diff --git a/.github/workflows/elder-partner-extremes.yml b/.github/workflows/elder-partner-extremes.yml new file mode 100644 index 000000000..77741c122 --- /dev/null +++ b/.github/workflows/elder-partner-extremes.yml @@ -0,0 +1,39 @@ +name: Elder-selected extreme partner checks + +'on': + pull_request: + paths: + - 'frontiers/elder_partner_extremes_20260930/**' + - '.github/workflows/elder-partner-extremes.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 commits + run: | + set -euo pipefail + git fetch --no-tags origin 80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337 + git fetch --no-tags origin 56a7904f9ad7d409c31e23195c0e7eeab230613a + git fetch --no-tags origin 79f18f0cb318fceccd5bdf4e6dad29815165bc7d + git fetch --no-tags origin 9c6a73489c5ed2465a3cf65dcc950eeeae074b05 + - name: Verify five source identities, packet and both Python modes + env: + PYTHONDONTWRITEBYTECODE: '1' + GIT_NO_REPLACE_OBJECTS: '1' + run: | + set -euo pipefail + python -B -S frontiers/elder_partner_extremes_20260930/verify.py + git diff --exit-code diff --git a/.github/workflows/remainder-rate.yml b/.github/workflows/remainder-rate.yml new file mode 100644 index 000000000..1aacac1c5 --- /dev/null +++ b/.github/workflows/remainder-rate.yml @@ -0,0 +1,98 @@ +name: Remainder rate exact controls + +'on': + pull_request: + paths: + - 'frontiers/remainder_rate_20260930/**' + - '.github/workflows/remainder-rate.yml' + # bound merged sources (consumed in SOURCES.json) and the pinned unmerged dependency (#191; verified by blob) + - 'frontiers/remainder_vanishing_20260930/PROOF.md' + - 'frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md' + - 'frontiers/c7_total_bounded_20260929/PROOF.md' + - 'frontiers/fixed_r_inverse_lifetime_20260930/PROOF.md' + - 'frontiers/c7_zero_gap_limit_20260929/PROOF.md' + - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' + - 'imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md' + - 'reviews/d1_section9_borel_repair_20260925/REPAIR.md' + - 'reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-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: Verify sources, replay both modes, reject mutants + env: + PYTHONDONTWRITEBYTECODE: '1' + GH_TOKEN: ${{ github.token }} + GH_REPOSITORY: ${{ github.repository }} + run: | + set -euo pipefail + python -B -S - <<'PY' + import base64, hashlib, json, os, pathlib, subprocess, sys, urllib.request + root = pathlib.Path('frontiers/remainder_rate_20260930').resolve() + manifest = json.loads((root / 'SOURCES.json').read_text()) + present = sorted(p.name for p in root.iterdir()) + expected_names = sorted([e['path'] for e in manifest['files']] + ['SOURCES.json']) + if present != expected_names: + raise ValueError('packet tree differs from manifest: ' + repr(present)) + for entry in manifest['files']: + path = root / entry['path'] + if path.is_symlink(): + raise ValueError('symlink: ' + entry['path']) + data = path.read_bytes() + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']: + raise ValueError('packet file identity mismatch: ' + entry['path']) + for entry in manifest['consumed'] + manifest['cited_only']: + data = pathlib.Path(entry['path']).read_bytes() + framed = b'blob ' + str(len(data)).encode() + b'\0' + data + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or hashlib.sha1(framed).hexdigest() != entry['git_blob']: + raise ValueError('merged source drifted: ' + entry['path']) + # the consumed unmerged dependency (#191) is verified by its pinned blob: from the tree if the dependency + # has landed (then it must be byte-identical to the pinned version, else this packet needs rebinding), + # otherwise fetched by blob id through the repository API; absence or drift fails the job + for entry in manifest.get('consumed_unmerged', []): + tree_path = pathlib.Path(entry['path']) + if tree_path.exists(): + data = tree_path.read_bytes() + where = 'tree' + else: + req = urllib.request.Request( + 'https://api.github.com/repos/' + os.environ['GH_REPOSITORY'] + '/git/blobs/' + entry['git_blob'], + headers={'Authorization': 'Bearer ' + os.environ['GH_TOKEN'], 'Accept': 'application/vnd.github+json'}) + with urllib.request.urlopen(req, timeout=60) as resp: + blob = json.load(resp) + if blob.get('encoding') != 'base64': + raise ValueError('unexpected blob encoding for ' + entry['path']) + data = base64.b64decode(blob['content']) + where = 'blob api' + framed = b'blob ' + str(len(data)).encode() + b'\0' + data + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or hashlib.sha1(framed).hexdigest() != entry['git_blob']: + raise ValueError('consumed unmerged dependency differs from the pinned blob (rebind required): ' + entry['path'] + ' via ' + where) + print(json.dumps({'consumed_unmerged_verified': entry['path'], 'git_blob': entry['git_blob'], 'via': where})) + script = str(root / 'rate_ledger_check.py') + expected = (root / 'RESULTS.json').read_bytes() + for flags in (['-B', '-S'], ['-B', '-O', '-S']): + out = subprocess.run([sys.executable, *flags, script], check=True, capture_output=True, timeout=600).stdout + if out != expected: + raise RuntimeError('output differs from RESULTS.json under ' + ' '.join(flags)) + for m in manifest['mutants']: + rc = subprocess.run([sys.executable, *flags, script, '--mutant', m], capture_output=True, timeout=600).returncode + if rc != 1: + raise RuntimeError('mutant ' + m + ' not rejected') + rc = subprocess.run([sys.executable, *flags, script, '--mutant', 'M9'], capture_output=True, timeout=600).returncode + if rc != 2: + raise RuntimeError('unknown mutant label not rejected with exit 2') + print(json.dumps({'scientific_effect': 'NONE', 'mathematical_acceptance': False})) + PY + git diff --exit-code diff --git a/.github/workflows/remainder-vanishing.yml b/.github/workflows/remainder-vanishing.yml new file mode 100644 index 000000000..e046d962d --- /dev/null +++ b/.github/workflows/remainder-vanishing.yml @@ -0,0 +1,73 @@ +name: Remainder vanishing exact controls + +'on': + pull_request: + paths: + - 'frontiers/remainder_vanishing_20260930/**' + - '.github/workflows/remainder-vanishing.yml' + # bound sources (consumed and cited_only in SOURCES.json) + - 'frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md' + - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' + - 'imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md' + - 'reviews/d1_section9_borel_repair_20260925/REPAIR.md' + - 'reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md' + - 'frontiers/c7_total_bounded_20260929/PROOF.md' + - 'frontiers/unrestricted_selection_difference_20260929/PROOF.md' + - 'frontiers/c7_zero_gap_limit_20260929/PROOF.md' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-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: Verify sources, replay both modes, reject mutants + env: + PYTHONDONTWRITEBYTECODE: '1' + run: | + set -euo pipefail + python -B -S - <<'PY' + import hashlib, json, pathlib, subprocess, sys + root = pathlib.Path('frontiers/remainder_vanishing_20260930').resolve() + manifest = json.loads((root / 'SOURCES.json').read_text()) + present = sorted(p.name for p in root.iterdir()) + expected_names = sorted([e['path'] for e in manifest['files']] + ['SOURCES.json']) + if present != expected_names: + raise ValueError('packet tree differs from manifest: ' + repr(present)) + for entry in manifest['files']: + path = root / entry['path'] + if path.is_symlink(): + raise ValueError('symlink: ' + entry['path']) + data = path.read_bytes() + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']: + raise ValueError('packet file identity mismatch: ' + entry['path']) + for entry in manifest['consumed'] + manifest['cited_only']: + data = pathlib.Path(entry['path']).read_bytes() + framed = b'blob ' + str(len(data)).encode() + b'\0' + data + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or hashlib.sha1(framed).hexdigest() != entry['git_blob']: + raise ValueError('merged source drifted: ' + entry['path']) + script = str(root / 'even_contact_check.py') + expected = (root / 'RESULTS.json').read_bytes() + for flags in (['-B', '-S'], ['-B', '-O', '-S']): + out = subprocess.run([sys.executable, *flags, script], check=True, capture_output=True, timeout=600).stdout + if out != expected: + raise RuntimeError('output differs from RESULTS.json under ' + ' '.join(flags)) + for m in manifest['mutants']: + rc = subprocess.run([sys.executable, *flags, script, '--mutant', m], capture_output=True, timeout=600).returncode + if rc != 1: + raise RuntimeError('mutant ' + m + ' not rejected') + rc = subprocess.run([sys.executable, *flags, script, '--mutant', 'M9'], capture_output=True, timeout=600).returncode + if rc != 2: + raise RuntimeError('unknown mutant label not rejected with exit 2') + print(json.dumps({'scientific_effect': 'NONE', 'mathematical_acceptance': False})) + PY + git diff --exit-code diff --git a/frontiers/concave_fibre_elder_20260930/HARD_FIBRE_LEMMA.md b/frontiers/concave_fibre_elder_20260930/HARD_FIBRE_LEMMA.md new file mode 100644 index 000000000..7365ed9bc --- /dev/null +++ b/frontiers/concave_fibre_elder_20260930/HARD_FIBRE_LEMMA.md @@ -0,0 +1,254 @@ +# Hard-negative fibre lift of the planar elder barrier + +Author-side deterministic contribution by OpenAI `concave_fibre_geometry`, 30 September 2026. This is a mathematical extension candidate, not a source acceptance decision. It changes no existing source, status, integration disposition or probability/count claim. Its planar input is §§2–6 of `local_elder_geometry_20260930/PROOF.md`, corresponding to the requested PR170 cut. Source-cut verification and acceptance provenance are separate obligations handled by the parent task. + +## 1. The planar input and the additional geometric condition + +Let + +\[ +P(X,Z)=2kX^3-\frac{3k}{2}X-\frac{k}{2} + +\frac{s}{2}Z^2+\frac a2(X^2-1/4)Z + +\frac\beta2XZ^2+\frac c6Z^3, +\] + +with \(k>0\), \(M=(-1/2,0)\), \(S=(1/2,0)\), +\[ +B=\beta-a^2/(12k),\qquad +D=(c-a\beta/(4k)+a^3/(72k^2))/2. +\] +Assume the strict typed condition \(s<-|B|/2\), and exclude +\[ +12kD^2+(s-B)^2(B+2s)=0,\qquad 12kD^2+B^3=0. +\] +Let \(\mathcal Y=\{S\}\cup C(P)\), where \(C(P)\) consists of the additional critical points in \(-kk\), the hard boundary obeys the relative drop estimate + \[ + g_i(x,z)\le G_i(x)-D_0\quad(x\in\overline B_R,\ z\in\partial F_i(x)). \tag{HD} + \] + A graph-centred ellipsoid tube, described below, is a concrete sufficient interpretation of this tube hypothesis. + +No uniform lower bound on the hard Hessian eigenvalues is needed if (HD) is given directly. No control is required outside this tube. Strict negativity by itself, without (HD) or an equivalent closed-barrier condition, does not suffice. + +The hard tube need not cover the whole of \(B_{R+1}\): the reduced height only needs to be defined there for the planar collar construction, whereas the full lift needs a tube over the smaller disk containing the resulting caps. The graph paths also remain in that smaller disk. + +## 2. Deterministic conclusion + +For each \(Y\in\mathcal Y\), let \(Y_i\) be the unique continued critical point of \(G_i\), taking \(S_i=S\), and let +\[ +h_i=\max_{Y\in\mathcal Y}G_i(Y_i),\qquad +\widehat Y_i=\Psi_i(Y_i,\zeta_i(Y_i)),\qquad +\widehat M_i=\Psi_i(M,\zeta_i(M)). +\] +For all sufficiently large \(i\), +\[ +\boxed{\quad d_{f_i}(\widehat M_i)=b_i+r_i^3h_i.\quad} \tag{EHD} +\] +Here \(d_f(M)=\sup_{\gamma(0)=M,\ f(\gamma(1))>f(M)}\min_t f(\gamma(t))\) is the ordinary global superlevel maximin. In particular, +\[ +\frac{d_{f_i}(\widehat M_i)-b_i}{r_i^3}\to h_*. +\] +On the actual global Morse distinct-value locus, the unique graph point \(\widehat Y_i\) attaining \(h_i\) is the actual elder merge saddle. If \(C(P)\) is empty, this is exactly the lifted pinned saddle. If \(C(P)\) is nonempty, a continued window saddle has height strictly above \(-k\) and preempts the pinned saddle. Limiting ties do not obstruct the exact statement: one compares the finitely many actual heights at index \(i\). + +The actual lifetime is therefore exactly +\[ +b_i-d_{f_i}(\widehat M_i)=-r_i^3h_i, +\] +and its normalized limit is \(-h_*\). This is a deterministic geometry/lifetime result. It supplies no higher-dimensional tilted-law exhaustion, intensity, coefficient or count statement. + +## 3. Graph regularity and full critical-point indices + +At each fibre stationary point, negative definiteness makes \(g_{zz}\) invertible. The ordinary implicit function argument gives \(\zeta_i\in C^1\) for \(g_i\in C^2\), and +\[ +D\zeta_i=-g_{zz}^{-1}g_{zx},\qquad +\nabla G_i=g_x,\qquad +\nabla^2G_i=g_{xx}-g_{xz}g_{zz}^{-1}g_{zx}. \tag{SC} +\] +Thus \(G_i\in C^2\) even though a \(C^2\) graph is unnecessary. Existence of an interior fibre maximum must be assumed or separately verified; strict concavity alone only gives uniqueness once such a point exists. + +A point of the tube is full critical if and only if it lies on the graph and its soft point is critical for \(G_i\). At such a point, for soft and hard tangent vectors \(v,w\), +\[ +(v,w)^T\nabla^2g_i(v,w) +=v^T\nabla^2G_i v + +(w+g_{zz}^{-1}g_{zx}v)^Tg_{zz}(w+g_{zz}^{-1}g_{zx}v). +\] +This is an explicit invertible triangular change of variables, so full inertia adds \(m\) negative directions to reduced inertia. A reduced maximum lifts to a full maximum; a reduced saddle lifts to a nondegenerate full saddle with one positive direction and \(d-1\) negative directions, the superlevel merge index. No parametric Morse normal-form theorem is invoked. + +## 4. A fixed soft collar works at every level above the actual highest saddle + +For completeness, this records the properties of the planar construction that the lift actually needs. + +The cubic gradient estimate off the second excluded algebraic set gives +\[ +|\nabla P(x)|\ge c_0|x|^2\quad(|x|\ge R_0). +\] +Choose \(R_0\) to contain all window roots, the pins, and a small oval about \(M\). Choose \(R\) sufficiently large to contain all doubled chords and satisfy +\[ +R-R_0>\frac{2k}{c_0R_0^2}+1. +\] +For all sufficiently large \(i\), reduced \(C^1\) convergence on \(\overline B_{R+1}\) gives +\[ +|\nabla G_i(x)|\ge(c_0/2)|x|^2 +\quad(R_0\le|x|\le R+1). +\] +There are no reduced critical points in \((h_i,0)\) inside \(B_{R+1}\). This is elementary finite-root stability: around any nondegenerate root \(Y\), the map +\(x\mapsto x-(\nabla^2P(Y))^{-1}\nabla G_i(x)\) +is a contraction on a sufficiently small closed ball and maps that ball into itself for large \(i\). It has exactly one root there. Outside these disjoint root balls, in a slightly enlarged compact height band containing \([-k,0]\), \(\nabla P\) has a positive lower bound; convergence excludes any other roots. The excluded transition set prevents an additional critical point at \(-k\), and the source proves that the only critical point at or above zero is \(M\). + +Fix \(t_0\in(h_*,0)\) sufficiently close to zero. Uniformly negative Hessians in a small ball about the exact pin give a small disk \(K_{i,0}\) with +\[ +G_i=t_0\text{ on }\partial K_{i,0},\qquad +t_0\le G_i\le0\text{ on }K_{i,0}. +\] +One can construct the disk without a Morse lemma: the radial derivative from \(M\) is strictly negative under the negative Hessian bound, so every ray meets level \(t_0\) once and the implicit function theorem gives its oval. + +For every \(h_ih_i\), choose \(h\in(h_i,\min(t,t_0))\): this is a compact full cap whose boundary is strictly below that target. These direct derivative checks supply all regularity used here. + +The barrel argument already proves global normalized death \(\le h\) for every \(h>h_i\) sufficiently close to \(h_i\). Letting \(h\downarrow h_i\) gives the exact upper bound \(\le h_i\). + +## 6. Exact lower bound on the graph, and actual pairing + +For each \(Y\in\mathcal Y\), use the moving soft critical chord +\[ +x_i(t)=M+t(Y_i-M),\qquad +q_i(t)=G_i(x_i(t)),\qquad 0\le t\le2. +\] +Exact criticality gives \(q_i'(0)=q_i'(1)=0\). Reduced \(C^2\) convergence gives convergence of these restrictions in \(C^2\) to +\[ +q(t)=P(Y)(3t^2-2t^3). +\] +The limit has \(q''(0)=6P(Y)<0\), \(q''(1)=-6P(Y)>0\), negative derivative on \((0,1)\), positive derivative on \((1,2)\), and \(q(2)=-4P(Y)>0\). Second-derivative signs near the two exact derivative zeros, followed by first-derivative convergence away from those zeros, give the same strict monotonicity for \(q_i\) for all sufficiently large \(i\). Therefore its exact minimum is \(G_i(Y_i)\), and its endpoint is older than birth. + +Lift the chord to the actual chart by +\[ +\gamma_{i,Y}(t)=\Psi_i(x_i(t),\zeta_i(x_i(t))). +\] +This path has exactly the same height restriction \(b_i+r_i^3q_i(t)\). The graph may curve sharply and its hard coordinates need not converge; no approximation by a full-dimensional straight chord is made. Its inclusion in the tube is all that is needed. There are finitely many such paths, so one sufficiently large index works for all of them. The path corresponding to an actual maximizer of \(G_i(Y_i)\) gives normalized global death \(\ge h_i\), proving (EHD). + +This also directly identifies the merge at that graph point. Just above \(h_i\), the chord's two sides lead respectively to the birth and to an older endpoint, and they are in different global superlevel components because death is \(h_i\). At level \(h_i\), the chord joins them at \(\widehat Y_i\). On the global distinct-critical-value locus, no other critical point has that same height. The Schur identity proves it is a full merge-index saddle. Hence it is the actual elder saddle under the same ordinary maximin convention as the planar theorem. No Morse–Smale assumption or separatrix identification is needed. + +## 7. A quantitative test for anisotropic hard extent + +A simple sufficient replacement for (HD) is a positive-definite matrix \(A_i(x)\), a fixed depth \(D_0>k\), and the graph-centred ellipsoid tube +\[ +F_i(x)=\{\zeta_i(x)+v:v^TA_i(x)v<2D_0\}. +\] +Assume this compact tube lies in the chart interior and, throughout every fibre, +\[ +g_{zz}(x,z)\preceq-A_i(x). +\] +Twice integrating along the hard segment from the stationary maximum gives +\[ +g_i(x,\zeta_i(x)+v)\le G_i(x)-\frac12v^TA_i(x)v. +\] +On the hard boundary the drop is at least \(D_0\); above a planar cap, \(G_i\le0\), so hard boundary height is at most \(-D_0<-k\). This proves the needed barrier. The theorem only needs this boundary estimate above the cap family; imposing it over all \(B_R\) is a convenient checkable condition. + +The semiaxes are \(\sqrt{2D_0/\lambda_j(A_i(x))}\). They may depend on \(i\), on the base point, and on direction. Thus anisotropic full-chart extent is harmless provided those ellipsoids actually fit. Uniform hard concavity combined with an arbitrarily thin chart is insufficient. In physical coordinates with straight hard fibres (or an affine map in the hard variables), if an unnormalized physical hard Hessian obeys \(f_{yy}\preceq-H_i(x)\), the corresponding physical requirement is a contained ellipsoid +\[ +v^TH_i(x)v\le2D_0r_i^3. +\] +When \(H_i\) has order-one eigenvalues, its physical semiaxes have order \(r_i^{3/2}\). This is a sufficient geometric scale, not an assertion that the actual stochastic model supplies it. + +For an arbitrary nonlinear chart, one must verify the normalized-coordinate hard Hessian/drop directly, or include the chart's extra second-derivative terms; a physical Hessian bound does not automatically transform by congruence away from a full critical point. + +The logically weaker boundary condition is that hard boundary height is at most \(h_i\) over every cap used for levels decreasing to \(h_i\). A fixed margin below \(-k\) avoids depending on the unknown actual winning height and verifies this at once. Some closed full-dimensional barrier condition is indispensable if the exterior is arbitrary; strict hard concavity is one useful means of producing its vertical part. + +## 8. Explicit vertical-escape counterexample without depth + +Choose +\[ +P(X,Z)=2X^3-\frac32X-\frac12-\frac12Z^2+\frac13Z^3. +\] +Here \(k=1,s=-1,a=\beta=0,c=2\), hence \(B=0,D=1\), and the excluded discriminants are \(10\) and \(12\). The four critical points have heights +\[ +P(-1/2,0)=0,\quad P(1/2,0)=-1,\quad +P(-1/2,1)=-1/6,\quad P(1/2,1)=-7/6. +\] +Thus \(h_*=-1/6\). + +Let \(\epsilon_i\downarrow0\), and let \(\chi\in C^\infty(\mathbb R,[0,1])\) be zero on \(( -\infty,1]\) and one on \([2,\infty)\). For definiteness use elder height \(A=\sqrt2\), and set +\[ +H_i(w)=(1-\chi(w/\epsilon_i))(-w^2) + +\chi(w/\epsilon_i)\,[A-(w-3\epsilon_i)^2], +\qquad F_i(X,Z,w)=P(X,Z)+H_i(w). +\] +On the chart \(\mathbb R^2\times(-\epsilon_i,\epsilon_i)\), +\[ +F_i=P-w^2, +\] +so the hard Hessian is the uniformly strictly negative scalar \(-2\), the unique fibre maximum graph is \(w=0\), and its reduced height is exactly \(P\) on every compact soft set. Yet the exterior has an older strict maximum at \((M,3\epsilon_i)\), of height \(A\). Along \(0\le w\le3\epsilon_i\), the first blend term vanishes for \(w\ge2\epsilon_i\) and is otherwise at least \(-4\epsilon_i^2\); the second term is nonnegative for sufficiently small \(\epsilon_i\). Hence +\[ +d_{F_i}(M,0)\ge-4\epsilon_i^2>-1/6. +\] +A maximin beginning at height zero is at most zero, so these actual death levels tend to zero, rather than to \(h_*\). The obstruction is already visible at the hard chart boundary: +\[ +F_i(M,\epsilon_i)=-\epsilon_i^2\to0. +\] +For a compact-manifold version, protect a soft neighbourhood containing the pins, the four critical points and the vertical path; for each \(i\) its normalized radius can be finite but increasing to infinity. Multiply/extend smoothly outside a larger neighbourhood into a coordinate ball of a compact manifold. This retains convergence on every fixed normalized soft disk and changes neither the chart hypotheses nor the exhibited path bound. Additional \(m-1\) hard directions can be appended as \(-|z'|^2\), giving the same failure in every \(d\ge3\). Generic arbitrarily small perturbations away from the protected chart and the protected older maximum can impose the global Morse distinct-value locus while preserving the vertical-path lower bound (with, for example, \(-5\epsilon_i^2\) in place of \(-4\epsilon_i^2\)). The height counterexample itself does not need this extra locus. + +It also realizes genuinely shrinking anisotropic charts. Choose any \(r_i\downarrow0\), take +\(\Psi_i(X,Z,w)=(r_iX,r_iZ,r_i^{3/2}w)\) +in a fixed coordinate ball, and set \(f_i\circ\Psi_i=b_i+r_i^3F_i\) on the protected region. For example, choose its normalized soft radius \(R_i=r_i^{-1/2}\); its physical radius is then \(r_i^{1/2}\), so it fits inside the fixed coordinate ball while containing every fixed normalized soft disk for all large \(i\). Extend outside the protected physical region. On the thin physical hard chart \(|y|\0` followed by the exact bytes. No source file or manifest was changed. + +## Deterministic graph, collar, cap and exact elder saddle + +**Pass, under the displayed fixed-target hypotheses.** SC's complete-jet coupling bounds the actual global C4 norm at a fixed spectral target. With positive hard eigenvalues, the physical hard Hessian stays uniformly negative on each needed soft disk of radius `Rr` and a hard ball of radius `c0r`. The hard-gradient forcing at zero is `O(r^2)`, including the `−tau_xxwj/8` term forced by the endpoint pins. Strict concavity and an inward radial derivative give the interior graph; uniqueness preserves its exact hard-coordinate pins. + +Implicit differentiation gives graph size `O(r^2)`, first soft derivatives `O(r)` and second derivatives `O(1)`. The graph substitution changes the physical potential and its first and second soft derivatives by `O(r^4)`, `O(r^3)` and `O(r^2)`. Rescaling two soft coordinates by `r` and height by `r^3` makes each correction `O(r)` in C2. No inverse hard bound from this pointwise argument is placed in the probability majorant. + +The essential geometric addition is verified: a graph-centered physical hard radius `rho r^(3/2)` fits since `Cr^2+rho r^(3/2)k`. The graph's own `r^2` width would only produce an `r^4` physical drop and would not close the `r^3` death window. + +Off Delta, the cubic gradient grows quadratically at infinity. On a large fixed annulus, C1 convergence retains a gradient floor. Finite-root stability, exact pins and the Sigma exclusion remove all active-strip critical points other than the finite continued set. The cutoff gradient flow therefore transports the small maximum oval to level `h`, and the annulus speed bound prevents its boundary reaching the fixed soft disk boundary within time less than `k`. The same sufficiently large index works for every `h_i=2`, `Lambda` is a retained hard eigenvalue. Holding `J,lambda2,...,lambdam` fixed leaves `U=J+Lambda` fixed during the `lambda1` integration. This does not assume that matrix entries or eigenvalues are independent. + +The actual second soft determinant factor is retained. The inner integral is exactly + +`integral_0^(DrU^2) lambda1(lambda1+CrU) d lambda1 = r^3[(D^3/3)U^6+(CD^2/2)U^5]`. + +Together with the original outer `r^2` weight, this gives an `r^5` unnormalized numerator. Enlarging the ordered `lambda1` interval and dropping its Gaussian factor are valid positive-majorant operations. Division once by the full `Z_r>=z_*r^2` yields the required `r^3` scale. + +Retaining `1{U>A}` in that same integral and using higher uniform residual moments gives the polynomial tail tending to zero. Retaining `1{lambda2<=epsilon}` gives the vanishing hard strip mass by Gaussian polynomial integrability, including higher-corank intersections. There is no inverse eigenvalue or inverse gap factor. The M4 exception has tilted mass `O(r^4)` and vanishes after `r^−3` scaling. + +Weyl's inequality and Hessian Lipschitz control convert endpoint tightness to actual midpoint jet tightness. At fixed `A,epsilon`, a midpoint `h2<=epsilon` on `U<=A` implies endpoint `lambda2<=2epsilon` for small radius. Sending the U tail to zero and then epsilon to zero proves E7, including negative midpoint hard values. This is exhaustion of the full selector-failure measure. No count event, near-occurrence tail, factorial estimate or remote-witness event is substituted. + +## Spectral kernel, null sets and actual marks + +**Pass.** The spectral change contributes one raw soft Jacobian `r`, the actual determinant weight is on the `r^4` soft-sector scale, and the original full normalizer is on the `r^2` scale. After failure rescaling the prefactor is exactly `c_m r^2/Z_r`. The Jacobian's one hard power and determinant weight's two hard powers give the displayed `product h_j^3`; all remaining hard Vandermonde factors are retained. + +The extended sign-frame construction is essential and is stated correctly. Normalized Haar is the reference angular measure with SC's multiplicity constant, while the anisotropic raw Gaussian density stays inside the integral. It is not silently replaced by an invariant or GOE law. Simultaneous changes of O, the complete tensor, and the soft root coordinate leave the physical location unchanged. + +On compact positive hard sectors, the two-short-column bound supplies integrable conditional domination. Exact H1 supplies pointwise convergence of the actual failure indicator on generic typed targets. The determinant-weight limit is continuous and zero on or outside the typed boundary, so no selector convergence is needed there. Sigma, Delta, repeated spectra and the additional position set `D=0` are null sets under the displayed absolute-continuity interface; none is a discarded positive leading mass. Repeated hard spectra need no inverse-gap estimate. + +Compact L1 convergence and E4–E7 give global total variation for the extended jet/failure measure. SC's deterministic integrability bound makes the limiting measure finite; open positive nonempty sectors make its mass positive. The mass is exactly SC's displayed local integral `a1+a2`, with no `beta_far` term. This consumes a deterministic kernel and integral, not SC's global count conclusion. + +The actual partner, normalized death and lifetime marks are claimed only weakly. On generic compact couplings H1 supplies the local winning root and older path; the graph's hard displacement divided by r vanishes. Conditional dominated convergence then gives the joint pushforward, and actual failure exhaustion removes sentinel/essential mass. Weak tightness with converging total masses implies actual failed-partner locality for every deterministic cutoff with `delta_r/r` tending to infinity. No growing-radius root enclosure is needed. + +The fixed-radius Morse null set is used correctly through an arbitrary fixed sequence tending to zero: disintegration and a countable intersection give the relevant joint almost-sure locus along that sequence. The deterministic graph/cap/chord argument then applies. Limits along every such sequence establish the stated limit without an unsupported common null set for uncountably many pinned laws. + +## Compact actual lifetime-density interface and retained boundaries + +**Pass for L1 at its exact population.