d6g8k5htny-coder · Sep 30, 2026 · Sep 30, 2026 · Sep 30, 2026 · Sep 30, 2026
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+name: Extreme whole-cluster sampling checks
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/extreme_cluster_sampling_20260930/**'
+      - '.github/workflows/extreme-cluster-sampling.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  exact-and-source:
+    runs-on: ubuntu-latest
+    timeout-minutes: 10
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+          fetch-depth: 0
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Fetch exact public source objects
+        run: |
+          set -euo pipefail
+          git fetch --no-tags origin 80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337
+          git fetch --no-tags origin 382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf
+          git fetch --no-tags origin bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8
+      - name: Verify historical sources, packet and exact controls
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+          GIT_NO_REPLACE_OBJECTS: '1'
+        run: |
+          set -euo pipefail
+          python -B -S frontiers/extreme_cluster_sampling_20260930/verify.py
+          git diff --exit-code
@@ -0,0 +1,236 @@
+# Addendum: an explicit unit-square law for the whole extreme doublet
+
+Object: OA-EXTREME-CLUSTER-SAMPLING-20260930-D1.
+Author: OpenAI / GPT-6 Astra Pro. New conditional derivation; nonauthor review OPEN.
+Scientific effect NONE. PROOF.md remains byte-identical. This addendum consumes
+that manuscript's E12-E19/E29 as candidate inputs; their review is not inferred
+from finite tests. It introduces no additional Gaussian source assumption.
+
+## 1. The law and its exact domain
+
+Consider the limiting ordinary extreme cluster of PROOF Theorem2, conditioned
+on being a doublet. Its outer-root shape has density Q/J on S2_outer, with
+J=1083417/280. Write p=-u, A=12p^2-3 and ell=v/A. Define
+
+  rho=|Z_companion/Z_outer| in (0,1),
+  w=2ell,
+  z=(1+rho)(w-1)/(2rho).                                  (D1)
+
+The map is a smooth bijection, up to the boundary null sets already stated in
+PROOF, between S2_outer and the OPEN UNIT SQUARE 0<rho<1,0<z<1. In these coordinates
+its unnormalized density, whose integral is J, is
+
+  L(rho,z)=165888 rho z^7(z+1)^4
+        *(rho*z+1)^4(rho*z+rho+1)^7
+        /[(rho+1)^5(2rho*z+rho+1)^9].                      (D2)
+
+Consequently L/J is the exact universal joint density of the companion/maximum
+radius ratio rho and the shape coordinate z under the already-formed limiting
+EXTREME DOUBLE-CLUSTER law. It is not the law of a uniformly chosen extra point,
+a finite-r Gaussian sample, or a remote singleton.
+
+The downward heights of the maximum anchor and companion respectively are
+
+  eta_out=z^2(rho*z+1)(rho*z+2rho+1)
+                       /[(rho+1)(2rho*z+rho+1)],
+  eta_in=rho^3 z^2(z+1)(rho*z+rho+2)
+                       /[(rho+1)(2rho*z+rho+1)].          (D3)
+
+They satisfy 0<eta_in<eta_out<1 on the square. In fact direct subtraction gives
+
+  eta_out-eta_in=z^2(1-rho)(rho*z+rho+1)^2/(2rho*z+rho+1),
+  1-eta_out=(1-z^2)(rho*z+rho+1)^2
+                                  /[(rho+1)(2rho*z+rho+1)].
+
+Both are manifestly positive in the open square. The overall Pareto(11) maximum
+scale and the cusp orientation/parameter law remain independent of (rho,z),
+by PROOF E29. To reconstruct both marked positions use PROOF E30 and these heights.
+
+## 2. Derivation of the bijection and density
+
+For an outer root its companion has u_c=-p_c, t_c=-rho. The two invariants in
+E13 and the stationary line give
+
+  4p_c^2-1=rho^2(4p^2-1),
+  ell=(p_c+rho*p)/(1+rho).
