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Hodge: source and evidence
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799 symbol spellings in 1280 mathematical passages across 18 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
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+Introduction passage · 101 passages
Search aliases: +
Introduction passage · august_development.tex, line 928
Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic (2,2)-class lies in the one-dimensional Fermat line V(widetilde a).
Exact mathematical passage
\label{eq:legacy-10.1} C^Tb+10\mathbf1=(6,18,27,9,12,21)=\widetilde a-\mathbf1.
\mathbf1Introduction passage · 25 passages
Search aliases: 1 mathbf
Introduction passage · august_development.tex, line 928
Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic (2,2)-class lies in the one-dimensional Fermat line V(widetilde a).
Exact mathematical passage
\label{eq:legacy-10.1} C^Tb+10\mathbf1=(6,18,27,9,12,21)=\widetilde a-\mathbf1.
=Introduction passage · 435 passages
Search aliases: =
Introduction passage · august_development.tex, line 689
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
U_m=(\mathbf Z/m\mathbf Z)^\times
AIntroduction passage · 77 passages
Search aliases: A
Introduction passage · august_development.tex, line 927
Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic (2,2)-class lies in the one-dimensional Fermat line V(widetilde a).
Exact mathematical passage
C=33A^{-1}
aIntroduction passage · 103 passages
Search aliases: a
Introduction passage · august_development.tex, line 693
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
a=(a_0,\ldots,a_5),\qquad 1\le a_i<m, \qquad \sum_i a_i\equiv0\pmod m.
A^{-1}Introduction passage · 15 passages
Search aliases: A^-1
Introduction passage · august_development.tex, line 927
Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic (2,2)-class lies in the one-dimensional Fermat line V(widetilde a).
Exact mathematical passage
C=33A^{-1}
a_0Introduction passage · 8 passages
Search aliases: a_0
Introduction passage · august_development.tex, line 693
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
a=(a_0,\ldots,a_5),\qquad 1\le a_i<m, \qquad \sum_i a_i\equiv0\pmod m.
a_5Introduction passage · 6 passages
Search aliases: a_5
Introduction passage · august_development.tex, line 693
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
a=(a_0,\ldots,a_5),\qquad 1\le a_i<m, \qquad \sum_i a_i\equiv0\pmod m.
a_iIntroduction passage · 15 passages
Search aliases: a_i
Introduction passage · august_development.tex, line 693
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
a=(a_0,\ldots,a_5),\qquad 1\le a_i<m, \qquad \sum_i a_i\equiv0\pmod m.
bIntroduction passage · 122 passages
Search aliases: b
Introduction passage · august_development.tex, line 765
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
a=g(a)b
\bar uIntroduction passage · 2 passages
Search aliases: bar u bar
Introduction passage · august_development.tex, line 1036
Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h)) · =L_sn(b)ρ_s-1(bar ub) +2^s-2D_sn(b)ε_s(u,b), where ε_s(u,b)∈{±1}. Repetition yields a Haar-like expansion whose detail packets carry visible powers of two. The Hodge balance equation becomes a binary carry relation between weighted details.
Exact mathematical passage
\label{eq:legacy-13.1} \boxed{ \begin{aligned} &n(b)\rho_s(ub)+n(b+h)\rho_s(u(b+h))\\ &\qquad=L_sn(b)\rho_{s-1}(\bar ub) +2^{s-2}D_sn(b)\varepsilon_s(u,b), \end{aligned} }
\bigwedgeIntroduction passage · 1 passage
Search aliases: bigwedge
Introduction passage · weil_audit.tex, line 19
Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.
Exact mathematical passage
\Omega_6=\bigwedge_{j=1}^{6}(x_j+iy_j),\qquad u=\Re\Omega_6,\qquad v=\Im\Omega_6.
\bigwedge_{j=1}^{6}Introduction passage · 1 passage
Search aliases: bigwedge_j=1^6 bigwedge
Introduction passage · weil_audit.tex, line 19
Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.
Exact mathematical passage
\Omega_6=\bigwedge_{j=1}^{6}(x_j+iy_j),\qquad u=\Re\Omega_6,\qquad v=\Im\Omega_6.
CIntroduction passage · 43 passages
Search aliases: C
Introduction passage · august_development.tex, line 927
Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic (2,2)-class lies in the one-dimensional Fermat line V(widetilde a).
Exact mathematical passage
C=33A^{-1}
cIntroduction passage · 78 passages
Search aliases: c
Introduction passage · august_development.tex, line 760
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
g(a)=\gcd(m,a_0,\ldots,a_5), \qquad c(a)=\frac{m}{g(a)}.
C^TIntroduction passage · 2 passages
Search aliases: C^T
Introduction passage · august_development.tex, line 928
Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic (2,2)-class lies in the one-dimensional Fermat line V(widetilde a).
Exact mathematical passage
\label{eq:legacy-10.1} C^Tb+10\mathbf1=(6,18,27,9,12,21)=\widetilde a-\mathbf1.
DIntroduction passage · 31 passages
Search aliases: D
Introduction passage · august_development.tex, line 1030
Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))
Exact mathematical passage
L_sn(b)=n(b)+n(b+h), \qquad D_sn(b)=n(b)-n(b+h).
dIntroduction passage · 71 passages
Search aliases: d
Introduction passage · august_development.tex, line 1529
Explicit carrier and remaining Chow map. For an audited CM type, the lower Fermat curve provides a candidate rational character-orbit factor of its Jacobian. Write e_orbJ(F_d) for that factor, up to isogeny. The idempotent selects the complete rational Galois orbit. A projector onto just one holomorphic CM half-plane would not be rational and is not an algebraic rational packet projector.
Exact mathematical passage
e_{\mathrm{orb}}J(F_d)
D_sIntroduction passage · 2 passages
Search aliases: D_s
Introduction passage · august_development.tex, line 1030
Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))
Exact mathematical passage
L_sn(b)=n(b)+n(b+h), \qquad D_sn(b)=n(b)-n(b+h).
\degIntroduction passage · 5 passages
Search aliases: deg
Introduction passage · capacity_cycles.tex, line 559
Let h denote the hyperplane class. Since deg Z_κ/m=2q, the primitive part
Exact mathematical passage
\deg Z_\kappa/m=2q
eIntroduction passage · 50 passages
Search aliases: e
Introduction passage · august_development.tex, line 1529
Explicit carrier and remaining Chow map. For an audited CM type, the lower Fermat curve provides a candidate rational character-orbit factor of its Jacobian. Write e_orbJ(F_d) for that factor, up to isogeny. The idempotent selects the complete rational Galois orbit. A projector onto just one holomorphic CM half-plane would not be rational and is not an algebraic rational packet projector.
Exact mathematical passage
e_{\mathrm{orb}}J(F_d)
e_{\mathrm{orb}}Introduction passage · 2 passages
Search aliases: e_orb mathrm
Introduction passage · august_development.tex, line 1529
Explicit carrier and remaining Chow map. For an audited CM type, the lower Fermat curve provides a candidate rational character-orbit factor of its Jacobian. Write e_orbJ(F_d) for that factor, up to isogeny. The idempotent selects the complete rational Galois orbit. A projector onto just one holomorphic CM half-plane would not be rational and is not an algebraic rational packet projector.
Exact mathematical passage
e_{\mathrm{orb}}J(F_d)
FIntroduction passage · 32 passages
Search aliases: F
Introduction passage · august_development.tex, line 1529
Explicit carrier and remaining Chow map. For an audited CM type, the lower Fermat curve provides a candidate rational character-orbit factor of its Jacobian. Write e_orbJ(F_d) for that factor, up to isogeny. The idempotent selects the complete rational Galois orbit. A projector onto just one holomorphic CM half-plane would not be rational and is not an algebraic rational packet projector.
Exact mathematical passage
e_{\mathrm{orb}}J(F_d)
F_dIntroduction passage · 2 passages
Search aliases: F_d
Introduction passage · august_development.tex, line 1529
Explicit carrier and remaining Chow map. For an audited CM type, the lower Fermat curve provides a candidate rational character-orbit factor of its Jacobian. Write e_orbJ(F_d) for that factor, up to isogeny. The idempotent selects the complete rational Galois orbit. A projector onto just one holomorphic CM half-plane would not be rational and is not an algebraic rational packet projector.
Exact mathematical passage
e_{\mathrm{orb}}J(F_d)
GIntroduction passage · 9 passages
Search aliases: G
Introduction passage · carrier70.tex, line 25
The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).
Exact mathematical passage
G_m=(\mu_m)^6/\mu_m
gIntroduction passage · 12 passages
Search aliases: g
Introduction passage · august_development.tex, line 760
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
g(a)=\gcd(m,a_0,\ldots,a_5), \qquad c(a)=\frac{m}{g(a)}.
G_mIntroduction passage · 4 passages
Search aliases: G_m
Introduction passage · carrier70.tex, line 25
The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).
Exact mathematical passage
G_m=(\mu_m)^6/\mu_m
\gcdIntroduction passage · 5 passages
Search aliases: gcd
Introduction passage · august_development.tex, line 760
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
g(a)=\gcd(m,a_0,\ldots,a_5), \qquad c(a)=\frac{m}{g(a)}.
HIntroduction passage · 56 passages
Search aliases: H
Introduction passage · carrier70.tex, line 28
The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).
Exact mathematical passage
H^4(X_m,\mathbf Q)
hIntroduction passage · 42 passages
Search aliases: h
Introduction passage · august_development.tex, line 1029
Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))
Exact mathematical passage
h=2^{s-1}
H^4Introduction passage · 7 passages
Search aliases: H^4
Introduction passage · carrier70.tex, line 28
The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).
Exact mathematical passage
H^4(X_m,\mathbf Q)
iIntroduction passage · 69 passages
Search aliases: i
Introduction passage · august_development.tex, line 693
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
a=(a_0,\ldots,a_5),\qquad 1\le a_i<m, \qquad \sum_i a_i\equiv0\pmod m.
\ImIntroduction passage · 2 passages
Search aliases: Im
Introduction passage · weil_audit.tex, line 19
Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.
Exact mathematical passage
\Omega_6=\bigwedge_{j=1}^{6}(x_j+iy_j),\qquad u=\Re\Omega_6,\qquad v=\Im\Omega_6.
JIntroduction passage · 11 passages
Search aliases: J
Introduction passage · august_development.tex, line 1529
Explicit carrier and remaining Chow map. For an audited CM type, the lower Fermat curve provides a candidate rational character-orbit factor of its Jacobian. Write e_orbJ(F_d) for that factor, up to isogeny. The idempotent selects the complete rational Galois orbit. A projector onto just one holomorphic CM half-plane would not be rational and is not an algebraic rational packet projector.
Exact mathematical passage
e_{\mathrm{orb}}J(F_d)
jIntroduction passage · 49 passages
Search aliases: j
Introduction passage · weil_audit.tex, line 19
Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.
Exact mathematical passage
\Omega_6=\bigwedge_{j=1}^{6}(x_j+iy_j),\qquad u=\Re\Omega_6,\qquad v=\Im\Omega_6.
kIntroduction passage · 30 passages
Search aliases: k
Introduction passage · august_development.tex, line 765
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
L_k
LIntroduction passage · 36 passages
Search aliases: L
Introduction passage · august_development.tex, line 765
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
L_k
L_kIntroduction passage · 1 passage
Search aliases: L_k
Introduction passage · august_development.tex, line 765
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
L_k
L_sIntroduction passage · 2 passages
Search aliases: L_s
Introduction passage · august_development.tex, line 1030
Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))
Exact mathematical passage
L_sn(b)=n(b)+n(b+h), \qquad D_sn(b)=n(b)-n(b+h).
mIntroduction passage · 174 passages
Search aliases: m
Introduction passage · august_development.tex, line 689
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
U_m=(\mathbf Z/m\mathbf Z)^\times
\midIntroduction passage · 9 passages
Search aliases: mid
Introduction passage · august_development.tex, line 692
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
m\mid r
nIntroduction passage · 32 passages
Search aliases: n
Introduction passage · august_development.tex, line 1030
Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))
Exact mathematical passage
L_sn(b)=n(b)+n(b+h), \qquad D_sn(b)=n(b)-n(b+h).
\mathrm{orb}Introduction passage · 2 passages
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Introduction passage · august_development.tex, line 1529
Explicit carrier and remaining Chow map. For an audited CM type, the lower Fermat curve provides a candidate rational character-orbit factor of its Jacobian. Write e_orbJ(F_d) for that factor, up to isogeny. The idempotent selects the complete rational Galois orbit. A projector onto just one holomorphic CM half-plane would not be rational and is not an algebraic rational packet projector.
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e_{\mathrm{orb}}J(F_d)
PIntroduction passage · 48 passages
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Introduction passage · august_development.tex, line 924
Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic (2,2)-class lies in the one-dimensional Fermat line V(widetilde a).
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P(y)=y_1y_2y_5.
\pmodIntroduction passage · 32 passages
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Introduction passage · august_development.tex, line 693
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
a=(a_0,\ldots,a_5),\qquad 1\le a_i<m, \qquad \sum_i a_i\equiv0\pmod m.
\mathbf QIntroduction passage · 9 passages
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Introduction passage · carrier70.tex, line 28
The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).
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H^4(X_m,\mathbf Q)
qIntroduction passage · 154 passages
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Introduction passage · capacity_cycles.tex, line 559
Let h denote the hyperplane class. Since deg Z_κ/m=2q, the primitive part
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\deg Z_\kappa/m=2q
rIntroduction passage · 105 passages
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Introduction passage · august_development.tex, line 692
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
\langle r\rangle_m\in\{1,\ldots,m\}
\ReIntroduction passage · 1 passage
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Introduction passage · weil_audit.tex, line 19
Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.
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\Omega_6=\bigwedge_{j=1}^{6}(x_j+iy_j),\qquad u=\Re\Omega_6,\qquad v=\Im\Omega_6.
sIntroduction passage · 49 passages
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Introduction passage · august_development.tex, line 1029
Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))
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s\ge3
TIntroduction passage · 44 passages
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Introduction passage · august_development.tex, line 928
Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic (2,2)-class lies in the one-dimensional Fermat line V(widetilde a).
Exact mathematical passage
\label{eq:legacy-10.1} C^Tb+10\mathbf1=(6,18,27,9,12,21)=\widetilde a-\mathbf1.
UIntroduction passage · 20 passages
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Introduction passage · august_development.tex, line 689
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
U_m=(\mathbf Z/m\mathbf Z)^\times
uIntroduction passage · 67 passages
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Introduction passage · august_development.tex, line 697
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
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u\in U_m
U_mIntroduction passage · 9 passages
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Introduction passage · august_development.tex, line 689
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
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U_m=(\mathbf Z/m\mathbf Z)^\times
\uplusIntroduction passage · 5 passages
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Introduction passage · transfer210.tex, line 81
Write ν(a) for its integer residue-multiplicity vector. Write auplus b for concatenation regarded as a multiset. Permuting coordinates is an algebraic automorphism, so multiset identities suffice for algebraicity transfer. Write -a for coordinatewise negation modulo m.
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a\uplus b
VIntroduction passage · 28 passages
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Introduction passage · august_development.tex, line 930
0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic (2,2)-class lies in the one-dimensional Fermat line V(widetilde a).
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V(\widetilde a)
vIntroduction passage · 65 passages
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Introduction passage · weil_audit.tex, line 19
Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.
Exact mathematical passage
\Omega_6=\bigwedge_{j=1}^{6}(x_j+iy_j),\qquad u=\Re\Omega_6,\qquad v=\Im\Omega_6.
WIntroduction passage · 40 passages
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Introduction passage · carrier70.tex, line 27
The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).
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W_{[a]}
W_4Introduction passage · 2 passages
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Introduction passage · carrier70.tex, line 394
Here W_4 denotes the weight filtration, not the Fermat packet notation. The descended classes have weight four, so belong to this image. The category of polarizable pure rational Hodge structures is semisimple. The surjection from H^4(widetilde Y) onto its image thus has a Hodge-theoretic section. Every descended rational (2,2)-class has a rational (2,2)-lift on widetilde Y.
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W_4
W_{[a]}Introduction passage · 4 passages
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Introduction passage · carrier70.tex, line 27
The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).
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W_{[a]}
\widetilde aIntroduction passage · 5 passages
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Introduction passage · august_development.tex, line 928
Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic (2,2)-class lies in the one-dimensional Fermat line V(widetilde a).
Exact mathematical passage
\label{eq:legacy-10.1} C^Tb+10\mathbf1=(6,18,27,9,12,21)=\widetilde a-\mathbf1.
\widetilde YIntroduction passage · 10 passages
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Introduction passage · carrier70.tex, line 397
Here W_4 denotes the weight filtration, not the Fermat packet notation. The descended classes have weight four, so belong to this image. The category of polarizable pure rational Hodge structures is semisimple. The surjection from H^4(widetilde Y) onto its image thus has a Hodge-theoretic section. Every descended rational (2,2)-class has a rational (2,2)-lift on widetilde Y.
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H^4(\widetilde Y)
XIntroduction passage · 103 passages
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Introduction passage · august_development.tex, line 766
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
\pi_{g(a)}:X_m^4\longrightarrow X_{c(a)}^4, \qquad [x_i]\longmapsto[x_i^{g(a)}],
xIntroduction passage · 38 passages
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Introduction passage · august_development.tex, line 766
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
\pi_{g(a)}:X_m^4\longrightarrow X_{c(a)}^4, \qquad [x_i]\longmapsto[x_i^{g(a)}],
x_1Introduction passage · 4 passages
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Introduction passage · weil_audit.tex, line 18
Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.
Exact mathematical passage
(x_1,y_1,\ldots,x_6,y_6)
x_6Introduction passage · 2 passages
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Introduction passage · weil_audit.tex, line 18
Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.
Exact mathematical passage
(x_1,y_1,\ldots,x_6,y_6)
X_{c(a)}^4Introduction passage · 1 passage
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Introduction passage · august_development.tex, line 766
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
\pi_{g(a)}:X_m^4\longrightarrow X_{c(a)}^4, \qquad [x_i]\longmapsto[x_i^{g(a)}],
x_iIntroduction passage · 5 passages
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Introduction passage · august_development.tex, line 766
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
\pi_{g(a)}:X_m^4\longrightarrow X_{c(a)}^4, \qquad [x_i]\longmapsto[x_i^{g(a)}],
x_i^{g(a)}Introduction passage · 1 passage
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Introduction passage · august_development.tex, line 766
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
\pi_{g(a)}:X_m^4\longrightarrow X_{c(a)}^4, \qquad [x_i]\longmapsto[x_i^{g(a)}],
x_jIntroduction passage · 1 passage
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Introduction passage · weil_audit.tex, line 19
Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.
Exact mathematical passage
\Omega_6=\bigwedge_{j=1}^{6}(x_j+iy_j),\qquad u=\Re\Omega_6,\qquad v=\Im\Omega_6.
X_mIntroduction passage · 3 passages
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Introduction passage · carrier70.tex, line 28
The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).
Exact mathematical passage
H^4(X_m,\mathbf Q)
X_m^4Introduction passage · 5 passages
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Introduction passage · august_development.tex, line 766
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
\pi_{g(a)}:X_m^4\longrightarrow X_{c(a)}^4, \qquad [x_i]\longmapsto[x_i^{g(a)}],
yIntroduction passage · 44 passages
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Introduction passage · august_development.tex, line 924
Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic (2,2)-class lies in the one-dimensional Fermat line V(widetilde a).
Exact mathematical passage
P(y)=y_1y_2y_5.
y_1Introduction passage · 7 passages
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Introduction passage · august_development.tex, line 924
Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic (2,2)-class lies in the one-dimensional Fermat line V(widetilde a).
Exact mathematical passage
P(y)=y_1y_2y_5.
y_2Introduction passage · 8 passages
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Introduction passage · august_development.tex, line 924
Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic (2,2)-class lies in the one-dimensional Fermat line V(widetilde a).
Exact mathematical passage
P(y)=y_1y_2y_5.
y_5Introduction passage · 8 passages
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Introduction passage · august_development.tex, line 924
Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic (2,2)-class lies in the one-dimensional Fermat line V(widetilde a).
Exact mathematical passage
P(y)=y_1y_2y_5.
y_6Introduction passage · 2 passages
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Introduction passage · weil_audit.tex, line 18
Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.
Exact mathematical passage
(x_1,y_1,\ldots,x_6,y_6)
y_jIntroduction passage · 1 passage
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Introduction passage · weil_audit.tex, line 19
Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.
Exact mathematical passage
\Omega_6=\bigwedge_{j=1}^{6}(x_j+iy_j),\qquad u=\Re\Omega_6,\qquad v=\Im\Omega_6.
ZIntroduction passage · 34 passages
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Introduction passage · capacity_cycles.tex, line 559
Let h denote the hyperplane class. Since deg Z_κ/m=2q, the primitive part
Exact mathematical passage
\deg Z_\kappa/m=2q
\mathbf ZIntroduction passage · 15 passages
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Introduction passage · august_development.tex, line 689
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
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U_m=(\mathbf Z/m\mathbf Z)^\times
Z_\kappaIntroduction passage · 12 passages
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Introduction passage · capacity_cycles.tex, line 559
Let h denote the hyperplane class. Since deg Z_κ/m=2q, the primitive part
Exact mathematical passage
\deg Z_\kappa/m=2q
\timesIntroduction passage · 13 passages
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Introduction passage · august_development.tex, line 689
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
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U_m=(\mathbf Z/m\mathbf Z)^\times
\varepsilonIntroduction passage · 2 passages
Search aliases: ε varepsilon
Introduction passage · august_development.tex, line 1036
Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h)) · =L_sn(b)ρ_s-1(bar ub) +2^s-2D_sn(b)ε_s(u,b), where ε_s(u,b)∈{±1}. Repetition yields a Haar-like expansion whose detail packets carry visible powers of two. The Hodge balance equation becomes a binary carry relation between weighted details.
Exact mathematical passage
\label{eq:legacy-13.1} \boxed{ \begin{aligned} &n(b)\rho_s(ub)+n(b+h)\rho_s(u(b+h))\\ &\qquad=L_sn(b)\rho_{s-1}(\bar ub) +2^{s-2}D_sn(b)\varepsilon_s(u,b), \end{aligned} }
\varepsilon_sIntroduction passage · 2 passages
Search aliases: ε_s varepsilon
Introduction passage · august_development.tex, line 1036
Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h)) · =L_sn(b)ρ_s-1(bar ub) +2^s-2D_sn(b)ε_s(u,b), where ε_s(u,b)∈{±1}. Repetition yields a Haar-like expansion whose detail packets carry visible powers of two. The Hodge balance equation becomes a binary carry relation between weighted details.
Exact mathematical passage
\label{eq:legacy-13.1} \boxed{ \begin{aligned} &n(b)\rho_s(ub)+n(b+h)\rho_s(u(b+h))\\ &\qquad=L_sn(b)\rho_{s-1}(\bar ub) +2^{s-2}D_sn(b)\varepsilon_s(u,b), \end{aligned} }
\kappaIntroduction passage · 42 passages
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Introduction passage · capacity_cycles.tex, line 559
Let h denote the hyperplane class. Since deg Z_κ/m=2q, the primitive part
Exact mathematical passage
\deg Z_\kappa/m=2q
\muIntroduction passage · 8 passages
Search aliases: μ mu
Introduction passage · carrier70.tex, line 25
The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).
Exact mathematical passage
G_m=(\mu_m)^6/\mu_m
\mu_mIntroduction passage · 2 passages
Search aliases: μ_m mu
Introduction passage · carrier70.tex, line 25
The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).
Exact mathematical passage
G_m=(\mu_m)^6/\mu_m
\nuIntroduction passage · 3 passages
Search aliases: ν nu
Introduction passage · transfer210.tex, line 80
Write ν(a) for its integer residue-multiplicity vector. Write auplus b for concatenation regarded as a multiset. Permuting coordinates is an algebraic automorphism, so multiset identities suffice for algebraicity transfer. Write -a for coordinatewise negation modulo m.
Exact mathematical passage
\nu(a)
\piIntroduction passage · 9 passages
Search aliases: π pi
Introduction passage · august_development.tex, line 766
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
\pi_{g(a)}:X_m^4\longrightarrow X_{c(a)}^4, \qquad [x_i]\longmapsto[x_i^{g(a)}],
\pi_{g(a)}Introduction passage · 1 passage
Search aliases: π_g(a) pi
Introduction passage · august_development.tex, line 766
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
\pi_{g(a)}:X_m^4\longrightarrow X_{c(a)}^4, \qquad [x_i]\longmapsto[x_i^{g(a)}],
\rhoIntroduction passage · 13 passages
Search aliases: ρ rho
Introduction passage · august_development.tex, line 1029
Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))
Exact mathematical passage
\rho_s=\rho_{2^s}
\rho_{2^s}Introduction passage · 1 passage
Search aliases: ρ_2^s rho
Introduction passage · august_development.tex, line 1029
Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))
Exact mathematical passage
\rho_s=\rho_{2^s}
\rho_sIntroduction passage · 2 passages
Search aliases: ρ_s rho
Introduction passage · august_development.tex, line 1029
Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))
Exact mathematical passage
\rho_s=\rho_{2^s}
\rho_{s-1}Introduction passage · 1 passage
Search aliases: ρ_s-1 rho
Introduction passage · august_development.tex, line 1036
Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h)) · =L_sn(b)ρ_s-1(bar ub) +2^s-2D_sn(b)ε_s(u,b), where ε_s(u,b)∈{±1}. Repetition yields a Haar-like expansion whose detail packets carry visible powers of two. The Hodge balance equation becomes a binary carry relation between weighted details.
Exact mathematical passage
\label{eq:legacy-13.1} \boxed{ \begin{aligned} &n(b)\rho_s(ub)+n(b+h)\rho_s(u(b+h))\\ &\qquad=L_sn(b)\rho_{s-1}(\bar ub) +2^{s-2}D_sn(b)\varepsilon_s(u,b), \end{aligned} }
\OmegaIntroduction passage · 6 passages
Search aliases: Ω Omega
Introduction passage · weil_audit.tex, line 19
Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.
Exact mathematical passage
\Omega_6=\bigwedge_{j=1}^{6}(x_j+iy_j),\qquad u=\Re\Omega_6,\qquad v=\Im\Omega_6.
\Omega_6Introduction passage · 3 passages
Search aliases: Ω_6 Omega
Introduction passage · weil_audit.tex, line 19
Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.
Exact mathematical passage
\Omega_6=\bigwedge_{j=1}^{6}(x_j+iy_j),\qquad u=\Re\Omega_6,\qquad v=\Im\Omega_6.
\ldotsIntroduction passage · 23 passages
Search aliases: … ldots
Introduction passage · august_development.tex, line 692
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
\langle r\rangle_m\in\{1,\ldots,m\}
\ellIntroduction passage · 14 passages
Search aliases: ℓ ell
Introduction passage · august_development.tex, line 698
terUniversal balanced-packet equations Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
\ell_a(u)=\frac1m\sum_{i=0}^5\langle ua_i\rangle_m.
\ell_aIntroduction passage · 3 passages
Search aliases: ℓ_a ell
Introduction passage · august_development.tex, line 698
terUniversal balanced-packet equations Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
\ell_a(u)=\frac1m\sum_{i=0}^5\langle ua_i\rangle_m.
\inIntroduction passage · 50 passages
Search aliases: ∈ in
Introduction passage · august_development.tex, line 692
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
\langle r\rangle_m\in\{1,\ldots,m\}
\sumIntroduction passage · 68 passages
Search aliases: ∑ sum
Introduction passage · august_development.tex, line 693
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
a=(a_0,\ldots,a_5),\qquad 1\le a_i<m, \qquad \sum_i a_i\equiv0\pmod m.
\sum_iIntroduction passage · 7 passages
Search aliases: ∑_i sum
Introduction passage · august_development.tex, line 693
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
a=(a_0,\ldots,a_5),\qquad 1\le a_i<m, \qquad \sum_i a_i\equiv0\pmod m.
\sum_{i=0}^5Introduction passage · 2 passages
Search aliases: ∑_i=0^5 sum
Introduction passage · august_development.tex, line 698
terUniversal balanced-packet equations Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
\ell_a(u)=\frac1m\sum_{i=0}^5\langle ua_i\rangle_m.
\equivIntroduction passage · 20 passages
Search aliases: ≡ equiv
Introduction passage · august_development.tex, line 693
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
a=(a_0,\ldots,a_5),\qquad 1\le a_i<m, \qquad \sum_i a_i\equiv0\pmod m.
\leIntroduction passage · 43 passages
Search aliases: ≤ le
Introduction passage · august_development.tex, line 693
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
a=(a_0,\ldots,a_5),\qquad 1\le a_i<m, \qquad \sum_i a_i\equiv0\pmod m.
\geIntroduction passage · 35 passages
Search aliases: ≥ ge
Introduction passage · august_development.tex, line 1029
Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))
Exact mathematical passage
s\ge3
\langleIntroduction passage · 28 passages
Search aliases: ⟨ langle
Introduction passage · august_development.tex, line 692
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
\langle r\rangle_m\in\{1,\ldots,m\}
\rangleIntroduction passage · 28 passages
Search aliases: ⟩ rangle
Introduction passage · august_development.tex, line 692
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
\langle r\rangle_m\in\{1,\ldots,m\}
\rangle_mIntroduction passage · 11 passages
Search aliases: ⟩_m rangle
Introduction passage · august_development.tex, line 692
Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
\langle r\rangle_m\in\{1,\ldots,m\}
\longrightarrowIntroduction passage · 7 passages
Search aliases: ⟶ longrightarrow
Introduction passage · august_development.tex, line 766
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
\pi_{g(a)}:X_m^4\longrightarrow X_{c(a)}^4, \qquad [x_i]\longmapsto[x_i^{g(a)}],
\longmapstoIntroduction passage · 3 passages
Search aliases: ⟼ longmapsto
Introduction passage · august_development.tex, line 766
Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.
Exact mathematical passage
\pi_{g(a)}:X_m^4\longrightarrow X_{c(a)}^4, \qquad [x_i]\longmapsto[x_i^{g(a)}],
*Use in source · 26 passages
Search aliases: *
Use in source · appendices.tex, line 9
Cycle frame · S=CC^*. Positivity spans a target packet when every supplied algebraic vector lies in it.
Exact mathematical passage
S=CC^*
\mathbf1^TUse in source · 7 passages
Search aliases: 1^T mathbf
Use in source · august_development.tex, line 717
To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebraic automorphisms, the count-vector descriptions correspond to U_m-orbits in B_m. The dimension of a fixed rational character packet is computed from di…
Exact mathematical passage
\label{eq:legacy-7.2} \boxed{ \mathcal B_m= \left\{ n\in\mathbf N^{m-1}: \mathbf1^Tn=6, \quad R_mn=3m\,\mathbf1 \right\}. }
A^TUse in source · 6 passages
Search aliases: A^T
Use in source · august_development.tex, line 933
The Laurent exponent match is a useful exact certificate, but by itself it does not prove that a rational class lifts through a compact graph. A complete manuscript argument instead uses the torus-carrier criterion developed in Chapter~[source reference]. For the reordered character, the exponent matrix A of the cubic satisfies A^Twidetilde a/33=β=(1,2,2,1,1,2), C=33A^-1. With r_0 the first row of A, put u=β-3r_0. Then 1^Tu=0 and C^Tu=widetilde a-99e_0. These identities establish triviality of t…
Exact mathematical passage
A^T\widetilde a/33=\beta=(1,2,2,1,1,2),\qquad C=33A^{-1}.
A^\perpUse in source · 1 passage
Search aliases: A^⊥ perp
Use in source · august_development.tex, line 114
[Finite positive cycle-frame certificate] Let A=Span_R{c_1,…,c_N}. Then im S=A, ker S=A^⊥. Under the stated containment c_j∈ V, the selected cycles span V if and only if S|_V is positive definite. Without containment, positivity only detects the absence of vectors in V orthogonal to the cycle span and does not prove that span contains V. The equality im(CC^*)=im C is standard. Moreover ⟨ v,Sv⟩= C^*v ^2=∑_j|⟨ c_j,v⟩|^2. This vanishes exactly when v is orthogonal to every cycle class.
Exact mathematical passage
\im S=A, \qquad \ker S=A^\perp.
A_1^pUse in source · 1 passage
Search aliases: A_1^p
Use in source · august_development.tex, line 206
Finite saturation and exact rational closure. For each degree cutoff D, let A_D^p(X) be the rational span of codimension-p cycle classes of degree at most D. Then A_1^p⊆ A_2^p⊆⋯⊆ H^2p(X,Q).
Exact mathematical passage
A_1^p\subseteq A_2^p\subseteq\cdots\subseteq H^{2p}(X,\Q).
A_2^pUse in source · 1 passage
Search aliases: A_2^p
Use in source · august_development.tex, line 206
Finite saturation and exact rational closure. For each degree cutoff D, let A_D^p(X) be the rational span of codimension-p cycle classes of degree at most D. Then A_1^p⊆ A_2^p⊆⋯⊆ H^2p(X,Q).
Exact mathematical passage
A_1^p\subseteq A_2^p\subseteq\cdots\subseteq H^{2p}(X,\Q).
A_{2^t}Use in source · 1 passage
Search aliases: A_2^t
Use in source · august_development.tex, line 1326
[Complete dyadic grammar] After exact conductor descent, every balanced packet at every dyadic modulus is generated by complementary pairs, SAT(2) surface packets, and coordinate-power pullbacks of the single SAT(3) family A_2^t. No second residual dyadic family occurs at any depth.
Exact mathematical passage
A_{2^t}
A_{b,c,d}Use in source · 2 passages
Search aliases: A_b,c,d
Use in source · august_development.tex, line 1385
K4 four-phase reduction. Choose nonzero residues b,c,dpmod q and put e≡-5^-1bpmod q. Define A_b,c,d= (λ_1(b),λ_2(b),λ_3(b),λ_4(b),5c,5d) and Γ_b,c,d=(b,c,d,e).
Exact mathematical passage
A_{b,c,d}= (\lambda_1(b),\lambda_2(b),\lambda_3(b),\lambda_4(b),5c,5d)
A_{\mathrm{conic}}Use in source · 2 passages
Search aliases: A_conic mathrm
Use in source · august_development.tex, line 570
A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equivalently prove T_tot =∑_rA_r^† A_r satisfies λ_min(T_tot)>0.
Exact mathematical passage
A_{\mathrm{line}}, A_{\mathrm{conic}}, A_{\mathrm{plane}}, A_{\mathrm{exceptional}}.
A_D^pUse in source · 4 passages
Search aliases: A_D^p
Use in source · august_development.tex, line 205
Finite saturation and exact rational closure. For each degree cutoff D, let A_D^p(X) be the rational span of codimension-p cycle classes of degree at most D. Then A_1^p⊆ A_2^p⊆⋯⊆ H^2p(X,Q).
Exact mathematical passage
A_D^p(X)
A_{D_0}^pUse in source · 1 passage
Search aliases: A_D_0^p
Use in source · august_development.tex, line 211
[Finite stabilization] There exists D_0 such that A_D^p(X)=A_D_0^p(X) for every D≥ D_0. The dimensions form a nondecreasing sequence of integers bounded by dim H^2p(X,Q). Only finitely many strict increases are possible.
Exact mathematical passage
A_D^p(X)=A_{D_0}^p(X)
A_{\mathrm{exceptional}}Use in source · 2 passages
Search aliases: A_exceptional mathrm
Use in source · august_development.tex, line 570
A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equivalently prove T_tot =∑_rA_r^† A_r satisfies λ_min(T_tot)>0.
Exact mathematical passage
A_{\mathrm{line}}, A_{\mathrm{conic}}, A_{\mathrm{plane}}, A_{\mathrm{exceptional}}.
a_jUse in source · 1 passage
Search aliases: a_j
Use in source · carrier70.tex, line 504
Here W·β=3D follows from ∑ a_j=3m. Thus deck invariance follows uniformly from the matrix congruence, it is not an extra conjecture. The positive weight sum gives negative canonical coefficient, normality and well-formedness justify the canonical computation, and the uniruledness, descent, lifting and return arguments above finish the proof. This corollary replaces an assumed cycle seed with a finite exponent matrix and checkable geometric hypotheses.
Exact mathematical passage
\sum a_j=3m
A_{\mathrm{line}}Use in source · 2 passages
Search aliases: A_line mathrm
Use in source · august_development.tex, line 570
A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equivalently prove T_tot =∑_rA_r^† A_r satisfies λ_min(T_tot)>0.
Exact mathematical passage
A_{\mathrm{line}}, A_{\mathrm{conic}}, A_{\mathrm{plane}}, A_{\mathrm{exceptional}}.
A_mUse in source · 5 passages
Search aliases: A_m
Use in source · august_development.tex, line 1298
abeleq:legacy-13.5 P_a= (8a,q-a,2q-a,2q-4a,3q-a,4q-a). The cases c<q/2 and c>q/2 give the same unordered tuple after the internal-edge cancellations. Choose an odd u satisfying ua≡1pmod2q. Since uq≡ q or 3qpmod4q, multiplication by u sends [source reference] to P_1=(8,q-1,2q-4,2q-1,3q-1,4q-1). Its conjugate is A_m= ( 1,frac m4+1,frac m2+1,frac m2+4, frac3m4+1,m-8 ). Thus every full-conductor star belongs to one U_m-orbit.
Exact mathematical passage
\label{eq:legacy-13.6} \boxed{ A_m= \left( 1,\frac m4+1,\frac m2+1,\frac m2+4, \frac{3m}4+1,m-8 \right). }
a_{n+1}Use in source · 1 passage
Search aliases: a_n+1
Use in source · transfer210.tex, line 65
A full-support character is an ordered tuple a=(a_0,…,a_n+1) of nonzero residues modulo m, with sum zero modulo m. Its eigenspace in primitive cohomology is a complex line. For even n, its rational Galois packet is of middle Hodge type precisely when
Exact mathematical passage
a=(a_0,\ldots,a_{n+1})
A_{\mathrm{plane}}Use in source · 2 passages
Search aliases: A_plane mathrm
Use in source · august_development.tex, line 570
A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equivalently prove T_tot =∑_rA_r^† A_r satisfies λ_min(T_tot)>0.
Exact mathematical passage
A_{\mathrm{line}}, A_{\mathrm{conic}}, A_{\mathrm{plane}}, A_{\mathrm{exceptional}}.
a_qUse in source · 4 passages
Search aliases: a_q
Use in source · transfer210.tex, line 382
a_q=(1,1+q,1+2q,1+3q,1+4q,5q-5)pmod 5q.
Exact mathematical passage
\label{eq:transfer210-7.1} a_q=(1,1+q,1+2q,1+3q,1+4q,5q-5)\pmod {5q}.
a_{q,b}Use in source · 6 passages
Search aliases: a_q,b
Use in source · capacity_cycles.tex, line 222
m=5q, a_q,b=(b,b+q,b+2q,b+3q,b+4q,5(q-b)).
Exact mathematical passage
\label{eq:capacity_cycles-3.1} m=5q,\qquad a_{q,b}=(b,b+q,b+2q,b+3q,b+4q,5(q-b)).
a_{q,b,i}Use in source · 1 passage
Search aliases: a_q,b,i
Use in source · capacity_cycles.tex, line 519
Q_b=∏_i=0^5x_i^m-a_q,b,i-1.
Exact mathematical passage
\label{eq:capacity_cycles-6.3} Q_b=\prod_{i=0}^5x_i^{m-a_{q,b,i}-1}.
A_rUse in source · 14 passages
Search aliases: A_r
Use in source · appendices.tex, line 13
Divisor transfer · T=∑_rB_rA_r with actual Chow correspondences, packet preservation and invertibility.
Exact mathematical passage
T=\sum_rB_rA_r
A_r^\daggerUse in source · 6 passages
Search aliases: A_r^† dagger
Use in source · appendices.tex, line 14
Positive transfer · T=∑_rA_r^† A_r only with algebraic realizations of those adjoints.
Exact mathematical passage
T=\sum_rA_r^\dagger A_r
\AlgUse in source · 11 passages
Search aliases: Alg
Use in source · appendices.tex, line 7
Algebraic space · Alg^p(X)=im(CH^p(X)_Q→ H^2p(X,Q)).
Exact mathematical passage
\Alg^p(X)=\im(\CH^p(X)_{\Q}\to H^{2p}(X,\Q))
\Alg^pUse in source · 7 passages
Search aliases: Alg^p Alg
Use in source · appendices.tex, line 7
Algebraic space · Alg^p(X)=im(CH^p(X)_Q→ H^2p(X,Q)).
Exact mathematical passage
\Alg^p(X)=\im(\CH^p(X)_{\Q}\to H^{2p}(X,\Q))
BUse in source · 26 passages
Search aliases: B
Use in source · appendices.tex, line 13
Divisor transfer · T=∑_rB_rA_r with actual Chow correspondences, packet preservation and invertibility.
