 remove the translated indicator, even for
+an arbitrary fixed Borel E. Therefore (I3) holds with
+
+    J_r=integral_E dx integral_S de c_r(x,e), 0<J_r<infinity.   (I17)
+
+In particular, for each fixed r there is delta_r>0 such that
+(J_r/2)delta<=j_r(delta)<=(3J_r/2)delta on (0,delta_r), and the ratio tends to
+one after division by J_r delta. At larger distances, fixed-r remote pair
+covariance and Gaussian moments are bounded on separated compact sets. Thus
+M_p(r) is finite iff integral_0^delta_r delta^(p+1)d delta is finite, which is
+precisely p>-2. This proves (I4), including its necessity, rather than merely a
+sufficient condition from an upper bound.
+
+To obtain the constants in (I5), fix eta>0 small so j_r(delta)/(J_r delta) lies
+between 1-epsilon_1 and 1+epsilon_1 for delta<eta. Integrate the power on
+(epsilon,eta); the contribution from (eta,delta_0) is finite and negligible
+relative to the divergent expression. First send epsilon->0, then
+ epsilon_1->0. For q=2 the antiderivative is log; for q>2 it is
+(delta^(2-q))/(2-q). This proves the two asymptotic equivalents in (I5).
+The fixed-r constants and cutoffs are existential, not numerically evaluated.
+
+## 5. Matching the fixed-r coefficient to the contact coefficient
+
+Disintegrating the remote height in (I15) gives a useful equivalent form:
+
+    c_r(x,e)=1/(4Z_r) integral_(b-k r^3)^b dh
+       p_(grad f(x),H_x e,f(x)|Qr)(0,0,h)
+       E_Qr[W_r T^2(det A)^2|grad f(x)=H_xe=0,f(x)=h].         (I18)
+
+Put h=b-k r^3 theta. The joint covariance of (U_r,grad f(x),H_xe,f(x)) has a
+uniform floor through r=0: it is a subvector of RP's contact collision frame,
+or follows directly from distinct derivative evaluations at 0 and remote x.
+The same Gaussian regression coupling as RP and RM applies uniformly in
+0<=theta<=1. Under these exact extra observations, original endpoint pin
+identities give W_r/r^2 -> w_0 in every fixed L^a, and Z_r/r^2 -> z_0>0.
+The remote A,T and the density also converge with bounded moments; no restricted
+normalizer replaces Z_r. Dominated convergence in (I18) now yields, uniformly
+on the fixed compact x,e,mark sets,
+
+    c_r(x,e)/r^3 -> c_0(x,e)
+       = k/(4z_0) p_(Y|Q0)(0,0,b)
+          E_Q0[w_0 T^2(det A)^2|Y=(0,0,b)].                  (I19)
+
+Integrating defines J_0=integral_E integral_S c_0 and proves J_r/r^3->J_0>0.
+This is a simultaneous contact-kernel estimate for c_r, not a statement that
+(I15)'s error is uniform for every joint pair (r,delta). Those are distinct
+claims, and the latter is unnecessary here.
+
+## 6. The limiting scaled law at zero and inverse-moment convergence
+
+RP's L1 convergence of the blown-up density, followed by its measurable
+pushforward s=dist/r, gives TV convergence of the normalized pair-weighted law
+of S_r to S. Define
+
+    Psi_E(t)=integral_E dx integral_S de p_(V0|Q0)(0,0,b,t)
+        E_Q0[w_0(det A)^2|V_0=(grad f,H e,f,-T/12)=(0,0,b,t)].