** P is consumed with ERR's corrected congruence, BOREL's whole-field replacement, and REC's convergence-in-law and embedded-chart reading rules. P's actual radial intensity is `r A_r dr`, with the dimension powers already cancelled. Pushing forward by `ell=kr^3` yields `ell^(-1/3)A_r/(3k^(2/3))`. Multiplication by failure probability `r^3[(1-p_r)/r^3]` gives exactly the density-difference factor `A_r[(1-p_r)/r^3]/(3k^(5/3))` after division by `ell^(2/3)`. + +P supplies the required compact common bounds; these are not inferred from pointwise topology. Dominated convergence therefore gives a finite positive coefficient C_fail for the compact candidate-minus-selected density. The original full z0 cancels against A0 if substituted, with one original pin density remaining. Transverse-frame independence follows from the physical probability and its unique limit, not from false torus rotational invariance or higher-dimensional birth cancellation. + +This density is not a replacement-bar intensity: a maximum may participate in multiple rejected candidates. It is not a standalone second-order selected-density expansion, an unrestricted refined difference, a numerical coefficient, a new count/moment theorem, or a statement uniform in vanishing k, growing dimension, volume or escaping marks. Those boundaries are retained. No numerical or engineering test was treated as mathematical evidence. + +## Recorded disposition + +Initial exact cut: **AMEND**, for the six literal/scope items above. Substantive H/F/M/L1 reasoning: no load-bearing defect found at the stated conditional source scope. All failures and requested corrections are preserved in this report. Publication or integration of a revised cut requires its own exact reread; this report does not accept author changes it has not seen. + +## Intermediate reread: residual coordinate label retained as AMEND + +The first author-corrected PROOF.md had 23415 bytes, SHA256 `fa405e286cb1cc337f57bffa671035a533818a0dc31196874005e1e300c4454c`, Git blob `8c1db93878253a3aa53f48cee1062ce9f9044cb8`. The corrected SOURCE_INTERFACE.md had 18098 bytes, SHA256 `cec41136dfac1fde2247f25dcf7601e1a01a6ec25c4d5f18e8bf1450e8360e48`, Git blob `7eb203bdf1311ff56b312962ba3f5ce6d5dd9766`. I reread both complete bodies and independently checked their identities. HARD_FIBRE_LEMMA.md and SOURCES.json were unchanged. + +The finite-r/contact derivative, exact/limiting lifetime, manifest locator, dependency roles, P lifetime scope, normalized pin arguments and cumulative-sentence removal were corrected. However, PROOF.md line 157 still called the normalized moving soft chord a chord “from M”, after explicitly reserving M for the physical pin. Both my read and the read-only geometric reviewer retained **AMEND** for that remaining instance of the original coordinate finding. The deterministic lemma itself remained correct, and the law reviewer accepted its corrected slice conditionally. This intermediate failed precision outcome is not erased by the final reread. + +## Final exact reread and source-scoped disposition + +The authors changed the remaining soft-chord label to M0. I independently verified and reread the complete final PROOF.md; the complete corrected SOURCE_INTERFACE.md was already reread at its unchanged final identity. The other load-bearing packet files are unchanged. The final reviewed cut is: + +| File | Bytes | Git blob | SHA256 | +|---|---:|---|---| +| PROOF.md | 23416 | `923d3236b5291a29bd162937f172190db84dcef5` | `5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e` | +| HARD_FIBRE_LEMMA.md | 18907 | `7365ed9bcf145c832d1c09f0b685fbe8d6695050` | `434ffd30ac96eb635211ea627459b78c1d39806f75b0634e0f4876dce0765b33` | +| SOURCE_INTERFACE.md | 18098 | `7eb203bdf1311ff56b312962ba3f5ce6d5dd9766` | `cec41136dfac1fde2247f25dcf7601e1a01a6ec25c4d5f18e8bf1450e8360e48` | +| SOURCES.json | 7186 | `f072e1bb5c79456be3c79451b6df5b287807edfa` | `b97a29ea09f8e8270bfddda8f364f0d61e431e30c1681af4b019f3fb48709a4d` | + +**Final disposition: ACCEPT at the displayed conditional source scope.** All initial literal precision findings and the intermediate residual label are resolved in this exact cut. No remaining mathematical or packet-precision AMEND was found. I did not edit any packet file, author any amendment to its proof, or make a remote write. + +This acceptance covers the contained negative hard-tube lift and its thin-domain falsifier; exact identification of the actual highest continued elder saddle with arbitrary exterior; full retained endpoint-matrix failure exhaustion and hard-boundary removal; the extended spectral jet/failure law in total variation with coefficient a1+a2; the weak actual joint partner/death/lifetime law and actual failed-partner locality; and the positive ell^(2/3) compact candidate-minus-selected lifetime-density coefficient L1 at its exact population. + +The hypotheses remain conditions. In particular the Gaussian/regression/cap/normalizer/Borel interfaces are consumed at the immutable identities above, with ERR, BOREL and REC's reading rules. LOCAL remains the exact public draft source identified by the manifest; its prior same-provider noncontributor review is not an external verdict on it or on this successor. SC's displayed analytic and deterministic spectral integral interfaces are consumed without promoting its global count theorem. Source verification, publication and finite checks supply no mathematical acceptance by themselves. + +The final acceptance supplies no replacement-bar intensity, standalone count or moment extension, register/status action, numerical enclosure, unrestricted refined lifetime difference, separate selected-density second-order expansion, or new parameter uniformity. It carries zero organizational-independence credit. The initial and intermediate AMEND records above remain part of the honest review history. diff --git a/frontiers/concave_fibre_elder_20260930/PROOF.md b/frontiers/concave_fibre_elder_20260930/PROOF.md new file mode 100644 index 000000000..923d3236b --- /dev/null +++ b/frontiers/concave_fibre_elder_20260930/PROOF.md @@ -0,0 +1,468 @@ +# Hard-negative fibres identify the actual elder law in every fixed dimension + +Author-side successor by OpenAI, 30 September 2026. Scientific effect NONE. +Disposition: AUTHOR_SIDE / HOLD pending qualified source-bound review. +The root, concave_fibre_geometry and higher_d_source_interface are contributors; +their source audits are not nonauthor acceptance. All source conditions below +are retained. No count moment or remote-count theorem is a premise. + +The complete deterministic lift, including its counterexample, is proved in +[HARD_FIBRE_LEMMA.md](HARD_FIBRE_LEMMA.md). The detailed analytic source +interface and retained matrix integral are in +[SOURCE_INTERFACE.md](SOURCE_INTERFACE.md). [SOURCES.json](SOURCES.json) +binds the complete public sources by immutable commit/path/blob/SHA256. +Together these files expose the full new argument; source hypotheses are not +accepted merely by being linked or having verified hashes. + +## 1. Model and the actual pairing estimand + +Fix an integer d>=3, torus side L>0, birth b in R, positive gap mark k, and an +orthonormal axial/transverse frame R. Use the exact variance-one periodized +Bargmann–Fock Gaussian field of [P], with strictly positive Fourier weights. +At M=(-r/2,0), S=(r/2,0), impose the original observations + + f(M)=b, f(S)=b-k r^3, grad f(M)=grad f(S)=0. + +Q_r is this original 2(d+1)-observation Gaussian regression. Set + + W_r=F_d(H_M) F_(d-1)(H_S), Z_r=E_(Q_r) W_r, + Q_r^W=(W_r/Z_r) Q_r. + +F_j is absolute Hessian determinant restricted to j negative eigenvalues. +There is one original weight and one FULL normalizer. No adjacency restriction, +extra pin density or good-event normalizer is inserted. [P] gives +Z_r/r^2 -> z0>0, with its congruence erratum [ERR] and convergence-in-law reading +rule [REC]. The ordinary global elder death is the superlevel maximin + + d_f(M)=sup_(gamma(0)=M, f(gamma(1))>b) min_t f(gamma(t)). + +An essential maximum has d_f(M)=-infinity. On the almost-sure global Morse +distinct-value locus of [P] §8, d_f(M) is the value of the actual elder merge +saddle when finite. Define + + F_r={actual ordinary elder partner of M is not S}, + p_r=Q_r^W(F_r^c), D_r=(d_f(M)-b)/r^3. + +The event is Borel by [P] §8; whole-field marked Kac–Rice uses the exact [BOREL] +repair, not an assumption that elder selection is a continuous local mark. +These are candidate pair-Palm estimands; counting every rejected candidate +does not count each replacement elder bar once. + +## 2. Original spectral coordinates and the soft cubic + +Put m=d-1. Write the signed ordered eigenvalues of -D_y^2 f(0) as +mu1<...0, -k P in C2 + +on every fixed soft disk. Write M0=(-1/2,0), S0=(1/2,0) for the normalized +soft pins, keeping M,S for the physical original pins. The graph retains +EXACT critical pins and values G_r(M0)=0, G_r(S0)=-k, since the original hard +gradients vanish there and the hard stationary point is unique. The proof of the supplied C2 interface, +including its derivative powers, is displayed in SOURCE_INTERFACE §2. + +The missing geometry is a full-dimensional barrier, not another root count. +For a fixed soft disk containing the planar collar and doubled critical chords, +let eta=min_j h_j>0. Taylor and the fixed C4 bound give + + f_ww <= -(eta/2) I + +on that soft disk at physical r scale and on a hard ball of radius c0 r, for +all sufficiently small r. This is a pointwise fixed-target bound; constants may +depend on eta. Twice integrating along a straight hard fibre gives + + f(t,zeta_r(t)+v) <= f(t,zeta_r(t))-(eta/4)|v|^2. + +Choose fixed rho with D0=eta rho^2/4>k. The centred hard tube +|v|<=rho r^(3/2) fits inside the c0 r ball, since zeta_r=O(r^2) and +r^(3/2)=o(r). On its hard boundary the normalized drop is at least D0. +This is the quantitative hard-depth hypothesis of HARD_FIBRE_LEMMA. The r^2 +scale of the critical graph alone would give a drop of order r^4 and would +not block escape across a death window of order r^3. + +For each continued soft window root Y_r, and the exact normalized S0, put + + h_r=max({-k} union {G_r(Y_r):Y in C(P)}). + +Theorem H (actual highest partner). Under the displayed fixed-target hypotheses, +for every sufficiently small r, + + d_f(M)=b+r^3 h_r. (H1) + +On the actual global Morse distinct-value locus, the unique highest continued +graph saddle is the actual ordinary elder partner. Its full Hessian has one +positive and d-1 negative directions. If n=0 the partner is EXACTLY S; if n>0 +a continued local window saddle preempts S. Limiting ties do not obstruct H1; +the comparison is between the finitely many ACTUAL heights at that r. + +Proof. [LOCAL] §§3–6 construct, in the reduced profile, compact disks K_(r,h) +for every h_rh_* pointwise in this coupling. When D!=0, the highest soft root +Y_*(Theta) is unique and the actual partner divided by r tends, in the original +physical axial frame, to + + T_*(Theta)=R(X_*, O(Z_*,0,...,0)). (H2) + +The hard graph coordinates divided by r vanish. Sign changes of an eigenframe +change Z_*,tau and O together, leaving H2 invariant. No separatrix or +Morse–Smale premise is needed. HARD_FIBRE_LEMMA §8 gives an explicit smooth +counterexample if the contained hard-depth tube is omitted, even when graph +convergence is exact and the physical hard Hessian is uniformly negative. + +## 4. Full failure-measure exhaustion from the endpoint matrix + +H1 is pointwise in fixed jets. It does not alone identify the full rare failure +law. This section supplies the required exhaustion from the actual endpoint +matrix, independently of any count or near-occurrence event. + +On typed support put B_M=-D_y^2 f(M)>0, with ordered eigenvalues + + 0=2, + v=m(m-1)/2. + +[P] (4.2)–(4.3) regress the whole field on the ENTIRE endpoint matrix and give +an independent residual g_r. Put J=1+||g_r||C4, U=J+Lambda. All fixed J moments +are uniformly bounded, J is independent of the whole matrix, and + + K=1+||f||C4 <= C(J+||B_M||F) <= C U. (E1) + +Here Lambda is among the retained eigenvalues lambda2,...,lambdam; its being +independent of lambda1 for this conditional integration is why m>=2 matters. +No independence among matrix entries, derivative suprema or internal residual +evaluations is asserted. + +The exact global good-cap implication [CAP]/[P] §8 gives F_r subset G_r^c, + + G_r={lambda1>[4/(3k)] r M3^2, r M4<=3k/10}. + +[P] (7.7), divided by the FULL Z floor once, bounds the M4 exception by O(r^4), +which is o(r^3). On the remaining depth branch, + + lambda1<=D r U^2, + W_r<=C r^2 U^(2m) lambda1(lambda1+C r U). (E2) + +The second factor is essential: it retains the saddle mixed-square term in +[P] (6.2). The endpoint matrix density has Gaussian bound [P] (3.5); its +spectral Jacobian is at most C Lambda^v. For every retained event E depending +only on J and the hard eigenvalues, the SAME integral gives + + E_Q[W_r;depth,E] + <= C r^2 E_J integral_hard exp(-c sum_hard lambda_j^2) + Lambda^v U^(2m) 1_E + integral_0^(D r U^2) lambda1(lambda1+C rU) d lambda1 + <= C r^5 E_J integral_hard exp(-c sum_hard lambda_j^2) + Lambda^v U^(2m+6) 1_E d lambda_hard. (E3) + +The extension of the lambda1 interval beyond its actual ordered interval is +an upper bound of a nonnegative majorant. U and E are retained while integrating +lambda1; its Gaussian factor may be dropped. Equation E3 is the retained-mark +version of [P] (7.4), not a new independence or cutoff inversion. + +Taking E={U>A}, using arbitrarily high J moments and Gaussian polynomial +integrability, and dividing once by Z_r>=z_*r^2, gives, for every fixed p>0, + + limsup_r r^-3 Q_r^W(F_r,U>A) <= C_p A^-p. (E4) + +Taking E={lambda2<=epsilon} in E3 gives + + lim_epsilon->0 limsup_r + r^-3 Q_r^W(F_r,lambda2<=epsilon)=0. (E5) + +Indeed integration of the hard Gaussian polynomial over this strip tends to +zero; for epsilon<=1 a constant times epsilon suffices after integrating all +other hard variables and the J moments. All higher-corank intersections are +included, with no inverse-hard-eigenvalue or inverse-gap factor. + +Weyl's eigenvalue inequality and Hessian Lipschitz control give + + |muj-lambdaj| <= r M3/2 <= C rU. + +On the depth branch this implies + + |s|<=D U^2+C U, |h|+|tau|+K<=C U. (E6) + +Consequently E4 is tightness of the FULL failure measure in the actual midpoint +jets and derivative norm, not just a compact-near count estimate. On U<=A, +midpoint h2<=epsilon implies endpoint lambda2<=2epsilon for sufficiently small +r at fixed epsilon,A. Equations E4–E5 and the M4 exception yield + + lim_epsilon->0 limsup_r + r^-3 Q_r^W(F_r,h2<=epsilon)=0. (E7) + +Midpoint hard eigenvalues of either sign near zero can carry no lost leading +failure mass. Inverse hard eigenvalues are used only at fixed positive targets +for graph convergence; they never enter E3. The complete detailed derivation +is SOURCE_INTERFACE §§4–5. No scalar shortcut is used: for d=2 one must retain +[LOCAL] §8's separate scalar near/far split, since Lambda=lambda1 there. + +## 5. The identified full elder-failure measure + +Let the limiting hard domain be 0=2) h_j^3 + product_(2<=i0} + ds dh dO d tau. (S1) + +It is extended by zero off the stated hard/typed domain. The deterministic +classifier bound [SC] (19) proves integrability: if x=-s>0 and n>0, then +x<=2|B|+(48kD^2)^(1/3), and + + integral_s w_+ 1{n>0} <=384k^2 |B|^3+2304k^3 D^2. + +The remaining hard spectral factor is a polynomial, while h0 has uniform +Gaussian decay in the ORIGINAL orthogonally rotated raw variables. All angular +variables remain inside the integral. Positive Gaussian density on open typed +nonempty sectors gives mu_fail(total)>0. In [SC] (20)'s notation its mass is + + a_fail=a1+a2. (S2) + +This consumes the displayed deterministic integral, not [SC]'s global count +limit. Its remote singleton coefficient beta_far does not enter S1 or S2. + +Theorem F (full fixed-dimensional selector-failure law). Under the exact +source interfaces and H1, in the extended spectral law of §2, + + r^-3 Q_r^W(Theta_r in dot, F_r) -> mu_fail in total variation, + (1-p_r)/r^3 -> a1+a2>0. (S3) + +Proof. On a fixed bounded spectral box with hard eigenvalues bounded below, +[SC] §2 disintegrates the actual additional full jet. Its soft eigenvalue +change has Jacobian r c_m J_r, where + + J_r=product_(j>=2)(h_j+rs) product_(2<=i product h_j^2 w_+. Its limiting +weight is zero off the typed domain and ON the typed boundary; selector +convergence is unnecessary there. For a typed generic target H1 gives the +actual indicator I=1_F eventually equal to 1{n>0}. The excluded algebraic +sets are null, not discarded positive mass. + +The independent complete-jet coupling [SC] (5)–(6) gives +K<=C_box(1+||F||C4). Its two-short-column bound [SC] (15) bounds W_r/r^4 by an +integrable fixed polynomial in that coupling norm on the box. Thus pointwise +convergence, the exact weight limit, and dominated convergence give compact +L1 convergence of S4 to S1, including near the typed boundary. Repeated hard +spectra are null; no uniform inverse-gap estimate is required on the box. +E4–E7 exhaust the ACTUAL failure measure, and S1 is integrable. First exhaust +bounded boxes and then let the positive hard lower bound decrease to zero. +The limsup of both excluded tails tends to zero. Compact L1 plus those two +tails is global L1, which is precisely the total variation statement S3. +Taking total masses proves its coefficient. QED. + +This establishes the pairing relevance of the microscopic soft geometry. +It does not equate two extra saddle witnesses with one maximum/saddle pair. +Neither a factorial moment nor fixed-remote mixing is used to infer selection. +When the fixed-r almost-sure global Morse/distinct-value locus is used in this +conditional coupling, take a countable intersection along an arbitrary fixed +sequence r_j->0. Dominated convergence along every such sequence proves the +stated limit. No common probability-one locus for uncountably many pinned laws +is assumed. + +## 6. Actual replacement location, death and lifetime + +On F_r let T_r be the actual elder partner's displacement from the midpoint, +in the local torus chart, divided by r. Use an isolated sentinel if the partner +is essential or outside that chart. For the same reason use a sentinel in D_r +when d_f(M) is not finite. These exceptions have r^-3 mass tending to zero: +on each regular compact jet sector H1 provides a local older path and actual +partner for all sufficiently small r; E4–E7 exhaust the remaining failure mass. + +Theorem M (actual marked law). The finite measures + + r^-3 Q_r^W((Theta_r,T_r,D_r) in dot, F_r) + +converge WEAKLY to the pushforward of S1 under + + Theta -> (Theta,T_*(Theta),h_*(Theta)). (M1) + +The conditional-on-failure law is M1 divided by a_fail. The lifetime mark +(b-d_f(M))/r^3 converges in that law to -h_* in (0,k), and its fraction of the +candidate lifetime converges to -h_*/k in (0,1). + +Proof. At a generic positive hard target H1 identifies the exact actual +winning continued root. Off the additional null set D=0, [LOCAL] §6's winning +soft root is unique; continuation and the vanishing r-scaled hard graph give +H2 and D_r->h_*. Insert any bounded continuous joint test function in S4. +The same compact dominated-convergence argument proves convergence of its +integral. E4–E7 remove the tails; sentinel mass has already been excluded. +No continuity across the polynomial transition sets is assumed. QED. + +Weak, not total variation, convergence is claimed for these real-valued marks. +Tightness of M1 also gives, for EVERY deterministic delta_r->0 with +delta_r/r->infinity, + + Q_r^W(F_r, dist(actual elder partner,midpoint)>delta_r)=o(r^3). (M2) + +For a given tolerance choose a fixed large radius with arbitrarily small +limiting marked tail, use weak tightness including the sentinel, then use +delta_r/r eventually larger than that radius. This is actual failed-partner +localization, rather than localization of a count-defined witness. + +## 7. Exact compact lifetime-density consequence + +Retain [P] §§10–12's exact compact population: birth b in an interval B, gap k +in an interval K, both of positive length and K bounded away from zero, axial +orientation u in S^(d-1) with ordinary area measure, per unit midpoint volume. +Choose a Borel transverse completion R(u) as in the source. The coefficient +a_fail is independent of that completion: the original selection probability +is physical and independent of a transverse coordinate choice, and S3 is its +unique limit. This does not assert torus rotational invariance or birth-factor +cancellation in higher dimensions. + +Use the source's ACTUAL radial ledger, in every fixed d: + + A_r=12 pi_r(R(u);v_r) Z_r/r^2, + nu_cand(ell)=ell^-1/3 integral_(B x K x S^(d-1)) + A_((ell/k)^(1/3))/(3k^(2/3)). + +The selected density has the additional p_r. The dimension powers have already +cancelled in [P] §10; no new spatial Jacobian or dimension-dependent lifetime +power may be inserted. [P] (7.8) supplies a common compact-mark bound on +(1-p_r)/r^3, and [P] (10.2)–(10.3) supply common bounds and A_r->A0. These are +existing domination interfaces, not uniformity inferred from pointwise H1. + +With r=(ell/k)^(1/3), the exact nonselected candidate density therefore has + + ell^-2/3 [nu_cand^(B,K)(ell)-nu_eld^(B,K)(ell)] + = integral_(B x K x S^(d-1)) + A_r/(3k^(5/3)) [(1-p_r)/r^3]. + +Dominated convergence and S3 identify + + nu_rej^(B,K)(ell) ~ C_fail^(B,K) ell^(2/3), + C_fail^(B,K)=integral_(B x K x S^(d-1)) + A0(b,k,u) a_fail(b,k,u)/(3k^(5/3)) db dk d sigma(u). (L1) + +C_fail is finite and strictly positive. The full z0 cancels if A0=12pi0z0 is +substituted into S1; there is still one original pin density and no additional +normalization. + +L1 is the compact candidate-minus-selected counting measure. It is not a +replacement-bar intensity obtained by counting all rejected candidates, nor a +standalone second-order selected-density expansion. An unrestricted refined +difference does not follow; its far rejected population is different. No new +k->0, d->infinity, volume, or escaping-mark uniformity is claimed. + +## 8. Hypotheses, explicit falsifiers and review boundary + +H1 assumes a generic typed cubic and its pinned reduced C2 convergence, an +embedded fixed soft collar, and a CONTAINED negative hard tube with drop D0>k. +The graph and tube are proved from [SC]'s actual smooth field on fixed positive +hard targets. The exact finite-r partner additionally uses the source's actual +global Morse distinct-value locus. A unique limiting position additionally +excludes D=0, a null set. The deterministic lemma does not require probability, +Morse–Smale, a separatrix assertion or control of the exterior. + +The stochastic statements require the original positive-spectrum field, +nonsingular contact/full-jet regression, independent endpoint-matrix residual +with all C4 moments, actual cap implication, double soft-factor weight bound, +full normalizer floor, and the original spectral density/coupling interfaces. +They are source-bound conditions, not consequences of tests or publication. +All inverse-hard constants are pointwise only. The actual rare failure tail +and multiple-soft exclusion are proved by E3–E7, rather than borrowed from a +near-count event. The full typed boundary is treated by its continuous zero +weight. The Haar-gauge extension is part of the precise measure statement. + +Falsifiers: + +- A full path bypassing a contained hard-depth cap at height above h_r, or a + graph-lifted moving chord whose exact minimum is not the continued root, + invalidates H1. The thin-tube counterexample in HARD_FIBRE_LEMMA §8 explains + why graph convergence alone is not a substitute for that hypothesis. +- A leading failure mass escaping every bounded midpoint jet set, or staying + at a nonvanishing fraction on h2<=epsilon as epsilon->0, contradicts E4/E7. + A defect in the parent cap implication, original residual independence or + retained matrix soft integral would reopen that analytic step. +- A dropped saddle mixed-square factor, extra restricted normalizer, omitted + hard/sign multiplicity or GOE substitution changes S1 and invalidates S3. +- A positive leading essential/sentinel mass contradicts M1. Total variation + for real-valued location/lifetime marks is outside the claim. +- Replacement-bar, unrestricted refined-density, new parameter-uniform or + numerical uses of L1 are outside this proof's interface. + +This is a new fixed-dimensional actual elder/failure/lifetime candidate, with +full proofs and source identities exposed. d=2 remains the separate [LOCAL] +scalar proof. No controlling acceptance, status, graph, prize, Boolean, parent +proof or source history is altered. Qualified nonauthor review must bind this +successor itself; acceptance of a count packet or of [LOCAL] alone is not an +acceptance of this higher-dimensional lift. diff --git a/frontiers/concave_fibre_elder_20260930/README.md b/frontiers/concave_fibre_elder_20260930/README.md new file mode 100644 index 000000000..cd78f41c8 --- /dev/null +++ b/frontiers/concave_fibre_elder_20260930/README.md @@ -0,0 +1,43 @@ +# From the soft plane to the actual elder partner in fixed d>=3 + +[PROOF.md](PROOF.md) extends the planar actual-pair theorem through negative +hard fibres and identifies the full higher-dimensional selector-failure law. +This is an additive mathematical candidate, AUTHOR_SIDE / HOLD, conditional +on the exact public source interfaces and pending qualified nonauthor review. + +The geometric step is a contained hard tube of width r^(3/2): its height drop +matches the r^3 death window and closes every transverse escape. The actual +critical graph itself has hard displacement O(r^2), which is too narrow to +provide that barrier. The full proof and an explicit thin-tube falsifier are +in [HARD_FIBRE_LEMMA.md](HARD_FIBRE_LEMMA.md). Graph-lifted critical chords and +thickened compact planar caps identify the actual global merge saddle as the +highest continued local window saddle for all sufficiently small r. + +Retaining the residual/hard-jet tail mark in the parent's endpoint-matrix +integral exhausts the ACTUAL failure measure. No count event or factorial bound +is substituted. It gives the full extended spectral-jet failure measure in +total variation, coefficient (1-p_r)/r^3 -> a1+a2, and the weak actual +replacement-location/death/lifetime law. The remote singleton count coefficient +does not enter. Actual failed partners localize inside every shrinking cutoff +whose ratio to r tends to infinity. + +The exact compact birth/gap candidate-minus-selected density consequently has +an identified positive ell^(2/3) coefficient in every fixed dimension. This is +the parent's exact universal radial ledger, not a new dimension-dependent +power, a replacement-bar counting formula, or a standalone second-order +selected-density expansion. + +[SOURCE_INTERFACE.md](SOURCE_INTERFACE.md) displays the analytic graph and full +matrix-tail derivations. [SOURCES.json](SOURCES.json) links all 11 full public +sources with immutable identities and separates required inputs from context. +[RECONNAISSANCE.md](RECONNAISSANCE.md) records the limited primary-source +comparison without a novelty claim. [NONAUTHOR_REVIEW.md](NONAUTHOR_REVIEW.md) +preserves the initial and intermediate AMEND findings and the final conditional +source-scoped ACCEPT of the corrected exact prose. It supplies zero +organizational-independence credit; qualified external review remains pending. + +All current contributors are OpenAI, with zero organizational-independence +credit. No existing proof, count packet, register, graph, source status, prize, +Boolean, workflow or active peer branch is changed. Uniformity in varying d, +volume or k->0, unrestricted refined lifetime differences and numerical +coefficients remain outside this note. diff --git a/frontiers/concave_fibre_elder_20260930/RECONNAISSANCE.md b/frontiers/concave_fibre_elder_20260930/RECONNAISSANCE.md new file mode 100644 index 000000000..b6511fb8a --- /dev/null +++ b/frontiers/concave_fibre_elder_20260930/RECONNAISSANCE.md @@ -0,0 +1,32 @@ +# Primary-source