Exact mathematical passage
T=\sum_rB_rA_r
\mathcal BUse in source · 6 passages
Search aliases: B mathcal
Use in source · august_development.tex, line 717
To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebraic automorphisms, the count-vector descriptions correspond to U_m-orbits in B_m. The dimension of a fixed rational character packet is computed from di…
Exact mathematical passage
\label{eq:legacy-7.2} \boxed{ \mathcal B_m= \left\{ n\in\mathbf N^{m-1}: \mathbf1^Tn=6, \quad R_mn=3m\,\mathbf1 \right\}. }
b'Use in source · 5 passages
Search aliases: b'
Use in source · capacity_cycles.tex, line 232
Let t be a unit modulo 5q, and let b'=tbbmod q, with 1≤ b'<q. Multiplication by t permutes the five residues in the fiber over b', because t is a unit modulo 5. The first five least-positive residues are therefore, in some order,
Exact mathematical passage
b'=tb\bmod q
B^TUse in source · 3 passages
Search aliases: B^T
Use in source · carrier70.tex, line 313
and hence B^Tβ=α. To obtain a genuine character of the projective torus, let r_0=(0,0,2,0,0,3) be the first row of A and set
Exact mathematical passage
B^T\beta=\alpha
b_0Use in source · 2 passages
Search aliases: b_0
Use in source · august_development.tex, line 960
The tower packet remains an old conductor-33 sector. A stronger construction appends a complementary pair (1,m-1) after scaling a solved seed to modulus m=kd. If b=(b_0,…,b_2p+1) is algebraic at degree d, then Θ_m,b=(kb_0,…,kb_2p+1,1,m-1) has full raw conductor m and middle Hodge type two dimensions higher. Its rational orbit has dimension φ(m). This produces full-raw-conductor algebraic packets of unbounded rank.
Exact mathematical passage
b=(b_0,\ldots,b_{2p+1})
B_{1,\overline{\chi^*}}Use in source · 1 passage
Search aliases: B_1,overlineχ^* overline chi
Use in source · august_development.tex, line 1180
_m is diagonalized by the odd Dirichlet characters, with eigenvalues λ_χ= ∑_a∈ U_mρ_m(a)overlineχ(a) = ∑_a∈ U_maoverlineχ(a). If χ is induced from a primitive odd character χ^* of conductor 2^t, then λ_χ,2^s=2^s-tλ_χ^*,2^t. The primitive value is, up to a nonzero scalar, the generalized Bernoulli value B_1,overlineχ^*, equivalently -L(0,overlineχ^*), and is nonzero for every odd primitive character [source reference]. Thus every odd Fourier eigenvalue is nonzero. The odd residual must vanish, so…
Exact mathematical passage
B_{1,\overline{\chi^*}}
B_{13}Use in source · 1 passage
Search aliases: B_13
Use in source · weil_audit.tex, line 138
The selected mixed tangent direction B_13=1 breaks the original factorization infinitesimally: the old surface factor loses type (1,1) and the old fourfold factor loses type (2,2). Nevertheless the full u,v remain type (3,3). At first this looked like a promising obstruction.
Exact mathematical passage
B_{13}=1
b_{2p+1}Use in source · 2 passages
Search aliases: b_2p+1
Use in source · august_development.tex, line 960
The tower packet remains an old conductor-33 sector. A stronger construction appends a complementary pair (1,m-1) after scaling a solved seed to modulus m=kd. If b=(b_0,…,b_2p+1) is algebraic at degree d, then Θ_m,b=(kb_0,…,kb_2p+1,1,m-1) has full raw conductor m and middle Hodge type two dimensions higher. Its rational orbit has dimension φ(m). This produces full-raw-conductor algebraic packets of unbounded rank.
Exact mathematical passage
b=(b_0,\ldots,b_{2p+1})
\mathcal B_c^{\mathrm{full}}Use in source · 1 passage
Search aliases: B_c^full mathcal mathrm
Use in source · august_development.tex, line 774
[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balance equation descends. The maximal common divisor is unique. The converse follows by scaling.
Exact mathematical passage
\mathcal B_m=\bigsqcup_{c\mid m}L_{m/c}\mathcal B_c^{\mathrm{full}}.
b_IUse in source · 2 passages
Search aliases: b_I
Use in source · weil_audit.tex, line 131
The product locus and replacement factorizations. Split the six directions into a four-dimensional Weil factor C and a two-dimensional factor B. The determinant forms factor: Ω_6=Ω_CwedgeΩ_B. Writing their real and imaginary parts as c_R,c_I,b_R,b_I gives u=c_Rb_R-c_Ib_I, v=c_Rb_I+c_Ib_R. On the product locus, the degree-four Weil classes on C are algebraic by the checked fourfold theorem, and the degree-two classes on B are divisors by Lefschetz (1,1). Their products give the degree-six classes…
Exact mathematical passage
c_R,c_I,b_R,b_I
b_iUse in source · 2 passages
Search aliases: b_i
Use in source · august_development.tex, line 780
orem[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balance equation descends. The maximal common divisor is unique. The converse follows by scaling.
Exact mathematical passage
\langle ugb_i\rangle_{gc}=g\langle \bar u b_i\rangle_c,
B_{ij}Use in source · 1 passage
Search aliases: B_ij
Use in source · carrier70.tex, line 148
On the projective tori define y_i=∏_jx_j^B_ij. Negative exponents are allowed there. The second identity makes this well-defined into the weighted projective torus, the first gives
Exact mathematical passage
y_i=\prod_jx_j^{B_{ij}}
b_kUse in source · 1 passage
Search aliases: b_k
Use in source · august_development.tex, line 262
Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.
Exact mathematical passage
M_{ki}=\int_X[Z_i]\smile\gamma_k, \qquad b_k=\int_X\alpha\smile\gamma_k.
\mathcal B_mUse in source · 5 passages
Search aliases: B_m mathcal
Use in source · august_development.tex, line 717
To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebraic automorphisms, the count-vector descriptions correspond to U_m-orbits in B_m. The dimension of a fixed rational character packet is computed from di…
Exact mathematical passage
\label{eq:legacy-7.2} \boxed{ \mathcal B_m= \left\{ n\in\mathbf N^{m-1}: \mathbf1^Tn=6, \quad R_mn=3m\,\mathbf1 \right\}. }
\mathcal B_{m,\chi}Use in source · 2 passages
Search aliases: B_m,χ mathcal chi
Use in source · august_development.tex, line 742
Centered Fourier form. Define ρ_m(r)=⟨ r⟩_m-frac m2, F_a(u)=∑_iρ_m(ua_i). Then a is balanced if and only if F_a(u)=0 for all units. Moreover, F_a(-u)=-F_a(u), so only odd characters occur. For a Dirichlet character χ on U_m, put B_m,χ(r)= ∑_u∈ U_mρ_m(ur)overlineχ(u). Finite Fourier inversion gives the exact criterion n(a)∈ B_m Longleftrightarrow ∑_i B_m,χ(a_i)=0 for every odd χ. Here n(a) is the residue count vector. This formulation underlies the prime-power filtration and the proposed conducto…
Exact mathematical passage
\mathcal B_{m,\chi}(r)= \sum_{u\in U_m}\rho_m(ur)\overline{\chi(u)}.
B_rUse in source · 4 passages
Search aliases: B_r
Use in source · appendices.tex, line 13
Divisor transfer · T=∑_rB_rA_r with actual Chow correspondences, packet preservation and invertibility.
Exact mathematical passage
T=\sum_rB_rA_r
b_RUse in source · 2 passages
Search aliases: b_R
Use in source · weil_audit.tex, line 131
The product locus and replacement factorizations. Split the six directions into a four-dimensional Weil factor C and a two-dimensional factor B. The determinant forms factor: Ω_6=Ω_CwedgeΩ_B. Writing their real and imaginary parts as c_R,c_I,b_R,b_I gives u=c_Rb_R-c_Ib_I, v=c_Rb_I+c_Ib_R. On the product locus, the degree-four Weil classes on C are algebraic by the checked fourfold theorem, and the degree-two classes on B are divisors by Lefschetz (1,1). Their products give the degree-six classes…
Exact mathematical passage
c_R,c_I,b_R,b_I
\bigsqcupUse in source · 1 passage
Search aliases: bigsqcup
Use in source · august_development.tex, line 774
[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balance equation descends. The maximal common divisor is unique. The converse follows by scaling.
Exact mathematical passage
\mathcal B_m=\bigsqcup_{c\mid m}L_{m/c}\mathcal B_c^{\mathrm{full}}.
\bigsqcup_{c\mid m}Use in source · 1 passage
Search aliases: bigsqcup_cmid m bigsqcup mid
Use in source · august_development.tex, line 774
[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balance equation descends. The maximal common divisor is unique. The converse follows by scaling.
Exact mathematical passage
\mathcal B_m=\bigsqcup_{c\mid m}L_{m/c}\mathcal B_c^{\mathrm{full}}.
\operatorname{Bl}Use in source · 1 passage
Search aliases: Bl operatorname
Use in source · august_development.tex, line 375
[Blow-up closure] If it holds for X and a smooth center Z, it holds for Bl_ZX. Use the standard blow-up cohomology decomposition. The new summands are Tate twists of cohomology from Z multiplied by powers of the algebraic exceptional divisor.
Exact mathematical passage
\operatorname{Bl}_ZX
\operatorname{Bl}_ZUse in source · 1 passage
Search aliases: Bl_Z operatorname
Use in source · august_development.tex, line 375
[Blow-up closure] If it holds for X and a smooth center Z, it holds for Bl_ZX. Use the standard blow-up cohomology decomposition. The new summands are Tate twists of cohomology from Z multiplied by powers of the algebraic exceptional divisor.
Exact mathematical passage
\operatorname{Bl}_ZX
\bmodUse in source · 2 passages
Search aliases: bmod
Use in source · capacity_cycles.tex, line 207
C(1,20,24,42,61,62bmod70)=11/10.
Exact mathematical passage
\label{eq:capacity_cycles-2.5} \boxed{C(1,20,24,42,61,62\bmod70)=11/10.}
\mathbf CUse in source · 3 passages
Search aliases: C mathbf
Use in source · capacity_cycles.tex, line 54
X_m={F_m=x_0^m+⋯+x_5^m=0}⊂ P^5_ C, G_m=(μ_m)^6/μ_m.
Exact mathematical passage
X_m=\{F_m=x_0^m+\cdots+x_5^m=0\}\subset\mathbf P^5_{\mathbf C}, \qquad G_m=(\mu_m)^6/\mu_m.
C^*Use in source · 4 passages
Search aliases: C^*
Use in source · appendices.tex, line 9
Cycle frame · S=CC^*. Positivity spans a target packet when every supplied algebraic vector lies in it.
Exact mathematical passage
S=CC^*
c^6Use in source · 2 passages
Search aliases: c^6
Use in source · carrier70.tex, line 187
F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.
Exact mathematical passage
\begin{aligned} F_u&=3u^2+s^2t^2z, &F_v&=5v^4,\\ F_s&=2sz^3+t^2c^7+2ust^2z, &F_t&=2t(z^3+sc^7+us^2z),\\ F_c&=7st^2c^6, &F_z&=3z^2(s^2+t^2)+us^2t^2. \end{aligned}
c^7Use in source · 2 passages
Search aliases: c^7
Use in source · carrier70.tex, line 181
F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.
Exact mathematical passage
F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.
c^tUse in source · 1 passage
Search aliases: c^t
Use in source · edition_audit.tex, line 67
Take V=Span_R{e_1} in R^2 and let the only cycle vector be c=e_1+e_2. Then S=cc^t=1 · 1 1 · 1. The compressed Gram matrix on V is the positive 1×1 matrix (1), while e_1 does not belong to Span(c). Positivity proves that the target direction is detected, not that synthesis returns it inside the original cycle span.
Exact mathematical passage
S=cc^t=\begin{pmatrix}1&1\\1&1\end{pmatrix}.
c_0Use in source · 3 passages
Search aliases: c_0
Use in source · august_development.tex, line 301
Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.
Exact mathematical passage
\chi_T(t)=t^d+c_{d-1}t^{d-1}+\cdots+c_0
c_1Use in source · 3 passages
Search aliases: c_1
Use in source · august_development.tex, line 99
Positive cycle-frame calculus. Analysis and synthesis. Fix a finite family of rational algebraic cycle classes c_1,…,c_N∈ H^2p(X,Q). Let V⊆ H^2p(X,R) be the real extension of the rational Hodge packet under study. For the positive frame certificate below, require every c_j to belong to V. This can be achieved by a rational algebraic packet projector when such a projector has been constructed. Projection through an arbitrary orthogonal Hodge projector does not establish that the projected vectors…
Exact mathematical passage
c_1,\ldots,c_N\in H^{2p}(X,\Q).
C_DUse in source · 2 passages
Search aliases: C_D
Use in source · august_development.tex, line 233
[Finite completion certificate] If H is a rational basis matrix of Hdg^p(X) and C_D is the cycle matrix, then the Hodge conjecture for (X,p) is equivalent to the existence of D with rank_Q C_D=rank_Q[C_D H]. A positive compressed frame is equivalent only when the supplied cycle vectors lie in the target packet, or when a separately verified invariant restriction supplies the required containment. The rank equality is valid without that extra hypothesis.
Exact mathematical passage
C_D
c_{d-1}Use in source · 2 passages
Search aliases: c_d-1
Use in source · august_development.tex, line 301
Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.
Exact mathematical passage
\chi_T(t)=t^d+c_{d-1}t^{d-1}+\cdots+c_0
c_IUse in source · 2 passages
Search aliases: c_I
Use in source · weil_audit.tex, line 131
The product locus and replacement factorizations. Split the six directions into a four-dimensional Weil factor C and a two-dimensional factor B. The determinant forms factor: Ω_6=Ω_CwedgeΩ_B. Writing their real and imaginary parts as c_R,c_I,b_R,b_I gives u=c_Rb_R-c_Ib_I, v=c_Rb_I+c_Ib_R. On the product locus, the degree-four Weil classes on C are algebraic by the checked fourfold theorem, and the degree-two classes on B are divisors by Lefschetz (1,1). Their products give the degree-six classes…
Exact mathematical passage
c_R,c_I,b_R,b_I
c_iUse in source · 4 passages
Search aliases: c_i
Use in source · capacity_cycles.tex, line 143
The infinitely many inequalities can be certified by a finite residue graph. Write y_i=c_i/L, where c_i,L are positive integers. On Z/m Z, an edge r→ r+a_i has cost c_i. An integer function d satisfying
Exact mathematical passage
y_i=c_i/L
c_jUse in source · 8 passages
Search aliases: c_j
Use in source · august_development.tex, line 102
Positive cycle-frame calculus. Analysis and synthesis. Fix a finite family of rational algebraic cycle classes c_1,…,c_N∈ H^2p(X,Q). Let V⊆ H^2p(X,R) be the real extension of the rational Hodge packet under study. For the positive frame certificate below, require every c_j to belong to V. This can be achieved by a rational algebraic packet projector when such a projector has been constructed. Projection through an arbitrary orthogonal Hodge projector does not establish that the projected vectors…
Exact mathematical passage
c_j
c_NUse in source · 2 passages
Search aliases: c_N
Use in source · august_development.tex, line 99
Positive cycle-frame calculus. Analysis and synthesis. Fix a finite family of rational algebraic cycle classes c_1,…,c_N∈ H^2p(X,Q). Let V⊆ H^2p(X,R) be the real extension of the rational Hodge packet under study. For the positive frame certificate below, require every c_j to belong to V. This can be achieved by a rational algebraic packet projector when such a projector has been constructed. Projection through an arbitrary orthogonal Hodge projector does not establish that the projected vectors…
Exact mathematical passage
c_1,\ldots,c_N\in H^{2p}(X,\Q).
c_{n,j}Use in source · 1 passage
Search aliases: c_n,j
Use in source · august_development.tex, line 497
A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge j _L^2^2+O(k^n-2j-1), c_n,j>0. This detects nonintegrability. It does not by itself remove the defect.
Exact mathematical passage
\Tr(P_{2j}\Pi_{A,k}) =c_{n,j}k^{n-2j}\|(F_A^{0,2})^{\wedge j}\|_{L^2}^2+O(k^{n-2j-1}), \qquad c_{n,j}>0.
C_PUse in source · 5 passages
Search aliases: C_P
Use in source · august_development.tex, line 1427
entatives c,d. Require c,d,e to be nonzero modulo q so that all three resulting character blocks have full support. Degenerate cases must leave this construction and be handled through pair removal or another channel. For example (1,14,2,9,14,5) at modulus 15 is balanced and admits the indicated phase split, but P=(1,14) gives c=0. The phase condition alone is insufficient. Define C_P=Psqcup{m-5c}, S_T=Tsqcup{m-5d}, and L=(5c,5d,5e).
Exact mathematical passage
C_P=P\sqcup\{m-5c\},
c_RUse in source · 2 passages
Search aliases: c_R
Use in source · weil_audit.tex, line 131
The product locus and replacement factorizations. Split the six directions into a four-dimensional Weil factor C and a two-dimensional factor B. The determinant forms factor: Ω_6=Ω_CwedgeΩ_B. Writing their real and imaginary parts as c_R,c_I,b_R,b_I gives u=c_Rb_R-c_Ib_I, v=c_Rb_I+c_Ib_R. On the product locus, the degree-four Weil classes on C are algebraic by the checked fourfold theorem, and the degree-two classes on B are divisors by Lefschetz (1,1). Their products give the degree-six classes…
Exact mathematical passage
c_R,c_I,b_R,b_I
\CHUse in source · 7 passages
Search aliases: CH
Use in source · appendices.tex, line 7
Algebraic space · Alg^p(X)=im(CH^p(X)_Q→ H^2p(X,Q)).
Exact mathematical passage
\Alg^p(X)=\im(\CH^p(X)_{\Q}\to H^{2p}(X,\Q))
\operatorname{ch}Use in source · 3 passages
Search aliases: ch operatorname
Use in source · august_development.tex, line 424
Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees is equivalent to algebraization of topological K-classes whose Chern-character components are rational Hodge classes [source reference].
Exact mathematical passage
\operatorname{ch}_{\mathrm{top}}:K^0_{\mathrm{top}}(X)\otimes\Q\xrightarrow{\sim}H^{\mathrm{even}}(X,\Q).
\CH^2Use in source · 1 passage
Search aliases: CH^2 CH
Use in source · august_development.tex, line 679
The constructive objective is [Packet-resolved construction target] For every rational packet W⊂ H^4_prim(X_m^4,Q)∩ H^2,2, construct Z_W∈CH^2(X_m^4)_Q with [Z_W]≠0 in W. Once one nonzero algebraic vector is produced in a rational irreducible diagonal-symmetry packet, the entire rational orbit is algebraic.
Exact mathematical passage
Z_W\in\CH^2(X_m^4)_\Q \qquad\text{with}\qquad [Z_W]\ne0\text{ in }W.
\CH^{\dim Y_r+p-1}Use in source · 1 passage
Search aliases: CH^dim Y_r+p-1 CH dim
Use in source · august_development.tex, line 274
Divisor-lowering correspondence channels. Degree bookkeeping. Let X have complex dimension n, let V⊆ H_prim^2p(X,Q) be a rational Hodge substructure, and let Y_r be smooth projective. Choose correspondences Γ_r∈CH^n-p+1(X× Y_r)_Q, Λ_r∈CH^dim Y_r+p-1(Y_r× X)_Q. They induce A_r=(Γ_r)_*:V→ H^2(Y_r,Q), B_r=(Λ_r)_*:H^2(Y_r,Q)→ H^2p(X,Q). Algebraic correspondences preserve rationality and shift Hodge type by the required bidegree.
Exact mathematical passage
\Gamma_r\in\CH^{n-p+1}(X\times Y_r)_{\Q}, \qquad \Lambda_r\in\CH^{\dim Y_r+p-1}(Y_r\times X)_{\Q}.
\CH^{n-p+1}Use in source · 1 passage
Search aliases: CH^n-p+1 CH
Use in source · august_development.tex, line 274
Divisor-lowering correspondence channels. Degree bookkeeping. Let X have complex dimension n, let V⊆ H_prim^2p(X,Q) be a rational Hodge substructure, and let Y_r be smooth projective. Choose correspondences Γ_r∈CH^n-p+1(X× Y_r)_Q, Λ_r∈CH^dim Y_r+p-1(Y_r× X)_Q. They induce A_r=(Γ_r)_*:V→ H^2(Y_r,Q), B_r=(Λ_r)_*:H^2(Y_r,Q)→ H^2p(X,Q). Algebraic correspondences preserve rationality and shift Hodge type by the required bidegree.
Exact mathematical passage
\Gamma_r\in\CH^{n-p+1}(X\times Y_r)_{\Q}, \qquad \Lambda_r\in\CH^{\dim Y_r+p-1}(Y_r\times X)_{\Q}.
\CH^pUse in source · 3 passages
Search aliases: CH^p CH
Use in source · appendices.tex, line 7
Algebraic space · Alg^p(X)=im(CH^p(X)_Q→ H^2p(X,Q)).
Exact mathematical passage
\Alg^p(X)=\im(\CH^p(X)_{\Q}\to H^{2p}(X,\Q))
\CH^rUse in source · 1 passage
Search aliases: CH^r CH
Use in source · appendices.tex, line 25
Correspondence degrees. Let X have dimension n and Y dimension m. A correspondence Γ∈CH^r(X× Y) acts by Γ_*:H^i(X)⟶ H^i+2(r-n)(Y) and shifts Hodge type by (r-n,r-n). To lower H^2p(X) to H^2(Y), take r=n-p+1. To return degree two to degree 2p on X, a correspondence on Y× X has codimension m+p-1.
Exact mathematical passage
\Gamma\in\CH^r(X\times Y)
\CH^{\ast}Use in source · 1 passage
Search aliases: CH^∗ CH ast
Use in source · august_development.tex, line 1532
The desired correspondence is schematically Γ∈CH^∗(F_d× C_P× S_T)_Q, Γ_*:e_orbH^1(F_d)⟶ e_diag(M(C_P)⊗ M(S_T))(1). This formula is a target specification, not a constructed map. Its coefficient field, diagonal projector, rational multiplicities, and dimensions must be fixed before an isomorphism can be asserted. The smaller CM conductor can have a smaller rational degree than the original packet. Matching sign functions alone does not resolve that difference.
Exact mathematical passage
\Gamma\in\CH^{\ast}(F_d\times C_P\times S_T)_{\Q},\qquad \Gamma_*:e_{\mathrm{orb}}H^1(F_d)\longrightarrow e_{\mathrm{diag}}\bigl(M(C_P)\otimes M(S_T)\bigr)(1).
\operatorname{ch}_pUse in source · 1 passage
Search aliases: ch_p operatorname
Use in source · august_development.tex, line 440
q≠ p fracψ^m-m^qIm^p-m^q. Then ch_q(e_p^(m)ξ)=δ_pqch_p(ξ). This exactly isolates codimension p inside a topological K-packet.
Exact mathematical passage
\operatorname{ch}_q(e_p^{(m)}\xi)=\delta_{pq}\operatorname{ch}_p(\xi).
\operatorname{ch}_qUse in source · 2 passages
Search aliases: ch_q operatorname
Use in source · august_development.tex, line 431
Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X
Exact mathematical passage
\operatorname{ch}_q(\psi^m\xi)=m^q\operatorname{ch}_q(\xi).
\operatorname{ch}_{\mathrm{top}}Use in source · 1 passage
Search aliases: ch_top operatorname mathrm
Use in source · august_development.tex, line 424
Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees is equivalent to algebraization of topological K-classes whose Chern-character components are rational Hodge classes [source reference].
Exact mathematical passage
\operatorname{ch}_{\mathrm{top}}:K^0_{\mathrm{top}}(X)\otimes\Q\xrightarrow{\sim}H^{\mathrm{even}}(X,\Q).
\clUse in source · 1 passage
Search aliases: cl
Use in source · august_development.tex, line 12
The rational Hodge target. Statement and status. Let X be a smooth projective complex variety. For each codimension p, define Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X) and Alg^p(X)=im(CH^p(X)_QxrightarrowclH^2p(X,Q)). Every algebraic cycle class has type (p,p), so Alg^p(X)⊆Hdg^p(X). The rational Hodge conjecture asserts equality for every smooth projective X and every p [source reference].
Exact mathematical passage
\Alg^p(X)=\im\bigl(\CH^p(X)_{\Q}\xrightarrow{\cl}H^{2p}(X,\Q)\bigr).
\mathrm{conic}Use in source · 2 passages
Search aliases: conic mathrm
Use in source · august_development.tex, line 570
A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equivalently prove T_tot =∑_rA_r^† A_r satisfies λ_min(T_tot)>0.
Exact mathematical passage
A_{\mathrm{line}}, A_{\mathrm{conic}}, A_{\mathrm{plane}}, A_{\mathrm{exceptional}}.
D_0Use in source · 3 passages
Search aliases: D_0
Use in source · august_development.tex, line 211
[Finite stabilization] There exists D_0 such that A_D^p(X)=A_D_0^p(X) for every D≥ D_0. The dimensions form a nondecreasing sequence of integers bounded by dim H^2p(X,Q). Only finitely many strict increases are possible.
Exact mathematical passage
D_0
d_nUse in source · 3 passages
Search aliases: d_n
Use in source · august_development.tex, line 1065
Complement defects and dyadic triangles. Put q=frac m4, h=2q, m=4q. For a packet count vector n, define its oriented complement defect ∂ n=∑_r=1^h-1d_n(r)[r], d_n(r)=n_r-n_m-r. The self-complementary residue h contributes zero. Since ρ_m(m-r)=-ρ_m(r), the balanced-packet equation is equivalent to ∑_r=1^h-1d_n(r)ρ_m(ur)=0 for every odd unit u. Moreover, ∂ n _1=∑_r=1^h-1|d_n(r)|≤ 6. Thus the dyadic problem is a short-vector problem in an integral defect lattice.
Exact mathematical passage
\partial n=\sum_{r=1}^{h-1}d_n(r)[r], \qquad d_n(r)=n_r-n_{m-r}.
D_rUse in source · 2 passages
Search aliases: D_r
Use in source · august_development.tex, line 295
[Invertible divisor-lowering transfer] Suppose T=∑_r=1^M B_rA_r preserves V and is invertible on V. Then every rational Hodge class in V is algebraic. Let α∈ V be of type (p,p). Each A_rα is rational of type (1,1) on Y_r, hence is the class of a rational divisor D_r by the Lefschetz (1,1) theorem. Therefore Tα=∑_r(Λ_r)_*[D_r] is algebraic on X.
Exact mathematical passage
D_r
D_\kappaUse in source · 6 passages
Search aliases: D_κ kappa
Use in source · capacity_cycles.tex, line 433
H=(f_1,g_1,f_2,g_2,f_3,g_3), D_κ=detJac_(x_0,…,x_4,z)H,
Exact mathematical passage
H=(f_1,g_1,f_2,g_2,f_3,g_3),\qquad D_\kappa=\det\operatorname{Jac}_{(x_0,\ldots,x_4,z)}H,
\dashrightarrowUse in source · 2 passages
Search aliases: dashrightarrow
Use in source · august_development.tex, line 914
Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.
Exact mathematical passage
\phi:X_{33}^4\dashrightarrow Y
\detUse in source · 3 passages
Search aliases: det
Use in source · capacity_cycles.tex, line 433
H=(f_1,g_1,f_2,g_2,f_3,g_3), D_κ=detJac_(x_0,…,x_4,z)H,
Exact mathematical passage
H=(f_1,g_1,f_2,g_2,f_3,g_3),\qquad D_\kappa=\det\operatorname{Jac}_{(x_0,\ldots,x_4,z)}H,
\mathrm{diag}Use in source · 2 passages
Search aliases: diag mathrm
Use in source · august_development.tex, line 1458
After fixing a common cyclotomic coefficient field and a diagonal character projector, write the selected factor schematically as N=e_diag(M(C_P)⊗ M(S_T)). An ordinary tensor product of rational packets contains off-diagonal embedding pairs and is generally not rank one. Rank one here refers only to a selected eigenspace after scalar extension. The coefficient field and projector are part of the unresolved correspondence specification. The Hodge-level identity implies that N(1) has weight one an…
Exact mathematical passage
N=e_{\mathrm{diag}}\bigl(M(C_P)\otimes M(S_T)\bigr).
\operatorname{diag}Use in source · 3 passages
Search aliases: diag operatorname
Use in source · weil_audit.tex, line 8
The lattice, polarization and seed. Let K=Q(i) and take homology lattice Z[i]^6. Use the conjugate-linear-first Hermitian form h(v,w)=overline v^ tdiag(1,1,1,-1,-1,-3)w, E=Im h. The seed complex structure is J_0=diag(i,i,i,-i,-i,-i). It preserves the alternating form E, and E(v,J_0v)>0 for nonzero v. These are the Riemann positivity conditions for the seed abelian variety. The sign depends on the declared Hermitian convention. Omitting that convention would leave the polarization claim ambiguous…
Exact mathematical passage
h(v,w)=\overline v^{\,t}\operatorname{diag}(1,1,1,-1,-1,-3)w, \qquad E=\Im h.
\dimUse in source · 5 passages
Search aliases: dim
Use in source · august_development.tex, line 214
[Finite stabilization] There exists D_0 such that A_D^p(X)=A_D_0^p(X) for every D≥ D_0. The dimensions form a nondecreasing sequence of integers bounded by dim H^2p(X,Q). Only finitely many strict increases are possible.
Exact mathematical passage
\dim H^{2p}(X,\Q)
EUse in source · 10 passages
Search aliases: E
Use in source · august_development.tex, line 368
Projective bundles, blow-ups, and descent. [Projective-bundle closure] If the rational Hodge conjecture holds for X, it holds for P(E)→ X. The projective-bundle formula expresses cohomology as a direct sum of powers of the algebraic hyperplane class times pullbacks of cohomology from X.
Exact mathematical passage
\mathbb P(E)\to X
e^{-tX}Use in source · 1 passage
Search aliases: e^-tX
Use in source · weil_audit.tex, line 145
The exact audit then found replacement divisor factors using the pairs (1,6), (2,5), (3,4). These replacement factors remain of divisor type along the selected tangent. With X=-tfrac12(Δ J)J_0, J(t)=e^tXJ_0e^-tX, the pair blocks are preserved by the integrating generator. Thus the alternate factorization defeats the tested counterexample path. It proves no universal result for every fully mixed point, and it invalidates the inference that failure of the original factorization establishes nonalge…
Exact mathematical passage
X=-\tfrac12(\Delta J)J_0,\qquad J(t)=e^{tX}J_0e^{-tX},
e^{2\pi ir/q}Use in source · 1 passage
Search aliases: e^2π ir/q pi
Use in source · august_development.tex, line 1499
For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overlineΣ_ℓ. This is the ri…
Exact mathematical passage
1-e^{2\pi ir/q}=-2i\,e^{\pi ir/q}\sin\frac{\pi r}{q},
e^{tX}Use in source · 1 passage
Search aliases: e^tX
Use in source · weil_audit.tex, line 145
The exact audit then found replacement divisor factors using the pairs (1,6), (2,5), (3,4). These replacement factors remain of divisor type along the selected tangent. With X=-tfrac12(Δ J)J_0, J(t)=e^tXJ_0e^-tX, the pair blocks are preserved by the integrating generator. Thus the alternate factorization defeats the tested counterexample path. It proves no universal result for every fully mixed point, and it invalidates the inference that failure of the original factorization establishes nonalge…
Exact mathematical passage
X=-\tfrac12(\Delta J)J_0,\qquad J(t)=e^{tX}J_0e^{-tX},
e^{\pi ir/q}Use in source · 1 passage
Search aliases: e^π ir/q pi
Use in source · august_development.tex, line 1499
For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overlineΣ_ℓ. This is the ri…
Exact mathematical passage
1-e^{2\pi ir/q}=-2i\,e^{\pi ir/q}\sin\frac{\pi r}{q},
e_0Use in source · 2 passages
Search aliases: e_0
Use in source · august_development.tex, line 936
does not prove that a rational class lifts through a compact graph. A complete manuscript argument instead uses the torus-carrier criterion developed in Chapter~[source reference]. For the reordered character, the exponent matrix A of the cubic satisfies A^Twidetilde a/33=β=(1,2,2,1,1,2), C=33A^-1. With r_0 the first row of A, put u=β-3r_0. Then 1^Tu=0 and C^Tu=widetilde a-99e_0. These identities establish triviality of the selected character on the torus deck group. Every selected character has…
Exact mathematical passage
C^Tu=\widetilde a-99e_0
e_1Use in source · 15 passages
Search aliases: e_1
Use in source · appendices.tex, line 19
Five-phase surface · e_1=e_2=P-κ z^5=0, with κ^q=-1/5.
Exact mathematical passage
e_1=e_2=P-\kappa z^5=0
e_1^2Use in source · 2 passages
Search aliases: e_1^2
Use in source · capacity_cycles.tex, line 337
p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.
Exact mathematical passage
\label{eq:capacity_cycles-4.2} p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.
e_1^4Use in source · 2 passages
Search aliases: e_1^4
Use in source · capacity_cycles.tex, line 337
p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.
Exact mathematical passage
\label{eq:capacity_cycles-4.2} p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.
e_2Use in source · 9 passages
Search aliases: e_2
Use in source · appendices.tex, line 19
Five-phase surface · e_1=e_2=P-κ z^5=0, with κ^q=-1/5.
Exact mathematical passage
e_1=e_2=P-\kappa z^5=0
e_2^2Use in source · 2 passages
Search aliases: e_2^2
Use in source · capacity_cycles.tex, line 337
p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.
Exact mathematical passage
\label{eq:capacity_cycles-4.2} p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.
e_3Use in source · 4 passages
Search aliases: e_3
Use in source · capacity_cycles.tex, line 337
p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.
Exact mathematical passage
\label{eq:capacity_cycles-4.2} p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.
e_4Use in source · 6 passages
Search aliases: e_4
Use in source · capacity_cycles.tex, line 337
p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.
Exact mathematical passage
\label{eq:capacity_cycles-4.2} p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.
e_5Use in source · 5 passages
Search aliases: e_5
Use in source · capacity_cycles.tex, line 337
p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.
Exact mathematical passage
\label{eq:capacity_cycles-4.2} p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.
e_{[\alpha]}Use in source · 2 passages
Search aliases: e_[α] alpha
Use in source · carrier70.tex, line 357
e_[α]H^4(X_210, Q) xrightarrow∼ e_[α]H^4(U_210, Q).
Exact mathematical passage
e_{[\alpha]}H^4(X_{210},\mathbf Q) \xrightarrow{\sim} e_{[\alpha]}H^4(U_{210},\mathbf Q).
e_dUse in source · 1 passage
Search aliases: e_d
Use in source · august_development.tex, line 332
[Finite Gram certificate] Choose a rational basis e_1,…,e_d of V. If G_ij=∑_r⟨ A_re_i,A_re_j⟩_Y_r has nonzero determinant, every rational Hodge class in V is algebraic.
Exact mathematical passage
e_1,\ldots,e_d
e_{\mathrm{diag}}Use in source · 2 passages
Search aliases: e_diag mathrm
Use in source · august_development.tex, line 1458
After fixing a common cyclotomic coefficient field and a diagonal character projector, write the selected factor schematically as N=e_diag(M(C_P)⊗ M(S_T)). An ordinary tensor product of rational packets contains off-diagonal embedding pairs and is generally not rank one. Rank one here refers only to a selected eigenspace after scalar extension. The coefficient field and projector are part of the unresolved correspondence specification. The Hodge-level identity implies that N(1) has weight one an…
Exact mathematical passage
N=e_{\mathrm{diag}}\bigl(M(C_P)\otimes M(S_T)\bigr).
e_iUse in source · 2 passages
Search aliases: e_i
Use in source · august_development.tex, line 333
[Finite Gram certificate] Choose a rational basis e_1,…,e_d of V. If G_ij=∑_r⟨ A_re_i,A_re_j⟩_Y_r has nonzero determinant, every rational Hodge class in V is algebraic.
Exact mathematical passage
G_{ij}=\sum_r\langle A_re_i,A_re_j\rangle_{Y_r}
e_jUse in source · 3 passages
Search aliases: e_j
Use in source · august_development.tex, line 103
bseteq H^2p(X,R) be the real extension of the rational Hodge packet under study. For the positive frame certificate below, require every c_j to belong to V. This can be achieved by a rational algebraic packet projector when such a projector has been constructed. Projection through an arbitrary orthogonal Hodge projector does not establish that the projected vectors are algebraic. Define C:R^N→ H^2p(X,R), Ce_j=c_j, and the positive frame operator S=CC^*=∑_j=1^N|c_j⟩⟨ c_j|.
Exact mathematical passage
C:\R^N\to H^{2p}(X,\R), \qquad Ce_j=c_j,
E_{m,d}Use in source · 4 passages
Search aliases: E_m,d
Use in source · august_development.tex, line 787
For dmid m, let E_m,d=frac1(m/d)^5π_m/d^*π_m/d*. On a character of exact conductor c, E_m,d=1, · cmid d,
Exact mathematical passage
E_{m,d}=\frac1{(m/d)^5}\pi_{m/d}^*\pi_{m/d*}.
E_{m,e}Use in source · 1 passage
Search aliases: E_m,e
Use in source · august_development.tex, line 795
0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is itself algebraic.
Exact mathematical passage
\label{eq:legacy-8.1} \boxed{E_{m,d}E_{m,e}=E_{m,\gcd(d,e)}.}
E_{m,\gcd(d,e)}Use in source · 1 passage
Search aliases: E_m,gcd(d,e) gcd
Use in source · august_development.tex, line 795
0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is itself algebraic.
Exact mathematical passage
\label{eq:legacy-8.1} \boxed{E_{m,d}E_{m,e}=E_{m,\gcd(d,e)}.}
e_{m-r}Use in source · 1 passage
Search aliases: e_m-r
Use in source · august_development.tex, line 825
Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.
Exact mathematical passage
p_r=e_r+e_{m-r}.
e_p^{(m)}Use in source · 2 passages
Search aliases: e_p^(m)
Use in source · august_development.tex, line 435
Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X q≠ p fracψ^m-m^qIm^p-m^q. Then ch_q(e_p^(m)ξ)=δ_pqch_p(ξ). This exactly isolates codimension p inside a topological K-packet.
Exact mathematical passage
e_p^{(m)}=\prod_{\substack{0\le q\le\dim X\\q\ne p}} \frac{\psi^m-m^qI}{m^p-m^q}.
e_rUse in source · 1 passage
Search aliases: e_r
Use in source · august_development.tex, line 825
Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.
Exact mathematical passage
p_r=e_r+e_{m-r}.
\mathrm{even}Use in source · 1 passage
Search aliases: even mathrm
Use in source · august_development.tex, line 424
Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees is equivalent to algebraization of topological K-classes whose Chern-character components are rational Hodge classes [source reference].
Exact mathematical passage
\operatorname{ch}_{\mathrm{top}}:K^0_{\mathrm{top}}(X)\otimes\Q\xrightarrow{\sim}H^{\mathrm{even}}(X,\Q).
\mathrm{exceptional}Use in source · 2 passages
Search aliases: exceptional mathrm
Use in source · august_development.tex, line 570
A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equivalently prove T_tot =∑_rA_r^† A_r satisfies λ_min(T_tot)>0.
Exact mathematical passage
A_{\mathrm{line}}, A_{\mathrm{conic}}, A_{\mathrm{plane}}, A_{\mathrm{exceptional}}.
\mathcal FUse in source · 5 passages
Search aliases: F mathcal
Use in source · august_development.tex, line 835
Balanced four-packets and saturation. Let F_m= { γ∈ N^m-1: 1^Tγ=4, R_mγ=2m 1 }. A saturated certificate is an exact nonnegative identity n+∑_j=1^2N-3p_r_j =∑_i=1^Nγ_i, γ_i∈ F_m. The case N=2 is one-step quasi-decomposition. Higher N allows several surface packets to reconstruct one six-packet after Tate stabilization.
Exact mathematical passage
\mathcal F_m= \left\{ \gamma\in\mathbf N^{m-1}: \mathbf1^T\gamma=4, \quad R_m\gamma=2m\,\mathbf1 \right\}.
fUse in source · 17 passages
Search aliases: f
Use in source · appendices.tex, line 33
For a finite generically finite morphism of degree d, the push-pull identity on rational cohomology is f_*f^*=did. Division by d transfers algebraicity downward. Pullback transfers it upward without division. Coordinate-power maps preserve the old conductor of an inflated character, even though the ambient Fermat degree grows.
Exact mathematical passage
f_*f^*=d\id
f^*Use in source · 3 passages
Search aliases: f^*
Use in source · appendices.tex, line 33
For a finite generically finite morphism of degree d, the push-pull identity on rational cohomology is f_*f^*=did. Division by d transfers algebraicity downward. Pullback transfers it upward without division. Coordinate-power maps preserve the old conductor of an inflated character, even though the ambient Fermat degree grows.
Exact mathematical passage
f_*f^*=d\id
f_*Use in source · 2 passages
Search aliases: f_*
Use in source · appendices.tex, line 33
For a finite generically finite morphism of degree d, the push-pull identity on rational cohomology is f_*f^*=did. Division by d transfers algebraicity downward. Pullback transfers it upward without division. Coordinate-power maps preserve the old conductor of an inflated character, even though the ambient Fermat degree grows.
Exact mathematical passage
f_*f^*=d\id
f_1Use in source · 4 passages
Search aliases: f_1
Use in source · capacity_cycles.tex, line 418
f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.
Exact mathematical passage
\begin{aligned} f_1&=e_1,&g_1&=e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4,\\ f_2&=e_2,&g_2&=-5e_3,\\ f_3&=P-\kappa z^5,&g_3&=5\sum_{j=0}^{q-1}P^{q-1-j}\kappa^jz^{5j}. \end{aligned}
f_2Use in source · 4 passages
Search aliases: f_2
Use in source · capacity_cycles.tex, line 418
f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.
Exact mathematical passage
\begin{aligned} f_1&=e_1,&g_1&=e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4,\\ f_2&=e_2,&g_2&=-5e_3,\\ f_3&=P-\kappa z^5,&g_3&=5\sum_{j=0}^{q-1}P^{q-1-j}\kappa^jz^{5j}. \end{aligned}
f_3Use in source · 4 passages
Search aliases: f_3
Use in source · capacity_cycles.tex, line 418
f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.
