D5: remote single-point configuration and conditional location law (review required)

.github/workflows/remote-singleton.yml

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+name: Remote single-point law finite controls
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/remote_singleton_law_20260928/**'
+      - '.github/workflows/remote-singleton.yml'
+  push:
+    branches: [main]
+    paths:
+      - 'frontiers/remote_singleton_law_20260928/**'
+      - '.github/workflows/remote-singleton.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  finite-singleton-controls:
+    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: Exact sources, two Python modes and semantic mutations
+        run: |
+          set -euo pipefail
+          python -B -S frontiers/remote_singleton_law_20260928/run_validation.py --output "${RUNNER_TEMP}/remote-singleton-${GITHUB_RUN_ID}-${GITHUB_RUN_ATTEMPT}"
+          git diff --exit-code

frontiers/remote_singleton_law_20260928/PROOF.md

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+# Remote rare critical points: the single-point configuration law
+
+Object: OA-D5-REMOTE-SINGLETON-LAW-20260928-v1.
+Author: OpenAI / ChatGPT, foreground continuation.
+Disposition: AUTHOR-SIDE COROLLARY; NONAUTHOR REVIEW REQUIRED.
+Scientific effect: NONE. No governing status, proof index, graph, prize or source body is changed.
+
+## 1. Intent and exact source dependencies
+
+The fixed-remote mean and second factorial moment say more than that a window
+critical point is rare: when one appears, its complete remote configuration is
+usually a singleton, and its location/index distribution can be identified. This
+note makes that consumption explicit. It proves an elementary general probability
+lemma and applies only already specified fixed-remote estimates. It introduces no
+new Gaussian covariance argument and no global witness-collision assertion.
+
+Repository: `d6g8k5htny-coder/Math-`.
+
+1. `frontiers/remote_window_20260924/PROOF.md`, Theorem A, read at commit
+   `541d4e9daf3f2ca8b5a485d6e96ecf3dbf958f17`, blob
+   `b383bfcc88ec4ad497dff01fb6640e429ba24a84`. This gives the uniform fixed-remote
+   mean density expansion, not by itself a matching event-probability law.
+2. `frontiers/remote_collision_20260928/PROOF.md`, Corollary D, v4 at commit
+   `66437d90f3b7222256b2b8aed62400c42a78a08a`, 26,002 bytes, SHA-256
+   `b9b8b58fd8266db7ffe6537078003888ef138b445d3289ec5588f116af9050c2`, blob
+   `7b48a88e2af54e759e89a8c8219573bb450a65ea`. This gives the second factorial
+   moment with BOTH witnesses in the same fixed remote region. OpenAI's
+   source-exposed review 5343164624 of that source is not a review of this note.
+
+The application below is conditional on the mathematical validity of these
+precise inputs. Publication or merger is not an additional proof premise. The
+exact generic lemma in Sections 2-4 is proved here. No earlier verdict is
+inherited for the new application. Math-#107's planar global first moment is
+NOT an input: the present statement is remote in every fixed d>=2.
+
+## 2. A finite point-process lemma with an exact error identity
+
+Let S be a standard Borel space. Give the space of finite counting measures on S
+its counting-map sigma algebra. A point is represented by its unit mass delta_x;
+the singleton embedding and its inverse on singleton configurations are
+measurable. Multiplicities are allowed in this lemma.
+
+Let Xi be a random finite counting measure, N=Xi(S), and
+
+    mu(A)=E Xi(A),  m=mu(S)<=1,
+    q=E[N(N-1)]<infinity,
+    h=E[N; N>=2].                                      (S1)
+
+Define Ber(mu) as a random counting measure: it is empty with probability 1-m,
+and assigns singleton delta_x with subprobability measure mu(dx). When m>0 this
+means sample a Bernoulli(m) indicator and, if nonempty, sample x from mu/m. For
+m=0 it is the deterministic empty configuration. The condition m<=1 is necessary
+for this definition; it is not silently replaced by min(m,1).
+
+Use probability total variation
+
+    dTV(P,Q)=sup_A |P(A)-Q(A)|=(1/2)||P-Q||_var,
+
+where || ||_var is the full mass norm of a signed measure.
+
+**Lemma 1 (exact identity).**
+
+    dTV(Law Xi, Ber(mu)) = h <= q.                       (S2)
+
+**Proof.** Define the singleton submeasure sigma on S by
+
+    sigma(A)=P(N=1 and the unique point belongs to A),
+
+and the nonnegative multiple-point mean measure eta by
+
+    eta(A)=E[Xi(A); N>=2].
+
+Then mu=sigma+eta, sigma(S)=s:=P(N=1), and eta(S)=h. Put
+alpha=P(N>=2) and p=P(N>=1)=s+alpha. Thus m=s+h and
+m-p=h-alpha>=0. On the three disjoint parts of configuration space:
+
+- empty: Law Xi minus Ber(mu) has mass m-p=h-alpha;
+- singletons: Ber(mu) minus Law Xi is the pushforward of eta, of mass h;