reconnaissance for the hard-fibre elder step + +30 September 2026, OpenAI author-side reconnaissance. No novelty or literature +priority claim. This is evidence of the actual access depth, not acceptance. + +The classical comparison is reduction along nondegenerate Hessian directions. +The primary abstract of Guangcun Lu, *The splitting lemmas for nonsmooth +functionals on Hilbert spaces I*, arXiv:1211.2127v2, +https://arxiv.org/abs/1211.2127v2, was read. It discusses a splitting theorem +and critical-group results near a critical point under weaker smoothness. Only +the author abstract and bibliographic metadata were inspected; its 68-page +proof was not read and no theorem from it is used in this packet. + +Searches also exposed Golubitsky–Marsden's *The Morse Lemma in Infinite +Dimensions via Singularity Theory*. The Ohio State reprint URL and Caltech +author-record URL returned restricted-fetch errors in the available public +retrieval service. No full-source exposure or proof consumption is claimed for +those attempts, and no alternate credential or access path was used. + +The inference guiding this packet is narrower than a splitting theorem: +critical-point reduction alone does not determine which labelled maximum +merges in the arbitrary exterior. The new argument proves its own Schur +completion, fibre concavity drop, compact full-dimensional cap and graph path. +HARD_FIBRE_LEMMA §8 explicitly falsifies elder transfer from reduced graph +convergence alone. No black-box splitting lemma supplies the hard-side depth. + +All mathematical dependencies actually consumed are the full public repository +sources in SOURCES.json, read at their exact immutable identities. SC supplies +the local analytic graph but explicitly excludes elder selection; the new +barrier is therefore a separately reviewed consumer step. The retained matrix +tail and actual-pair lifetime pushforward are proved in PROOF.md and +SOURCE_INTERFACE.md rather than inferred from a count theorem or a paper title. diff --git a/frontiers/concave_fibre_elder_20260930/SOURCES.json b/frontiers/concave_fibre_elder_20260930/SOURCES.json new file mode 100644 index 000000000..f072e1bb5 --- /dev/null +++ b/frontiers/concave_fibre_elder_20260930/SOURCES.json @@ -0,0 +1,121 @@ +{ + "repository": "d6g8k5htny-coder/Math-", + "public_main": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "planar_pr": 170, + "planar_head": "79f18f0cb318fceccd5bdf4e6dad29815165bc7d", + "scientific_effect": "NONE", + "review_independence_credit": 0, + "sources": [ + { + "id": "SC", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", + "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/spectral_cluster_closure_20260929/PROOF.md", + "bytes": 25006, + "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", + "role": "required: \u00a7\u00a72,5,6 raw/full jet disintegration, orthogonal spectral Jacobian, actual hard graph C2 convergence, continuous typed weight, deterministic integral; global count law NOT consumed" + }, + { + "id": "CUB", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md", + "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/planar_cubic_cluster_20260929/PROOF.md", + "bytes": 19889, + "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0", + "role": "required: displayed original planar cubic, exact classifier and polynomial null sets; count expectations NOT consumed" + }, + { + "id": "EDL", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "frontiers/elder_dimension_lift_20260928/PROOF.md", + "git_blob": "7303bd791a68a1139251f0f6e403a9f7cc89b006", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/elder_dimension_lift_20260928/PROOF.md", + "bytes": 23971, + "sha256": "b529fe3780e1909014b3cb144b9a74156d976bbdc19b6711d0c5b1a1da9e5669", + "role": "context only: preceding actual root/index interface; no count or pairing conclusion consumed" + }, + { + "id": "DL", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "frontiers/d5_dimension_lift_20260929/PROOF.md", + "git_blob": "9d82c707fdb17d3072a8930f26dabedf59e456fc", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/d5_dimension_lift_20260929/PROOF.md", + "bytes": 53727, + "sha256": "6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80", + "role": "scope audit only: dimension-lift count result excludes elder selection; no count conclusion consumed" + }, + { + "id": "LOCAL", + "commit": "79f18f0cb318fceccd5bdf4e6dad29815165bc7d", + "path": "frontiers/local_elder_geometry_20260930/PROOF.md", + "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/79f18f0cb318fceccd5bdf4e6dad29815165bc7d/frontiers/local_elder_geometry_20260930/PROOF.md", + "bytes": 39722, + "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b", + "role": "required: Theorem E compact planar collar and moving critical chords; position uniqueness off D=0; draft at exact stated source scope" + }, + { + "id": "P", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "role": "required: endpoint matrix density/residual, double soft-factor bound, full Z, global cap-failure inclusion, genericity/Borel and compact lifetime ledger" + }, + { + "id": "ELDERD", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "frontiers/elder_lower_all_d_20260929/PROOF.md", + "git_blob": "7f41c9e315be0a7985d4c770a11a2cdf717bd46a", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/elder_lower_all_d_20260929/PROOF.md", + "bytes": 16485, + "sha256": "88c6bfc7973fe832918f91e2ed1f47da3e51987ed9efc1a99648e82b2eb119e5", + "role": "context only: preceding elder-failure lower event; not used as a coefficient or upper barrier" + }, + { + "id": "ERR", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md", + "git_blob": "213594d6ca6a86fb938110f4d166d9ce275a02d0", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md", + "bytes": 1782, + "sha256": "bad7ef609c4ad8c41ad6af562c1b6807921e19a9d556ed793ad1a0db6e202028", + "role": "required reading rule: endpoint congruence correction" + }, + { + "id": "BOREL", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "reviews/d1_section9_borel_repair_20260925/REPAIR.md", + "git_blob": "fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/reviews/d1_section9_borel_repair_20260925/REPAIR.md", + "bytes": 9062, + "sha256": "845abf9f9c99d672c2a10a887b5a2e7206a3d2de3d876f35f75ff6e2dc13e62f", + "role": "required reading rule: whole-field Borel weighted Kac-Rice repair" + }, + { + "id": "REC", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md", + "git_blob": "75da2597971510f843f8d90c743950cb8c177342", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md", + "bytes": 23312, + "sha256": "451b9d7ffee072a73fc904cab891b6693e3f233808b89df10cce1b57036b65da", + "role": "required reading rules and reviewed source scope: W1 convergence-in-law and embedded-chart radius" + }, + { + "id": "CAP", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "git_blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "bytes": 15160, + "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc", + "role": "required: deterministic global good-cap selection implication" + } + ], + "source_exposure": "Full public immutable sources. Required interfaces remain conditional on their displayed assumptions and actual review scope. Context sources are not mathematical premises." +} diff --git a/frontiers/concave_fibre_elder_20260930/SOURCE_INTERFACE.md b/frontiers/concave_fibre_elder_20260930/SOURCE_INTERFACE.md new file mode 100644 index 000000000..7eb203bdf --- /dev/null +++ b/frontiers/concave_fibre_elder_20260930/SOURCE_INTERFACE.md @@ -0,0 +1,191 @@ +# Source-interface audit for a higher-dimensional elder bridge + +Read-only source audit, 30 September 2026. Same-provider OpenAI contributor; zero organizational-independence credit. No mathematical acceptance, scientific-status transition, existing-proof modification, PR action, or count-law promotion is asserted. The analytic lemmas reconstructed below are new author-side consumer steps requiring source-bound review. + +## 1. Exact public source cut + +Public `Math-/main` readback was `ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f`. Public PR170 is an open draft at head `79f18f0cb318fceccd5bdf4e6dad29815165bc7d`, branch `codex/local-planar-elder-f68038be`. All listed files were fetched in full from the exact public commit. The scratch copies in `sources/` have been checked against the public Git blob identities, including UTF-8 byte lengths. Existing mutable source maps were not changed. + +| Tag | Exact path | Git blob | Bytes | Interface consumed | +|---|---|---|---:|---| +| SC | `frontiers/spectral_cluster_closure_20260929/PROOF.md` | `16c56821b52fd76b0be791622b9c3809eafde75a` | 25006 | §§2,5,6: raw midpoint regression, orthogonal spectral chart, physical anisotropic normal form, hard slaving, full weight, local spectral kernel and its integrability. Its count exhaustions are not premises. | +| CUB | `frontiers/planar_cubic_cluster_20260929/PROOF.md` | `bb446d08db8a944537a743ad550b88c1c2ad5758` | 19889 | C1–C15, Theorem C and Lemma S: original-coordinate cubic, classifier, stable roots, shear only for algebra. | +| EDL | `frontiers/elder_dimension_lift_20260928/PROOF.md` | `7303bd791a68a1139251f0f6e403a9f7cc89b006` | 23971 | Explanatory context only: A7–A28 compare physical jets, bounded shear charts and actual root/index constructions. SC supplies the orthogonal graph interface used here; no EDL count or pairing conclusion is a premise. | +| ELDERD | `frontiers/elder_lower_all_d_20260929/PROOF.md` | `7f41c9e315be0a7985d4c770a11a2cdf717bd46a` | 16485 | Context only: soft eigenplane path gives an elder-failure lower event, without an upper barrier or actual replacement partner. | +| DL | `frontiers/d5_dimension_lift_20260929/PROOF.md` | `9d82c707fdb17d3072a8930f26dabedf59e456fc` | 53727 | Scope audit only: §1.3 expressly excludes elder selection. No count/first-moment conclusion is consumed. | +| P | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` | `dfed3b8d318a3ab1950957f393307733a4bef3f2` | 40261 | §§3–8: endpoint matrix density, independent full-field residual, original weight and full normalizer, cap-failure inclusion, genericity and maximin; §§10–12: exact compact radial/lifetime ledger and its domination interfaces. | +| CAP | `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md` | `0633aca3c2a2882b0de4399da0a75d64c2e6b2e1` | 15160 | §§1–5: deterministic all-dimensional good-cap implication. | +| ERR | `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md` | `213594d6ca6a86fb938110f4d166d9ce275a02d0` | 1782 | Corrected endpoint congruence in P §5. | +| BOREL | `reviews/d1_section9_borel_repair_20260925/REPAIR.md` | `fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a` | 9062 | Required whole-field Borel marked Kac–Rice replacement for P §9. | +| REC | `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md` | `75da2597971510f843f8d90c743950cb8c177342` | 23312 | §1 reading rule; §5 W1 convergence-in-law clarification and embedding radius. | +| LOCAL | `frontiers/local_elder_geometry_20260930/PROOF.md` at PR170 head | `ef2aa57959ea9f721bbf2316ce94cf616c1c9113` | 39722 | Theorem E, §§3–6: planar compact collar and exact highest continued partner; §8 scalar failure exhaustion. SHA256 `f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b`. | + +For the first ten rows the immutable URL is `https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/` followed by the path. For LOCAL replace the commit with `79f18f0cb318fceccd5bdf4e6dad29815165bc7d`. `SOURCES.json` gives complete URLs and SHA256 values. + +## 2. The supplied analytic graph interface + +Fix d≥3, m=d−1, one original axial frame, b∈R, k>0, and a midpoint spectral target. In SC's orthogonal coordinates + + A_mid = O diag(rs,−h2,…,−hm) Oᵀ, + 00 and fixed coupling C4 bound K, Taylor gives + + f_ww ≤ −h_min I/2, + |f_w(t,0)| ≤ C r², + |f_tw(t,w)| ≤ C r. + +The graph follows from strict concavity and the inward hard radial derivative. Implicit differentiation gives `|zeta_r|≤Cr²`, `|D_t zeta_r|≤Cr`, and `|D_t² zeta_r|≤C`. For `psi_r(t)=f(t,zeta_r(t))`, define `delta_r=psi_r(t)−f(t,0)`. Then + + |delta_r|≤Cr⁴, + |D_t delta_r|≤Cr³, + |D_t² delta_r|≤Cr². + +The Hessian identity used in the last bound is + + D_t² psi_r = f_tt−f_tw f_ww^(−1) f_wt + +evaluated on the graph. Each rescaled soft derivative contributes one factor r. Hence `(psi_r(rX,rZ)−b)/r³−(f(rX,rZ,0)−b)/r³=O(r)` in C². Inverse-hard constants are allowed at a fixed positive hard target; they must not be placed in the failure-tail majorant below. + +Since the original hard gradients vanish at each pin and the hard graph is unique, `zeta_r(M)=zeta_r(S)=0`. The reduced potential retains the exact critical pins and values 0,−k. Nondegenerate reduced critical points are actual full roots on the graph. Block congruence gives + + index(H_full)=(d−2)+index(H_reduced). + +Thus CUB's window saddles lift to actual index-(d−1) roots, and their hard coordinates are r² times `diag(h)^(−1)Q+o(1)`. Their physical coordinates divided by r tend to `(X,O(Z,0))` in the original axial/transverse frame. + +EDL's distinct construction uses the bounded determinant-one shear `y=(z,w−qz)` with q=D^(−1)v. Its original physical r-scaled location is `(X,Z,−qZ)` after the axial frame, not `(X,Z,0)`. In that chart sigma, not the raw a entry of A, is the soft curvature. Both SC's orthogonal chart and EDL's determinant-one shear preserve criticality, inertia and Hessian determinants; a general invertible linear change would introduce its squared determinant factor. SC's orthogonal chart is the appropriate global norm-controlled chart here. CUB's separate planar shear `u=X+aZ/(12k)` is only an algebraic change; reported physical X must be recovered as `u−aZ/(12k)`. + +## 3. Exact missing topological consumer step + +LOCAL Theorem E is stated on surfaces. It cannot simply be cited as a theorem about the full d-dimensional field. The new consumer needs to thicken its compact planar collar and control every hard boundary exit. + +The graph profile alone on an r²V box is insufficient. Hard side boundaries at that scale have a height drop of order r⁴, smaller than the r³ death window. A correct thickening uses a larger hard radius, for example `rho_r=A r^(3/2)` centered on zeta_r. Strict hard concavity yields + + f(t,w) ≤ psi_r(t)−(h_min/4)|w−zeta_r(t)|². + +Choose A with `h_min A²/4>k`. For every planar collar disk K_(r,h) supplied by the LOCAL §§4–6 construction for the actual reduced profile, psi_r≤b inside and psi_r=b+r³h on its boundary. The graph-centered hard tube over K_(r,h), of radius rho_r, is contained in the local chart for small r. Its soft boundary has height ≤b+r³h and its hard side boundary has height 0`, a witness event, or any remote-count estimate. + +Let `F_r={actual global ordinary partner of M is not S}` and let `B=−D_y²f(M)>0` on typed support. Order its eigenvalues + + 0[4/(3k)]r M3², r M4≤3k/10}. + +The fourth-derivative exception has tilted mass O(r⁴) by P (7.7) and the full floor (5.5), hence has zero r^(−3) limit. On the remaining depth branch, + + lambda1≤D rU². + +P (6.2), with its second soft factor retained, gives + + W_r ≤ C r² U^(2m) lambda1(lambda1+C rU). + +The endpoint matrix density has the Gaussian bound P (3.5). Its spectral Jacobian is bounded by `C Lambda^v`; the hard angular volume is finite. For fixed J and lambda2,…,lambdam, U is independent of lambda1. Thus for any retained event E depending only on J and the hard eigenvalues, + + E_Q[W_r;depth,E] + ≤ C r² E_J ∫ e^(−c sum_hard lambda_j²) Lambda^v U^(2m) + 1_E ∫_0^(DrU²) lambda1(lambda1+C rU) d lambda1 d lambda_hard + ≤ C r⁵ E_J ∫ e^(−c sum_hard lambda_j²) + Lambda^v U^(2m+6) 1_E d lambda_hard. (T1) + +The positive majorant permits extension of the lambda1 domain beyond its actual ordered interval; the Gaussian lambda1 factor can be dropped. The integral is exactly the r³ soft integral before the outer r² weight. Divide by the original full `Z_r≥z_*r²` once. + +With `E={U>A}`, every higher J moment and Gaussian polynomial integrability give, for every fixed p>0, + + limsup_(r→0) r^(−3) Q_r^W(F_r,U>A) ≤ C_p A^(−p). (T2) + +This follows directly from `U^q 1{U>A}≤A^(−p)U^(q+p)`, q=2m+6. It requires no common residual law for every r; the uniform moment bounds suffice. + +With `E={lambda2≤eta}`, averaging J's moments leaves an ordinary integrable Gaussian polynomial over the ordered positive hard eigenvalues. Its mass on the lambda2 strip tends to zero (a bound C eta for eta≤1 suffices). Consequently, + + lim_(eta→0) limsup_(r→0) + r^(−3) Q_r^W(F_r,lambda2≤eta)=0. (T3) + +This handles all higher-corank intersections in measure and never divides by a hard eigenvalue. It is a failure-measure statement, stronger for this purpose than SC §4's near-occurrence tail. + +## 5. Endpoint-to-midpoint conversion + +Let `mu1≤…≤mum` be the signed eigenvalues of `−D_y²f(0)`. Hessian Lipschitz control and Weyl's inequality give + + |muj−lambdaj|≤r M3/2≤C rU. + +With SC's midpoint variables `s=−mu1/r`, `h_j=mu_j` for j≥2 and rotated complete tensor tau, the depth branch satisfies + + |s|≤D U²+C U, + |h|+|tau|+K≤C U. (T4) + +T2 therefore implies full failure tightness in the actual midpoint spectral jets and in K. No replacement of the endpoint density by a midpoint density occurred in this tail argument. + +On U≤A, midpoint `h2≤eta` implies endpoint `lambda2≤2eta` for all sufficiently small r at each fixed eta,A. T3 then gives + + lim_(eta→0) limsup_(r→0) + r^(−3) Q_r^W(F_r,h2≤eta)=0. (T5) + +The error from U>A is sent to zero by T2 and the M4 exception is o(r³). In particular midpoint hard eigenvalues that are negative can carry no unidentified leading failure mass. Repeated hard eigenvalues need no inverse-gap estimates: the spectral chart boundary is null; on each compact sector its Jacobian is bounded and vanishes at a collision. Positive hard lower bounds are used only for pointwise graph convergence and removed using T5. + +For d=2/m=1, retain LOCAL §8's existing scalar near/far split and tail proof; T1 cannot be used with Lambda fixed because Lambda=lambda1. + +## 6. Exact resulting marked-law interface, conditional on the new geometry + +SC §2 disintegrates the raw physical midpoint (A,T) law, with density h_r→h_0. Its ordered orthogonal spectral transformation contributes `r c_m J_r`, where + + J_r=prod_(j≥2)(h_j+rs) prod_(2≤i0 the original typed weight has + + W_r/r⁴ → prod_(j≥2)h_j² w_+(s,a,beta), + w_+=9k²[B−2s]_+[−B−2s]_+, B=beta−a²/(12k). + +The continuous determinant-weight limit is zero outside the typed domain and on its boundary; no selector convergence is needed there. The full normalizer is P (5.4), with ERR and REC's W1 reading rule. For any bounded failure multiplier the exact scaled disintegration is + + (c_m r²/Z_r) J_r h_r(O diag(rs,−h)Oᵀ,Rot_O tau) + E[(W_r/r⁴) multiplier]. (T6) + +There is one raw soft-coordinate Jacobian r, one original W_r, and one full Z_r. On compact jets SC (6) and the two short-column determinant bound SC (15) provide an integrable coupling envelope without using a count event. Conditional on the deterministic fibre bridge, pointwise failure convergence is `1_F→1{n>0}` off LOCAL's polynomial null sets. The actual death and location converge to h_* and the winning physical root. Dominated convergence proves the compact marked failure law. T2–T5 exhaust the actual failure measure, so no near-occurrence exhaustion is substituted. + +The candidate finite limiting failure measure is + + dmu_fail = 1{n(s,a,beta,c)>0} (c_m/z0) + h_0(O diag(0,−h)Oᵀ,Rot_O tau) + prod_(j≥2)h_j³ prod_(2≤i=2, L>0, b real, k>0 and an axial/transverse orthonormal frame. Use the +exact variance-one periodized field, original endpoint observations and the +one full determinant-tilted pair-Palm law Q_r^W of the sources. Write F_r for +failure of the prescribed saddle to be the ordinary elder partner of the +prescribed maximum. Let T_r be the ACTUAL replacement-partner displacement +from the midpoint divided by r and let + + lambda_r=(b-d_f(M))/(k r^3). (1) + +Out-of-chart or essential partners may be given an isolated sentinel. The +source interface says their probability on F_r is o(r^3). We do not identify +an arbitrary witness or farthest root with an elder partner. + +The two exact input types are deliberately separate: + +- **ELDER2:** PR170 at79f18f0cb318fceccd5bdf4e6dad29815165bc7d, + PROOF.md, Theorem E and Sections8.6-8.8. In d=2 it supplies the actual highest + local saddle, the full failure measure and its weak location/lifetime marks. +- **ELDERD:** PR175 at9c6a73489c5ed2465a3cf65dcc950eeeae074b05, + PROOF.md, H1/S1-S3/M1. In d>=3 it supplies the hard-negative-fibre bridge, + the extended spectral failure measure, and the actual weak marked law. +- **R:** PR166 RADIAL_TAIL.md at80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337, + R4-R13, supplies the exact root-resolved density of the limiting microscopic + measure and its Gaussian majorant. This is used to integrate the already + identified selector, NOT to establish actual elder pairing from counts. +- **C:** PR169 PROOF.md at56a7904f9ad7d409c31e23195c0e7eeab230613a, + E10-E25, supplies the deterministic companion/height map, inner/outer sectors + and exact shape integrals. Its whole-cluster maximum theorem is not used as + an elder theorem. +- **D:** COMPANION_RATIO.md at that same PR169 head, D1-D7, supplies the exact + unit-square density/height formulas. Only Section6 below needs this input. + +All complete commit/path/blob/SHA256 bindings are in SOURCES.json. ELDER2 and +ELDERD are distinct author-side source candidates with source-bound reviews; +this consumer is conditional on their displayed mathematical interfaces. It +neither promotes their status nor discharges a missing premise by citation. +The prior source proofs and review bodies remain unchanged. + +The common limiting microscopic measure is denoted M. Its restriction to +n>0, where n is the number of extra strict-window cubic saddles, is the finite +failure measure mu_F with mass a_F=a1+a2>0. It is NOT the global nonempty count +measure: remote singleton mass is absent. With theta=(s,h,O,tau), + + dM=(c_m/z0) H(h) h0(A0,Rot_O tau) + 9k^2(4s^2-B^2) ds dh dO dtau, (2) + +on the positive hard/typed domain. Here H is the ordered hard spectral +polynomial, A0=O diag(0,-h)O^T, B=beta-a^2/(12k), and s<-|B|/2. +The scalar case has empty hard products; a normalized soft sign-frame +augmentation changes no physical integral. There is one full z0 and no +additional pin density or root multiplicity. + +The actual interface is finite-measure weak convergence, with converging masses, + + r^-3 Q_r^W((T_r,lambda_r) in dot,F_r) + -> pushforward of mu_F by (T_*,eta_*), (3) + +where T_* is the root with GREATEST cubic height P, and eta_*=-P(T_*)/k. +Thus eta_* is the SMALLEST downward height among the window roots. +The assertion that this is the actual elder partner is precisely the elder +source premise, not a conclusion of the polynomial computations below. + +**Order:** first r->0. Then the radius threshold t->infinity inside (3)'s limit. +Any further small-lifetime or large-companion limit is taken third. No uniform +simultaneous limit, finite-r inverse moment, or full lifetime-intensity claim +is made. + +## 2. A once-per-elder selector with a fixed shape domain + +Use the root variables of R/C: + + u=X+aZ/(12k), b0=3-12u^2, v=(-s)Z^2/k, + S={-3/20 on S. (4) + +A counted companion is given, off an algebraic null set, by + + Lc=4v^2-8b0uv+b0^2, + t_c=(4v^2-b0^2)/Lc, + u_c=[2b0v-u(4v^2+b0^2)]/Lc, v_c=v t_c^2. (5) + +C proves that the map is an involution, t_c<0 on doublets, and that for an +outer-|Z| root eta_ct). Then + + **Theorem 1.** + G_eld(t) ~ C_cusp E0 t^-11, + mu_F(n=2,|T_*|>t) ~ C_cusp D t^-11. (9) + +For every fixed t>0 the radius event is a continuity set of the limiting +selected measure, and therefore + + lim_(t->infinity) t^11 lim_(r->0) + r^-3 Q_r^W(F_r, |T_r|>t, T_r finite)=C_cusp E0. (10) + +Conditional on a distant actual failed partner, in that same iterated order, +its lifetime fraction and selected shape have the universal laws in Section4. +This is candidate pair-Palm sampling given failure; it does NOT count each +replacement persistence bar once across different rejected candidates. + +Proof: inserting (7) into R's root integral gives exactly + + (108 k^7 c_m/z0) Q(u,v)1_E H(h)|Z|^-12 + h0(A0,Rot_O tau(a,a^2/(12k)+k b0/Z^2, + a^3/(144k^2)+a b0/(4Z^2)+2k(v-b0u)/Z^3,xi)) + du dv dZ da dxi dh dO. (11) + +The selector 1_E is independent of Z. The radius is still +|T_*|^2=(u-aZ/(12k))^2+Z^2. Under Z=t z it tends, after division by t, to +gamma(a)|z|. For t>=3 the radius condition implies |z|>1/(2gamma(a)). +The source majorant is a polynomial in h times + + Q 1_E |z|^-12 exp[-c(|h|^2+a^2+|xi|^2)] + 1{|z|>1/(2gamma(a))}. + +Integrating z costs only a factor proportional to gamma^11; the remaining +integral is finite. Dominated convergence yields the common C_cusp times the +shape mass E0. Restriction to counted doublets replaces E by the inner sector +and gives D. No hard-eigenvalue inverse or unproved radius ordering enters. + +For each t>0 and fixed u,a, the radius equation is a nonconstant quadratic in +Z with at most two solutions, hence null in (11). Thus |T_*| has no positive +atoms. Finite-mass weak convergence in (3), including negligible sentinels, +allows the unbounded exterior-radius indicator by Portmanteau. A bounded +continuous function of the other marks can be inserted as well, since its +only new discontinuity is that sphere. The denominator G_eld(t)>0 for all t, +by the positive tail at large t and monotonicity. This proves the fixed-t +conditional transfer and then (10). No uniformity in a moving t(r) is used. +QED. + +Some exact coefficient ratios, independent of the model amplitude, are + + G_eld(t)/F_point(t) -> E0/I=296269/657408, + G_eld(t)/G_max(t) -> E0/(I-D), + mu_F(n=2,|T_*|>t)/G_eld(t) -> D/E0. (12) + +They are approximately0.450662298,0.457796706,0.0345807201, respectively. +These are three different denominators. The final doublet probability concerns +the local cubic configuration of an extreme elder, not the global torus count. +The role of n here is entirely inside the limiting measure; no global-count +moment theorem is needed to identify the actual partner. + +For nonnegative powers, the radius of this selected limiting law has moment +threshold11, and + + integral min(|T_*|,t)^11 dmu_F ~11 C_cusp E0 log t. (13) + +The proof is Tonelli applied to (9). It is not uniform integrability of the +finite-r spatial moments in (3). + +## 4. An explicit universal lifetime-fraction density + +In the extreme-selected limit, the shape density is Q1_E/E0. The normalized +radius q=|T_*|/t is independently Pareto(11), with density11q^-12 on q>1; +its sign is symmetric and the remaining cusp parameters have their original +H gamma^11 h0/J_cusp density. This follows from (11) with q=gamma(a)|z|, +using the same majorant. Projecting to physical spatial marks gives WEAK +convergence. No assertion of spatial total variation is required. + +**Theorem 2.** Let e in(0,1). Define + + p_lo(e)=sqrt(1-e)-1/2, + p_hi(e)=cos((1/3) arccos(2e-1)). (14) + +Equivalently p_hi is the unique root in(1/2,1) of +2p^3-(3/2)p+1/2=e. The limiting fraction lambda=(b-death)/(k r^3) of a distant +failed elder partner has density + + f_eld(e)=(6/E0) integral_(p_lo(e))^(p_hi(e)) + Q(-p,6(e+p-1/2)) dp. (15) + +The integrand is a polynomial, so (15) is an explicit algebraic density with +one radical and one specified cubic root. It is strictly positive on(0,1). +This is a density of the LIMITING conditional mark, not an assertion that the +original fixed-r persistence intensity has this density or converges in TV. + +Proof: eta=e sets v=6(e+p-1/2), dv=6 de. The full shape region at fixed e gives + + sqrt(1-e)-1/2 < p < 1/2+sqrt(e). + +For p>1/2 write x=p-1/2. The lower shape condition becomes x^2=1 the entire +shape interval is outer. It follows that O2 removes exactly + + p_hi(e)1/2; +4cos^3(theta)-3cos(theta)=cos(3theta) gives (14). Nonnegativity, normalization +and strict positivity follow from the original selected shape integral. + +Every nonnegative integer moment is rational. To see this directly, integrate +Q(u,v)(u+1/2+v/6)^j over S and subtract the same polynomial integral over the +two strips (6), all of which have polynomial bounds and rational endpoints. +The first three unnormalized moments are + + E0=888807/280, + E1=3524743221/1361360, + E2=190490806059/87127040. + +Thus + + E[lambda]=1174914407/1440459878, + E[lambda^2]=63496935353/92189432192. (16) + +They are approximately0.8156522961481611 and0.6887658795940618. +Neither is the point-intensity nor the maximum-anchor lifetime mean. All +shape factors are universal in the fixed model parameters; directions and +C_cusp need not be universal. QED. + +For reproducible evaluation, with x=p-1/2 the unnormalized density integrand is + + 10368(e-x^2)(e+x^2+2x)(e+2ex+x^2). (17) + +Its integration endpoints are sqrt(1-e)-1 and the nonnegative root of +3x^2+2x^3=e. The companion uses rational bisection plus interval polynomial +evaluation to enclose (15), not unconstrained floating quadrature. + +## 5. A fourth-power small-lifetime law and its inverse-moment threshold + +**Theorem 3.