+
+This nonnegative function is bounded, continuous, and decays faster than any
+fixed inverse polynomial after appropriate polynomial moment factors. Its
+weighted integral against t^2 is positive. The all-index limiting density is
+
+    g_S(s)=36s/(z_0 A_0) integral_R t^2(k-s^3|t|)_+Psi_E(t)dt. (I20)
+
+At s->0, dominate the integrand by k t^2 Psi_E(t). Then
+
+    lim g_S(s)/s = 36k/(z_0 A_0) integral t^2 Psi_E(t)dt
+                 = J_0/A_0,                                (I21)
+
+because T=-12t converts 36t^2 to T^2/4. Integrating the positive density near
+zero proves the small-separation cumulative statement in (I6). Its inverse
+q-moment is finite near zero iff q<2; for q>=0 there is no inverse-moment
+obstruction at infinity, since s^-q<=1 for s>=1. This proves the limiting-law
+threshold, without using RD's upper-tail exponent.
+
+For 0<q<2, apply (I2) at p=-q and at p=0:
+
+    E[S_r^(-q)] = r^q M_(-q)(r)/M_0(r) -> A_(-q)/A_0.
+
+The same weighted dominated-convergence argument computes E[S^(-q)] from
+(I20), giving that value. TV convergence alone would not justify this step for
+an unbounded inverse-distance observable. At q>=2, Section 4 proves the finite-r
+expectation is already infinite, and (I21) proves the limit expectation is
+infinite as well. All statements are about pair-weighted means, not probabilities
+of the existence of a very close pair or a persistence-lifetime law.
+
+## 7. Scope and review obligations
+
+The positive diagonal coefficient, not a finite numerical fit, makes the inverse
+threshold sharp. The two different collision frames must not be mixed: the
+height-divided frame supplies RP's r-blow-up; the gradient-only frame supplies
+the fixed-r diagonal with Jacobian delta^-d. Local curvature and polar volume
+cancel the dimension in (I16). This does not assert isotropy of the torus field.
+
+All claimed new limits keep E fixed and remote. Fixed-r divergence, normalized
+contact inverse moments, and an arbitrary coupled cutoff epsilon(r) are three
+different assertions; only the first two are proved here. No rate for remainders,
+leading event-probability constant, same-index inverse threshold, p>=1
+r-asymptotic, global pin-neighborhood bound or scientific-status promotion is
+provided. RP's new coefficient/height law has an actual Anthropic discussion
+review; its source remains on the original branch at author publication time.
+
+The finite runner checks powers, moment domains, cutoff integrals, determinant
+and height signs on explicit polynomial fields, and coefficients. It cannot
+certify Gaussian support, uniform regression, Kac-Rice applicability, or the
+Borel translation argument. Those displayed analytic steps require a new
+source-bound nonauthor review. No unavailable author runner is reconstructed.
diff --git a/frontiers/remote_inverse_separation_20260928/README.md b/frontiers/remote_inverse_separation_20260928/README.md
new file mode 100644
index 000000000..83167187c
--- /dev/null
+++ b/frontiers/remote_inverse_separation_20260928/README.md
@@ -0,0 +1,26 @@
+# Remote inverse separation — author-side mathematical candidate
+
+[PROOF.md](PROOF.md) extends the microscopic moment law to -2<p<1 and proves a
+fixed-r positive diagonal intensity j_r(delta)~J_r delta. Inverse-separation
+moments are finite exactly below power2, with logarithmic divergence at2.
+The scaled factorial-pair separation has small-value probability asymptotic to
+(J_0/(2A_0))epsilon^2, and its finite inverse moments converge. No event-conditioned
+pair sampling, persistence-partner law, same-index threshold, shrinking rho,
+or arbitrary coupled-cutoff limit is asserted.
+
+Scientific effect NONE. Source-bound nonauthor analytic review is required.
+The note uses the original RP pair-law source; it does not duplicate or discharge
+RD's pending positive-distance moment transition. Its code is a new finite
+power/threshold companion, not a recovery of any unavailable author executable.
+
+Run from this directory:
+
+    python -B -S verify.py --output /path/to/new-output
+    python -B -O -S verify.py --output /path/to/other-new-output
+
+Both entries run12 tests and8 semantic negatives in EACH Python mode, checking
+flat source membership, bytes, SHA256/Git identities and intended assertion
+failures. Output must be new and outside the packet. These finite controls do
+not prove the Gaussian support, conditioning, Kac-Rice or limiting arguments.