+- configurations of size >=2: Law Xi minus Ber(mu) has mass alpha.
+
+The full variation is (h-alpha)+h+alpha=2h, proving equality in (S2).
+Finally n<=n(n-1) for integers n>=2 gives h<=q. This also shows all relevant
+measures are finite. No point independence, stationarity, Poisson hypothesis,
+Gaussian conditioning or pair-density formula is used. QED.
+
+The constant one in h<=q is sharp: take Xi=delta_a+delta_b with probability
+epsilon<=1/2, otherwise empty. Then m=h=q=2epsilon and dTV=2epsilon.
+The points can coincide in the generic example; choosing distinct a,b gives a
+simple-process example as well.
+
+**Why a first moment alone is insufficient.** The same example with
+epsilon=r^3/2 has m=r^3 but dTV=r^3, not O(r^5). The factorial input is essential
+to the stronger approximation asserted below.
+
+## 3. Conditional laws and single-witness location
+
+Suppose m>0. Necessarily p>0 since Xi is nonnegative. Denote by Delta(nu) the law
+of a singleton at an S-valued point with probability distribution nu.
+
+**Lemma 2.**
+
+    dTV(Law(Xi | N>=1), Delta(mu/m)) <= h/m <= q/m.       (S3)
+
+If s=P(N=1)>0, then also
+
+    dTV(Law(unique point | N=1), mu/m) <= h/m.           (S4)
+
+**Proof.** The common submeasure Delta(sigma/m) is dominated by both laws in
+(S3). In the first law the singleton part is Delta(sigma/p), and p<=m; in the
+second it is Delta((sigma+eta)/m). Its total mass is s/m, so the probability-TV
+distance is at most 1-s/m=h/m. This domination argument proves the claim for
+arbitrary measurable configuration events, not just events determined by N.
+For (S4), write mu/m=(s/m)(sigma/s)+eta/m; the usual mixture difference has
+TV at most h/m. This argument identifies the unique point, not an arbitrary
+selection rule from a configuration with several points. QED.
+
+The elementary count inequalities also give
+
+    0<=m-p=E[(N-1);N>=2]<=q/2,
+    P(N>=2 | N>=1)<=q/(2p),
+    s=m-h>=m-q.                                        (S5)
+
+The factors 1/2 arise from 1<=n(n-1)/2 and n-1<=n(n-1)/2 for n>=2. Thus
+q=o(m) ensures both a positive singleton event and conditional uniqueness with
+probability tending to one. All these claims count multiplicity, not merely
+distinct occupied sites.
+
+## 4. Replacing the exact mean measure by a leading approximation
+
+Let mu_0 be another nonnegative measure with 0<m_0=mu_0(S)<=1. Set
+D=||mu-mu_0||_var. Disjoint empty and singleton parts yield the exact formula
+
+    dTV(Ber(mu),Ber(mu_0))=(|m-m_0|+D)/2 <= D.           (S6)
+
+For m,m_0>0, direct normalization gives
+
+    dTV(mu/m,mu_0/m_0)<=D/m.                            (S7)
+
+Indeed mu/m-mu_0/m_0=(mu-mu_0)/m+mu_0(1/m-1/m_0), whose variation mass is at
+most (D+|m-m_0|)/m<=2D/m. Combining with (S2)-(S4),
+
+    dTV(Law Xi,Ber(mu_0)) <= q+D,
+    dTV(Law(Xi|N>=1),Delta(mu_0/m_0)) <= (q+D)/m.        (S8)
+
+The unique-point version has the same upper bound when s>0. These bounds retain
+the normalization denominator. An absolute O(r^4) mean error becomes O(r)
+after conditioning on an event of order r^3; it does not remain O(r^4).
+
+## 5. Exact Gaussian application, keeping the remote scope
+
+Use exactly the normalized periodized field and original endpoint law in the
+source Theorem A. In particular d>=2, L and 0<rho<L/4 are fixed, marks b lie in
+a fixed compact interval, 0<k_-<=k<=k_+, frames range over O(d), and
+
+    M=-(r/2)u, S=(r/2)u,
+    f(M)=b, f(S)=b-k r^3, grad f(M)=grad f(S)=0,
+    W_r=F_d(H_M)F_(d-1)(H_S), Z_r=E_Qr W_r,
+    dQ_r^W=(W_r/Z_r)dQ_r.
+
+Q_r has the original 2(d+1) observations (six only when d=2). No adjacency
+condition, second normalizer, or endpoint Jacobian is introduced. The law Q_r^W
+need not be Gaussian; Lemmas 1-2 apply to any probability law.
+
+Let E be a deterministic Borel subset of D_rho={dist(x,0)>=rho}. Define Xi_r on
+S_E=E x {0,...,d} by placing one atom (x,j) at each nonsingular index-j critical
+point with height in I_r=(b-k r^3,b). Thus N_r=Xi_r(S_E) is the TOTAL window count
+on E. Positions and indices are joint marks; their independence is not assumed.
+The height is restricted and then discarded as a mark. This note makes no claim
+about the height's conditional distribution.
+
+The mean input and factorial input, summing finitely many indices where needed,
+are
+
+    mu_r(dx,{j}) = [k r^3 Lambda_j(x;b,k,u)+e_(r,j)(x)] dx,
+    |e_(r,j)(x)| <= C r^4 almost everywhere,
+    q_r:=E_(Q_r^W)[N_r(N_r-1)] <= C r^5 |E|.            (S9)
+
+Theorem A already states the mean-density formulation. Alternatively its error
+bound for every Borel set implies absolute continuity and the stated a.e. density
+bound. Do not infer that statement from a single value of E. The factorial
+estimate is Corollary D, including its uniform-in-E absorption of the separated
+Cr^6|E|^2 term using finite torus volume and bounded small r.
