Complete Philippon paper — all results and corrections

For each universe u, PhilipponMultiplicity.paper_results asserts, with no top-level mathematical hypotheses, the conjunction of all 26 clauses expanded below. Each field-parametrized clause is universally closed internally over every specified field structure and every stated datum, with the displayed hypotheses and existential conclusions in their stated order. It does not assume any of the 26 propositions as an argument. Here are the conventions and data used in the complete expansion below. A field K means any type in the fixed universe u equipped with a nontrivially normed field structure and an isometric ring isomorphism either to the complex numbers or to the p-adic complex field for some natural prime p; existence of either isomorphism is a hypothesis, and no particular one is selected. A multiprojective space M has a positive number p of factors, natural ambient dimensions N_i, points in $\prod_{i<p}\mathbf P^{N_i}(K)$, and coordinate ring $R=K[X_{ij}:i<p,\ 0\le j\le N_i]$. Evaluation uses the chosen nonzero vector representing each projective point. A polynomial has multidegree D if every monomial in its support has degree D_i in block i; the zero polynomial satisfies this for every D. Degree at most D means existence of one such multidegree bounded coordinatewise by D. A homogeneous ideal is closed under every projection to a block-homogeneous component. The Zariski topology is generated by the nonvanishing sets of homogeneous polynomials, and $I_M(V)$ is the ideal spanned by homogeneous polynomials vanishing on V. An embedded commutative group E is a locally closed subset of a single projective space, with an additive commutative group structure on its points and regular addition and negation; regularity means local representation in each target projective factor by a nonzero tuple of homogeneous polynomials of a common multidegree. G is a product of a positive number of such groups $E_i$, with its induced Zariski topology; it need not be connected or positive dimensional. $I_G(V)$ is the corresponding homogeneous vanishing ideal of a subset of G, $I_G=I_G(G)$, $Z_G(J)$ is the common zero set in G of all polynomials in an ideal J, and $Z_G(P)$ is the zero set of a single polynomial. An algebraic subgroup H is an additive subgroup closed in this topology; connectedness is topological connectedness, and $H^\circ$ is the connected component of 0 within H. A group subvariety is merely a locally closed subset of G, and a projective subvariety is likewise a locally closed subset of a single projective space; neither nonemptiness nor irreducibility is part of these two definitions. Translation is $g+V=\{g+v:v\in V\}$. For a finite sample $\Gamma$, $\Gamma(t)$ is the finite set of sums of exactly t elements of $\Gamma$, with repetitions, so $\Gamma(0)=\{0\}$; $N_\Gamma(H)$ counts the distinct sets $g+H$ with $g\in\Gamma$. For any ideal J of a multiprojective coordinate ring, $h_J\in\mathbf Q[t_0,\ldots,t_{p-1}]$ is the polynomial chosen if there exists a polynomial whose value at every sufficiently large coordinatewise natural multidegree d equals the K-dimension of the image of the degree-d polynomial subspace in $R/J$; if none exists, $h_J$ is defined to be zero. Put $a(J)=\deg h_J$, using total degree, with degree zero for the zero polynomial; put $\Delta_J(d)=a(J)!\,(h_J)_{a(J)}(d)$, where the subscript denotes the top total-homogeneous part. These values are rational. Put $\mu_J(\alpha)=[t^\alpha]h_J\prod_i\alpha_i!$ if $\sum_i\alpha_i=a(J)$, and put it equal to zero otherwise. For a locus V, its dimension is $a(I(V))$ and its degree value is $\Delta_{I(V)}$; for G write $H_G(V;D)$ for this degree value regarded as a real number, and $\mu_G(V;\alpha)$ for the real version of $\mu_{I_G(V)}(\alpha)$. A single projective subvariety has degree $\Delta_{I(V)}(1)$. Each factor dimension $n_i$ is its single-projective Hilbert-polynomial degree, and $n=\dim G$ in the statements means $\sum_i n_i$. The irrelevant ideal B is the intersection of the ideals generated by all coordinates of each individual block. An ideal J is called nontrivial precisely when $B\nsubseteq\sqrt J$, and a prime q is relevant precisely when $B\nsubseteq q$. For a minimal prime q of J, $Q_J(q)$ is the contraction of $JR_q$, and $\lambda_J(q)$ is the module length of $R_q/JR_q$ over $R_q$, an extended