Exact mathematical passage
\begin{aligned} f_1&=e_1,&g_1&=e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4,\\ f_2&=e_2,&g_2&=-5e_3,\\ f_3&=P-\kappa z^5,&g_3&=5\sum_{j=0}^{q-1}P^{q-1-j}\kappa^jz^{5j}. \end{aligned}
F_{35}^3Use in source · 1 passage
Search aliases: F_35^3
Use in source · capacity_cycles.tex, line 628
∫_Z_κres(fracQ_1ΩF_35^3) =frac(2π i)^2κ^6350≠0.
Exact mathematical passage
\boxed{ \int_{Z_\kappa}\operatorname{res}\left(\frac{Q_1\Omega}{F_{35}^3}\right) =\frac{(2\pi i)^2\kappa^6}{350}\ne0. }
F_aUse in source · 3 passages
Search aliases: F_a
Use in source · august_development.tex, line 732
Centered Fourier form. Define ρ_m(r)=⟨ r⟩_m-frac m2, F_a(u)=∑_iρ_m(ua_i). Then a is balanced if and only if F_a(u)=0 for all units. Moreover, F_a(-u)=-F_a(u), so only odd characters occur. For a Dirichlet character χ on U_m, put B_m,χ(r)= ∑_u∈ U_mρ_m(ur)overlineχ(u). Finite Fourier inversion gives the exact criterion n(a)∈ B_m Longleftrightarrow ∑_i B_m,χ(a_i)=0 for every odd χ. Here n(a) is the residue count vector. This formulation underlies the prime-power filtration and the proposed conducto…
Exact mathematical passage
\rho_m(r)=\langle r\rangle_m-\frac m2, \qquad F_a(u)=\sum_i\rho_m(ua_i).
F_A^{0,2}Use in source · 3 passages
Search aliases: F_A^0,2
Use in source · august_development.tex, line 496
A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge j _L^2^2+O(k^n-2j-1), c_n,j>0. This detects nonintegrability. It does not by itself remove the defect.
Exact mathematical passage
F_A^{0,2}
F_cUse in source · 1 passage
Search aliases: F_c
Use in source · carrier70.tex, line 187
F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.
Exact mathematical passage
\begin{aligned} F_u&=3u^2+s^2t^2z, &F_v&=5v^4,\\ F_s&=2sz^3+t^2c^7+2ust^2z, &F_t&=2t(z^3+sc^7+us^2z),\\ F_c&=7st^2c^6, &F_z&=3z^2(s^2+t^2)+us^2t^2. \end{aligned}
F_mUse in source · 7 passages
Search aliases: F_m
Use in source · capacity_cycles.tex, line 54
X_m={F_m=x_0^m+⋯+x_5^m=0}⊂ P^5_ C, G_m=(μ_m)^6/μ_m.
Exact mathematical passage
X_m=\{F_m=x_0^m+\cdots+x_5^m=0\}\subset\mathbf P^5_{\mathbf C}, \qquad G_m=(\mu_m)^6/\mu_m.
\mathcal F_mUse in source · 5 passages
Search aliases: F_m mathcal
Use in source · august_development.tex, line 835
Balanced four-packets and saturation. Let F_m= { γ∈ N^m-1: 1^Tγ=4, R_mγ=2m 1 }. A saturated certificate is an exact nonnegative identity n+∑_j=1^2N-3p_r_j =∑_i=1^Nγ_i, γ_i∈ F_m. The case N=2 is one-step quasi-decomposition. Higher N allows several surface packets to reconstruct one six-packet after Tate stabilization.
Exact mathematical passage
\mathcal F_m= \left\{ \gamma\in\mathbf N^{m-1}: \mathbf1^T\gamma=4, \quad R_m\gamma=2m\,\mathbf1 \right\}.
F_m^3Use in source · 1 passage
Search aliases: F_m^3
Use in source · capacity_cycles.tex, line 545
∫_Z_κres(fracQ_bΩF_m^3) =frac(2π i)^250q κ^q-b≠0.
Exact mathematical passage
\label{eq:capacity_cycles-6.5} \boxed{ \int_{Z_\kappa}\operatorname{res}\left(\frac{Q_b\Omega}{F_m^3}\right) =\frac{(2\pi i)^2}{50q}\,\kappa^{q-b}\ne0. }
F_sUse in source · 2 passages
Search aliases: F_s
Use in source · carrier70.tex, line 187
F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.
Exact mathematical passage
\begin{aligned} F_u&=3u^2+s^2t^2z, &F_v&=5v^4,\\ F_s&=2sz^3+t^2c^7+2ust^2z, &F_t&=2t(z^3+sc^7+us^2z),\\ F_c&=7st^2c^6, &F_z&=3z^2(s^2+t^2)+us^2t^2. \end{aligned}
F_tUse in source · 2 passages
Search aliases: F_t
Use in source · carrier70.tex, line 187
F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.
Exact mathematical passage
\begin{aligned} F_u&=3u^2+s^2t^2z, &F_v&=5v^4,\\ F_s&=2sz^3+t^2c^7+2ust^2z, &F_t&=2t(z^3+sc^7+us^2z),\\ F_c&=7st^2c^6, &F_z&=3z^2(s^2+t^2)+us^2t^2. \end{aligned}
F_uUse in source · 1 passage
Search aliases: F_u
Use in source · carrier70.tex, line 187
F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.
Exact mathematical passage
\begin{aligned} F_u&=3u^2+s^2t^2z, &F_v&=5v^4,\\ F_s&=2sz^3+t^2c^7+2ust^2z, &F_t&=2t(z^3+sc^7+us^2z),\\ F_c&=7st^2c^6, &F_z&=3z^2(s^2+t^2)+us^2t^2. \end{aligned}
F_vUse in source · 1 passage
Search aliases: F_v
Use in source · carrier70.tex, line 187
F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.
Exact mathematical passage
\begin{aligned} F_u&=3u^2+s^2t^2z, &F_v&=5v^4,\\ F_s&=2sz^3+t^2c^7+2ust^2z, &F_t&=2t(z^3+sc^7+us^2z),\\ F_c&=7st^2c^6, &F_z&=3z^2(s^2+t^2)+us^2t^2. \end{aligned}
F_zUse in source · 2 passages
Search aliases: F_z
Use in source · carrier70.tex, line 187
F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.
Exact mathematical passage
\begin{aligned} F_u&=3u^2+s^2t^2z, &F_v&=5v^4,\\ F_s&=2sz^3+t^2c^7+2ust^2z, &F_t&=2t(z^3+sc^7+us^2z),\\ F_c&=7st^2c^6, &F_z&=3z^2(s^2+t^2)+us^2t^2. \end{aligned}
\mathrm{Frob}Use in source · 1 passage
Search aliases: Frob mathrm
Use in source · august_development.tex, line 199
Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods to solve different geometric sectors before exact synthesis.
Exact mathematical passage
\lambda=(r_{\mathrm{Lef}},r_{\deg},\chi_{\mathrm{mon}},s_{\mathrm{supp}},\nu_{\mathrm{Frob}}).
\mathrm{full}Use in source · 1 passage
Search aliases: full mathrm
Use in source · august_development.tex, line 774
[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balance equation descends. The maximal common divisor is unique. The converse follows by scaling.
Exact mathematical passage
\mathcal B_m=\bigsqcup_{c\mid m}L_{m/c}\mathcal B_c^{\mathrm{full}}.
g_1Use in source · 5 passages
Search aliases: g_1
Use in source · capacity_cycles.tex, line 418
f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.
Exact mathematical passage
\begin{aligned} f_1&=e_1,&g_1&=e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4,\\ f_2&=e_2,&g_2&=-5e_3,\\ f_3&=P-\kappa z^5,&g_3&=5\sum_{j=0}^{q-1}P^{q-1-j}\kappa^jz^{5j}. \end{aligned}
g_2Use in source · 4 passages
Search aliases: g_2
Use in source · capacity_cycles.tex, line 418
f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.
Exact mathematical passage
\begin{aligned} f_1&=e_1,&g_1&=e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4,\\ f_2&=e_2,&g_2&=-5e_3,\\ f_3&=P-\kappa z^5,&g_3&=5\sum_{j=0}^{q-1}P^{q-1-j}\kappa^jz^{5j}. \end{aligned}
G_{210}Use in source · 2 passages
Search aliases: G_210
Use in source · carrier70.tex, line 328
Thus u is a character of the weighted projective torus. Its pullback restricts on G_210 to the Fermat character α, since 630=3·210.
Exact mathematical passage
G_{210}
g_3Use in source · 4 passages
Search aliases: g_3
Use in source · capacity_cycles.tex, line 418
f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.
Exact mathematical passage
\begin{aligned} f_1&=e_1,&g_1&=e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4,\\ f_2&=e_2,&g_2&=-5e_3,\\ f_3&=P-\kappa z^5,&g_3&=5\sum_{j=0}^{q-1}P^{q-1-j}\kappa^jz^{5j}. \end{aligned}
G_{ij}Use in source · 1 passage
Search aliases: G_ij
Use in source · august_development.tex, line 333
[Finite Gram certificate] Choose a rational basis e_1,…,e_d of V. If G_ij=∑_r⟨ A_re_i,A_re_j⟩_Y_r has nonzero determinant, every rational Hodge class in V is algebraic.
Exact mathematical passage
G_{ij}=\sum_r\langle A_re_i,A_re_j\rangle_{Y_r}
H^*Use in source · 1 passage
Search aliases: H^*
Use in source · august_development.tex, line 130
If H is a matrix whose columns form an orthonormal basis of V, the finite Gram certificate is G=H^*SH. Under A⊆ V, we have A=V if and only if λ_min(G)>0. Without that containment, use a rational basis matrix H_Q for the target and the exact certificate rank_Q[C H_Q]=rank_QC to test V⊆ A.
Exact mathematical passage
G=H^*SH.
H^1Use in source · 1 passage
Search aliases: H^1
Use in source · august_development.tex, line 1532
The desired correspondence is schematically Γ∈CH^∗(F_d× C_P× S_T)_Q, Γ_*:e_orbH^1(F_d)⟶ e_diag(M(C_P)⊗ M(S_T))(1). This formula is a target specification, not a constructed map. Its coefficient field, diagonal projector, rational multiplicities, and dimensions must be fixed before an isomorphism can be asserted. The smaller CM conductor can have a smaller rational degree than the original packet. Matching sign functions alone does not resolve that difference.
Exact mathematical passage
\Gamma\in\CH^{\ast}(F_d\times C_P\times S_T)_{\Q},\qquad \Gamma_*:e_{\mathrm{orb}}H^1(F_d)\longrightarrow e_{\mathrm{diag}}\bigl(M(C_P)\otimes M(S_T)\bigr)(1).
H^{1,1}Use in source · 1 passage
Search aliases: H^1,1
Use in source · august_development.tex, line 65
For a fourfold and p=2, α=Lβ+α_prim, β∈ H^1,1(X)∩ H^2(X,Q). The Lefschetz (1,1) theorem makes β algebraic. Only α_prim remains.
Exact mathematical passage
\alpha=L\beta+\alpha_{\mathrm{prim}}, \qquad \beta\in H^{1,1}(X)\cap H^2(X,\Q).
H^2Use in source · 5 passages
Search aliases: H^2
Use in source · appendices.tex, line 29
Correspondence degrees. Let X have dimension n and Y dimension m. A correspondence Γ∈CH^r(X× Y) acts by Γ_*:H^i(X)⟶ H^i+2(r-n)(Y) and shifts Hodge type by (r-n,r-n). To lower H^2p(X) to H^2(Y), take r=n-p+1. To return degree two to degree 2p on X, a correspondence on Y× X has codimension m+p-1.
Exact mathematical passage
H^2(Y)
h^2Use in source · 3 passages
Search aliases: h^2
Use in source · august_development.tex, line 903
r after permutation, widetilde a=(7,19,28,10,13,22). Every unit conjugate has residue sum 99=3·33, so the orbit is rational of type (2,2). It is not totally pair-decomposable. The previously proposed symmetric surface W={p_1(x^11)=p_2(x^11)=p_3(x^11)=0} does not carry the primitive packet: Newton's identity makes the third equation redundant on X_33^4, and [W]=242h^2. Thus W is purely ambient.
Exact mathematical passage
[W]=242h^2.
H^{2,2}Use in source · 1 passage
Search aliases: H^2,2
Use in source · august_development.tex, line 675
The constructive objective is [Packet-resolved construction target] For every rational packet W⊂ H^4_prim(X_m^4,Q)∩ H^2,2, construct Z_W∈CH^2(X_m^4)_Q with [Z_W]≠0 in W. Once one nonzero algebraic vector is produced in a rational irreducible diagonal-symmetry packet, the entire rational orbit is algebraic.
Exact mathematical passage
W\subset H^4_{\mathrm{prim}}(X_m^4,\Q)\cap H^{2,2},
H^2_{\mathrm{prim}}Use in source · 1 passage
Search aliases: H^2_prim mathrm
Use in source · august_development.tex, line 396
Cubic fourfolds as a validation of the transfer theorem. Let X⊂ P^5 be a smooth cubic fourfold and let F(X) be its Fano variety of lines. The universal incidence variety P={(ℓ,x):x∈ℓ}⊂ F(X)× X induces the Abel-Jacobi map A=p_*q^*:H^4_prim(X,Q)→ H^2_prim(F(X),Q). Beauville and Donagi proved that this is a polarized Hodge isomorphism up to the standard twist [source reference]. The polarized adjoint is represented by an algebraic incidence return after inserting the appropriate power of the Plueck…
Exact mathematical passage
A=p_*q^*:H^4_{\mathrm{prim}}(X,\Q)\to H^2_{\mathrm{prim}}(F(X),\Q).
H^{2p}Use in source · 19 passages
Search aliases: H^2p
Use in source · appendices.tex, line 6
Rational Hodge space · Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X).
Exact mathematical passage
\Hdg^p(X)=H^{2p}(X,\Q)\cap H^{p,p}(X)
H^{4,0}Use in source · 2 passages
Search aliases: H^4,0
Use in source · transfer210.tex, line 452
The residue description places the c-line in H^4,0 under the direct-action convention (or its conjugate under the inverse-character convention). Its rational Galois span contains a nonzero (4,0) part and is also K-invariant.
Exact mathematical passage
H^{4,0}
H^4_{\mathrm{prim}}Use in source · 3 passages
Search aliases: H^4_prim mathrm
Use in source · august_development.tex, line 396
Cubic fourfolds as a validation of the transfer theorem. Let X⊂ P^5 be a smooth cubic fourfold and let F(X) be its Fano variety of lines. The universal incidence variety P={(ℓ,x):x∈ℓ}⊂ F(X)× X induces the Abel-Jacobi map A=p_*q^*:H^4_prim(X,Q)→ H^2_prim(F(X),Q). Beauville and Donagi proved that this is a polarized Hodge isomorphism up to the standard twist [source reference]. The polarized adjoint is represented by an algebraic incidence return after inserting the appropriate power of the Plueck…
Exact mathematical passage
A=p_*q^*:H^4_{\mathrm{prim}}(X,\Q)\to H^2_{\mathrm{prim}}(F(X),\Q).
H^6Use in source · 2 passages
Search aliases: H^6
Use in source · august_development.tex, line 414
item For suitable GM fourfolds, the vanishing H^4 packet is lowered by universal lines to divisor degree. item For suitable GM sixfolds, the vanishing H^6 packet is lowered by universal planes to divisor degree. item Ambient and tautological classes are treated separately and are algebraic.
Exact mathematical passage
H^6
H^{\mathrm{even}}Use in source · 1 passage
Search aliases: H^even mathrm
Use in source · august_development.tex, line 424
Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees is equivalent to algebraization of topological K-classes whose Chern-character components are rational Hodge classes [source reference].
Exact mathematical passage
\operatorname{ch}_{\mathrm{top}}:K^0_{\mathrm{top}}(X)\otimes\Q\xrightarrow{\sim}H^{\mathrm{even}}(X,\Q).
H^iUse in source · 1 passage
Search aliases: H^i
Use in source · appendices.tex, line 26
Correspondence degrees. Let X have dimension n and Y dimension m. A correspondence Γ∈CH^r(X× Y) acts by Γ_*:H^i(X)⟶ H^i+2(r-n)(Y) and shifts Hodge type by (r-n,r-n). To lower H^2p(X) to H^2(Y), take r=n-p+1. To return degree two to degree 2p on X, a correspondence on Y× X has codimension m+p-1.
Exact mathematical passage
\Gamma_*:H^i(X)\longrightarrow H^{i+2(r-n)}(Y)
H^{i+2(r-n)}Use in source · 1 passage
Search aliases: H^i+2(r-n)
Use in source · appendices.tex, line 26
Correspondence degrees. Let X have dimension n and Y dimension m. A correspondence Γ∈CH^r(X× Y) acts by Γ_*:H^i(X)⟶ H^i+2(r-n)(Y) and shifts Hodge type by (r-n,r-n). To lower H^2p(X) to H^2(Y), take r=n-p+1. To return degree two to degree 2p on X, a correspondence on Y× X has codimension m+p-1.
Exact mathematical passage
\Gamma_*:H^i(X)\longrightarrow H^{i+2(r-n)}(Y)
H^{p,p}Use in source · 2 passages
Search aliases: H^p,p
Use in source · appendices.tex, line 6
Rational Hodge space · Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X).
Exact mathematical passage
\Hdg^p(X)=H^{2p}(X,\Q)\cap H^{p,p}(X)
h^rUse in source · 1 passage
Search aliases: h^r
Use in source · august_development.tex, line 61
[Primitive reduction] If every primitive rational Hodge class is algebraic, then every rational Hodge class is algebraic. Write α=∑_rL^rα_r with each α_r primitive and of Hodge type. The class h is algebraic. If α_r is algebraic, then L^rα_r=h^rsmileα_r is algebraic. Summing gives an algebraic representative of α.
Exact mathematical passage
L^r\alpha_r=h^r\smile\alpha_r
h_BUse in source · 1 passage
Search aliases: h_B
Use in source · august_development.tex, line 1438
For any nonzero-entry block B at modulus m, put h_B(u)=m^-1∑_r∈ B⟨ ur⟩_m, with multiplicities retained.
Exact mathematical passage
h_B(u)=m^{-1}\sum_{r\in B}\langle ur\rangle_m
h_{C_P}Use in source · 1 passage
Search aliases: h_C_P
Use in source · august_development.tex, line 1447
[Exact K23 packet identity] One has the exact multiset equality A+(5c,m-5c)+(5d,m-5d)=C_P+S_T+L. For every unit upmod m, h_C_P(u)+h_S_T(u)+h_L(u)=5, and c+d+e≡0pmod q.
Exact mathematical passage
h_{C_P}(u)+h_{S_T}(u)+h_L(u)=5,
h_LUse in source · 1 passage
Search aliases: h_L
Use in source · august_development.tex, line 1447
[Exact K23 packet identity] One has the exact multiset equality A+(5c,m-5c)+(5d,m-5d)=C_P+S_T+L. For every unit upmod m, h_C_P(u)+h_S_T(u)+h_L(u)=5, and c+d+e≡0pmod q.
Exact mathematical passage
h_{C_P}(u)+h_{S_T}(u)+h_L(u)=5,
H_{\mathrm{prim}}^{2p}Use in source · 1 passage
Search aliases: H_prim^2p mathrm
Use in source · august_development.tex, line 273
Divisor-lowering correspondence channels. Degree bookkeeping. Let X have complex dimension n, let V⊆ H_prim^2p(X,Q) be a rational Hodge substructure, and let Y_r be smooth projective. Choose correspondences Γ_r∈CH^n-p+1(X× Y_r)_Q, Λ_r∈CH^dim Y_r+p-1(Y_r× X)_Q. They induce A_r=(Γ_r)_*:V→ H^2(Y_r,Q), B_r=(Λ_r)_*:H^2(Y_r,Q)→ H^2p(X,Q). Algebraic correspondences preserve rationality and shift Hodge type by the required bidegree.
Exact mathematical passage
V\subseteq H_{\mathrm{prim}}^{2p}(X,\Q)
H_{\mathrm{prim}}^{2p-2r}Use in source · 1 passage
Search aliases: H_prim^2p-2r mathrm
Use in source · august_development.tex, line 52
Primitive and coniveau reductions. Lefschetz packet decomposition. Let h∈ H^2(X,Q) be the class of an ample divisor and let L(α)=hsmileα. Hard Lefschetz and primitive decomposition yield H^2p(X,Q)=⊕_r≥0L^rH_prim^2p-2r(X,Q). The decomposition respects Hodge type.
Exact mathematical passage
H^{2p}(X,\Q)=\bigoplus_{r\ge0}L^rH_{\mathrm{prim}}^{2p-2r}(X,\Q).
H_{\Q}Use in source · 5 passages
Search aliases: H_Q Q
Use in source · appendices.tex, line 10
Unconditional rank test · rank_Q[C H_Q]=rank_QC tests target containment in the actual cycle span.
Exact mathematical passage
\rank_{\Q}[C\ H_{\Q}]=\rank_{\Q}C
h_{S_T}Use in source · 1 passage
Search aliases: h_S_T
Use in source · august_development.tex, line 1447
[Exact K23 packet identity] One has the exact multiset equality A+(5c,m-5c)+(5d,m-5d)=C_P+S_T+L. For every unit upmod m, h_C_P(u)+h_S_T(u)+h_L(u)=5, and c+d+e≡0pmod q.
Exact mathematical passage
h_{C_P}(u)+h_{S_T}(u)+h_L(u)=5,
H_Z^{2p}Use in source · 1 passage
Search aliases: H_Z^2p
Use in source · august_development.tex, line 244
Support verification and multiplicity recovery. Let Z⊆ X be a closed algebraic subset and j:X∖ Z↪ X. The support exact sequence contains H_Z^2p(X,Q)⟶ H^2p(X,Q)xrightarrowj^*H^2p(X∖ Z,Q).
Exact mathematical passage
H_Z^{2p}(X,\Q)\longrightarrow H^{2p}(X,\Q)\xrightarrow{j^*}H^{2p}(X\setminus Z,\Q).
h_\ellUse in source · 2 passages
Search aliases: h_ℓ ell
Use in source · august_development.tex, line 1495
For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overlineΣ_ℓ. This is the ri…
Exact mathematical passage
h_\ell(u)=\frac{\langle uc\rangle_q+\langle ud\rangle_q+\langle ue\rangle_q}{q}\in\{1,2\}.
\HdgUse in source · 12 passages
Search aliases: Hdg
Use in source · appendices.tex, line 6
Rational Hodge space · Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X).
Exact mathematical passage
\Hdg^p(X)=H^{2p}(X,\Q)\cap H^{p,p}(X)
\Hdg^pUse in source · 8 passages
Search aliases: Hdg^p Hdg
Use in source · appendices.tex, line 6
Rational Hodge space · Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X).
Exact mathematical passage
\Hdg^p(X)=H^{2p}(X,\Q)\cap H^{p,p}(X)
\operatorname{Hess}Use in source · 1 passage
Search aliases: Hess operatorname
Use in source · capacity_cycles.tex, line 532
detHess(F_m)=[m(m-1)]^6S_m.
Exact mathematical passage
\det\operatorname{Hess}(F_m)=[m(m-1)]^6S_m.
IUse in source · 13 passages
Search aliases: I
Use in source · august_development.tex, line 148
Packet assembly without false orthogonality. Partition the cycle indices into packets I_λ. Define S_λ=∑_j∈ I_λ|c_j⟩⟨ c_j|. Then S=∑_λ S_λ. No orthogonality of the child spans is assumed.
Exact mathematical passage
I_\lambda
I_{M,r}Use in source · 1 passage
Search aliases: I_M,r
Use in source · august_development.tex, line 177
Degree-residue polyphase refinement. Exact parent-child frame law. Fix a projective embedding and assign each cycle its degree. For modulus M, define the residue packet I_M,r={j:deg Z_j≡ rpmod M}, S_M,r=∑_j∈ I_M,r|c_j⟩⟨ c_j|.
Exact mathematical passage
I_{M,r}=\{j:\deg Z_j\equiv r\pmod M\}, \qquad S_{M,r}=\sum_{j\in I_{M,r}}|c_j\rangle\langle c_j|.
I_\lambdaUse in source · 2 passages
Search aliases: I_λ lambda
Use in source · august_development.tex, line 148
Packet assembly without false orthogonality. Partition the cycle indices into packets I_λ. Define S_λ=∑_j∈ I_λ|c_j⟩⟨ c_j|. Then S=∑_λ S_λ. No orthogonality of the child spans is assumed.
Exact mathematical passage
I_\lambda
\idUse in source · 1 passage
Search aliases: id
Use in source · appendices.tex, line 33
For a finite generically finite morphism of degree d, the push-pull identity on rational cohomology is f_*f^*=did. Division by d transfers algebraicity downward. Pullback transfers it upward without division. Coordinate-power maps preserve the old conductor of an inflated character, even though the ambient Fermat degree grows.
Exact mathematical passage
f_*f^*=d\id
\operatorname{id}Use in source · 1 passage
Search aliases: id operatorname
Use in source · carrier70.tex, line 437
π_*π^*=(degπ)id
Exact mathematical passage
\pi_*\pi^*=(\deg\pi)\operatorname{id}
\imUse in source · 5 passages
Search aliases: im
Use in source · appendices.tex, line 7
Algebraic space · Alg^p(X)=im(CH^p(X)_Q→ H^2p(X,Q)).
Exact mathematical passage
\Alg^p(X)=\im(\CH^p(X)_{\Q}\to H^{2p}(X,\Q))
\operatorname{im}Use in source · 2 passages
Search aliases: im operatorname
Use in source · august_development.tex, line 809
endcases Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is itself algebraic.
Exact mathematical passage
\label{eq:legacy-8.3} \boxed{ H^4_{\mathrm{prim}}(X_m^4,\Q) =\bigoplus_{c\mid m}\operatorname{im}Q_{m,c}. }
j^*Use in source · 3 passages
Search aliases: j^*
Use in source · appendices.tex, line 12
Support test · j^*α=0. Purity turns it into cycle algebraicity when the support has codimension at least p.
Exact mathematical passage
j^*\alpha=0
J_0Use in source · 4 passages
Search aliases: J_0
Use in source · weil_audit.tex, line 13
The lattice, polarization and seed. Let K=Q(i) and take homology lattice Z[i]^6. Use the conjugate-linear-first Hermitian form h(v,w)=overline v^ tdiag(1,1,1,-1,-1,-3)w, E=Im h. The seed complex structure is J_0=diag(i,i,i,-i,-i,-i). It preserves the alternating form E, and E(v,J_0v)>0 for nonzero v. These are the Riemann positivity conditions for the seed abelian variety. The sign depends on the declared Hermitian convention. Omitting that convention would leave the polarization claim ambiguous…
Exact mathematical passage
J_0=\operatorname{diag}(i,i,i,-i,-i,-i).
J_{F_m}Use in source · 3 passages
Search aliases: J_F_m
Use in source · capacity_cycles.tex, line 82
J_F_m=(x_0^m-1,…,x_5^m-1), S_m=∏_i=0^5x_i^m-2.
Exact mathematical passage
\label{eq:capacity_cycles-1.2} J_{F_m}=(x_0^{m-1},\ldots,x_5^{m-1}),\qquad S_m=\prod_{i=0}^5x_i^{m-2}.
\operatorname{Jac}Use in source · 1 passage
Search aliases: Jac operatorname
Use in source · capacity_cycles.tex, line 433
H=(f_1,g_1,f_2,g_2,f_3,g_3), D_κ=detJac_(x_0,…,x_4,z)H,
Exact mathematical passage
H=(f_1,g_1,f_2,g_2,f_3,g_3),\qquad D_\kappa=\det\operatorname{Jac}_{(x_0,\ldots,x_4,z)}H,
\operatorname{Jac}_{(x_0,\ldots,x_4,z)}Use in source · 1 passage
Search aliases: Jac_(x_0,…,x_4,z) operatorname ldots
Use in source · capacity_cycles.tex, line 433
H=(f_1,g_1,f_2,g_2,f_3,g_3), D_κ=detJac_(x_0,…,x_4,z)H,
Exact mathematical passage
H=(f_1,g_1,f_2,g_2,f_3,g_3),\qquad D_\kappa=\det\operatorname{Jac}_{(x_0,\ldots,x_4,z)}H,
KUse in source · 25 passages
Search aliases: K
Use in source · august_development.tex, line 424
Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees is equivalent to algebraization of topological K-classes whose Chern-character components are rational Hodge classes [source reference].
Exact mathematical passage
\operatorname{ch}_{\mathrm{top}}:K^0_{\mathrm{top}}(X)\otimes\Q\xrightarrow{\sim}H^{\mathrm{even}}(X,\Q).
\mathcal KUse in source · 6 passages
Search aliases: K mathcal
Use in source · august_development.tex, line 1122
The integral triangle basis. Let K_m= { d∈ Z^h-1: ∑_r=1^h-1d(r)ρ_m(ur)=0 for every odd u }. The triangle relations do not merely generate examples, they form the whole balanced defect lattice.
Exact mathematical passage
\mathcal K_m= \left\{ d\in\mathbf Z^{h-1}: \sum_{r=1}^{h-1}d(r)\rho_m(ur)=0 \text{ for every odd }u \right\}.
K^0_{\mathrm{top}}Use in source · 1 passage
Search aliases: K^0_top mathrm
Use in source · august_development.tex, line 424
Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees is equivalent to algebraization of topological K-classes whose Chern-character components are rational Hodge classes [source reference].
Exact mathematical passage
\operatorname{ch}_{\mathrm{top}}:K^0_{\mathrm{top}}(X)\otimes\Q\xrightarrow{\sim}H^{\mathrm{even}}(X,\Q).
k^{n-2j}Use in source · 1 passage
Search aliases: k^n-2j
Use in source · august_development.tex, line 497
A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge j _L^2^2+O(k^n-2j-1), c_n,j>0. This detects nonintegrability. It does not by itself remove the defect.
Exact mathematical passage
\Tr(P_{2j}\Pi_{A,k}) =c_{n,j}k^{n-2j}\|(F_A^{0,2})^{\wedge j}\|_{L^2}^2+O(k^{n-2j-1}), \qquad c_{n,j}>0.
k^{n-2j-1}Use in source · 1 passage
Search aliases: k^n-2j-1
Use in source · august_development.tex, line 497
A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge j _L^2^2+O(k^n-2j-1), c_n,j>0. This detects nonintegrability. It does not by itself remove the defect.
Exact mathematical passage
\Tr(P_{2j}\Pi_{A,k}) =c_{n,j}k^{n-2j}\|(F_A^{0,2})^{\wedge j}\|_{L^2}^2+O(k^{n-2j-1}), \qquad c_{n,j}>0.
K_{1,3}Use in source · 2 passages
Search aliases: K_1,3
Use in source · august_development.tex, line 1252
[Exact SAT(2) criterion] Let n be a balanced six-packet at modulus m=2^s. Then n is pair-decomposable or has an SAT(2) certificate unless its gauged defect is ±1 on a four-vertex star K_1,3 in the internal tree T_m. A four-star has no SAT(2) certificate. Every nonstar shape in the preceding proposition splits into at most two four-packet atoms: item a vertex or edge is paired with a zero atom item a three-vertex path is an edge plus its remaining terminal vertex item a four-vertex path is the un…
Exact mathematical passage
K_{1,3}
\mathcal K_mUse in source · 6 passages
Search aliases: K_m mathcal
Use in source · august_development.tex, line 1122
The integral triangle basis. Let K_m= { d∈ Z^h-1: ∑_r=1^h-1d(r)ρ_m(ur)=0 for every odd u }. The triangle relations do not merely generate examples, they form the whole balanced defect lattice.
Exact mathematical passage
\mathcal K_m= \left\{ d\in\mathbf Z^{h-1}: \sum_{r=1}^{h-1}d(r)\rho_m(ur)=0 \text{ for every odd }u \right\}.
K_{\widetilde Y}Use in source · 2 passages
Search aliases: K_widetilde Y widetilde
Use in source · carrier70.tex, line 277
K_widetilde Y·widetilde C=K_Y· C=-3H· C<0.
Exact mathematical passage
K_{\widetilde Y}\cdot\widetilde C=K_Y\cdot C=-3H\cdot C<0.
K_YUse in source · 2 passages
Search aliases: K_Y
Use in source · carrier70.tex, line 256
K_Y∼_ Q-3H.
Exact mathematical passage
K_Y\sim_{\mathbf Q}-3H.
\kerUse in source · 6 passages
Search aliases: ker
Use in source · august_development.tex, line 114
[Finite positive cycle-frame certificate] Let A=Span_R{c_1,…,c_N}. Then im S=A, ker S=A^⊥. Under the stated containment c_j∈ V, the selected cycles span V if and only if S|_V is positive definite. Without containment, positivity only detects the absence of vectors in V orthogonal to the cycle span and does not prove that span contains V. The equality im(CC^*)=im C is standard. Moreover ⟨ v,Sv⟩= C^*v ^2=∑_j|⟨ c_j,v⟩|^2. This vanishes exactly when v is orthogonal to every cycle class.
Exact mathematical passage
\im S=A, \qquad \ker S=A^\perp.
\mathcal LUse in source · 3 passages
Search aliases: L mathcal
Use in source · capacity_cycles.tex, line 101
L_a={v∈ Z_≥0^6∖{0}:a· v≡0pmod m}.
Exact mathematical passage
\mathcal L_a=\{v\in\mathbf Z_{\ge0}^6\setminus\{0\}:a\cdot v\equiv0\pmod m\}.
lUse in source · 1 passage
Search aliases: l
Use in source · capacity_cycles.tex, line 602
a=(1,8,15,22,29,30), Z_κ={ l ∑_i=0^4x_i^7=0, ∑_0≤ i<j≤4x_i^7x_j^7=0, x_0x_1x_2x_3x_4=κ x_5^5 }⊂ X_35.
Exact mathematical passage
a=(1,8,15,22,29,30),\quad Z_\kappa=\left\{ \begin{array}{l} \sum_{i=0}^4x_i^7=0,\\ \sum_{0\le i<j\le4}x_i^7x_j^7=0,\\ x_0x_1x_2x_3x_4=\kappa x_5^5 \end{array}\right\}\subset X_{35}.
L^2Use in source · 1 passage
Search aliases: L^2
Use in source · august_development.tex, line 497
A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge j _L^2^2+O(k^n-2j-1), c_n,j>0. This detects nonintegrability. It does not by itself remove the defect.
Exact mathematical passage
\Tr(P_{2j}\Pi_{A,k}) =c_{n,j}k^{n-2j}\|(F_A^{0,2})^{\wedge j}\|_{L^2}^2+O(k^{n-2j-1}), \qquad c_{n,j}>0.
L^rUse in source · 3 passages
Search aliases: L^r
Use in source · august_development.tex, line 52
Primitive and coniveau reductions. Lefschetz packet decomposition. Let h∈ H^2(X,Q) be the class of an ample divisor and let L(α)=hsmileα. Hard Lefschetz and primitive decomposition yield H^2p(X,Q)=⊕_r≥0L^rH_prim^2p-2r(X,Q). The decomposition respects Hodge type.
Exact mathematical passage
H^{2p}(X,\Q)=\bigoplus_{r\ge0}L^rH_{\mathrm{prim}}^{2p-2r}(X,\Q).
\mathcal L_aUse in source · 2 passages
Search aliases: L_a mathcal
Use in source · capacity_cycles.tex, line 101
L_a={v∈ Z_≥0^6∖{0}:a· v≡0pmod m}.
Exact mathematical passage
\mathcal L_a=\{v\in\mathbf Z_{\ge0}^6\setminus\{0\}:a\cdot v\equiv0\pmod m\}.
\mathcal L_{a_{q,b}}Use in source · 1 passage
Search aliases: L_a_q,b mathcal
Use in source · capacity_cycles.tex, line 269
If v∈ L_a_q,b, reduction modulo q, followed by cancellation of the unit b, gives
Exact mathematical passage
v\in\mathcal L_{a_{q,b}}
L_{m/c}Use in source · 1 passage
Search aliases: L_m/c
Use in source · august_development.tex, line 774
[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balance equation descends. The maximal common divisor is unique. The converse follows by scaling.
Exact mathematical passage
\mathcal B_m=\bigsqcup_{c\mid m}L_{m/c}\mathcal B_c^{\mathrm{full}}.
\mathrm{Lef}Use in source · 1 passage
Search aliases: Lef mathrm
Use in source · august_development.tex, line 199
Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods to solve different geometric sectors before exact synthesis.
Exact mathematical passage
\lambda=(r_{\mathrm{Lef}},r_{\deg},\chi_{\mathrm{mon}},s_{\mathrm{supp}},\nu_{\mathrm{Frob}}).
\limUse in source · 1 passage
Search aliases: lim
Use in source · august_development.tex, line 219
[Approximation becomes exact] Let P_D be the orthogonal projector onto A_D^p(X)⊗R. For a rational class α, lim_D→∞ (I-P_D)α =0 if and only if α∈ A_D^p(X) for some finite D. After stabilization, the residual is constant. A constant tail converging to zero is zero.
Exact mathematical passage
\lim_{D\to\infty}\|(I-P_D)\alpha\|=0
\lim_{D\to\infty}Use in source · 1 passage
Search aliases: lim_D→∞ lim to infty
Use in source · august_development.tex, line 219
[Approximation becomes exact] Let P_D be the orthogonal projector onto A_D^p(X)⊗R. For a rational class α, lim_D→∞ (I-P_D)α =0 if and only if α∈ A_D^p(X) for some finite D. After stabilization, the residual is constant. A constant tail converging to zero is zero.
Exact mathematical passage
\lim_{D\to\infty}\|(I-P_D)\alpha\|=0
\mathrm{line}Use in source · 2 passages
Search aliases: line mathrm
Use in source · august_development.tex, line 570
A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equivalently prove T_tot =∑_rA_r^† A_r satisfies λ_min(T_tot)>0.
Exact mathematical passage
A_{\mathrm{line}}, A_{\mathrm{conic}}, A_{\mathrm{plane}}, A_{\mathrm{exceptional}}.
\LongleftrightarrowUse in source · 2 passages
Search aliases: Longleftrightarrow
Use in source · august_development.tex, line 28
[Master obstruction projector] The operator O_X^p=P_Hdg^p-P_Alg^p is the orthogonal projector onto (Hdg^p(X)⊗_QR)∩(Alg^p(X)⊗_QR)^⊥. Therefore O_X^p=0 Longleftrightarrow Hdg^p(X)=Alg^p(X). For nested orthogonal projectors Q≤ P, the difference P-Q is self-adjoint and (P-Q)^2=P-PQ-QP+Q=P-Q. Its range is im P∩(im Q)^⊥.
Exact mathematical passage
\mathcal O_X^p=0 \quad\Longleftrightarrow\quad \Hdg^p(X)=\Alg^p(X).
MUse in source · 23 passages
Search aliases: M
Use in source · appendices.tex, line 11
Polyphase assembly · S_M,r=∑_ℓ=0^L-1S_LM,r+ℓ M. These are frame sums, not sums of span projectors.
Exact mathematical passage
S_{M,r}=\sum_{\ell=0}^{L-1}S_{LM,r+\ell M}
m^{-1}Use in source · 1 passage
Search aliases: m^-1
Use in source · august_development.tex, line 1438
For any nonzero-entry block B at modulus m, put h_B(u)=m^-1∑_r∈ B⟨ ur⟩_m, with multiplicities retained.
Exact mathematical passage
h_B(u)=m^{-1}\sum_{r\in B}\langle ur\rangle_m
M^2Use in source · 1 passage
Search aliases: M^2
Use in source · august_development.tex, line 460
[Range-packet contraction] Assume R ≤ M, B(a,b) ≤ C a b , 4CM^2 Qf <1. Then the map T(a)=-RQ(f+B(a,a)) has a unique fixed point in the gauge-fixed ball a ≤2M Qf . If Qf=0, the ball is the singleton {0} and the range solution is immediate. At the fixed point, the range component of F(a) vanishes. The remaining obstruction is Φ=Π(f+B(a,a)). The full integrability equation holds if and only if Φ=0. For a ≤ r=2M Qf , T(a) ≤ M Qf +MC r^2≤ r under the stated bound. Moreover T(a)- T(b) ≤ MC( a + b ) a-…
Exact mathematical passage
\|R\|\le M, \qquad \|B(a,b)\|\le C\|a\|\|b\|, \qquad 4CM^2\|Qf\|<1.
m^pUse in source · 1 passage
Search aliases: m^p
Use in source · august_development.tex, line 435
Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X q≠ p fracψ^m-m^qIm^p-m^q. Then ch_q(e_p^(m)ξ)=δ_pqch_p(ξ). This exactly isolates codimension p inside a topological K-packet.
Exact mathematical passage
e_p^{(m)}=\prod_{\substack{0\le q\le\dim X\\q\ne p}} \frac{\psi^m-m^qI}{m^p-m^q}.
m^qUse in source · 2 passages
Search aliases: m^q
Use in source · august_development.tex, line 431
Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X
Exact mathematical passage
\operatorname{ch}_q(\psi^m\xi)=m^q\operatorname{ch}_q(\xi).
M_{ki}Use in source · 1 passage
Search aliases: M_ki
Use in source · august_development.tex, line 262
Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.
Exact mathematical passage
M_{ki}=\int_X[Z_i]\smile\gamma_k, \qquad b_k=\int_X\alpha\smile\gamma_k.
