D2: sharp joint-mass Schur floor, attaining law and rational certificate

reviews/d2_joint_mass_schur_20261006/NOTE.md

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+# D2: the sharp joint-mass Schur floor reduces to one scalar minimum
+
+Object: D2-JOINT-MASS-SCHUR-20261006-r6.
+Dylan Roy — delegated AI work. Actual author: OpenAI / GPT-6 Astra Pro,
+`canonical-joint-mass-schur-20261006-r6`, pickup Math-#328/6016666735.
+**NEW AUTHOR-SIDE MATHEMATICAL CANDIDATE; independent source review pending.**
+Scientific effect NONE; organizational-independence credit 0. Prior sources and
+reviews are exposed. No earlier source, review, status, premise or flag changes.
+
+## 1. What is additional, and what is not
+
+#326 is the primary source for the common moment floors under squared-radius
+mass lower bounds. #328 gives the conservative upper-moment-free Schur bound.
+Part B of #328 comment6008747995 gives a better bound from an independent lower
+input pair; comment6016072968 reviews that exact observation, and separate PR371
+is its source-publication route. None of those verdicts reviews this note.
+
+The question left outside those scopes is the *joint* constraint: the smallest
+second moment and smallest fourth-moment gap need not occur in one law. Here the
+exact lower envelope of the feasible moment pairs is computed. This reduces the
+optimal all-direction Schur constant over the full prescribed probability-law
+class to a compact one-variable minimum and supplies an attaining law. A concrete
+example has an exact algebraic minimizer and a rational outward certificate.
+
+The residual square identity is elementary variance decomposition, not a claim
+of inventing a general inequality. No global literature novelty is asserted.
+This is not a consolidation of the earlier packets or a prerequisite for their
+integration. It is not a Gaussian/Palm covariance identification or field result.
+
+## 2. Probability class and exact scalar expressions
+
+Fix real numbers p,q>0, P=p+q<=1, and distinct u,v>0. Let Y be a nonnegative
+measurable random variable on a probability space satisfying
+
+    Prob(Y=u)>=p, Prob(Y=v)>=q, E[Y^2]<infinity.          (2.1)
+
+Equivalently one may take Y=X^2 with E[X^4]<infinity. A sixth moment is not needed
+for this Schur-only statement. It is not a tau/Delta theorem with a dropped
+sixth-moment premise. Set
+
+    s=E[Y], g=E[Y^2]-s^2, a=pu+qv, c=a/P,
+    w=1-P, g0=pq(u-v)^2/P.                              (2.2)
+
+The original mass lower bounds, not necessarily actual atom masses, are p,q.
+Since u!=v, the specified atom events are disjoint. We will prove s>=a>0,g>=g0>0.
+For x=s^2 and t in the closed interval [0,1/4], define
+
+    D0=x(g+x), D1=g(g+4x)/4, N=xg(g+2x),
+    D(t)=(1-4t)D0+4tD1, S(t)=N/D(t),
+    F0(x,g)=g(g+2x)/(g+x),
+    F1(x,g)=4x(g+2x)/(g+4x).                             (2.3)
+
+These are the exact existing D2Schur numerator/denominator expressions after
+m2=s,m4=g+s^2. They are scalar expressions. Their identification with an actual
+conditional covariance is not supplied by this note. All denominators below
+are positive in the stated domain; no totalized division at zero is used.
+
+## 3. Exact feasible lower variance curve
+
+Suppose 0<P<1. Let mu be the law of Y and subtract the prescribed atom masses:
+
+    rho=mu-p delta_u-q delta_v.
+
+This is a nonnegative measure of total mass w, even when mu has larger masses
+at u or v. Its first moment is s-a>=0 and its second moment is finite. Put
+z=(s-a)/w. Expansion of the nonnegative square integral gives
+
+    integral (y-z)^2 d rho
+      = E[Y^2]-pu^2-qv^2-(s-a)^2/w >=0.                (3.1)
+
+Consequently
+
+    g >= h(s):=pu^2+qv^2+(s-a)^2/w-s^2
+              =g0+(P/w)(s-c)^2,   s>=a.               (3.2)
+
+For the last equality use pu^2+qv^2=g0+a^2/P and w=1-P.
+Thus s>=a>0 and g>=g0>0 follow without using an earlier result as an assumption.
+
+For EVERY finite s>=a equality in (3.2) is realized by the probability law
+
+    mu_s=p delta_u+q delta_v+w delta_((s-a)/w).         (3.3)
+
+The residual radius is nonnegative. Repeated atom locations simply combine
+masses and still satisfy (2.1). Equality in (3.1) means rho is concentrated at z,
+because a nonnegative integrable square has integral zero only when it vanishes
+rho-almost everywhere. This is also a characterization of lower-envelope laws.
+The construction has finite moments of every order.