natural number. In the top-component sum its natural-number conversion is used; that conversion would send infinity to zero, although finiteness at these minimal primes is established in the imported definitions. For an open U of the maximal spectrum of R, a prime meets U when it is contained in some maximal ideal belonging to U. Write $\mathcal S_U(J;d)$ for the sum of $\Delta_{Q_J(q)}(d)$ over the relevant minimal primes q of J meeting U. Thus this sum uses the canonical primary components with their scheme multiplicities, and U is an open set of the maximal spectrum of the coordinate ring, not an open subset of the multiprojective point space. J is locally Cohen--Macaulay on U when, at every maximal ideal in U, the localized quotient is either the zero ring or has a regular sequence of nonunits whose length equals its ring Krull dimension. A regular element means that multiplication by it is injective; a nonunit condition is not included when regularity of a single element is used. Empty intersections of ideals are R. Define $J_{=b}$, respectively $J_{\ge b}$, to be the intersection of $Q_J(q)$ over relevant minimal primes of J having dimension b, respectively at least b. The relevant radical core is the intersection of the relevant minimal primes themselves. A primary decomposition is a finite family of homogeneous primary ideals, with intersection J, irredundant after removal of any member, and with distinct radicals. Its isolated members have radicals minimal over J; all other members are called embedded, including any irrelevant ones satisfying this condition. Its isolated and embedded intersections are the intersections over those respective classes. Given $J\subseteq I$, let $N_1(J,I)$, respectively $N_2(J,I)$, be the intersection of $Q_J(q)$ over minimal primes q of J not associated, respectively associated, to $R/I$. Also put $\mathcal B_J(D;d)=\sum_{\substack{0\le\alpha_i\le N_i\\\sum_i\alpha_i=a(J)}}\mu_J(\alpha)\left(\sum_i\alpha_iD_i/d_i\right)(a(J)-1)!\,\prod_i d_i^{\alpha_i}/\prod_i\alpha_i!$, where subtraction in $a(J)-1$ is natural-number subtraction and the rational divisions are total operations. An analytic subgroup datum A consists of a positive natural parameter dimension q, a positive real radius $\rho$, the open ball $B(0,\rho)\subset K^q$, a map $\phi:B(0,\rho)\to G$ taking 0 to 0 and satisfying $\phi(x+y)=\phi(x)+\phi(y)$ whenever all three arguments lie in the ball, and homogeneous-coordinate lifts $L_g(z)$ for every $g\in G$. Each coordinate of every lift is analytic at 0; the lift at $g=0$ has a power-series expansion on the full specified ball and represents $\phi$ at every point of that ball, whereas for general g the lift has nonzero blocks representing $g+\phi(z)$ on a neighborhood of 0. The carrier $|A|$ is the additive subgroup generated by $\phi(B(0,\rho))$, not a topological closure. No injectivity of $\phi$ is imposed. For $V\subseteq G$, let $T_A(V)\subset K^q$ be the intersection, over all $P\in I_G(V)$, of the kernels of the first derivatives at 0 of $z\mapsto P(L_0(z))$. Put $\delta_A(V)=q-\dim_K T_A(V)$ and $\dim A=\delta_A(\{0\})$, with natural-number subtraction. The order $\operatorname{ord}_{A,g}P$ is the infimum in $\mathbf N\cup\{\infty\}$ of all natural j for which the full j-th Fréchet derivative at 0 of $P(L_g(z))$ is nonzero; an empty set gives infinity. A transverse coordinate family for H consists of $\delta_A(H)$ standard coordinate vectors in $K^q$ whose images in $K^q/T_A(H)$ are linearly independent and span that quotient; the indexing map is supplied as part of the datum. Write $\operatorname{Inc}_G(J,V)$ for the conjunction that $V=Z_G(I_G(V))$ and that every minimal prime of $I_G(V)$ whose equations vanish at some point of G is minimal over $I_G+J$. Its multiplicity-at-least-$\ell$ version additionally requires $\lambda_{I_G+J}(q)\ge\ell$ for every such minimal prime q. Saying that V has an incomplete definition bounded by D means that there is a homogeneous ideal J satisfying $\operatorname{Inc}_G(J,V)$ and a finite set of homogeneous polynomials of respective multidegrees at most D whose generated ideal has the same sum with $I_G$ as J does. For $\gamma=(\gamma_0,\ldots,\gamma_{l-1})$ and real S, the sampling grid is $\mathcal G_\gamma(S)=\{\sum_{i<l}a_i\gamma_i:a_i\in\mathbf N,\ a_i\le S\}$. Its quotient rank $r_\gamma(H)$ is the integer-module rank of the submodule