** In the limiting probability law of Section4, + + f_eld(e) ~ (3328/E0)e^3, + P(lambda<=e) ~ (832/E0)e^4=(232960/888807)e^4, + f_eld(1-) =1312200/296269. (18) + +Consequently E[lambda^-p] is finite exactly for nonnegative p<4. At the critical +order, the capped inverse moment has asymptotic + + E[min(lambda^-1,T)^4] ~ (3328/E0)log T. (19) + +All three limits occur INSIDE the already-formed extreme-elder mark law: r->0, +then distant-partner radius->infinity, then e->0 (or T->infinity). Weak marked +convergence alone would not justify analogous finite-r inverse moments. + +Proof: use (17), put x=sqrt(e)z and divide the density numerator by e^3. The +lower endpoint tends to0, the upper to1/sqrt(3), and the integrand tends to + + 10368*2z(1-z^4). + +The scaled intervals lie in a fixed compact set for all small e and the +integrands have a common polynomial bound. Hence their limit integral is +10368(1/3-1/81)=3328. Integrating this density asymptotic gives the CDF formula. +At e=1 the integral of (17) over x in[-1,1/2] is98415/7; division by E0 gives +the endpoint in (18). Continuity and strict positivity away from zero then +show that only the lower endpoint controls inverse-moment integrability. +Finally lambda^-1 has tail coefficient832/E0 at exponent4; Tonelli gives +4*(832/E0)log T in (19). QED. + +## 6. The more distant companion has a thirteenth-power ratio tail + +This section conditions the limiting extreme-ELDER law further on a doublet. +Let rho=|Z_inner/Z_outer| in(0,1) and z in(0,1) be D's unit-square coordinates +of the OUTER root. Write L(rho,z) for D2's unnormalized density: + + L=165888 rho z^7(z+1)^4 (rho z+1)^4(rho z+rho+1)^7 + /[(rho+1)^5(2rho z+rho+1)^9]. (20) + +Its total mass is J. The elder is the inner root. C E18-E19 give the deterministic +change-of-variables identity + + Q(T)|det DT|=Q|t_c|^11, integral rho^11 L =D. + +**Theorem 4.** Under the limiting extreme elder law given a doublet, (rho,z) +has density + + rho^11 L(rho,z)/D. (21) + +It is NOT L/J. The scalar radial ratio q=|T_*|/threshold remains independent +Pareto(11). In physical positions divided by that threshold the configuration is + + { epsilon q e_*, -epsilon q rho^-1 e_* }, (22) + +where the first point is the actual limiting elder and the second the farther, +lower saddle. Its heights are respectively + + eta_eld=rho^3 z^2(z+1)(rho z+rho+2) + /[(rho+1)(2rho z+rho+1)], + eta_other=z^2(rho z+1)(rho z+2rho+1) + /[(rho+1)(2rho z+rho+1)]. (23) + +Let V=R_other/R_eld=1/rho in this limit, and H0=6401/3960. Then + + P(V>T | extreme elder, doublet) ~ C13 T^-13, + C13=165888 H0/(13D)>0. (24) + +Its nonnegative moments are finite exactly below13. Conditional further on +V>T, (T/V,z) converges in total variation on the fixed scalar-coordinate +unit square to independent densities13y^12 and z^7(z+1)^4/H0. In particular + + eta_eld V^3 -> 2z^2(z+1), eta_other -> z^2 (25) + +as a further weak-mark consequence. No spatial TV is claimed for (22). + +Proof: the inner-selected shape measure is Q on the inner sector. Pull it +back to the outer sector through the companion involution. The weighted +Jacobian gives exactly the factor rho^11, and then D's unit-square bijection +gives (21). Equivalently the physical inner-radius cutoff has asymptotic +rho gamma(a)|Z_outer|>threshold, giving the same rho^11 after radial integration. +It is not legitimate to retain the old maximum-conditioned doublet density. +The linear transformation of the sign and the radial change q=rho gamma|Z| +yield independence and (22)-(23). + +The rational factor multiplying165888 rho z^7(z+1)^4 in (20) converges uniformly +to1 as rho->0 and is bounded on the closed square. Thus the numerator of (21) +is asymptotic to165888 rho^12 z^7(z+1)^4, whose integral over rho<1/T gives +165888 H0/(13D) T^-13. Substituting rho=y/T and dividing by that mass proves +the normalized L1/TV claim. The uniform height limits of (23) give (25). +Moment integrability follows since V>=1 and the tail coefficient is positive. +The density L/J from maximum selection instead has a second-power tail for +1/rho; that is a different measure, not a contradiction. QED. + +## 7. Review obligations and excluded conclusions + +A: actual-selector composition (1)-(11), matching ELDER2/ELDERD's failure +measure to the root density without a count substitute; no lost sign/Haar/ +normalizer; fixed-t continuity and the iterated-limit order. +B: selected shape interval, all polynomial integrals and moments, density +(15), fourth-power endpoint and inverse-moment threshold (18)-(19). +C: rho^11 reweighting, thirteenth-power companion ratio, scalar TV versus +moving spatial marks, and the independence/height statements (21)-(25). + +The tests corroborate exact algebra, root domains and interval arithmetic. +They do not prove the Gaussian or actual-pairing sources, grant the new +consumer independent acceptance, or formalize these theorems in Lean. +No original r-limit rate, growing-volume/vanishing-k uniformity, finite-r +inverse moments, unrestricted lifetime density, replacement-bar intensity, +remote singleton elder coefficient or across-pair independence is claimed. +The finite physical-radius ordering counterexample in C is retained: the +elder selector is based on height before taking any radius limit. diff --git a/frontiers/elder_partner_extremes_20260930/README.md b/frontiers/elder_partner_extremes_20260930/README.md new file mode 100644 index 000000000..8c3d59674 --- /dev/null +++ b/frontiers/elder_partner_extremes_20260930/README.md @@ -0,0 +1,73 @@ +# Distant elder partners: a selected lifetime law, not a maximum-cluster law + +OpenAI / GPT-6 Astra Pro. **Author-side conditional candidate; new nonauthor +review OPEN. Scientific effect NONE. No self-merge.** + +The actual elder rule selects the HIGHEST saddle. On a cubic doublet that is +the inner-|Z| saddle, the opposite selector to the extreme-cluster maximum +anchor. This packet composes the actual failure/partner interfaces of PR170 +(d=2) and PR175 (d>=3) with the exact microscopic root measure of PR166/169. +It does not establish actual pairing merely from a window-count law. + +## Four new conditional results + +1. The limiting actual selected partner has radius tail + `C_cusp (888807/280) t^-11`. Its coefficient is I-J, not the point coefficient + I or the maximum-cluster coefficient I-D. Only about3.458% of these distant + selected partners belong to doublets. The nonempty failure measure excludes + the remote singleton coefficient. +2. Its extreme-selected lifetime fraction e has an explicit algebraic density: + + f(e)=(6/E0) integral_[sqrt(1-e)-1/2]^[p_hi(e)] + Q(-p,6(e+p-1/2)) dp, + 2p_hi^3-(3/2)p_hi+1/2=e, p_hi in(1/2,1), E0=888807/280. + + All nonnegative integer moments are rational. The mean is exactly + 1174914407/1440459878, about0.8156522961481611. +3. The small-lifetime CDF is asymptotic to `(232960/888807)e^4`, so inverse + lifetime moments have threshold4 in THIS limiting probability law. No + finite-r inverse-moment convergence is inferred from weak convergence. +4. Given a doublet, selecting for large elder radius reweights the former + companion-ratio square density by rho^11. The other/elder radius ratio has + tail exponent13, rather than exponent2 under maximum-cluster selection. + +PROOF.md gives the full arguments, exact scope and fixed-threshold transfer. +The order is **first r->0, then the selected microscopic radius t->infinity**; +small-lifetime and relative-companion limits occur inside that second limit. +No uniform simultaneous limit, unrestricted persistence-lifetime density, +replacement-bar intensity, original finite-r rate or Gaussian coefficient +numerical enclosure is claimed. + +## Source and review boundary + +SOURCES.json binds five complete historical commit/path/Git-blob/SHA256 identities. +ELDER2/ELDERD remain separate author-side source candidates with their own +source-scoped reviews; this new consumer is conditional on their stated actual +selector/failure-measure interfaces. Linking them neither promotes their status +nor revalidates their Gaussian/topological proof. C and D were authored by this +conversation. Same account/provider is not independent scientific acceptance. + +Requested scopes: A actual-elder source composition and selected tail; B explicit +lifetime density/moments/endpoints; C rho^11 reweighting and exponent13. No new +analytic review is inferred from a code pass or an old source review. + +## Replay + +In a full checkout containing the pinned source objects: + + python -B -S frontiers/elder_partner_extremes_20260930/verify.py + +Standalone packet: + + python -B -S verify.py --local-only + +The latter reports source_pins_checked=false. The hosted workflow must fetch and +verify all five actual sources. It checks declared sizes where supplied, all +SHA256/blob identities and regular Git path modes, with replacements disabled. +Local Git fixtures test behavior; they do not authenticate project sources. + +The standard-library suite has41 methods, run under ordinary and optimized +Python. It includes893 exact rational root-shape cases, direct unsheared cubic +checks,12 independent inner-root Jacobians, polynomial interpolation, rational +root-isolation density intervals and13 real-Git/inventory tests. Six semantic +mutants must reject in each mode. No Lean formalization of these new theorems. diff --git a/frontiers/elder_partner_extremes_20260930/RECONNAISSANCE.md b/frontiers/elder_partner_extremes_20260930/RECONNAISSANCE.md new file mode 100644 index 000000000..8a4f041bc --- /dev/null +++ b/frontiers/elder_partner_extremes_20260930/RECONNAISSANCE.md @@ -0,0 +1,33 @@ +# Narrow primary-source reconnaissance — 30 September 2026 + +Actual web queries in this pass included: +- site.arxiv.org persistence elder rule Gaussian random fields critical points +- site.arxiv.org Palm theory extremes stationary regularly varying random fields anchored clusters Planinic +- site.arxiv.org "elder rule" "Gaussian" persistence + +Primary abstract/metadata pages opened: + +1. Hrvoje Planinic, Palm theory for extremes of stationary regularly varying + time series and random fields, arXiv:2104.03810. + https://arxiv.org/abs/2104.03810 + Existing framework: typical exceedances, typical clusters and their different + biases. This packet does not claim to invent anchored/Palm sampling. + +2. Bojan Basrak, Ilya Molchanov and Hrvoje Planinic, Extremal behavior of + stationary marked point processes, arXiv:2210.01474. + https://arxiv.org/abs/2210.01474 + Related marked extreme-process framework. Its hypotheses are not silently + imported as a theorem about our Gaussian pin pairs. + +Access depth: abstract/metadata only, not full-text theorem comparison. No +exhaustive novelty/priority claim or absence-from-literature conclusion is made. +No result from these pages is a missing proof step. The actual mathematical +inputs are the exact source-bound elder and root-density interfaces in SOURCES. + +The new point is source-specific: highest-height selection removes the OUTER +root sector, rather than the inner sector removed by maximum-radius anchoring. +The explicit lifetime density and companion exponent follow by recalculating +that selector's measure, not by renaming the maximum-cluster distribution. + +Consensus was not invoked in this pass (the current request names GitHub and +Superpowers). No changed quota or connector result is claimed. diff --git a/frontiers/elder_partner_extremes_20260930/RESULTS.json b/frontiers/elder_partner_extremes_20260930/RESULTS.json new file mode 100644 index 000000000..7c4015952 --- /dev/null +++ b/frontiers/elder_partner_extremes_20260930/RESULTS.json @@ -0,0 +1,62 @@ +{ + "companion_over_elder_moment_threshold": 13, + "constants": { + "double_given_extreme_elder": { + "lower": "0.034580720107731639", + "upper": "0.034580720107731640" + }, + "elder_to_maximum_tail_ratio": { + "lower": "0.457796705679599976", + "upper": "0.457796705679599977" + }, + "elder_to_point_tail_ratio": { + "lower": "0.450662297994548286", + "upper": "0.450662297994548287" + }, + "mean_lifetime_fraction": { + "lower": "0.815652296148161094", + "upper": "0.815652296148161095" + }, + "outer_over_elder_ratio_tail_constant": { + "lower": "187.906062596204997257", + "upper": "187.906062596204997258" + }, + "second_lifetime_fraction_moment": { + "lower": "0.688765879594061834", + "upper": "0.688765879594061835" + } + }, + "height_density_values": { + "1/100": { + "lower": "0.000001295538683393", + "upper": "0.000001295538683394" + }, + "1/2": { + "lower": "0.411333720068101484", + "upper": "0.411333720068101485" + }, + "1/4": { + "lower": "0.039478044779395017", + "upper": "0.039478044779395018" + }, + "3/4": { + "lower": "1.649750418945205199", + "upper": "1.649750418945205200" + }, + "99/100": { + "lower": "4.279300396681162358", + "upper": "4.279300396681162359" + } + }, + "inverse_lifetime_moment_threshold": 4, + "mathematical_acceptance": false, + "rational_shape_cases": 893, + "scientific_effect": "NONE", + "scope": "finite algebra, not validation of Gaussian or elder premises", + "selected_shape_moments": { + "0": "888807/280", + "1": "3524743221/1361360", + "2": "190490806059/87127040" + }, + "small_lifetime_cdf_coefficient": "232960/888807" +} diff --git a/frontiers/elder_partner_extremes_20260930/SOURCES.json b/frontiers/elder_partner_extremes_20260930/SOURCES.json new file mode 100644 index 000000000..e32cf6444 --- /dev/null +++ b/frontiers/elder_partner_extremes_20260930/SOURCES.json @@ -0,0 +1,50 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "scope": "Conditional exact interfaces; no parent revalidation or source-status change", + "sources": [ + { + "id": "R", + "commit": "80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337", + "path": "frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md", + "bytes": 12533, + "git_blob": "0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4", + "sha256": "250897858c8b314c0ff85ec1860a725efacfdae0ff7409fced4cadf7bfb85973", + "use": "R4-R13 exact root integral and Gaussian majorant; not pairing from counts." + }, + { + "id": "C", + "commit": "56a7904f9ad7d409c31e23195c0e7eeab230613a", + "path": "frontiers/extreme_cluster_sampling_20260930/PROOF.md", + "bytes": 21891, + "git_blob": "19a6044f2066643b5baab34ce5b6b6efb431e84a", + "sha256": "b90cf745e2e67bc9b501c481552f7080d27c28245580668cf72aefa5b7f04715", + "use": "E10-E25 companion/height map, sectors, exact integrals. Author exposure." + }, + { + "id": "D", + "commit": "56a7904f9ad7d409c31e23195c0e7eeab230613a", + "path": "frontiers/extreme_cluster_sampling_20260930/COMPANION_RATIO.md", + "bytes": 10032, + "git_blob": "56348f6765793cd8115041ba367b821eb173f695", + "sha256": "74ff3af5ec071f57d40938fbeca6bca1c189272d5c1bc0440e41cbe764c04d23", + "use": "D1-D7 exact square-coordinate density and heights only; reweighted anew here. Author exposure." + }, + { + "id": "ELDER2", + "commit": "79f18f0cb318fceccd5bdf4e6dad29815165bc7d", + "path": "frontiers/local_elder_geometry_20260930/PROOF.md", + "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", + "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b", + "use": "d=2 Theorem E, failure measure and actual weak partner/lifetime law in Sections8.6-8.8. Conditional candidate interface, not promoted." + }, + { + "id": "ELDERD", + "commit": "9c6a73489c5ed2465a3cf65dcc950eeeae074b05", + "path": "frontiers/concave_fibre_elder_20260930/PROOF.md", + "git_blob": "923d3236b5291a29bd162937f172190db84dcef5", + "sha256": "5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e", + "use": "d>=3 H1/S1-S3/M1 actual selector/failure measure/weak marks. Conditional candidate interface, not promoted." + } + ] +} diff --git a/frontiers/elder_partner_extremes_20260930/VALIDATION.md b/frontiers/elder_partner_extremes_20260930/VALIDATION.md new file mode 100644 index 000000000..938346370 --- /dev/null +++ b/frontiers/elder_partner_extremes_20260930/VALIDATION.md @@ -0,0 +1,45 @@ +# Author-side validation, not analytic acceptance + +The initial eight-method mathematics scaffold failed all eight assertions. +After implementation those eight passed. Expanded mathematics tests now total28. +The separate13-method source/inventory scaffold produced20 failing assertions, +including subtests. A local attempt to reuse an older verifier found no mounted +file at that path and left the scaffold unchanged; its failed replay was not +called success. The completed verifier was then written and all13 tests passed. + +Fresh final baseline:41 tests PASS in normal Python and41 PASS with -O, both +also under the hosted GIT_NO_REPLACE_OBJECTS=1 environment. Each mode executes +all six named semantic mutants (return1) and an unknown label (return2), with +identical deterministic baseline stdout and no stderr. Mathematical core: +-893 rational shape/selector/companion checks; +-40 direct original-cubic gradient/value/Hessian cases; +-12 independent exact inner-root Jacobians from the square coordinates; +-independent dense polynomial interpolation of moment integrals; +-rational bisection/interval enclosures for the algebraic height density; +-small-height and small-ratio finite diagnostics, strict boundaries and ties; +-the exact finite physical-radius-order counterexample retained from PR169. + +Custody tests use real temporary Git repositories: full identity fields, +missing/unsafe paths, omitted/duplicated IDs, noncommit tree/blob objects, +symlink payloads, historical replacement forgery and packet drift. These are +simulated failures, not observed tampering of the user's sources. + +All code is Python standard library. Runtime Python3.13.5; the target workflow +uses Python3.11.16 as in the existing repository. Exploratory symbolic algebra +was used privately to cross-check strip constants, not as a shipped dependency. +The final manuscript's rounded second-moment decimal was corrected against the +exact rational result before publication; no source file or earlier review changed. + +No complete project checkout could be obtained locally (the public raw-host +request failed DNS). Hence local-only replay explicitly reports +source_pins_checked=false. Historical authentication of the five ACTUAL source +objects remains the hosted requirement. The two elder source bindings supply +SHA256 and Git blob; the other three also include declared byte size. No size +was invented for an unread full-byte copy. A matching content hash is not a +proof that the source's mathematics or external review is correct. + +The new actual-source composition, infinite-dimensional domination and weak +transfer require source-bound nonauthor analytic review. Neither these finite +tests nor existing Lean workflow success formalizes this manuscript. Any +hosted success must be recorded against its actual head/run, not anticipated +in this author-time record. No self-merge or scientific register transition. diff --git a/frontiers/elder_partner_extremes_20260930/elder.py b/frontiers/elder_partner_extremes_20260930/elder.py new file mode 100644 index 000000000..9a97c39e6 --- /dev/null +++ b/frontiers/elder_partner_extremes_20260930/elder.py @@ -0,0 +1,250 @@ +"""Exact controls for elder-selected microscopic extreme laws. + +Standard library only. This is finite algebra and interval arithmetic, not a +proof of the actual Gaussian elder-selector interface or the limiting theorem. +""" +import argparse +from fractions import Fraction as F +import json +from math import comb + +MUTANTS=('choose-outer','count-both','drop-half','wrong-lifetime-sign', + 'forget-radius-bias','wrong-height-power') +_MUTANT=None + + +def rational(x): + if type(x) not in (int,F): raise TypeError('exact int/Fraction required') + return F(x) + + +def b0(u): return 3-12*u*u + +def height(u,v): return u+F(1,2)+v/6 + +def shape(u,v): + return -F(3,2) 10368 int_0^(1/sqrt(3)) 2z(1-z^4) dz. + return F(0) if _MUTANT=='wrong-height-power' else F(10368)*(F(1,3)-F(1,81)) + + +def outer_over_elder_tail_exponent(): + return 2 if _MUTANT=='forget-radius-bias' else 1+11+1 + + +def imul(a,b): + vals=[x*y for x in a for y in b] + return min(vals),max(vals) + +def iadd(a,b):return a[0]+b[0],a[1]+b[1] + +def ipoly(coefs,x): + ans=(F(0),F(0)) + for j in range(max(coefs),-1,-1): + c=coefs.get(j,F(0));ans=iadd(imul(ans,x),(c,c)) + return ans + + +def _isolate(target,func,left,right,bits): + for _ in range(bits): + mid=(left+right)/2;value=func(mid) + if value==target:return mid,mid + if value=hi:continue + for j in range(1,20): + v=lo+(hi-lo)*F(j,20);sel=selection(u,v);pair=companion(u,v) + if pair and shape(pair[0],pair[1]): + if sel+selection(pair[0],pair[1])!=1:raise ValueError('one partner per configuration') + if sel in (0,1) and sel!=1-int(outer(u,v)):raise ValueError('height/strip selector mismatch') + elif sel!=1:raise ValueError('singleton must select its one root') + n+=1 + return n + + +def result(): + n=checks() + return {'scientific_effect':'NONE','mathematical_acceptance':False, + 'scope':'finite algebra, not validation of Gaussian or elder premises', + 'rational_shape_cases':n, + 'selected_shape_moments':{str(j):str(moment(j)) for j in range(3)}, + 'small_lifetime_cdf_coefficient':str(F(832)/moment(0)), + 'inverse_lifetime_moment_threshold':4, + 'companion_over_elder_moment_threshold':13, + 'constants':{key:decimal_interval(val) for key,val in constants().items()}, + 'height_density_values':{str(e):decimal_interval(height_density_bounds(e)) for e in (F(1,100),F(1,4),F(1,2),F(3,4),F(99,100))}} + + +def main(): + parser=argparse.ArgumentParser(description=__doc__);parser.add_argument('--mutant',choices=MUTANTS);args=parser.parse_args() + global _MUTANT + _MUTANT=args.mutant + try:ans=result() + except ValueError as exc: + print(json.dumps({'passed':False,'error':str(exc)},sort_keys=True));raise SystemExit(1) + print(json.dumps(ans,indent=2,sort_keys=True)) + +if __name__=='__main__':main() diff --git a/frontiers/elder_partner_extremes_20260930/test_elder.py b/frontiers/elder_partner_extremes_20260930/test_elder.py new file mode 100644 index 000000000..6f9670e88 --- /dev/null +++ b/frontiers/elder_partner_extremes_20260930/test_elder.py @@ -0,0 +1,265 @@ +from fractions import Fraction as F +import unittest +import elder + +class InitialTests(unittest.TestCase): + def test_selected_mass(self): + self.assertEqual(elder.moment(0), F(888807,280)) + def test_selected_mean(self): + self.assertEqual(elder.moment(1)/elder.moment(0), F(1174914407,1440459878)) + def test_selected_second(self): + self.assertEqual(elder.moment(2)/elder.moment(0), F(63496935353,92189432192)) + def test_outer_not_elder(self): + self.assertEqual(elder.selection(F(-1), F(6)), 0) + def test_inner_is_elder(self): + self.assertEqual(elder.selection(F(-13,23), F(294,529)), 1) + def test_near_zero_density(self): + self.assertEqual(elder.small_height_density_coefficient(),F(3328)) + def test_upper_density(self): + lo,hi=elder.height_density_bounds(F(1)) + self.assertEqual((lo,hi),(F(1312200,296269),F(1312200,296269))) + def test_new_ratio_exponent(self): + self.assertEqual(elder.outer_over_elder_tail_exponent(),13) + + + +class GeometryTests(unittest.TestCase): + def test_full_shape_grid(self): + self.assertEqual(elder.checks(),893) + + def test_selector_matches_cubic_boundary(self): + for ip in range(-7,24): + p=F(ip,16) + for j in range(1,15): + e=F(j,15); v=6*(e+p-F(1,2)); u=-p + if not elder.shape(u,v):continue + boundary=2*p**3-F(3,2)*p+F(1,2) + expected=F(1) if p<=F(1,2) or e>boundary else F(1,2) if e==boundary else F(0) + self.assertEqual(elder.selection(u,v),expected) + + def test_actual_cubic_gradient_height_and_type(self): + # Directly differentiate the unsheared cubic, independently of selector. + cases=0 + for p in (F(1,4),F(3,5),F(3,4),F(1),F(5,4)): + u=-p; b=3-12*u*u;lo,hi=abs(b)/2,3-6*u + for j in range(1,5): + v=lo+(hi-lo)*F(j,5) + for k,a,Z in ((F(1),F(-12),F(1)),(F(3,2),F(4),F(-2))): + s=-k*v/Z**2; B=k*b/Z**2; Dc=k*(v-b*u)/Z**3 + beta=B+a*a/(12*k);c=2*Dc+a*B/(4*k)+a**3/(144*k*k) + roots=[(u,v,F(1))] + other=elder.companion(u,v) + if other and elder.shape(other[0],other[1]):roots.append(other) + es=[] + for uu,vv,t in roots: + z=t*Z;x=uu-a*z/(12*k) + gx=6*k*x*x-F(3,2)*k+a*x*z+beta*z*z/2 + gz=s*z+a*(x*x-F(1,4))/2+beta*x*z+c*z*z/2 + val=2*k*x**3-F(3,2)*k*x-k/2+s*z*z/2+a*(x*x-F(1,4))*z/2+beta*x*z*z/2+c*z**3/6 + hxx=12*k*x+a*z;hzz=s+beta*x+c*z;hxz=a*x+beta*z + self.assertEqual((gx,gz),(0,0)) + self.assertEqual(-val/k,elder.height(uu,vv)) + self.assertTrue(-k1/2 and ell>1/2 imply 4p ell>1. The denominator in candidate (6) is a nonzero square on the open generic doublet sector. Thus sign(eta_c−eta)=sign(ell−p). Outer seeds have vAp and their companions have larger eta. Consequently the inner seed is the highest-P root. Equal-height v=Ap and the degree-drop/boundary curves are null. The companion involution exchanges the open outer and inner sectors. Singletons have no second qualifying root. Therefore S_E=S\O2 contains exactly one selected root of every generic nonempty configuration. + +The RADIAL R4–R10 map is locally a diffeomorphism with the exact root multiplicity. Restricting that root integral to S_E counts each nonempty configuration once. It introduces neither a factor two nor a restricted jet normalizer. Restricting further to I2 gives the selected doublet event; removing both doublet sectors gives the selected singleton event. These are exact cubic selections before the radius limit, independent of a and Z. This is the substantive new selector operation. + +## Tail and marked-law audit + +The root density (8) matches RADIAL R11 term by term: raw jet Jacobian 2, stationary-point Jacobian 6k^3, typed weight 9k^4, physical X=u−aZ/(12k), full tensor, normalized Haar, ordered hard spectrum, original c_m/z0. The cusp entries beta=a^2/(12k), c=a^3/(144k^2) and physical shear factor gamma are retained. + +After Z=t z, the power is t^-11. The exact physical threshold implies |z|>1/(2gamma) for t>=3. SC (4),(10) gives Gaussian control in raw orthogonally rotated coordinates; dropping beta,c leaves a valid upper bound in h,a,xi. Integrating |z|^-12 on that threshold majorant costs only a constant times gamma^11. The bounded shape region/Q and polynomial hard factor are then Gaussian-integrable. This is sufficient domination on all retained variables and avoids inverse-hard constants. The pointwise limiting threshold boundary gamma|z|=1 is null. + +The two signs contribute 2gamma^11/11, giving exactly C0. The selected shape integrals are integral_(S\O2)Q=I−J, integral_I2 Q=D, and the singleton integral I−J−D. The imported values are assigned to the correct sectors. Positivity follows analytically from open positive-density inner-doublet and singleton sectors. The singleton example near u=0,v=2 is valid because B>0 excludes a doublet. Hence (10) and the selected doublet frequency D/(I−J) are correct at this scope. + +For bounded continuous marked tests the same majorant applies. The q=gamma|z| substitution factors the limit into shape Q/(I−J) on S_E, Pareto(11) radius, uniform sign and the cusp parameter law. The height fraction is the pushforward eta=u+1/2+v/6 of this selected shape law, independent of q and latent parameters. It is not the point-selected or maximum-anchor height law. + +On I2, t_c<0 and |t_c|>1. Thus the selected extreme position tends to epsilon q e_* and its deeper companion to −epsilon q |t_c| e_*. The rational companion map may diverge near exceptional boundaries, but it is continuous on compact regular subsets and the limiting shape density is integrable. Exhausting those subsets proves the claimed marked weak convergence. No spatial total-variation transfer is made. + +In the outer chart, the involution identity Q(T)|det DT|=Q rho^11 yields the selected-doublet shape density Q rho^11/D. Comparison with Q/J for a maximum-radius outer anchor gives exactly the claimed additional rho^11 bias. This is a deterministic coordinate transformation, not independence or temporal Palm sampling. + +## Exact physical-order counterexample + +Candidate (12) is consistent with (2)–(4). For k=1, a=−12 and Z=1, B=−15/4, s=−14/5 satisfies s<−|B|/2. The seed has b0=−15/4, L=−7031/400 and t_c=−6919/7031. Its physical X=u+Z=1/4 gives R^2=17/16. The companion has X_c=u_c+Z_c=−48583/28124, so + + R_c^2=(48583^2+27676^2)/28124^2 + =3126268865/790959376. + +Its eta_c is the stated rational value and is less than 13/60; both shapes lie in the strict window. Hence the companion is selected and is physically farther in this finite cubic example. Thresholds strictly between the two radii falsify replacement of the selected event by the finite-t minimum-radius event. The asymptotic coincidence of their leading doublet coefficient does not repair that false finite-t identity. + +## Actual elder interface and sequential quantifiers + +PR175 S1 is exactly 1{n>0}dM: same raw Gaussian density, c_m, full z0, normalized Haar/sign extension, ordered hard eigenvalues, full cubic tensor, H and w on the typed domain. Its M1 mark is the highest-P additional saddle on failure, with death h_*=−k eta_E. The original-coordinate location agrees with (2). No count conclusion, remote singleton coefficient, new adjacency weight or good-event normalizer is used. + +The imported theorem is explicitly at fixed d>=3 in its original pair-Palm model. Its contained hard-depth tube, graph/pinned C2 interface, actual global genericity/Borel repair, endpoint-residual/full-weight estimates, full normalization and actual failure exhaustion remain hypotheses. Reading its full companions confirms that these are the stated source obligations. This review does not establish them independently. The d=2 microscopic calculation therefore does not acquire a planar finite-r identification from this citation. + +For fixed t>0, the limiting selected radius has no atom at t: at fixed other root coordinates R^2 is a quadratic in Z with positive leading coefficient gamma^2, so R=t has finitely many Z solutions. Absolute continuity of the restricted coarea integral makes that boundary null. The isolated sentinel is excluded from the real-radius event; its mass is negligible by the explicitly imported exhaustion. Finite-measure weak convergence and Portmanteau consequently apply to the radius event and bounded continuous joint tests restricted to it. E(t)>0 follows from open singleton shapes and arbitrarily large |Z| with positive raw Gaussian density. Normalization of the restricted measures is legitimate. + +Thus the r->0 conditional limit at each fixed t follows from M1, and only then does Theorem 2 give t->infinity. Equation (13) is an iterated limit. The argument supplies no t(r), finite-r uniform tail, spatial moment uniform integrability, essential-class probability, or replacement-bar intensity. Retained shape marks are obtained on regular jet sectors where root selection is continuous; transition/tie sets are null and compact exhaustion removes the remainder. + +## Review boundary + +No mandatory amendment or obstruction was found for the conditional implications in the exact candidate. The briefly missing SOURCES.json at the start of the review was supplied without changing the proof; the final source map was read and bound above, so that initial packaging gap is resolved. + +This acceptance is source-bound analytic assessment by a same-provider nonauthor. It is not source acceptance, organizational independence, a scientific-status promotion or proof by hashes/tests. No finite-r lifetime-moment or essential-probability claim was reviewed; those remain outside this assignment. diff --git a/frontiers/elder_selected_extremes_20260930/PROOF.md b/frontiers/elder_selected_extremes_20260930/PROOF.md new file mode 100644 index 000000000..1310dc11b --- /dev/null +++ b/frontiers/elder_selected_extremes_20260930/PROOF.md @@ -0,0 +1,311 @@ +# The elder-selected microscopic radius–lifetime law + +OpenAI author-side candidate, 30 September 2026. **AUTHOR_SIDE / HOLD.