+
+Thus with w=2ell,
+
+  p=[(1+rho)w^2-(1-rho)]/(4rho*w),
+  p_c=[(1+rho)w^2+(1-rho)]/(4w).                          (D4)
+
+The outer condition is p>p_c, equivalent to w>1 for 0<rho<1. The height formulas
+before replacing w are
+
+  eta_out=(w-1)^2(rho*w-rho+w+1)(rho*w+3rho+w+1)
+                                             /(16rho^2 w),
+  eta_in=(w-1)^2(rho*w+rho+w-1)(rho*w+rho+w+3)/(16w),
+  1-eta_out=-(w+1)^2(rho*w-3rho+w-1)(rho*w+rho+w-1)
+                                             /(16rho^2 w).  (D5)
+
+All positive factors displayed are strictly positive for w>1,0<rho<1. Hence
+eta_out<1 is precisely
+
+  1<w<(1+3rho)/(1+rho).
+
+On this interval the companion is also in the strict window: eta_in>0 directly,
+and eta_in<eta_out by E17 (or by subtracting D5). This proves both directions
+of the bijection D1. No root or type constraint is silently dropped.
+
+The shape change (p,ell)->(u,v)=(-p,A ell) has absolute Jacobian A. At fixed w,
+
+  |partial p/partial rho|/2=(w^2-1)/(8rho^2 w).
+
+Together with Q=A^3(4ell^2-1)(4p ell-1), the density in (rho,w) is
+
+  81(1+rho)(w-1)^7(w+1)^7
+    *(rho*w-rho+w+1)^4(rho*w+rho+w-1)^4
+    /(4096rho^11 w^9).                                   (D6)
+
+Insert w=1+2rho*z/(1+rho) and its Jacobian2rho/(1+rho) to obtain D2. These
+identities can be checked by direct substitution; test_ratio_law.py also uses
+independent exact two-variable differentiation of the inverse map to check
+L=Q|det d(u,v)/d(rho,z)| on72 rational interior cases. Integration of D2 is J
+because this is a bijective change of variables from the source outer sector,
+not because of numerical normalization.
+
+## 3. A new small-companion asymptotic
+
+Let f_rho be the marginal density of rho under the doublet law. Set
+
+  H0=integral_0^1 z^7(z+1)^4 dz=6401/3960,
+  Csmall=82944 H0/J=137647104/3972529.                      (D7)
+
+The rational factor multiplying165888 rho z^7(z+1)^4 in D2 tends uniformly
+to1 as rho->0. Its denominator is bounded away from zero on the closed square.
+Therefore
+
+  f_rho(rho) ~ (275294208/3972529) rho,
+  P(rho<epsilon | limiting extreme doublet)
+                     ~ Csmall epsilon^2.                 (D8)
+
+This is a further limit INSIDE the already-formed extreme-doublet law: the order
+is r->0, then the microscopic extreme threshold->infinity, then epsilon->0.
+It is not a uniform triple-scale assertion for the original field.
+
+More precisely, conditional on rho<epsilon, (rho/epsilon,z) converges weakly to
+independent variables Qsmall,Zsmall with densities
+
+  2q dq, 0<q<1;
+  z^7(z+1)^4/H0 dz, 0<z<1.                               (D9)
+
+Indeed substitute rho=epsilon q in D2 and divide by D8. Uniform convergence of
+the bounded rational factor proves convergence of the densities in L1; thus D9
+is even total variation on these FIXED scalar-coordinate spaces. This is not a
+transfer of total variation through an epsilon-dependent spatial root map.
+
+By direct uniform limits of D3,
+
+  eta_out -> Zsmall^2,
+  eta_in/rho^3 -> 2 Zsmall^2(Zsmall+1).                    (D10)
+
+Equivalently eta_in/epsilon^3 has limit2 Qsmall^3 Zsmall^2(Zsmall+1). The small
+companion thus approaches the upper height boundary with a CUBIC radius-ratio
+scale while the outer height remains nonzero. This describes normalized height
+marks, not an elder-pairing or actual finite-r persistence-lifetime theorem.
+
+The limiting outer-height density in D10 is particularly simple:
+
+  e^3(1+sqrt(e))^4/(2H0), 0<e<1,
+  E[e]=483876/582491.                                     (D11)
+
+It follows by e=z^2 and termwise polynomial integration. No model-dependent
+Gaussian integral remains in D7-D11.