of $G/H$ generated by the images of the $\gamma_i$. A mixed codimension index for H is a vector $r_i\in\mathbf N$ with $r_i\le n_i$ and $\sum_i r_i+\dim_G H=n$; its complement is $\bar r_i=n_i-r_i$ and its degree monomial is $\prod_iD_i^{r_i}$. Let $d_i(H)$ be $n_i$ minus the single-projective Hilbert-polynomial dimension of the projection of H to $E_i$, again with natural subtraction. Disjoint factors means that every algebraic subgroup H is exactly the product of its projections and that each projection is closed in its factor. For any r, let $\mathcal O_A(r)$ consist of all algebraic subgroups H with $|A|\nsubseteq H$ and $r_i\le d_i(H)$ for every i. The two minima used below are separately $\eta_A(r)=\inf_{H\in\mathcal O_A(r)}\delta_A(H)$ and $\rho_{A,\gamma}(r)=\inf_{H\in\mathcal O_A(r)}r_\gamma(H)$ in the natural numbers. Either is zero when the indexing family is empty; there is no requirement that the two minima have a common minimizer. A translation chart C for A and g consists of a Zariski open domain in G, natural degrees $b_i$, and for every homogeneous output coordinate a polynomial in the group homogeneous coordinates with functions $K^q\to K$ as coefficients. All its finitely many coefficients are analytic at 0. An output coordinate in factor i is homogeneous of degree $b_i$ in input block i and degree zero in all other blocks. At every group point, on a parameter neighborhood of 0, the coordinate tuple is cross-proportional to $L_{g+x}(z)$; outside the chart it may be the zero tuple. At each point of the chart, on a parameter neighborhood that may depend on the point, its blocks are nonzero and represent $x+g+\phi(z)$. A translation atlas is an arbitrary family of these charts indexed by a type in universe u and covering G; it need not be finite. It is bounded by c if every chart degree in block i is at most $c_i$. For a chart C, order t, and an ordered list $\nu$ of t standard coordinate directions, repeated directions being allowed, $\mathcal L_{C,t,\nu}(P)$ substitutes the chart coordinate polynomials into P, applies the t-th Fréchet derivative at 0 in those directions to each parameter-function coefficient, and reassembles the resulting polynomial. Let $\mathcal R_G(J)$ be the intersection of the contractions of $JR_{\mathfrak m}$ for the evaluation maximal ideals $\mathfrak m$ of every nonzero homogeneous representative, in every block, of every group point. The retained operator ideal $F_{\mathcal A,T}(I)$ is $\mathcal R_G$ of the sum of $I_G$ with the ideal generated by all $\mathcal L_{C,t,\nu}(P)$, where P is any homogeneous member of I, C is any chart of the atlas $\mathcal A$, and $t\le T$. A polynomial translation chart, without analytic parameters, has a Zariski open domain and ordinary coordinate polynomials of the same pure block-degree form representing $g+x$ on that domain; a polynomial translation atlas is a cover by such charts. A translation degree bound c means that for every g there exists such a polynomial translation atlas with every degree at most $c_i$. The intrinsic differential ideal $D_{A,g,T}(I)$ is defined from local sections as follows. For each homogeneous $P\in I$, each choice b of a nonzero target coordinate in every block, and each ordered derivative in at most T standard coordinate directions, take the derivative at 0 of $z\mapsto P((L_{g+x}(z)_{ij}/L_{g+x}(z)_{i,b_i})_{ij})$ as a section on the set of x for which $g+x$ belongs to that coordinate chart. The ideal is spanned by all homogeneous Q locally generated by these sections: at every x one must find a Zariski neighborhood U contained in some source coordinate chart, finitely many of the sections all defined on U, and coefficients each a ratio of equal-multidegree homogeneous polynomials with denominator nonzero everywhere on U, such that the source-chart normalization of Q is their linear combination throughout U. The number of sections may be zero. The translated ideal $\tau_g(I)$ has the identical local-generation definition with sections the normalized values of homogeneous $P\in I$ at $g+x$, without differentiation. A section-five input consists of A, G, a finite $\Gamma$ containing 0, $T\in\mathbf N$, a natural degree vector D, a nonzero polynomial P of exact multidegree D with $\operatorname{ord}_{A,g}P\ge nT+1$ for $g\in\Gamma(n)$, positive natural coordinate bounds $b_i$, and a selected translation atlas for A