** +Scientific effect NONE. This is a new selector consumer, not another count +estimate. The author contributed PR170/175 and reviewed PR169 Slice A. A +same-provider noncontributor review gives no organizational-independence credit. +All mathematical implications below retain their exact imported hypotheses. + +## 1. Interfaces and scope + +Fix d>=2, k>0 and the fixed model parameters of the sources. Work first with +the already-formed microscopic finite measure on nonempty configurations. +[RADIAL] R4–R12 supplies the root-resolved density and Gaussian domination; +[CUB] supplies the typed strict-window classifier; [EXT] E12–E19 supplies the +companion map, height ordering and shape partition, and E20–E26 its integrals. +[MICRO] M1–M2 and [SC] supply the original-frame measure and embedding only. +Their global count/moment conclusions are not used. Full immutable public +texts and byte/blob identities are in [SOURCES.json](SOURCES.json). + +Only §6 uses an actual finite-r elder theorem: [ELDER] S3 and M1 at its exact +fixed-d>=3 original pair-Palm scope. Its deterministic contained barrier, +whole-field genericity/Borel repair, original full normalization and exhaustion +remain hypotheses of this consumer; linking it does not accept its proof. +This consumer does not independently close PR170/175's composition review. +The d=2 microscopic conclusion does not automatically identify the planar +finite-r law: an exact frame/normalization interface is needed for that use. +No uniformity in dimension, k, hard gaps, t or a simultaneous r,t limit is claimed. + +Write theta=(s,h,O,tau). The ordered hard eigenvalues are +00) is finite and strictly positive. +The probability used below is pi=1{n>0}dM/A_micro. + +The soft cubic, original position and downward height are + + P(X,Z)=2kX^3-3kX/2-k/2+sZ^2/2+(a/2)(X^2-1/4)Z + +(beta/2)XZ^2+(c/6)Z^3, + z=X e_axis+Z e_soft(O), R=|z|, eta=-P(X,Z)/k. (2) + +Every additional counted root has limiting full index d-1 and 00. (3) + +For a doublet define + + L=4v^2-8b0uv+b0^2, + t_c=(4v^2-b0^2)/L, + u_c=[2b0v-u(4v^2+b0^2)]/L, + v_c=v t_c^2, Z_c=t_c Z. (4) + +This map preserves the jet, swaps the two roots and is involutive off the +null exceptional sets. [EXT] proves its determinant/intensity identity + + Q(u_c,v_c)|det D(u_c,v_c)/D(u,v)|=Q(u,v)|t_c|^11. (5) + +For completeness, the explicit doublet partition needed here is as follows. +Put p=-u, A=12p^2-3 and + + V_+(p)=[36p^2-12p-3+3sqrt((2p-1)^3(18p+7))]/4. + +The outer sector O2 has .51. + +One can verify the partition directly from the strict doublet inequality: + + (v-A)^2(2v+A)-12(v-Ap)^2=(3+6p-v) H_p(v), + H_p(v)=-2v^2+(36p^2-12p-3)v-72p^3+36p^2+18p-9. + +H_p(A/2)>0, its upper root is V_+, and for p<1 its zero lies +between Ap and A; for p>=1 the height cap 3+6p lies below A. +The two admissible roots are on the negative branch of the same hyperbola. +Strict convexity of line-minus-branch, negative at Z=0, puts zero between +the roots, hence t_c<0. Substitution in (4) yields the |t_c| split at v=Ap. +These are the deterministic interfaces of [EXT] E14–E16, not a stochastic +assumption about two independent witnesses. + +On a doublet let ell=v/A. Exact substitution gives + + eta_c-eta=2(ell-p)(4p ell-1)^3/(4ell^2-8p ell+1)^2. (6) + +Here p>1/2 and ell>1/2, so 4p ell>1. Consequently the outer seed +has eta_ceta and +IS selected. The equal-height curve v=Ap is null. Therefore the exact +highest-height seed domain is + + S_E=S minus O2, (7) + +containing all singleton seeds and precisely the inner seed of every doublet. +This equality holds at every nondegenerate cubic jet, before any large-radius +limit, and is independent of a and Z. Coarea restricted to S_E consequently +counts one selected root per nonempty configuration. It does not renormalize +the jet measure, double a bar, or equate point intensity with cluster intensity. + +## 3. Full root density and selected-tail coefficients + +The original density in the root coordinates of [RADIAL] is + + K0 Q(u,v) H(h)|Z|^-12 + h0(A0,Rot_O tau(a, + a^2/(12k)+k b0/Z^2, + a^3/(144k^2)+a b0/(4Z^2)+2k(v-b0u)/Z^3,xi)) + du dv dZ da dxi dh dO, + K0=108 k^7 c_m/z0. (8) + +The physical radius is exactly sqrt((u-aZ/(12k))^2+Z^2). +Multiplying (8) by 1{(u,v) in S_E} and 1{R>t} is an EXACT +representation of E(t)=M(n>0,R_E>t), since (7) selects precisely one seed. +The doublet version E2(t)=M(n=2,R_E>t) restricts the shape to I2. +The singleton version E1(t) restricts it to S minus (O2 union I2). + +Define + + gamma(a)=sqrt(1+(a/(12k))^2), + J_cusp=integral H(h) gamma(a)^11 + h0(A0,Rot_O tau(a,a^2/(12k),a^3/(144k^2),xi)) + da dxi dh dO, + C0=(216/11) k^7(c_m/z0) J_cusp, + I=246528/35, J=1083417/280, + D=27066286003/223205220-(79298560/4782969) log(2). (9) + +[EXT] E20–E26 gives integral_S Q=I, integral_O2 Q=J, +integral_I2 Q=D. The last integral is the inner-root integral, not a hard +curvature or the cubic coefficient sometimes denoted D. These exact +integrals are imported, not inferred from a numeric PASS label. + +**Theorem 1.** For the fixed microscopic measure (1), + + E(t) ~ C0(I-J)t^-11, + E2(t) ~ C0 D t^-11, + E1(t) ~ C0(I-J-D)t^-11. (10) + +All three coefficients are strictly positive. In particular, conditional on +R_E>t under pi, the doublet probability tends to D/(I-J). This differs from +the cluster-MAXIMUM doublet frequency J/(I-D). + +Proof. Put Z=t z in (8), retaining the selector (7). The Jacobian contributes +t^-11 |z|^-12 dz. At each fixed nonzero z, beta and c tend to the cusp +entries in (9), and R/t tends to gamma(a)|z|. The shape selector remains +exactly S_E; it need not converge from a radius-based selector. +For t>=3, R>t and |u|<1.5 imply |z|>1/(2gamma(a)) by +t1/(2gamma) costs C gamma^11. S is bounded, Q is bounded on S, +H is a nonnegative polynomial and Haar measure is finite. Thus the remaining +Gaussian integral is finite. This proves domination over all unbounded +retained variables without an inverse hard eigenvalue. The limiting integral +on each sign is gamma^11/11. Both signs and K0 give C0 times the shape +integral I-J, D or I-J-D. This proves (10). +Strict positivity follows from open inner-doublet and singleton sectors, +Q>0 and positive raw Gaussian density. For example a neighborhood of +u=0, v=2 is a singleton sector: b0>0 excludes a doublet. J_cusp is positive +and finite by the same Gaussian argument. QED. + +## 4. The selected joint radius, location and lifetime-fraction law + +**Theorem 2.** Under pi conditional on R_E>t, retain the selected seed +shape (u,v), q=R_E/t, epsilon=sign Z and (a,h,O,xi). Their weak limit is +the product probability + + Q(u,v)1{S_E}/(I-J) du dv; + 11q^-12 dq on q>1; + uniform epsilon in {-1,1}; + H(h)gamma(a)^11 h0(A0,Rot_O tau_cusp)/J_cusp + da dxi dh dO. (11) + +The selected height fraction is eta_E=u+1/2+v/6. Its limiting law is the +pushforward of the FIRST factor in (11), independent of q and the remaining +factors. Thus (11) specifies the selected lifetime-fraction distribution, +without substituting the point-selected or maximum-radius height law. + +With e_*=[-a e_axis/(12k)+e_soft(O)]/gamma(a), the selected position +divided by t tends to epsilon q e_*. For doublets the companion position +divided by t tends to -epsilon q |t_c| e_*, where |t_c|>1 on I2; +its height fraction eta_c exceeds eta_E by (6). The selected root is therefore +asymptotically the higher, nearer root of this extreme doublet, while the +other root is deeper and farther. These are SECOND limits of the microscopic +geometry, not finite-r collinearity or independence of two roots. + +Proof. Insert a bounded continuous test function in the exact selected +integral (8). The majorant in Theorem 1 is unchanged. Dominated convergence +and q=gamma(a)|z| give gamma^11 q^-12 dq; normalization by (10) +produces (11). The position map is continuous away from already null +exceptional sets and has the limits just displayed. One may exhaust compact +shape/latent subsets to avoid the rational companion denominators on the +boundary, then let the excluded mass tend to zero under the integrable +limit density. This proves joint weak convergence of the marked finite +configurations, not moving-map TV convergence of positions. QED. + +An equivalent description of the doublet shape is useful for consumers of +the outer-root chart. If rho=|t_c|<1 on O2, swapping inner to outer in (5) +gives outer density Q rho^11/D. The ordinary maximum-radius outer density +is Q/J. Hence elder-selected extreme doublets have an additional rho^11 +size bias. This is a deterministic root-swap identity, not a temporal Palm +or independent-root hypothesis. It also makes clear why the maximum-radius +law cannot be copied as the actual elder-selected law. + +## 5. Physical-order falsifier retained at finite radius + +Do not replace E2(t) by the doublet minimum-physical-radius tail at finite t. +They have the same leading coefficient C0 D, but their selectors can differ. +Here is an exact typed cubic example, k=1: + + Z=1, a=-12, s=-14/5, beta=33/4, c=-31/40, + u=-3/4, v=14/5, eta=13/60, R^2=17/16, + u_c=-20907/28124, Z_c=-6919/7031, + v_c=670215854/247174805, + eta_c=618523783/2966097660, + R_c^2=3126268865/790959376. (12) + +Both roots are strict-window saddles. eta_cR. Substitution in (2)–(4) verifies every +entry. The typed inequality holds since B=-15/4 and s=-14/5<-15/8; +the seed is in O2 and its companion in I2. For thresholds between these +two radii, the selected doublet exceeds the threshold while the minimum +radius does not. Thus a finite-t physical-minimum identity is false. +Only after Z=t z and t->infinity does the bounded u offset vanish. A +proof that silently identifies the selectors before that limit is invalid. + +## 6. Sequential consequence for the actual elder partner + +Fix d>=3 and the exact source parameters and original Q_r^W of [ELDER]. +Let F_r be failure of pairing M with S; T_r the actual partner displacement +divided by r, and L_r=(b-d_f(M))/r^3. As in [ELDER], chart/essential +exceptions use an isolated sentinel; only REAL T_r enter a radius event. +Assume [ELDER] S3/M1, including its full exhaustion and sentinel exclusion. +Its limit measure is exactly 1{n>0}dM in (1), with selected position z_E +and death value -k eta_E. The equality of measures is termwise: original +c_m/z0, normalized Haar, ordered hard spectrum, full raw tensor and w on +the same typed domain. No new normalization or point-count factor occurs. + +**Corollary 3 (sequential actual selector/lifetime law).** For every fixed t>0, + + lim_(r->0) r^-3 Q_r^W(F_r, T_r real, |T_r|>t)=E(t). + +Consequently + + lim_(t->infinity) t^11 lim_(r->0) + r^-3 Q_r^W(F_r,T_r real,|T_r|>t)=C0(I-J). (13) + +Conditional on this actual event, first let r->0 at fixed t, then let +t->infinity. The limit of (T_r/t,L_r/k), and the retained microscopic +marks when desired, is the selected law of Theorem 2: position epsilon +q e_* and lifetime fraction eta_E with the FIRST shape factor of (11). + +Proof. [ELDER] M1 is weak convergence of finite measures, with mass limit +from S3. The boundary {|z_E|=t} has limiting measure zero. Indeed, in (8) +restricted to S_E, for fixed other coordinates R^2=(u-aZ/(12k))^2+Z^2 +is a quadratic in Z with strictly positive leading coefficient gamma^2. +The equation R=t has finitely many Z solutions and hence zero mass in the +absolutely continuous root integral. Portmanteau therefore gives the +fixed-t event mass and bounded continuous marked tests restricted to that +event. E(t)>0: an open singleton shape and arbitrarily large |Z| in (8) +have positive density. Divide by E(t) to obtain the first conditional limit, +then apply Theorems 1–2 for the second. Sentinel mass is o(r^3) by the +imported exhaustion. No moment uniform integrability is used. QED. + +Equation (13) is an ITERATED limit. It proves neither a uniform finite-r +radius bound nor a limit for t=t(r), nor finite-r lifetime moments, essential +class probabilities, unconditioned bar intensity, or identification outside +the exact imported elder interface. The active lifetime-moment review is +not consumed by this proof. The present step advances which saddle pairs +and its lifetime mark; the original source composition remains conditional. + +## 7. Falsifiers and remaining acceptance obligations + +This consumer fails if: (6) selects the wrong height root on an open doublet +set; S_E does not contain exactly one seed; the original unsheared density +(8) loses a Jacobian or type weight; the Gaussian majorant is unavailable +for the fixed source; or [ELDER]'s selected limit/normalization does not +match (1). Failure of [ELDER] invalidates §6, not the separately formed +microscopic implications of §§2–5. Example (12) falsifies the tempting +finite physical-minimum substitution, not (10). + +A qualified source-bound noncontributor review of this new composition is +still required. Same-provider review, arithmetic checks, hashes, publication, +CI and engineering merge do not establish organizationally independent +mathematical acceptance. No scientific Boolean, predecessor disposition or +prize is changed. The next unresolved issue for this candidate is review of +the exact elder-selected interface, not another count packet. diff --git a/frontiers/elder_selected_extremes_20260930/README.md b/frontiers/elder_selected_extremes_20260930/README.md new file mode 100644 index 000000000..c87b4f94e --- /dev/null +++ b/frontiers/elder_selected_extremes_20260930/README.md @@ -0,0 +1,18 @@ +# Elder-selected radius and lifetime + +[Full proof](PROOF.md): the higher additional saddle, selected before any +extreme limit, has microscopic tail coefficient `C0(I-J)` and an explicit +joint radius/lifetime-fraction law. On doublets its seed is the inner soft +root; a finite physical-radius counterexample is retained in proof §5. + +This is an author-side conditional candidate. The actual finite-r conclusion +uses PR175's exact fixed-d>=3 elder interface and only the sequential limit +`r->0` followed by `t->infinity`. No additional count estimate is asserted. + +- [Immutable full source cut](SOURCES.json), including the complete elder + dependency cut and all direct byte/blob identities. +- [Full noncontributor review](NONAUTHOR_REVIEW.md), with source exposure and + zero same-provider organizational-independence credit stated explicitly. + +Qualified source-bound composition review remains required. 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Required interfaces remain conditional on their displayed assumptions and actual review scope. Context sources are not mathematical premises." + } +} diff --git a/frontiers/remainder_rate_20260930/PROOF.md b/frontiers/remainder_rate_20260930/PROOF.md new file mode 100644 index 000000000..abfb98ae4 --- /dev/null +++ b/frontiers/remainder_rate_20260930/PROOF.md @@ -0,0 +1,378 @@ +# The remainder of Theorem R has a rate: `ν_eld(ℓ) = cℓ^{−1/3} + O(ℓ^{1/4})` + +Object: CL-D2-REMAINDER-RATE-20260930-v1.1 (v1 → v1.1, same day, before any nonauthor read: the exponent `1/6` of v1 +is raised to `1/4` by the sign-window majorant Lemma W, which the clean-context referee pass on v1 suggested as a +heuristic and which turns out to need only the sign structure of the typed weight and Lemma E's Steps 1–3; Lemma B, +Lemma F′ and the assembly are otherwise unchanged; v1's ledger and its `1/6` survive as a parenthetical in §4. A second +referee pass, on v1.1's §§3–5, found Lemma W proved and the assembly correct, and corrected the dependency +statements, Remark 1 and the term count of §4). +Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 30 September 2026. +Disposition: AUTHOR-SIDE PROOF CANDIDATE; NONAUTHOR REVIEW REQUIRED. Scientific effect: NONE — no register, +graph, STATUS, PROOF_INDEX, prize or Boolean change; no numerical constant is claimed. Same GitHub account as +every lane; zero organizational independence. **Dependency:** this note consumes Theorem R+ (R+.3), all of Lemma E +(Steps 1–5 through (E.1); Steps 1–3 alone for Lemma W (W.1), (W.3), (W.4)), and §2 ((2.1), the domination, the +`k = 0` identity) of Math- #191 (`frontiers/remainder_vanishing_20260930/PROOF.md`, v1.1, blob `441152df`; an unmerged +author-side candidate of this session with one nonauthor read, xAI 5369140697, findings applied). It cannot be integrated before #191 and +must be rebound if #191 changes. Everything else consumed is merged and reviewed: [R], [C7-K], [182], [Z] and the +reconciled [P] chain. + +## 0. Statement + +Setting and notation are those of [R] (`frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md`, blob `247b3ecf`) +and of #191 §0: fixed `d ≥ 2`, `L > 0`, the [P] field on `X = R^d/(LZ^d)`; the elder lifetime density `ν_eld(ℓ)` +(the manuscript's `ν_{3,24}` at `d = 3`, `L = 24`), the candidate density `ν_cand(ℓ)`, the rejected density +`ρ_rej = ν_cand − ν_eld`, all per unit volume ([C7-K] §0); the near pins `M = −ru/2`, `S = ru/2` at heights `b`, +`b − kr³`, the regression law `Q = Q_{r,b,k}`, the typed weight `W_r = |det H_M det H_S|1{H_M < 0, index H_S = d − 1}`, +`W_r/r² = F_d(K_M)F_{d−1}(K_S)` in the scaled Hessians of [R] §4, the pin density `π_r(v_r)`, `P = 1 + |b| + k`; the +elder mark `e = 1{d_f(M) = f(S)}` with the maximin `d_f(M) = sup{min_t f(γ(t)) : γ(0) = M, f(γ(1)) > f(M)}` ([P] §8; +Borel, [E2] §9); the leading coefficient `c = c_{d,L}` of [P] (15.2); superscripts `near,ρ` / `far,ρ` for the +separation ranges `dist(M, S) < ρ` / `≥ ρ` ([R] §7, [C7-K] §4); and the admissible radius +`r_0^* = min{r_0^{[R]}, L/(4√2), r_0^{[K]}, r_0^{[Z]}}` of #191 §0. `C, N` denote constants depending on `d, L` only, +changing from line to line. + +**Theorem R++ (rate for the elder remainder).** For the canonical marked Kac–Rice version of the elder density +([P] §9 read as [E2]; the version every input below concerns, and the pointwise representative the manuscript's +supplement C.4 fixes), there are `C < ∞` and `ℓ_0 ∈ (0, 1]` depending on `d, L` only such that + + |ν_eld(ℓ) − c ℓ^{−1/3}| ≤ C ℓ^{1/4}, 0 < ℓ ≤ ℓ_0; (0.1) + +equivalently + + ν_eld(ℓ) = c ℓ^{−1/3} (1 + O(ℓ^{7/12})) as ℓ ↓ 0. (0.2) + +The proof has three new estimates. Two are elementary consequences of the deterministic barrier of [182] §3 +(Lemmas B, F′); the third (Lemma W) is a majorant for the unmarked near-pair weight that improves [C7-K] (K1) at +small gap marks, using only the sign structure of the typed weight and the expansions of #191 Lemma E, Steps 1–3: + +**Lemma B (short elder bars at near separations are born at soft maxima).** For `0 < r ≤ r_0^{[R]}`, `b ∈ R`, +`k > 0`, with `ℓ = kr³`, + + E_Q[(W_r/r²) e] ≤ C ℓ^{1/3} r^{−1} (k + r) P^N, (B.1) + +and consequently, for `0 < ρ ≤ r_0 ≤ r_0^{[R]}`, the elder density at separations in `[ρ, r_0)` satisfies (an empty +range, and `0`, when `ρ = r_0`) + + ν_eld^{[ρ,r_0)}(ℓ) := ∫_ρ^{r_0}∫_R∫_{S^{d−1}} r^{−2} A_r^{eld}(b, ℓ/r³, u) dσ(u) db dr ≤ C ℓ^{1/3} (ρ^{−1} + ℓρ^{−5}), (B.2) + +where `A_r^{eld} := 12π_r(v_r)E_Q[(W_r/r²)e]` (the elder-marked version of [R]'s `A_r`; [C7-K] §4, [Z] (Z13)–(Z14)). + +**Lemma F′ (far elder density, crude rate).** For every fixed `r_0 ∈ (0, L/2]` there is `C = C(d, L, r_0)` with + + ν_eld^{far,r_0}(ℓ) ≤ C ℓ^{1/3}, 0 < ℓ ≤ 1. (F′.1) + +(Math- #187, unmerged, proves `Cℓ^{2/3}` by keeping the determinant weight through a spectral integration; (F′.1) +is the one-line version that needs no Hessian density, and it suffices here because the far term is not the +bottleneck of the ledger in §4.) + +**Lemma W (sign-window majorant for the near-pair weight).** For `0 < r ≤ r_0^{[R]}`, `b ∈ R`, `k ≥ 0` (the pinned +family at `k = 0` included: nothing below divides by `k`, and (R2)–(R5) do not depend on the target), + + E_Q[W_r/r²] ≤ C (k² + r⁴(1 + k)²) P^N. (W.1) + +(Compare [C7-K] (K1): `C(k + r)²P^N`. Since `k² + r⁴(1 + k)² ≤ 2(1 + r_0)²(k + r)²`, (W.1) is never worse; for `k < r` +it is better by the factor `k²/r² + r²`.) Consequently, for `0 < ρ ≤ r_0^*` and `0 < ℓ ≤ 1`: + + |ν_cand^{near,ρ}(ℓ) − cℓ^{−1/3}| ≤ C (min(ℓ^{1/4}, ρ) + ρ³ + ℓ²ρ^{−7}), (W.2) + 0 ≤ ρ_rej^{near,ρ}(ℓ) ≤ C (ℓ^{1/4} + ρ³), (W.3) + ∫_{0 < dist(0,y) < ρ} ∫_R Ψ_0(b, y) db dy ≤ C ρ³ (W.4) + +— (W.2) sharpens #191 (R+.3) (`C(ρ + ℓ²ρ^{−7})`) for `ρ ≳ ℓ^{1/4}`, (W.3) sharpens [C7-K] §4's near rejected bound +(`C(ℓ^{1/4} + ρ)`), and (W.4) says the equal-height kernel mass of [Z] (Z3) below separation `ρ` is `O(ρ³)`, sharpening +[Z] (Z18)'s `Cδ` for the limit object (the radial density `r^{d−1}Ψ_0(b, ru)` is `O(r²)`). (W.1), (W.3), (W.4) use only +Steps 1–3 of Lemma E; (W.2) uses (E.1), i.e. all of Lemma E. + +**What is not claimed.** No sharpness of the exponent `1/4` in either direction (§5 explains what fixes it — the +transition `r = k` at separation `ℓ^{1/4}` — and records, as unproved, a formal leading-order computation from the +second referee pass suggesting that `ℓ^{1/4}` is the true order, with a negative coefficient); no numerical `C`, `ℓ_0`; no rate for the *candidate* remainder (`ν_cand = cℓ^{−1/3} + B_{d,L} + o(1)` of +#191 stays without a rate: the far candidate kernel's dependence on `ℓ` is not quantified anywhere); no rate for +`ρ_rej(ℓ) → B_{d,L}` (Theorem Z); no uniformity in `d`, `L`; no finite-radius band; no RN / 24-jet closure; nothing +beyond the existential scope of Theorem R, Theorem R+ and [C7-K]. + +## 1. Proof of Lemma B + +Under `Q` the field is almost surely `C^∞` (`Q` is the law of `F_r` in the coupling [R] (R3); [E2] §9.1). Fix a +realisation `f` in the event `{W_r > 0} ∩ {e = 1}`. Then `∇f(M) = 0`, `f(M) = b`, `H_M = D²f(M) < 0`, and, by the +definition of the elder mark, `d_f(M) = f(S) = b − kr³ = b − ℓ`, i.e. `b − d_f(M) = ℓ`. This is exactly the +situation of [182] §3, whose display (9) reads, for any `C³` field with `f(M) = b`, `∇f(M) = 0`, `H_M < 0`, +`λ := λ_min(−H_M)`, `K := 1 + ‖D²f‖_{∞,op} + ‖D³f‖_{∞,op}` (suprema over the torus), `a_L = min(1, L/4)`, +`c_L = a_L²/3`: + + b − d_f(M) ≥ c_L λ³ / K². (1.1) + +(Its proof, [182] §3 (8): for a unit vector `v` and `0 < t ≤ R := a_Lλ/K`, Taylor's theorem gives +`f(M + tv) ≤ b − (λ/2)t² + (K/6)t³ ≤ b − (λ/3)t²`, since `t ≤ λ/K`; the ball `B(M, R)` is embedded (`R ≤ L/4`), no +point of it lies above `b`, every path from `M` to a point above `b` crosses its boundary sphere, on which +`f ≤ b − λR²/3`; hence `d_f(M) ≤ b − λR²/3 = b − c_Lλ³/K²`.) So on the elder event + + λ_min(−H_M) ≤ (ℓ K² / c_L)^{1/3}. (1.2) + +Since `−H_M > 0` with eigenvalues `λ = λ_1 ≤ … ≤ λ_d ≤ ‖H_M‖_op ≤ K`, + + |det H_M| = ∏_j λ_j ≤ λ_1 K^{d−1} ≤ c_L^{−1/3} ℓ^{1/3} K^{d − 1/3}. (1.3) + +By the scaling dictionary of [R] §4 (`|det H_i|/r = |det K_i|`), `W_r/r² = (|det H_M|/r)·F_{d−1}(K_S)` on the support +of `W_r`, and [R] §4 bounds the typed scaled determinant at `S` through the exact block identity of Lemma R3.2 with +(R8)–(R9) — quoted: "The exact block determinant also gives `F_j(K_i) <= C(k+r) T^d`" — where `T ≥ 1 + k` is [R]'s +dominating variable with `‖T‖_p ≤ C_pP` for every `p` ((R9), from (R4)). Therefore, on `{W_r > 0} ∩ {e = 1}`, + + W_r/r² ≤ C ℓ^{1/3} r^{−1} (k + r) K^{d − 1/3} T^d, (1.4) + +and `(W_r/r²)e ≤` the right side of (1.4) everywhere (`e ∈ {0, 1}`, `W_r ≥ 0`). Taking `E_Q`: `K ≤ C_d(1 + ‖F_r‖_{C³})` +has `‖K‖_p ≤ C_pP` by [R] (R4) with `q = 3` ("for finite p>=1 and each finite derivative order q, +`||1+||F_r||_{C^q}+||F_0||_{C^q}||_p <= C_{p,q} P`"), so Hölder gives `E_Q[K^{d−1/3}T^d] ≤ CP^N`, which is (B.1). + +For (B.2): by [R] (R5), `π_r(v_r) ≤ Ce^{−c(b²+k²)}` for `r ≤ r_0^{[R]}`, so with `k = ℓ/r³` + + r^{−2} A_r^{eld}(b, ℓ/r³, u) ≤ C ℓ^{1/3} r^{−3} (ℓr^{−3} + r) P^N e^{−c(b²+k²)} ≤ C ℓ^{1/3} (ℓ r^{−6} + r^{−2}) (1 + |b|)^N e^{−cb²}, + +using `(1 + |b| + k)^N ≤ 2^N(1 + |b|)^N(1 + k)^N` and `(1 + k)^Ne^{−ck²} ≤ C`. Integrating over `S^{d−1}`, `b ∈ R` and +`r ∈ [ρ, r_0)`: + + ∫_ρ^{r_0} (ℓ r^{−6} + r^{−2}) dr = ℓ(ρ^{−5} − r_0^{−5})/5 + (ρ^{−1} − r_0^{−1}) ≤ ℓρ^{−5}/5 + ρ^{−1}, + +which is (B.2). ∎ + +The identity behind (B.2) — that the elder density at lifetime `ℓ` from separations in `[ρ, r_0)` is +`∫_ρ^{r_0}∫∫ r^{−2}A_r^{eld}(b, ℓ/r³, u)` — is the marked Kac–Rice disintegration of [P] §§9–10 read as [E2]: [P] +(10.2) gives the elder intensity `r A_r p_r dr db dk dσ` per unit volume with `A_r p_r = 12π_r(v_r)E_Q[(W_r/r²)e]`; +pushing forward at fixed `r` with `dk = dℓ/r³` gives the `ℓ`-density `r^{−2}A_r^{eld} dr db dσ`. In the `k` variable +this is [C7-K] §4's rejected intensity identity with the mark `e` in place of `1 − e`, and in the `r` variable it is +[Z] (Z13)–(Z14) with `e` in place of `1 − e`; for the unmarked weight the same display is [R] §6 and #191 (2.1). + +## 2. Proof of Lemma F′ + +Write `D_{r_0} = {y ∈ X : dist(0, y) ≥ r_0}` (compact) and, as in [Z] §1 and [C7-K] §4, `O_y = (f(0), ∇f(0), f(y), ∇f(y))`, +`v_{b,ℓ} = (b, 0, b − ℓ, 0)`, `p_y` the density of `O_y`, `Q_{y,b,ℓ}` the regression law given `O_y = v_{b,ℓ}`, +`W = |det H_0 det H_y|1{H_0 < 0, index H_y = d − 1}`, `e_y = 1{d_f(0) = f(y)}`. The far elder density is the marked +Kac–Rice integral on this domain ([P] §9 as [E2], §14: the height Jacobian is `1` there), + + ν_eld^{far,r_0}(ℓ) = ∫_{D_{r_0}} ∫_R p_y(v_{b,ℓ}) E_{Q_{y,b,ℓ}}[W e_y] db dy. (2.1) + +*Uniform Gaussian facts on `D_{r_0}`.