+SOURCE_MAP.json pins the analytic inputs but the packet runner does not replay
+or independently validate those complete upstream manuscripts.
diff --git a/frontiers/remote_inverse_separation_20260928/RECONNAISSANCE.md b/frontiers/remote_inverse_separation_20260928/RECONNAISSANCE.md
new file mode 100644
index 000000000..1e67139c1
--- /dev/null
+++ b/frontiers/remote_inverse_separation_20260928/RECONNAISSANCE.md
@@ -0,0 +1,28 @@
+# Dependency-focused primary-source reconnaissance — 28 September 2026
+
+Read scope: primary arXiv abstract records and author institutional metadata;
+not a full-text novelty audit or a claim of historical priority.
+
+- Jean-Marc Azaïs and Céline Delmas, *Mean number and correlation function of
+  critical points of isotropic Gaussian fields and some results on GOE random
+  matrices*, arXiv:1911.02300, https://arxiv.org/abs/1911.02300 . The paper already
+  treats dimension-dependent short-range correlations and adjacent-index
+  attraction. The underlying collision mechanism is not claimed as a new
+  invention. The present theorem keeps a particular pinned full-determinant
+  tilt and a shrinking height window on a fixed remote region; those hypotheses
+  must be proved rather than attributed to the abstract.
+- Stephen Muirhead, *A second moment bound for critical points of planar Gaussian
+  fields in shrinking height windows*, arXiv:1901.11336,
+  https://arxiv.org/abs/1901.11336 . Published in Statistics & Probability Letters
+  160 (2020), article108698, as confirmed by the author's Monash publication
+  record. This is relevant shrinking-window context, not a direct assertion of
+  the inverse-separation threshold under this conditioned law.
+
+The new argument is a source-consumer extension of the exact RP/RM/RC manuscripts
+listed in SOURCE_MAP.json. The gradient-only fixed-r frame and the value-divided
+r-blow-up frame are deliberately separated. No external abstract is used as a
+substitute for a covariance floor or a weighted Kac-Rice theorem.
+
+Consensus's monthly quota was exhausted earlier in this same campaign. No new
+Consensus result or full-text read is claimed in this continuation. Broad search
+would be required before any novelty or first-discovery assertion.
diff --git a/frontiers/remote_inverse_separation_20260928/RESULTS.json b/frontiers/remote_inverse_separation_20260928/RESULTS.json
new file mode 100644
index 000000000..c67569f16
--- /dev/null
+++ b/frontiers/remote_inverse_separation_20260928/RESULTS.json
@@ -0,0 +1,8 @@
+{
+  "tests_per_mode": 12,
+  "semantic_mutants_per_mode": 8,
+  "identical_output_pairs": 9,
+  "all_intended_failures": true,
+  "scientific_effect": "NONE",
+  "continuum_analysis_verified_by_code": false
+}
diff --git a/frontiers/remote_inverse_separation_20260928/SOURCE_FILES.json b/frontiers/remote_inverse_separation_20260928/SOURCE_FILES.json
new file mode 100644
index 000000000..077ab5280
--- /dev/null
+++ b/frontiers/remote_inverse_separation_20260928/SOURCE_FILES.json