\minUse in source · 3 passages
Search aliases: min
Use in source · august_development.tex, line 133
If H is a matrix whose columns form an orthonormal basis of V, the finite Gram certificate is G=H^*SH. Under A⊆ V, we have A=V if and only if λ_min(G)>0. Without that containment, use a rational basis matrix H_Q for the target and the exact certificate rank_Q[C H_Q]=rank_QC to test V⊆ A.
Exact mathematical passage
\lambda_{\min}(G)>0
\mathrm{mon}Use in source · 1 passage
Search aliases: mon mathrm
Use in source · august_development.tex, line 199
Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods to solve different geometric sectors before exact synthesis.
Exact mathematical passage
\lambda=(r_{\mathrm{Lef}},r_{\deg},\chi_{\mathrm{mon}},s_{\mathrm{supp}},\nu_{\mathrm{Frob}}).
NUse in source · 23 passages
Search aliases: N
Use in source · august_development.tex, line 75
Support localization and coniveau. Let N^pH^2p(X,Q) be the geometric coniveau-p subspace. In degree 2p, purity forces support contributions of codimension p to be generated by fundamental classes of codimension-p components.
Exact mathematical passage
N^pH^{2p}(X,\Q)
\mathbf NUse in source · 3 passages
Search aliases: N mathbf
Use in source · august_development.tex, line 708
To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebraic automorphisms, the count-vector descriptions correspond to U_m-orbits in B_m. The dimension of a fixed rational character packet is computed from di…
Exact mathematical passage
n(a)=(n_1,\ldots,n_{m-1})\in\mathbf N^{m-1}, \qquad n_r=\#\{i:a_i=r\}.
N^3Use in source · 1 passage
Search aliases: N^3
Use in source · august_development.tex, line 507
0 · 0 · 0) is nonzero, while N^3=0 and Tr(N^r)=0 for every positive integer r. This is the elementary negative control.
Exact mathematical passage
N^3=0
\mathbf N^{m-1}Use in source · 3 passages
Search aliases: N^m-1 mathbf
Use in source · august_development.tex, line 708
To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebraic automorphisms, the count-vector descriptions correspond to U_m-orbits in B_m. The dimension of a fixed rational character packet is computed from di…
Exact mathematical passage
n(a)=(n_1,\ldots,n_{m-1})\in\mathbf N^{m-1}, \qquad n_r=\#\{i:a_i=r\}.
N^pUse in source · 3 passages
Search aliases: N^p
Use in source · august_development.tex, line 75
Support localization and coniveau. Let N^pH^2p(X,Q) be the geometric coniveau-p subspace. In degree 2p, purity forces support contributions of codimension p to be generated by fundamental classes of codimension-p components.
Exact mathematical passage
N^pH^{2p}(X,\Q)
N^rUse in source · 1 passage
Search aliases: N^r
Use in source · august_development.tex, line 507
0 · 0 · 0) is nonzero, while N^3=0 and Tr(N^r)=0 for every positive integer r. This is the elementary negative control.
Exact mathematical passage
\operatorname{Tr}(N^r)=0
n_1Use in source · 1 passage
Search aliases: n_1
Use in source · august_development.tex, line 708
To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebraic automorphisms, the count-vector descriptions correspond to U_m-orbits in B_m. The dimension of a fixed rational character packet is computed from di…
Exact mathematical passage
n(a)=(n_1,\ldots,n_{m-1})\in\mathbf N^{m-1}, \qquad n_r=\#\{i:a_i=r\}.
n_jUse in source · 1 passage
Search aliases: n_j
Use in source · august_development.tex, line 976
Valuation strata and orbit mass. Let m=p^r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonzero odd kernel-orbit defect has mass at least 2p.
Exact mathematical passage
\delta_j(b)=n_j(b)-n_j(-b).
n_{m-1}Use in source · 1 passage
Search aliases: n_m-1
Use in source · august_development.tex, line 708
To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebraic automorphisms, the count-vector descriptions correspond to U_m-orbits in B_m. The dimension of a fixed rational character packet is computed from di…
Exact mathematical passage
n(a)=(n_1,\ldots,n_{m-1})\in\mathbf N^{m-1}, \qquad n_r=\#\{i:a_i=r\}.
n_{m-r}Use in source · 1 passage
Search aliases: n_m-r
Use in source · august_development.tex, line 1065
Complement defects and dyadic triangles. Put q=frac m4, h=2q, m=4q. For a packet count vector n, define its oriented complement defect ∂ n=∑_r=1^h-1d_n(r)[r], d_n(r)=n_r-n_m-r. The self-complementary residue h contributes zero. Since ρ_m(m-r)=-ρ_m(r), the balanced-packet equation is equivalent to ∑_r=1^h-1d_n(r)ρ_m(ur)=0 for every odd unit u. Moreover, ∂ n _1=∑_r=1^h-1|d_n(r)|≤ 6. Thus the dyadic problem is a short-vector problem in an integral defect lattice.
Exact mathematical passage
\partial n=\sum_{r=1}^{h-1}d_n(r)[r], \qquad d_n(r)=n_r-n_{m-r}.
n_rUse in source · 3 passages
Search aliases: n_r
Use in source · august_development.tex, line 708
To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebraic automorphisms, the count-vector descriptions correspond to U_m-orbits in B_m. The dimension of a fixed rational character packet is computed from di…
Exact mathematical passage
n(a)=(n_1,\ldots,n_{m-1})\in\mathbf N^{m-1}, \qquad n_r=\#\{i:a_i=r\}.
\nmidUse in source · 2 passages
Search aliases: nmid
Use in source · august_development.tex, line 791
For dmid m, let E_m,d=frac1(m/d)^5π_m/d^*π_m/d*. On a character of exact conductor c, E_m,d=1, · cmid d, 0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is itself algebraic.
Exact mathematical passage
E_{m,d}=\begin{cases}1,&c\mid d,\\0,&c\nmid d.\end{cases}
\operatorname{Norm}Use in source · 1 passage
Search aliases: Norm operatorname
Use in source · weil_audit.tex, line 80
This also corrects a source-level eigencharacter confusion. The polarization cube has eigenvalue [2+i]^*(θ^3)=125 θ^3=Norm(2+i)^3θ^3, while the Weil directions have eigenvalues -117±44i. The norm-cubed character therefore identifies the polarization cube, and does not identify the two-dimensional Weil subspace. The September audit recorded this local failure in Section 4.3 of the cited secant-cycle preprint. That specific calculation is not a full audit of the preprint [source reference].
Exact mathematical passage
[2+i]^*(\theta^3)=125\,\theta^3=\operatorname{Norm}(2+i)^3\theta^3,
OUse in source · 3 passages
Search aliases: O
Use in source · august_development.tex, line 497
A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge j _L^2^2+O(k^n-2j-1), c_n,j>0. This detects nonintegrability. It does not by itself remove the defect.
Exact mathematical passage
\Tr(P_{2j}\Pi_{A,k}) =c_{n,j}k^{n-2j}\|(F_A^{0,2})^{\wedge j}\|_{L^2}^2+O(k^{n-2j-1}), \qquad c_{n,j}>0.
\mathcal OUse in source · 5 passages
Search aliases: O mathcal
Use in source · appendices.tex, line 8
Real obstruction · O=P_Hdg⊗R-P_Alg⊗R. No algebraic realization of these orthogonal projectors is presumed.
Exact mathematical passage
\mathcal O=P_{\Hdg\otimes\R}-P_{\Alg\otimes\R}
O_bUse in source · 2 passages
Search aliases: O_b
Use in source · august_development.tex, line 995
The three-primary branch. For p=3 and r≥2, a higher-level kernel orbit has three elements. Define O_b={b,b+q,b+2q}, q=3^r-1, and Γ_b=O_bsqcup{3(q-b)}. Take 1≤ b<q with 3nmid b, ensuring all entries are nonzero. Then Γ_b is a balanced four-packet. A case analysis on the number of oriented kernel orbits gives exact identities expressing every full-conductor nonpair packet as a pair/surface packet or a one-step saturated sum of two balanced four-packets.
Exact mathematical passage
O_b=\{b,b+q,b+2q\}, \qquad q=3^{r-1},
\mathcal O_{S_q}Use in source · 1 passage
Search aliases: O_S_q mathcal
Use in source · capacity_cycles.tex, line 387
Forgetting z defines a finite flat cover Z_κ→ S_q of degree five. On a chart trivializing O_S_q(1), its equation is z^5=P/κ, with the corresponding trivializations understood. At a generic coordinate curve its right side has valuation one. The degree-five polynomial is Eisenstein over that discrete valuation ring and is therefore irreducible over the function field of S_q. Finite flatness makes the cover generically integral without vertical or embedded components, locally its monic algebra inje…
Exact mathematical passage
\mathcal O_{S_q}(1)
\mathcal O_X^pUse in source · 2 passages
Search aliases: O_X^p mathcal
Use in source · august_development.tex, line 24
[Master obstruction projector] The operator O_X^p=P_Hdg^p-P_Alg^p is the orthogonal projector onto (Hdg^p(X)⊗_QR)∩(Alg^p(X)⊗_QR)^⊥. Therefore O_X^p=0 Longleftrightarrow Hdg^p(X)=Alg^p(X). For nested orthogonal projectors Q≤ P, the difference P-Q is self-adjoint and (P-Q)^2=P-PQ-QP+Q=P-Q. Its range is im P∩(im Q)^⊥.
Exact mathematical passage
\mathcal O_X^p=P_{\Hdg}^p-P_{\Alg}^p
\mathcal O_YUse in source · 1 passage
Search aliases: O_Y mathcal
Use in source · carrier70.tex, line 261
Here H is the ample rational hyperplane class, not an assertion that O_Y(1) is everywhere a line bundle. This argument establishes the adjunction formula in the present non-quasismooth example directly.
Exact mathematical passage
\mathcal O_Y(1)
\overline vUse in source · 1 passage
Search aliases: overline v overline
Use in source · weil_audit.tex, line 8
The lattice, polarization and seed. Let K=Q(i) and take homology lattice Z[i]^6. Use the conjugate-linear-first Hermitian form h(v,w)=overline v^ tdiag(1,1,1,-1,-1,-3)w, E=Im h. The seed complex structure is J_0=diag(i,i,i,-i,-i,-i). It preserves the alternating form E, and E(v,J_0v)>0 for nonzero v. These are the Riemann positivity conditions for the seed abelian variety. The sign depends on the declared Hermitian convention. Omitting that convention would leave the polarization claim ambiguous…
Exact mathematical passage
h(v,w)=\overline v^{\,t}\operatorname{diag}(1,1,1,-1,-1,-3)w, \qquad E=\Im h.
\overline v^{\,t}Use in source · 1 passage
Search aliases: overline v^ t overline
Use in source · weil_audit.tex, line 8
The lattice, polarization and seed. Let K=Q(i) and take homology lattice Z[i]^6. Use the conjugate-linear-first Hermitian form h(v,w)=overline v^ tdiag(1,1,1,-1,-1,-3)w, E=Im h. The seed complex structure is J_0=diag(i,i,i,-i,-i,-i). It preserves the alternating form E, and E(v,J_0v)>0 for nonzero v. These are the Riemann positivity conditions for the seed abelian variety. The sign depends on the declared Hermitian convention. Omitting that convention would leave the polarization claim ambiguous…
Exact mathematical passage
h(v,w)=\overline v^{\,t}\operatorname{diag}(1,1,1,-1,-1,-3)w, \qquad E=\Im h.
\overline zUse in source · 1 passage
Search aliases: overline z overline
Use in source · weil_audit.tex, line 87
The seed graph supplies both rational Weil classes. At the seed, take the graph of (z_1,z_2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety graph and is algebraic at J_0. On its three source factors the polarization restricts to weights (2,2,4), giving ∫_Zθ^3=3!·2·2·4=96. Pulling the explic…
Exact mathematical passage
(z_1,z_2,z_3)\longmapsto (z_1,z_2,z_3,\overline z_1,\overline z_2,\overline z_3).
\overline z_1Use in source · 1 passage
Search aliases: overline z_1 overline
Use in source · weil_audit.tex, line 87
The seed graph supplies both rational Weil classes. At the seed, take the graph of (z_1,z_2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety graph and is algebraic at J_0. On its three source factors the polarization restricts to weights (2,2,4), giving ∫_Zθ^3=3!·2·2·4=96. Pulling the explic…
Exact mathematical passage
(z_1,z_2,z_3)\longmapsto (z_1,z_2,z_3,\overline z_1,\overline z_2,\overline z_3).
\overline z_2Use in source · 1 passage
Search aliases: overline z_2 overline
Use in source · weil_audit.tex, line 87
The seed graph supplies both rational Weil classes. At the seed, take the graph of (z_1,z_2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety graph and is algebraic at J_0. On its three source factors the polarization restricts to weights (2,2,4), giving ∫_Zθ^3=3!·2·2·4=96. Pulling the explic…
Exact mathematical passage
(z_1,z_2,z_3)\longmapsto (z_1,z_2,z_3,\overline z_1,\overline z_2,\overline z_3).
\overline z_3Use in source · 1 passage
Search aliases: overline z_3 overline
Use in source · weil_audit.tex, line 87
The seed graph supplies both rational Weil classes. At the seed, take the graph of (z_1,z_2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety graph and is algebraic at J_0. On its three source factors the polarization restricts to weights (2,2,4), giving ∫_Zθ^3=3!·2·2·4=96. Pulling the explic…
Exact mathematical passage
(z_1,z_2,z_3)\longmapsto (z_1,z_2,z_3,\overline z_1,\overline z_2,\overline z_3).
\overline{\Delta_\ell}Use in source · 1 passage
Search aliases: overlineΔ_ℓ overline Delta ell
Use in source · august_development.tex, line 1479
[Pure imaginary selector] overlineΔ_ℓ=-Δ_ℓ. Since 1-ζ_q^-a=-ζ_q^-a(1-ζ_q^a), complex conjugation multiplies the product by (-1)^3ζ_q^-(c+d+e)=-1.
Exact mathematical passage
\overline{\Delta_\ell}=-\Delta_\ell.
\overline{\Sigma_\ell}Use in source · 1 passage
Search aliases: overlineΣ_ℓ overline Sigma ell
Use in source · august_development.tex, line 1509
1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overlineΣ_ℓ. This is the rigorous form of the statement that the half-twist occurs on the imaginary plane.
Exact mathematical passage
\label{eq:legacy-16.3} \boxed{ \Sigma_N=\overline{\Sigma_\ell}. }
\overline{\chi(a)}Use in source · 1 passage
Search aliases: overlineχ(a) overline chi
Use in source · august_development.tex, line 1170
ecover the unique integers z_b from its even coordinates, and subtract ∑_bz_bκ_b. The residual balanced vector is supported only on the odd residues 1≤ r<h. The corresponding square sawtooth matrix is nonsingular. Indeed, extend an odd-support vector to an odd function on U_m. Convolution with ρ_m is diagonalized by the odd Dirichlet characters, with eigenvalues λ_χ= ∑_a∈ U_mρ_m(a)overlineχ(a) = ∑_a∈ U_maoverlineχ(a). If χ is induced from a primitive odd character χ^* of conductor 2^t, then λ_χ,…
Exact mathematical passage
\lambda_\chi= \sum_{a\in U_m}\rho_m(a)\overline{\chi(a)} = \sum_{a\in U_m}a\overline{\chi(a)}.
\overline{\chi(u)}Use in source · 1 passage
Search aliases: overlineχ(u) overline chi
Use in source · august_development.tex, line 742
Centered Fourier form. Define ρ_m(r)=⟨ r⟩_m-frac m2, F_a(u)=∑_iρ_m(ua_i). Then a is balanced if and only if F_a(u)=0 for all units. Moreover, F_a(-u)=-F_a(u), so only odd characters occur. For a Dirichlet character χ on U_m, put B_m,χ(r)= ∑_u∈ U_mρ_m(ur)overlineχ(u). Finite Fourier inversion gives the exact criterion n(a)∈ B_m Longleftrightarrow ∑_i B_m,χ(a_i)=0 for every odd χ. Here n(a) is the residue count vector. This formulation underlies the prime-power filtration and the proposed conducto…
Exact mathematical passage
\mathcal B_{m,\chi}(r)= \sum_{u\in U_m}\rho_m(ur)\overline{\chi(u)}.
\overline{\chi^*}Use in source · 2 passages
Search aliases: overlineχ^* overline chi
Use in source · august_development.tex, line 1180
_m is diagonalized by the odd Dirichlet characters, with eigenvalues λ_χ= ∑_a∈ U_mρ_m(a)overlineχ(a) = ∑_a∈ U_maoverlineχ(a). If χ is induced from a primitive odd character χ^* of conductor 2^t, then λ_χ,2^s=2^s-tλ_χ^*,2^t. The primitive value is, up to a nonzero scalar, the generalized Bernoulli value B_1,overlineχ^*, equivalently -L(0,overlineχ^*), and is nonzero for every odd primitive character [source reference]. Thus every odd Fourier eigenvalue is nonzero. The odd residual must vanish, so…
Exact mathematical passage
B_{1,\overline{\chi^*}}
\mathbb PUse in source · 2 passages
Search aliases: P mathbb
Use in source · august_development.tex, line 368
Projective bundles, blow-ups, and descent. [Projective-bundle closure] If the rational Hodge conjecture holds for X, it holds for P(E)→ X. The projective-bundle formula expresses cohomology as a direct sum of powers of the algebraic hyperplane class times pullbacks of cohomology from X.
Exact mathematical passage
\mathbb P(E)\to X
\mathbf PUse in source · 10 passages
Search aliases: P mathbf
Use in source · august_development.tex, line 655
Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finite Galois orbits of those lines. This permits exact symbolic analysis of individual addresses, conductors, local prime phases, and candidate algebraic …
Exact mathematical passage
X_m^4=\{x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0\}\subset\mathbf P^5, \qquad m\ge 3.
pUse in source · 105 passages
Search aliases: p
Use in source · appendices.tex, line 6
Rational Hodge space · Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X).
Exact mathematical passage
\Hdg^p(X)=H^{2p}(X,\Q)\cap H^{p,p}(X)
\mathbf P^1Use in source · 2 passages
Search aliases: P^1 mathbf
Use in source · carrier70.tex, line 295
For context, this theorem does not use the unknown general fourfold case: a uniruled fourfold is dominated generically finitely by a resolution of T× P^1, with dim T=3. The relevant Hodge classes on the product are algebraic by the threefold case and Künneth, blowup centers have dimension at most two, push-pull transfers algebraicity.
Exact mathematical passage
T\times\mathbf P^1
\mathbf P^4Use in source · 1 passage
Search aliases: P^4 mathbf
Use in source · capacity_cycles.tex, line 356
S_q={e_1(x_0^q,…,x_4^q)=e_2(x_0^q,…,x_4^q)=0}⊂ P^4.
Exact mathematical passage
S_q=\{e_1(x_0^q,\ldots,x_4^q)=e_2(x_0^q,\ldots,x_4^q)=0\}\subset\mathbf P^4.
\mathbb P^5Use in source · 1 passage
Search aliases: P^5 mathbb
Use in source · august_development.tex, line 391
Cubic fourfolds as a validation of the transfer theorem. Let X⊂ P^5 be a smooth cubic fourfold and let F(X) be its Fano variety of lines. The universal incidence variety P={(ℓ,x):x∈ℓ}⊂ F(X)× X induces the Abel-Jacobi map A=p_*q^*:H^4_prim(X,Q)→ H^2_prim(F(X),Q). Beauville and Donagi proved that this is a polarized Hodge isomorphism up to the standard twist [source reference]. The polarized adjoint is represented by an algebraic incidence return after inserting the appropriate power of the Plueck…
Exact mathematical passage
X\subset\mathbb P^5
\mathbf P^5Use in source · 2 passages
Search aliases: P^5 mathbf
Use in source · august_development.tex, line 655
Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finite Galois orbits of those lines. This permits exact symbolic analysis of individual addresses, conductors, local prime phases, and candidate algebraic …
Exact mathematical passage
X_m^4=\{x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0\}\subset\mathbf P^5, \qquad m\ge 3.
\mathbf P^5_\mathbfUse in source · 1 passage
Search aliases: P^5_ mathbf mathbf
Use in source · carrier70.tex, line 21
X_m={x_0^m+⋯+x_5^m=0}⊂ P^5_ C.
Exact mathematical passage
X_m=\{x_0^m+\cdots+x_5^m=0\}\subset\mathbf P^5_\mathbf C.
\mathbf P^5_{\mathbf C}Use in source · 1 passage
Search aliases: P^5_ C mathbf mathbf
Use in source · capacity_cycles.tex, line 54
X_m={F_m=x_0^m+⋯+x_5^m=0}⊂ P^5_ C, G_m=(μ_m)^6/μ_m.
Exact mathematical passage
X_m=\{F_m=x_0^m+\cdots+x_5^m=0\}\subset\mathbf P^5_{\mathbf C}, \qquad G_m=(\mu_m)^6/\mu_m.
P^{b-1}Use in source · 1 passage
Search aliases: P^b-1
Use in source · capacity_cycles.tex, line 448
D_κ=625q^5Δ(x_0^q,…,x_4^q) ∑_b=1^q-1κ^q-bP^b-1z^5(q-b)-1.
Exact mathematical passage
\label{eq:capacity_cycles-5.2} \boxed{ D_\kappa=625q^5\Delta(x_0^q,\ldots,x_4^q) \sum_{b=1}^{q-1}\kappa^{q-b}P^{b-1}z^{5(q-b)-1}. }
p^jUse in source · 1 passage
Search aliases: p^j
Use in source · august_development.tex, line 975
Valuation strata and orbit mass. Let m=p^r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonzero odd kernel-orbit defect has mass at least 2p.
Exact mathematical passage
a=p^jb
\mathbf P^{n+1}_{\mathbf C}Use in source · 1 passage
Search aliases: P^n+1_ C mathbf mathbf
Use in source · transfer210.tex, line 20
X_m^n={x_0^m+⋯+x_n+1^m=0}⊂ P^n+1_ C.
Exact mathematical passage
X_m^n=\{x_0^m+\cdots+x_{n+1}^m=0\}\subset\mathbf P^{n+1}_{\mathbf C}.
P^qUse in source · 3 passages
Search aliases: P^q
Use in source · capacity_cycles.tex, line 340
Since e_5=P^q, on Z_κ one has
Exact mathematical passage
e_5=P^q
P^{q-1}Use in source · 1 passage
Search aliases: P^q-1
Use in source · capacity_cycles.tex, line 482
Use e_5=P^q and dt_0wedge⋯wedge dt_4=q^5P^q-1 dx_0wedge⋯wedge dx_4. Cancellation in the polynomial ring gives
Exact mathematical passage
dt_0\wedge\cdots\wedge dt_4=q^5P^{q-1}\,dx_0\wedge\cdots\wedge dx_4
P^{q-1-j}Use in source · 2 passages
Search aliases: P^q-1-j
Use in source · capacity_cycles.tex, line 418
f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.
Exact mathematical passage
\begin{aligned} f_1&=e_1,&g_1&=e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4,\\ f_2&=e_2,&g_2&=-5e_3,\\ f_3&=P-\kappa z^5,&g_3&=5\sum_{j=0}^{q-1}P^{q-1-j}\kappa^jz^{5j}. \end{aligned}
p^rUse in source · 1 passage
Search aliases: p^r
Use in source · august_development.tex, line 975
Valuation strata and orbit mass. Let m=p^r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonzero odd kernel-orbit defect has mass at least 2p.
Exact mathematical passage
m=p^r
p^{r-j}Use in source · 1 passage
Search aliases: p^r-j
Use in source · august_development.tex, line 975
Valuation strata and orbit mass. Let m=p^r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonzero odd kernel-orbit defect has mass at least 2p.
Exact mathematical passage
b\in U_{p^{r-j}}
p^sUse in source · 1 passage
Search aliases: p^s
Use in source · august_development.tex, line 983
r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonzero odd kernel-orbit defect has mass at least 2p.
Exact mathematical passage
U_{p^s}\to U_{p^{s-1}}.
p^{s-1}Use in source · 1 passage
Search aliases: p^s-1
Use in source · august_development.tex, line 983
r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonzero odd kernel-orbit defect has mass at least 2p.
Exact mathematical passage
U_{p^s}\to U_{p^{s-1}}.
p_*Use in source · 1 passage
Search aliases: p_*
Use in source · august_development.tex, line 396
Cubic fourfolds as a validation of the transfer theorem. Let X⊂ P^5 be a smooth cubic fourfold and let F(X) be its Fano variety of lines. The universal incidence variety P={(ℓ,x):x∈ℓ}⊂ F(X)× X induces the Abel-Jacobi map A=p_*q^*:H^4_prim(X,Q)→ H^2_prim(F(X),Q). Beauville and Donagi proved that this is a polarized Hodge isomorphism up to the standard twist [source reference]. The polarized adjoint is represented by an algebraic incidence return after inserting the appropriate power of the Plueck…
Exact mathematical passage
A=p_*q^*:H^4_{\mathrm{prim}}(X,\Q)\to H^2_{\mathrm{prim}}(F(X),\Q).
P_1Use in source · 1 passage
Search aliases: P_1
Use in source · august_development.tex, line 1294
its four triangle atoms gives the unordered packet P_a= (8a,q-a,2q-a,2q-4a,3q-a,4q-a). The cases c<q/2 and c>q/2 give the same unordered tuple after the internal-edge cancellations. Choose an odd u satisfying ua≡1pmod2q. Since uq≡ q or 3qpmod4q, multiplication by u sends [source reference] to P_1=(8,q-1,2q-4,2q-1,3q-1,4q-1). Its conjugate is A_m= ( 1,frac m4+1,frac m2+1,frac m2+4, frac3m4+1,m-8 ). Thus every full-conductor star belongs to one U_m-orbit.
Exact mathematical passage
P_1=(8,q-1,2q-4,2q-1,3q-1,4q-1).
p_1Use in source · 1 passage
Search aliases: p_1
Use in source · august_development.tex, line 899
The exceptional packet. A representative is a=(7,10,13,19,22,28), or after permutation, widetilde a=(7,19,28,10,13,22). Every unit conjugate has residue sum 99=3·33, so the orbit is rational of type (2,2). It is not totally pair-decomposable. The previously proposed symmetric surface W={p_1(x^11)=p_2(x^11)=p_3(x^11)=0} does not carry the primitive packet: Newton's identity makes the third equation redundant on X_33^4, and [W]=242h^2. Thus W is purely ambient.
Exact mathematical passage
W=\{p_1(x^{11})=p_2(x^{11})=p_3(x^{11})=0\}
P_{102}Use in source · 1 passage
Search aliases: P_102
Use in source · transfer210.tex, line 240
buplus P_24uplus P_27uplus P_105uplus Q_3uplus S_5uplus S_7 = · suplus P_12uplus P_30uplus P_42uplus P_102 · uplus Q_9uplus Q_24uplus Q_27uplus R_2.
Exact mathematical passage
\label{eq:transfer210-5.1} \boxed{ \begin{aligned} b\uplus P_{24}\uplus P_{27}\uplus P_{105}\uplus Q_3\uplus S_5\uplus S_7 ={}&s\uplus P_{12}\uplus P_{30}\uplus P_{42}\uplus P_{102}\\ &\uplus Q_9\uplus Q_{24}\uplus Q_{27}\uplus R_2. \end{aligned}}
P_{105}Use in source · 1 passage
Search aliases: P_105
Use in source · transfer210.tex, line 240
buplus P_24uplus P_27uplus P_105uplus Q_3uplus S_5uplus S_7 = · suplus P_12uplus P_30uplus P_42uplus P_102 · uplus Q_9uplus Q_24uplus Q_27uplus R_2.
Exact mathematical passage
\label{eq:transfer210-5.1} \boxed{ \begin{aligned} b\uplus P_{24}\uplus P_{27}\uplus P_{105}\uplus Q_3\uplus S_5\uplus S_7 ={}&s\uplus P_{12}\uplus P_{30}\uplus P_{42}\uplus P_{102}\\ &\uplus Q_9\uplus Q_{24}\uplus Q_{27}\uplus R_2. \end{aligned}}
P_{12}Use in source · 1 passage
Search aliases: P_12
Use in source · transfer210.tex, line 240
buplus P_24uplus P_27uplus P_105uplus Q_3uplus S_5uplus S_7 = · suplus P_12uplus P_30uplus P_42uplus P_102 · uplus Q_9uplus Q_24uplus Q_27uplus R_2.
Exact mathematical passage
\label{eq:transfer210-5.1} \boxed{ \begin{aligned} b\uplus P_{24}\uplus P_{27}\uplus P_{105}\uplus Q_3\uplus S_5\uplus S_7 ={}&s\uplus P_{12}\uplus P_{30}\uplus P_{42}\uplus P_{102}\\ &\uplus Q_9\uplus Q_{24}\uplus Q_{27}\uplus R_2. \end{aligned}}
p_2Use in source · 1 passage
Search aliases: p_2
Use in source · august_development.tex, line 899
The exceptional packet. A representative is a=(7,10,13,19,22,28), or after permutation, widetilde a=(7,19,28,10,13,22). Every unit conjugate has residue sum 99=3·33, so the orbit is rational of type (2,2). It is not totally pair-decomposable. The previously proposed symmetric surface W={p_1(x^11)=p_2(x^11)=p_3(x^11)=0} does not carry the primitive packet: Newton's identity makes the third equation redundant on X_33^4, and [W]=242h^2. Thus W is purely ambient.
Exact mathematical passage
W=\{p_1(x^{11})=p_2(x^{11})=p_3(x^{11})=0\}
P_{24}Use in source · 1 passage
Search aliases: P_24
Use in source · transfer210.tex, line 240
buplus P_24uplus P_27uplus P_105uplus Q_3uplus S_5uplus S_7 = · suplus P_12uplus P_30uplus P_42uplus P_102 · uplus Q_9uplus Q_24uplus Q_27uplus R_2.
Exact mathematical passage
\label{eq:transfer210-5.1} \boxed{ \begin{aligned} b\uplus P_{24}\uplus P_{27}\uplus P_{105}\uplus Q_3\uplus S_5\uplus S_7 ={}&s\uplus P_{12}\uplus P_{30}\uplus P_{42}\uplus P_{102}\\ &\uplus Q_9\uplus Q_{24}\uplus Q_{27}\uplus R_2. \end{aligned}}
P_{27}Use in source · 1 passage
Search aliases: P_27
Use in source · transfer210.tex, line 240
buplus P_24uplus P_27uplus P_105uplus Q_3uplus S_5uplus S_7 = · suplus P_12uplus P_30uplus P_42uplus P_102 · uplus Q_9uplus Q_24uplus Q_27uplus R_2.
Exact mathematical passage
\label{eq:transfer210-5.1} \boxed{ \begin{aligned} b\uplus P_{24}\uplus P_{27}\uplus P_{105}\uplus Q_3\uplus S_5\uplus S_7 ={}&s\uplus P_{12}\uplus P_{30}\uplus P_{42}\uplus P_{102}\\ &\uplus Q_9\uplus Q_{24}\uplus Q_{27}\uplus R_2. \end{aligned}}
P_{2j}Use in source · 1 passage
Search aliases: P_2j
Use in source · august_development.tex, line 497
A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge j _L^2^2+O(k^n-2j-1), c_n,j>0. This detects nonintegrability. It does not by itself remove the defect.
Exact mathematical passage
\Tr(P_{2j}\Pi_{A,k}) =c_{n,j}k^{n-2j}\|(F_A^{0,2})^{\wedge j}\|_{L^2}^2+O(k^{n-2j-1}), \qquad c_{n,j}>0.
p_3Use in source · 1 passage
Search aliases: p_3
Use in source · august_development.tex, line 899
The exceptional packet. A representative is a=(7,10,13,19,22,28), or after permutation, widetilde a=(7,19,28,10,13,22). Every unit conjugate has residue sum 99=3·33, so the orbit is rational of type (2,2). It is not totally pair-decomposable. The previously proposed symmetric surface W={p_1(x^11)=p_2(x^11)=p_3(x^11)=0} does not carry the primitive packet: Newton's identity makes the third equation redundant on X_33^4, and [W]=242h^2. Thus W is purely ambient.
Exact mathematical passage
W=\{p_1(x^{11})=p_2(x^{11})=p_3(x^{11})=0\}
P_{30}Use in source · 1 passage
Search aliases: P_30
Use in source · transfer210.tex, line 240
buplus P_24uplus P_27uplus P_105uplus Q_3uplus S_5uplus S_7 = · suplus P_12uplus P_30uplus P_42uplus P_102 · uplus Q_9uplus Q_24uplus Q_27uplus R_2.
Exact mathematical passage
\label{eq:transfer210-5.1} \boxed{ \begin{aligned} b\uplus P_{24}\uplus P_{27}\uplus P_{105}\uplus Q_3\uplus S_5\uplus S_7 ={}&s\uplus P_{12}\uplus P_{30}\uplus P_{42}\uplus P_{102}\\ &\uplus Q_9\uplus Q_{24}\uplus Q_{27}\uplus R_2. \end{aligned}}
P_{42}Use in source · 1 passage
Search aliases: P_42
Use in source · transfer210.tex, line 240
buplus P_24uplus P_27uplus P_105uplus Q_3uplus S_5uplus S_7 = · suplus P_12uplus P_30uplus P_42uplus P_102 · uplus Q_9uplus Q_24uplus Q_27uplus R_2.
Exact mathematical passage
\label{eq:transfer210-5.1} \boxed{ \begin{aligned} b\uplus P_{24}\uplus P_{27}\uplus P_{105}\uplus Q_3\uplus S_5\uplus S_7 ={}&s\uplus P_{12}\uplus P_{30}\uplus P_{42}\uplus P_{102}\\ &\uplus Q_9\uplus Q_{24}\uplus Q_{27}\uplus R_2. \end{aligned}}
p_5Use in source · 2 passages
Search aliases: p_5
Use in source · capacity_cycles.tex, line 337
p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.
Exact mathematical passage
\label{eq:capacity_cycles-4.2} p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.
P_AUse in source · 1 passage
Search aliases: P_A
Use in source · august_development.tex, line 140
The orthogonal synthesis of a class α from the cycle span is P_Aα=S S^†α. If α∈ A, this is exact.
Exact mathematical passage
P_A\alpha=S S^\dagger\alpha.
P_aUse in source · 1 passage
Search aliases: P_a
Use in source · august_development.tex, line 1286
Assume now s≥5 and let a full-conductor star have center c=2a with a odd. Up to global conjugation, expanding its four triangle atoms gives the unordered packet P_a= (8a,q-a,2q-a,2q-4a,3q-a,4q-a). The cases c<q/2 and c>q/2 give the same unordered tuple after the internal-edge cancellations. Choose an odd u satisfying ua≡1pmod2q. Since uq≡ q or 3qpmod4q, multiplication by u sends [source reference] to P_1=(8,q-1,2q-4,2q-1,3q-1,4q-1). Its conjugate is A_m= ( 1,frac m4+1,frac m2+1,frac m2+4, frac3m…
Exact mathematical passage
\label{eq:legacy-13.5} P_a= (8a,q-a,2q-a,2q-4a,3q-a,4q-a).
P_{\Alg}^pUse in source · 3 passages
Search aliases: P_Alg^p Alg
Use in source · august_development.tex, line 17
Choose a positive inner product on H^2p(X,R). The orthogonal projectors P_Hdg^p and P_Alg^p act on the real scalar extensions Hdg^p(X)⊗_QR and Alg^p(X)⊗_QR. These real orthogonal projectors provide a reformulation. They are not asserted to be rational matrices or algebraic correspondences. Since the algebraic space is nested in the Hodge space, P_Hdg^pP_Alg^p=P_Alg^p.
Exact mathematical passage
P_{\Alg}^p
P_{\Alg\otimes\R}Use in source · 1 passage
Search aliases: P_Alg⊗R Alg otimes R
Use in source · appendices.tex, line 8
Real obstruction · O=P_Hdg⊗R-P_Alg⊗R. No algebraic realization of these orthogonal projectors is presumed.
Exact mathematical passage
\mathcal O=P_{\Hdg\otimes\R}-P_{\Alg\otimes\R}
P_DUse in source · 2 passages
Search aliases: P_D
Use in source · august_development.tex, line 218
[Approximation becomes exact] Let P_D be the orthogonal projector onto A_D^p(X)⊗R. For a rational class α, lim_D→∞ (I-P_D)α =0 if and only if α∈ A_D^p(X) for some finite D. After stabilization, the residual is constant. A constant tail converging to zero is zero.
Exact mathematical passage
P_D
p_hUse in source · 1 passage
Search aliases: p_h
Use in source · august_development.tex, line 1263
. A zero four-packet absorbs that pair and the stabilizing pair. Cutting a three- or four-vertex path produces exactly one oppositely oriented boundary edge, hence exactly one complementary pair in the union of the two atom packets, this is the SAT(2) stabilizer. In the three-vertex case, the vertex atom also supplies the unique neutral h. For two vertex atoms, their two neutral entries are (h,h)=p_h. The zero-defect case is already a union of three complementary pairs. Thus in every nonstar cas…
Exact mathematical passage
(h,h)=p_h
P_{\Hdg}^pUse in source · 3 passages
Search aliases: P_Hdg^p Hdg
Use in source · august_development.tex, line 17
Choose a positive inner product on H^2p(X,R). The orthogonal projectors P_Hdg^p and P_Alg^p act on the real scalar extensions Hdg^p(X)⊗_QR and Alg^p(X)⊗_QR. These real orthogonal projectors provide a reformulation. They are not asserted to be rational matrices or algebraic correspondences. Since the algebraic space is nested in the Hodge space, P_Hdg^pP_Alg^p=P_Alg^p.
Exact mathematical passage
P_{\Hdg}^p
P_{\Hdg\otimes\R}Use in source · 1 passage
Search aliases: P_Hdg⊗R Hdg otimes R
Use in source · appendices.tex, line 8
Real obstruction · O=P_Hdg⊗R-P_Alg⊗R. No algebraic realization of these orthogonal projectors is presumed.
Exact mathematical passage
\mathcal O=P_{\Hdg\otimes\R}-P_{\Alg\otimes\R}
P_kUse in source · 1 passage
Search aliases: P_k
Use in source · transfer210.tex, line 196
P_k=(k,210-k)
Exact mathematical passage
P_k=(k,210-k)
p_rUse in source · 3 passages
Search aliases: p_r
Use in source · august_development.tex, line 825
Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.
Exact mathematical passage
p_r=e_r+e_{m-r}.
p_{r_1}Use in source · 1 passage
Search aliases: p_r_1
Use in source · august_development.tex, line 829
Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.
Exact mathematical passage
\label{eq:legacy-9.1} n=p_{r_1}+p_{r_2}+p_{r_3}.
p_{r_2}Use in source · 1 passage
Search aliases: p_r_2
Use in source · august_development.tex, line 829
Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.
Exact mathematical passage
\label{eq:legacy-9.1} n=p_{r_1}+p_{r_2}+p_{r_3}.
p_{r_3}Use in source · 1 passage
Search aliases: p_r_3
Use in source · august_development.tex, line 829
Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.
Exact mathematical passage
\label{eq:legacy-9.1} n=p_{r_1}+p_{r_2}+p_{r_3}.
p_{r_j}Use in source · 1 passage
Search aliases: p_r_j
Use in source · august_development.tex, line 844
Balanced four-packets and saturation. Let F_m= { γ∈ N^m-1: 1^Tγ=4, R_mγ=2m 1 }. A saturated certificate is an exact nonnegative identity n+∑_j=1^2N-3p_r_j =∑_i=1^Nγ_i, γ_i∈ F_m. The case N=2 is one-step quasi-decomposition. Higher N allows several surface packets to reconstruct one six-packet after Tate stabilization.
Exact mathematical passage
\label{eq:legacy-9.2} \boxed{ n+\sum_{j=1}^{2N-3}p_{r_j} =\sum_{i=1}^N\gamma_i, \qquad \gamma_i\in\mathcal F_m. }
P_WUse in source · 6 passages
Search aliases: P_W
Use in source · appendices.tex, line 20
Weil projection · P_W(c)=-(⟨ c,u⟩ u+⟨ c,v⟩ v)/32.
Exact mathematical passage
P_W(c)=-(\langle c,u\rangle u+\langle c,v\rangle v)/32
\mathrm{plane}Use in source · 2 passages
Search aliases: plane mathrm
Use in source · august_development.tex, line 570
A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equivalently prove T_tot =∑_rA_r^† A_r satisfies λ_min(T_tot)>0.
Exact mathematical passage
A_{\mathrm{line}}, A_{\mathrm{conic}}, A_{\mathrm{plane}}, A_{\mathrm{exceptional}}.
\mathrm{prim}Use in source · 7 passages
Search aliases: prim mathrm
Use in source · august_development.tex, line 52
Primitive and coniveau reductions. Lefschetz packet decomposition. Let h∈ H^2(X,Q) be the class of an ample divisor and let L(α)=hsmileα. Hard Lefschetz and primitive decomposition yield H^2p(X,Q)=⊕_r≥0L^rH_prim^2p-2r(X,Q). The decomposition respects Hodge type.
Exact mathematical passage
H^{2p}(X,\Q)=\bigoplus_{r\ge0}L^rH_{\mathrm{prim}}^{2p-2r}(X,\Q).
QUse in source · 26 passages
Search aliases: Q
Use in source · august_development.tex, line 35
[Master obstruction projector] The operator O_X^p=P_Hdg^p-P_Alg^p is the orthogonal projector onto (Hdg^p(X)⊗_QR)∩(Alg^p(X)⊗_QR)^⊥. Therefore O_X^p=0 Longleftrightarrow Hdg^p(X)=Alg^p(X). For nested orthogonal projectors Q≤ P, the difference P-Q is self-adjoint and (P-Q)^2=P-PQ-QP+Q=P-Q. Its range is im P∩(im Q)^⊥.