+
+Put
+
+    Lambda(x,j)=Lambda_j(x;b,k,u),
+    A_E=sum_j integral_E Lambda_j(x;b,k,u) dx,
+    mu_(0,r)(dx,{j})=k r^3 Lambda_j(x;b,k,u) dx.
+
+The fixed-remote Lambda_j are continuous, uniformly positive and bounded on the
+source's compact parameter ranges. Consequently for |E|>0 and sufficiently small
+r, uniformly over these deterministic E,
+
+    c r^3 |E| <= m_r=mu_r(S_E) <= C r^3 |E| <=1,
+    D_r:=||mu_r-mu_(0,r)||_var <= C r^4 |E|,
+    q_r/m_r <= C r^2.                                  (S10)
+
+The cutoff ensuring m_r<=1 uses |E|<=L^d and is uniform in E. Reducing it further
+gives q_r<m_r and m_(0,r)<=1. For |E|=0 the source mean is zero, so Xi_r is empty
+a.s.; the conditional assertions are deliberately not made in that case.
+
+**Corollary 3 (complete remote configuration).**
+
+    dTV(Law_(Q_r^W) Xi_r,Ber(mu_r)) <= C r^5 |E|,
+    dTV(Law_(Q_r^W) Xi_r,Ber(mu_(0,r))) <= C r^4 |E|.    (S11)
+
+The first is a complete-configuration approximation at the exact mean; the second
+uses the explicit leading contact intensity. Neither assumes point independence.
+
+**Corollary 4 (location/index given a rare occurrence).** For |E|>0 define
+
+    nu_E(dx,{j}) = Lambda_j(x;b,k,u) dx / A_E.
+
+Then
+
+    Q_r^W(N_r=1 | N_r>=1) >= 1-C r^2,
+    dTV(Law(Xi_r | N_r>=1),Delta(mu_r/m_r)) <= C r^2,
+    dTV(Law(Xi_r | N_r>=1),Delta(nu_E)) <= C r.          (S12)
+
+The distribution of the unique point conditional on N_r=1 obeys the same Cr
+bound to nu_E. Equations (S11)-(S12) follow directly from (S2)-(S10), and their
+constants have exactly the fixed d,L,rho and compact-mark dependencies of the
+inputs. Conditional uniqueness uses p_r>=m_r-q_r/2 and q_r/m_r=O(r^2).
+
+These bounds hold uniformly even for positive-volume deterministic E=E_r shrinking
+inside the SAME D_rho, since |E| cancels from (S10). They do not permit a random
+region chosen from the realized field, or rho=rho(r) tending to zero.
+
+The joint density nu_E need not factor into an independent spatial law and index
+law. Restricting to one chosen index gives its corresponding single-type version;
+it is not necessary to pretend that several indices occur independently.
+
+## 6. What this contributes and what remains open
+
+The new output is a usable rare-event configuration and conditional spatial/index
+law, obtained from source-bound mean and pair estimates. It can compare a sampled
+remote witness distribution with the normalized contact profile; it cannot certify
+an elder-pairing law or the location of a globally selected persistence partner.
+
+It does not prove a torus-wide second factorial moment, control pairs with either
+witness near the pins or at intermediate distances, remove the height window,
+produce numerical constants, change any governing status, or prove a nondegenerate
+Poisson limit. Ber(mu_r) itself tends to the empty configuration; (S12), obtained
+after conditioning on a rare occurrence, is the nontrivial spatial statement.
+No Poisson approximation theorem is imported or needed here.
+
+No mathematical proof is inferred from test counts. Sections 2-4 provide complete
+measure-theoretic proofs of the probability statements; Section 5 spells out
+exactly which analytic inputs are still required from the Gaussian sources.
+Nonauthor review should check the TV convention, empty/single/multiple signed
+measure split, normalization factors, arbitrary-Borel mean-error implication,
+index marking, and fixed-remote versus global boundaries.
+
+## 7. Finite controls and limited reconnaissance
+
+The standard-library test suite computes exact rational probability laws on two
+atoms with up to three-point configurations, retaining multiplicities. It checks
+(S2)-(S8), sharpness, invalid mean>1 and zero conditioning events, the failure of
+first-moment-only inference, and an explicit r-dependent family with r^3 mean,
+r^5 factorial moment and r-sized leading conditional correction. Deliberate
+mutations distinguish h from P(N>=2), mean from event probability, and TV from
+the full variation norm. None of this tests Gaussian field simulation or infinite
+configuration spaces.
+
+Consensus discovery query in this session concerned total variation, factorial
+moments and rare single-point processes. A fetched record was Breton--Privault,
+*Factorial moments of point processes*, Stochastic Processes and their Applications
+124, 3412-3428 (record lists 2013). The fetch supplied metadata only, not an abstract
+or theorem. No result from it is consumed. This note's generic probability proof
+is elementary, and no novelty or priority claim is made for it. The contribution
+claimed here is the explicit, correctly scoped application to the project's remote
+critical-point inputs, not invention of Bernoulli approximations.