and every g, each bounded by b. Its scaled degrees are $b_iD_i$. Its ideals are $J_0=I_G$, $J_1=I_G+(P)$, and, for $r\ge0$, $J_{r+2}=I_G+\langle\mathcal L_{C,t,\nu}P:g\in\Gamma(r+1),\ C\text{ in the selected atlas at }g,\ t\le(r+1)T\rangle$. A section-five construction extends that same input by an index $j\le n$, equality $\dim_G Z_G(J_j)=\dim_G Z_G(J_{j+1})$, a prime q minimal over both $J_j$ and $J_{j+1}$, a group point on its zero locus, equality $\dim_G Z_G(q)=\dim_G Z_G(J_j)$, and closedness of the stabilizer of $V=Z_G(q)$. Its chosen ideal is $J_j$, its component prime is q, its component is V, and its stabilizer is $\operatorname{Stab}(V)=\{g:g+V=V\}$, an algebraic subgroup because of the supplied closedness. These are hypotheses when a construction is quantified; none of the three conclusions in clause (13) is a field of the construction. The projective closure of G is the Zariski closure of its embedded points in its multiprojective ambient space. Translations extend to the closure if each g admits a map from the entire ambient point space to itself that preserves this closure, is regular when restricted to it in the local homogeneous-coordinate sense above, and agrees with $x\mapsto g+x$ on G. An algebraic reembedding is an additive group equivalence between two embedded groups, regular in both directions. The bound $\mathrm{MW}(G,a)$ means that for every finitely generated integer submodule $\Lambda\subseteq G$ and every $g\in\Lambda$, one polynomial translation chart for g contains every point of $\Lambda$ in its domain and has all degrees at most a. A set S is defined by equations bounded by a real B when it is exactly the common zero set in G of finitely many homogeneous polynomials, each with some natural multidegree whose every entry, cast to a real number, is at most B. The 26 complete clauses are as follows. (1) For every K there exists a function $c$ from embedded commutative groups over K to natural numbers, with $c(E)\ge1$ for every E, such that for every product G, analytic datum A, finite sample $\Gamma$ containing 0, $T\in\mathbf N$, natural degree vector D, and nonzero P of exact multidegree D, if $\operatorname{ord}_{A,g}P\ge nT+1$ for every $g\in\Gamma(n)$, there exists a connected algebraic subgroup H having an incomplete definition bounded by $(c(E_i)D_i)_i$, for which $H\subseteq t+Z_G(P)$ for some $t\in G$, and $\binom{T+\delta_A(H)}{\delta_A(H)}N_\Gamma(H)H_G(H;D)\le H_G(G;(c(E_i)D_i)_i)$. There is no positive-dimension or positive-D assumption in this clause. (2) For every K and G there is a real $c>0$ such that for every A with $\dim A=1$, every finite $\Gamma$ containing 0, every natural T, every D with $D_i\ge1$, and every nonzero P of exact degree D satisfying the same order bound $nT+1$ on $\Gamma(n)$, the following implication holds: if for every connected algebraic subgroup H not containing $|A|$ there exists a mixed codimension index r for H with $c\prod_iD_i^{r_i}\le (T+1)N_\Gamma(H)\mu_G(H;\bar r)$, then $t+|A|\subseteq Z_G(P)$ for some $t\in G$. The numerical obstruction condition is an additional hypothesis; no assertion that it can be satisfied is included. (3) For every K and G with disjoint factors there is a real $c>0$ such that, for every analytic datum A, natural l, tuple $\gamma\in G^l$, real $S\ge0$, natural T, natural degree vector D, and nonzero P of exact degree D, if $\operatorname{ord}_{A,g}P\ge nT+1$ at every $g\in\mathcal G_\gamma(nS)$ and if, for every natural vector r with $r_i\le n_i$, $c\prod_iD_i^{r_i}\le(T+1)^{\eta_A(r)}S^{\rho_{A,\gamma}(r)}$, then $t+|A|\subseteq Z_G(P)$ for some t. Here all bounded r are tested, without a condition on their sum; neither positivity of D nor a positive dimension for A or G nor $l>0$ is required. (4) For every K, M, homogeneous nontrivial ideal I with $a(I)>0$, natural degree vector D with all $D_i\ge1$, and polynomial P homogeneous of exact degree D whose class is regular in $R/I$, set $J=I+(P)$. Then for every d with all $d_i\ge1$, $\Delta_J(d)=\mathcal B_I(D;d)$, and also $\Delta_J(D)=\Delta_I(D)$. Regularity of P is required, but a separate assertion that its class is a nonunit is not. (5) For every K, M, homogeneous nontrivial I, and d with all $d_i\ge1$, $\Delta_I(d)=\sum_{\substack{q\text{ minimal over }I\\B\nsubseteq q,\ a(q)=a(I)}}\lambda_I(q)_{\mathrm{nat}}\Delta_q(d)$, where the summation uses exactly the relevant minimal