Exact mathematical passage
Q\le P
\QUse in source · 41 passages
Search aliases: Q
Use in source · appendices.tex, line 6
Rational Hodge space · Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X).
Exact mathematical passage
\Hdg^p(X)=H^{2p}(X,\Q)\cap H^{p,p}(X)
q^*Use in source · 3 passages
Search aliases: q^*
Use in source · august_development.tex, line 396
Cubic fourfolds as a validation of the transfer theorem. Let X⊂ P^5 be a smooth cubic fourfold and let F(X) be its Fano variety of lines. The universal incidence variety P={(ℓ,x):x∈ℓ}⊂ F(X)× X induces the Abel-Jacobi map A=p_*q^*:H^4_prim(X,Q)→ H^2_prim(F(X),Q). Beauville and Donagi proved that this is a polarized Hodge isomorphism up to the standard twist [source reference]. The polarized adjoint is represented by an algebraic incidence return after inserting the appropriate power of the Plueck…
Exact mathematical passage
A=p_*q^*:H^4_{\mathrm{prim}}(X,\Q)\to H^2_{\mathrm{prim}}(F(X),\Q).
q^2Use in source · 1 passage
Search aliases: q^2
Use in source · capacity_cycles.tex, line 401
dim Z_κ=2, deg Z_κ=q(2q)5=10q^2.
Exact mathematical passage
\dim Z_\kappa=2,\qquad\deg Z_\kappa=q(2q)5=10q^2.
q^4Use in source · 1 passage
Search aliases: q^4
Use in source · capacity_cycles.tex, line 485
de_1wedge⋯wedge de_4wedge dP =q^4Δ(x^q) dx_0wedge⋯wedge dx_4.
Exact mathematical passage
\label{eq:capacity_cycles-5.5} de_1\wedge\cdots\wedge de_4\wedge dP =q^4\Delta(x^q)\,dx_0\wedge\cdots\wedge dx_4.
q^5Use in source · 6 passages
Search aliases: q^5
Use in source · capacity_cycles.tex, line 448
D_κ=625q^5Δ(x_0^q,…,x_4^q) ∑_b=1^q-1κ^q-bP^b-1z^5(q-b)-1.
Exact mathematical passage
\label{eq:capacity_cycles-5.2} \boxed{ D_\kappa=625q^5\Delta(x_0^q,\ldots,x_4^q) \sum_{b=1}^{q-1}\kappa^{q-b}P^{b-1}z^{5(q-b)-1}. }
Q_1Use in source · 2 passages
Search aliases: Q_1
Use in source · capacity_cycles.tex, line 622
Q_1=x_0^33x_1^26x_2^19x_3^12x_4^5x_5^4,
Exact mathematical passage
Q_1=x_0^{33}x_1^{26}x_2^{19}x_3^{12}x_4^5x_5^4,
Q_{24}Use in source · 2 passages
Search aliases: Q_24
Use in source · transfer210.tex, line 202
Q_3 · =(3,105,108,204), Q_9 · =(9,105,114,192), Q_24 · =(24,105,129,162), Q_27 · =(27,105,132,156), R_2 · =(2,72,142,204).
Exact mathematical passage
\begin{aligned} Q_3&=(3,105,108,204),\\ Q_9&=(9,105,114,192),\\ Q_{24}&=(24,105,129,162),\\ Q_{27}&=(27,105,132,156),\\ R_2&=(2,72,142,204). \end{aligned}
Q_{27}Use in source · 2 passages
Search aliases: Q_27
Use in source · transfer210.tex, line 202
Q_3 · =(3,105,108,204), Q_9 · =(9,105,114,192), Q_24 · =(24,105,129,162), Q_27 · =(27,105,132,156), R_2 · =(2,72,142,204).
Exact mathematical passage
\begin{aligned} Q_3&=(3,105,108,204),\\ Q_9&=(9,105,114,192),\\ Q_{24}&=(24,105,129,162),\\ Q_{27}&=(27,105,132,156),\\ R_2&=(2,72,142,204). \end{aligned}
Q_3Use in source · 2 passages
Search aliases: Q_3
Use in source · transfer210.tex, line 202
Q_3 · =(3,105,108,204), Q_9 · =(9,105,114,192), Q_24 · =(24,105,129,162), Q_27 · =(27,105,132,156), R_2 · =(2,72,142,204).
Exact mathematical passage
\begin{aligned} Q_3&=(3,105,108,204),\\ Q_9&=(9,105,114,192),\\ Q_{24}&=(24,105,129,162),\\ Q_{27}&=(27,105,132,156),\\ R_2&=(2,72,142,204). \end{aligned}
Q_9Use in source · 2 passages
Search aliases: Q_9
Use in source · transfer210.tex, line 202
Q_3 · =(3,105,108,204), Q_9 · =(9,105,114,192), Q_24 · =(24,105,129,162), Q_27 · =(27,105,132,156), R_2 · =(2,72,142,204).
Exact mathematical passage
\begin{aligned} Q_3&=(3,105,108,204),\\ Q_9&=(9,105,114,192),\\ Q_{24}&=(24,105,129,162),\\ Q_{27}&=(27,105,132,156),\\ R_2&=(2,72,142,204). \end{aligned}
\mathbf Q_{>0}^6Use in source · 1 passage
Search aliases: Q_>0^6 mathbf
Use in source · capacity_cycles.tex, line 128
Suppose y∈ Q_>0^6 and y· v≥1 for every v∈ L_a. Then
Exact mathematical passage
y\in\mathbf Q_{>0}^6
Q_bUse in source · 4 passages
Search aliases: Q_b
Use in source · capacity_cycles.tex, line 519
Q_b=∏_i=0^5x_i^m-a_q,b,i-1.
Exact mathematical passage
\label{eq:capacity_cycles-6.3} Q_b=\prod_{i=0}^5x_i^{m-a_{q,b,i}-1}.
q_iUse in source · 1 passage
Search aliases: q_i
Use in source · august_development.tex, line 258
Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.
Exact mathematical passage
\alpha=\sum_{i=1}^r q_i[Z_i].
Q_{m,c}Use in source · 4 passages
Search aliases: Q_m,c
Use in source · august_development.tex, line 798
0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is itself algebraic.
Exact mathematical passage
\label{eq:legacy-8.2} \boxed{ Q_{m,c}=\sum_{d\mid c}\mu(c/d)E_{m,d}. }
Q_{m,e}Use in source · 1 passage
Search aliases: Q_m,e
Use in source · august_development.tex, line 803
0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is itself algebraic.
Exact mathematical passage
Q_{m,c}Q_{m,e}=\delta_{ce}Q_{m,c}, \qquad \sum_{c\mid m}Q_{m,c}=I.
RUse in source · 17 passages
Search aliases: R
Use in source · august_development.tex, line 456
ction. After gauge fixing, write the integrability equation abstractly as F(a)=f+La+B(a,a)=0, where L is the linearized operator and B is a bounded bilinear term. Assume a Fredholm splitting of the target I=Q+Π, where Q projects onto the closed range of L and Π onto a finite-dimensional obstruction space. Choose a gauge-fixed complement to ker L as the domain and let R be a right inverse of L taking values in that complement. Use norms on the projected data for which QB(a,b) ≤ C a b .
Exact mathematical passage
R
\RUse in source · 11 passages
Search aliases: R
Use in source · appendices.tex, line 8
Real obstruction · O=P_Hdg⊗R-P_Alg⊗R. No algebraic realization of these orthogonal projectors is presumed.
Exact mathematical passage
\mathcal O=P_{\Hdg\otimes\R}-P_{\Alg\otimes\R}
\R^2Use in source · 1 passage
Search aliases: R^2 R
Use in source · edition_audit.tex, line 66
Take V=Span_R{e_1} in R^2 and let the only cycle vector be c=e_1+e_2. Then S=cc^t=1 · 1
Exact mathematical passage
\R^2
r^2Use in source · 1 passage
Search aliases: r^2
Use in source · august_development.tex, line 479
(f+B(a,a)) has a unique fixed point in the gauge-fixed ball a ≤2M Qf . If Qf=0, the ball is the singleton {0} and the range solution is immediate. At the fixed point, the range component of F(a) vanishes. The remaining obstruction is Φ=Π(f+B(a,a)). The full integrability equation holds if and only if Φ=0. For a ≤ r=2M Qf , T(a) ≤ M Qf +MC r^2≤ r under the stated bound. Moreover T(a)- T(b) ≤ MC( a + b ) a-b . The Lipschitz constant is strictly less than one on the stated ball. Banach's fixed-poin…
Exact mathematical passage
\|\mathcal T(a)\|\le M\|Qf\|+MC r^2\le r
\R^NUse in source · 1 passage
Search aliases: R^N R
Use in source · august_development.tex, line 103
bseteq H^2p(X,R) be the real extension of the rational Hodge packet under study. For the positive frame certificate below, require every c_j to belong to V. This can be achieved by a rational algebraic packet projector when such a projector has been constructed. Projection through an arbitrary orthogonal Hodge projector does not establish that the projected vectors are algebraic. Define C:R^N→ H^2p(X,R), Ce_j=c_j, and the positive frame operator S=CC^*=∑_j=1^N|c_j⟩⟨ c_j|.
Exact mathematical passage
C:\R^N\to H^{2p}(X,\R), \qquad Ce_j=c_j,
r_0Use in source · 13 passages
Search aliases: r_0
Use in source · august_development.tex, line 936
e Laurent exponent match is a useful exact certificate, but by itself it does not prove that a rational class lifts through a compact graph. A complete manuscript argument instead uses the torus-carrier criterion developed in Chapter~[source reference]. For the reordered character, the exponent matrix A of the cubic satisfies A^Twidetilde a/33=β=(1,2,2,1,1,2), C=33A^-1. With r_0 the first row of A, put u=β-3r_0. Then 1^Tu=0 and C^Tu=widetilde a-99e_0. These identities establish triviality of the…
Exact mathematical passage
r_0
r_1Use in source · 1 passage
Search aliases: r_1
Use in source · august_development.tex, line 829
Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.
Exact mathematical passage
\label{eq:legacy-9.1} n=p_{r_1}+p_{r_2}+p_{r_3}.
R_2Use in source · 2 passages
Search aliases: R_2
Use in source · transfer210.tex, line 202
Q_3 · =(3,105,108,204), Q_9 · =(9,105,114,192), Q_24 · =(24,105,129,162), Q_27 · =(27,105,132,156), R_2 · =(2,72,142,204).
Exact mathematical passage
\begin{aligned} Q_3&=(3,105,108,204),\\ Q_9&=(9,105,114,192),\\ Q_{24}&=(24,105,129,162),\\ Q_{27}&=(27,105,132,156),\\ R_2&=(2,72,142,204). \end{aligned}
r_2Use in source · 1 passage
Search aliases: r_2
Use in source · august_development.tex, line 829
Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.
Exact mathematical passage
\label{eq:legacy-9.1} n=p_{r_1}+p_{r_2}+p_{r_3}.
r_3Use in source · 1 passage
Search aliases: r_3
Use in source · august_development.tex, line 829
Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.
Exact mathematical passage
\label{eq:legacy-9.1} n=p_{r_1}+p_{r_2}+p_{r_3}.
r_{\deg}Use in source · 1 passage
Search aliases: r_deg deg
Use in source · august_development.tex, line 199
Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods to solve different geometric sectors before exact synthesis.
Exact mathematical passage
\lambda=(r_{\mathrm{Lef}},r_{\deg},\chi_{\mathrm{mon}},s_{\mathrm{supp}},\nu_{\mathrm{Frob}}).
r_iUse in source · 2 passages
Search aliases: r_i
Use in source · carrier70.tex, line 497
Indeed, choose M divisible by m and all denominators of A^-1. Put B=MA^-1, α=(M/m)a, and β=A^Ta/m. For any row r_i of A,
Exact mathematical passage
r_i
r_jUse in source · 1 passage
Search aliases: r_j
Use in source · august_development.tex, line 844
Balanced four-packets and saturation. Let F_m= { γ∈ N^m-1: 1^Tγ=4, R_mγ=2m 1 }. A saturated certificate is an exact nonnegative identity n+∑_j=1^2N-3p_r_j =∑_i=1^Nγ_i, γ_i∈ F_m. The case N=2 is one-step quasi-decomposition. Higher N allows several surface packets to reconstruct one six-packet after Tate stabilization.
Exact mathematical passage
\label{eq:legacy-9.2} \boxed{ n+\sum_{j=1}^{2N-3}p_{r_j} =\sum_{i=1}^N\gamma_i, \qquad \gamma_i\in\mathcal F_m. }
r_{\mathrm{Lef}}Use in source · 1 passage
Search aliases: r_Lef mathrm
Use in source · august_development.tex, line 199
Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods to solve different geometric sectors before exact synthesis.
Exact mathematical passage
\lambda=(r_{\mathrm{Lef}},r_{\deg},\chi_{\mathrm{mon}},s_{\mathrm{supp}},\nu_{\mathrm{Frob}}).
R_mUse in source · 3 passages
Search aliases: R_m
Use in source · august_development.tex, line 713
To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebraic automorphisms, the count-vector descriptions correspond to U_m-orbits in B_m. The dimension of a fixed rational character packet is computed from di…
Exact mathematical passage
R_m(u,r)=\langle ur\rangle_m.
\operatorname{rank}Use in source · 1 passage
Search aliases: rank operatorname
Use in source · august_development.tex, line 133
If H is a matrix whose columns form an orthonormal basis of V, the finite Gram certificate is G=H^*SH. Under A⊆ V, we have A=V if and only if λ_min(G)>0. Without that containment, use a rational basis matrix H_Q for the target and the exact certificate rank_Q[C H_Q]=rank_QC to test V⊆ A.
Exact mathematical passage
\operatorname{rank}_{\Q}[C\ H_{\Q}]=\operatorname{rank}_{\Q}C
\rankUse in source · 3 passages
Search aliases: rank
Use in source · appendices.tex, line 10
Unconditional rank test · rank_Q[C H_Q]=rank_QC tests target containment in the actual cycle span.
Exact mathematical passage
\rank_{\Q}[C\ H_{\Q}]=\rank_{\Q}C
\operatorname{rank}_{\Q}Use in source · 1 passage
Search aliases: rank_Q operatorname Q
Use in source · august_development.tex, line 133
If H is a matrix whose columns form an orthonormal basis of V, the finite Gram certificate is G=H^*SH. Under A⊆ V, we have A=V if and only if λ_min(G)>0. Without that containment, use a rational basis matrix H_Q for the target and the exact certificate rank_Q[C H_Q]=rank_QC to test V⊆ A.
Exact mathematical passage
\operatorname{rank}_{\Q}[C\ H_{\Q}]=\operatorname{rank}_{\Q}C
\rank_{\Q}Use in source · 3 passages
Search aliases: rank_Q rank Q
Use in source · appendices.tex, line 10
Unconditional rank test · rank_Q[C H_Q]=rank_QC tests target containment in the actual cycle span.
Exact mathematical passage
\rank_{\Q}[C\ H_{\Q}]=\rank_{\Q}C
\operatorname{red}Use in source · 2 passages
Search aliases: red operatorname
Use in source · august_development.tex, line 1247
SAT(2) closure and the four-star obstruction. For a defect d, let red(d) be the unmatched residue multiset: a positive coefficient at r contributes copies of r, while a negative coefficient contributes copies of m-r. Its length is d _1. Every packet with defect d is red(d) together with complementary pairs and copies of h.
Exact mathematical passage
\operatorname{red}(d)
\operatorname{res}Use in source · 2 passages
Search aliases: res operatorname
Use in source · capacity_cycles.tex, line 545
∫_Z_κres(fracQ_bΩF_m^3) =frac(2π i)^250q κ^q-b≠0.
Exact mathematical passage
\label{eq:capacity_cycles-6.5} \boxed{ \int_{Z_\kappa}\operatorname{res}\left(\frac{Q_b\Omega}{F_m^3}\right) =\frac{(2\pi i)^2}{50q}\,\kappa^{q-b}\ne0. }
SUse in source · 46 passages
Search aliases: S
Use in source · appendices.tex, line 9
Cycle frame · S=CC^*. Positivity spans a target packet when every supplied algebraic vector lies in it.
Exact mathematical passage
S=CC^*
s^2Use in source · 4 passages
Search aliases: s^2
Use in source · carrier70.tex, line 181
F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.
Exact mathematical passage
F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.
S^\daggerUse in source · 1 passage
Search aliases: S^† dagger
Use in source · august_development.tex, line 140
The orthogonal synthesis of a class α from the cycle span is P_Aα=S S^†α. If α∈ A, this is exact.
Exact mathematical passage
P_A\alpha=S S^\dagger\alpha.
s_0Use in source · 2 passages
Search aliases: s_0
Use in source · august_development.tex, line 343
[Propagation of invertibility] If a relative transfer T_s is invertible on one fiber s_0, it is invertible on every fiber in the connected base on which the local systems and correspondences are defined. The determinant of the constant rational matrix representing the local-system morphism is locally constant. It is nonzero at s_0, hence everywhere on the connected base.
Exact mathematical passage
s_0
S_5Use in source · 3 passages
Search aliases: S_5
Use in source · transfer210.tex, line 218
S_5=(30,60,72,114,156,198), S_7=(12,42,72,102,126,132,162,192).
Exact mathematical passage
S_5=(30,60,72,114,156,198), \qquad S_7=(12,42,72,102,126,132,162,192).
S_7Use in source · 3 passages
Search aliases: S_7
Use in source · transfer210.tex, line 218
S_5=(30,60,72,114,156,198), S_7=(12,42,72,102,126,132,162,192).
Exact mathematical passage
S_5=(30,60,72,114,156,198), \qquad S_7=(12,42,72,102,126,132,162,192).
S_{LM,r+\ell M}Use in source · 2 passages
Search aliases: S_LM,r+ℓ M ell
Use in source · appendices.tex, line 11
Polyphase assembly · S_M,r=∑_ℓ=0^L-1S_LM,r+ℓ M. These are frame sums, not sums of span projectors.
Exact mathematical passage
S_{M,r}=\sum_{\ell=0}^{L-1}S_{LM,r+\ell M}
S_mUse in source · 3 passages
Search aliases: S_m
Use in source · capacity_cycles.tex, line 82
J_F_m=(x_0^m-1,…,x_5^m-1), S_m=∏_i=0^5x_i^m-2.
Exact mathematical passage
\label{eq:capacity_cycles-1.2} J_{F_m}=(x_0^{m-1},\ldots,x_5^{m-1}),\qquad S_m=\prod_{i=0}^5x_i^{m-2}.
S_{M,r}Use in source · 3 passages
Search aliases: S_M,r
Use in source · appendices.tex, line 11
Polyphase assembly · S_M,r=∑_ℓ=0^L-1S_LM,r+ℓ M. These are frame sums, not sums of span projectors.
Exact mathematical passage
S_{M,r}=\sum_{\ell=0}^{L-1}S_{LM,r+\ell M}
S_qUse in source · 7 passages
Search aliases: S_q
Use in source · capacity_cycles.tex, line 356
S_q={e_1(x_0^q,…,x_4^q)=e_2(x_0^q,…,x_4^q)=0}⊂ P^4.
Exact mathematical passage
S_q=\{e_1(x_0^q,\ldots,x_4^q)=e_2(x_0^q,\ldots,x_4^q)=0\}\subset\mathbf P^4.
s_{\mathrm{supp}}Use in source · 1 passage
Search aliases: s_supp mathrm
Use in source · august_development.tex, line 199
Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods to solve different geometric sectors before exact synthesis.
Exact mathematical passage
\lambda=(r_{\mathrm{Lef}},r_{\deg},\chi_{\mathrm{mon}},s_{\mathrm{supp}},\nu_{\mathrm{Frob}}).
S_TUse in source · 5 passages
Search aliases: S_T
Use in source · august_development.tex, line 1430
c,d,e to be nonzero modulo q so that all three resulting character blocks have full support. Degenerate cases must leave this construction and be handled through pair removal or another channel. For example (1,14,2,9,14,5) at modulus 15 is balanced and admits the indicated phase split, but P=(1,14) gives c=0. The phase condition alone is insufficient. Define C_P=Psqcup{m-5c}, S_T=Tsqcup{m-5d}, and L=(5c,5d,5e).
Exact mathematical passage
S_T=T\sqcup\{m-5d\},
S_\lambdaUse in source · 5 passages
Search aliases: S_λ lambda
Use in source · august_development.tex, line 149
Packet assembly without false orthogonality. Partition the cycle indices into packets I_λ. Define S_λ=∑_j∈ I_λ|c_j⟩⟨ c_j|. Then S=∑_λ S_λ. No orthogonality of the child spans is assumed.
Exact mathematical passage
S_\lambda=\sum_{j\in I_\lambda}|c_j\rangle\langle c_j|.
\sinUse in source · 2 passages
Search aliases: sin
Use in source · august_development.tex, line 1499
For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overlineΣ_ℓ. This is the ri…
Exact mathematical passage
1-e^{2\pi ir/q}=-2i\,e^{\pi ir/q}\sin\frac{\pi r}{q},
\smileUse in source · 3 passages
Search aliases: smile
Use in source · august_development.tex, line 51
Primitive and coniveau reductions. Lefschetz packet decomposition. Let h∈ H^2(X,Q) be the class of an ample divisor and let L(α)=hsmileα. Hard Lefschetz and primitive decomposition yield H^2p(X,Q)=⊕_r≥0L^rH_prim^2p-2r(X,Q). The decomposition respects Hodge type.
Exact mathematical passage
L(\alpha)=h\smile\alpha
\SpanUse in source · 4 passages
Search aliases: Span
Use in source · august_development.tex, line 113
[Finite positive cycle-frame certificate] Let A=Span_R{c_1,…,c_N}. Then im S=A, ker S=A^⊥. Under the stated containment c_j∈ V, the selected cycles span V if and only if S|_V is positive definite. Without containment, positivity only detects the absence of vectors in V orthogonal to the cycle span and does not prove that span contains V. The equality im(CC^*)=im C is standard. Moreover ⟨ v,Sv⟩= C^*v ^2=∑_j|⟨ c_j,v⟩|^2. This vanishes exactly when v is orthogonal to every cycle class.
Exact mathematical passage
A=\Span_{\R}\{c_1,\ldots,c_N\}
\Span_{\Q}Use in source · 1 passage
Search aliases: Span_Q Span Q
Use in source · weil_audit.tex, line 26
For complex structures in the Weil domain with three positive and three negative K-embedding directions, Ω_6 has type (3,3). Indeed, the K-determinant contains three one-forms from each Hodge side. Thus the rational span W=Span_Q{u,v} is a Hodge substructure of degree six across the domain. The first-order computation confirms that persistence in all 18 real tangent directions at the seed. The all-domain assertion comes from the determinant Hodge-type argument, rather than extrapolation from tho…
Exact mathematical passage
W=\Span_{\Q}\{u,v\}
\Span_{\R}Use in source · 2 passages
Search aliases: Span_R Span R
Use in source · august_development.tex, line 113
[Finite positive cycle-frame certificate] Let A=Span_R{c_1,…,c_N}. Then im S=A, ker S=A^⊥. Under the stated containment c_j∈ V, the selected cycles span V if and only if S|_V is positive definite. Without containment, positivity only detects the absence of vectors in V orthogonal to the cycle span and does not prove that span contains V. The equality im(CC^*)=im C is standard. Moreover ⟨ v,Sv⟩= C^*v ^2=∑_j|⟨ c_j,v⟩|^2. This vanishes exactly when v is orthogonal to every cycle class.
Exact mathematical passage
A=\Span_{\R}\{c_1,\ldots,c_N\}
\sqcupUse in source · 5 passages
Search aliases: sqcup
Use in source · august_development.tex, line 868
Curve, Delsarte, and incidence channels. A curve certificate splits the six entries into triples a=Asqcup B, ∑ A=m, ∑ B=2m, B=uApmod m. The resulting (1,1) class on a product of Fermat curves is algebraic by Lefschetz (1,1), and the join correspondence returns a surface class.
Exact mathematical passage
a=A\sqcup B, \qquad \sum A=m, \quad \sum B=2m, \quad B=uA\pmod m.
\squareUse in source · 2 passages
Search aliases: square
Use in source · capacity_cycles.tex, line 288
Conversely v=1 is invariant, because ∑ a_i=3m, and putting weight 1 on it is a feasible point of [source reference]. Thus C(a_q,b)=1, proving [source reference]. square
Exact mathematical passage
\square
\supUse in source · 2 passages
Search aliases: sup
Use in source · appendices.tex, line 17
Matrix capacity · C(a)=sup{∑ w_v:∑ w_vv=1, w_v≥0}.
Exact mathematical passage
C(a)=\sup\{\sum w_v:\sum w_vv=\mathbf1,\ w_v\ge0\}
\mathrm{supp}Use in source · 1 passage
Search aliases: supp mathrm
Use in source · august_development.tex, line 199
Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods to solve different geometric sectors before exact synthesis.
Exact mathematical passage
\lambda=(r_{\mathrm{Lef}},r_{\deg},\chi_{\mathrm{mon}},s_{\mathrm{supp}},\nu_{\mathrm{Frob}}).
\mathcal TUse in source · 3 passages
Search aliases: T mathcal
Use in source · august_development.tex, line 468
[Range-packet contraction] Assume R ≤ M, B(a,b) ≤ C a b , 4CM^2 Qf <1. Then the map T(a)=-RQ(f+B(a,a)) has a unique fixed point in the gauge-fixed ball a ≤2M Qf . If Qf=0, the ball is the singleton {0} and the range solution is immediate. At the fixed point, the range component of F(a) vanishes. The remaining obstruction is Φ=Π(f+B(a,a)). The full integrability equation holds if and only if Φ=0. For a ≤ r=2M Qf , T(a) ≤ M Qf +MC r^2≤ r under the stated bound. Moreover T(a)- T(b) ≤ MC( a + b ) a-…
Exact mathematical passage
\mathcal T(a)=-RQ(f+B(a,a))
tUse in source · 31 passages
Search aliases: t
Use in source · august_development.tex, line 301
Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.
Exact mathematical passage
\chi_T(t)=t^d+c_{d-1}t^{d-1}+\cdots+c_0
T^{-1}Use in source · 3 passages
Search aliases: T^-1
Use in source · august_development.tex, line 302
Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.
Exact mathematical passage
T^{-1}=-\frac{1}{c_0}\bigl(T^{d-1}+c_{d-1}T^{d-2}+\cdots+c_1I\bigr).
t^2Use in source · 5 passages
Search aliases: t^2
Use in source · carrier70.tex, line 181
F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.
Exact mathematical passage
F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.
t^dUse in source · 1 passage
Search aliases: t^d
Use in source · august_development.tex, line 301
Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.
Exact mathematical passage
\chi_T(t)=t^d+c_{d-1}t^{d-1}+\cdots+c_0
T^{d-1}Use in source · 1 passage
Search aliases: T^d-1
Use in source · august_development.tex, line 302
Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.
Exact mathematical passage
T^{-1}=-\frac{1}{c_0}\bigl(T^{d-1}+c_{d-1}T^{d-2}+\cdots+c_1I\bigr).
t^{d-1}Use in source · 1 passage
Search aliases: t^d-1
Use in source · august_development.tex, line 301
Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.
Exact mathematical passage
\chi_T(t)=t^d+c_{d-1}t^{d-1}+\cdots+c_0
T^{d-2}Use in source · 1 passage
Search aliases: T^d-2
Use in source · august_development.tex, line 302
Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.
Exact mathematical passage
T^{-1}=-\frac{1}{c_0}\bigl(T^{d-1}+c_{d-1}T^{d-2}+\cdots+c_1I\bigr).
t_0Use in source · 3 passages
Search aliases: t_0
Use in source · capacity_cycles.tex, line 324
and let e_j be the elementary symmetric polynomial of degree j in t_0,…,t_4. Define
Exact mathematical passage
t_0,\ldots,t_4
t_4Use in source · 3 passages
Search aliases: t_4
Use in source · capacity_cycles.tex, line 324
and let e_j be the elementary symmetric polynomial of degree j in t_0,…,t_4. Define
Exact mathematical passage
t_0,\ldots,t_4
t_iUse in source · 2 passages
Search aliases: t_i
Use in source · capacity_cycles.tex, line 319
z=x_5, t_i=x_i^q (0≤ i≤4), P=x_0x_1x_2x_3x_4,
Exact mathematical passage
z=x_5,\quad t_i=x_i^q\ (0\le i\le4),\quad P=x_0x_1x_2x_3x_4,
t_jUse in source · 1 passage
Search aliases: t_j
Use in source · capacity_cycles.tex, line 440
Δ(t)=∏_0≤ i<j≤4(t_i-t_j).
Exact mathematical passage
\Delta(t)=\prod_{0\le i<j\le4}(t_i-t_j).
T_mUse in source · 1 passage
Search aliases: T_m
Use in source · august_development.tex, line 1252
[Exact SAT(2) criterion] Let n be a balanced six-packet at modulus m=2^s. Then n is pair-decomposable or has an SAT(2) certificate unless its gauged defect is ±1 on a four-vertex star K_1,3 in the internal tree T_m. A four-star has no SAT(2) certificate. Every nonstar shape in the preceding proposition splits into at most two four-packet atoms: item a vertex or edge is paired with a zero atom item a three-vertex path is an edge plus its remaining terminal vertex item a four-vertex path is the un…
Exact mathematical passage
T_m
T_sUse in source · 1 passage
Search aliases: T_s
Use in source · august_development.tex, line 343
[Propagation of invertibility] If a relative transfer T_s is invertible on one fiber s_0, it is invertible on every fiber in the connected base on which the local systems and correspondences are defined. The determinant of the constant rational matrix representing the local-system morphism is locally constant. It is nonzero at s_0, hence everywhere on the connected base.
Exact mathematical passage
T_s
T_{\mathrm{tot}}Use in source · 1 passage
Search aliases: T_tot mathrm
Use in source · august_development.tex, line 584
Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equivalently prove T_tot =∑_rA_r^† A_r satisfies λ_min(T_tot)>0.
Exact mathematical passage
T_{\mathrm{tot}} =\sum_rA_r^\dagger A_r \quad\text{satisfies}\quad \lambda_{\min}(T_{\mathrm{tot}})>0.
\mathrm{top}Use in source · 1 passage
Search aliases: top mathrm
Use in source · august_development.tex, line 424
Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees is equivalent to algebraization of topological K-classes whose Chern-character components are rational Hodge classes [source reference].
Exact mathematical passage
\operatorname{ch}_{\mathrm{top}}:K^0_{\mathrm{top}}(X)\otimes\Q\xrightarrow{\sim}H^{\mathrm{even}}(X,\Q).
\mathrm{tot}Use in source · 1 passage
Search aliases: tot mathrm
Use in source · august_development.tex, line 584
Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equivalently prove T_tot =∑_rA_r^† A_r satisfies λ_min(T_tot)>0.
Exact mathematical passage
T_{\mathrm{tot}} =\sum_rA_r^\dagger A_r \quad\text{satisfies}\quad \lambda_{\min}(T_{\mathrm{tot}})>0.
\TrUse in source · 1 passage
Search aliases: Tr
Use in source · august_development.tex, line 497
A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge j _L^2^2+O(k^n-2j-1), c_n,j>0. This detects nonintegrability. It does not by itself remove the defect.
Exact mathematical passage
\Tr(P_{2j}\Pi_{A,k}) =c_{n,j}k^{n-2j}\|(F_A^{0,2})^{\wedge j}\|_{L^2}^2+O(k^{n-2j-1}), \qquad c_{n,j}>0.
\operatorname{Tr}Use in source · 1 passage
Search aliases: Tr operatorname
Use in source · august_development.tex, line 507
0 · 0 · 0) is nonzero, while N^3=0 and Tr(N^r)=0 for every positive integer r. This is the elementary negative control.
Exact mathematical passage
\operatorname{Tr}(N^r)=0
\operatorname{TV}Use in source · 1 passage
Search aliases: TV operatorname
Use in source · august_development.tex, line 1221
For an integer-valued y, discrete coarea gives TV(y) = ∑_k≥1|∂{y≥ k}| + ∑_k≥1|∂{y≤-k}|. If S is a finite forest of internal vertices of a trivalent tree and c(S) is its number of connected components, then |∂ S|=|S|+2c(S), because 3|S|=2(|S|-c(S))+|∂ S|.
Exact mathematical passage
\label{eq:legacy-13.3} \operatorname{TV}(y) = \sum_{k\ge1}|\partial\{y\ge k\}| + \sum_{k\ge1}|\partial\{y\le-k\}|.
u^2Use in source · 1 passage
Search aliases: u^2
Use in source · carrier70.tex, line 187
F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.
Exact mathematical passage
\begin{aligned} F_u&=3u^2+s^2t^2z, &F_v&=5v^4,\\ F_s&=2sz^3+t^2c^7+2ust^2z, &F_t&=2t(z^3+sc^7+us^2z),\\ F_c&=7st^2c^6, &F_z&=3z^2(s^2+t^2)+us^2t^2. \end{aligned}
u^3Use in source · 1 passage
Search aliases: u^3
Use in source · carrier70.tex, line 181
F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.
Exact mathematical passage
F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.
U_{210}Use in source · 5 passages
Search aliases: U_210
Use in source · carrier70.tex, line 159
q:U_210⟶ U_Y,
Exact mathematical passage
q:U_{210}\longrightarrow U_Y,
U_{p^{r-j}}Use in source · 1 passage
Search aliases: U_p^r-j
Use in source · august_development.tex, line 975
Valuation strata and orbit mass. Let m=p^r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonzero odd kernel-orbit defect has mass at least 2p.
Exact mathematical passage
b\in U_{p^{r-j}}
U_{p^s}Use in source · 1 passage
Search aliases: U_p^s
Use in source · august_development.tex, line 983
r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonzero odd kernel-orbit defect has mass at least 2p.
Exact mathematical passage
U_{p^s}\to U_{p^{s-1}}.
U_{p^{s-1}}Use in source · 1 passage
Search aliases: U_p^s-1
Use in source · august_development.tex, line 983
r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonzero odd kernel-orbit defect has mass at least 2p.
Exact mathematical passage
U_{p^s}\to U_{p^{s-1}}.
U_YUse in source · 6 passages
Search aliases: U_Y
Use in source · carrier70.tex, line 159
q:U_210⟶ U_Y,
Exact mathematical passage
q:U_{210}\longrightarrow U_Y,
v^4Use in source · 1 passage
Search aliases: v^4
Use in source · carrier70.tex, line 187
F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.
Exact mathematical passage
\begin{aligned} F_u&=3u^2+s^2t^2z, &F_v&=5v^4,\\ F_s&=2sz^3+t^2c^7+2ust^2z, &F_t&=2t(z^3+sc^7+us^2z),\\ F_c&=7st^2c^6, &F_z&=3z^2(s^2+t^2)+us^2t^2. \end{aligned}
v^5Use in source · 1 passage
Search aliases: v^5
Use in source · carrier70.tex, line 181
F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.
Exact mathematical passage
F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.
v_0Use in source · 3 passages
Search aliases: v_0
Use in source · capacity_cycles.tex, line 272
v_0+v_1+v_2+v_3+v_4-5v_5≡0pmod q.
Exact mathematical passage
v_0+v_1+v_2+v_3+v_4-5v_5\equiv0\pmod q.
v_1Use in source · 2 passages
Search aliases: v_1
Use in source · capacity_cycles.tex, line 272
v_0+v_1+v_2+v_3+v_4-5v_5≡0pmod q.
Exact mathematical passage
v_0+v_1+v_2+v_3+v_4-5v_5\equiv0\pmod q.
v_2Use in source · 4 passages
Search aliases: v_2
Use in source · august_development.tex, line 1275
[Star conductor] Let a four-star have center c, and write j=v_2(c). The gcd of m with its six packet entries is 2^j-1. Hence a star has full conductor exactly when v_2(c)=1. Use the boundary-edge labels of widetilde T_m. The two children of c have valuation j-1. The four boundary edges leaving those children have labels of exact valuation j-1: for j=1 they are the odd leaf labels, and for j>1 they are labels 2b coming from grandchildren of valuation j-2. The two boundary edges leaving the parent…
Exact mathematical passage
j=v_2(c)
v_3Use in source · 2 passages
Search aliases: v_3
Use in source · capacity_cycles.tex, line 272
v_0+v_1+v_2+v_3+v_4-5v_5≡0pmod q.
Exact mathematical passage
v_0+v_1+v_2+v_3+v_4-5v_5\equiv0\pmod q.
v_4Use in source · 3 passages
Search aliases: v_4
Use in source · capacity_cycles.tex, line 272
v_0+v_1+v_2+v_3+v_4-5v_5≡0pmod q.
Exact mathematical passage
v_0+v_1+v_2+v_3+v_4-5v_5\equiv0\pmod q.
v_5Use in source · 4 passages
Search aliases: v_5
Use in source · capacity_cycles.tex, line 272
v_0+v_1+v_2+v_3+v_4-5v_5≡0pmod q.
Exact mathematical passage
v_0+v_1+v_2+v_3+v_4-5v_5\equiv0\pmod q.
v_jUse in source · 5 passages
Search aliases: v_j
Use in source · carrier70.tex, line 469
For searching, choose nonnegative integral columns v_j satisfying a· v_j≡0pmod m. Seek a nonsingular six-column matrix A with
Exact mathematical passage
v_j
V_mUse in source · 1 passage
Search aliases: V_m
Use in source · august_development.tex, line 1141
[Integral dyadic triangle basis] For m=2^s, s≥3, the vectors {κ_b:1≤ b<q} form a Z-basis of K_m. Set r_0=q/2. On the vertex set V_m={1,2,…,q-1}, define, for b≠ r_0, p(b)= 2b, · b<r_0,
Exact mathematical passage
V_m=\{1,2,\ldots,q-1\},
wUse in source · 15 passages
Search aliases: w
Use in source · appendices.tex, line 17
Matrix capacity · C(a)=sup{∑ w_v:∑ w_vv=1, w_v≥0}.
Exact mathematical passage
C(a)=\sup\{\sum w_v:\sum w_vv=\mathbf1,\ w_v\ge0\}
W_{[3a]}Use in source · 1 passage
Search aliases: W_[3a]
Use in source · carrier70.tex, line 435
Its pullback identifies W_[a] with W_[3a]. The identity
Exact mathematical passage
W_{[3a]}
W_{[b]}Use in source · 2 passages
Search aliases: W_[b]
Use in source · transfer210.tex, line 258
Assuming the inherited degree-70 carrier proposition, every rational class in W_[b] is algebraic.
Exact mathematical passage
W_{[b]}
W_{[ka]}Use in source · 1 passage
Search aliases: W_[ka]
Use in source · carrier70.tex, line 39
all classes in the 24-dimensional rational space W_[a] are rational linear combinations of classes of algebraic surfaces on X_70. For every positive integer k, the corresponding pulled-back packet W_[ka] on X_70k is also algebraic.
Exact mathematical passage
W_{[ka]}
W_{[\alpha]}Use in source · 1 passage
Search aliases: W_[α] alpha
Use in source · carrier70.tex, line 426
The resulting class and the prescribed class agree on U_210, their difference lies in the packet and restricts to zero. The preceding boundary isomorphism proves that difference is zero. This establishes algebraicity of the entire W_[α] packet on X_210.
Exact mathematical passage
W_{[\alpha]}
W_iUse in source · 1 passage
Search aliases: W_i
Use in source · carrier70.tex, line 114
D-∑ W_i=30-33=-3.
Exact mathematical passage
D-\sum W_i=30-33=-3.
w_jUse in source · 4 passages
Search aliases: w_j
Use in source · capacity_cycles.tex, line 258
∑_j w_j≤1.
Exact mathematical passage
\label{eq:capacity_cycles-3.2} \boxed{\sum_j w_j\le1.}
w_vUse in source · 3 passages
Search aliases: w_v
Use in source · appendices.tex, line 17
Matrix capacity · C(a)=sup{∑ w_v:∑ w_vv=1, w_v≥0}.
Exact mathematical passage
C(a)=\sup\{\sum w_v:\sum w_vv=\mathbf1,\ w_v\ge0\}
\wedgeUse in source · 9 passages
Search aliases: wedge
Use in source · august_development.tex, line 497
A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge j _L^2^2+O(k^n-2j-1), c_n,j>0. This detects nonintegrability. It does not by itself remove the defect.
Exact mathematical passage
\Tr(P_{2j}\Pi_{A,k}) =c_{n,j}k^{n-2j}\|(F_A^{0,2})^{\wedge j}\|_{L^2}^2+O(k^{n-2j-1}), \qquad c_{n,j}>0.
\widehat{dx_i}Use in source · 1 passage
Search aliases: widehatdx_i widehat
Use in source · capacity_cycles.tex, line 88
Ω=∑_i=0^5(-1)^ix_i dx_0wedge⋯widehatdx_i⋯wedge dx_5
Exact mathematical passage
\Omega=\sum_{i=0}^5(-1)^ix_i\,dx_0\wedge\cdots\widehat{dx_i}\cdots\wedge dx_5
\widetilde CUse in source · 1 passage
Search aliases: widetilde C widetilde
Use in source · carrier70.tex, line 277
K_widetilde Y·widetilde C=K_Y· C=-3H· C<0.
Exact mathematical passage
K_{\widetilde Y}\cdot\widetilde C=K_Y\cdot C=-3H\cdot C<0.