* `Γ_y := Cov(O_y)` is positive definite for every `y ≠ 0` ([P] §2, distinct-site +rank) and continuous in `y`, hence `λ_min(Γ_y) ≥ c_0 > 0` on the compact `D_{r_0}` — the compactness argument of +[P] §14, used again in [Z] (Z16). Consequently: + +(a) `p_y(v) ≤ (2πc_0)^{−(d+1)} exp(−|v|²/(2 tr Γ_y)) ≤ Ce^{−cb²}` at `v = v_{b,ℓ}` (`|v_{b,ℓ}|² ≥ b²`, `tr Γ_y ≤ C`); + +(b) (Gaussian regression at two fixed sites, [P] §4's argument; [Z] §2, (Z10).) With +`ψ^y := Γ_y^{−1}Cov(O_y, f(·))` — `2d + 2` smooth functions on `X` with `‖ψ^y_j‖_{C³} ≤ C` uniformly on `D_{r_0}` +(covariance derivatives are bounded, `‖Γ_y^{−1}‖ ≤ 1/c_0`) — the field decomposes as `f = Σ_j O_y^j ψ^y_j + g_y` +with `g_y` a centred Gaussian field independent of `O_y`; under `Q_{y,b,ℓ}` the field has the law of +`Σ_j v_{b,ℓ}^j ψ^y_j + g_y`. Since `‖g_y‖_{C³} ≤ ‖f‖_{C³} + C|O_y|` and `E‖f‖_{C³}^p + E|O_y|^p ≤ C_p` ([P] §2, quoted: +"the field has a smooth version and all finite moments of its global derivative suprema exist"), `E‖g_y‖_{C³}^p ≤ C_p` +uniformly on `D_{r_0}`. Hence, for +`0 < ℓ ≤ 1` (`|v_{b,ℓ}| ≤ 2|b| + 1`), `K = 1 + ‖D²f‖_{∞,op} + ‖D³f‖_{∞,op} ≤ C(1 + |b| + ‖g_y‖_{C³})` and + + E_{Q_{y,b,ℓ}}[K^p] ≤ C_p (1 + |b|)^p for every p ≥ 1, uniformly for y ∈ D_{r_0}, 0 < ℓ ≤ 1. (2.2) + +*The barrier.* On `{W > 0} ∩ {e_y = 1}` the point `0` is a nondegenerate maximum with `f(0) = b`, and +`d_f(0) = f(y) = b − ℓ` by the definition of the mark; [182] §3 (9), i.e. (1.1) with `M = 0`, gives +`λ_min(−H_0) ≤ (ℓK²/c_L)^{1/3}`, hence `|det H_0| ≤ c_L^{−1/3}ℓ^{1/3}K^{d−1/3}` as in (1.3), while `|det H_y| ≤ ‖H_y‖^d ≤ K^d`. +So `W e_y ≤ Cℓ^{1/3}K^{2d−1/3}` and, by (2.2), `E_{Q_{y,b,ℓ}}[W e_y] ≤ Cℓ^{1/3}(1 + |b|)^{2d}`. With (a) in (2.1), + + ν_eld^{far,r_0}(ℓ) ≤ C ℓ^{1/3} · vol(D_{r_0}) · ∫_R (1 + |b|)^{2d} e^{−cb²} db ≤ C ℓ^{1/3}. ∎ + +## 3. Proof of Lemma W and of (W.2)–(W.4) + +*The structure of the two scaled determinants.* Work under the coupling (R3) of [R] as in #191 §1 (`f = F_r`, `T` the +dominating variable with `‖T‖_p ≤ C_pP`, `A_0 = D_y²F_0(0)`, `Δ := det A_0`). #191 Lemma E, Steps 1–3, give — pathwise, +for `0 < r ≤ r_0^{[R]}` — the expansions #191-(1.1), #191-(1.3), #191-(1.5) and hence, multiplying out +`det K_i = α_i det A_i − rq_i` (Lemma R3.2 of [R]), + + det K_M = −6kΔ + rY + R_M, det K_S = +6kΔ + rY + R_S, |R_M| + |R_S| ≤ C r² (1 + k) T^{N_1}, (3.1) + +with the *same* `Y = (f_4/12)Δ + 3k tr(adj(A_0)B) − q` (#191 §1, the display after (1.6): "`det K_M = −6k det A_0 + rY + +O(r²)` and `det K_S = +6k det A_0 + rY + O(r²)` with the same `Y`"). The `(1 + k)`-quantified remainder is derived here, +not quoted: with `τ := tr(adj(A_0)B)`, `det A_i = Δ ∓ (r/2)τ + E′_i`, `|E′_i| ≤ Cr²T^m` (from #191-(1.3)), `α_i = ∓6k + +(r/12)f_4 + ρ_i`, `|ρ_i| ≤ Cr²T` (#191-(1.1)), `q_i = q + O(rT^{m+1})` (#191-(1.5)), multiplying out gives + + R_M = −6kE′_M − (r²/24)f_4τ + (r/12)f_4E′_M + ρ_M det A_M − r(q_M − q), + R_S = +6kE′_S + (r²/24)f_4τ + (r/12)f_4E′_S + ρ_S det A_S − r(q_S − q), + +each term `≤ Cr²(1 + k)T^{N_1}` (the `k`-linear term is `6kE′_i`; the `O(r)` error of `q_i` enters as `r·O(rT^{m+1})`); +the `r`-linear terms — `(r/12)f_4Δ` from `α_i det A_i`, `+3krτ` from `(∓6k)(∓(r/2)τ)`, and `−rq` — are the same at +both endpoints, which is (3.1). + +*The sign window.* On the support of `W_r`, `index K_M = d` and `index K_S = d − 1` (`K_i` is congruent to `H_i`), so +`det K_M` has the sign `(−1)^d` and `det K_S` the sign `(−1)^{d−1}`: `det K_M · det K_S < 0`. Put `w := rY`. By (3.1) +the product is `(w − 6kΔ + R_M)(w + 6kΔ + R_S)`, negative only when `w` lies strictly between the two roots +`6kΔ − R_M` and `−6kΔ − R_S`; hence + + |w| ≤ 6k|Δ| + max(|R_M|, |R_S|), |det K_i| ≤ |w| + 6k|Δ| + |R_i| ≤ 12k|Δ| + 2 max(|R_M|, |R_S|) (i = M, S), + +and therefore, on the support of `W_r`, + + W_r/r² = |det K_M det K_S| ≤ (12k|Δ| + 2Cr²(1 + k)T^{N_1})² ≤ C (k²Δ² + r⁴(1 + k)²T^{2N_1}). (3.2) + +Off the support `W_r = 0`. Taking expectations: `E[Δ²] ≤ E[T^{2m}] ≤ CP^{2m}` and `E[T^{2N_1}] ≤ CP^{2N_1}` (`|Δ| ≤ ‖A_0‖^m ≤ +T^m`; (R4) bounds the moments of `F_0` as well as of `F_r`, and `W_r` is a functional of `F_r` alone, whose marginal on +the coupling space is `Q`, so a bound that holds pathwise on the coupling space bounds `E_Q`), which is (W.1). ∎ + +*Proof of (W.2).* Put `r_1 := ℓ^{1/4}` (so `r ≤ r_1 ⇔ k = ℓ/r³ ≥ r`) and use #191 (2.1): `ν_cand^{near,ρ} − cℓ^{−1/3} = +∫_0^ρ∫∫ r^{−2}(A_r − A_0)(b, ℓ/r³, u) − ∫_ρ^∞∫∫ r^{−2}A_0`. On `0 < r < min(r_1, ρ)`: #191 Lemma E (E.1) — Steps 1–5 — +`|r^{−2}(A_r − A_0)| ≤ H(b, k) ≤ Ce^{−cb²}`, contributes at most `C min(r_1, ρ)`. On `r_1 ≤ r < ρ` (empty if `ρ ≤ r_1`): +`|A_r − A_0| ≤ A_r + A_0` with `A_r = 12π_r(v_r)E_Q[W_r/r²] ≤ C(k² + r⁴(1 + k)²)P^Ne^{−c(b²+k²)}` by (W.1) and (R5), and +`A_0 ≤ k²H` (#191 §2); so `r^{−2}|A_r − A_0| ≤ C(ℓ²r^{−8} + r²)(1 + |b|)^Ne^{−cb²}` (the factor `(1 + k)^{N}e^{−ck²}` is +bounded), which integrates to `C(ℓ²r_1^{−7}/7 + ρ³/3) = C(ℓ^{1/4}/7 + ρ³/3)`. The tail is `≤ Cℓ²ρ^{−7}` (#191 §2). ∎ + +*Proof of (W.3).* `ρ_rej^{near,ρ} = ∫_0^ρ∫∫ r^{−2}A_r^{rej}` with `A_r^{rej} = 12π_r(v_r)E_Q[(W_r/r²)(1 − e)]` ([C7-K] §4, [Z] +(Z13)). On `0 < r ≤ min(r_1, ρ)`: `1 − e ≤ 1_{G_r^c}` on the generic locus ([C7-K] §4) and (K2) (valid as `r ≤ k` and +`r ≤ ρ ≤ r_0^{[K]}`) give `r^{−2}A_r^{rej} ≤ C r^{−2}(r³/k)P^Ne^{−c(b²+k²)} = C(r⁴/ℓ)(…)`, which integrates to +`≤ Cr_1⁵/(5ℓ) = Cℓ^{1/4}/5` — this is [C7-K] §4's first piece in the `r` variable. On `r_1 < r ≤ ρ`: `1 − e ≤ 1` and (W.1) +as in the proof of (W.2) give `≤ C(ℓ^{1/4} + ρ³)`. ∎ + +*Proof of (W.4).* By #191 §2 (the pointwise limit), `r^{d−1}Ψ_0(b, ru) = r^{−2}A_r(b, 0, u)` ((Z2) at `y = ru`, the `k = 0` +member of the pinned family: the pin Jacobian `12r^{−(d+3)}`, the polar factor `r^{d−1}`, the determinant scale `r²`), +and (W.1) at `k = 0` with (R5) gives `A_r(b, 0, u) ≤ Cr⁴(1 + |b|)^Ne^{−cb²}`; so +`∫_{0 0`, [P] (15.2)) gives (0.2). ∎ + +(With the v1 inputs — (R+.3) and [C7-K] §4 in place of (W.2)–(W.3) — the same assembly at `ρ = ℓ^{1/6}` gives the +ledger `ℓ^{1/6} + ℓ^{5/6} + ℓ^{1/4} + ℓ^{1/6} + ℓ^{1/6} + ℓ^{1/2} + ℓ^{1/3}`, i.e. the rate `ℓ^{1/6}` of v1; checker E1 keeps +both ledgers.) + +## 5. Remarks + +1. **What fixes the exponent, and whether it is sharp.** In (4.1) with (W.2)–(W.3) the leading `ρ`-dependence is `ρ³` + (the candidate excess and the rejected mass at separations in `(ℓ^{1/4}, ρ)`, both `O(r²)` per unit separation by + Lemma W) against `ℓ^{1/3}ρ^{−1}` (Lemma B above `ρ`); the balance is `ρ = ℓ^{1/12}`, where both equal `ℓ^{1/4}`. But + `ℓ^{1/4}` also appears three times *independently of `ρ`*, all from the transition `r = k` at separation `ℓ^{1/4}` + where the pinned pair stops being a "cap" pair: the (E.1) excess `C min(ℓ^{1/4}, ρ)` below `ℓ^{1/4}`, the (K2) loss at + `r ≤ k` ([C7-K] §4's first piece), and the tail `∫_{r > ℓ^{1/4}} r^{−2}k²` of the leading law `A_0 ≍ k²`. So within this + ledger `1/4` is optimal (checker E1), trivially: the ledger has `ρ`-free `ℓ^{1/4}` terms. Whether `1/4` is the true + order is **open**; the second referee pass on this note gave a formal leading-order computation (not proved, + recorded here as a route) indicating that it is, with a *negative* coefficient: at `k = κr` (the cusp scale + `r ≍ k ≍ ℓ^{1/4}`), (3.1) gives `det K_{M,S} = r(Y ∓ 6κΔ) + O(r²)`, so to leading order the typed event is + `{A_0 < 0, |Y| < 6κ|Δ|}` (verified exactly on 160 pinned sextics in `d = 2` by the referee) and on it + `W_r/r² = r²(36κ²Δ² − Y²) + O(r³) < 36k²Δ²`: the candidate kernel sits strictly *below* the contact kernel throughout + the cusp scale, and integrating over `s = rℓ^{−1/4}` gives a contribution `I^{cand}ℓ^{1/4}` with + `I^{cand} = −(96/7)·6^{1/4}∫∫ π_0(v_0(b, 0)) E_0[|Y_0|^{7/4}|det A_0|^{1/4}1{A_0 < 0}] dσ db < 0`, while the rejected part + at `κ ≍ 1` is `O(1)` so `I^{eld} ≤ I^{cand}`. Unless elder mass at separations `ℓ^{1/4} ≪ r ≲ ℓ^{1/12}` (which the ledger + only bounds by `ℓ^{1/4}`) compensates at that order, `ν_eld(ℓ) = cℓ^{−1/3} + I^{eld}ℓ^{1/4} + o(ℓ^{1/4})` with + `I^{eld} < 0` would be the next term — the contact law *overcounts* at `k < r`, where `|A_r − A_0| ≍ k²` + (Lemma W's window makes `A_r = o(k²) + O(r⁴)` as `(r, k/r) → 0`, against `A_0 ≥ c(b)k²`), so no `k < r` refinement of + Lemma E of the form `O(k²r² + r⁴)` can exist, and the "k²-tail" is not an artifact of the bounds. Nothing about + this is claimed. What *would* sharpen Lemma B (the spectral refinement of Math- #187 §3 in the near regime, which + needs a lower bound on the conditional variance of the axial curvature under the pins — a routine extension of + [P] §3 / [R] (R13) with `f_xxxx(0)` appended to `U_r`) would not touch the three `ρ`-free `ℓ^{1/4}` terms. Note + finally that at separations `s ≲ ℓ^{1/4}` the elder intensity is *not* small — it carries the leading law + `cℓ^{−1/3}` itself; and that (0.1) does not use Theorem Z (Z4), so (0.1) with (R+.1) re-derives (Z4) independently. +2. **Consistency.** (0.1) implies (R+.2) of #191 (`ν_eld = cℓ^{−1/3} + o(1)`) and is consistent with Theorem R + ((R1): `|ν_eld − cℓ^{−1/3}| ≤ C`), with [C7-K] (T1)–(T2) (`ρ_rej = Θ(1)`: `ν_cand − cℓ^{−1/3} = ρ_rej + O(ℓ^{1/4})` is + `Θ(1)`), with Theorem Z (Z4) and (R+.1) (`ν_cand − cℓ^{−1/3} → B_{d,L}`), with [Z] (Z18) (rejected mass below + separation `δ` is `≤ Cδ`; (W.3)–(W.4) give `C(ℓ^{1/4} + δ³)` and `Cδ³` for the limit), and with #187 (Theorem F, + `ℓ^{2/3}` for the far elder density, sharper than (F′.1)) and #188 (`ℓ^N`). It supersedes #191 Remark 2's conditional + route: no far-elder rate with a polynomial `ρ`-dependence is needed, because the near-regime Lemma B (whose + constants are those of [R] (R4)–(R5), uniform in `r ≤ r_0^{[R]}`) covers the separations down to `ρ = ℓ^{1/12}`, + and the far part is taken at the fixed radius `r_0^*`. (W.4) also says that the equal-height kernel's radial + density `r^{d−1}Ψ_0(b, ru)` is `O(r²)` near the diagonal, which supersedes #191 Remark 1's "`O(1)` per unit + separation" there; [Z] itself proves only that there is no diagonal atom and (Z18)'s `Cδ`, and the picture of + `B_{d,L}` as a macroscopic-separation mass is [C7-K] §4's remark, with which (W.4) agrees. +3. **For the manuscript.** Once #191 (Lemma E) and this note both have nonauthor ACCEPTs, Abstract / Theorem 2.1 / + §3.8 can state, for the pointwise Kac–Rice representative that supplement C.4 fixes, + `ν_{3,24}(ℓ) = c_{3,24}ℓ^{−1/3}(1 + O(ℓ^{7/12}))`, i.e. `ν_{3,24}(ℓ) = c_{3,24}ℓ^{−1/3} + O(ℓ^{1/4})`, in place of + `(1 + o(1))`. Until then E10 of the V3 edit pack (`(1 + O(ℓ^{1/3}))`, resting on Theorem R with two reads, `[R·2P]`) + is the safe statement, and E12 (`(1 + o(ℓ^{1/3}))`, #191 at candidate status) would be strengthened by this + note, also at candidate status. The candidate density keeps `c_{3,24}ℓ^{−1/3} + B_{3,24} + o(1)` without a rate. +4. **Inputs by status.** Merged and reviewed: [R] (xAI R1–R6; Anthropic full depth #186 integrated), [C7-K] (one + reviewing provider), [Z] (Anthropic A/B/C reads), [P]/[E1]/[E2]/[REC]. Merged with a caveat: [182] (OpenAI; + integrated `2a10ed3`) — its §3 barrier was accepted at Slice A by this same author and session and by no other + provider, so reviewers should treat §1's re-proof of (1.1) as new material. Unmerged, consumed: #191 — (R+.3), + all of Lemma E ((E.1) in (W.2); Steps 1–3 in (3.1)), §2 ((2.1), the domination, the `k = 0` identity) — this + note stands or falls with Lemma E of #191 (one nonauthor read, xAI, findings applied, no ACCEPT yet). Unmerged, + cited only: #187, #188. + +## 6. Sources (exact identities in `SOURCES.json`) + +| Tag | Path | Role | +|---|---|---| +| [R] | `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md` (blob `247b3ecf`) | §§2, 4, 6–7: (R3)–(R5), the scaling dictionary and the block determinant bound `F_j(K_i) ≤ C(k + r)T^d`, (R8)–(R9), the near/far decomposition — consumed | +| [C7-K] | `frontiers/c7_total_bounded_20260929/PROOF.md` (blob `28748b08`) | (K1), (K2), §4 (the rejected intensity identity; the proof of (T1) with cutoff `ρ`) — consumed | +| [182] | `frontiers/fixed_r_inverse_lifetime_20260930/PROOF.md` (blob `0d401877`) | §3 (8)–(9), the deterministic barrier — consumed (re-proved in §1) | +| [Z] | `frontiers/c7_zero_gap_limit_20260929/PROOF.md` (blob `5b6328ea`) | §1 notation, §2 (Z10) regression at fixed sites, (Z13)–(Z14), (Z16) — consumed | +| [P] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (blob `dfed3b8d`) | §2 distinct-site rank, §4 regression, §8 maximin and elder mark, §§9–10 as [E2], §14, (15.2) — consumed | +| [E1] | `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md` (blob `213594d6`) | reading rule | +| [E2] | `reviews/d1_section9_borel_repair_20260925/REPAIR.md` (blob `fe9b9ce4`) | §9 replacement: regular conditional law, Borel marks, (9.1) | +| [REC] | `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md` (blob `75da2597`) | reading rule, consumption contract | +| #191 | `frontiers/remainder_vanishing_20260930/PROOF.md` (unmerged; v1.1 blob `441152df`, head `abae6d0`; the workflow verifies this blob — from the tree once landed, else by blob id — and fails on drift) | Theorem R+ (R+.3) and its §2 (domination, pointwise limit, (2.1)); Lemma E ((E.1) and Steps 1–3: (1.1), (1.3), (1.5), the structural display after (1.6)); the admissible radius `r_0^*` — **consumed, unmerged** | +| #187 | `frontiers/far_elder_rate_20260930/PROOF.md` (unmerged) | Theorem F (`ℓ^{2/3}` far rate, sharper than (F′.1)) and §4 — cited only (Lemma F′ and Remarks 1–2) | +| #188 | `frontiers/far_elder_flat_ridge_20260930/PROOF.md` (unmerged) | `ℓ^N` far rate — cited only (Remark 2) | + +## 7. Exact controls (`rate_ledger_check.py`; stdlib; exact rationals; byte-identical under `-O`) + +E1 the exponent ledgers of §4: with `ρ = ℓ^θ` the v1.1 terms have exponents `1/4, 3θ, 2 − 7θ, 1/4, 3θ, 1/3 − θ, +4/3 − 5θ, 1/3` and the v1 terms `θ, 2 − 7θ, 1/4, θ, 1/3 − θ, 4/3 − 5θ, 1/3`; at `θ = 1/12` (v1.1) and `θ = 1/6` (v1) the +minima are `1/4` and `1/6`; over a fine rational grid of `θ ∈ (0, 1/3)` and at every breakpoint neither minimum exceeds +its value (optimality within each ledger); and at rational twelfth powers `ℓ = q^{12}` each term is `≤ ℓ^{1/4}` +(resp. `ℓ^{1/6}`) exactly. Mutant M1 (claims the rate `1/3` for v1.1) fails: the ledger's minimum is `1/4` at every `θ`. +E2 the barrier arithmetic of [182] (8)–(9): for rational `0 < λ ≤ K` and `t ≤ a_Lλ/K`, `(K/6)t³ ≤ (λ/6)t²` and +`b − (λ/2)t² + (K/6)t³ ≤ b − (λ/3)t²` exactly; `b − d ≥ c_Lλ³/K² ⇔ λ ≤ ((b − d)K²/c_L)^{1/3}` at rational cubes. +Mutant M2 (radius `2λ/K`) fails. +E3 the determinant/soft-eigenvalue inequality (1.3): for rational symmetric positive definite matrices +`O diag(λ)Oᵀ` with rational orthogonal `O` (Cayley transforms of rational skew matrices), `det ≤ λ_min λ_max^{d−1}` +and `λ_max ≤ ‖·‖_F`; mutant M3 (exponent `d − 2`) fails. +E4 the integrals and the cutoff ledger: `∫_ρ^{r_0}(ℓr^{−6} + r^{−2})dr = ℓ(ρ^{−5} − r_0^{−5})/5 + ρ^{−1} − r_0^{−1} ≤ ℓρ^{−5}/5 + ρ^{−1}` +exactly at rationals; [C7-K] §4's second piece `ℓ^{−1/3}·ℓ^{2/3}·3(ℓ/ρ³)^{−1/3} = 3ρ` and first piece +`ℓ^{−1/3}(3/5)ℓk_*^{−5/3} = (3/5)ℓ^{1/4}` at `k_* = ℓ^{1/4}`, at rational twelfth powers; the (W.2)–(W.3) integrals +`∫_{r_1}^ρ(ℓ²r^{−8} + r²)dr ≤ ℓ²r_1^{−7}/7 + ρ³/3` and `∫_0^{r_1} r⁴/ℓ dr = r_1⁵/(5ℓ) = ℓ^{1/4}/5` at `r_1 = ℓ^{1/4}`. +E5 the sign window of Lemma W: for rational `(k, D, Y, R_M, R_S, r)` on a grid including `k = 0`, whenever +`(rY − 6kD + R_M)(rY + 6kD + R_S) < 0` (the typed sign pattern) one has `|rY| ≤ 6k|D| + max(|R_M|, |R_S|)` and both +`|det K_i| ≤ 12k|D| + 2 max(|R_M|, |R_S|)`, exactly; the bound is sharp (approached as `rY → ±6kD` with `R_M = R_S = 0`, +where the product tends to `0` from below); and it is *not* implied without the sign condition (same-sign instances +violate it). Mutant M4 (drops the factor `2` on the remainder) fails. +Mutants `M1`–`M4` exit 1; an unknown label exits 2. + +**What the controls do not test.** They check exponent bookkeeping, the barrier's algebra, a linear-algebra +inequality and the sign-window inequality at rational points. They do not test Lemma B, Lemma F′ or Lemma W as +statements about the Gaussian laws (the expansions (3.1) imported from #191, the moment imports (R4)/(R9)/(2.2), the +density bound (R5), the Kac–Rice disintegrations) nor the assembly's use of #191 and [C7-K] — those are prose only, +and they are the review. The controls are not acceptance. + +## 8. Review slices + +A: Lemma B — the use of the elder mark's definition and of [182] (9), the determinant bound (1.3), the block bound + for `K_S`, the moment imports, and (B.2)'s integration; the identity behind (B.2). +B: Lemma F′ — the uniform far facts (a), (b), (2.2), and the barrier at `0`. +C: Lemma W and (W.2)–(W.4) — the expansions (3.1) against #191 Lemma E Steps 1–3 (in particular the remainder bound + `Cr²(1 + k)T^{N_1}`), the sign window, the moments, and the three integrations. +D: §4 — the decomposition (4.1), the four inputs with their exact provenance, the choice `ρ = ℓ^{1/12}`, the scope + and Remarks 1–4. diff --git a/frontiers/remainder_rate_20260930/README.md b/frontiers/remainder_rate_20260930/README.md new file mode 100644 index 000000000..4758c9144 --- /dev/null +++ b/frontiers/remainder_rate_20260930/README.md @@ -0,0 +1,61 @@ +# The remainder of Theorem R has a rate: `ν_eld(ℓ) = cℓ^{−1/3} + O(ℓ^{1/4})` (CL-D2-REMAINDER-RATE-20260930-v1.1) + +Author-side proof candidate, Anthropic Claude, 30 September 2026. Scientific effect NONE. Nonauthor review required. +**Depends on Math- #191** (Theorem R+ (R+.3), all of Lemma E through (E.1), and its §2; unmerged author-side candidate +of this session, v1.1 blob `441152df`): +this packet cannot be integrated before #191 and must be rebound if #191 changes. Everything else consumed is +merged and reviewed: [R] Theorem R, [C7-K], [182] (the deterministic barrier), [Z], the reconciled [P] chain. + +**Statement (Theorem R++).** For the periodized Gaussian field on the fixed torus (every `d ≥ 2`, `L > 0`) and the +canonical marked Kac–Rice version of the elder lifetime density, there are `C, ℓ_0` depending on `d, L` only with + + |ν_eld(ℓ) − cℓ^{−1/3}| ≤ C ℓ^{1/4} (0 < ℓ ≤ ℓ_0), i.e. ν_eld(ℓ) = cℓ^{−1/3}(1 + O(ℓ^{7/12})). + +Theorem R (D2) gave `O(1)`; #191 gave `o(1)` with a rate only conditionally (on unmerged far-elder packets and on +a polynomial `ρ`-dependence of their constants). This packet gives the rate unconditionally, modulo #191's Lemma E. + +**Mechanism.** Split the elder density by separation at `ρ = ℓ^{1/12}` and at the fixed radius `r_0^*`. Below `ρ`: +**Lemma W** (sign window) — on the support of the typed weight the two scaled endpoint determinants have opposite +signs, and by #191 Lemma E Steps 1–3 they are `∓6k det A_0 + rY + O(r²)` with the *same* `rY`, so `|rY| ≤ 6k|det A_0| ++ O(r²)` and `E_Q[W_r/r²] ≤ C(k² + r⁴(1 + k)²)P^N` — improving [C7-K] (K1)'s `(k + r)²` at `k < r`; with it the +candidate excess over the contact law below `ρ` is `C(ℓ^{1/4} + ρ³ + ℓ²ρ^{−7})` (sharpening #191 (R+.3)) and the +rejected mass below `ρ` is `C(ℓ^{1/4} + ρ³)` (sharpening [C7-K] §4), and the equal-height kernel mass of Theorem Z +below separation `ρ` is `O(ρ³)`. Between `ρ` and `r_0^*`: **Lemma B** +— every elder pair of lifetime `ℓ` is born at a maximum whose weakest curvature is `≤ (ℓK²/c_L)^{1/3}` ([182] §3's +deterministic barrier, applied under the near-pair regression law of [R] whose moment and density bounds +(R4), (R5), (R9) are uniform in `r ≤ r_0^{[R]}`), so the typed weight is `≤ Cℓ^{1/3}r^{−1}(k + r)K^{d−1/3}T^d` and the +elder mass at separations in `[ρ, r_0^*)` is `≤ Cℓ^{1/3}(ρ^{−1} + ℓρ^{−5})`. Beyond `r_0^*`: **Lemma F′**, the same barrier +under the far regression law of [P] §14 / [Z] §2, `≤ Cℓ^{1/3}` (Math- #187 has `ℓ^{2/3}`; not needed). The ledger at +`ρ = ℓ^{1/12}` has eight terms, five of them `ℓ^{1/4}`, the others `ℓ^{17/12}`, `ℓ^{11/12}`, `ℓ^{1/3}`; `1/4` is now fixed by +the transition `r = k` at separation `ℓ^{1/4}` (the (E.1) excess below it, (K2)'s loss, and the tail of the leading +law), independently of `ρ`. + +v1 → v1.1 (same day, before any nonauthor read): v1 had the rate `1/6` from (R+.3) and [C7-K] §4 directly (ledger at +`ρ = ℓ^{1/6}`); the clean-context referee pass on v1 suggested the sign window as a heuristic; Lemma W is its rigorous +form and needs only the sign structure of the typed weight plus Lemma E's Steps 1–3 (the assembly still needs (E.1) +below separation `ℓ^{1/4}`). v1's ledger survives as a parenthetical in §4 and in checker E1. A second referee pass +on v1.1's §§3–5 found Lemma W proved and the assembly correct; its findings (the dependency on all of Lemma E, the +eight-term count, `k ≥ 0` in Lemma W, the `min(ℓ^{1/4}, ρ)` form of (W.2), notation, and the rewriting of Remark 1 — +its own formal computation replaces v1.1's wrong "not expected to be the true order") were applied. + +**Not claimed:** sharpness of `1/4` in either direction — Remark 1 records, as unproved, the second referee pass's +formal leading-order computation indicating that `ℓ^{1/4}` is the true order with a *negative* coefficient (the +contact law overcounts at the cusp scale `r ≍ k ≍ ℓ^{1/4}`); numerical constants; a rate for the candidate remainder +or for `ρ_rej → B_{d,L}`; uniformity in `d, L`. + +Files: `PROOF.md`; `rate_ledger_check.py` (stdlib exact controls; `RESULTS.json` its output, `-O` identical; mutants +`M1`–`M4` exit 1, unknown label exit 2: both exponent ledgers and their optimality within the ledger, the barrier's +algebra, the soft-eigenvalue determinant inequality on rational orthogonal conjugates, the integrals and the +[C7-K] cutoff ledger, the sign-window inequality of Lemma W); `SOURCES.json` (exact identities of every merged source; #191 recorded as consumed-unmerged +with its blob — the workflow verifies that blob too, from the tree once #191 has landed or through the repository's +blob API before, and fails on absence or drift; #187/#188 as cited-unmerged). A clean-context referee agent (same provider; not a nonauthor +review) read a first version before landing: no error found in Lemma B, Lemma F′, the decomposition or the ledger; +it confirmed Lemma E's Steps 1–3 symbolically on pinned quintics in `d = 2, 3`; its thirteen findings (the +canonical-version wording of the statement, a false sentence and an overclaim in Remark 1, Remark 3's gating, +citation precision for [R] §4 / [Z] (Z14) / [P] §2, the range `[r_0^*, diam X]`, `ρ ≤ r_0^*`, the explicit uniformity of +(R+.3)'s constant, the provenance of [182]'s barrier review) were applied. + +**What the controls do not test.** Lemma B, Lemma F′ and Lemma W as statements about the Gaussian laws (the +expansions imported from #191 Lemma E, the moments, the Kac–Rice disintegrations), and the assembly's use of #191 +and [C7-K] §4, are prose only. Review slices (PROOF.md §8): A Lemma B; B Lemma F′; C Lemma W and (W.2)–(W.4); D the +assembly, provenance and remarks. diff --git a/frontiers/remainder_rate_20260930/RESULTS.json b/frontiers/remainder_rate_20260930/RESULTS.json new file mode 100644 index 000000000..88d97cc6a --- /dev/null +++ b/frontiers/remainder_rate_20260930/RESULTS.json @@ -0,0 +1,62 @@ +{ + "checks": { + "E1_exponent_ledger": { + "info": { + "grid_points": 208, + "relative_exponents": { + "v1": "1/2", + "v1.1": "7/12" + }, + "twelfth_power_points": 4, + "v1": { + "rate": "1/6", + "theta": "1/6" + }, + "v1.1": { + "rate": "1/4", + "theta": "1/12" + } + }, + "passed": true + }, + "E2_barrier_arithmetic": { + "info": { + "barrier_cases": 48, + "solved_form_points": 4 + }, + "passed": true + }, + "E3_soft_eigenvalue_determinant": { + "info": { + "matrices": 5, + "sizes": [ + 2, + 3, + 4 + ] + }, + "passed": true + }, + "E4_integrals_and_cutoff_ledger": { + "info": { + "cutoff_ledger_points": 3, + "integral_points": 3, + "window_integral_points": 3 + }, + "passed": true + }, + "E5_sign_window": { + "info": { + "exact_equality_cases": 0, + "near_boundary_case": true, + "same_sign_instances": 13858, + "typed_instances": 15962 + }, + "passed": true + } + }, + "mutant": null, + "object": "CL-D2-REMAINDER-RATE-20260930-v1.1", + "passed": true, + "scientific_effect": "NONE" +} diff --git a/frontiers/remainder_rate_20260930/SOURCES.json b/frontiers/remainder_rate_20260930/SOURCES.json new file mode 100644 index 000000000..d8634fd83 --- /dev/null +++ b/frontiers/remainder_rate_20260930/SOURCES.json @@ -0,0 +1,144 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "object": "CL-D2-REMAINDER-RATE-20260930-v1.1", + "kind": "author-side proof candidate: Theorem R++ (nu_eld = c ell^-1/3 + O(ell^1/4), i.e. (1 + O(ell^7/12))) via Lemma B (the [182] barrier under the near-pair law of [R]), Lemma F-prime (the barrier under the far law) and Lemma W (sign-window majorant from Math-#191 Lemma E Steps 1-3, sharpening (K1), (R+.3) and [C7-K] section 4), assembled at the cutoff rho = ell^1/12; 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not a nonauthor review)", + "findings": "two passes. On v1 (rate 1/6): no error in Lemma B, Lemma F-prime, the decomposition or the ledger; suggested the sign window heuristically (made rigorous as Lemma W in v1.1). On v1.1 sections 3-5: Lemma W (W.1) proved from #191 Steps 1-3 with the (1+k)-quantified remainder re-derived exactly; (W.2)-(W.4) and the assembly correct; findings applied: the dependency on all of Lemma E through (E.1) (not Steps 1-3 alone), eight terms in section 4 (8C), k >= 0 in Lemma W, (W.2) with min(ell^1/4, rho), notation (w, Delta, #191-(1.x)), Remark 1 rewritten around three rho-free ell^1/4 sources with the referee formal cusp-scale computation recorded as unproved (route (b) of v1.1 cannot exist), Remark 2 wording. Earlier v1 findings: Lemma E Steps 1-3 confirmed symbolically on pinned quintics in d = 2, 3; thirteen findings applied: canonical Kac-Rice version named in the statement; Remark 1 rewritten (a false sentence on near-separation elder mass, the unproved \"of order rho\", the misdiagnosed bottleneck, the (K1)/(E.1) majorant route); Remark 3 gating (E10 stays until both notes are accepted); [R] section 4 quoted as the filtered bound; (Z14)/[C7-K] attribution; [P] section 2 for the derivative-suprema moments; range [r_0^*, diam X]; rho <= r_0^*; the uniformity of (R+.3) constant quoted from #191 section 2; the [182] barrier review provenance; SOURCES.json added" + }, + "files": [ + { + "path": "PROOF.md", + "bytes": 33131, + "sha256": "0b690df6ac40707ca06783f5641d8b480277c716df75b1031d84ee53a0575f72" + }, + { + "path": "rate_ledger_check.py", + "bytes": 18229, + "sha256": "5ad2b0ca9e7a131040252db5f23a3dc6444c3b0120912fbede4a593fbe7cb5d2" + }, + { + "path": "RESULTS.json", + "bytes": 1074, + "sha256": "c2c8ed1ccf0e15c992d88aff04caae1a99291f6d71b75f59db943cadbf5dc7e0" + }, + { + "path": "README.md", + "bytes": 5845, + "sha256": "c5a5fc8d1c108c664366c8aabd0cd0683908407524486cf0260a4d46709cfd9d" + } + ] +} diff --git a/frontiers/remainder_rate_20260930/rate_ledger_check.py b/frontiers/remainder_rate_20260930/rate_ledger_check.py new file mode 100644 index 000000000..3293ca22a --- /dev/null +++ b/frontiers/remainder_rate_20260930/rate_ledger_check.py @@ -0,0 +1,389 @@ +#!