@@ -0,0 +1,54 @@
+{
+  "schema": 1,
+  "scientific_effect": "NONE",
+  "files": [
+    {
+      "path": "PROOF.md",
+      "bytes": 18054,
+      "sha256": "95a9aa75e4a254a8f50efc16ee00a78a2ea1481f4013a708dfcbb77dcad4c393",
+      "git_blob": "1b24d0f6d07f84706a0a8e9b25e111e6eff93a2b"
+    },
+    {
+      "path": "README.md",
+      "bytes": 1502,
+      "sha256": "5213369ccafd2cc65992a181eca9d30d874e8ad5c2445516a56bef0fc4b5d06b",
+      "git_blob": "83167187c809f7cb6723bea8369dc29a2eaf8f73"
+    },
+    {
+      "path": "RECONNAISSANCE.md",
+      "bytes": 1795,
+      "sha256": "51d0f1132965fb0f0e2207b85a9f23835e9cda078c9784a0da5cfe2a82ade67a",
+      "git_blob": "1e67139c1cf90edc5c72c282a46294f2e68d1386"
+    },
+    {
+      "path": "RESULTS.json",
+      "bytes": 204,
+      "sha256": "21578b27a2fdd655c0cff256c26bd76babb5661ae7df79e56009ce3965e719bb",
+      "git_blob": "c67569f164239d6af1299a8c5112a9ee092cc962"
+    },
+    {
+      "path": "SOURCE_MAP.json",
+      "bytes": 2384,
+      "sha256": "54b2259bb54430391b5ff9529003beec4fa39186c6440f3de9cae0184c0814b1",
+      "git_blob": "e6a9299daed339e4ec30e99816950398b785c705"
+    },
+    {
+      "path": "inverse.py",
+      "bytes": 2446,
+      "sha256": "e37c380eafcafad1dd815f3ecebd47b1fa9e41be93f5bd5ffe8dfbe2311ec175",
+      "git_blob": "d63833225f11715270889a0093da7c3313b9754e"
+    },
+    {
+      "path": "test_inverse.py",
+      "bytes": 3590,
+      "sha256": "ff421fac99b83f0434af293968676c151b144c97edca8a69c21805fd0961d49e",
+      "git_blob": "3c22788edd413151797196e9224bea5aa8935cc3"
+    },
+    {
+      "path": "verify.py",
+      "bytes": 4860,
+      "sha256": "21b54f9156096d6c0cecb1d521474ab8c5c3b51ad817f5889cd6f617c24bf060",
+      "git_blob": "d74dd57fd6dfd1c2ff104d9b3728ea5e0ab85a45"
+    }
+  ]
+}
diff --git a/frontiers/remote_inverse_separation_20260928/SOURCE_MAP.json b/frontiers/remote_inverse_separation_20260928/SOURCE_MAP.json
new file mode 100644
index 000000000..e6a9299da
--- /dev/null
+++ b/frontiers/remote_inverse_separation_20260928/SOURCE_MAP.json
@@ -0,0 +1,52 @@
+{
+  "scientific_effect": "NONE",
+  "source_repository": "d6g8k5htny-coder/Math-",
+  "sources": [
+    {
+      "id": "RM",
+      "commit": "5a5b97d2c518dd5c04c89d17029ec96e721d3b78",
+      "path": "frontiers/remote_window_20260924/PROOF.md",
+      "bytes": 18355,
+      "sha256": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7",
+      "git_blob": "b383bfcc88ec4ad497dff01fb6640e429ba24a84",
+      "use": "Exact law, weighted Kac-Rice/field marks, conditional moments and full normalizer."
+    },
+    {
+      "id": "RC",
+      "commit": "5a5b97d2c518dd5c04c89d17029ec96e721d3b78",
+      "path": "frontiers/remote_collision_20260928/PROOF.md",
+      "bytes": 26002,
+      "sha256": "b9b8b58fd8266db7ffe6537078003888ef138b445d3289ec5588f116af9050c2",
+      "git_blob": "7b48a88e2af54e759e89a8c8219573bb450a65ea",
+      "use": "Pair Kac-Rice, collision regression floor and moments, separated-pair bound."