\widetilde TUse in source · 3 passages
Search aliases: widetilde T widetilde
Use in source · august_development.tex, line 1192
The trivalent defect tree. The integral basis turns defect mass into exact tree variation. Define a sign gauge by τ_r_0=1, τ_b=ϵ_bτ_p(b), y_b=τ_bz_b. Attach one zero-valued boundary leaf to the root and two zero-valued boundary leaves to every odd vertex. Every vertex of the resulting augmented tree widetilde T_m has degree three. The boundary edges correspond bijectively to defect coordinates: the root edge is labelled q, the edge from b to p(b) is labelled 2b, and the two leaves at an odd b ar…
Exact mathematical passage
\widetilde T_m
\widetilde T_mUse in source · 3 passages
Search aliases: widetilde T_m widetilde
Use in source · august_development.tex, line 1192
The trivalent defect tree. The integral basis turns defect mass into exact tree variation. Define a sign gauge by τ_r_0=1, τ_b=ϵ_bτ_p(b), y_b=τ_bz_b. Attach one zero-valued boundary leaf to the root and two zero-valued boundary leaves to every odd vertex. Every vertex of the resulting augmented tree widetilde T_m has degree three. The boundary edges correspond bijectively to defect coordinates: the root edge is labelled q, the edge from b to p(b) is labelled 2b, and the two leaves at an odd b ar…
Exact mathematical passage
\widetilde T_m
\mathcal XUse in source · 1 passage
Search aliases: X mathcal
Use in source · august_development.tex, line 340
Family propagation. Let π: X→ S and ρ_r: Y_r→ S be smooth projective families over a connected base. Relative algebraic correspondences induce morphisms of Gauss-Manin local systems. In a flat rational trivialization, their transfer matrix is constant.
Exact mathematical passage
\pi:\mathcal X\to S
x^{11}Use in source · 1 passage
Search aliases: x^11
Use in source · august_development.tex, line 899
The exceptional packet. A representative is a=(7,10,13,19,22,28), or after permutation, widetilde a=(7,19,28,10,13,22). Every unit conjugate has residue sum 99=3·33, so the orbit is rational of type (2,2). It is not totally pair-decomposable. The previously proposed symmetric surface W={p_1(x^11)=p_2(x^11)=p_3(x^11)=0} does not carry the primitive packet: Newton's identity makes the third equation redundant on X_33^4, and [W]=242h^2. Thus W is purely ambient.
Exact mathematical passage
W=\{p_1(x^{11})=p_2(x^{11})=p_3(x^{11})=0\}
x^2Use in source · 1 passage
Search aliases: x^2
Use in source · weil_audit.tex, line 158
Modulo 3, the equation x^2+y^2=0 has only the solution (0,0). This is a finite anisotropy control for the residual discriminant. It does not construct a semistable algebraic family with this nilpotent as monodromy. An abstract rational nilpotent satisfying Lie-algebra identities is weaker than a geometric degeneration with the required integral, polarization and admissibility data.
Exact mathematical passage
x^2+y^2=0
x^qUse in source · 1 passage
Search aliases: x^q
Use in source · capacity_cycles.tex, line 485
de_1wedge⋯wedge de_4wedge dP =q^4Δ(x^q) dx_0wedge⋯wedge dx_4.
Exact mathematical passage
\label{eq:capacity_cycles-5.5} de_1\wedge\cdots\wedge de_4\wedge dP =q^4\Delta(x^q)\,dx_0\wedge\cdots\wedge dx_4.
x_0Use in source · 6 passages
Search aliases: x_0
Use in source · capacity_cycles.tex, line 88
Ω=∑_i=0^5(-1)^ix_i dx_0wedge⋯widehatdx_i⋯wedge dx_5
Exact mathematical passage
\Omega=\sum_{i=0}^5(-1)^ix_i\,dx_0\wedge\cdots\widehat{dx_i}\cdots\wedge dx_5
x_0^{210}Use in source · 1 passage
Search aliases: x_0^210
Use in source · carrier70.tex, line 152
F(y(x))=x_0^210+⋯+x_5^210.
Exact mathematical passage
F(y(x))=x_0^{210}+\cdots+x_5^{210}.
x_0^{33}Use in source · 1 passage
Search aliases: x_0^33
Use in source · capacity_cycles.tex, line 622
Q_1=x_0^33x_1^26x_2^19x_3^12x_4^5x_5^4,
Exact mathematical passage
Q_1=x_0^{33}x_1^{26}x_2^{19}x_3^{12}x_4^5x_5^4,
x_0^{a_0-1}Use in source · 1 passage
Search aliases: x_0^a_0-1
Use in source · capacity_cycles.tex, line 67
The Griffiths residue basis of the Fermat hypersurface diagonalizes the action of G_m. The monomial x_0^a_0-1⋯ x_5^a_5-1 at pole order three gives the character of middle Hodge type when ∑ a_i=3m. The entire rational orbit has type (2,2) when
Exact mathematical passage
x_0^{a_0-1}\cdots x_5^{a_5-1}
x_0^{b-1}Use in source · 1 passage
Search aliases: x_0^b-1
Use in source · capacity_cycles.tex, line 502
x_0^b-1x_1^b+q-1x_2^b+2q-1x_3^b+3q-1x_4^b+4q-1 z^5(q-b)-1
Exact mathematical passage
\label{eq:capacity_cycles-6.1} x_0^{b-1}x_1^{b+q-1}x_2^{b+2q-1}x_3^{b+3q-1}x_4^{b+4q-1} z^{5(q-b)-1}
x_0^mUse in source · 4 passages
Search aliases: x_0^m
Use in source · august_development.tex, line 655
Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finite Galois orbits of those lines. This permits exact symbolic analysis of individual addresses, conductors, local prime phases, and candidate algebraic …
Exact mathematical passage
X_m^4=\{x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0\}\subset\mathbf P^5, \qquad m\ge 3.
x_0^{m-1}Use in source · 1 passage
Search aliases: x_0^m-1
Use in source · capacity_cycles.tex, line 82
J_F_m=(x_0^m-1,…,x_5^m-1), S_m=∏_i=0^5x_i^m-2.
Exact mathematical passage
\label{eq:capacity_cycles-1.2} J_{F_m}=(x_0^{m-1},\ldots,x_5^{m-1}),\qquad S_m=\prod_{i=0}^5x_i^{m-2}.
x_0^qUse in source · 2 passages
Search aliases: x_0^q
Use in source · capacity_cycles.tex, line 356
S_q={e_1(x_0^q,…,x_4^q)=e_2(x_0^q,…,x_4^q)=0}⊂ P^4.
Exact mathematical passage
S_q=\{e_1(x_0^q,\ldots,x_4^q)=e_2(x_0^q,\ldots,x_4^q)=0\}\subset\mathbf P^4.
x_1^{26}Use in source · 1 passage
Search aliases: x_1^26
Use in source · capacity_cycles.tex, line 622
Q_1=x_0^33x_1^26x_2^19x_3^12x_4^5x_5^4,
Exact mathematical passage
Q_1=x_0^{33}x_1^{26}x_2^{19}x_3^{12}x_4^5x_5^4,
x_1^{b+q-1}Use in source · 1 passage
Search aliases: x_1^b+q-1
Use in source · capacity_cycles.tex, line 502
x_0^b-1x_1^b+q-1x_2^b+2q-1x_3^b+3q-1x_4^b+4q-1 z^5(q-b)-1
Exact mathematical passage
\label{eq:capacity_cycles-6.1} x_0^{b-1}x_1^{b+q-1}x_2^{b+2q-1}x_3^{b+3q-1}x_4^{b+4q-1} z^{5(q-b)-1}
x_1^mUse in source · 1 passage
Search aliases: x_1^m
Use in source · august_development.tex, line 655
Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finite Galois orbits of those lines. This permits exact symbolic analysis of individual addresses, conductors, local prime phases, and candidate algebraic …
Exact mathematical passage
X_m^4=\{x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0\}\subset\mathbf P^5, \qquad m\ge 3.
x_2Use in source · 2 passages
Search aliases: x_2
Use in source · capacity_cycles.tex, line 319
z=x_5, t_i=x_i^q (0≤ i≤4), P=x_0x_1x_2x_3x_4,
Exact mathematical passage
z=x_5,\quad t_i=x_i^q\ (0\le i\le4),\quad P=x_0x_1x_2x_3x_4,
X_{210}Use in source · 6 passages
Search aliases: X_210
Use in source · carrier70.tex, line 163
where U_210=X_210∩{∏ x_i≠0} and U_Y=Y∩{∏ y_i≠0}.
Exact mathematical passage
U_{210}=X_{210}\cap\{\prod x_i\ne0\}
X_{210}^4Use in source · 3 passages
Search aliases: X_210^4
Use in source · transfer210.tex, line 168
Pullback under X_210^4→ X_70^4 gives the algebraic source packet
Exact mathematical passage
X_{210}^4\to X_{70}^4
X_{210}^{52}Use in source · 1 passage
Search aliases: X_210^52
Use in source · transfer210.tex, line 280
The right-hand side is algebraic by concatenation, because the source and every auxiliary component are algebraic. The left-hand auxiliary part Luplus(-L) is a multiset of exactly 24 complementary pairs. The full character has 54 entries and is a character on X_210^52. Complementary-pair removal therefore gives algebraicity of b on X_210^4.
Exact mathematical passage
X_{210}^{52}
X_{210k}^4Use in source · 1 passage
Search aliases: X_210k^4
Use in source · transfer210.tex, line 288
For every k≥1, coordinate-power pullback gives the associated packet on X_210k^4. It does not establish the Hodge conjecture for all packets on any of these fourfolds.
Exact mathematical passage
X_{210k}^4
x_2^{19}Use in source · 1 passage
Search aliases: x_2^19
Use in source · capacity_cycles.tex, line 622
Q_1=x_0^33x_1^26x_2^19x_3^12x_4^5x_5^4,
Exact mathematical passage
Q_1=x_0^{33}x_1^{26}x_2^{19}x_3^{12}x_4^5x_5^4,
x_2^{b+2q-1}Use in source · 1 passage
Search aliases: x_2^b+2q-1
Use in source · capacity_cycles.tex, line 502
x_0^b-1x_1^b+q-1x_2^b+2q-1x_3^b+3q-1x_4^b+4q-1 z^5(q-b)-1
Exact mathematical passage
\label{eq:capacity_cycles-6.1} x_0^{b-1}x_1^{b+q-1}x_2^{b+2q-1}x_3^{b+3q-1}x_4^{b+4q-1} z^{5(q-b)-1}
x_2^mUse in source · 1 passage
Search aliases: x_2^m
Use in source · august_development.tex, line 655
Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finite Galois orbits of those lines. This permits exact symbolic analysis of individual addresses, conductors, local prime phases, and candidate algebraic …
Exact mathematical passage
X_m^4=\{x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0\}\subset\mathbf P^5, \qquad m\ge 3.
x_3Use in source · 2 passages
Search aliases: x_3
Use in source · capacity_cycles.tex, line 319
z=x_5, t_i=x_i^q (0≤ i≤4), P=x_0x_1x_2x_3x_4,
Exact mathematical passage
z=x_5,\quad t_i=x_i^q\ (0\le i\le4),\quad P=x_0x_1x_2x_3x_4,
X_{33}^4Use in source · 3 passages
Search aliases: X_33^4
Use in source · august_development.tex, line 902
13,19,22,28), or after permutation, widetilde a=(7,19,28,10,13,22). Every unit conjugate has residue sum 99=3·33, so the orbit is rational of type (2,2). It is not totally pair-decomposable. The previously proposed symmetric surface W={p_1(x^11)=p_2(x^11)=p_3(x^11)=0} does not carry the primitive packet: Newton's identity makes the third equation redundant on X_33^4, and [W]=242h^2. Thus W is purely ambient.
Exact mathematical passage
X_{33}^4
X_{33k}^4Use in source · 1 passage
Search aliases: X_33k^4
Use in source · august_development.tex, line 952
[Infinite packet tower] Let W be an algebraic rational character packet on X_d^2p. For every k≥1, its coordinate-power pullback is an algebraic packet on X_kd^2p. Applied to the preceding degree-33 construction under its geometric hypotheses, this gives a 20-dimensional algebraic packet on every X_33k^4.
Exact mathematical passage
X_{33k}^4
X_{35}Use in source · 1 passage
Search aliases: X_35
Use in source · capacity_cycles.tex, line 602
a=(1,8,15,22,29,30), Z_κ={ l ∑_i=0^4x_i^7=0, ∑_0≤ i<j≤4x_i^7x_j^7=0, x_0x_1x_2x_3x_4=κ x_5^5 }⊂ X_35.
Exact mathematical passage
a=(1,8,15,22,29,30),\quad Z_\kappa=\left\{ \begin{array}{l} \sum_{i=0}^4x_i^7=0,\\ \sum_{0\le i<j\le4}x_i^7x_j^7=0,\\ x_0x_1x_2x_3x_4=\kappa x_5^5 \end{array}\right\}\subset X_{35}.
x_3^{12}Use in source · 1 passage
Search aliases: x_3^12
Use in source · capacity_cycles.tex, line 622
Q_1=x_0^33x_1^26x_2^19x_3^12x_4^5x_5^4,
Exact mathematical passage
Q_1=x_0^{33}x_1^{26}x_2^{19}x_3^{12}x_4^5x_5^4,
x_3^{b+3q-1}Use in source · 1 passage
Search aliases: x_3^b+3q-1
Use in source · capacity_cycles.tex, line 502
x_0^b-1x_1^b+q-1x_2^b+2q-1x_3^b+3q-1x_4^b+4q-1 z^5(q-b)-1
Exact mathematical passage
\label{eq:capacity_cycles-6.1} x_0^{b-1}x_1^{b+q-1}x_2^{b+2q-1}x_3^{b+3q-1}x_4^{b+4q-1} z^{5(q-b)-1}
x_3^mUse in source · 1 passage
Search aliases: x_3^m
Use in source · august_development.tex, line 655
Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finite Galois orbits of those lines. This permits exact symbolic analysis of individual addresses, conductors, local prime phases, and candidate algebraic …
Exact mathematical passage
X_m^4=\{x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0\}\subset\mathbf P^5, \qquad m\ge 3.
x_4Use in source · 5 passages
Search aliases: x_4
Use in source · capacity_cycles.tex, line 319
z=x_5, t_i=x_i^q (0≤ i≤4), P=x_0x_1x_2x_3x_4,
Exact mathematical passage
z=x_5,\quad t_i=x_i^q\ (0\le i\le4),\quad P=x_0x_1x_2x_3x_4,
x_4^5Use in source · 1 passage
Search aliases: x_4^5
Use in source · capacity_cycles.tex, line 622
Q_1=x_0^33x_1^26x_2^19x_3^12x_4^5x_5^4,
Exact mathematical passage
Q_1=x_0^{33}x_1^{26}x_2^{19}x_3^{12}x_4^5x_5^4,
x_4^{b+4q-1}Use in source · 1 passage
Search aliases: x_4^b+4q-1
Use in source · capacity_cycles.tex, line 502
x_0^b-1x_1^b+q-1x_2^b+2q-1x_3^b+3q-1x_4^b+4q-1 z^5(q-b)-1
Exact mathematical passage
\label{eq:capacity_cycles-6.1} x_0^{b-1}x_1^{b+q-1}x_2^{b+2q-1}x_3^{b+3q-1}x_4^{b+4q-1} z^{5(q-b)-1}
x_4^mUse in source · 1 passage
Search aliases: x_4^m
Use in source · august_development.tex, line 655
Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finite Galois orbits of those lines. This permits exact symbolic analysis of individual addresses, conductors, local prime phases, and candidate algebraic …
Exact mathematical passage
X_m^4=\{x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0\}\subset\mathbf P^5, \qquad m\ge 3.
x_4^qUse in source · 2 passages
Search aliases: x_4^q
Use in source · capacity_cycles.tex, line 356
S_q={e_1(x_0^q,…,x_4^q)=e_2(x_0^q,…,x_4^q)=0}⊂ P^4.
Exact mathematical passage
S_q=\{e_1(x_0^q,\ldots,x_4^q)=e_2(x_0^q,\ldots,x_4^q)=0\}\subset\mathbf P^4.
x_5Use in source · 2 passages
Search aliases: x_5
Use in source · capacity_cycles.tex, line 88
Ω=∑_i=0^5(-1)^ix_i dx_0wedge⋯widehatdx_i⋯wedge dx_5
Exact mathematical passage
\Omega=\sum_{i=0}^5(-1)^ix_i\,dx_0\wedge\cdots\widehat{dx_i}\cdots\wedge dx_5
x_5^{210}Use in source · 1 passage
Search aliases: x_5^210
Use in source · carrier70.tex, line 152
F(y(x))=x_0^210+⋯+x_5^210.
Exact mathematical passage
F(y(x))=x_0^{210}+\cdots+x_5^{210}.
x_5^4Use in source · 1 passage
Search aliases: x_5^4
Use in source · capacity_cycles.tex, line 622
Q_1=x_0^33x_1^26x_2^19x_3^12x_4^5x_5^4,
Exact mathematical passage
Q_1=x_0^{33}x_1^{26}x_2^{19}x_3^{12}x_4^5x_5^4,
x_5^5Use in source · 1 passage
Search aliases: x_5^5
Use in source · capacity_cycles.tex, line 602
a=(1,8,15,22,29,30), Z_κ={ l ∑_i=0^4x_i^7=0, ∑_0≤ i<j≤4x_i^7x_j^7=0, x_0x_1x_2x_3x_4=κ x_5^5 }⊂ X_35.
Exact mathematical passage
a=(1,8,15,22,29,30),\quad Z_\kappa=\left\{ \begin{array}{l} \sum_{i=0}^4x_i^7=0,\\ \sum_{0\le i<j\le4}x_i^7x_j^7=0,\\ x_0x_1x_2x_3x_4=\kappa x_5^5 \end{array}\right\}\subset X_{35}.
x_5^{a_5-1}Use in source · 1 passage
Search aliases: x_5^a_5-1
Use in source · capacity_cycles.tex, line 67
The Griffiths residue basis of the Fermat hypersurface diagonalizes the action of G_m. The monomial x_0^a_0-1⋯ x_5^a_5-1 at pole order three gives the character of middle Hodge type when ∑ a_i=3m. The entire rational orbit has type (2,2) when
Exact mathematical passage
x_0^{a_0-1}\cdots x_5^{a_5-1}
x_5^mUse in source · 3 passages
Search aliases: x_5^m
Use in source · august_development.tex, line 655
Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finite Galois orbits of those lines. This permits exact symbolic analysis of individual addresses, conductors, local prime phases, and candidate algebraic …
Exact mathematical passage
X_m^4=\{x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0\}\subset\mathbf P^5, \qquad m\ge 3.
x_5^{m-1}Use in source · 1 passage
Search aliases: x_5^m-1
Use in source · capacity_cycles.tex, line 82
J_F_m=(x_0^m-1,…,x_5^m-1), S_m=∏_i=0^5x_i^m-2.
Exact mathematical passage
\label{eq:capacity_cycles-1.2} J_{F_m}=(x_0^{m-1},\ldots,x_5^{m-1}),\qquad S_m=\prod_{i=0}^5x_i^{m-2}.
X_{5q}Use in source · 2 passages
Search aliases: X_5q
Use in source · capacity_cycles.tex, line 347
Thus Z_κ⊂ X_5q. Its defining equations have degrees q,2q,5.
Exact mathematical passage
Z_\kappa\subset X_{5q}
X_{70}Use in source · 4 passages
Search aliases: X_70
Use in source · carrier70.tex, line 38
all classes in the 24-dimensional rational space W_[a] are rational linear combinations of classes of algebraic surfaces on X_70. For every positive integer k, the corresponding pulled-back packet W_[ka] on X_70k is also algebraic.
Exact mathematical passage
X_{70}
X_{70}^4Use in source · 1 passage
Search aliases: X_70^4
Use in source · transfer210.tex, line 168
Pullback under X_210^4→ X_70^4 gives the algebraic source packet
Exact mathematical passage
X_{210}^4\to X_{70}^4
X_{70k}Use in source · 2 passages
Search aliases: X_70k
Use in source · carrier70.tex, line 39
all classes in the 24-dimensional rational space W_[a] are rational linear combinations of classes of algebraic surfaces on X_70. For every positive integer k, the corresponding pulled-back packet W_[ka] on X_70k is also algebraic.
Exact mathematical passage
X_{70k}
X_d^{2p}Use in source · 1 passage
Search aliases: X_d^2p
Use in source · august_development.tex, line 952
[Infinite packet tower] Let W be an algebraic rational character packet on X_d^2p. For every k≥1, its coordinate-power pullback is an algebraic packet on X_kd^2p. Applied to the preceding degree-33 construction under its geometric hypotheses, this gives a 20-dimensional algebraic packet on every X_33k^4.
Exact mathematical passage
X_d^{2p}
x_i^{2q}Use in source · 1 passage
Search aliases: x_i^2q
Use in source · capacity_cycles.tex, line 364
∑ x_i^q=0, ∑ x_i^2q=0.
Exact mathematical passage
\sum x_i^q=0,\qquad\sum x_i^{2q}=0.
x_i^3Use in source · 1 passage
Search aliases: x_i^3
Use in source · carrier70.tex, line 430
π:X_210⟶ X_70, [x_i]⟼[x_i^3].
Exact mathematical passage
\pi:X_{210}\longrightarrow X_{70},\qquad [x_i]\longmapsto[x_i^3].
x_i^7Use in source · 1 passage
Search aliases: x_i^7
Use in source · capacity_cycles.tex, line 602
a=(1,8,15,22,29,30), Z_κ={ l ∑_i=0^4x_i^7=0, ∑_0≤ i<j≤4x_i^7x_j^7=0, x_0x_1x_2x_3x_4=κ x_5^5 }⊂ X_35.
Exact mathematical passage
a=(1,8,15,22,29,30),\quad Z_\kappa=\left\{ \begin{array}{l} \sum_{i=0}^4x_i^7=0,\\ \sum_{0\le i<j\le4}x_i^7x_j^7=0,\\ x_0x_1x_2x_3x_4=\kappa x_5^5 \end{array}\right\}\subset X_{35}.
x_i^{m-2}Use in source · 1 passage
Search aliases: x_i^m-2
Use in source · capacity_cycles.tex, line 82
J_F_m=(x_0^m-1,…,x_5^m-1), S_m=∏_i=0^5x_i^m-2.
Exact mathematical passage
\label{eq:capacity_cycles-1.2} J_{F_m}=(x_0^{m-1},\ldots,x_5^{m-1}),\qquad S_m=\prod_{i=0}^5x_i^{m-2}.
x_i^{m-a_{q,b,i}-1}Use in source · 1 passage
Search aliases: x_i^m-a_q,b,i-1
Use in source · capacity_cycles.tex, line 519
Q_b=∏_i=0^5x_i^m-a_q,b,i-1.
Exact mathematical passage
\label{eq:capacity_cycles-6.3} Q_b=\prod_{i=0}^5x_i^{m-a_{q,b,i}-1}.
x_i^qUse in source · 4 passages
Search aliases: x_i^q
Use in source · capacity_cycles.tex, line 319
z=x_5, t_i=x_i^q (0≤ i≤4), P=x_0x_1x_2x_3x_4,
Exact mathematical passage
z=x_5,\quad t_i=x_i^q\ (0\le i\le4),\quad P=x_0x_1x_2x_3x_4,
x_j^7Use in source · 1 passage
Search aliases: x_j^7
Use in source · capacity_cycles.tex, line 602
a=(1,8,15,22,29,30), Z_κ={ l ∑_i=0^4x_i^7=0, ∑_0≤ i<j≤4x_i^7x_j^7=0, x_0x_1x_2x_3x_4=κ x_5^5 }⊂ X_35.
Exact mathematical passage
a=(1,8,15,22,29,30),\quad Z_\kappa=\left\{ \begin{array}{l} \sum_{i=0}^4x_i^7=0,\\ \sum_{0\le i<j\le4}x_i^7x_j^7=0,\\ x_0x_1x_2x_3x_4=\kappa x_5^5 \end{array}\right\}\subset X_{35}.
x_j^{B_{ij}}Use in source · 1 passage
Search aliases: x_j^B_ij
Use in source · carrier70.tex, line 148
On the projective tori define y_i=∏_jx_j^B_ij. Negative exponents are allowed there. The second identity makes this well-defined into the weighted projective torus, the first gives
Exact mathematical passage
y_i=\prod_jx_j^{B_{ij}}
X_{kd}^{2p}Use in source · 1 passage
Search aliases: X_kd^2p
Use in source · august_development.tex, line 952
[Infinite packet tower] Let W be an algebraic rational character packet on X_d^2p. For every k≥1, its coordinate-power pullback is an algebraic packet on X_kd^2p. Applied to the preceding degree-33 construction under its geometric hypotheses, this gives a 20-dimensional algebraic packet on every X_33k^4.
Exact mathematical passage
X_{kd}^{2p}
X_{km}^nUse in source · 1 passage
Search aliases: X_km^n
Use in source · transfer210.tex, line 127
Power transport. The finite coordinate-power morphism X_km^n→ X_m^n transports algebraic cycle classes by pullback. The character is multiplied by k.
Exact mathematical passage
X_{km}^n\to X_m^n
X_m^nUse in source · 2 passages
Search aliases: X_m^n
Use in source · transfer210.tex, line 20
X_m^n={x_0^m+⋯+x_n+1^m=0}⊂ P^n+1_ C.
Exact mathematical passage
X_m^n=\{x_0^m+\cdots+x_{n+1}^m=0\}\subset\mathbf P^{n+1}_{\mathbf C}.
x_{n+1}^mUse in source · 1 passage
Search aliases: x_n+1^m
Use in source · transfer210.tex, line 20
X_m^n={x_0^m+⋯+x_n+1^m=0}⊂ P^n+1_ C.
Exact mathematical passage
X_m^n=\{x_0^m+\cdots+x_{n+1}^m=0\}\subset\mathbf P^{n+1}_{\mathbf C}.
\xrightarrowUse in source · 5 passages
Search aliases: xrightarrow
Use in source · august_development.tex, line 12
The rational Hodge target. Statement and status. Let X be a smooth projective complex variety. For each codimension p, define Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X) and Alg^p(X)=im(CH^p(X)_QxrightarrowclH^2p(X,Q)). Every algebraic cycle class has type (p,p), so Alg^p(X)⊆Hdg^p(X). The rational Hodge conjecture asserts equality for every smooth projective X and every p [source reference].
Exact mathematical passage
\Alg^p(X)=\im\bigl(\CH^p(X)_{\Q}\xrightarrow{\cl}H^{2p}(X,\Q)\bigr).
YUse in source · 31 passages
Search aliases: Y
Use in source · appendices.tex, line 25
Correspondence degrees. Let X have dimension n and Y dimension m. A correspondence Γ∈CH^r(X× Y) acts by Γ_*:H^i(X)⟶ H^i+2(r-n)(Y) and shifts Hodge type by (r-n,r-n). To lower H^2p(X) to H^2(Y), take r=n-p+1. To return degree two to degree 2p on X, a correspondence on Y× X has codimension m+p-1.
Exact mathematical passage
Y
\mathcal YUse in source · 1 passage
Search aliases: Y mathcal
Use in source · august_development.tex, line 340
Family propagation. Let π: X→ S and ρ_r: Y_r→ S be smooth projective families over a connected base. Relative algebraic correspondences induce morphisms of Gauss-Manin local systems. In a flat rational trivialization, their transfer matrix is constant.
Exact mathematical passage
\rho_r:\mathcal Y_r\to S
y^2Use in source · 1 passage
Search aliases: y^2
Use in source · weil_audit.tex, line 158
Modulo 3, the equation x^2+y^2=0 has only the solution (0,0). This is a finite anisotropy control for the residual discriminant. It does not construct a semistable algebraic family with this nilpotent as monodromy. An abstract rational nilpotent satisfying Lie-algebra identities is weaker than a geometric degeneration with the required integral, polarization and admissibility data.
Exact mathematical passage
x^2+y^2=0
y_0Use in source · 7 passages
Search aliases: y_0
Use in source · august_development.tex, line 910
Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.
Exact mathematical passage
Y:\quad y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0.
y_0^2Use in source · 1 passage
Search aliases: y_0^2
Use in source · august_development.tex, line 910
Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.
Exact mathematical passage
Y:\quad y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0.
y_0^3Use in source · 2 passages
Search aliases: y_0^3
Use in source · carrier70.tex, line 107
F=y_0^3+y_1^5+y_2^2y_5^3+y_2y_3^2y_4^7 +y_0y_2^2y_3^2y_5+y_3^2y_5^3.
Exact mathematical passage
F=y_0^3+y_1^5+y_2^2y_5^3+y_2y_3^2y_4^7 +y_0y_2^2y_3^2y_5+y_3^2y_5^3.
y_1^2Use in source · 1 passage
Search aliases: y_1^2
Use in source · august_development.tex, line 910
Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.
Exact mathematical passage
Y:\quad y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0.
y_1^4Use in source · 1 passage
Search aliases: y_1^4
Use in source · carrier70.tex, line 228
y_0^3, y_1^4, (y_2y_5)^3, (y_3y_4)^13, (y_3y_5)^3
Exact mathematical passage
y_0^3,\quad y_1^4,\quad (y_2y_5)^3,\quad (y_3y_4)^{13},\quad (y_3y_5)^3
y_1^5Use in source · 2 passages
Search aliases: y_1^5
Use in source · carrier70.tex, line 107
F=y_0^3+y_1^5+y_2^2y_5^3+y_2y_3^2y_4^7 +y_0y_2^2y_3^2y_5+y_3^2y_5^3.
Exact mathematical passage
F=y_0^3+y_1^5+y_2^2y_5^3+y_2y_3^2y_4^7 +y_0y_2^2y_3^2y_5+y_3^2y_5^3.
y_2^2Use in source · 2 passages
Search aliases: y_2^2
Use in source · august_development.tex, line 910
Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.
Exact mathematical passage
Y:\quad y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0.
y_3Use in source · 5 passages
Search aliases: y_3
Use in source · august_development.tex, line 910
Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.
Exact mathematical passage
Y:\quad y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0.
y_3^2Use in source · 3 passages
Search aliases: y_3^2
Use in source · august_development.tex, line 910
Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.
Exact mathematical passage
Y:\quad y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0.
y_4Use in source · 5 passages
Search aliases: y_4
Use in source · august_development.tex, line 910
Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.
Exact mathematical passage
Y:\quad y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0.
y_4^2Use in source · 1 passage
Search aliases: y_4^2
Use in source · august_development.tex, line 910
Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.
Exact mathematical passage
Y:\quad y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0.
y_4^7Use in source · 2 passages
Search aliases: y_4^7
Use in source · carrier70.tex, line 107
F=y_0^3+y_1^5+y_2^2y_5^3+y_2y_3^2y_4^7 +y_0y_2^2y_3^2y_5+y_3^2y_5^3.
Exact mathematical passage
F=y_0^3+y_1^5+y_2^2y_5^3+y_2y_3^2y_4^7 +y_0y_2^2y_3^2y_5+y_3^2y_5^3.
y_5^3Use in source · 2 passages
Search aliases: y_5^3
Use in source · august_development.tex, line 910
Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.
Exact mathematical passage
Y:\quad y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0.
y_bUse in source · 3 passages
Search aliases: y_b
Use in source · august_development.tex, line 1185
The trivalent defect tree. The integral basis turns defect mass into exact tree variation. Define a sign gauge by τ_r_0=1, τ_b=ϵ_bτ_p(b), y_b=τ_bz_b. Attach one zero-valued boundary leaf to the root and two zero-valued boundary leaves to every odd vertex. Every vertex of the resulting augmented tree widetilde T_m has degree three. The boundary edges correspond bijectively to defect coordinates: the root edge is labelled q, the edge from b to p(b) is labelled 2b, and the two leaves at an odd b ar…
Exact mathematical passage
\tau_{r_0}=1, \qquad \tau_b=\epsilon_b\tau_{p(b)}, \qquad y_b=\tau_bz_b.
y_iUse in source · 4 passages
Search aliases: y_i
Use in source · capacity_cycles.tex, line 143
The infinitely many inequalities can be certified by a finite residue graph. Write y_i=c_i/L, where c_i,L are positive integers. On Z/m Z, an edge r→ r+a_i has cost c_i. An integer function d satisfying
Exact mathematical passage
y_i=c_i/L
y_{p(b)}Use in source · 2 passages
Search aliases: y_p(b)
Use in source · august_development.tex, line 1196
[Exact trivalent-tree norm] If d=∑_bz_bκ_b∈ K_m, then d _1 =|y_r_0| +∑_b≠ r_0|y_b-y_p(b)| +2∑_substack1≤ b<q b odd|y_b|. The right side is the total variation of the integer function y on widetilde T_m with zero boundary values. The root coordinate is d(q)=z_r_0. For b≠ r_0, multiplying d(2b)=z_b-ϵ_bz_p(b) by τ_b gives τ_bd(2b)=y_b-y_p(b). For odd b, the two odd coordinates are d(b)=-z_b, d(h-b)=z_b. Taking absolute values and summing all defect coordinates gives [source reference].
Exact mathematical passage
\label{eq:legacy-13.2} \boxed{ \|d\|_1 =|y_{r_0}| +\sum_{b\ne r_0}|y_b-y_{p(b)}| +2\sum_{\substack{1\le b<q\\ b\text{ odd}}}|y_b|. }
y_qUse in source · 2 passages
Search aliases: y_q
Use in source · capacity_cycles.tex, line 265
y_q=frac1q(1,1,1,1,1,q-5), y_q·1=1.
Exact mathematical passage
y_q=\frac1q(1,1,1,1,1,q-5),\qquad y_q\cdot\mathbf1=1.
Y_rUse in source · 6 passages
Search aliases: Y_r
Use in source · august_development.tex, line 273
Divisor-lowering correspondence channels. Degree bookkeeping. Let X have complex dimension n, let V⊆ H_prim^2p(X,Q) be a rational Hodge substructure, and let Y_r be smooth projective. Choose correspondences Γ_r∈CH^n-p+1(X× Y_r)_Q, Λ_r∈CH^dim Y_r+p-1(Y_r× X)_Q. They induce A_r=(Γ_r)_*:V→ H^2(Y_r,Q), B_r=(Λ_r)_*:H^2(Y_r,Q)→ H^2p(X,Q). Algebraic correspondences preserve rationality and shift Hodge type by the required bidegree.
Exact mathematical passage
Y_r
\mathcal Y_rUse in source · 1 passage
Search aliases: Y_r mathcal
Use in source · august_development.tex, line 340
Family propagation. Let π: X→ S and ρ_r: Y_r→ S be smooth projective families over a connected base. Relative algebraic correspondences induce morphisms of Gauss-Manin local systems. In a flat rational trivialization, their transfer matrix is constant.
Exact mathematical passage
\rho_r:\mathcal Y_r\to S
y_{r_0}Use in source · 1 passage
Search aliases: y_r_0
Use in source · august_development.tex, line 1196
[Exact trivalent-tree norm] If d=∑_bz_bκ_b∈ K_m, then d _1 =|y_r_0| +∑_b≠ r_0|y_b-y_p(b)| +2∑_substack1≤ b<q b odd|y_b|. The right side is the total variation of the integer function y on widetilde T_m with zero boundary values. The root coordinate is d(q)=z_r_0. For b≠ r_0, multiplying d(2b)=z_b-ϵ_bz_p(b) by τ_b gives τ_bd(2b)=y_b-y_p(b). For odd b, the two odd coordinates are d(b)=-z_b, d(h-b)=z_b. Taking absolute values and summing all defect coordinates gives [source reference].
Exact mathematical passage
\label{eq:legacy-13.2} \boxed{ \|d\|_1 =|y_{r_0}| +\sum_{b\ne r_0}|y_b-y_{p(b)}| +2\sum_{\substack{1\le b<q\\ b\text{ odd}}}|y_b|. }
\ZUse in source · 2 passages
Search aliases: Z
Use in source · weil_audit.tex, line 7
The lattice, polarization and seed. Let K=Q(i) and take homology lattice Z[i]^6. Use the conjugate-linear-first Hermitian form h(v,w)=overline v^ tdiag(1,1,1,-1,-1,-3)w, E=Im h. The seed complex structure is J_0=diag(i,i,i,-i,-i,-i). It preserves the alternating form E, and E(v,J_0v)>0 for nonzero v. These are the Riemann positivity conditions for the seed abelian variety. The sign depends on the declared Hermitian convention. Omitting that convention would leave the polarization claim ambiguous…
Exact mathematical passage
\Z[i]^6
zUse in source · 36 passages
Search aliases: z
Use in source · appendices.tex, line 19
Five-phase surface · e_1=e_2=P-κ z^5=0, with κ^q=-1/5.
Exact mathematical passage
e_1=e_2=P-\kappa z^5=0
z^2Use in source · 4 passages
Search aliases: z^2
Use in source · carrier70.tex, line 187
F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.
Exact mathematical passage
\begin{aligned} F_u&=3u^2+s^2t^2z, &F_v&=5v^4,\\ F_s&=2sz^3+t^2c^7+2ust^2z, &F_t&=2t(z^3+sc^7+us^2z),\\ F_c&=7st^2c^6, &F_z&=3z^2(s^2+t^2)+us^2t^2. \end{aligned}
z^3Use in source · 2 passages
Search aliases: z^3
Use in source · carrier70.tex, line 181
F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.
Exact mathematical passage
F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.
z^5Use in source · 4 passages
Search aliases: z^5
Use in source · appendices.tex, line 19
Five-phase surface · e_1=e_2=P-κ z^5=0, with κ^q=-1/5.
Exact mathematical passage
e_1=e_2=P-\kappa z^5=0
z^{5(q-b)-1}Use in source · 2 passages
Search aliases: z^5(q-b)-1
Use in source · capacity_cycles.tex, line 448
D_κ=625q^5Δ(x_0^q,…,x_4^q) ∑_b=1^q-1κ^q-bP^b-1z^5(q-b)-1.
Exact mathematical passage
\label{eq:capacity_cycles-5.2} \boxed{ D_\kappa=625q^5\Delta(x_0^q,\ldots,x_4^q) \sum_{b=1}^{q-1}\kappa^{q-b}P^{b-1}z^{5(q-b)-1}. }
z^{5j}Use in source · 1 passage
Search aliases: z^5j
Use in source · capacity_cycles.tex, line 418
f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.
Exact mathematical passage
\begin{aligned} f_1&=e_1,&g_1&=e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4,\\ f_2&=e_2,&g_2&=-5e_3,\\ f_3&=P-\kappa z^5,&g_3&=5\sum_{j=0}^{q-1}P^{q-1-j}\kappa^jz^{5j}. \end{aligned}
z^{5j-1}Use in source · 1 passage
Search aliases: z^5j-1
Use in source · capacity_cycles.tex, line 465
df_3wedge dg_3= 25q∑_j=1^q-1κ^jP^q-1-jz^5j-1 dPwedge dz.
Exact mathematical passage
\label{eq:capacity_cycles-5.4} df_3\wedge dg_3= 25q\sum_{j=1}^{q-1}\kappa^jP^{q-1-j}z^{5j-1}\,dP\wedge dz.
z^{5q}Use in source · 1 passage
Search aliases: z^5q
Use in source · capacity_cycles.tex, line 342
F_m=p_5+z^5q=5P^q+z^5q =(5κ^q+1)z^5q=0.
Exact mathematical passage
F_m=p_5+z^{5q}=5P^q+z^{5q} =(5\kappa^q+1)z^{5q}=0.
\mathbf Z^6Use in source · 3 passages
Search aliases: Z^6 mathbf
Use in source · capacity_cycles.tex, line 113
Every nonsingular nonnegative integral matrix A satisfying A^Ta∈ m Z^6, with w=A^-11>0, gives one feasible point. Hence
Exact mathematical passage
A^Ta\in m\mathbf Z^6
\mathbf Z^{h-1}Use in source · 1 passage
Search aliases: Z^h-1 mathbf
Use in source · august_development.tex, line 1122
The integral triangle basis. Let K_m= { d∈ Z^h-1: ∑_r=1^h-1d(r)ρ_m(ur)=0 for every odd u }. The triangle relations do not merely generate examples, they form the whole balanced defect lattice.
Exact mathematical passage
\mathcal K_m= \left\{ d\in\mathbf Z^{h-1}: \sum_{r=1}^{h-1}d(r)\rho_m(ur)=0 \text{ for every odd }u \right\}.
Z_1Use in source · 1 passage
Search aliases: Z_1
Use in source · august_development.tex, line 257
Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.
Exact mathematical passage
Z_1,\ldots,Z_r
z_1Use in source · 1 passage
Search aliases: z_1
Use in source · weil_audit.tex, line 87
The seed graph supplies both rational Weil classes. At the seed, take the graph of (z_1,z_2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety graph and is algebraic at J_0. On its three source factors the polarization restricts to weights (2,2,4), giving ∫_Zθ^3=3!·2·2·4=96. Pulling the explic…
Exact mathematical passage
(z_1,z_2,z_3)\longmapsto (z_1,z_2,z_3,\overline z_1,\overline z_2,\overline z_3).
\mathbf Z_2Use in source · 1 passage
Search aliases: Z_2 mathbf
Use in source · august_development.tex, line 1024
Weighted dyadic reconstruction. Prime 2 cannot be handled by inserting p=2 into the odd-prime orbit-mass proof. The unit group has the separate form Z_2^×≅{±1}×(1+4 Z_2). The first address is a sign packet, only afterward does the principal-unit refinement begin.