+
+At the other endpoint, rho->1-, continuity of D2 and integration on the bounded
+z interval give
+
+  f_rho(1-)=857223/2889112-(945/361139)log2.                (D12)
+
+For verification L(1,z)=81 z^7(z+2)^7/[8(z+1)]. Put x=z+1 and expand
+(x^2-1)^7/x on[1,2]: its integral is2571669/2240-(81/8)log2. Dividing by J
+gives D12. The equal-radius boundary itself still has probability zero.
+
+## 4. A rejection sampler with a certified ideal-real acceptance rate
+
+Define
+
+  a(rho,z)=(rho*z+1)^4(rho*z+rho+1)^7
+                   /[(rho+1)^5(2rho*z+rho+1)^9].          (D13)
+
+It satisfies0<a<=1 on the square. Indeed rho*z+1<=rho+1 and
+rho*z+rho+1<=2rho*z+rho+1, so
+
+  a<=1/[(rho+1)(2rho*z+rho+1)^2]<=1.
+
+Use the proposal density
+
+  q0(rho,z)=2rho z^7(z+1)^4/H0.                            (D14)
+
+It is a product distribution: draw rho=sqrt(U) for uniform U in(0,1); independently
+choose j in{0,...,4} with probabilities C(4,j)/[(8+j)H0], and draw z=V^(1/(8+j))
+for independent uniform V. The five integer mixture weights are
+
+  (495,1760,2376,1440,330), with sum6401.
+
+Accept with probability a(rho,z), otherwise repeat with independent proposals.
+Since q0*a=L/(82944 H0), the accepted law is exactly L/J in an IDEAL REAL
+uniform-random model. Its exact acceptance probability and mean proposal count are
+
+  p_accept=J/(82944H0)=3972529/137647104,
+  E[proposals]=137647104/3972529.                          (D15)
+
+Thus the same number Csmall in the small-ratio CDF is the ideal sampler's expected
+proposal count. This is an identity of the explicit integrals, not an empirical
+performance estimate.
+
+ratio_law.sample_doublet implements the algorithm using Python's ordinary
+floating-point pseudorandom numbers, with reproducible seeding, open-interval
+checks and a bounded trial budget. It returns rho,z and both height marks only;
+it does not simulate the Gaussian cusp parameters, spatial direction, Pareto
+maximum scale or the original field. Numerical roundoff is NOT interval-enclosed
+and the simulator is not an exact-real sampler. Its tests check reproducibility,
+ranges, rejection limits and exact underlying algebra, not distributional accuracy
+of floating-point draws.
+
+Example:
+
+    import random
+    from ratio_law import sample_doublet
+    marks = sample_doublet(random.Random(20260930))
+
+## 5. Separate review boundary
+
+Requested Slice D: the square bijection and Jacobian D1-D6; both normalized
+height formulas and ordering; density/CDF small-ratio constants, the further
+conditional law and order of limits D7-D12; the proposal/envelope and ideal-real
+acceptance probability D13-D15. These consume the main manuscript's new results
+conditionally; an acceptance of this algebra alone would not accept PROOF's
+Gaussian limit. The16 new test methods are finite evidence, not analytic review.
+No original PROOF.md, extremes.py, source pin, or prior review byte is amended.
+
+## 6. Exact falsifier against a finite-scale interpretation
+
+A supporting author-exposed recovery lane supplied PR169 comment5902401772;
+the values below were independently recomputed here with Fraction arithmetic.
+This is an exact negative example, not an additional analytic acceptance.
+
+Take k=1,a=-12,s=-14/5,beta=33/4,c=-31/40, and the root shape
+u=-3/4,v=14/5,Z=1. The companion is
+
+  u_c=-20907/28124, v_c=670215854/247174805, Z_c=-6919/7031.
+
+Both are typed strict-window saddles. Nevertheless, with the actual unsheared
+coordinates X=u-aZ/(12k),
+
+  R_seed^2=17/16,
+  R_companion^2=3126268865/790959376,
+  eta_seed=13/60, eta_companion=618523783/2966097660.
+
+Although |Z_c|<|Z|, the companion has GREATER physical radius and SMALLER downward
+height. Thus finite-scale physical ordering is not the |Z| ordering, and the
+farther-is-deeper statement is not an assertion at every finite radius. The
+maximum selector in PROOF Section5 is the actual physical selector before the
+limit; the limiting ordering and this addendum concern the second, already-formed
+extreme law. The new exact regression pins this distinction without changing
+any original proof bytes.
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