+
+At s=a, h(a)=g0+wa^2/P>g0; at s=c, h(c)=g0 but c>a. Thus when w>0 no law attains
+both s=a and g=g0. This joint obstruction concerns s and g, not the already
+separate g and Delta extremizers in #326/#328.
+
+If P=1, rho has mass zero, mu=p delta_u+q delta_v, s=a=c and g=g0. Formula (3.2)
+with division by w is NOT used. This case is retained separately throughout.
+
+## 4. Exact sharp all-direction lower constant
+
+Both endpoints D0,D1 and N are positive. Since D(t) is a convex interpolation,
+
+    min_(0<=t<=1/4) S(t)
+      =N/max(D0,D1)=min(F0(s^2,g),F1(s^2,g)).           (4.1)
+
+The minimum is attained at t=0 when D0>=D1, and at t=1/4 when D1>=D0.
+Equality of the denominators makes every t equivalent. Endpoint inclusion is
+material; no bound is asserted outside [0,1/4].
+
+Both Fi are STRICTLY increasing in each positive coordinate. For X>=x>0,
+G>=g>0, the exact differences (including zero increments) are
+
+    F0(X,g)-F0(x,g)=g^2(X-x)/((g+X)(g+x));
+    F0(x,G)-F0(x,g)=(G-g)+x^2(G-g)/((G+x)(g+x));
+    F1(X,g)-F1(x,g)=2(X-x)+2g^2(X-x)/((g+4X)(g+4x));
+    F1(x,G)-F1(x,g)=8x^2(G-g)/((G+4x)(g+4x)).          (4.2)
+
+These are the credited Part B identities, included explicitly. Every denominator
+is positive and a strictly positive increment makes each displayed difference
+strictly positive. Therefore (3.2) and (4.1) give, for each law with mean s,
+
+    S(t)>=Phi(s):=min(F0(s^2,h(s)),F1(s^2,h(s))).        (4.3)
+
+Every value Phi(s), s>=a, is attained by (3.3) and a suitable endpoint t. For
+s>=c, both s^2 and h(s) are nondecreasing in s, so each endpoint value, hence
+Phi(s), is nondecreasing. Phi is continuous on [a,c], a nonempty compact
+interval. Its minimum is attained. It follows that
+
+    C(p,q,u,v):=min_(a<=s<=c) Phi(s)                    (4.4)
+      = inf over ALL laws (2.1) AND all t in [0,1/4] of S(t).
+
+This is an EXACT SHARP constant for the declared class, not merely a lower
+bound obtained by sampling s. An extremizer is mu_(s_star) from (3.3), with
+s_star any minimizer in (4.4) and t its maximizing-denominator endpoint.
+
+For X with specified individual nonzero atoms b1,b2, b1^2=u!=v=b2^2, the same
+infimum holds: choose p delta_b1+q delta_b2+w delta_sqrt((s_star-a)/w).
+No sign symmetry or independence is necessary. For the broader squared-radius
+class any signs may be chosen without changing the scalar moments.
+
+When P=1 the exact constant is min(F0(a^2,g0),F1(a^2,g0)); the preceding
+one-variable interval collapses to a point, without division by w.
+
+## 5. Strict improvement over independent lower inputs
+
+For 0<P<1, let B=min(F0(a^2,g0),F1(a^2,g0)), the credited Part B lower-input
+bound. Every s in [a,c] has s^2>=a^2 and h(s)>=g0, but these cannot both be
+equalities: equality of s^2 requires s=a whereas equality of h requires s=c>a.
+By (4.2), BOTH Fi(s^2,h(s))>Fi(a^2,g0). Each is therefore greater than B, so
+Phi(s)>B everywhere on the compact interval. Taking its attained minimum gives
+
+    C(p,q,u,v)>B>=min(g0,2a^2),   when 0<P<1.          (5.1)
+
+The strict positive gain depends on the fixed inputs. No uniform gain is
+claimed as w tends to zero, a mass vanishes, or the radii coalesce. At P=1,
+C=B, consistently with simultaneous attainment. Scaling u,v by k>0 scales
+s by k, g by k^2 and C by k^2, equivalently degree four under X->lambda X.
+
+## 6. A fully certified sharp example
+
+Take p=q=1/4,u=1,v=4. Then P=w=1/2, a=5/4,c=5/2,g0=9/8 and
+
+    h(s)=9/8+(s-5/2)^2,     5/4<=s<=5/2.
+
+Write f0(s)=F0(s^2,h(s)), f1(s)=F1(s^2,h(s)). Direct simplification gives
+
+    f0=(8s^2-40s+59)(24s^2-40s+59)/(8(16s^2-40s+59)),
+    f1=4s^2(24s^2-40s+59)/(40s^2-40s+59).