Exact mathematical passage
\mathbf Z_2^\times\cong\{\pm1\}\times(1+4\mathbf Z_2).
z_2Use in source · 1 passage
Search aliases: z_2
Use in source · weil_audit.tex, line 87
The seed graph supplies both rational Weil classes. At the seed, take the graph of (z_1,z_2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety graph and is algebraic at J_0. On its three source factors the polarization restricts to weights (2,2,4), giving ∫_Zθ^3=3!·2·2·4=96. Pulling the explic…
Exact mathematical passage
(z_1,z_2,z_3)\longmapsto (z_1,z_2,z_3,\overline z_1,\overline z_2,\overline z_3).
\mathbf Z_2^\timesUse in source · 1 passage
Search aliases: Z_2^× mathbf times
Use in source · august_development.tex, line 1024
Weighted dyadic reconstruction. Prime 2 cannot be handled by inserting p=2 into the odd-prime orbit-mass proof. The unit group has the separate form Z_2^×≅{±1}×(1+4 Z_2). The first address is a sign packet, only afterward does the principal-unit refinement begin.
Exact mathematical passage
\mathbf Z_2^\times\cong\{\pm1\}\times(1+4\mathbf Z_2).
z_3Use in source · 1 passage
Search aliases: z_3
Use in source · weil_audit.tex, line 87
The seed graph supplies both rational Weil classes. At the seed, take the graph of (z_1,z_2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety graph and is algebraic at J_0. On its three source factors the polarization restricts to weights (2,2,4), giving ∫_Zθ^3=3!·2·2·4=96. Pulling the explic…
Exact mathematical passage
(z_1,z_2,z_3)\longmapsto (z_1,z_2,z_3,\overline z_1,\overline z_2,\overline z_3).
z_bUse in source · 9 passages
Search aliases: z_b
Use in source · august_development.tex, line 1160
Suppose d=∑_b=1^q-1z_bκ_b. Its even coordinates are d(q)=z_r_0, d(2b)=z_b-ϵ_bz_p(b) (b≠ r_0). Starting at the root and moving toward the leaves, these equations recover every z_b integrally. Hence the triangle vectors are linearly independent and their even-coordinate map is unimodular.
Exact mathematical passage
d=\sum_{b=1}^{q-1}z_b\kappa_b
Z_iUse in source · 2 passages
Search aliases: Z_i
Use in source · august_development.tex, line 258
Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.
Exact mathematical passage
\alpha=\sum_{i=1}^r q_i[Z_i].
Z_jUse in source · 1 passage
Search aliases: Z_j
Use in source · august_development.tex, line 177
Degree-residue polyphase refinement. Exact parent-child frame law. Fix a projective embedding and assign each cycle its degree. For modulus M, define the residue packet I_M,r={j:deg Z_j≡ rpmod M}, S_M,r=∑_j∈ I_M,r|c_j⟩⟨ c_j|.
Exact mathematical passage
I_{M,r}=\{j:\deg Z_j\equiv r\pmod M\}, \qquad S_{M,r}=\sum_{j\in I_{M,r}}|c_j\rangle\langle c_j|.
z_{p(b)}Use in source · 2 passages
Search aliases: z_p(b)
Use in source · august_development.tex, line 1161
Suppose d=∑_b=1^q-1z_bκ_b. Its even coordinates are d(q)=z_r_0, d(2b)=z_b-ϵ_bz_p(b) (b≠ r_0). Starting at the root and moving toward the leaves, these equations recover every z_b integrally. Hence the triangle vectors are linearly independent and their even-coordinate map is unimodular.
Exact mathematical passage
d(q)=z_{r_0}, \qquad d(2b)=z_b-\epsilon_bz_{p(b)} \quad(b\ne r_0).
Z_rUse in source · 1 passage
Search aliases: Z_r
Use in source · august_development.tex, line 257
Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.
Exact mathematical passage
Z_1,\ldots,Z_r
z_{r_0}Use in source · 2 passages
Search aliases: z_r_0
Use in source · august_development.tex, line 1161
Suppose d=∑_b=1^q-1z_bκ_b. Its even coordinates are d(q)=z_r_0, d(2b)=z_b-ϵ_bz_p(b) (b≠ r_0). Starting at the root and moving toward the leaves, these equations recover every z_b integrally. Hence the triangle vectors are linearly independent and their even-coordinate map is unimodular.
Exact mathematical passage
d(q)=z_{r_0}, \qquad d(2b)=z_b-\epsilon_bz_{p(b)} \quad(b\ne r_0).
Z_WUse in source · 1 passage
Search aliases: Z_W
Use in source · august_development.tex, line 679
The constructive objective is [Packet-resolved construction target] For every rational packet W⊂ H^4_prim(X_m^4,Q)∩ H^2,2, construct Z_W∈CH^2(X_m^4)_Q with [Z_W]≠0 in W. Once one nonzero algebraic vector is produced in a rational irreducible diagonal-symmetry packet, the entire rational orbit is algebraic.
Exact mathematical passage
Z_W\in\CH^2(X_m^4)_\Q \qquad\text{with}\qquad [Z_W]\ne0\text{ in }W.
\mathbf Z_{\ge0}^6Use in source · 2 passages
Search aliases: Z_≥0^6 mathbf ge
Use in source · capacity_cycles.tex, line 101
L_a={v∈ Z_≥0^6∖{0}:a· v≡0pmod m}.
Exact mathematical passage
\mathcal L_a=\{v\in\mathbf Z_{\ge0}^6\setminus\{0\}:a\cdot v\equiv0\pmod m\}.
|Use in source · 20 passages
Search aliases: |
Use in source · august_development.tex, line 108
tudy. For the positive frame certificate below, require every c_j to belong to V. This can be achieved by a rational algebraic packet projector when such a projector has been constructed. Projection through an arbitrary orthogonal Hodge projector does not establish that the projected vectors are algebraic. Define C:R^N→ H^2p(X,R), Ce_j=c_j, and the positive frame operator S=CC^*=∑_j=1^N|c_j⟩⟨ c_j|.
Exact mathematical passage
S=CC^*=\sum_{j=1}^N|c_j\rangle\langle c_j|.
|^2Use in source · 1 passage
Search aliases: |^2
Use in source · august_development.tex, line 123
dots,c_N}. Then im S=A, ker S=A^⊥. Under the stated containment c_j∈ V, the selected cycles span V if and only if S|_V is positive definite. Without containment, positivity only detects the absence of vectors in V orthogonal to the cycle span and does not prove that span contains V. The equality im(CC^*)=im C is standard. Moreover ⟨ v,Sv⟩= C^*v ^2=∑_j|⟨ c_j,v⟩|^2. This vanishes exactly when v is orthogonal to every cycle class.
Exact mathematical passage
\langle v,Sv\rangle=\|C^*v\|^2=\sum_j|\langle c_j,v\rangle|^2.
|_KUse in source · 3 passages
Search aliases: |_K
Use in source · carrier70.tex, line 335
Let K be the kernel of the torus isogeny. The relation AB=210I shows that K embeds in G_210: an element in the kernel has representatives whose 210th powers are all the same, and a common scalar makes them roots of unity. The displayed character identity proves χ_α|_K=1. The same holds for every rational conjugate. Therefore the whole rational packet is K-invariant.
Exact mathematical passage
\chi_\alpha|_K=1
|_VUse in source · 2 passages
Search aliases: |_V
Use in source · august_development.tex, line 119
[Finite positive cycle-frame certificate] Let A=Span_R{c_1,…,c_N}. Then im S=A, ker S=A^⊥. Under the stated containment c_j∈ V, the selected cycles span V if and only if S|_V is positive definite. Without containment, positivity only detects the absence of vectors in V orthogonal to the cycle span and does not prove that span contains V. The equality im(CC^*)=im C is standard. Moreover ⟨ v,Sv⟩= C^*v ^2=∑_j|⟨ c_j,v⟩|^2. This vanishes exactly when v is orthogonal to every cycle class.
Exact mathematical passage
S|_V
\pmUse in source · 9 passages
Search aliases: ± pm
Use in source · august_development.tex, line 1024
Weighted dyadic reconstruction. Prime 2 cannot be handled by inserting p=2 into the odd-prime orbit-mass proof. The unit group has the separate form Z_2^×≅{±1}×(1+4 Z_2). The first address is a sign packet, only afterward does the principal-unit refinement begin.
Exact mathematical passage
\mathbf Z_2^\times\cong\{\pm1\}\times(1+4\mathbf Z_2).
\cdotUse in source · 25 passages
Search aliases: · cdot
Use in source · appendices.tex, line 18
Global dual · y· v≥1 for every invariant column implies C(a)≤ y·1.
Exact mathematical passage
y\cdot v\ge1
\alphaUse in source · 37 passages
Search aliases: α alpha
Use in source · appendices.tex, line 12
Support test · j^*α=0. Purity turns it into cycle algebraicity when the support has codimension at least p.
Exact mathematical passage
j^*\alpha=0
\alpha_{5^r}Use in source · 1 passage
Search aliases: α_5^r alpha
Use in source · august_development.tex, line 1007
The five-primary branch. For p=5 and r≥2, a nonzero odd defect consumes exactly five unit entries. The base case r=1 was already handled by the base-prime Fourier argument. The sixth entry is forced by balance. Up to the unit action there is one full-conductor exceptional orbit α_5^r= (1,1+q,1+2q,1+3q,1+4q,5q-5), q=5^r-1. All other packets lie in the pair/surface grammar. The family [source reference] is the pure five-primary Hasse--Davenport branch.
Exact mathematical passage
\label{eq:legacy-12.2} \boxed{ \alpha_{5^r}= (1,1+q,1+2q,1+3q,1+4q,5q-5), \qquad q=5^{r-1}. }
\alpha_{\mathrm{prim}}Use in source · 2 passages
Search aliases: α_prim alpha mathrm
Use in source · august_development.tex, line 65
For a fourfold and p=2, α=Lβ+α_prim, β∈ H^1,1(X)∩ H^2(X,Q). The Lefschetz (1,1) theorem makes β algebraic. Only α_prim remains.
Exact mathematical passage
\alpha=L\beta+\alpha_{\mathrm{prim}}, \qquad \beta\in H^{1,1}(X)\cap H^2(X,\Q).
\alpha_rUse in source · 4 passages
Search aliases: α_r alpha
Use in source · august_development.tex, line 61
[Primitive reduction] If every primitive rational Hodge class is algebraic, then every rational Hodge class is algebraic. Write α=∑_rL^rα_r with each α_r primitive and of Hodge type. The class h is algebraic. If α_r is algebraic, then L^rα_r=h^rsmileα_r is algebraic. Summing gives an algebraic representative of α.
Exact mathematical passage
\alpha=\sum_rL^r\alpha_r
\betaUse in source · 10 passages
Search aliases: β beta
Use in source · august_development.tex, line 65
For a fourfold and p=2, α=Lβ+α_prim, β∈ H^1,1(X)∩ H^2(X,Q). The Lefschetz (1,1) theorem makes β algebraic. Only α_prim remains.
Exact mathematical passage
\alpha=L\beta+\alpha_{\mathrm{prim}}, \qquad \beta\in H^{1,1}(X)\cap H^2(X,\Q).
\GammaUse in source · 12 passages
Search aliases: Γ Gamma
Use in source · appendices.tex, line 25
Correspondence degrees. Let X have dimension n and Y dimension m. A correspondence Γ∈CH^r(X× Y) acts by Γ_*:H^i(X)⟶ H^i+2(r-n)(Y) and shifts Hodge type by (r-n,r-n). To lower H^2p(X) to H^2(Y), take r=n-p+1. To return degree two to degree 2p on X, a correspondence on Y× X has codimension m+p-1.
Exact mathematical passage
\Gamma\in\CH^r(X\times Y)
\gammaUse in source · 7 passages
Search aliases: γ gamma
Use in source · august_development.tex, line 261
Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.
Exact mathematical passage
\gamma_k
\Gamma_*Use in source · 2 passages
Search aliases: Γ_* Gamma
Use in source · appendices.tex, line 26
Correspondence degrees. Let X have dimension n and Y dimension m. A correspondence Γ∈CH^r(X× Y) acts by Γ_*:H^i(X)⟶ H^i+2(r-n)(Y) and shifts Hodge type by (r-n,r-n). To lower H^2p(X) to H^2(Y), take r=n-p+1. To return degree two to degree 2p on X, a correspondence on Y× X has codimension m+p-1.
Exact mathematical passage
\Gamma_*:H^i(X)\longrightarrow H^{i+2(r-n)}(Y)
\gamma_1Use in source · 2 passages
Search aliases: γ_1 gamma
Use in source · august_development.tex, line 1263
oppositely oriented boundary edge, hence exactly one complementary pair in the union of the two atom packets, this is the SAT(2) stabilizer. In the three-vertex case, the vertex atom also supplies the unique neutral h. For two vertex atoms, their two neutral entries are (h,h)=p_h. The zero-defect case is already a union of three complementary pairs. Thus in every nonstar case there are γ_1,γ_2∈ F_m and a complementary pair p_r such that n+p_r=γ_1+γ_2.
Exact mathematical passage
\gamma_1,\gamma_2\in\mathcal F_m
\gamma_2Use in source · 2 passages
Search aliases: γ_2 gamma
Use in source · august_development.tex, line 1263
oppositely oriented boundary edge, hence exactly one complementary pair in the union of the two atom packets, this is the SAT(2) stabilizer. In the three-vertex case, the vertex atom also supplies the unique neutral h. For two vertex atoms, their two neutral entries are (h,h)=p_h. The zero-defect case is already a union of three complementary pairs. Thus in every nonstar case there are γ_1,γ_2∈ F_m and a complementary pair p_r such that n+p_r=γ_1+γ_2.
Exact mathematical passage
\gamma_1,\gamma_2\in\mathcal F_m
\Gamma_bUse in source · 5 passages
Search aliases: Γ_b Gamma
Use in source · august_development.tex, line 1000
The three-primary branch. For p=3 and r≥2, a higher-level kernel orbit has three elements. Define O_b={b,b+q,b+2q}, q=3^r-1, and Γ_b=O_bsqcup{3(q-b)}. Take 1≤ b<q with 3nmid b, ensuring all entries are nonzero. Then Γ_b is a balanced four-packet. A case analysis on the number of oriented kernel orbits gives exact identities expressing every full-conductor nonpair packet as a pair/surface packet or a one-step saturated sum of two balanced four-packets.
Exact mathematical passage
\Gamma_b=O_b\sqcup\{3(q-b)\}.
\Gamma_{b,c,d}Use in source · 2 passages
Search aliases: Γ_b,c,d Gamma
Use in source · august_development.tex, line 1390
K4 four-phase reduction. Choose nonzero residues b,c,dpmod q and put e≡-5^-1bpmod q. Define A_b,c,d= (λ_1(b),λ_2(b),λ_3(b),λ_4(b),5c,5d) and Γ_b,c,d=(b,c,d,e).
Exact mathematical passage
\Gamma_{b,c,d}=(b,c,d,e).
\gamma_iUse in source · 1 passage
Search aliases: γ_i gamma
Use in source · august_development.tex, line 844
Balanced four-packets and saturation. Let F_m= { γ∈ N^m-1: 1^Tγ=4, R_mγ=2m 1 }. A saturated certificate is an exact nonnegative identity n+∑_j=1^2N-3p_r_j =∑_i=1^Nγ_i, γ_i∈ F_m. The case N=2 is one-step quasi-decomposition. Higher N allows several surface packets to reconstruct one six-packet after Tate stabilization.
Exact mathematical passage
\label{eq:legacy-9.2} \boxed{ n+\sum_{j=1}^{2N-3}p_{r_j} =\sum_{i=1}^N\gamma_i, \qquad \gamma_i\in\mathcal F_m. }
\gamma_kUse in source · 2 passages
Search aliases: γ_k gamma
Use in source · august_development.tex, line 261
Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.
Exact mathematical passage
\gamma_k
\Gamma_rUse in source · 2 passages
Search aliases: Γ_r Gamma
Use in source · august_development.tex, line 274
Divisor-lowering correspondence channels. Degree bookkeeping. Let X have complex dimension n, let V⊆ H_prim^2p(X,Q) be a rational Hodge substructure, and let Y_r be smooth projective. Choose correspondences Γ_r∈CH^n-p+1(X× Y_r)_Q, Λ_r∈CH^dim Y_r+p-1(Y_r× X)_Q. They induce A_r=(Γ_r)_*:V→ H^2(Y_r,Q), B_r=(Λ_r)_*:H^2(Y_r,Q)→ H^2p(X,Q). Algebraic correspondences preserve rationality and shift Hodge type by the required bidegree.
Exact mathematical passage
\Gamma_r\in\CH^{n-p+1}(X\times Y_r)_{\Q}, \qquad \Lambda_r\in\CH^{\dim Y_r+p-1}(Y_r\times X)_{\Q}.
\gamma_rUse in source · 1 passage
Search aliases: γ_r gamma
Use in source · august_development.tex, line 860
Dual charges. An antisymmetric charge is a rational function w on residues satisfying w(m-r)=-w(r) and ∑_rγ_r w(r)=0 ∀ γ∈ F_m. Such a charge vanishes on every complementary pair and on every saturated identity. Writing w(n)=∑_r n_r w(r), if w(n)≠0, the entire lower-packet saturation semigroup is excluded. This is a universal obstruction to a grammar, not a nonalgebraicity theorem.
Exact mathematical passage
\sum_r\gamma_r w(r)=0 \qquad\forall \gamma\in\mathcal F_m.
\DeltaUse in source · 9 passages
Search aliases: Δ Delta
Use in source · august_development.tex, line 1473
Imaginary CM selector. Write the lower curve packet as ℓ=(c,d,e)pmod q, c+d+e≡0pmod q. Let K=Q(ζ_q) and define Δ_ℓ=(1-ζ_q^c)(1-ζ_q^d)(1-ζ_q^e).
Exact mathematical passage
\label{eq:legacy-16.1} \boxed{ \Delta_\ell=(1-\zeta_q^c)(1-\zeta_q^d)(1-\zeta_q^e). }
\deltaUse in source · 5 passages
Search aliases: δ delta
Use in source · august_development.tex, line 440
q≠ p fracψ^m-m^qIm^p-m^q. Then ch_q(e_p^(m)ξ)=δ_pqch_p(ξ). This exactly isolates codimension p inside a topological K-packet.
Exact mathematical passage
\operatorname{ch}_q(e_p^{(m)}\xi)=\delta_{pq}\operatorname{ch}_p(\xi).
\delta_{ce}Use in source · 1 passage
Search aliases: δ_ce delta
Use in source · august_development.tex, line 803
0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is itself algebraic.
Exact mathematical passage
Q_{m,c}Q_{m,e}=\delta_{ce}Q_{m,c}, \qquad \sum_{c\mid m}Q_{m,c}=I.
\delta_jUse in source · 3 passages
Search aliases: δ_j delta
Use in source · august_development.tex, line 976
Valuation strata and orbit mass. Let m=p^r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonzero odd kernel-orbit defect has mass at least 2p.
Exact mathematical passage
\delta_j(b)=n_j(b)-n_j(-b).
\delta_{pq}Use in source · 1 passage
Search aliases: δ_pq delta
Use in source · august_development.tex, line 440
q≠ p fracψ^m-m^qIm^p-m^q. Then ch_q(e_p^(m)ξ)=δ_pqch_p(ξ). This exactly isolates codimension p inside a topological K-packet.
Exact mathematical passage
\operatorname{ch}_q(e_p^{(m)}\xi)=\delta_{pq}\operatorname{ch}_p(\xi).
\Delta_\ellUse in source · 4 passages
Search aliases: Δ_ℓ Delta ell
Use in source · august_development.tex, line 1473
Imaginary CM selector. Write the lower curve packet as ℓ=(c,d,e)pmod q, c+d+e≡0pmod q. Let K=Q(ζ_q) and define Δ_ℓ=(1-ζ_q^c)(1-ζ_q^d)(1-ζ_q^e).
Exact mathematical passage
\label{eq:legacy-16.1} \boxed{ \Delta_\ell=(1-\zeta_q^c)(1-\zeta_q^d)(1-\zeta_q^e). }
\epsilonUse in source · 4 passages
Search aliases: ϵ epsilon
Use in source · august_development.tex, line 1145
[Integral dyadic triangle basis] For m=2^s, s≥3, the vectors {κ_b:1≤ b<q} form a Z-basis of K_m. Set r_0=q/2. On the vertex set V_m={1,2,…,q-1}, define, for b≠ r_0, p(b)= 2b, · b<r_0, 2(q-b), · b>r_0, ϵ_b= +1, · b<r_0, -1, · b>r_0. The map p raises the two-adic valuation by one, so repeated application reaches r_0. It therefore defines a rooted tree.
Exact mathematical passage
p(b)= \begin{cases} 2b,&b<r_0,\\ 2(q-b),&b>r_0, \end{cases} \qquad \epsilon_b= \begin{cases} +1,&b<r_0,\\ -1,&b>r_0. \end{cases}
\epsilon_bUse in source · 4 passages
Search aliases: ϵ_b epsilon
Use in source · august_development.tex, line 1145
[Integral dyadic triangle basis] For m=2^s, s≥3, the vectors {κ_b:1≤ b<q} form a Z-basis of K_m. Set r_0=q/2. On the vertex set V_m={1,2,…,q-1}, define, for b≠ r_0, p(b)= 2b, · b<r_0, 2(q-b), · b>r_0, ϵ_b= +1, · b<r_0, -1, · b>r_0. The map p raises the two-adic valuation by one, so repeated application reaches r_0. It therefore defines a rooted tree.
Exact mathematical passage
p(b)= \begin{cases} 2b,&b<r_0,\\ 2(q-b),&b>r_0, \end{cases} \qquad \epsilon_b= \begin{cases} +1,&b<r_0,\\ -1,&b>r_0. \end{cases}
\zetaUse in source · 5 passages
Search aliases: ζ zeta
Use in source · august_development.tex, line 1472
Imaginary CM selector. Write the lower curve packet as ℓ=(c,d,e)pmod q, c+d+e≡0pmod q. Let K=Q(ζ_q) and define Δ_ℓ=(1-ζ_q^c)(1-ζ_q^d)(1-ζ_q^e).
Exact mathematical passage
K=\Q(\zeta_q)
\zeta_qUse in source · 2 passages
Search aliases: ζ_q zeta
Use in source · august_development.tex, line 1472
Imaginary CM selector. Write the lower curve packet as ℓ=(c,d,e)pmod q, c+d+e≡0pmod q. Let K=Q(ζ_q) and define Δ_ℓ=(1-ζ_q^c)(1-ζ_q^d)(1-ζ_q^e).
Exact mathematical passage
K=\Q(\zeta_q)
\zeta_q^{-(c+d+e)}Use in source · 1 passage
Search aliases: ζ_q^-(c+d+e) zeta
Use in source · august_development.tex, line 1489
[Pure imaginary selector] overlineΔ_ℓ=-Δ_ℓ. Since 1-ζ_q^-a=-ζ_q^-a(1-ζ_q^a), complex conjugation multiplies the product by (-1)^3ζ_q^-(c+d+e)=-1.
Exact mathematical passage
(-1)^3\zeta_q^{-(c+d+e)}=-1.
\zeta_q^{-a}Use in source · 1 passage
Search aliases: ζ_q^-a zeta
Use in source · august_development.tex, line 1485
[Pure imaginary selector] overlineΔ_ℓ=-Δ_ℓ. Since 1-ζ_q^-a=-ζ_q^-a(1-ζ_q^a), complex conjugation multiplies the product by (-1)^3ζ_q^-(c+d+e)=-1.
Exact mathematical passage
1-\zeta_q^{-a}=-\zeta_q^{-a}(1-\zeta_q^a),
\zeta_q^aUse in source · 1 passage
Search aliases: ζ_q^a zeta
Use in source · august_development.tex, line 1485
[Pure imaginary selector] overlineΔ_ℓ=-Δ_ℓ. Since 1-ζ_q^-a=-ζ_q^-a(1-ζ_q^a), complex conjugation multiplies the product by (-1)^3ζ_q^-(c+d+e)=-1.
Exact mathematical passage
1-\zeta_q^{-a}=-\zeta_q^{-a}(1-\zeta_q^a),
\zeta_q^cUse in source · 1 passage
Search aliases: ζ_q^c zeta
Use in source · august_development.tex, line 1473
Imaginary CM selector. Write the lower curve packet as ℓ=(c,d,e)pmod q, c+d+e≡0pmod q. Let K=Q(ζ_q) and define Δ_ℓ=(1-ζ_q^c)(1-ζ_q^d)(1-ζ_q^e).
Exact mathematical passage
\label{eq:legacy-16.1} \boxed{ \Delta_\ell=(1-\zeta_q^c)(1-\zeta_q^d)(1-\zeta_q^e). }
\zeta_q^dUse in source · 1 passage
Search aliases: ζ_q^d zeta
Use in source · august_development.tex, line 1473
Imaginary CM selector. Write the lower curve packet as ℓ=(c,d,e)pmod q, c+d+e≡0pmod q. Let K=Q(ζ_q) and define Δ_ℓ=(1-ζ_q^c)(1-ζ_q^d)(1-ζ_q^e).
Exact mathematical passage
\label{eq:legacy-16.1} \boxed{ \Delta_\ell=(1-\zeta_q^c)(1-\zeta_q^d)(1-\zeta_q^e). }
\zeta_q^eUse in source · 1 passage
Search aliases: ζ_q^e zeta
Use in source · august_development.tex, line 1473
Imaginary CM selector. Write the lower curve packet as ℓ=(c,d,e)pmod q, c+d+e≡0pmod q. Let K=Q(ζ_q) and define Δ_ℓ=(1-ζ_q^c)(1-ζ_q^d)(1-ζ_q^e).
Exact mathematical passage
\label{eq:legacy-16.1} \boxed{ \Delta_\ell=(1-\zeta_q^c)(1-\zeta_q^d)(1-\zeta_q^e). }
\zeta_q^uUse in source · 1 passage
Search aliases: ζ_q^u zeta
Use in source · august_development.tex, line 1494
For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overline\
Exact mathematical passage
\sigma_u:\zeta_q\mapsto\zeta_q^u
\etaUse in source · 6 passages
Search aliases: η eta
Use in source · transfer210.tex, line 315
η=frac1420(31,87,57,101,84,60).
Exact mathematical passage
\label{eq:transfer210-6.2} \eta=\frac1{420}(31,87,57,101,84,60).
\eta^TUse in source · 1 passage
Search aliases: η^T eta
Use in source · transfer210.tex, line 351
1=η^T1=η^TAw =∑_jw_j(η· v_j)≥∑_jw_j.
Exact mathematical passage
1=\eta^T\mathbf1=\eta^TAw =\sum_jw_j(\eta\cdot v_j)\ge\sum_jw_j.
\eta_qUse in source · 2 passages
Search aliases: η_q eta
Use in source · transfer210.tex, line 405
η_q=frac1q(1,1,1,1,1,q-5).
Exact mathematical passage
\eta_q=\frac1q(1,1,1,1,1,q-5).
\ThetaUse in source · 1 passage
Search aliases: Θ Theta
Use in source · august_development.tex, line 964
The tower packet remains an old conductor-33 sector. A stronger construction appends a complementary pair (1,m-1) after scaling a solved seed to modulus m=kd. If b=(b_0,…,b_2p+1) is algebraic at degree d, then Θ_m,b=(kb_0,…,kb_2p+1,1,m-1) has full raw conductor m and middle Hodge type two dimensions higher. Its rational orbit has dimension φ(m). This produces full-raw-conductor algebraic packets of unbounded rank.
Exact mathematical passage
\Theta_{m,b}=(kb_0,\ldots,kb_{2p+1},1,m-1)
\thetaUse in source · 7 passages
Search aliases: θ theta
Use in source · weil_audit.tex, line 36
The intersection form and primitive directions. Use the orientation ∫ x_1y_1⋯ x_6y_6=-1 for this seed. If θ is the polarization class specified by E, the exact Gram matrix on (θ^3,u,v) is 2160 · 0 · 0
Exact mathematical passage
\theta
\theta^3Use in source · 5 passages
Search aliases: θ^3 theta
Use in source · weil_audit.tex, line 36
The intersection form and primitive directions. Use the orientation ∫ x_1y_1⋯ x_6y_6=-1 for this seed. If θ is the polarization class specified by E, the exact Gram matrix on (θ^3,u,v) is 2160 · 0 · 0
Exact mathematical passage
(\theta^3,u,v)
\Theta_{m,b}Use in source · 1 passage
Search aliases: Θ_m,b Theta
Use in source · august_development.tex, line 964
The tower packet remains an old conductor-33 sector. A stronger construction appends a complementary pair (1,m-1) after scaling a solved seed to modulus m=kd. If b=(b_0,…,b_2p+1) is algebraic at degree d, then Θ_m,b=(kb_0,…,kb_2p+1,1,m-1) has full raw conductor m and middle Hodge type two dimensions higher. Its rational orbit has dimension φ(m). This produces full-raw-conductor algebraic packets of unbounded rank.
Exact mathematical passage
\Theta_{m,b}=(kb_0,\ldots,kb_{2p+1},1,m-1)
\kappa^6Use in source · 2 passages
Search aliases: κ^6 kappa
Use in source · capacity_cycles.tex, line 616
10,504,375 κ^6.
Exact mathematical passage
10,504,375\,\kappa^6.
\kappa^7Use in source · 1 passage
Search aliases: κ^7 kappa
Use in source · capacity_cycles.tex, line 600
Take q=7, b=1, κ^7=-1/5. Then
Exact mathematical passage
\kappa^7=-1/5
\kappa^jUse in source · 2 passages
Search aliases: κ^j kappa
Use in source · capacity_cycles.tex, line 418
f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.
Exact mathematical passage
\begin{aligned} f_1&=e_1,&g_1&=e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4,\\ f_2&=e_2,&g_2&=-5e_3,\\ f_3&=P-\kappa z^5,&g_3&=5\sum_{j=0}^{q-1}P^{q-1-j}\kappa^jz^{5j}. \end{aligned}
\kappa^qUse in source · 3 passages
Search aliases: κ^q kappa
Use in source · appendices.tex, line 19
Five-phase surface · e_1=e_2=P-κ z^5=0, with κ^q=-1/5.
Exact mathematical passage
\kappa^q=-1/5
\kappa^{q-b}Use in source · 5 passages
Search aliases: κ^q-b kappa
Use in source · capacity_cycles.tex, line 448
D_κ=625q^5Δ(x_0^q,…,x_4^q) ∑_b=1^q-1κ^q-bP^b-1z^5(q-b)-1.
Exact mathematical passage
\label{eq:capacity_cycles-5.2} \boxed{ D_\kappa=625q^5\Delta(x_0^q,\ldots,x_4^q) \sum_{b=1}^{q-1}\kappa^{q-b}P^{b-1}z^{5(q-b)-1}. }
\kappa_1Use in source · 1 passage
Search aliases: κ_1 kappa
Use in source · august_development.tex, line 1316
· +(4,m/2,m/2+4,m-8). The first and third packets on the right have defects -κ_1 and -κ_4. The middle packet has defect -κ_2+κ_q-1, an edge atom. Hence all three are balanced four-packets, and [source reference] is an identity of multisets.
Exact mathematical passage
-\kappa_1
\kappa_2Use in source · 1 passage
Search aliases: κ_2 kappa
Use in source · august_development.tex, line 1316
· +(4,m/2,m/2+4,m-8). The first and third packets on the right have defects -κ_1 and -κ_4. The middle packet has defect -κ_2+κ_q-1, an edge atom. Hence all three are balanced four-packets, and [source reference] is an identity of multisets.
Exact mathematical passage
-\kappa_2+\kappa_{q-1}
\kappa_4Use in source · 1 passage
Search aliases: κ_4 kappa
Use in source · august_development.tex, line 1316
· +(4,m/2,m/2+4,m-8). The first and third packets on the right have defects -κ_1 and -κ_4. The middle packet has defect -κ_2+κ_q-1, an edge atom. Hence all three are balanced four-packets, and [source reference] is an identity of multisets.
Exact mathematical passage
-\kappa_4
\kappa_bUse in source · 8 passages
Search aliases: κ_b kappa
Use in source · august_development.tex, line 1083
For 1≤ b<q, define the triangle defect κ_b=[2b]-[b]+[h-b]. It is realized by the four-packet Γ_b=(2b,h-b,h,m-b).
Exact mathematical passage
\kappa_b=[2b]-[b]+[h-b].
\kappa_{q-1}Use in source · 1 passage
Search aliases: κ_q-1 kappa
Use in source · august_development.tex, line 1316
· +(4,m/2,m/2+4,m-8). The first and third packets on the right have defects -κ_1 and -κ_4. The middle packet has defect -κ_2+κ_q-1, an edge atom. Hence all three are balanced four-packets, and [source reference] is an identity of multisets.
Exact mathematical passage
-\kappa_2+\kappa_{q-1}
\LambdaUse in source · 3 passages
Search aliases: Λ Lambda
Use in source · august_development.tex, line 274
Divisor-lowering correspondence channels. Degree bookkeeping. Let X have complex dimension n, let V⊆ H_prim^2p(X,Q) be a rational Hodge substructure, and let Y_r be smooth projective. Choose correspondences Γ_r∈CH^n-p+1(X× Y_r)_Q, Λ_r∈CH^dim Y_r+p-1(Y_r× X)_Q. They induce A_r=(Γ_r)_*:V→ H^2(Y_r,Q), B_r=(Λ_r)_*:H^2(Y_r,Q)→ H^2p(X,Q). Algebraic correspondences preserve rationality and shift Hodge type by the required bidegree.
Exact mathematical passage
\Gamma_r\in\CH^{n-p+1}(X\times Y_r)_{\Q}, \qquad \Lambda_r\in\CH^{\dim Y_r+p-1}(Y_r\times X)_{\Q}.
\lambdaUse in source · 18 passages
Search aliases: λ lambda
Use in source · august_development.tex, line 133
If H is a matrix whose columns form an orthonormal basis of V, the finite Gram certificate is G=H^*SH. Under A⊆ V, we have A=V if and only if λ_min(G)>0. Without that containment, use a rational basis matrix H_Q for the target and the exact certificate rank_Q[C H_Q]=rank_QC to test V⊆ A.
Exact mathematical passage
\lambda_{\min}(G)>0
\lambda_1Use in source · 1 passage
Search aliases: λ_1 lambda
Use in source · august_development.tex, line 1385
K4 four-phase reduction. Choose nonzero residues b,c,dpmod q and put e≡-5^-1bpmod q. Define A_b,c,d= (λ_1(b),λ_2(b),λ_3(b),λ_4(b),5c,5d) and Γ_b,c,d=(b,c,d,e).
Exact mathematical passage
A_{b,c,d}= (\lambda_1(b),\lambda_2(b),\lambda_3(b),\lambda_4(b),5c,5d)
\lambda_2Use in source · 1 passage
Search aliases: λ_2 lambda
Use in source · august_development.tex, line 1385
K4 four-phase reduction. Choose nonzero residues b,c,dpmod q and put e≡-5^-1bpmod q. Define A_b,c,d= (λ_1(b),λ_2(b),λ_3(b),λ_4(b),5c,5d) and Γ_b,c,d=(b,c,d,e).
Exact mathematical passage
A_{b,c,d}= (\lambda_1(b),\lambda_2(b),\lambda_3(b),\lambda_4(b),5c,5d)
\lambda_3Use in source · 1 passage
Search aliases: λ_3 lambda
Use in source · august_development.tex, line 1385
K4 four-phase reduction. Choose nonzero residues b,c,dpmod q and put e≡-5^-1bpmod q. Define A_b,c,d= (λ_1(b),λ_2(b),λ_3(b),λ_4(b),5c,5d) and Γ_b,c,d=(b,c,d,e).
Exact mathematical passage
A_{b,c,d}= (\lambda_1(b),\lambda_2(b),\lambda_3(b),\lambda_4(b),5c,5d)
\lambda_4Use in source · 1 passage
Search aliases: λ_4 lambda
Use in source · august_development.tex, line 1385
K4 four-phase reduction. Choose nonzero residues b,c,dpmod q and put e≡-5^-1bpmod q. Define A_b,c,d= (λ_1(b),λ_2(b),λ_3(b),λ_4(b),5c,5d) and Γ_b,c,d=(b,c,d,e).
Exact mathematical passage
A_{b,c,d}= (\lambda_1(b),\lambda_2(b),\lambda_3(b),\lambda_4(b),5c,5d)
\lambda_{\min}Use in source · 3 passages
Search aliases: λ_min lambda min
Use in source · august_development.tex, line 133
If H is a matrix whose columns form an orthonormal basis of V, the finite Gram certificate is G=H^*SH. Under A⊆ V, we have A=V if and only if λ_min(G)>0. Without that containment, use a rational basis matrix H_Q for the target and the exact certificate rank_Q[C H_Q]=rank_QC to test V⊆ A.
Exact mathematical passage
\lambda_{\min}(G)>0
\Lambda_rUse in source · 3 passages
Search aliases: Λ_r Lambda
Use in source · august_development.tex, line 274
Divisor-lowering correspondence channels. Degree bookkeeping. Let X have complex dimension n, let V⊆ H_prim^2p(X,Q) be a rational Hodge substructure, and let Y_r be smooth projective. Choose correspondences Γ_r∈CH^n-p+1(X× Y_r)_Q, Λ_r∈CH^dim Y_r+p-1(Y_r× X)_Q. They induce A_r=(Γ_r)_*:V→ H^2(Y_r,Q), B_r=(Λ_r)_*:H^2(Y_r,Q)→ H^2p(X,Q). Algebraic correspondences preserve rationality and shift Hodge type by the required bidegree.
Exact mathematical passage
\Gamma_r\in\CH^{n-p+1}(X\times Y_r)_{\Q}, \qquad \Lambda_r\in\CH^{\dim Y_r+p-1}(Y_r\times X)_{\Q}.
\lambda_rUse in source · 2 passages
Search aliases: λ_r lambda
Use in source · august_development.tex, line 1359
Local mod-5 phase grammar. Assume m=5q and first suppose (5,q)=1. For bpmod q and a phase r∈{0,1,2,3,4}, let λ_r(b) be the CRT lift satisfying λ_r(b)≡ rpmod5, λ_r(b)≡ bpmod q. The finite audit reveals four canonical multiplicity profiles, up to multiplication by a unit modulo five: (1,0,0,0,5), (2,1,1,1,1), (1,0,2,1,2), (1,0,1,3,1). The first is the complete-five-phase K5 packet, the second is a four-phase K4 lift, and the final two are one K23 pair--triple mechanism.
Exact mathematical passage
\lambda_r(b)
\lambda_\chiUse in source · 1 passage
Search aliases: λ_χ lambda chi
Use in source · august_development.tex, line 1170
ecover the unique integers z_b from its even coordinates, and subtract ∑_bz_bκ_b. The residual balanced vector is supported only on the odd residues 1≤ r<h. The corresponding square sawtooth matrix is nonsingular. Indeed, extend an odd-support vector to an odd function on U_m. Convolution with ρ_m is diagonalized by the odd Dirichlet characters, with eigenvalues λ_χ= ∑_a∈ U_mρ_m(a)overlineχ(a) = ∑_a∈ U_maoverlineχ(a). If χ is induced from a primitive odd character χ^* of conductor 2^t, then λ_χ,…
Exact mathematical passage
\lambda_\chi= \sum_{a\in U_m}\rho_m(a)\overline{\chi(a)} = \sum_{a\in U_m}a\overline{\chi(a)}.
\lambda_{\chi,2^s}Use in source · 1 passage
Search aliases: λ_χ,2^s lambda chi
Use in source · august_development.tex, line 1177
ing square sawtooth matrix is nonsingular. Indeed, extend an odd-support vector to an odd function on U_m. Convolution with ρ_m is diagonalized by the odd Dirichlet characters, with eigenvalues λ_χ= ∑_a∈ U_mρ_m(a)overlineχ(a) = ∑_a∈ U_maoverlineχ(a). If χ is induced from a primitive odd character χ^* of conductor 2^t, then λ_χ,2^s=2^s-tλ_χ^*,2^t. The primitive value is, up to a nonzero scalar, the generalized Bernoulli value B_1,overlineχ^*, equivalently -L(0,overlineχ^*), and is nonzero for eve…
Exact mathematical passage
\lambda_{\chi,2^s}=2^{s-t}\lambda_{\chi^*,2^t}.
\lambda_{\chi^*,2^t}Use in source · 1 passage
Search aliases: λ_χ^*,2^t lambda chi
Use in source · august_development.tex, line 1177
ing square sawtooth matrix is nonsingular. Indeed, extend an odd-support vector to an odd function on U_m. Convolution with ρ_m is diagonalized by the odd Dirichlet characters, with eigenvalues λ_χ= ∑_a∈ U_mρ_m(a)overlineχ(a) = ∑_a∈ U_maoverlineχ(a). If χ is induced from a primitive odd character χ^* of conductor 2^t, then λ_χ,2^s=2^s-tλ_χ^*,2^t. The primitive value is, up to a nonzero scalar, the generalized Bernoulli value B_1,overlineχ^*, equivalently -L(0,overlineχ^*), and is nonzero for eve…
Exact mathematical passage
\lambda_{\chi,2^s}=2^{s-t}\lambda_{\chi^*,2^t}.
\mu_{210}Use in source · 1 passage
Search aliases: μ_210 mu
Use in source · carrier70.tex, line 348
The coordinate boundary of X_210 is a simple normal crossing divisor. Every nonempty coordinate stratum is fixed pointwise by at least one coordinate scaling subgroup μ_210. Our full-support character is nontrivial on that subgroup. The same is true for every character in its rational orbit.