/usr/bin/env python3 +"""Exact finite controls for PROOF.md (CL-D2-REMAINDER-RATE-20260930-v1.1). Standard library only; exact rationals. + + E1 the exponent ledgers of section 4: with rho = ell^theta the v1.1 terms of (4.1)(i)-(iv) have exponents + 1/4, 3 theta, 2 - 7 theta, 1/4, 3 theta, 1/3 - theta, 4/3 - 5 theta, 1/3 (minimum 1/4 at theta = 1/12) and the + v1 terms theta, 2 - 7 theta, 1/4, theta, 1/3 - theta, 4/3 - 5 theta, 1/3 (minimum 1/6 at theta = 1/6); over a + rational grid of theta in (0, 1/3) and at every breakpoint neither minimum exceeds its value (optimality within + each ledger); at rational twelfth powers ell = q^12 every term is <= ell^(1/4) (resp. ell^(1/6)) exactly, so the + assembled bounds are 8 C ell^(1/4) (eight terms, five of them ell^(1/4)) and 7 C ell^(1/6); mutant M1 claims the + rate 1/3 for the v1.1 ledger; + E2 the barrier arithmetic of [182] section 3 (8)-(9): for rational 0 < lambda <= K, a_L <= 1 and 0 < t <= a_L + lambda / K, (K/6) t^3 <= (lambda/6) t^2 and b - (lambda/2) t^2 + (K/6) t^3 <= b - (lambda/3) t^2 exactly; the + solved form b - d >= c_L lambda^3 / K^2 <=> lambda <= ((b - d) K^2 / c_L)^(1/3) at rational cubes; mutant M2 + uses the radius 2 lambda / K; + E3 the determinant / soft-eigenvalue inequality (1.3): for rational symmetric positive definite matrices + O diag(lambda) O^T with rational orthogonal O (Cayley transforms of rational skew-symmetric matrices), + det <= lambda_min lambda_max^(d-1), and lambda_max <= Frobenius norm; mutant M3 uses the exponent d - 2; + E4 the integrals of Lemma B and the cutoff ledger of [C7-K] section 4: int_rho^r0 (ell r^-6 + r^-2) dr = + ell (rho^-5 - r0^-5)/5 + rho^-1 - r0^-1 <= ell rho^-5 / 5 + rho^-1 exactly at rationals; the second piece + ell^(-1/3) ell^(2/3) 3 (ell/rho^3)^(-1/3) = 3 rho and the first piece ell^(-1/3) (3/5) ell k_*^(-5/3) = + (3/5) ell^(1/4) at k_* = ell^(1/4), at rational twelfth powers; the (W.2)-(W.3) integrals + int_(r1)^rho (ell^2 r^-8 + r^2) dr <= ell^2 r1^-7 / 7 + rho^3 / 3 and int_0^(r1) r^4 / ell dr = r1^5 / (5 ell) = + ell^(1/4) / 5 at r1 = ell^(1/4), at rational twelfth powers; + E5 the sign window of Lemma W: for rational (k, D, Y, R_M, R_S, r) on a grid including k = 0, whenever the typed + sign pattern (r Y - 6 k D + R_M)(r Y + 6 k D + R_S) < 0 holds one has |r Y| <= 6 k |D| + max(|R_M|, |R_S|) and + |det K_i| <= 12 k |D| + 2 max(|R_M|, |R_S|) for both i, exactly; the bound is sharp (a near-boundary case); same-sign instances + violate the bound (so the sign condition is used); mutant M4 drops the factor 2 on the remainder. + M mutants (--mutant M1|M2|M3|M4) exit 1; an unknown label exits 2. + Scope: E1-E5 check exponent bookkeeping, the barrier's algebra, a linear-algebra inequality and the sign-window + inequality at rational points. They do NOT test Lemma B, Lemma F' or Lemma W as statements about the Gaussian + laws (the expansions (3.1) imported from Math-#191, the moment imports, the density bound, the Kac-Rice + disintegrations), nor the assembly's use of Math-#191 and [C7-K] section 4; those are prose only. The controls + are not acceptance. +""" +import argparse +import json +import sys +from fractions import Fraction as F + + +# ----------------------------------------------------------------------------------------------------- helpers + +def det(A): + n = len(A) + M = [row[:] for row in A] + d = F(1) + for i in range(n): + p = None + for r in range(i, n): + if M[r][i] != 0: + p = r + break + if p is None: + return F(0) + if p != i: + M[i], M[p] = M[p], M[i] + d = -d + d *= M[i][i] + for r in range(i + 1, n): + f = M[r][i] / M[i][i] + for c in range(i, n): + M[r][c] -= f * M[i][c] + return d + + +def matmul(A, B): + n, m, p = len(A), len(B), len(B[0]) + return [[sum(A[i][k] * B[k][j] for k in range(m)) for j in range(p)] for i in range(n)] + + +def transpose(A): + return [list(r) for r in zip(*A)] + + +def identity(n): + return [[F(int(i == j)) for j in range(n)] for i in range(n)] + + +def inverse(A): + n = len(A) + M = [row[:] + identity(n)[i] for i, row in enumerate(A)] + for i in range(n): + p = next(r for r in range(i, n) if M[r][i] != 0) + M[i], M[p] = M[p], M[i] + piv = M[i][i] + M[i] = [x / piv for x in M[i]] + for r in range(n): + if r != i and M[r][i] != 0: + f = M[r][i] + M[r] = [x - f * y for x, y in zip(M[r], M[i])] + return [row[n:] for row in M] + + +def cayley(S): + """Rational orthogonal matrix (I - S)(I + S)^{-1} for rational skew-symmetric S.""" + n = len(S) + I = identity(n) + A = [[I[i][j] - S[i][j] for j in range(n)] for i in range(n)] + B = [[I[i][j] + S[i][j] for j in range(n)] for i in range(n)] + return matmul(A, inverse(B)) + + +def is_cube(x): + """Exact rational cube root if x is a rational cube, else None.""" + if x < 0: + r = is_cube(-x) + return None if r is None else -r + n, d = x.numerator, x.denominator + + def iroot3(m): + lo, hi = 0, 1 + while hi ** 3 <= m: + hi *= 2 + while lo < hi: + mid = (lo + hi) // 2 + if mid ** 3 < m: + lo = mid + 1 + else: + hi = mid + return lo if lo ** 3 == m else None + + a, b = iroot3(n), iroot3(d) + return None if a is None or b is None else F(a, b) + + +# ---------------------------------------------------------------------------------------------------- checks + +def ledger_v11(theta): + """Exponents of the eight terms of (4.1)(i)-(iv) with rho = ell^theta (v1.1: (W.2), (W.3), (B.2), (F'.1)).""" + return [F(1, 4), 3 * theta, 2 - 7 * theta, F(1, 4), 3 * theta, F(1, 3) - theta, F(4, 3) - 5 * theta, F(1, 3)] + + +def ledger_v1(theta): + """Exponents of the seven terms with the v1 inputs (R+.3) and [C7-K] section 4 in place of (W.2)-(W.3).""" + return [theta, 2 - 7 * theta, F(1, 4), theta, F(1, 3) - theta, F(4, 3) - 5 * theta, F(1, 3)] + + +def check_E1(mutant): + grid = [F(i, 600) for i in range(1, 200)] + breakpoints = [F(1, 12), F(1, 6), F(1, 4), F(1, 8), F(2, 9), F(5, 24), F(7, 30), F(1, 24), F(1, 16)] + out = {} + for name, ledger, theta_star, rate in (('v1.1', ledger_v11, F(1, 12), F(1, 3) if mutant == 'M1' else F(1, 4)), + ('v1', ledger_v1, F(1, 6), F(1, 6))): + m = min(ledger(theta_star)) + if m != rate: + return False, '%s ledger minimum at theta = %s is %s, not the claimed rate %s' % (name, theta_star, m, rate) + for theta in grid + breakpoints: + if min(ledger(theta)) > rate: + return False, '%s ledger minimum exceeds %s at theta = %s' % (name, rate, theta) + out[name] = {'theta': str(theta_star), 'rate': str(rate)} + # exact assembled bounds at rational twelfth powers ell = q^12 + checked = 0 + for q in (F(1, 2), F(1, 3), F(2, 7), F(1, 10)): + ell = q ** 12 + # v1.1: rho = ell^(1/12) = q; terms ell^(1/4)=q^3, rho^3=q^3, ell^2 rho^-7=q^17, ell^(1/3)/rho=q^3, + # ell^(4/3) rho^-5 = q^11, ell^(1/3) = q^4 + rho = q + terms = [q ** 3, rho ** 3, ell ** 2 / rho ** 7, q ** 3, rho ** 3, q ** 4 / rho, q ** 16 / rho ** 5, q ** 4] + if ell ** 2 / rho ** 7 != q ** 17 or q ** 4 / rho != q ** 3 or q ** 16 / rho ** 5 != q ** 11: + return False, 'monomial substitution (v1.1)' + if any(t > q ** 3 for t in terms) or sum(terms) > 8 * q ** 3: + return False, 'a v1.1 ledger term exceeds ell^(1/4) at q = %s' % q + # v1: rho = ell^(1/6) = q^2 + rho = q ** 2 + terms = [rho, ell ** 2 / rho ** 7, q ** 3, rho, q ** 4 / rho, q ** 16 / rho ** 5, q ** 4] + if ell ** 2 / rho ** 7 != q ** 10 or q ** 4 / rho != q ** 2 or q ** 16 / rho ** 5 != q ** 6: + return False, 'monomial substitution (v1)' + if any(t > rho for t in terms) or sum(terms) > 7 * rho: + return False, 'a v1 ledger term exceeds ell^(1/6) at q = %s' % q + checked += 1 + # relative forms: ell^(1/4) / ell^(-1/3) = ell^(7/12); ell^(1/6) / ell^(-1/3) = ell^(1/2) + if F(1, 4) + F(1, 3) != F(7, 12) or F(1, 6) + F(1, 3) != F(1, 2): + return False, 'relative exponent' + out['grid_points'] = len(grid) + len(breakpoints) + out['twelfth_power_points'] = checked + out['relative_exponents'] = {'v1.1': '7/12', 'v1': '1/2'} + return True, out + + +def check_E2(mutant): + cases = 0 + for L in (F(1), F(24), F(1, 2)): + a_L = min(F(1), L / 4) + c_L = a_L * a_L / 3 + for lam, K in ((F(1, 5), F(3)), (F(2), F(2)), (F(1, 100), F(7, 2)), (F(3, 4), F(1))): + if not (0 < lam <= K): + return False, 'test data' + R = (2 * lam / K) if mutant == 'M2' else a_L * lam / K + for frac in (F(1, 7), F(1, 2), F(9, 10), F(1)): + t = frac * R + if not ((K / 6) * t ** 3 <= (lam / 6) * t ** 2): + return False, 'cubic term not dominated at t = %s (lambda %s, K %s)' % (t, lam, K) + b = F(5, 3) + if not (b - (lam / 2) * t ** 2 + (K / 6) * t ** 3 <= b - (lam / 3) * t ** 2): + return False, 'barrier inequality (8)' + cases += 1 + # (9): d_f <= b - lambda R^2 / 3, i.e. b - d >= c_L lambda^3 / K^2 with R = a_L lambda / K + if mutant != 'M2' and lam * R * R / 3 != c_L * lam ** 3 / K ** 2: + return False, 'lambda R^2 / 3 != c_L lambda^3 / K^2' + # solved form at rational cubes: b - d >= c_L lambda^3 / K^2 <=> lambda <= ((b - d) K^2 / c_L)^(1/3) + a_L = F(1) + c_L = a_L * a_L / 3 + solved = 0 + for lam, K, gap in ((F(1, 2), F(2), F(2, 3)), (F(1), F(1), F(1, 3)), (F(2), F(1), F(8, 3)), (F(3), F(3), F(1, 27))): + lhs = gap >= c_L * lam ** 3 / K ** 2 + cube = is_cube(gap * K * K / c_L) + if cube is None: + return False, 'test point is not a rational cube' + rhs = lam <= cube + if lhs != rhs: + return False, 'solved form of (9) fails' + solved += 1 + return True, {'barrier_cases': cases, 'solved_form_points': solved} + + +def check_E3(mutant): + exp = -2 if mutant == 'M3' else -1 # det <= lambda_min * lambda_max^(d + exp) + skews = [ + [[F(0), F(1, 2)], [F(-1, 2), F(0)]], + [[F(0), F(1, 3), F(-2)], [F(-1, 3), F(0), F(1, 5)], [F(2), F(-1, 5), F(0)]], + [[F(0), F(1), F(0), F(1, 2)], [F(-1), F(0), F(1, 3), F(0)], [F(0), F(-1, 3), F(0), F(2)], [F(-1, 2), F(0), F(-2), F(0)]], + ] + spectra = [ + [F(1, 100), F(2)], + [F(1, 10), F(3, 2), F(5)], + [F(1, 7), F(1, 3), F(2), F(9, 2)], + [F(3), F(3)], + [F(1, 50), F(1, 50), F(4)], + ] + cases = 0 + for S in skews: + n = len(S) + O = cayley(S) + # orthogonality, exactly + if matmul(transpose(O), O) != identity(n): + return False, 'Cayley transform not orthogonal' + for lam in spectra: + if len(lam) != n: + continue + D = [[lam[i] if i == j else F(0) for j in range(n)] for i in range(n)] + A = matmul(matmul(O, D), transpose(O)) + if transpose(A) != A: + return False, 'not symmetric' + dA = det(A) + prod = F(1) + for x in lam: + prod *= x + if dA != prod: + return False, 'determinant != product of eigenvalues' + lmin, lmax = min(lam), max(lam) + if not (dA <= lmin * lmax ** (n + exp)): + return False, 'det <= lambda_min lambda_max^(d-1) fails (spectrum %s)' % [str(x) for x in lam] + frob2 = sum(A[i][j] ** 2 for i in range(n) for j in range(n)) + if not (lmax ** 2 <= frob2): + return False, 'lambda_max <= Frobenius fails' + cases += 1 + if cases == 0: + return False, 'no cases' + return True, {'matrices': cases, 'sizes': [2, 3, 4]} + + +def check_E4(mutant): + # Lemma B integral, exact antiderivative + pts = 0 + for ell, rho, r0 in ((F(1, 1000), F(1, 10), F(1, 2)), (F(1, 64), F(1, 4), F(1)), (F(3, 1000), F(1, 5), F(2, 5))): + exact = ell * (rho ** -5 - r0 ** -5) / 5 + (rho ** -1 - r0 ** -1) + # antiderivative of ell r^-6 + r^-2 is -ell r^-5 / 5 - r^-1 + anti = lambda r: -ell * r ** -5 / 5 - r ** -1 + if anti(r0) - anti(rho) != exact: + return False, 'antiderivative' + if not (exact <= ell * rho ** -5 / 5 + rho ** -1): + return False, 'integral bound' + pts += 1 + # [C7-K] section 4 pieces at rational twelfth powers ell = q^12, rho = q^3 * s (so ell / rho^3 = (q / s)^3) + pieces = 0 + for q, s in ((F(1, 2), F(1)), (F(1, 3), F(2)), (F(2, 5), F(3, 2))): + ell = q ** 12 + rho = s * q ** 3 + ell13 = q ** 4 # ell^(1/3) + ell23 = q ** 8 # ell^(2/3) + ell14 = q ** 3 # ell^(1/4) = k_* + # second piece: ell^(-1/3) * ell^(2/3) * 3 * (ell/rho^3)^(-1/3) = 3 rho + cube = is_cube(ell / rho ** 3) + if cube is None: + return False, 'ell/rho^3 not a cube' + second = ell23 / ell13 * 3 / cube + if second != 3 * rho: + return False, 'second piece != 3 rho' + # first piece: ell^(-1/3) * (3/5) * ell * k_*^(-5/3) with k_* = ell^(1/4): k_*^(5/3) = q^5 + first = (F(3, 5) * ell / q ** 5) / ell13 + if first != F(3, 5) * ell14: + return False, 'first piece != (3/5) ell^(1/4)' + pieces += 1 + # (W.2)-(W.3) integrals at rational twelfth powers ell = q^12, r1 = ell^(1/4) = q^3, rho = s q^3 with s >= 1 + w = 0 + for q, s in ((F(1, 2), F(2)), (F(1, 3), F(5)), (F(2, 5), F(3, 2))): + ell = q ** 12 + r1 = q ** 3 + rho = s * r1 + # int_(r1)^rho (ell^2 r^-8 + r^2) dr = ell^2 (r1^-7 - rho^-7)/7 + (rho^3 - r1^3)/3 + val = ell ** 2 * (r1 ** -7 - rho ** -7) / 7 + (rho ** 3 - r1 ** 3) / 3 + anti = lambda r: -ell ** 2 * r ** -7 / 7 + r ** 3 / 3 + if anti(rho) - anti(r1) != val: + return False, '(W.2) antiderivative' + if not (val <= ell ** 2 * r1 ** -7 / 7 + rho ** 3 / 3): + return False, '(W.2) integral bound' + if ell ** 2 * r1 ** -7 != q ** 3: # ell^2 r1^-7 = ell^(1/4) + return False, 'ell^2 r1^-7 != ell^(1/4)' + # int_0^(r1) r^4 / ell dr = r1^5 / (5 ell) = ell^(1/4) / 5 + if r1 ** 5 / (5 * ell) != q ** 3 / 5: + return False, '(W.3) first piece' + w += 1 + return True, {'integral_points': pts, 'cutoff_ledger_points': pieces, 'window_integral_points': w} + + +def check_E5(mutant): + factor = F(1) if mutant == 'M4' else F(2) + ks = [F(0), F(1, 100), F(1, 2), F(3)] + Ds = [F(-2), F(-1, 3), F(1, 5), F(4)] + Ys = [F(n, 4) for n in range(-12, 13)] + Rs = [F(0), F(1, 50), F(-1, 50), F(1, 3), F(-2, 5)] + rs = [F(1, 10), F(1, 2), F(1)] + typed = same = 0 + attained = 0 + for k in ks: + for D in Ds: + for Y in Ys: + for RM in Rs: + for RS in Rs: + for r in rs: + u = r * Y + dM = u - 6 * k * D + RM + dS = u + 6 * k * D + RS + bound_u = 6 * k * abs(D) + max(abs(RM), abs(RS)) + bound_d = 12 * k * abs(D) + factor * max(abs(RM), abs(RS)) + if dM * dS < 0: + typed += 1 + if abs(u) > bound_u: + return False, '|rY| bound fails at %s' % ((k, D, Y, RM, RS, r),) + if abs(dM) > bound_d or abs(dS) > bound_d: + return False, '|det K_i| bound fails at %s' % ((k, D, Y, RM, RS, r),) + if abs(dM) == bound_d or abs(dS) == bound_d: + attained += 1 # not expected under the strict sign condition + elif dM * dS > 0: + same += 1 + if typed == 0 or same == 0: + return False, 'grid does not contain both sign patterns' + # the sign condition is used: a same-sign instance violating the |det K_i| bound + k, D, Y, RM, RS, r = F(1, 10), F(1), F(20), F(0), F(0), F(1) + dM, dS = r * Y - 6 * k * D + RM, r * Y + 6 * k * D + RS + if not (dM * dS > 0 and abs(dM) > 12 * k * abs(D)): + return False, 'same-sign counterexample missing' + # exact equality case: R_M = R_S = 0, rY = 6kD exactly is the boundary (product 0, not typed); just inside, |u| < 6k|D| + k, D, RM, RS, r = F(1), F(2), F(0), F(0), F(1) + Y = 6 * k * D - F(1, 1000) + dM, dS = r * Y - 6 * k * D + RM, r * Y + 6 * k * D + RS + if not (dM * dS < 0 and abs(dS) <= 12 * k * abs(D) and abs(dS) > 12 * k * abs(D) - F(1, 100)): + return False, 'near-equality case' + return True, {'typed_instances': typed, 'same_sign_instances': same, 'exact_equality_cases': attained, 'near_boundary_case': True} + + +CHECKS = [('E1_exponent_ledger', check_E1), ('E2_barrier_arithmetic', check_E2), + ('E3_soft_eigenvalue_determinant', check_E3), ('E4_integrals_and_cutoff_ledger', check_E4), + ('E5_sign_window', check_E5)] +MUTANTS = ('M1', 'M2', 'M3', 'M4') + + +def main(): + ap = argparse.ArgumentParser() + ap.add_argument('--mutant', default=None) + args = ap.parse_args() + mutant = args.mutant + if mutant is not None and mutant not in MUTANTS: + sys.stderr.write('unknown mutant label\n') + return 2 + results = {} + ok_all = True + for name, fn in CHECKS: + ok, info = fn(mutant) + results[name] = {'passed': ok, 'info': info} + ok_all = ok_all and ok + out = {'object': 'CL-D2-REMAINDER-RATE-20260930-v1.1', 'scientific_effect': 'NONE', 'passed': ok_all, + 'mutant': mutant, 'checks': results} + sys.stdout.write(json.dumps(out, indent=1, sort_keys=True) + '\n') + return 0 if ok_all else 1 + + +if __name__ == '__main__': + sys.exit(main()) diff --git a/frontiers/remainder_vanishing_20260930/PROOF.md b/frontiers/remainder_vanishing_20260930/PROOF.md new file mode 100644 index 000000000..441152dfc --- /dev/null +++ b/frontiers/remainder_vanishing_20260930/PROOF.md @@ -0,0 +1,309 @@ +# The bounded remainder of Theorem R: `B_{d,L}` for candidates, `0` for elder pairs + +Object: CL-D2-REMAINDER-VANISHING-20260930-v1.1 (v1 → v1.1: xAI review 5369140697 on Math- #191 — A1 the scaling +dictionary and Lemma R3.2 quoted verbatim, A2 the controls' scope stated, A3 the (R4) import quoted, B2 the signed +near bound proved as a separate remark; no change to Lemma E, Theorem R+ or (R+.1)–(R+.3)). +Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 30 September 2026. +Disposition: AUTHOR-SIDE PROOF CANDIDATE; NONAUTHOR REVIEW REQUIRED. Scientific effect: NONE — no register, +graph, STATUS, PROOF_INDEX, prize or Boolean change; no numerical constant is claimed. Same GitHub account as +every lane; zero organizational independence. Everything below is relative to merged, reviewed sources ([R], +[Z], [C7-K] and the reconciled [P] chain); the unmerged Math- #187/#188 are cited only, for the quantitative +Remark 2. + +## 0. Statement + +Setting, notation and every display number `(Rn)` are those of [R] (`frontiers/three_fronts_20260924/ +LIFETIME_REMAINDER.md`, blob `247b3ecf`, Theorem R; xAI R1–R6 on main #67, Anthropic full depth Math- #186): +fixed `d ≥ 2`, `L > 0`, the [P] field on `X = R^d/(LZ^d)`; near pins `M = −ru/2`, `S = ru/2` with heights `b`, +`b − kr³` and zero gradients; the observation rows `U_r`, target `v_r`, density `π_r`, the regression law `Q`, +the coupling `F_r` of [R] (R3)–(R4), `m = d − 1`, `P = 1 + |b| + k`; the scaled endpoint Hessians of [R] §4 +(*scaling dictionary*, quoted): "`alpha_i=F_r,xx(i)/r` and `beta_i=grad_y F_r,x(i)/r`. Let `D_r=diag(sqrt(r),I)`. +Then `K_i=D_r^(-1) H_i D_r^(-1) =[[alpha_i,sqrt(r) beta_i^T],[sqrt(r) beta_i,A_i]]`, `|det H_i|/r=|det K_i|`" — so +`H_i` (the full Hessian of `F_r` at the pin `i`, in the coordinates `(x, y)`) has entries `rα_i`, `rβ_i`, `A_i`, and +`K_i` has the scalar `α_i`, the off-diagonal `√r β_i` and the transverse block `A_i = D_y²F_r(i)` (`i = M, S`); the +filtered determinants `F_j`; `Z_r/r² = E_Q[F_d(K_M)F_{d−1}(K_S)]`; `A_0 = D_y²F_0(0)`, +`z_0 = E[(6k)²(det A_0)²1{A_0 < 0}]`; `A_r = 12π_r(v_r)Z_r/r²`, `A_0 = 12π_0(v_0)z_0`; the majorant +`H(b,k) = CP^Ne^{−c(b²+k²)}` (`N` enlarged when needed); the near/far split at a separation `r_0`, `a = ℓ/r_0³`; and +`c = c_{d,L}` of [P] (15.2). Admissible radii: `r_0 ≤ r_0^* := min{r_0^{[R]}, L/(4√2), r_0^{[K]}, r_0^{[Z]}}` ([R]'s compactness band on +which (R2)–(R5) hold, the cap-cylinder embedding of [REC] §5, and the radii of [C7-K] (K2) and [Z] §4). [R] proves +`|A_r − A_0| ≤ r(k + r)H` ((R11), first order in `r`) and from it Theorem R, `|ν_eld(ℓ) − cℓ^{−1/3}| ≤ C`, and states +that neither convergence of the remainder nor a second coefficient is identified. + +From [Z] (`frontiers/c7_zero_gap_limit_20260929/PROOF.md`, blob `5b6328ea`; Anthropic A/B/C reads on Math- #155): +the unmarked candidate kernel and the equal-height kernel + + Ψ_ℓ^{cand}(b, y) = p_y(v_{b,ℓ}) E_{Q_{y,b,ℓ}}[W], Ψ_0(b, y) = p_y(v_{b,0}) E_{Q_{y,b,0}}[W] ((Z2)), + +`B_{d,L} = ∫_{X∖{0}}∫_R Ψ_0(b, y) db dy ∈ (0, ∞)` ((Z3)), and Theorem Z (Z4): the rejected density +`ρ_rej(ℓ) = ν_cand(ℓ) − ν_eld(ℓ) → B_{d,L}` as `ℓ ↓ 0`. + +**Lemma E (even contact expansion).** There are `C, N` depending on `d, L` only such that for all `b ∈ R`, `k > 0` +and `0 < r ≤ r_0^*` + + |Z_r/r² − z_0(b, k, u)| ≤ C r² P^N, |A_r − A_0| ≤ r² H(b, k). (E.1) + +The first-order term of (R11) is absent: the two endpoint corrections enter the typed product with opposite signs +and cancel identically (§1, Steps 1–3), and on the inertia-flip layers the product of the two typed determinants is +`O(r²)` (Step 4). + +**Theorem R+.** For every fixed `d ≥ 2`, `L > 0`, as `ℓ ↓ 0`, + + ν_cand(ℓ) = c ℓ^{−1/3} + B_{d,L} + o(1), (R+.1) + ν_eld(ℓ) = c ℓ^{−1/3} + o(1), i.e. ν_eld(ℓ) = c ℓ^{−1/3} (1 + o(ℓ^{1/3})). (R+.2) + +So the bounded remainder of Theorem R has the constant term `B_{d,L}` (the equal-height kernel mass of [Z]) for the +candidate density and `0` for the elder density. Moreover, for every `r_0 ∈ (0, r_0^*]`, + + ν_cand^{near,r_0}(ℓ) − cℓ^{−1/3} → ∫_{0=1 and each +finite derivative order q**, `||F_r-F_0||_{L^p(C^q)} <= C_{p,q} r^2 P`, `||1+||F_r||_{C^q}+||F_0||_{C^q}||_p <= C_{p,q} P`, +`P=1+|b|+k`" — the second line with `q = 7` bounds the `C^7` norms in `L^p`, the first with `q = 2` bounds +`r^{−2}‖F_r − F_0‖_{C^2}` in `L^p`, so `T := 1 + k + ‖F_r‖_{C^7} + ‖F_0‖_{C^7} + r^{−2}‖F_r − F_0‖_{C^2}` has +`‖T‖_p ≤ C_pP` for every `p` (`[R]` §4 chose `q = 4` for its `T` because (R8)–(R11) need four derivatives; the +`C^q` moments exist for every `q` because `F_r` is the Gaussian regression of the smooth [P] field on finitely many +rows, [R] §2). Write `f = F_r`, `f_j = ∂_x^jf(0)`, and `O(x)` for a quantity bounded by `C|x|` with `C = C(d)`. + +*Step 1 (axial pins to second order).* The third and fourth rows of `U_r` at their pinned values `0` and `12k` +give, by Taylor's formula with integral remainder (the rules are centred, so the odd orders cancel), + + (f_x(S) − f_x(M))/r = f_2 + (r²/24) f_4 + O(r⁴T) = 0, + (6/r²)[f_x(M) + f_x(S) − 2(f(S) − f(M))/r] = f_3 + (r²/40) f_5 + O(r⁴T) = 12k. + +Taylor expansion of `f_xx` at `∓r/2` to second order, `f_xx(∓r/2) = f_2 ∓ (r/2)f_3 + (r²/8)f_4 + O(r³T)`, and +substitution of the two identities give + + α_M = −6k + (r/12) f_4 + ρ_M, α_S = 6k + (r/12) f_4 + ρ_S, |ρ_M| + |ρ_S| ≤ C r² T. (1.1) + +(For a polynomial profile of degree five the expansions are exact, `α_{M,S} = ∓6k + (r/12)f_4 ∓ (r²/120)f_5`; +checker E1.) The first-order coefficients coincide, so + + α_M α_S = −36k² + 6k(ρ_M − ρ_S) + r²(f_4/12)² + (rf_4/12)(ρ_M + ρ_S) + ρ_Mρ_S = −36k² + O(r²(1 + k)T²). (1.2) + +*Step 2 (transverse blocks).* With `B := ∂_xD_y²f(0)` and `‖D_y²f(0) − A_0‖ ≤ r²T` (coupling), + + A_M = A_0 − (r/2)B + O(r²T), A_S = A_0 + (r/2)B + O(r²T), (1.3) + +and `det(A_0 + X) = det A_0 + tr(adj(A_0)X) + O(‖X‖²T^{m−2})`, so + + det A_M det A_S = (det A_0)² + det A_0 · [−(r/2) + (r/2)] tr(adj(A_0)B) + O(r²T^{2m}) = (det A_0)² + O(r²T^{2m}). (1.4) + +The transverse difference row `(f_y(S) − f_y(M))/r = f_{xy}(0) + (r²/24)f_{xxxy}(0) + O(r⁴T) = 0` gives +`f_{xy}(0) = O(r²T)`, hence with `γ := ∇_yf_{xx}(0)`, + + β_M = −γ/2 + O(rT), β_S = γ/2 + O(rT), q_i := β_iᵀadj(A_i)β_i = q + O(rT^{m+1}), q := γᵀadj(A_0)γ/4. (1.5) + +(The sign of `β_i` enters `q_i` quadratically. That `f_{xy}(0)` is `O(r²)` and not `O(r)` is essential: an `O(r)` +value would shift `β_M` and `β_S` by a common `O(1)` and make `q_M − q_S = O(1)`.) + +*Step 3 (the product of determinants).* [R] Lemma R3.2, quoted: "For a symmetric (n-1) by (n-1) matrix A, scalar +alpha, vector beta and t>=0, set `K_t=[[alpha,sqrt(t) beta^T],[sqrt(t) beta,A]]`. … Indeed +`det K_t=alpha det A-t beta^T adj(A) beta`, INCLUDING singular A." With `t = r` and `(alpha, beta, A) = (α_i, β_i, A_i)`, +`K_r` is the scaled Hessian `K_i` of the dictionary in §0 (off-diagonal entries `√r β_i`), so the exact identity is + + det K_i = α_i det A_i − r β_iᵀadj(A_i)β_i = α_i det A_i − r q_i, + +`q_i` as in (1.5): the parameter multiplying `q_i` is the first power of `r` because the off-diagonal entries of `K_i` +carry `√r` (with `β_i = ∇_yF_{r,x}(i)/r` already containing the `1/r` of the dictionary; in terms of the unscaled +Hessian, `det H_i = r det K_i = rα_i det A_i − r²q_i`). Hence + + det K_M det K_S = α_Mα_S det A_M det A_S − r(α_M det A_M q_S + α_S det A_S q_M) + r² q_M q_S. + +By (1.2), (1.4) the first term is `−36k²(det A_0)² + O(r²(1 + k)²T^{N_1})`. In the second, by (1.1), (1.3), (1.5), +`α_M det A_M q_S = −6k det A_0 q + O(r(1 + k)T^{N_1})` and `α_S det A_S q_M = +6k det A_0 q + O(r(1 + k)T^{N_1})`, so +the bracket is `O(r(1 + k)T^{N_1})` and the term is `O(r²(1 + k)T^{N_1})`. The third is `O(r²T^{N_1})`. Hence + + | det K_M det K_S + 36k²(det A_0)² | ≤ C r² (1 + k)² T^{N_1}. (1.6) + +Structurally, `det K_M = −6k det A_0 + rY + O(r²)` and `det K_S = +6k det A_0 + rY + O(r²)` with the **same** +`Y = (f_4/12)det A_0 + 3k tr(adj(A_0)B) − q`; the product's `r`-term is `6k det A_0(Y − Y) = 0`. This is the +reflection symmetry of the pinned family: reversing the axis exchanges `M` and `S`, and the rows of `U_r` are even +in `r` (checker E2 verifies, on a general pinned quintic in two variables, the exact identity +`det H_M(−r) = det H_S(r)`, hence that `det H_M det H_S/r²` is an even polynomial in `r` with constant term +`−36k²A_0²`, while each single factor is not even). + +*Step 4 (inertia).* For `A_i` invertible, Haynsworth's inertia additivity for the block matrix `K_i` gives +`inertia(K_i) = inertia(A_i) + inertia(σ_i)`, `σ_i := α_i − rβ_iᵀA_i^{−1}β_i`. So `index K_M = d` iff +`A_M < 0` and `σ_M < 0`; `index K_S = d − 1` iff (`A_S < 0` and `σ_S > 0`) or (`index A_S = m − 1` and `σ_S < 0`). +Put `u := C_1 r T (1 + T/k)` with `C_1 = C_1(d) ≥ 2` large, and the good event + + G := { λ_max(A_0) ≤ −u } ∩ { k ≥ C_1 r T }. + +On `G`: `‖A_i − A_0‖ ≤ rT ≤ u/C_1`, so `A_M, A_S < 0`; `α_S ≥ 6k − CrT > 0` and `rβ_Sᵀ(−A_S)^{−1}β_S ≥ 0`, so +`σ_S > 0`; `α_M ≤ −6k + CrT ≤ −5k` and `rβ_Mᵀ(−A_M)^{−1}β_M ≤ rT²/λ_min(−A_M) ≤ 2rT²/u ≤ 2k/C_1`, so `σ_M < 0`. Hence +on `G` the indices are `d` and `d − 1`, `1{A_0 < 0} = 1`, and + + F_d(K_M)F_{d−1}(K_S) − 36k²(det A_0)²1{A_0<0} = |det K_M det K_S| − 36k²(det A_0)², + +whose absolute value is at most the left side of (1.6). Off `G` there are three cases, on each of which both typed +products are `O(r²)` times a polynomial: + +- on `{k < C_1rT}`: `F_d(K_M)F_{d−1}(K_S) ≤ C(k + r)²T^{2d} ≤ Cr²T^{2d+2}` ([R] §4's bound `F_j(K_i) ≤ C(k + r)T^d`, + `T ≥ 1`) and `36k²(det A_0)² ≤ Cr²T^{2m+2}`; +- on `{λ_max(A_0) ≥ u} ∩ {k ≥ C_1rT}`: `λ_max(A_M) ≥ u − rT > 0`, so `K_M` is not negative definite, `F_d(K_M) = 0`, + and `1{A_0 < 0} = 0` — both products vanish (this case also contains the pattern `index A_S = m − 1`, `σ_S < 0`, + which `F_d(K_M) = 0` kills); +- on `{|λ_max(A_0)| < u} ∩ {k ≥ C_1rT}`: by Weyl, `|det A_i| ≤ (|λ_max(A_0)| + rT)T^{m−1} ≤ 2uT^{m−1}`, so + `|det K_i| ≤ (6k + CrT)·2uT^{m−1} + rT^{m+1} ≤ C(ku + rT²)T^{m−1} ≤ CkuT^{m−1}` (as `rT² ≤ ku/C_1`, i.e. + `T ≤ k + T`), hence `F_d(K_M)F_{d−1}(K_S) ≤ Ck²u²T^{2m−2}`, and likewise `36k²(det A_0)²1{A_0<0} ≤ 36k²u²T^{2m−2}`; + with `k²u² = C_1²r²T²(k + T)²` both are `≤ Cr²(k + T)²T^{2m}`. + +No smallness of the layers' probability is needed: each typed determinant on the last layer is `O(r)` and their +product `O(r²)`. + +*Step 5 (assembly).* Taking expectations, `|Z_r/r² − z_0| ≤ E[Cr²(1 + k)²T^{N_1} + Cr²(k + T)²T^{2m+2}] ≤ Cr²P^N`. +For `A_r − A_0 = 12[π_r(v_r) − π_0(v_0)]Z_r/r² + 12π_0(v_0)[Z_r/r² − z_0]`, use (R5) (`|π_r(v_r) − π_0(v_0)| ≤ +Cr²P²e^{−c(b²+k²)}`, `π_0(v_0) ≤ Ce^{−c(b²+k²)}`), (R10) (`Z_r/r² ≤ C(k + r)²P^N`) and the first bound; every term is +`r²` times a polynomial in `P` times the Gaussian factor, i.e. `≤ r²H`. ∎ + +## 2. Proof of Theorem R+ + +*The separation variable.* As in [R] §6 and [Z] (Z14), with `r = (ℓ/k)^{1/3}`, `k = ℓ/r³`, `dk = 3ℓr^{−4}dr`, + + ν_cand^{near,r_0}(ℓ) = ∫_0^{r_0}∫_R∫_{S^{d−1}} r^{−2} A_r(b, ℓ/r³, u) dσ(u) db dr, + c ℓ^{−1/3} = ∫_0^{∞}∫_R∫_{S^{d−1}} r^{−2} A_0(b, ℓ/r³, u) dσ(u) db dr, + +and, by the pin-density and determinant normalizations of [P] (10.1)–(11.1) exactly as in (Z14) but without the +mark, `r^{−2}A_r(b, ℓ/r³, u) = r^{d−1}Ψ_ℓ^{cand}(b, ru)` (the polar factor `r^{d−1}`, the pin density +`12r^{−(d+3)}π_r(v_r)` and the determinant scale `r²`). Hence + + ν_cand^{near,r_0}(ℓ) − cℓ^{−1/3} = ∫_0^{r_0}∫∫ r^{−2}(A_r − A_0)(b, ℓ/r³, u) − ∫_{r_0}^{∞}∫∫ r^{−2}A_0(b, ℓ/r³, u). (2.1) + +*Domination.* By (E.1), `|r^{−2}(A_r − A_0)(b, ℓ/r³, u)| ≤ H(b, ℓ/r³) ≤ Ce^{−cb²}`, an integrable majorant on +`(0, r_0) × R × S^{d−1}` **independent of `ℓ`** — the near-diagonal domination that [Z] §5 records as unavailable +for candidates from (R11) alone (with `r(k + r)H` the integrand is only `≤ (1 + ℓ/r⁴)H`). The second term of (2.1) +is at most `∫_{r_0}^∞ r^{−2}(ℓ/r³)²H ≤ Cℓ²r_0^{−7}` (`A_0 ≤ k²H`). + +*Pointwise limit.* Fix `r, b, u`. As `ℓ ↓ 0`, `k = ℓ/r³ ↓ 0` and `A_r(b, k, u) → A_r(b, 0, u)`: the pinned law +`Q_{r,b,k}` is the regression law with target `v_r(b,k)`, affine in the target with the `k`-free covariance of +[R] §2, so `F_r` depends continuously on `k` in `C²`, the typed product is continuous off the type boundaries +(Gaussian-null by nondegeneracy, [Z] §2), and it is dominated by a polynomial in the endpoint Hessians; `π_r(v_r)` +is continuous in the target. Also `A_0(b, k, u) → 0` (`A_0 ≤ k²H`). Since `r^{−2}A_r(b, 0, u) = r^{d−1}Ψ_0(b, ru)` +((Z2) at `y = ru`), dominated convergence in (2.1) gives + + ν_cand^{near,r_0}(ℓ) − cℓ^{−1/3} → ∫_0^{r_0}∫∫ r^{d−1}Ψ_0(b, ru) dσ db dr = ∫_{0 0`, follows as + soon as the far constant is known to be polynomial in `1/ρ` (take `ρ = ℓ^{θ}`, `θ` small, `N` larger than + `θ` times that degree); the constant's `ρ`-dependence enters through the far-pair nondegeneracy at separation + `ρ` and, for #188, the `N`-site conditional density and `ℓ_1 = ρ²/(64N²)`. None of this is proved here. +3. **Consistency.** (R+.1) with (Z4) is consistent with [C7-K] (T2) and [U] (U1) (`ρ_rej = Θ(1)`); (R+.3) with + [P] (14.1) (candidate density `O(1)` at fixed separation) and with [Z] (Z18) (rejected mass below separation `δ` + is `≤ Cδ`); Lemma E with the remark in Math- #186 ("Where the `O(1)` comes from": at fixed marks the contact error + is `O(r²)`; that remark also noted that this alone does not improve Theorem R, which is why Theorem R+ needs the + domination-plus-limit argument of §2 rather than the ledger of (R16)). The `2r_0` of #186's C7 model check is + the (R17)–(R18) loss term, i.e. the rejected mass at separations `≍ r_0`, in agreement with Remark 1. +4. **For the manuscript.