+
+Their derivatives are
+
+    f0'(s)=P5(s)/(16s^2-40s+59)^2,
+    f1'(s)=8s R4(s)/(40s^2-40s+59)^2,
+    P5=768s^5-5440s^4+18464s^3-36320s^2+37524s-17405,
+    R4=960s^4-2240s^3+4432s^2-4720s+3481.             (6.1)
+
+The denominators are positive because they are positive multiples of the
+positive endpoint-ratio denominators in (2.3).
+
+### 6.1 Exact sign certificate, not sampled derivatives
+
+Put z=(s-5/4)/(5/4), so z in [0,1]. For a degree-n polynomial write
+sum_(j=0..n) b_j binomial(n,j) z^j(1-z)^(n-j). The EXACT Bernstein coefficients
+of P5 on this interval, from low index to high, are
+
+    (-2125, -8351/4, -10127/4, -5639/2, -2626, 405).
+
+All are negative except the last. For 0<z<=1 divide by z^5 and put r=(1-z)/z.
+The result is 405 minus a polynomial in r with strictly positive coefficients
+in every degree 1 through 5. It is strictly decreasing in r>=0, equals405 at
+r=0 and tends to minus infinity. Since r decreases as z increases, P5 has
+exactly ONE zero s_star in (5/4,5/2), is negative before it and positive after
+it. This elementary sign proof does not assume P5 itself is increasing.
+
+The exact degree-four Bernstein coefficients of R4 are
+
+    (9899/4, 14099/4, 35311/6, 10806, 21881),
+
+all strictly positive. The basis is nonnegative and sums to one, so R4>0 on the
+whole interval. Hence f0 has its unique minimum at s_star and f1 is increasing.
+Furthermore f0(5/2)<f1(5/4), an exact rational comparison. Consequently the
+minimum of min(f0,f1) is f0(s_star), rather than the f1 endpoint minimum.
+At s_star, f0<f1, so the attaining direction is t=0.
+
+### 6.2 Rational outward enclosure
+
+The checker derives the Bernstein coefficients by exact rational affine
+substitution, verifies their signs, and performs80 rational bisections of P5.
+This yields rational lo<s_star<hi inside(5/4,5/2), with P5(lo)<0<P5(hi).
+On that interval h is decreasing and s^2 is increasing. Therefore (4.2) gives
+
+    F0(lo^2,h(hi)) < f0(s_star) < F0(hi^2,h(lo)).        (6.2)
+
+These are rational endpoint evaluations, not floating-point quadrature. Integer
+floor/ceiling of the rational bounds at12 decimal places certifies
+
+    2.464780131951 < s_star < 2.464780131952,
+    2.076345574173 < C(1/4,1/4,1,4) < 2.076345574174.   (6.3)
+
+The exact defining polynomial, rational brackets, Bernstein coefficients and
+rounding directions are stored in RESULTS.json. The extremizing law is
+
+    X ~ (1/4)delta_1 + (1/4)delta_2
+          + (1/2)delta_sqrt(2s_star-5/2),   t=0.        (6.4)
+
+This is a finite probability law, not the actual periodic spectral law. The
+previous independent-lower-input bound is153/86; the conservative bound is9/8.
+All remain correct; (6.3) is sharper because it retains joint feasibility.
+
+## 7. Exact sources, verification and limits
+
+Source correspondence, not inherited review or runtime dependencies:
+
+- Math-#326, commit544499292f1cf09fe1bd030ce25d7cd0d80fc38c,
+  `reviews/d2_two_mass_floors_20261006/NOTE.md`,
+  blob b3e3b337697e4bf8c15bdab3666e49dc8833ca80, §§1-3/5.
+- Math-#328, commit31f7a6b5a36045be7795a6461da067a8ada96343,
+  `reviews/d2_two_atom_floors_20261006/NOTE.md`,
+  blob cedbb44a237d1a158111dca81c30983f7efd37f1, §§2-6.
+- Math- commit2f8b6f372be383d752e9dd30d38234faa977243c,
+  `formal/ResearchFormalCoreR1/D2Schur.lean`,
+  blob b95460c0d263a32ea274b347079cca6aaab3d2e9: definitions, cancellation,
+  ratio and endpoint interpolation. Reading it is not new Lean execution.
+- #328 comment6008747995 Part B, created/updated2026-10-06T03:29:55Z;
+  its own technical comment review6016072968 and separate source PR371 retain
+  their existing scope. This note neither replaces nor silently incorporates
+  changes into either original packet.
+
+Run from the repository root:
+
+    python -B -S -m unittest discover -s tests -p test_d2_joint_mass_schur.py -v
+    python -B -O -S -m unittest discover -s tests -p test_d2_joint_mass_schur.py -v
+    python -B -S reviews/d2_joint_mass_schur_20261006/joint_mass.py
+
+The twelve methods test exact square identities, attaining laws, endpoint
+reduction, boundaries, finite strict-improvement examples, homogeneity, derivative
+coefficient identities, Bernstein conversion, root/constant enclosure and source/
+result identities. Grids are finite diagnostics; the continuum proofs are above.