frontiers/remote_singleton_law_20260928/README.md

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+# Remote single-point law — bounded distributional corollary
+
+Scientific effect: NONE. AUTHOR_SIDE_REVIEW_REQUIRED.
+
+Read PROOF.md. It proves the exact generic identity
+
+    dTV(Law Xi, Ber(E Xi)) = E[N;N>=2] <= E[N(N-1)],
+
+when E N<=1, with probability-TV convention. It then consumes exactly the
+fixed-remote mean theorem and remote factorial bound to obtain complete-
+configuration error O(r^5), conditional singleton error O(r^2) to the exact mean
+profile, and O(r) to the leading location/index density. The region must remain
+inside fixed D_rho, and the original between-pin height window is retained.
+
+This is not a new Gaussian covariance proof, global collision bound, elder-rule
+claim, numerical certificate, or nondegenerate Poisson limit. Prior reviews are
+not inherited as reviews of this corollary. The full generic proof is separate
+from finite control fixtures and from the Gaussian application premises.
+
+Run (new output directory outside this folder):
+
+    python3 -B -S run_validation.py --output /tmp/new-singleton-validation
+
+The runner checks SOURCE_FILES membership and hashes, runs 14 named tests in each
+normal/optimized Python mode and six semantic mutants, and compares its stored
+summary. Its fixtures are finite rational laws, not a continuum proof checker.
+No archived research script, network request or status mutation is performed.

frontiers/remote_singleton_law_20260928/RESULTS.json

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+{
+  "all_mutants_assertion_rejected": true,
+  "baseline_passed": true,
+  "continuum_verified": false,
+  "distinct_mutants": 6,
+  "modes": [
+    "normal",
+    "optimized"
+  ],
+  "named_tests_per_mode": 14,
+  "scientific_effect": "NONE"
+}

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+  ],
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+  "schema": 1,
+  "scientific_effect": "NONE"
+}