+    },
+    {
+      "id": "RP",
+      "commit": "0507e3a1dbd3dede84cbeb805947c70aa16ebc36",
+      "path": "frontiers/window_multiplicity_laws_20260928/REMOTE_PAIR_LAW.md",
+      "bytes": 15878,
+      "sha256": "b818a11f1abcf37616d592ecda6341b0d99365cf3ac5047bc2dbaede95da4c5e",
+      "git_blob": "3fb602040b37158b16ba61bdfc62cc8df1ccc616",
+      "use": "Blown-up L1 density and determinant limits, fixed-Borel translation, exact A0 and sampling convention.",
+      "review": "Math-#116 comment5880900642, Anthropic analytic review; published review-code packet separate."
+    },
+    {
+      "id": "RD",
+      "role": "comparison_only_not_consumed",
+      "commit": "0507e3a1dbd3dede84cbeb805947c70aa16ebc36",
+      "path": "frontiers/window_multiplicity_laws_20260928/REMOTE_DISTANCE_MOMENTS.md",
+      "bytes": 11097,
+      "sha256": "b40dc67a0c4760d3db796914bdc50cbbd5fd1371786772fa919a3a100467b41d",
+      "git_blob": "d077743b7fda8b1bb2e1bafdab97d01bb9e321b1",
+      "use": "Only identifies the prior p>=0 scope. Its p=1 transition and upper tail are not premises."
+    }
+  ],
+  "new_obligations": [
+    "Fixed-r gradient-only collision coefficient and positive support",
+    "Height-indicator convergence and arbitrary fixed-Borel radial asymptotic",
+    "Contact matching J_r/r^3 and inverse-moment weighted domination"
+  ],
+  "blocked_executable_consumed_or_retried": false,
+  "upstream_full_replay_by_packet_runner": false,
+  "nonauthor_acceptance": false
+}
diff --git a/frontiers/remote_inverse_separation_20260928/inverse.py b/frontiers/remote_inverse_separation_20260928/inverse.py
new file mode 100644
index 000000000..d63833225
--- /dev/null
+++ b/frontiers/remote_inverse_separation_20260928/inverse.py
@@ -0,0 +1,66 @@
+"""Exact scalar companions; not a Gaussian covariance or Kac-Rice verifier."""
+from fractions import Fraction as F
+
+MUTANT = None
+
+def gradient_density_power(d: int) -> int:
+    if type(d) is not int or d<2:
+        raise ValueError('fixed integer dimension at least two required')
+    return -(d+3) if MUTANT=='height-jacobian-in-gradient-frame' else -d
+
+def radial_power(d: int) -> int:
+    determinants=0 if MUTANT=='drop-soft-determinants' else 2
+    return d-1+gradient_density_power(d)+determinants
+
+def envelope_integral(p: F) -> F:
+    p=F(p)
+    if not -2<p<1:
+        raise ValueError('weighted envelope requires -2<p<1')
+    return 1/(p+2)+1/(1-p)
+
+def overlap_coefficient(p: F) -> F:
+    p=F(p)
+    if p<=-2:
+        raise ValueError('origin integral requires p>-2')
+    numerator=1 if MUTANT=='lose-overlap-three' else 3
+    return numerator/((p+2)*(p+5))
+
+def moment_finite(p: F) -> bool:
+    return F(p)>=-2 if MUTANT=='include-divergent-endpoint' else F(p)>-2
+
+def inverse_cutoff(q: int, lo: F, hi: F):
+    if type(q) is not int or q<0 or not F(0)<lo<hi:
+        raise ValueError('nonnegative integer inverse power and 0<lo<hi required')
+    if q==2:
+        return {'kind':'log','ratio':hi/lo,'coefficient':F(1)}
+    return (hi**(2-q)-lo**(2-q))/(2-q)
+
+def small_cdf_coefficient(J: F, A: F) -> F:
+    if J<=0 or A<=0:
+        raise ValueError('positive intensity and mass required')
+    return J/A if MUTANT=='lose-cdf-half' else J/(2*A)
+
+def fold_data(delta: F, T: F, transverse: list[F]) -> dict:
+    if delta<=0 or not transverse:
+        raise ValueError('positive separation and nonempty transverse block required')
+    detA=F(1)
+    for a in transverse:detA*=a
+    gradient=lambda x:[T*x*x/2-T*delta*x/2]+[F(0)]*len(transverse)
+    value=lambda x:T*x**3/6-T*delta*x*x/4
+    left=-T*delta*detA/2;right=T*delta*detA/2
+    gap=value(delta)-value(F(0))
+    if MUTANT=='reverse-height-sign':gap=-gap
+    return {'gradients':(gradient(F(0)),gradient(delta)),
+            'height_difference':gap,'determinants':(left,right),
+            'absolute_product':abs(left*right)}
+
+def height_width_power() -> int:
+    return 6 if MUTANT=='two-height-windows-at-fixed-r' else 3
+
+def microscopic_power(p: F) -> F:
+    return F(5)+F(p)
+
+def pair_weight(n: int) -> int:
+    if type(n) is not int or n<0:
+        raise ValueError('nonnegative integer count required')
+    return int(n>=2) if MUTANT=='event-not-factorial-weight' else n*(n-1)
diff --git a/frontiers/remote_inverse_separation_20260928/test_inverse.py b/frontiers/remote_inverse_separation_20260928/test_inverse.py
new file mode 100644
index 000000000..3c22788ed
--- /dev/null
+++ b/frontiers/remote_inverse_separation_20260928/test_inverse.py
@@ -0,0 +1,79 @@
+"""Exact finite controls; the continuum argument is in PROOF.md."""
+from fractions import Fraction as F
+import unittest
+import inverse as m
+
+class InverseTests(unittest.TestCase):
+    def test_gradient_only_density_jacobian(self):
+        for d in range(2,10):
+            self.assertEqual(m.gradient_density_power(d),-d)
+
+    def test_radial_power_independent_of_dimension(self):
+        for d in range(2,10):
+            self.assertEqual(m.radial_power(d),1)
+
+    def test_microscopic_envelope_domain(self):
+        for p in (F(-7,4), F(-1), F(0), F(1,2)):
+            self.assertEqual(m.envelope_integral(p),1/(p+2)+1/(1-p))
+        for p in (F(-3),F(-2),F(1),F(2)):
+            with self.assertRaises(ValueError):m.envelope_integral(p)
+
+    def test_integrated_overlap_coefficient(self):
+        for p in (F(-7,4),F(-1),F(0),F(1,2),F(2)):
+            self.assertEqual(m.overlap_coefficient(p),1/(p+2)-1/(p+5))
+        self.assertEqual(m.overlap_coefficient(F(0)),F(3,10))
+        with self.assertRaises(ValueError):m.overlap_coefficient(F(-2))
+
+    def test_inverse_endpoint_is_not_finite(self):
+        for q in (F(0),F(1),F(199,100)):
+            self.assertTrue(m.moment_finite(-q))
+        for q in (F(2),F(3),F(10)):
+            self.assertFalse(m.moment_finite(-q))
+
+    def test_fixed_r_cutoff_logarithm(self):
+        self.assertEqual(m.inverse_cutoff(2,F(1,100),F(1,2)),
+                         {'kind':'log','ratio':F(50),'coefficient':F(1)})
+
+    def test_fixed_r_cutoff_powers(self):
+        lo,hi=F(1,100),F(1,2)
+        self.assertEqual(m.inverse_cutoff(0,lo,hi),(hi*hi-lo*lo)/2)
+        self.assertEqual(m.inverse_cutoff(1,lo,hi),hi-lo)
+        self.assertEqual(m.inverse_cutoff(3,lo,hi),1/lo-1/hi)
+        self.assertEqual(m.inverse_cutoff(4,lo,hi),(lo**-2-hi**-2)/2)
+        with self.assertRaises(ValueError):m.inverse_cutoff(3,F(0),hi)
+
+    def test_small_separation_cdf_has_half(self):
+        self.assertEqual(m.small_cdf_coefficient(F(7),F(3)),F(7,6))
+
+    def test_exact_polynomial_fold_data(self):
+        for d in (2,3,4):
+            for delta in (F(1,3),F(1,11)):