primes of the same Hilbert-polynomial dimension as I and the indicated natural-number value of the localized length. (6) For every K, M, homogeneous ideal $I_0$, natural m, tuple of polynomials $P_0,\ldots,P_{m-1}$, and natural degree vector D such that each $P_j$ is homogeneous of some multidegree at most D, set $I=I_0+(P_0,\ldots,P_{m-1})$. Then $\mathcal S_{\mathrm{MaxSpec}(R)}(\sqrt I;D)\le\mathcal S_{\mathrm{MaxSpec}(R)}(\sqrt{I_0};D)$; and, for every open $U\subseteq\mathrm{MaxSpec}(R)$ on which $I_0$ is locally Cohen--Macaulay, $\mathcal S_U(I;D)\le\mathcal S_U(I_0;D)$. No nontriviality of $I_0$, positive entries of D, nonzero equations, or positive m are assumed. (7) For every K, M, choice of locally closed projective subsets $V_i\subseteq\mathbf P^{N_i}(K)$, and d with all $d_i\ge1$, put $V=\prod_iV_i$, $a_V=\dim_M V$, $a_i=\dim V_i$, and $e_i=\deg V_i$. Then $\Delta_{I_M(V)}(d)=\dfrac{a_V!}{\prod_i a_i!}\left(\prod_i e_i\right)\left(\prod_i d_i^{a_i}\right)$. The factors may be empty or reducible; the numerator uses the defined dimension of the product locus. (8) For every K, G, A, $g,g'\in G$, natural $k,k'$, homogeneous ideal I, and any two translation atlases $\mathcal A,\mathcal A'$ for A and g, $F_{\mathcal A,k}(I)=F_{\mathcal A',k}(I)$; moreover for every translation atlas $\mathcal B$ for A and $g'$ and every translation atlas $\mathcal C$ for A and $g+g'$, $F_{\mathcal A,k}(F_{\mathcal B,k'}(I))=F_{\mathcal C,k+k'}(I)$. No degree bound on these atlases is a hypothesis. (9) For every K, G, A, points $g,h\in G$, natural k, homogeneous ideal I, and translation atlas $\mathcal A$ for A and g, all polynomials in $F_{\mathcal A,k}(I)$ vanish at the embedded point h if and only if $\operatorname{ord}_{A,g+h}Q>k$ for every $Q\in I$. Both universal quantifiers here range over all polynomials of the specified ideals, without limiting Q to homogeneous members. (10) For every K and G, every natural vector c with $c_i\ge1$ that is a polynomial translation degree bound, every $g\in G$, and every locally closed $V\subseteq G$, one has $\dim_G V=\dim_G(g+V)$; furthermore, for every D with $D_i\ge1$, $H_G(V;D)\le H_G(g+V;(c_iD_i)_i)$. (11) For every K, G, A, connected algebraic subgroup H, point $g\in G$, transverse coordinate family $\nu$ for H, and translation chart C for A and 0 whose domain meets $g+H$, put $s=\delta_A(H)$. There exist polynomials $Q_j$, indexed by $0\le j<s$, all in $I_G(g+H)$, such that, for every $i,j<s$, $\mathcal L_{C,1,\nu_i}(Q_j)\notin I_G(g+H)\ \Longleftrightarrow\ i=j$. No separate homogeneity or degree requirement is imposed on the $Q_j$; if $s=0$, the family and its conditions are empty. (12) For every K, G, A, connected algebraic subgroup H, $g\in G$, homogeneous ideal I, and natural T, if both $\operatorname{Inc}_G(I,g+H)$ and $\operatorname{Inc}_G(D_{A,0,T}(I),g+H)$ hold, then I incompletely defines $g+H$ with multiplicity at least $\binom{T+\delta_A(H)}{\delta_A(H)}$: it satisfies the incomplete-definition condition, and every minimal prime of $I_G(g+H)$ meeting G has localized length for $I_G+I$ at least that binomial coefficient. (13) For every K, G, A, and section-five construction C, with the input $(\Gamma,T,D,P,b,\text{atlases})$, chosen ideal $J_j$, component prime q, component $V=Z_G(q)$, stabilizer H, and $H^\circ$ as defined above, all three assertions hold: for every $g\in\Gamma$, $D_{A,g,T}(J_j)\subseteq q$; with $s=\delta_A(H^\circ)$, $\binom{T+s}{s}N_\Gamma(H^\circ)H_G(H^\circ;D)\le H_G(G;(b_iD_i)_i)$; and $\operatorname{Inc}_G(\sum_{v\in V}\tau_v(J_j),H)$, where the sum is the ideal supremum over all points of V. All input, dimension-stabilization, common-minimal-prime, maximal-dimension, meeting, and closed-stabilizer fields of C are hypotheses of this universal clause. (14) For every K there exists a function c from embedded commutative groups over K to natural numbers with $c(E)\ge1$ for every E such that for every G with $n>0$, every A, every finite $\Gamma$ containing 0, natural T and D, and nonzero P of exact degree D with $\operatorname{ord}_{A,g}P\ge nT+1$ for every $g\in\Gamma(n)$, there exists a connected algebraic subgroup H having an incomplete definition bounded by $(c(E_i)D_i)_i$, satisfying $g+H\subseteq Z_G(P)$ for every $g\in\Gamma$, and satisfying $\binom{T+\delta_A(H)}{\delta_A(H)}N_\Gamma(H)H_G(H;D)\le H_G(G;(c(E_i)D_i)_i)$. The positive dimension assumption is present here, while positive entries of D are not required. (15) For every K, G with $n>0$, A, finite $\Gamma$ containing 0, natural T and D satisfying $H_G(G;\mathbf1)\le D_i$ for every i, and algebraic subgroup H, if $\binom{T+\delta_A(H)}{\delta_A(H)}N_\Gamma(H)H_G(H;D)\le\dfrac{H_G(G;D)}{4^n n!