+The source inventory is an identity record, not an independent authenticity root.
+The initial twelve missing-implementation assertions and later missing-inventory
+assertion are retained in the author's evidence, not rewritten as successful runs.
+
+The new root test is eligible for existing repository test discovery. No new
+workflow, primary formal source, dependency pin, gate or acceptance flag is edited.
+Actual hosted execution must be observed on the new candidate, not inferred from
+older green checks. Author execution, source review, kernel checking, statement
+alignment and scientific acceptance remain separate. Full exact-source review of
+this new reduction, sharpness proof and certificate is required before integration.
+No fixed-field rate, lifetime/elder/capture theorem, all-parameter uniform floor,
+PSD matrix generalization or new Lean target is supplied.

reviews/d2_joint_mass_schur_20261006/RESULTS.json

@@ -0,0 +1,57 @@
+{
+  "P_Bernstein_on_5over4_5over2": [
+    "-2125",
+    "-8351/4",
+    "-10127/4",
+    "-5639/2",
+    "-2626",
+    "405"
+  ],
+  "P_low_to_high": [
+    -17405,
+    37524,
+    -36320,
+    18464,
+    -5440,
+    768
+  ],
+  "R_Bernstein_on_5over4_5over2": [
+    "9899/4",
+    "14099/4",
+    "35311/6",
+    "10806",
+    "21881"
+  ],
+  "R_low_to_high": [
+    3481,
+    -4720,
+    4432,
+    -2240,
+    960
+  ],
+  "attainment": "X law: (1/4) delta_1 + (1/4) delta_2 + (1/2) delta_sqrt(2*s_star-5/2); t=0",
+  "bisection_steps": 80,
+  "conservative_bound": "9/8",
+  "independent_lower_input_bound": "153/86",
+  "independently_reviewed": false,
+  "minimizer_decimal_outer": [
+    "2.464780131951",
+    "2.464780131952"
+  ],
+  "minimizer_rational_interval": [
+    "5959472682378079742711305/2417851639229258349412352",
+    "11918945364756159485422615/4835703278458516698824704"
+  ],
+  "other_branch_minimum": "2325/572",
+  "schema_version": 1,
+  "schur_decimal_outer": [
+    "2.076345574173",
+    "2.076345574174"
+  ],
+  "schur_rational_interval": [
+    "8176247611270519455439377274381378278602008301002078059013263634404576793595181582248512734974046249/3937806747089372899591395466138308282178432065239123997106996595610664321010636370754249041419173888",
+    "1022030951408814931929923015742655872995741342258406235618917401649186913883002558234025917262282403/492225843386171612448924786636701490949169924814390648222077739948112415250121170665521388701351936"
+  ],
+  "scientific_effect": "NONE",
+  "scope": "sharp real scalar infimum for p=q=1/4, u=1, v=4; not a field certificate"
+}

reviews/d2_joint_mass_schur_20261006/SOURCES.json

@@ -0,0 +1,56 @@
+{
+  "author": "OpenAI / GPT-6 Astra Pro",
+  "author_task": "canonical-joint-mass-schur-20261006-r6",
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+  ],
+  "review_status": "PENDING_NONAUTHOR_SOURCE_REVIEW",
+  "schema_version": 1,
+  "scientific_effect": "NONE",
+  "upstream_correspondence": [
+    {
+      "commit": "544499292f1cf09fe1bd030ce25d7cd0d80fc38c",
+      "git_blob": "b3e3b337697e4bf8c15bdab3666e49dc8833ca80",
+      "path": "reviews/d2_two_mass_floors_20261006/NOTE.md",
+      "role": "prior floors and separate attainment, not inherited verdict"
+    },
+    {
+      "commit": "31f7a6b5a36045be7795a6461da067a8ada96343",
+      "git_blob": "cedbb44a237d1a158111dca81c30983f7efd37f1",
+      "path": "reviews/d2_two_atom_floors_20261006/NOTE.md",
+      "role": "scalar formulas and example"
+    },
+    {
+      "commit": "2f8b6f372be383d752e9dd30d38234faa977243c",
+      "git_blob": "b95460c0d263a32ea274b347079cca6aaab3d2e9",
+      "path": "formal/ResearchFormalCoreR1/D2Schur.lean",
+      "role": "scalar-definition correspondence, no new Lean execution"
+    },
+    {
+      "comment_id": 6008747995,
+      "pr": 328,
+      "repository": "d6g8k5htny-coder/Math-",
+      "role": "Part B monotonicity identities; independently derived joint-envelope extension here",
+      "updated_at": "2026-10-06T03:29:55Z"
+    }
+  ]
+}