+                for T in (F(-5),F(7,3)):
+                    transverse=[F((-1)**i*(i+1)) for i in range(d-1)]
+                    detA=F(1)
+                    for a in transverse:detA*=a
+                    result=m.fold_data(delta,T,transverse)
+                    self.assertEqual(result['gradients'],([F(0)]*d,[F(0)]*d))
+                    self.assertEqual(result['height_difference'],-T*delta**3/12)
+                    self.assertEqual(result['determinants'],(-T*delta*detA/2,T*delta*detA/2))
+                    self.assertEqual(result['absolute_product'],delta**2*T**2*detA**2/4)
+
+    def test_t_to_T_and_coefficient_conversion(self):
+        self.assertEqual(36*F(1,12)**2,F(1,4))
+        self.assertEqual(108/F(12)**2,F(3,4))
+        self.assertEqual(m.height_width_power(),3)
+
+    def test_negative_moment_scaling(self):
+        for q in (F(1,2),F(1),F(3,2)):
+            self.assertEqual(m.microscopic_power(-q),5-q)
+            self.assertEqual(q+m.microscopic_power(-q)-m.microscopic_power(0),0)
+            self.assertGreater(6-m.microscopic_power(-q),0)
+
+    def test_probability_size_bias_not_event_sampling(self):
+        # P(N=2)=1/4, P(N=3)=1/4: pair weighting differs from event weighting.
+        self.assertEqual(m.pair_weight(2),2)
+        self.assertEqual(m.pair_weight(3),6)
+        self.assertEqual(F(1,4)*m.pair_weight(2)/(F(1,4)*m.pair_weight(2)+F(1,4)*m.pair_weight(3)),F(1,4))
+        self.assertNotEqual(F(1,4),F(1,2))
+
+if __name__=='__main__':unittest.main()
diff --git a/frontiers/remote_inverse_separation_20260928/verify.py b/frontiers/remote_inverse_separation_20260928/verify.py
new file mode 100644
index 000000000..d74dd57fd
--- /dev/null
+++ b/frontiers/remote_inverse_separation_20260928/verify.py
@@ -0,0 +1,94 @@
+"""Exact packet and both-mode finite verification. No network or proof-status writes."""
+import argparse
+import hashlib
+import io
+import json
+from pathlib import Path
+import subprocess
+import sys
+import unittest
+
+ROOT=Path(__file__).resolve().parent
+MUTANTS={
+ 'height-jacobian-in-gradient-frame':'test_gradient_only_density_jacobian',
+ 'drop-soft-determinants':'test_radial_power_independent_of_dimension',
+ 'lose-overlap-three':'test_integrated_overlap_coefficient',
+ 'include-divergent-endpoint':'test_inverse_endpoint_is_not_finite',
+ 'lose-cdf-half':'test_small_separation_cdf_has_half',
+ 'reverse-height-sign':'test_exact_polynomial_fold_data',
+ 'two-height-windows-at-fixed-r':'test_t_to_T_and_coefficient_conversion',
+ 'event-not-factorial-weight':'test_probability_size_bias_not_event_sampling',
+}
+
+def require(ok, message):
+    if not ok:raise ValueError(message)
+
+def decode(text):
+    def pairs(items):
+        out={}
+        for k,v in items:
+            require(k not in out,'duplicate JSON key: '+k);out[k]=v
+        return out
+    return json.loads(text,object_pairs_hook=pairs)
+
+def identity(path):
+    require(path.is_file() and not path.is_symlink(),'regular source required')
+    data=path.read_bytes()
+    return {'bytes':len(data),'sha256':hashlib.sha256(data).hexdigest(),
+            'git_blob':hashlib.sha1(f'blob {len(data)}\0'.encode()+data).hexdigest()}
+
+def source_check():
+    identity(ROOT/'SOURCE_FILES.json')
+    entries=decode((ROOT/'SOURCE_FILES.json').read_text())['files']
+    names=[e['path'] for e in entries]