** Once reviewed, (R+.2) supports reading the abstract's `(1 + o(1))` as + `(1 + o(ℓ^{1/3}))` and, for the candidate density, `cℓ^{−1/3} + B_{3,24} + o(1)`. Until then E10 of the V3 edit + pack (`(1 + O(ℓ^{1/3}))`, which rests on Theorem R and its two reads, the second still unmerged in Math- #186) + is the safe statement, and E3's "do not write an `o(1)` remainder for `ν_eld` alone" and E10's "not known to be + improvable" would be revised by this note when it lands. No `+ c_1 + o(1)` with `c_1 ≠ 0` is claimed for `ν_eld`; + the claim is precisely that its constant term is `0`. + +## 4. Sources (exact identities in `SOURCES.json`) + +| Tag | Path | Role | +|---|---|---| +| [R] | `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md` (blob `247b3ecf`) | Theorem R; §§2–6: coupling (R2)–(R5), Lemma R3.2, (R8)–(R11), the separation-variable identity — consumed | +| [Z] | `frontiers/c7_zero_gap_limit_20260929/PROOF.md` (blob `5b6328ea`) | (Z2)–(Z4), (Z10)–(Z12), (Z14), (Z16)–(Z18) — consumed | +| [C7-K] | `frontiers/c7_total_bounded_20260929/PROOF.md` (blob `28748b08`) | §4 (T1) proof with cutoff `ρ`, (T2) — Remark 2, Remark 3 | +| [P] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (blob `dfed3b8d`) | §§1–4, 7, 9 (as [E2]), (10.1), (11.1), 14, (15.2) — through [R], [Z] | +| [E1] | `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md` (blob `213594d6`) | reading rule | +| [E2] | `reviews/d1_section9_borel_repair_20260925/REPAIR.md` (blob `fe9b9ce4`) | §9 replacement | +| [REC] | `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md` (blob `75da2597`) | W1, embedding radius, consumption contract | +| [U] | `frontiers/unrestricted_selection_difference_20260929/PROOF.md` (blob `5a55b179`) | (U1) — cited only (Remark 3) | +| #187, #188 | `frontiers/far_elder_rate_20260930/PROOF.md`, `frontiers/far_elder_flat_ridge_20260930/PROOF.md` (unmerged) | far elder rates — cited only (Remark 2) | +| #186 | `reviews/d2_remainder_full_depth_claude_20260930/REVIEW.md` (unmerged) | the review remark that motivated Lemma E — cited only | + +## 5. Exact controls (`even_contact_check.py`; stdlib; exact rationals; byte-identical under `-O`) + +E1 the axial pin identities on a quintic profile: with `f_2 = −(r²/24)f_4`, `f_3 = 12k − (r²/40)f_5` the pins +`f(∓r/2) = b, b − kr³`, `f_x(∓r/2) = 0` hold exactly and `α_M = −6k + (r/12)f_4 − (r²/120)f_5`, +`α_S = 6k + (r/12)f_4 + (r²/120)f_5`, `α_Mα_S = −36k² + r²(f_4²/144 − kf_5/10) − r⁴f_5²/14400`, at rational +`(r, k, f_4, f_5, b)` — the coefficients `1/24, 1/40, 1/12, 1/120` are the content of Step 1; mutant M1 (drop the +`(r²/40)f_5` pin term) fails. +E2 the reflection symmetry on a general pinned quintic in two variables (six pinned coefficients solved exactly, +fifteen free ones fixed at rationals): `det H_M(−r) = det H_S(r)` exactly at rational `r`; consequently +`det H_M det H_S/r²`, recovered as the polynomial of degree ten in `r` by exact interpolation at eleven rational +`r`, has only even powers, constant term `−36k²A_0²` (`A_0 = f_yy(0)`), and the recorded `r²` coefficient. The parity +is forced by the symmetry, so this is a check of the mechanism of Step 3, not of its bookkeeping; mutant M2 +applies the parity test to the single factor `det H_M/r`, which has `r¹, r³, r⁵` terms, and fails. +E3 the block identity `det K = α det A − r βᵀadj(A)β` for `K = [[α, √r βᵀ],[√r β, A]]` (`r` a rational square, +`s = √r` rational, entries `s·β` — [R] Lemma R3.2 with `t = r`, the dictionary of §0) and Haynsworth's inertia count +on rational `3×3` and `4×4` examples with `A` negative definite, of index `m − 1`, and indefinite, and the scalar +Schur complement of both signs (inertia by exact congruence diagonalisation); the Step 4 case split at rational +points: `F_d(K_M) = 0` when `λ_max(A_M) > 0`, and the layer inequality `|det A| ≤ (|λ_max(A)|)·‖A‖^{m−1}` for +`|λ_max(A)|` small. +E4 the ledger of §2 and Remark 2 by exact monomial substitution `r = (ℓ/k)^{1/3}`: `r^{−2}(ℓ/r³)² = ℓ²r^{−8}` +(so `∫_{r_0}^∞ ≍ ℓ²r_0^{−7}`), `k^{−2/3}r^{3}/k = ℓk^{−8/3}`, `ℓκ^{−5/3} = κ^{7/3} = ℓ^{7/12}` at `κ = ℓ^{1/4}`, +`ℓ^{2/3}a^{−1/3} = r_0ℓ^{1/3}`, and `r ≤ k ⇔ ℓ ≤ k⁴`; mutant M3 (claims the loss exponent `1/3`) fails. +E5 the layer algebra of Step 4: `k²u² = C_1²r²T²(k + T)²` and `rT² ≤ ku/C_1` for `u = C_1rT(1 + T/k)`, exactly. +Mutants `M1`–`M3` exit 1; an unknown label exits 2. + +**What the controls do not test.** They check exact identities and finitely many rational instances of Steps 1–3 +(E1, E2), of the block/inertia facts and the case split of Step 4 (E3), of the ledger (E4) and of the layer algebra +(E5). They do **not** test Step 4 as a statement about the random matrices — the Weyl and Hadamard bounds on the +layer `{|λ_max(A_0)| < u}`, the choice of `C_1`, and the claim that the product of the two typed determinants is +`O(r²)` times a polynomial in `T` there — nor Step 5 (the `C^7` moment import from (R4) and the passage to +expectations), nor anything in §2 (the majorant, the pointwise limit, the far limit, the use of (Z4)). Those are +proved in prose only and are review slice A (Steps 4–5) and slice B (§2). The controls are not acceptance. + +## 6. Review slices + +A: Lemma E — Steps 1–3 (the expansions and the product cancellation) and Step 4 (Haynsworth, the good event, the + three layers), Step 5. +B: §2 — the separation-variable identity against [R] §6 / [Z] (Z14), the `ℓ`-uniform majorant from (E.1), the + pointwise limit `A_r(b,k,u) → A_r(b,0,u)`, the far candidate limit from [Z], and the use of (Z4). +C: the statement's scope, Remarks 1–4, and the sources' consumption (in particular that nothing unmerged is used). diff --git a/frontiers/remainder_vanishing_20260930/README.md b/frontiers/remainder_vanishing_20260930/README.md new file mode 100644 index 000000000..097ed0e6e --- /dev/null +++ b/frontiers/remainder_vanishing_20260930/README.md @@ -0,0 +1,52 @@ +# The remainder of Theorem R: `B_{d,L}` for candidates, `0` for elder pairs (CL-D2-REMAINDER-VANISHING-20260930-v1.1) + +Author-side proof candidate, Anthropic Claude, 30 September 2026. Scientific effect NONE. Nonauthor review required. +Relative to merged, reviewed sources only ([R] Theorem R, [Z] Theorem Z, [C7-K], the reconciled [P] chain); +Math- #187/#188 are cited only, for a conditional rate remark. + +**Statement (Theorem R+).** For the periodized Gaussian field on the fixed torus (every `d ≥ 2`, `L > 0`), as +`ℓ ↓ 0`, + + ν_cand(ℓ) = cℓ^{−1/3} + B_{d,L} + o(1), ν_eld(ℓ) = cℓ^{−1/3} + o(1), + +where `B_{d,L} = ∫∫Ψ_0` is the equal-height kernel mass of Theorem Z. Theorem R's bounded remainder is thus +identified: its constant term is the rejected mass `B_{d,L}` for candidate pairs and `0` for elder pairs, so +`ν_eld = cℓ^{−1/3}(1 + o(ℓ^{1/3}))`. Also `ν_cand^{near,r_0} − cℓ^{−1/3} → ∫_{|y|=1 and each finite derivative order q" quoted at the definition +of `T`; B2 the signed near bound proved in §2 as a separate remark (from `A_r ≥ 0` for every `r` and `A_0 ≤ k²H`), +not part of (R+.3); B1 the not-claimed list extended (no RN/24-jet closure; R+ does not enlarge (Z4)). Lemma E, +Theorem R+ and (R+.1)–(R+.3) unchanged. + +Review slices (§6): A Lemma E; B the separation-variable limit argument and the use of Theorem Z; C scope and +remarks. diff --git a/frontiers/remainder_vanishing_20260930/RESULTS.json b/frontiers/remainder_vanishing_20260930/RESULTS.json new file mode 100644 index 000000000..4d636012e --- /dev/null +++ b/frontiers/remainder_vanishing_20260930/RESULTS.json @@ -0,0 +1,85 @@ +{ + "checks": { + "E1_axial_pins_quintic": { + "info": { + "alpha_M": "-6k + (r/12) f4 - (r^2/120) f5", + "points": 3, + "product": "-36k^2 + r^2 (f4^2/144 - k f5/10) - r^4 f5^2/14400" + }, + "passed": true + }, + "E2_parity_pinned_quintic_2d": { + "info": { + "constant": "-1296", + "degrees_present": [ + 0, + 2, + 4, + 6, + 8, + 10 + ], + "r2_coefficient": "-3851051/2025", + "reflection_identity": "det H_M(-r) = det H_S(r) at 3 rational r" + }, + "passed": true + }, + "E3_block_and_inertia": { + "info": { + "cases": 64, + "positive_eigenvalue_cases": 12 + }, + "passed": true + }, + "E4_ledger": { + "info": { + "a^7/3 ell^-1/3": [ + "2", + "0", + "-7" + ], + "ell kappa^-5/3": [ + "7/12", + "0", + "0" + ], + "ell^2/3 a^-1/3": [ + "1/3", + "0", + "1" + ], + "k^-2/3 r^2": [ + "2/3", + "-4/3", + "0" + ], + "k^-2/3 r^3/k": [ + "1", + "-8/3", + "0" + ], + "kappa^7/3": [ + "7/12", + "0", + "0" + ], + "r^-2 (ell/r^3)^2 = ell^2 r^-8": [ + "2", + "0", + "-8" + ] + }, + "passed": true + }, + "E5_layer_algebra": { + "info": { + "identity": "k^2 u^2 = C1^2 r^2 T^2 (k+T)^2", + "inequality": "r T^2 <= k u / C1" + }, + "passed": true + } + }, + "object": "CL-D2-REMAINDER-VANISHING-20260930-v1.1", + "passed": true, + "scientific_effect": "NONE" +} diff --git a/frontiers/remainder_vanishing_20260930/SOURCES.json b/frontiers/remainder_vanishing_20260930/SOURCES.json new file mode 100644 index 000000000..73732cec0 --- /dev/null +++ b/frontiers/remainder_vanishing_20260930/SOURCES.json @@ -0,0 +1,142 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "object": "CL-D2-REMAINDER-VANISHING-20260930-v1.1", + "kind": "author-side proof candidate: Lemma E (even contact expansion) and Theorem R+ (nu_cand = c ell^-1/3 + B_{d,L} + o(1), nu_eld = c ell^-1/3 + o(1)) relative to the merged reviewed sources [R], [Z], [C7-K] and the reconciled [P] chain; 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not a nonauthor review)", + "findings": "no blocking; applied: Step 4 layer inequality false on the indefinite part (three-case split: lambda_max(A_0) >= u kills F_d(K_M); |lambda_max| < u via Weyl); Theorem R+ restated without any dependence on Math-#188 (Theorem Z (Z4) plus the ell-uniform candidate majorant from Lemma E identify the constant B_{d,L}); Proposition L removed as a proposition (it is [C7-K] section 4), Proposition C folded into (R+.3) with its limit identified; T dominates C^7; admissible radii stated; remarks corrected (attribution of the #186 model check, the rate remark, the manuscript remark, no-second-coefficient wording); controls: E2 recorded as symmetry-forced with the exact reflection identity added, E3 exercises the indefinite layer, E4 tautology removed" + }, + "files": [ + { + "path": "PROOF.md", + "bytes": 25353, + "sha256": "0cb5b949261476f10e539677e2744e3288752474393af25ba787e67b1350f32a" + }, + { + "path": "even_contact_check.py", + "bytes": 20220, + "sha256": "473ae49f2d4b3a319b5bd90a28ef34c7c9ae8a6f7af953d75ebc39969ea02f74" + }, + { + "path": "RESULTS.json", + "bytes": 1393, + "sha256": "b58b5e90419415c69aa16f1a13e09920147db9e24b131bd53864beb67fb6e428" + }, + { + "path": "README.md", + "bytes": 4393, + "sha256": "f3078a42edc8a0dbe88b6cca645ddff1518366756c2423b3ffc898bb283cd02f" + } + ] +} diff --git a/frontiers/remainder_vanishing_20260930/even_contact_check.py b/frontiers/remainder_vanishing_20260930/even_contact_check.py new file mode 100644 index 000000000..5c1cd64ae --- /dev/null +++ b/frontiers/remainder_vanishing_20260930/even_contact_check.py @@ -0,0 +1,422 @@ +#!/usr/bin/env python3 +"""Exact finite controls for PROOF.md (CL-D2-REMAINDER-VANISHING-20260930-v1.1). Standard library only; exact rationals. + + E1 axial pin identities on a quintic profile f(x) = b0 + f1 x + f2 x^2/2 + f3 x^3/6 + f4 x^4/24 + f5 x^5/120: with + b0, f1, f2, f3 solved from the four pins f(-r/2) = b, f(r/2) = b - k r^3, f'(-r/2) = f'(r/2) = 0 one finds + f2 = -(r^2/24) f4, f3 = 12k - (r^2/40) f5, and alpha_M = f''(-r/2)/r = -6k + (r/12) f4 - (r^2/120) f5, + alpha_S = f''(r/2)/r = 6k + (r/12) f4 + (r^2/120) f5, alpha_M alpha_S = -36k^2 + r^2 (f4^2/144 - k f5/10) + - r^4 f5^2/14400 -- all exact at rational (r, k, f4, f5, b); mutant M1 drops the (r^2/40) f5 term; + E2 reflection symmetry: for a general quintic in (x, y) with fifteen free rational coefficients, the six pinned + coefficients (c00, c10, c20, c30, c01, c11) are solved exactly at rational r and at -r from f(M) = b, + f(S) = b - k r^3, grad f(M) = grad f(S) = 0, M = (-r/2, 0), S = (r/2, 0); the identity det H_M(-r) = det H_S(r) + holds exactly; consequently det H_M det H_S / r^2, recovered by exact Lagrange interpolation at eleven rational r + as the polynomial of degree <= 10 in r, has only even powers, constant term -36 k^2 A0^2 (A0 = f_yy(0) = 2 c02) + and the recorded r^2 coefficient (the parity is forced by the symmetry: a check of the mechanism, not of the + bookkeeping); mutant M2 applies the parity test to the single factor det H_M / r, which is not even; + E3 the block identity det K = alpha det A - r beta^T adj(A) beta for K = [[alpha, sqrt(r) beta^T], [sqrt(r) beta, A]] + ([R] Lemma R3.2 with t = r; r a rational square, entries sqrt(r) beta), and Haynsworth's inertia additivity + inertia(K) = inertia(A) + inertia(alpha - r beta^T A^{-1} beta), on rational 2x2/3x3 blocks A that are negative + definite, of index m - 1, and indefinite, with the scalar Schur complement of both signs (inertia by exact + congruence diagonalisation); the Step 4 case split: F_d(K_M) = 0 whenever lambda_max(A_M) > 0 (A_M has a + positive eigenvalue), and |det A| <= |lambda_max(A)| * ||A||_op^(m-1) when |lambda_max(A)| is the smallest + absolute eigenvalue, at rational points; + E4 the ledger of section 2 and Remark 2 by exact monomial substitution r = (ell/k)^(1/3): r^-2 (ell/r^3)^2 = + ell^2 r^-8 (so the far-cutoff tail is ell^2 r0^-7), k^(-2/3) r^3 / k = ell k^(-8/3), ell kappa^(-5/3) = kappa^(7/3) + = ell^(7/12) at kappa = ell^(1/4), ell^(2/3) a^(-1/3) = r0 ell^(1/3) for a = ell r0^-3, and r <= k iff ell <= k^4 at + rational (k, r) with ell = k r^3; mutant M3 claims the loss exponent 1/3; + E5 the layer algebra of Step 4: k^2 u^2 = C1^2 r^2 T^2 (k + T)^2 and r T^2 <= k u / C1 for u = C1 r T (1 + T/k), exactly. + M mutants (--mutant M1|M2|M3) exit 1; an unknown label exits 2. + Scope: E1-E5 check exact identities and finitely many rational instances. They do NOT test Step 4 of PROOF.md as a + statement about random matrices (the Weyl/Hadamard layer bound, the choice of C1, the O(r^2) product on the flip + layer), nor Step 5 (the C^7 moment import from [R] (R4) and the expectations), nor section 2 (majorant, pointwise + and far limits, the use of (Z4)); those are proved in prose only. The controls are not acceptance. +""" +import argparse +import json +import sys +from fractions import Fraction as F + + +# ---------------------------------------------------------------------------------------------------- linear algebra +def solve(Mx, rhs): + n = len(Mx) + A = [list(r) + [b] for r, b in zip(Mx, rhs)] + for col in range(n): + piv = next((r for r in range(col, n) if A[r][col] != 0), None) + if piv is None: + return None + A[col], A[piv] = A[piv], A[col] + inv = 1/A[col][col] + A[col] = [v*inv for v in A[col]] + for r in range(n): + if r != col and A[r][col] != 0: + f = A[r][col] + A[r] = [a - f*b for a, b in zip(A[r], A[col])] + return [A[r][n] for r in range(n)] + + +def det(M): + n = len(M) + if n == 1: + return M[0][0] + total = F(0) + for j in range(n): + minor = [row[:j] + row[j+1:] for row in M[1:]] + total += (-1)**j*M[0][j]*det(minor) + return total + + +def adj(M): + n = len(M) + out = [[F(0)]*n for _ in range(n)] + for i in range(n): + for j in range(n): + minor = [row[:j] + row[j+1:] for r_, row in enumerate(M) if r_ != i] + out[j][i] = (-1)**(i+j)*det(minor) if n > 1 else F(1) + return out + + +def inertia(M): + """(negatives, zeros, positives) of a symmetric rational matrix by congruence diagonalisation.""" + n = len(M) + A = [row[:] for row in M] + neg = zero = pos = 0 + for i in range(n): + if A[i][i] == 0: + piv = next((j for j in range(i + 1, n) if A[j][j] != 0), None) + if piv is not None: + A[i], A[piv] = A[piv], A[i] + for row in A: + row[i], row[piv] = row[piv], row[i] + else: + j = next((j for j in range(i + 1, n) if A[i][j] != 0), None) + if j is None: + zero += 1 + continue + # row_i += row_j, col_i += col_j (congruence) makes the (i,i) entry 2 A[i][j] + A[j][j] = 2 A[i][j] + for c in range(n): + A[i][c] += A[j][c] + for r_ in range(n): + A[r_][i] += A[r_][j] + p = A[i][i] + if p == 0: + zero += 1 + continue + if p < 0: + neg += 1 + else: + pos += 1 + for r_ in range(i + 1, n): + if A[r_][i] != 0: + f = A[r_][i]/p + for c in range(n): + A[r_][c] -= f*A[i][c] + for c in range(n): + A[c][r_] -= f*A[c][i] + return neg, zero, pos + + +# ---------------------------------------------------------------------------------------------------- E1 +def check_E1(mutant): + n = 0 + for (r, k, f4, f5, b) in ((F(1, 10), F(2), F(3), F(-7), F(1, 3)), (F(1, 3), F(1, 5), F(-11), F(13), F(0)), + (F(2, 7), F(4), F(1, 2), F(1, 3), F(-5))): + f2 = -(r*r/24)*f4 + f3 = 12*k - (r*r/40)*f5 + if mutant == 'M1': + f3 = 12*k + # f1, b0 from the gradient and height pins + def fx(x, f1): + return f1 + f2*x + f3*x*x/2 + f4*x**3/6 + f5*x**4/24 + # f'(-r/2) = 0 gives f1 + f1 = -(f2*(-r/2) + f3*(r*r/4)/2 + f4*(-r/2)**3/6 + f5*(r/2)**4/24) + if fx(r/2, f1) != 0: + return False, 'gradient pin at S fails' + def f(x, b0): + return b0 + f1*x + f2*x*x/2 + f3*x**3/6 + f4*x**4/24 + f5*x**5/120 + b0 = b - (f(-r/2, F(0))) + if f(-r/2, b0) != b or f(r/2, b0) != b - k*r**3: + return False, 'height pins fail' + def fxx(x): + return f2 + f3*x + f4*x*x/2 + f5*x**3/6 + aM, aS = fxx(-r/2)/r, fxx(r/2)/r + if aM != -6*k + (r/12)*f4 - (r*r/120)*f5 or aS != 6*k + (r/12)*f4 + (r*r/120)*f5: + return False, 'alpha expansions' + if aM*aS != -36*k*k + r*r*(f4*f4/144 - k*f5/10) - r**4*f5*f5/14400: + return False, 'alpha product' + n += 1 + return True, {'points': n, 'alpha_M': '-6k + (r/12) f4 - (r^2/120) f5', 'product': '-36k^2 + r^2 (f4^2/144 - k f5/10) - r^4 f5^2/14400'} + + +# ---------------------------------------------------------------------------------------------------- E2 +MONOS = [(i, j) for i in range(6) for j in range(6 - i)] # 21 monomials x^i y^j, i + j <= 5 +PINNED = [(0, 0), (1, 0), (2, 0), (3, 0), (0, 1), (1, 1)] +FREE = [m for m in MONOS if m not in PINNED] + + +def d_coef(i, j, dx, dy, x, y): + """value of d^dx_x d^dy_y (x^i y^j) at (x, y).""" + c = F(1) + for t in range(dx): + if i - t <= 0: + return F(0) + c *= (i - t) + for t in range(dy): + if j - t <= 0: + return F(0) + c *= (j - t) + return c*x**(i - dx)*y**(j - dy) + + +def pinned_poly(r, k, b, free, sign): + """Return the full coefficient dict of the pinned quintic (sign = +1: f(S) = b - k r^3; sign = -1: b + k r^3).""" + M = (-r/2, F(0)); S = (r/2, F(0)) + conds = [((0, 0), M, b), ((0, 0), S, b - sign*k*r**3), ((1, 0), M, F(0)), ((1, 0), S, F(0)), ((0, 1), M, F(0)), ((0, 1), S, F(0))] + rows, rhs = [], [] + for (dx, dy), pt, val in conds: + rows.append([d_coef(i, j, dx, dy, *pt) for (i, j) in PINNED]) + rhs.append(val - sum(free[m]*d_coef(m[0], m[1], dx, dy, *pt) for m in FREE)) + sol = solve(rows, rhs) + coef = dict(free) + for m, v in zip(PINNED, sol): + coef[m] = v + return coef + + +def hess(coef, x, y): + fxx = sum(c*d_coef(i, j, 2, 0, x, y) for (i, j), c in coef.items()) + fxy = sum(c*d_coef(i, j, 1, 1, x, y) for (i, j), c in coef.items()) + fyy = sum(c*d_coef(i, j, 0, 2, x, y) for (i, j), c in coef.items()) + return fxx, fxy, fyy + + +def interpolate(xs, ys): + """Coefficients (low to high) of the unique polynomial of degree < len(xs) through (xs, ys): exact Lagrange.""" + n = len(xs) + coeffs = [F(0)]*n + for i in range(n): + # basis polynomial l_i + num = [F(1)] + denom = F(1) + for j in range(n): + if j != i: + num = [F(0)] + num + for t in range(len(num) - 1): + num[t] -= xs[j]*num[t + 1] + denom *= (xs[i] - xs[j]) + for t in range(n): + coeffs[t] += ys[i]*num[t]/denom + return coeffs + + +def check_E2(mutant): + k, b = F(3, 2), F(-1, 4) + free = {} + vals = [F(2), F(-1), F(1, 2), F(3), F(-2, 3), F(5), F(-1, 5), F(7, 3), F(1), F(-4), F(2, 7), F(-3, 2), F(4, 5), F(-1, 7), F(6)] + for m, v in zip(FREE, vals): + free[m] = v + sign = 1 + def dets(r): + coef = pinned_poly(r, k, b, free, sign) + fxxM, fxyM, fyyM = hess(coef, -r/2, F(0)) + fxxS, fxyS, fyyS = hess(coef, r/2, F(0)) + return fxxM*fyyM - fxyM*fxyM, fxxS*fyyS - fxyS*fxyS + # the reflection symmetry: the pinned family at -r has M and S exchanged, so det H_M(-r) = det H_S(r) + for r in (F(1, 10), F(3, 7), F(1)): + dMm, dSm = dets(-r) + dMp, dSp = dets(r) + if dMm != dSp or dSm != dMp: + return False, 'reflection identity det H_M(-r) = det H_S(r) fails' + def value(r): + dM, dS = dets(r) + if mutant == 'M2': + return dM/r # a single endpoint: NOT even in r + return dM*dS/(r*r) + xs = [F(n, 10) for n in range(1, 12)] # eleven rational r values + ys = [value(r) for r in xs] + poly = interpolate(xs, ys) + # verify at a twelfth value that the degree-10 interpolant is the polynomial (degree <= 10 in r) + r12 = F(13, 10) + if sum(c*r12**t for t, c in enumerate(poly)) != value(r12): + return False, 'not a polynomial of degree <= 10 in r' + odd = [t for t in range(1, 11, 2) if poly[t] != 0] + if odd: + return False, 'odd powers present: %s' % odd + A0 = 2*free[(0, 2)] + if poly[0] != -36*k*k*A0*A0: + return False, 'constant term is not -36 k^2 A0^2' + c02, c12, c21, c22, c31, c40, c50 = (free[(0, 2)], free[(1, 2)], free[(2, 1)], free[(2, 2)], free[(3, 1)], free[(4, 0)], free[(5, 0)]) + r2 = (16*c02**2*c40**2 - 48*c02**2*c50*k - 8*c02*c21**2*c40 + 24*c02*c21*c31*k - 72*c02*c22*k*k + + 36*c12**2*k*k - 12*c12*c21**2*k + c21**4) + if poly[2] != r2: + return False, 'r^2 coefficient differs from the displayed polynomial' + return True, {'reflection_identity': 'det H_M(-r) = det H_S(r) at 3 rational r', 'degrees_present': [t for t in range(11) if poly[t] != 0], + 'constant': str(poly[0]), 'r2_coefficient': str(poly[2])} + + +# ---------------------------------------------------------------------------------------------------- E3 +def block(alpha, beta, A, r): + n = len(A) + 1 + K = [[F(0)]*n for _ in range(n)] + K[0][0] = alpha + for i in range(len(A)): + K[0][i+1] = K[i+1][0] = r*beta[i] # sqrt(r) beta entries squared give r; with r rational we use t = r directly + for j in range(len(A)): + K[i+1][j+1] = A[i][j] + return K + + +def check_E3(mutant): + # [R] Lemma R3.2 with t = r: K = [[alpha, sqrt(r) beta^T], [sqrt(r) beta, A]] and det K = alpha det A - r beta^T adj(A) beta. + # Exact rationals: r is chosen a rational square, s = sqrt(r) rational, and the off-diagonal entries are s*beta, + # so the identity tested is det K = alpha det A - s^2 q = alpha det A - r q with q = beta^T adj(A) beta. + cases = 0 + A_list = [ + [[F(-2), F(1, 2)], [F(1, 2), F(-3)]], # negative definite + [[F(-2), F(0)], [F(0), F(1)]], # index m-1 = 1 + [[F(-3), F(1), F(0)], [F(1), F(-2), F(1, 2)], [F(0), F(1, 2), F(-4)]], # negative definite 3x3 + [[F(-3), F(1), F(0)], [F(1), F(2), F(1, 2)], [F(0), F(1, 2), F(-4)]], # indefinite 3x3 + ] + for A in A_list: + m = len(A) + adjA = adj(A); dA = det(A) + for beta in ([F(1), F(-2)] if m == 2 else [F(1), F(-2), F(1, 3)], [F(0)]*m): + for alpha in (F(-5), F(-1, 10), F(1, 10), F(5)): + for s in (F(1, 2), F(2)): # sqrt(r) = s, r = s^2 + r = s*s + n = m + 1 + K = [[F(0)]*n for _ in range(n)] + K[0][0] = alpha + for i in range(m): + K[0][i+1] = K[i+1][0] = s*beta[i] + for j in range(m): + K[i+1][j+1] = A[i][j] + q = sum(beta[i]*sum(adjA[i][j]*beta[j] for j in range(m)) for i in range(m)) + if det(K) != alpha*dA - r*q: + return False, 'block determinant identity' + if dA != 0: + sigma = alpha - r*q/dA # alpha - r beta^T A^{-1} beta (A^{-1} = adj/det) + negK = inertia(K)[0] + negA = inertia(A)[0] + if negK != negA + (1 if sigma < 0 else 0): + return False, 'Haynsworth inertia additivity fails' + cases += 1 + # good-event inequalities: A_S negative definite and alpha_S > 0 give sigma_S > 0 + A = A_list[0]; adjA = adj(A); dA = det(A) + for beta in ([F(1), F(-2)], [F(3), F(1, 7)]): + q = sum(beta[i]*sum(adjA[i][j]*beta[j] for j in range(2)) for i in range(2)) + for alpha in (F(1, 100), F(3)): + for r in (F(1, 10), F(1)): + if not (alpha - r*q/dA > 0): + return False, 'sigma_S > 0 fails' + # sigma_M < 0 under the margins: alpha_M <= -5k, r beta^T(-A)^{-1}beta <= 2rT^2/u <= 2k/C1 + k, T, C1, r = F(1), F(4), F(100), F(1, 1000) + u = C1*r*T*(1 + T/k) + if not (2*r*T*T/u <= 2*k/C1 and -5*k + 2*k/C1 < 0): + return False, 'sigma_M margin' + # Step 4, second layer: A_M with a positive eigenvalue => K_M has a positive eigenvalue => F_d(K_M) = 0 + layer2 = 0 + for A in ([[F(-2), F(0)], [F(0), F(1, 50)]], [[F(-3), F(1)], [F(1), F(1, 10)]], [[F(-3), F(1), F(0)], [F(1), F(1, 20), F(1, 2)], [F(0), F(1, 2), F(-4)]]): + m = len(A) + if inertia(A)[2] == 0: + return False, 'test matrix has no positive eigenvalue' + for alpha in (F(-5), F(-1, 10)): + for s_ in (F(1, 2), F(1, 10)): + beta = [F(1)]*m + n = m + 1 + K = [[F(0)]*n for _ in range(n)] + K[0][0] = alpha + for i in range(m): + K[0][i+1] = K[i+1][0] = s_*beta[i] + for j in range(m): + K[i+1][j+1] = A[i][j] + if inertia(K)[2] == 0: # K negative semidefinite would give index d + return False, 'K_M negative definite although A_M has a positive eigenvalue' + layer2 += 1 + # Step 4, third layer: |det A| <= |lambda_max(A)| ||A||_op^(m-1) when lambda_max is the eigenvalue of least modulus: + # diagonal examples, exact + for diag in ([F(-3), F(-1, 100)], [F(-5), F(-2), F(1, 1000)], [F(-4), F(-3), F(-1, 10)]): + A = [[F(0)]*len(diag) for _ in diag] + for i, dv in enumerate(diag): + A[i][i] = dv + lam_max = max(diag) + opnorm = max(abs(v) for v in diag) + if not (abs(det(A)) <= abs(lam_max)*opnorm**(len(diag) - 1)): + return False, 'layer determinant bound' + return True, {'cases': cases, 'positive_eigenvalue_cases': layer2} + + +# ---------------------------------------------------------------------------------------------------- E4 +class Mono: + def __init__(self, ell=F(0), k=F(0), r0=F(0)): + self.e = (F(ell), F(k), F(r0)) + def __mul__(self, o): + return Mono(*[a + b for a, b in zip(self.e, o.e)]) + def __pow__(self, p): + p = F(p); return Mono(*[a*p for a in self.e]) + + +def check_E4(mutant): + ell, k, r0 = Mono(1, 0, 0), Mono(0, 1, 0), Mono(0, 0, 1) + r = (ell*k**(-1))**F(1, 3) + a = ell*r0**(-3) + kappa = ell**F(1, 4) + rr = Mono(0, 0, 1) # a separation variable, reusing the third slot + claims = { + 'r^-2 (ell/r^3)^2 = ell^2 r^-8': (((rr**(-2))*(ell*rr**(-3))**2).e, (F(2), F(0), F(-8))), + 'k^-2/3 r^2': ((k**F(-2, 3)*r**2).e, (F(2, 3), F(-4, 3), F(0))), + 'k^-2/3 r^3/k': ((k**F(-2, 3)*r**3*k**(-1)).e, (F(1), F(-8, 3), F(0))), + 'ell^2/3 a^-1/3': (((ell**F(2, 3))*a**F(-1, 3)).e, (F(1, 3), F(0), F(1))), + 'ell kappa^-5/3': ((ell*kappa**F(-5, 3)).e, (F(7, 12), F(0), F(0))), + 'kappa^7/3': ((kappa**F(7, 3)).e, (F(7, 12), F(0), F(0))), + 'a^7/3 ell^-1/3': ((a**F(7, 3)*ell**F(-1, 3)).e, (F(2), F(0), F(-7))), + } + for name, (got, want) in claims.items(): + if got != want: + return False, 'monomial claim fails: %s got %s' % (name, got) + # absolute loss exponent: ell^(7/12) scaled -> ell^(7/12 - 1/3) = ell^(1/4) + loss = F(7, 12) - F(1, 3) + if mutant == 'M3': + loss = F(1, 3) + if loss != F(1, 4): + return False, 'loss exponent' + # r <= k <=> ell <= k^4, with r = (ell/k)^(1/3) computed exactly where ell/k is a rational cube + for (K, rv) in ((F(1, 10), F(1, 10)), (F(1, 10), F(1, 5)), (F(1, 2), F(1, 4)), (F(1, 16), F(1, 8))): + L = K*rv**3 # ell = k r^3, so r = (ell/k)^(1/3) exactly + if (rv <= K) != (L <= K**4): + return False, 'threshold r <= k <=> ell <= k^4 fails' + return True, {name: [str(x) for x in got] for name, (got, want) in claims.items()} + + +# ---------------------------------------------------------------------------------------------------- E5 +def check_E5(mutant): + for (k, T, C1, r) in ((F(1, 3), F(5), F(50), F(1, 100)), (F(4), F(2), F(10), F(1, 7)), (F(1, 100), F(30), F(1000), F(1, 10**6))): + u = C1*r*T*(1 + T/k) + if k*k*u*u != C1*C1*r*r*T*T*(k + T)**2: + return False, 'k^2 u^2 identity' + if not (r*T*T <= k*u/C1): + return False, 'rT^2 <= ku/C1' + return True, {'identity': 'k^2 u^2 = C1^2 r^2 T^2 (k+T)^2', 'inequality': 'r T^2 <= k u / C1'} + + +def main(): + ap = argparse.ArgumentParser() + ap.add_argument('--mutant', default=None) + args = ap.parse_args() + mutant = args.mutant + if mutant is not None and mutant not in ('M1', 'M2', 'M3'): + sys.stderr.write('unknown mutant label\n'); sys.exit(2) + results = {} + ok_all = True + for name, fn in (('E1_axial_pins_quintic', check_E1), ('E2_parity_pinned_quintic_2d', check_E2), ('E3_block_and_inertia', check_E3), + ('E4_ledger', check_E4), ('E5_layer_algebra', check_E5)): + ok, info = fn(mutant) + results[name] = {'passed': bool(ok), 'info': info} + ok_all = ok_all and ok + out = {'object': 'CL-D2-REMAINDER-VANISHING-20260930-v1.1', 'scientific_effect': 'NONE', 'passed': ok_all, 'checks': results} + print(json.dumps(out, indent=1, sort_keys=True)) + sys.exit(0 if ok_all else 1) + + +if __name__ == '__main__': + main()