}$, then there exists a nonzero polynomial P of exact multidegree D such that $\operatorname{ord}_{A,g+h}P\ge T+1$ for every $g\in\Gamma$ and $h\in H$, and P is nonzero at some point of G. H is not assumed connected. (16) For every K and M all of the following statements hold. (16a) For every homogeneous ideal I, an eventual quotient Hilbert polynomial exists; $h_I=0$ if and only if I is not nontrivial; if I is nontrivial, the ring Krull dimension of R modulo the relevant radical core of I equals $a(I)+p$, regarded as an element of the extended-natural-with-bottom dimension type; for every natural multi-index $\alpha$, $\mu_I(\alpha)$ is a natural number regarded as rational; it is zero whenever some $\alpha_i>N_i$; for every natural d, including zero entries, $\Delta_I(d)=\sum_{0\le\alpha_i\le N_i}\mu_I(\alpha)\,a(I)!\,\prod_i d_i^{\alpha_i}/\prod_i\alpha_i!$; and I has a finite irredundant homogeneous primary decomposition with distinct radicals. (16b) For every pair of homogeneous nontrivial ideals $I\subseteq J$ with $a(I)=a(J)$, and every d with $d_i\ge1$, $\Delta_J(d)\le\Delta_I(d)$. (16c) For every ideal I and every primary decomposition $(Q_i)_{i<t}$ of it, the associated primes of $R/I$ are exactly the set $\{\sqrt{Q_i}:i<t\}$. (16d) For every homogeneous nontrivial pair $J\subseteq J'$ and every natural b, if $\sqrt{J_{\ge b+1}}=\sqrt{J'_{\ge b+1}}$, then $J_{=b}\subseteq J'_{=b}$. (16e) For every pair of ideals $J\subseteq I$, without additional homogeneity assumptions, each associated prime q of $R/I$ contains some minimal prime p of J; each minimal prime p of J not associated to $R/I$ fails to contain I; and for every $P\in I$, $\sqrt{J+(P)}=\sqrt{N_1(J,I)+(P)}\cap\sqrt{N_2(J,I)}$, while every minimal prime q of J that is associated to $R/I$ remains a minimal prime of $J+(P)$. (16f) For every ideal I and every primary decomposition of I, its embedded intersection contains an element Q whose class is regular modulo its isolated intersection. (16g) For every ideal I, primary decomposition $(Q_i)_{i<t}$, and open U on which I is locally Cohen--Macaulay, no embedded member's radical meets U; furthermore, for every ideal L all of whose associated primes meet U, no embedded radical $\sqrt{Q_i}$ is contained in any associated prime of $R/L$, and the embedded intersection contains an element Q whose class is regular in $R/L$. In (16f) and (16g), isolated and embedded intersections include components according to minimality of their primes, without imposing relevance. (17) For every K, M, locally closed irreducible subset V of its point space, and natural multi-index $\alpha$ with $\alpha_i\le N_i$ and $\sum_i\alpha_i=\dim_M V$, every family of vector subspaces $L_i\subseteq K^{N_i+1}$ satisfying $\dim_K L_i+\alpha_i=N_i+1$ has the following property: if the slice $V\cap\prod_i\mathbf P(L_i)$, defined by requiring each chosen representing vector to belong to $L_i$, is finite, its number of points is at most $\mu_{I_M(V)}(\alpha)$. There also exists such a family $L_i$ for which that slice is finite and its number of points equals $\mu_{I_M(V)}(\alpha)$. Irreducibility here includes nonemptiness; the finite slice itself is permitted to be empty. (18) Over the complex numbers in $\mathbf P^4$ with homogeneous coordinates $x_0,\ldots,x_4$, define $I_0=(x_1^2x_3-x_2^2x_0,\ x_1x_4-x_2x_3,\ x_3^3-x_4^2x_0)$, $I=I_0+(x_3,x_1-x_4)$, $Q_1=(x_1^2,x_2^2,x_3,x_1-x_4)$, $Q_2=(x_0,x_1^2,x_3,x_1-x_4)$, and $w=x_1x_3^2-x_2x_4x_0$. The assertions are that $I_0$ is homogeneous and nontrivial; $x_3$ and $x_1-x_4$ have degree 1; $w\notin I_0$, $w^2\in I_0$, $w\in\sqrt{I_0}$, $\sqrt{I_0}\ne I_0$, and $I_0$ is not prime; $a(I_0)=2$, $h_{I_0}(t)=2t^2+3t+1$, and $\Delta_{I_0}(1)=4$; $I=(x_1^2,x_2^2x_0,x_3,x_1-x_4)=Q_1\cap Q_2$; both $Q_1,Q_2$ are primary, their radicals are distinct, the minimal primes of I are exactly those two radicals, and both radicals are relevant; $a(Q_1)=a(Q_2)=0$, $h_{Q_1}=4$, $h_{Q_2}=2$, $\Delta_{Q_1}(1)=4$, and $\Delta_{Q_2}(1)=2$; $\mathcal S_{\mathrm{MaxSpec}(R)}(I;1)=6$, $\mathcal S_{\mathrm{MaxSpec}(R)}(I_0;1)=4$, and the