+    require(len(names)==len(set(names)) and all(Path(n).name==n and n!='SOURCE_FILES.json' for n in names),'unique flat paths required')
+    require(sorted(p.name for p in ROOT.iterdir())==sorted(names+['SOURCE_FILES.json']),'source membership differs')
+    for e in entries:
+        require(identity(ROOT/e['path'])=={k:e[k] for k in ['bytes','sha256','git_blob']},'source identity differs: '+e['path'])
+    return {p.name:identity(p) for p in sorted(ROOT.iterdir())}
+
+def child(mutant):
+    import inverse
+    import test_inverse
+    inverse.MUTANT=mutant
+    result=unittest.TextTestRunner(stream=io.StringIO()).run(unittest.defaultTestLoader.loadTestsFromModule(test_inverse))
+    out={'tests':result.testsRun,'passed':result.wasSuccessful(),
+         'failures':sorted(t.id().rsplit('.',1)[-1] for t,_ in result.failures),
+         'errors':sorted(t.id().rsplit('.',1)[-1] for t,_ in result.errors)}
+    print(json.dumps(out,sort_keys=True))
+    return 0 if result.wasSuccessful() else 1
+
+def replay(output):
+    output=output.resolve()
+    require(not output.exists() and not output.is_relative_to(ROOT),'new output directory outside packet required')
+    before=source_check();output.mkdir(parents=True)
+    records={}
+    for case in ['baseline',*MUTANTS]:
+        outputs=[]
+        for mode,flags in [('normal',[]),('optimized',['-O'])]:
+            cmd=[sys.executable,'-B',*flags,'-S',str(ROOT/'verify.py'),'--child']
+            if case!='baseline':cmd+=['--mutant',case]
+            p=subprocess.run(cmd,capture_output=True,timeout=30)
+            (output/(case+'_'+mode+'.stdout')).write_bytes(p.stdout)
+            (output/(case+'_'+mode+'.stderr')).write_bytes(p.stderr)
+            r=decode(p.stdout.decode())
+            require(not p.stderr and not r['errors'],'execution error rather than semantic test: '+case)
+            require(r['tests']==12,'wrong finite test count')
+            if case=='baseline':require(p.returncode==0 and r['passed'] and not r['failures'],'baseline failed')
+            else:require(p.returncode==1 and not r['passed'] and MUTANTS[case] in r['failures'],'intended assertion did not reject: '+case)
+            outputs.append(p.stdout)
+        require(outputs[0]==outputs[1],'Python modes disagree: '+case)
+        records[case]=decode(outputs[0].decode())
+    summary={'tests_per_mode':12,'semantic_mutants_per_mode':8,'identical_output_pairs':9,
+             'all_intended_failures':True,'scientific_effect':'NONE','continuum_analysis_verified_by_code':False}
+    require(summary==decode((ROOT/'RESULTS.json').read_text()),'stored summary differs')
+    require(source_check()==before,'source changed during execution')
+    (output/'REPORT.json').write_text(json.dumps({**summary,'sources_unchanged':True,'source_identities':before,'cases':records},sort_keys=True,indent=2)+'\n')
+    print(json.dumps(summary,sort_keys=True,indent=2));return 0
+
+if __name__=='__main__':
+    p=argparse.ArgumentParser(description=__doc__);g=p.add_mutually_exclusive_group(required=True)
+    g.add_argument('--child',action='store_true');g.add_argument('--output',type=Path)
+    p.add_argument('--mutant',choices=MUTANTS);a=p.parse_args()
+    require(a.child or a.mutant is None,'mutant requires child mode')
+    raise SystemExit(child(a.mutant) if a.child else replay(a.output))