latter is strictly smaller than the former; finally, $I_0$ is not locally Cohen--Macaulay on the entire maximal spectrum. (19) Over the complex numbers in $\mathbf P^2\times\mathbf P^1$, with coordinates $X_0,X_1,X_2$ and $Y_0,Y_1$, set $I=(Y_1X_1,Y_1X_2)$, $P=Y_0$, $D=(0,1)$, and $J=I+(P)$. Then $I=(Y_1)\cap(X_1,X_2)$; I is homogeneous and nontrivial; P has exact degree D and its class in $R/I$ is regular; $a(I)=2$; $h_I(t_0,t_1)=\tfrac12t_0^2+\tfrac32t_0+1+t_1$ and $h_J=1$; for every natural degree vector d, $\Delta_I(d)=d_0^2$ and $\Delta_J(d)=1$; $\mathcal B_I(D;\mathbf1)=0$, $\Delta_J(\mathbf1)\ne\mathcal B_I(D;\mathbf1)$, and $\Delta_J(D)\ne\Delta_I(D)$. (20) For every K all of the following hold. There exists a function c assigning a natural $c(E)\ge1$ to every embedded commutative group E such that, for every G, A, and $g\in G$, some translation atlas for A and g is bounded by $(c(E_i))_i$. For every G, A, g, translation chart C, natural order t, and ordered list $\nu$ of t parameter coordinate directions, the operator obeys $\mathcal L_{C,t,\nu}(P+Q)=\mathcal L_{C,t,\nu}P+\mathcal L_{C,t,\nu}Q$ and $\mathcal L_{C,t,\nu}(aP)=a\mathcal L_{C,t,\nu}P$ for all P,Q and $a\in K$; for every homogeneous P of multidegree D its image is homogeneous of exact multidegree $(b_iD_i)_i$, where $b_i$ are C's degrees. For every such G,A,g,C and the empty ordered direction list, there exists a K-algebra endomorphism f of the coordinate ring with $f(P)=\mathcal L_{C,0,\varnothing}P$ for all P. For every G,A,g, atlas $\mathcal A$ at g, natural T, and homogeneous ideals I,J, $F_{\mathcal A,T}(I)=D_{A,g,T}(I)$; if $I\subseteq J$, then $F_{\mathcal A,T}(I)\subseteq F_{\mathcal A,T}(J)$; and $F_{\mathcal A,T}(I+J)=\mathcal R_G(F_{\mathcal A,T}(I)+F_{\mathcal A,T}(J))$. Finally, for every G,A,g, atlas $\mathcal A$, and natural T, $F_{\mathcal A,T}(I_G)=I_G$. (21) For every K, the following three statements hold. For every G, locally closed $V\subseteq G$, and $g\in G$, $\tau_g(I_G(V))=I_G(-g+V)$. For every connected G whose translations extend to its projective closure, every locally closed V, every g, and every D with $D_i\ge1$, $H_G(V;D)=H_G(g+V;D)$. For every connected single embedded group E, there exists another embedded commutative group F with an algebraic reembedding of E into F, in the sense of an additive equivalence regular in both directions, such that for every analytic datum A on the one-factor product F and every $g\in F$, there exists a translation atlas bounded by 2 in its single factor. (22) For every K, G, and A, the following holds. For every $g\in G$, every P homogeneous of an arbitrary natural degree vector D, and every family $f(z)$ of homogeneous-coordinate vectors whose individual coordinates are analytic at 0 and whose nonzero blocks represent $g+\phi(z)$ on a neighborhood of 0, with z in the specified parameter ball there, $\operatorname{ord}_{A,g}P$ equals the infimum in $\mathbf N\cup\{\infty\}$ of the orders of nonzero Fréchet derivatives at 0 of $z\mapsto P(f(z))$. Furthermore, for every algebraic subgroup H, $\delta_A(H)\le\dim A$, $\delta_A(H)=0$ if and only if $|A|\subseteq H$, and a transverse coordinate family for H exists. No connectedness of H is required. (23) For every K and G all four assertions hold: for every A, algebraic subgroup H, and natural T, with $s=\delta_A(H)$, $(T+1)^s/s!\le\binom{T+s}{s}$; for every A and algebraic subgroup H with $|A|\nsubseteq H$, every natural m, tuple $\gamma\in G^m$, and real $S\ge0$, $S^{r_\gamma(H)}\le\#\{x+H:x\in\mathcal G_\gamma(S)\}$; for every algebraic subgroup H, every mixed codimension index r for H, and every D with $D_i\ge1$, $\mu_G(H;\bar r)\,\dfrac{(\dim_G H)!}{\prod_i\bar r_i!}\prod_iD_i^{\bar r_i}\le H_G(H;D)$; and, if G has disjoint factors, there exists a real $c>0$ such that for every algebraic subgroup H and every D with $D_i\ge1$, $H_G(H;D)>0$ and $H_G(G;D)/H_G(H;D)\le c\prod_iD_i^{d_i(H)}$. The subgroup quantifiers in these four assertions do not require connectedness. (24) For every K, G, and A, every section-five input I admits a section-five construction C whose entire inherited input record equals I; for every algebraic subgroup H there is an algebraic subgroup $H_0$ with carrier $H^\circ$, both connected and irreducible, and a finite set R of points of H such that $H=\bigcup_{x\in R}(x+H_0)$ and distinct representatives in R give disjoint translates; every connected algebraic subgroup H is irreducible; and the set stabilizer of every locally closed subset V of G is closed in G. The first assertion is existence of all the additional construction fields listed above, not an assumption that those selections already exist. (25) For every K, single embedded group E with positive dimension $n=\dim E$ and connected carrier, natural $a,b\ge1$ with $\mathrm{MW}(E,a)$, and a finite set of homogeneous equations of degrees at most b whose common zero set in the ambient $\mathbf P^N$ is exactly the projective closure of E, the following implication holds for every pair of natural $m,D\ge1$, tuple $\gamma\in E^m$, real $\theta\ge n/m$, and polynomial P homogeneous of some degree at most D. Set $c=1/(a^n b^{N\mathbin{\dot-}n})$, where $N\mathbin{\dot-}n$ denotes truncated natural subtraction. If P vanishes at every point of $\mathcal G_\gamma(n(D/c)^\theta)$ and is nonzero at some point of E, there exist natural k,r with $1\le k\le m$, $1\le r\le n$, and $m<k+r/\theta$, an integer submodule $Z\subseteq\mathbf Z^m$ of integer-module rank at least k, and an algebraic subgroup H with $\dim_E H\le n-r$, such that $\sum_i\sigma_i\gamma_i\in H$ for every $\sigma\in Z$; there exist $k$ vectors $\sigma^{(j)}\in Z$, indexed by $0\le j<k$, whose underlying vectors in $\mathbf Z^m$ are linearly independent over $\mathbf Z$ and satisfy $|\sigma^{(j)}_i|\le(D/c)^{r/(m-j)}$ for every j and $i<m$; and there exists a locally closed subset S of E containing H, with $\dim_E S\le n-r$, exactly defined in E by finitely many homogeneous equations of natural degrees at most the real number $D/c$. Exponents $\theta$ and $r/(m-j)$ are real powers. H need not be connected, and Z is only required to have rank at least k. (26) Three independent existence statements over the complex numbers are conjoined. First, there exist G,A,H such that G has only one point, $n=0$, $H=G$, $|A|=\{0\}$, $\delta_A(H)=0$, $H_G(G;\mathbf1)=H_G(H;\mathbf1)=1$, and $\binom{0+\delta_A(H)}{\delta_A(H)}N_{\{0\}}(H)H_G(H;\mathbf1)\le H_G(G;\mathbf1)/(4^n n!)$, but there is no polynomial P, of any degree and without any nonzero or homogeneity restriction, satisfying both $\operatorname{ord}_{A,h}P\ge1$ for every $h\in H$ and nonvanishing at some point of G. Second, there exist G,A, a finite sample $\Gamma$, and P with $n=0$, $\#\Gamma=2$, $0\in\Gamma$, $|A|=\{0\}$, P nonzero of multidegree $\mathbf1$, and $\operatorname{ord}_{A,g}P=\infty$ for every $g\in\Gamma(n)$, but no algebraic subgroup H has $g+H\subseteq Z_G(P)$ for every $g\in\Gamma$. Third, there exist a two-factor G whose factors each have dimension 1, an A of dimension 1, and a nonzero P of exact multidegree $D=(1,0)$, with $\operatorname{ord}_{A,0}P=1$, such that for every real $c>0$ and every connected algebraic subgroup H not containing $|A|$, some mixed codimension index r for H satisfies $c\prod_iD_i^{r_i}\le N_{\{0\}}(H)\mu_G(H;\bar r)$, yet no translate $g+|A|$ is contained in $Z_G(P)$. Each numbered clause has its own quantifiers; constants in different clauses are independent, even when given the same letter. In particular, the functions in (1), (14), and (20) are each chosen after the field but before the universally quantified products and analytic data, the constants in (2) and (3) after the field and fixed product, and the last constant in (23) after the field and product with its disjointness hypothesis. Unless explicitly restricted, natural parameters and degree entries include zero; powers with natural exponent zero are 1, including $0^0$, natural subtraction truncates at zero, and division in the fields is total with $x/0=0$. The stated positive-degree and positive-dimension restrictions apply only where written. Finite empty families and empty open loci are allowed where not ruled out, and conditional conclusions do not assert existence of data satisfying their antecedents. The three concrete clauses (18), (19), and (26) use the complex field, while the other clauses quantify over the permitted fields in universe u. No universal multiplicity estimate, Hilbert-degree identity, atlas independence, differential-ideal comparison, or other numbered conclusion is included as a premise merely by naming these proposition-valued definitions; specific numerical inequalities and the explicit structural and geometric hypotheses inside the clauses remain genuine premises.