Exact mathematical passage
\mu_{210}
\mu_3Use in source · 1 passage
Search aliases: μ_3 mu
Use in source · carrier70.tex, line 243
The ambient weighted projective space is well formed. Its largest singular stratum is the μ_3-stratum y_0=y_5=0, of dimension three. The restriction of F there is y_1^5+y_2y_3^2y_4^7, not identically zero. All other ambient singular strata have dimension at most two. Thus Y meets the ambient singular locus in codimension at least two. The verifier checks all coordinate strata, not just the largest one.
Exact mathematical passage
\mu_3
\nu_{\mathrm{Frob}}Use in source · 1 passage
Search aliases: ν_Frob nu mathrm
Use in source · august_development.tex, line 199
Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods to solve different geometric sectors before exact synthesis.
Exact mathematical passage
\lambda=(r_{\mathrm{Lef}},r_{\deg},\chi_{\mathrm{mon}},s_{\mathrm{supp}},\nu_{\mathrm{Frob}}).
\xiUse in source · 4 passages
Search aliases: ξ xi
Use in source · august_development.tex, line 431
Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X
Exact mathematical passage
\operatorname{ch}_q(\psi^m\xi)=m^q\operatorname{ch}_q(\xi).
\xi_\kappaUse in source · 2 passages
Search aliases: ξ_κ xi kappa
Use in source · capacity_cycles.tex, line 562
ξ_κ=[Z_κ]-2q h^2
Exact mathematical passage
\xi_\kappa=[Z_\kappa]-2q h^2
\PiUse in source · 6 passages
Search aliases: Π Pi
Use in source · august_development.tex, line 453
Nonlinear Dolbeault obstruction packets. Let E be a smooth complex bundle and consider a candidate (0,1)-connection. After gauge fixing, write the integrability equation abstractly as F(a)=f+La+B(a,a)=0, where L is the linearized operator and B is a bounded bilinear term. Assume a Fredholm splitting of the target I=Q+Π, where Q projects onto the closed range of L and Π onto a finite-dimensional obstruction space. Choose a gauge-fixed complement to ker L as the domain and let R be a right inverse…
Exact mathematical passage
I=Q+\Pi,
\pi^*Use in source · 1 passage
Search aliases: π^* pi
Use in source · carrier70.tex, line 437
π_*π^*=(degπ)id
Exact mathematical passage
\pi_*\pi^*=(\deg\pi)\operatorname{id}
\pi_*Use in source · 1 passage
Search aliases: π_* pi
Use in source · carrier70.tex, line 437
π_*π^*=(degπ)id
Exact mathematical passage
\pi_*\pi^*=(\deg\pi)\operatorname{id}
\Pi_{A,k}Use in source · 1 passage
Search aliases: Π_A,k Pi
Use in source · august_development.tex, line 497
A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge j _L^2^2+O(k^n-2j-1), c_n,j>0. This detects nonintegrability. It does not by itself remove the defect.
Exact mathematical passage
\Tr(P_{2j}\Pi_{A,k}) =c_{n,j}k^{n-2j}\|(F_A^{0,2})^{\wedge j}\|_{L^2}^2+O(k^{n-2j-1}), \qquad c_{n,j}>0.
\pi_{m/d*}Use in source · 1 passage
Search aliases: π_m/d* pi
Use in source · august_development.tex, line 787
For dmid m, let E_m,d=frac1(m/d)^5π_m/d^*π_m/d*. On a character of exact conductor c, E_m,d=1, · cmid d,
Exact mathematical passage
E_{m,d}=\frac1{(m/d)^5}\pi_{m/d}^*\pi_{m/d*}.
\pi_{m/d}^*Use in source · 1 passage
Search aliases: π_m/d^* pi
Use in source · august_development.tex, line 787
For dmid m, let E_m,d=frac1(m/d)^5π_m/d^*π_m/d*. On a character of exact conductor c, E_m,d=1, · cmid d,
Exact mathematical passage
E_{m,d}=\frac1{(m/d)^5}\pi_{m/d}^*\pi_{m/d*}.
\rho_mUse in source · 10 passages
Search aliases: ρ_m rho
Use in source · august_development.tex, line 732
Centered Fourier form. Define ρ_m(r)=⟨ r⟩_m-frac m2, F_a(u)=∑_iρ_m(ua_i). Then a is balanced if and only if F_a(u)=0 for all units. Moreover, F_a(-u)=-F_a(u), so only odd characters occur. For a Dirichlet character χ on U_m, put B_m,χ(r)= ∑_u∈ U_mρ_m(ur)overlineχ(u). Finite Fourier inversion gives the exact criterion n(a)∈ B_m Longleftrightarrow ∑_i B_m,χ(a_i)=0 for every odd χ. Here n(a) is the residue count vector. This formulation underlies the prime-power filtration and the proposed conducto…
Exact mathematical passage
\rho_m(r)=\langle r\rangle_m-\frac m2, \qquad F_a(u)=\sum_i\rho_m(ua_i).
\rho_rUse in source · 1 passage
Search aliases: ρ_r rho
Use in source · august_development.tex, line 340
Family propagation. Let π: X→ S and ρ_r: Y_r→ S be smooth projective families over a connected base. Relative algebraic correspondences induce morphisms of Gauss-Manin local systems. In a flat rational trivialization, their transfer matrix is constant.
Exact mathematical passage
\rho_r:\mathcal Y_r\to S
\SigmaUse in source · 1 passage
Search aliases: Σ Sigma
Use in source · august_development.tex, line 1509
1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overlineΣ_ℓ. This is the rigorous form of the statement that the half-twist occurs on the imaginary plane.
Exact mathematical passage
\label{eq:legacy-16.3} \boxed{ \Sigma_N=\overline{\Sigma_\ell}. }
\sigmaUse in source · 5 passages
Search aliases: σ sigma
Use in source · august_development.tex, line 1494
For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overline\
Exact mathematical passage
\sigma_u:\zeta_q\mapsto\zeta_q^u
\sigma_{5,1}Use in source · 1 passage
Search aliases: σ_5,1 sigma
Use in source · transfer210.tex, line 424
This is a proof for every q>5, not an extrapolation from finite runs. The verifier's checks for 6≤ q≤80 are benchmarks only. These packets are themselves the standard family σ_5,1, so the obstruction is to the method, not to algebraicity.
Exact mathematical passage
\sigma_{5,1}
\sigma_{5,5}Use in source · 1 passage
Search aliases: σ_5,5 sigma
Use in source · transfer210.tex, line 226
S_5/6 · =(5,10,12,19,26,33)=σ_5,5 at degree 35, S_7/6 · =(2,7,12,17,21,22,27,32)=σ_7,2 at degree 35,
Exact mathematical passage
\begin{aligned} S_5/6&=(5,10,12,19,26,33)=\sigma_{5,5}\quad\text{at degree }35,\\ S_7/6&=(2,7,12,17,21,22,27,32)=\sigma_{7,2}\quad\text{at degree }35, \end{aligned}
\sigma_{7,2}Use in source · 1 passage
Search aliases: σ_7,2 sigma
Use in source · transfer210.tex, line 226
S_5/6 · =(5,10,12,19,26,33)=σ_5,5 at degree 35, S_7/6 · =(2,7,12,17,21,22,27,32)=σ_7,2 at degree 35,
Exact mathematical passage
\begin{aligned} S_5/6&=(5,10,12,19,26,33)=\sigma_{5,5}\quad\text{at degree }35,\\ S_7/6&=(2,7,12,17,21,22,27,32)=\sigma_{7,2}\quad\text{at degree }35, \end{aligned}
\Sigma_NUse in source · 1 passage
Search aliases: Σ_N Sigma
Use in source · august_development.tex, line 1509
1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overlineΣ_ℓ. This is the rigorous form of the statement that the half-twist occurs on the imaginary plane.
Exact mathematical passage
\label{eq:legacy-16.3} \boxed{ \Sigma_N=\overline{\Sigma_\ell}. }
\sigma_{p,i}Use in source · 1 passage
Search aliases: σ_p,i sigma
Use in source · transfer210.tex, line 118
σ_p,i=(i,i+d,…,i+(p-1)d,m-pi)
Exact mathematical passage
\sigma_{p,i}=(i,i+d,\ldots,i+(p-1)d,m-pi)
\sigma_uUse in source · 2 passages
Search aliases: σ_u sigma
Use in source · august_development.tex, line 1494
For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overline\
Exact mathematical passage
\sigma_u:\zeta_q\mapsto\zeta_q^u
\tauUse in source · 3 passages
Search aliases: τ tau
Use in source · august_development.tex, line 1185
The trivalent defect tree. The integral basis turns defect mass into exact tree variation. Define a sign gauge by τ_r_0=1, τ_b=ϵ_bτ_p(b), y_b=τ_bz_b. Attach one zero-valued boundary leaf to the root and two zero-valued boundary leaves to every odd vertex. Every vertex of the resulting augmented tree widetilde T_m has degree three. The boundary edges correspond bijectively to defect coordinates: the root edge is labelled q, the edge from b to p(b) is labelled 2b, and the two leaves at an odd b ar…
Exact mathematical passage
\tau_{r_0}=1, \qquad \tau_b=\epsilon_b\tau_{p(b)}, \qquad y_b=\tau_bz_b.
\tau_bUse in source · 3 passages
Search aliases: τ_b tau
Use in source · august_development.tex, line 1185
The trivalent defect tree. The integral basis turns defect mass into exact tree variation. Define a sign gauge by τ_r_0=1, τ_b=ϵ_bτ_p(b), y_b=τ_bz_b. Attach one zero-valued boundary leaf to the root and two zero-valued boundary leaves to every odd vertex. Every vertex of the resulting augmented tree widetilde T_m has degree three. The boundary edges correspond bijectively to defect coordinates: the root edge is labelled q, the edge from b to p(b) is labelled 2b, and the two leaves at an odd b ar…
Exact mathematical passage
\tau_{r_0}=1, \qquad \tau_b=\epsilon_b\tau_{p(b)}, \qquad y_b=\tau_bz_b.
\tau_{p(b)}Use in source · 1 passage
Search aliases: τ_p(b) tau
Use in source · august_development.tex, line 1185
The trivalent defect tree. The integral basis turns defect mass into exact tree variation. Define a sign gauge by τ_r_0=1, τ_b=ϵ_bτ_p(b), y_b=τ_bz_b. Attach one zero-valued boundary leaf to the root and two zero-valued boundary leaves to every odd vertex. Every vertex of the resulting augmented tree widetilde T_m has degree three. The boundary edges correspond bijectively to defect coordinates: the root edge is labelled q, the edge from b to p(b) is labelled 2b, and the two leaves at an odd b ar…
Exact mathematical passage
\tau_{r_0}=1, \qquad \tau_b=\epsilon_b\tau_{p(b)}, \qquad y_b=\tau_bz_b.
\tau_{r_0}Use in source · 1 passage
Search aliases: τ_r_0 tau
Use in source · august_development.tex, line 1185
The trivalent defect tree. The integral basis turns defect mass into exact tree variation. Define a sign gauge by τ_r_0=1, τ_b=ϵ_bτ_p(b), y_b=τ_bz_b. Attach one zero-valued boundary leaf to the root and two zero-valued boundary leaves to every odd vertex. Every vertex of the resulting augmented tree widetilde T_m has degree three. The boundary edges correspond bijectively to defect coordinates: the root edge is labelled q, the edge from b to p(b) is labelled 2b, and the two leaves at an odd b ar…
Exact mathematical passage
\tau_{r_0}=1, \qquad \tau_b=\epsilon_b\tau_{p(b)}, \qquad y_b=\tau_bz_b.
\PhiUse in source · 8 passages
Search aliases: Φ Phi
Use in source · august_development.tex, line 472
em[Range-packet contraction] Assume R ≤ M, B(a,b) ≤ C a b , 4CM^2 Qf <1. Then the map T(a)=-RQ(f+B(a,a)) has a unique fixed point in the gauge-fixed ball a ≤2M Qf . If Qf=0, the ball is the singleton {0} and the range solution is immediate. At the fixed point, the range component of F(a) vanishes. The remaining obstruction is Φ=Π(f+B(a,a)). The full integrability equation holds if and only if Φ=0. For a ≤ r=2M Qf , T(a) ≤ M Qf +MC r^2≤ r under the stated bound. Moreover T(a)- T(b) ≤ MC( a + b ) …
Exact mathematical passage
\Phi=\Pi(f+B(a,a)).
\phiUse in source · 1 passage
Search aliases: ϕ phi
Use in source · august_development.tex, line 914
Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.
Exact mathematical passage
\phi:X_{33}^4\dashrightarrow Y
\varphiUse in source · 2 passages
Search aliases: φ varphi
Use in source · august_development.tex, line 967
The tower packet remains an old conductor-33 sector. A stronger construction appends a complementary pair (1,m-1) after scaling a solved seed to modulus m=kd. If b=(b_0,…,b_2p+1) is algebraic at degree d, then Θ_m,b=(kb_0,…,kb_2p+1,1,m-1) has full raw conductor m and middle Hodge type two dimensions higher. Its rational orbit has dimension φ(m). This produces full-raw-conductor algebraic packets of unbounded rank.
Exact mathematical passage
\varphi(m)
\Phi_{\lambda}Use in source · 1 passage
Search aliases: Φ_λ Phi lambda
Use in source · august_development.tex, line 519
This creates a practical research program. One can refine the finite obstruction space by symmetry, monodromy, support, or arithmetic phase, obtaining Φ=∑_λΦ_λ. Since the decomposition is complete, integrability is equivalent to the simultaneous vanishing of all children. Different geometric arguments may eliminate different children. A symmetry packet may vanish by representation theory, a support packet by localization, and an arithmetic packet by compatibility across reductions. The analytic …
Exact mathematical passage
\Phi=\sum_{\lambda}\Phi_{\lambda}.
\chiUse in source · 12 passages
Search aliases: χ chi
Use in source · august_development.tex, line 199
Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods to solve different geometric sectors before exact synthesis.
Exact mathematical passage
\lambda=(r_{\mathrm{Lef}},r_{\deg},\chi_{\mathrm{mon}},s_{\mathrm{supp}},\nu_{\mathrm{Frob}}).
\chi^*Use in source · 2 passages
Search aliases: χ^* chi
Use in source · august_development.tex, line 1176
residues 1≤ r<h. The corresponding square sawtooth matrix is nonsingular. Indeed, extend an odd-support vector to an odd function on U_m. Convolution with ρ_m is diagonalized by the odd Dirichlet characters, with eigenvalues λ_χ= ∑_a∈ U_mρ_m(a)overlineχ(a) = ∑_a∈ U_maoverlineχ(a). If χ is induced from a primitive odd character χ^* of conductor 2^t, then λ_χ,2^s=2^s-tλ_χ^*,2^t. The primitive value is, up to a nonzero scalar, the generalized Bernoulli value B_1,overlineχ^*, equivalently -L(0,overl…
Exact mathematical passage
\chi^*
\chi_aUse in source · 1 passage
Search aliases: χ_a chi
Use in source · transfer210.tex, line 446
Proof. If K is a diagonal deck subgroup and χ_a|_K=1, then
Exact mathematical passage
\chi_a|_K=1
\chi_a^kUse in source · 1 passage
Search aliases: χ_a^k chi
Use in source · transfer210.tex, line 448
χ_c|_K=χ_a^k|_K=1.
Exact mathematical passage
\chi_c|_K=\chi_a^k|_K=1.
\chi_cUse in source · 1 passage
Search aliases: χ_c chi
Use in source · transfer210.tex, line 448
χ_c|_K=χ_a^k|_K=1.
Exact mathematical passage
\chi_c|_K=\chi_a^k|_K=1.
\chi_{\mathrm{mon}}Use in source · 1 passage
Search aliases: χ_mon chi mathrm
Use in source · august_development.tex, line 199
Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods to solve different geometric sectors before exact synthesis.
Exact mathematical passage
\lambda=(r_{\mathrm{Lef}},r_{\deg},\chi_{\mathrm{mon}},s_{\mathrm{supp}},\nu_{\mathrm{Frob}}).
\chi_TUse in source · 1 passage
Search aliases: χ_T chi
Use in source · august_development.tex, line 301
Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.
Exact mathematical passage
\chi_T(t)=t^d+c_{d-1}t^{d-1}+\cdots+c_0
\chi_\alphaUse in source · 1 passage
Search aliases: χ_α chi alpha
Use in source · carrier70.tex, line 335
Let K be the kernel of the torus isogeny. The relation AB=210I shows that K embeds in G_210: an element in the kernel has representatives whose 210th powers are all the same, and a common scalar makes them roots of unity. The displayed character identity proves χ_α|_K=1. The same holds for every rational conjugate. Therefore the whole rational packet is K-invariant.
Exact mathematical passage
\chi_\alpha|_K=1
\psiUse in source · 2 passages
Search aliases: ψ psi
Use in source · august_development.tex, line 431
Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X
Exact mathematical passage
\operatorname{ch}_q(\psi^m\xi)=m^q\operatorname{ch}_q(\xi).
\psi^mUse in source · 2 passages
Search aliases: ψ^m psi
Use in source · august_development.tex, line 431
Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X
Exact mathematical passage
\operatorname{ch}_q(\psi^m\xi)=m^q\operatorname{ch}_q(\xi).
\Omega_BUse in source · 1 passage
Search aliases: Ω_B Omega
Use in source · weil_audit.tex, line 128
The product locus and replacement factorizations. Split the six directions into a four-dimensional Weil factor C and a two-dimensional factor B. The determinant forms factor: Ω_6=Ω_CwedgeΩ_B. Writing their real and imaginary parts as c_R,c_I,b_R,b_I gives u=c_Rb_R-c_Ib_I, v=c_Rb_I+c_Ib_R. On the product locus, the degree-four Weil classes on C are algebraic by the checked fourfold theorem, and the degree-two classes on B are divisors by Lefschetz (1,1). Their products give the degree-six classes…
Exact mathematical passage
\Omega_6=\Omega_C\wedge\Omega_B.
\Omega_CUse in source · 1 passage
Search aliases: Ω_C Omega
Use in source · weil_audit.tex, line 128
The product locus and replacement factorizations. Split the six directions into a four-dimensional Weil factor C and a two-dimensional factor B. The determinant forms factor: Ω_6=Ω_CwedgeΩ_B. Writing their real and imaginary parts as c_R,c_I,b_R,b_I gives u=c_Rb_R-c_Ib_I, v=c_Rb_I+c_Ib_R. On the product locus, the degree-four Weil classes on C are algebraic by the checked fourfold theorem, and the degree-two classes on B are divisors by Lefschetz (1,1). Their products give the degree-six classes…
Exact mathematical passage
\Omega_6=\Omega_C\wedge\Omega_B.
\daggerUse in source · 7 passages
Search aliases: † dagger
Use in source · appendices.tex, line 14
Positive transfer · T=∑_rA_r^† A_r only with algebraic realizations of those adjoints.
Exact mathematical passage
T=\sum_rA_r^\dagger A_r
\ell_1Use in source · 1 passage
Search aliases: ℓ_1 ell
Use in source · august_development.tex, line 1508
For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overlineΣ_ℓ. This is the ri…
Exact mathematical passage
(\ell_1,\ell_2,\ell_3)=(c,d,e)
\ell_2Use in source · 1 passage
Search aliases: ℓ_2 ell
Use in source · august_development.tex, line 1508
For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overlineΣ_ℓ. This is the ri…
Exact mathematical passage
(\ell_1,\ell_2,\ell_3)=(c,d,e)
\ell_3Use in source · 1 passage
Search aliases: ℓ_3 ell
Use in source · august_development.tex, line 1508
For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overlineΣ_ℓ. This is the ri…
Exact mathematical passage
(\ell_1,\ell_2,\ell_3)=(c,d,e)
\ell_jUse in source · 1 passage
Search aliases: ℓ_j ell
Use in source · august_development.tex, line 1503
For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overlineΣ_ℓ. This is the ri…
Exact mathematical passage
\label{eq:legacy-16.2} \sigma_u(\Delta_\ell) =8i(-1)^{h_\ell(u)} \prod_{j=1}^3 \sin\frac{\pi\langle u\ell_j\rangle_q}{q}.
\toUse in source · 19 passages
Search aliases: → to
Use in source · appendices.tex, line 7
Algebraic space · Alg^p(X)=im(CH^p(X)_Q→ H^2p(X,Q)).
Exact mathematical passage
\Alg^p(X)=\im(\CH^p(X)_{\Q}\to H^{2p}(X,\Q))
\mapstoUse in source · 2 passages
Search aliases: ↦ mapsto
Use in source · august_development.tex, line 1494
For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overline\
Exact mathematical passage
\sigma_u:\zeta_q\mapsto\zeta_q^u
\hookrightarrowUse in source · 1 passage
Search aliases: ↪ hookrightarrow
Use in source · august_development.tex, line 243
Support verification and multiplicity recovery. Let Z⊆ X be a closed algebraic subset and j:X∖ Z↪ X. The support exact sequence contains H_Z^2p(X,Q)⟶ H^2p(X,Q)xrightarrowj^*H^2p(X∖ Z,Q).
Exact mathematical passage
j:X\setminus Z\hookrightarrow X
\forallUse in source · 2 passages
Search aliases: ∀ forall
Use in source · august_development.tex, line 702
when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.
Exact mathematical passage
\label{eq:legacy-7.1} \boxed{ \ell_a(u)=3\qquad\forall u\in U_m. }
\partialUse in source · 5 passages
Search aliases: ∂ partial
Use in source · august_development.tex, line 1065
Complement defects and dyadic triangles. Put q=frac m4, h=2q, m=4q. For a packet count vector n, define its oriented complement defect ∂ n=∑_r=1^h-1d_n(r)[r], d_n(r)=n_r-n_m-r. The self-complementary residue h contributes zero. Since ρ_m(m-r)=-ρ_m(r), the balanced-packet equation is equivalent to ∑_r=1^h-1d_n(r)ρ_m(ur)=0 for every odd unit u. Moreover, ∂ n _1=∑_r=1^h-1|d_n(r)|≤ 6. Thus the dyadic problem is a short-vector problem in an integral defect lattice.
Exact mathematical passage
\partial n=\sum_{r=1}^{h-1}d_n(r)[r], \qquad d_n(r)=n_r-n_{m-r}.
\prodUse in source · 8 passages
Search aliases: ∏ prod
Use in source · august_development.tex, line 435
Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X q≠ p fracψ^m-m^qIm^p-m^q. Then ch_q(e_p^(m)ξ)=δ_pqch_p(ξ). This exactly isolates codimension p inside a topological K-packet.
Exact mathematical passage
e_p^{(m)}=\prod_{\substack{0\le q\le\dim X\\q\ne p}} \frac{\psi^m-m^qI}{m^p-m^q}.
\prod_{0\le i<j\le4}Use in source · 1 passage
Search aliases: ∏_0≤ i<j≤4 prod le le
Use in source · capacity_cycles.tex, line 440
Δ(t)=∏_0≤ i<j≤4(t_i-t_j).
Exact mathematical passage
\Delta(t)=\prod_{0\le i<j\le4}(t_i-t_j).
\prod_{i=0}^5Use in source · 2 passages
Search aliases: ∏_i=0^5 prod
Use in source · capacity_cycles.tex, line 82
J_F_m=(x_0^m-1,…,x_5^m-1), S_m=∏_i=0^5x_i^m-2.
Exact mathematical passage
\label{eq:capacity_cycles-1.2} J_{F_m}=(x_0^{m-1},\ldots,x_5^{m-1}),\qquad S_m=\prod_{i=0}^5x_i^{m-2}.
\prod_jUse in source · 1 passage
Search aliases: ∏_j prod
Use in source · carrier70.tex, line 148
On the projective tori define y_i=∏_jx_j^B_ij. Negative exponents are allowed there. The second identity makes this well-defined into the weighted projective torus, the first gives
Exact mathematical passage
y_i=\prod_jx_j^{B_{ij}}
\prod_{j=1}^3Use in source · 1 passage
Search aliases: ∏_j=1^3 prod
Use in source · august_development.tex, line 1503
For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overlineΣ_ℓ. This is the ri…
Exact mathematical passage
\label{eq:legacy-16.2} \sigma_u(\Delta_\ell) =8i(-1)^{h_\ell(u)} \prod_{j=1}^3 \sin\frac{\pi\langle u\ell_j\rangle_q}{q}.
\prod_{\substack{0\le q\le\dim X\\q\ne p}}Use in source · 1 passage
Search aliases: ∏_substack0≤ q≤dim X q≠ p prod substack le le dim q ne
Use in source · august_development.tex, line 435
Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X q≠ p fracψ^m-m^qIm^p-m^q. Then ch_q(e_p^(m)ξ)=δ_pqch_p(ξ). This exactly isolates codimension p inside a topological K-packet.
Exact mathematical passage
e_p^{(m)}=\prod_{\substack{0\le q\le\dim X\\q\ne p}} \frac{\psi^m-m^qI}{m^p-m^q}.
\sum_{0\le i<j\le4}Use in source · 1 passage
Search aliases: ∑_0≤ i<j≤4 sum le le
Use in source · capacity_cycles.tex, line 602
a=(1,8,15,22,29,30), Z_κ={ l ∑_i=0^4x_i^7=0, ∑_0≤ i<j≤4x_i^7x_j^7=0, x_0x_1x_2x_3x_4=κ x_5^5 }⊂ X_35.
Exact mathematical passage
a=(1,8,15,22,29,30),\quad Z_\kappa=\left\{ \begin{array}{l} \sum_{i=0}^4x_i^7=0,\\ \sum_{0\le i<j\le4}x_i^7x_j^7=0,\\ x_0x_1x_2x_3x_4=\kappa x_5^5 \end{array}\right\}\subset X_{35}.
\sum_{a\in U_m}Use in source · 1 passage
Search aliases: ∑_a∈ U_m sum in
Use in source · august_development.tex, line 1170
ecover the unique integers z_b from its even coordinates, and subtract ∑_bz_bκ_b. The residual balanced vector is supported only on the odd residues 1≤ r<h. The corresponding square sawtooth matrix is nonsingular. Indeed, extend an odd-support vector to an odd function on U_m. Convolution with ρ_m is diagonalized by the odd Dirichlet characters, with eigenvalues λ_χ= ∑_a∈ U_mρ_m(a)overlineχ(a) = ∑_a∈ U_maoverlineχ(a). If χ is induced from a primitive odd character χ^* of conductor 2^t, then λ_χ,…
Exact mathematical passage
\lambda_\chi= \sum_{a\in U_m}\rho_m(a)\overline{\chi(a)} = \sum_{a\in U_m}a\overline{\chi(a)}.
\sum_bUse in source · 2 passages
Search aliases: ∑_b sum
Use in source · august_development.tex, line 1169
Now take d∈ K_m, recover the unique integers z_b from its even coordinates, and subtract ∑_bz_bκ_b. The residual balanced vector is supported only on the odd residues 1≤ r<h. The corresponding square sawtooth matrix is nonsingular. Indeed, extend an odd-support vector to an odd function on U_m. Convolution with ρ_m is diagonalized by the odd Dirichlet characters, with eigenvalues λ_χ= ∑_a∈ U_mρ_m(a)overlineχ(a) = ∑_a∈ U_maoverlineχ(a). If χ is induced from a primitive odd character χ^* of conduc…
Exact mathematical passage
\sum_bz_b\kappa_b
\sum_{b=1}^{q-1}Use in source · 2 passages
Search aliases: ∑_b=1^q-1 sum
Use in source · august_development.tex, line 1160
Suppose d=∑_b=1^q-1z_bκ_b. Its even coordinates are d(q)=z_r_0, d(2b)=z_b-ϵ_bz_p(b) (b≠ r_0). Starting at the root and moving toward the leaves, these equations recover every z_b integrally. Hence the triangle vectors are linearly independent and their even-coordinate map is unimodular.
Exact mathematical passage
d=\sum_{b=1}^{q-1}z_b\kappa_b
\sum_{b\ne r_0}Use in source · 1 passage
Search aliases: ∑_b≠ r_0 sum ne
Use in source · august_development.tex, line 1196
[Exact trivalent-tree norm] If d=∑_bz_bκ_b∈ K_m, then d _1 =|y_r_0| +∑_b≠ r_0|y_b-y_p(b)| +2∑_substack1≤ b<q b odd|y_b|. The right side is the total variation of the integer function y on widetilde T_m with zero boundary values. The root coordinate is d(q)=z_r_0. For b≠ r_0, multiplying d(2b)=z_b-ϵ_bz_p(b) by τ_b gives τ_bd(2b)=y_b-y_p(b). For odd b, the two odd coordinates are d(b)=-z_b, d(h-b)=z_b. Taking absolute values and summing all defect coordinates gives [source reference].
Exact mathematical passage
\label{eq:legacy-13.2} \boxed{ \|d\|_1 =|y_{r_0}| +\sum_{b\ne r_0}|y_b-y_{p(b)}| +2\sum_{\substack{1\le b<q\\ b\text{ odd}}}|y_b|. }
\sum_{c\mid m}Use in source · 1 passage
Search aliases: ∑_cmid m sum mid
Use in source · august_development.tex, line 803
0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is itself algebraic.
Exact mathematical passage
Q_{m,c}Q_{m,e}=\delta_{ce}Q_{m,c}, \qquad \sum_{c\mid m}Q_{m,c}=I.
\sum_{d\mid c}Use in source · 1 passage
Search aliases: ∑_dmid c sum mid
Use in source · august_development.tex, line 798
0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is itself algebraic.
Exact mathematical passage
\label{eq:legacy-8.2} \boxed{ Q_{m,c}=\sum_{d\mid c}\mu(c/d)E_{m,d}. }
\sum_{i=0}^4Use in source · 1 passage
Search aliases: ∑_i=0^4 sum
Use in source · capacity_cycles.tex, line 602
a=(1,8,15,22,29,30), Z_κ={ l ∑_i=0^4x_i^7=0, ∑_0≤ i<j≤4x_i^7x_j^7=0, x_0x_1x_2x_3x_4=κ x_5^5 }⊂ X_35.
Exact mathematical passage
a=(1,8,15,22,29,30),\quad Z_\kappa=\left\{ \begin{array}{l} \sum_{i=0}^4x_i^7=0,\\ \sum_{0\le i<j\le4}x_i^7x_j^7=0,\\ x_0x_1x_2x_3x_4=\kappa x_5^5 \end{array}\right\}\subset X_{35}.
\sum_{i=0}^{5}Use in source · 1 passage
Search aliases: ∑_i=0^5 sum
Use in source · carrier70.tex, line 58
∑_i=0^5⟨ ta_i⟩_70=210.
Exact mathematical passage
\sum_{i=0}^{5}\langle ta_i\rangle_{70}=210.
\sum_{i=1}^NUse in source · 1 passage
Search aliases: ∑_i=1^N sum
Use in source · august_development.tex, line 844
Balanced four-packets and saturation. Let F_m= { γ∈ N^m-1: 1^Tγ=4, R_mγ=2m 1 }. A saturated certificate is an exact nonnegative identity n+∑_j=1^2N-3p_r_j =∑_i=1^Nγ_i, γ_i∈ F_m. The case N=2 is one-step quasi-decomposition. Higher N allows several surface packets to reconstruct one six-packet after Tate stabilization.
Exact mathematical passage
\label{eq:legacy-9.2} \boxed{ n+\sum_{j=1}^{2N-3}p_{r_j} =\sum_{i=1}^N\gamma_i, \qquad \gamma_i\in\mathcal F_m. }
\sum_{i=1}^rUse in source · 1 passage
Search aliases: ∑_i=1^r sum
Use in source · august_development.tex, line 258
Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.
Exact mathematical passage
\alpha=\sum_{i=1}^r q_i[Z_i].
\sum_jUse in source · 5 passages
Search aliases: ∑_j sum
Use in source · august_development.tex, line 123
dots,c_N}. Then im S=A, ker S=A^⊥. Under the stated containment c_j∈ V, the selected cycles span V if and only if S|_V is positive definite. Without containment, positivity only detects the absence of vectors in V orthogonal to the cycle span and does not prove that span contains V. The equality im(CC^*)=im C is standard. Moreover ⟨ v,Sv⟩= C^*v ^2=∑_j|⟨ c_j,v⟩|^2. This vanishes exactly when v is orthogonal to every cycle class.
Exact mathematical passage
\langle v,Sv\rangle=\|C^*v\|^2=\sum_j|\langle c_j,v\rangle|^2.
\sum_{j=0}^{q-1}Use in source · 1 passage
Search aliases: ∑_j=0^q-1 sum
Use in source · capacity_cycles.tex, line 418
f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.
Exact mathematical passage
\begin{aligned} f_1&=e_1,&g_1&=e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4,\\ f_2&=e_2,&g_2&=-5e_3,\\ f_3&=P-\kappa z^5,&g_3&=5\sum_{j=0}^{q-1}P^{q-1-j}\kappa^jz^{5j}. \end{aligned}
\sum_{j=1}^{2N-3}Use in source · 1 passage
Search aliases: ∑_j=1^2N-3 sum
Use in source · august_development.tex, line 844
Balanced four-packets and saturation. Let F_m= { γ∈ N^m-1: 1^Tγ=4, R_mγ=2m 1 }. A saturated certificate is an exact nonnegative identity n+∑_j=1^2N-3p_r_j =∑_i=1^Nγ_i, γ_i∈ F_m. The case N=2 is one-step quasi-decomposition. Higher N allows several surface packets to reconstruct one six-packet after Tate stabilization.
Exact mathematical passage
\label{eq:legacy-9.2} \boxed{ n+\sum_{j=1}^{2N-3}p_{r_j} =\sum_{i=1}^N\gamma_i, \qquad \gamma_i\in\mathcal F_m. }
\sum_{j=1}^NUse in source · 1 passage
Search aliases: ∑_j=1^N sum
Use in source · august_development.tex, line 108
tudy. For the positive frame certificate below, require every c_j to belong to V. This can be achieved by a rational algebraic packet projector when such a projector has been constructed. Projection through an arbitrary orthogonal Hodge projector does not establish that the projected vectors are algebraic. Define C:R^N→ H^2p(X,R), Ce_j=c_j, and the positive frame operator S=CC^*=∑_j=1^N|c_j⟩⟨ c_j|.
Exact mathematical passage
S=CC^*=\sum_{j=1}^N|c_j\rangle\langle c_j|.
\sum_{j=1}^{q-1}Use in source · 1 passage
Search aliases: ∑_j=1^q-1 sum
Use in source · capacity_cycles.tex, line 465
df_3wedge dg_3= 25q∑_j=1^q-1κ^jP^q-1-jz^5j-1 dPwedge dz.
Exact mathematical passage
\label{eq:capacity_cycles-5.4} df_3\wedge dg_3= 25q\sum_{j=1}^{q-1}\kappa^jP^{q-1-j}z^{5j-1}\,dP\wedge dz.
\sum_{j\in I_{M,r}}Use in source · 1 passage
Search aliases: ∑_j∈ I_M,r sum in
Use in source · august_development.tex, line 177
Degree-residue polyphase refinement. Exact parent-child frame law. Fix a projective embedding and assign each cycle its degree. For modulus M, define the residue packet I_M,r={j:deg Z_j≡ rpmod M}, S_M,r=∑_j∈ I_M,r|c_j⟩⟨ c_j|.
Exact mathematical passage
I_{M,r}=\{j:\deg Z_j\equiv r\pmod M\}, \qquad S_{M,r}=\sum_{j\in I_{M,r}}|c_j\rangle\langle c_j|.
\sum_{j\in I_\lambda}Use in source · 1 passage
Search aliases: ∑_j∈ I_λ sum in lambda
Use in source · august_development.tex, line 149
Packet assembly without false orthogonality. Partition the cycle indices into packets I_λ. Define S_λ=∑_j∈ I_λ|c_j⟩⟨ c_j|. Then S=∑_λ S_λ. No orthogonality of the child spans is assumed.
Exact mathematical passage
S_\lambda=\sum_{j\in I_\lambda}|c_j\rangle\langle c_j|.
\sum_{k\ge1}Use in source · 1 passage
Search aliases: ∑_k≥1 sum ge
Use in source · august_development.tex, line 1221
For an integer-valued y, discrete coarea gives TV(y) = ∑_k≥1|∂{y≥ k}| + ∑_k≥1|∂{y≤-k}|. If S is a finite forest of internal vertices of a trivalent tree and c(S) is its number of connected components, then |∂ S|=|S|+2c(S), because 3|S|=2(|S|-c(S))+|∂ S|.
Exact mathematical passage
\label{eq:legacy-13.3} \operatorname{TV}(y) = \sum_{k\ge1}|\partial\{y\ge k\}| + \sum_{k\ge1}|\partial\{y\le-k\}|.
\sum_rUse in source · 12 passages
Search aliases: ∑_r sum
Use in source · appendices.tex, line 13
Divisor transfer · T=∑_rB_rA_r with actual Chow correspondences, packet preservation and invertibility.
Exact mathematical passage
T=\sum_rB_rA_r
\sum_{r=1}^{h-1}Use in source · 4 passages
Search aliases: ∑_r=1^h-1 sum
Use in source · august_development.tex, line 1065
Complement defects and dyadic triangles. Put q=frac m4, h=2q, m=4q. For a packet count vector n, define its oriented complement defect ∂ n=∑_r=1^h-1d_n(r)[r], d_n(r)=n_r-n_m-r. The self-complementary residue h contributes zero. Since ρ_m(m-r)=-ρ_m(r), the balanced-packet equation is equivalent to ∑_r=1^h-1d_n(r)ρ_m(ur)=0 for every odd unit u. Moreover, ∂ n _1=∑_r=1^h-1|d_n(r)|≤ 6. Thus the dyadic problem is a short-vector problem in an integral defect lattice.
Exact mathematical passage
\partial n=\sum_{r=1}^{h-1}d_n(r)[r], \qquad d_n(r)=n_r-n_{m-r}.
\sum_{r=1}^MUse in source · 1 passage
Search aliases: ∑_r=1^M sum
Use in source · august_development.tex, line 289
[Invertible divisor-lowering transfer] Suppose T=∑_r=1^M B_rA_r preserves V and is invertible on V. Then every rational Hodge class in V is algebraic. Let α∈ V be of type (p,p). Each A_rα is rational of type (1,1) on Y_r, hence is the class of a rational divisor D_r by the Lefschetz (1,1) theorem. Therefore Tα=∑_r(Λ_r)_*[D_r] is algebraic on X.
Exact mathematical passage
T=\sum_{r=1}^M B_rA_r
\sum_{r\in B}Use in source · 1 passage
Search aliases: ∑_r∈ B sum in
Use in source · august_development.tex, line 1438
For any nonzero-entry block B at modulus m, put h_B(u)=m^-1∑_r∈ B⟨ ur⟩_m, with multiplicities retained.
Exact mathematical passage
h_B(u)=m^{-1}\sum_{r\in B}\langle ur\rangle_m
\sum_{\substack{1\le b<q\\ b\text{ odd}}}Use in source · 1 passage
Search aliases: ∑_substack1≤ b<q b odd sum substack le text
Use in source · august_development.tex, line 1196
[Exact trivalent-tree norm] If d=∑_bz_bκ_b∈ K_m, then d _1 =|y_r_0| +∑_b≠ r_0|y_b-y_p(b)| +2∑_substack1≤ b<q b odd|y_b|. The right side is the total variation of the integer function y on widetilde T_m with zero boundary values. The root coordinate is d(q)=z_r_0. For b≠ r_0, multiplying d(2b)=z_b-ϵ_bz_p(b) by τ_b gives τ_bd(2b)=y_b-y_p(b). For odd b, the two odd coordinates are d(b)=-z_b, d(h-b)=z_b. Taking absolute values and summing all defect coordinates gives [source reference].
Exact mathematical passage
\label{eq:legacy-13.2} \boxed{ \|d\|_1 =|y_{r_0}| +\sum_{b\ne r_0}|y_b-y_{p(b)}| +2\sum_{\substack{1\le b<q\\ b\text{ odd}}}|y_b|. }
\sum_{u\in U_m}Use in source · 1 passage
Search aliases: ∑_u∈ U_m sum in
Use in source · august_development.tex, line 742
Centered Fourier form. Define ρ_m(r)=⟨ r⟩_m-frac m2, F_a(u)=∑_iρ_m(ua_i). Then a is balanced if and only if F_a(u)=0 for all units. Moreover, F_a(-u)=-F_a(u), so only odd characters occur. For a Dirichlet character χ on U_m, put B_m,χ(r)= ∑_u∈ U_mρ_m(ur)overlineχ(u). Finite Fourier inversion gives the exact criterion n(a)∈ B_m Longleftrightarrow ∑_i B_m,χ(a_i)=0 for every odd χ. Here n(a) is the residue count vector. This formulation underlies the prime-power filtration and the proposed conducto…
Exact mathematical passage
\mathcal B_{m,\chi}(r)= \sum_{u\in U_m}\rho_m(ur)\overline{\chi(u)}.
\sum_vUse in source · 2 passages
Search aliases: ∑_v sum
Use in source · capacity_cycles.tex, line 107
C(a)=sup{∑_vw_v: ∑_vw_vv=1, w_v≥0, only finitely many w_v≠0}.
Exact mathematical passage
\label{eq:capacity_cycles-2.1} C(a)=\sup\left\{\sum_vw_v: \sum_vw_vv=\mathbf1,\quad w_v\ge0, \quad\text{only finitely many }w_v\ne0\right\}.
\sum_\lambdaUse in source · 2 passages
Search aliases: ∑_λ sum lambda
Use in source · august_development.tex, line 153
Packet assembly without false orthogonality. Partition the cycle indices into packets I_λ. Define S_λ=∑_j∈ I_λ|c_j⟩⟨ c_j|. Then S=∑_λ S_λ. No orthogonality of the child spans is assumed.
Exact mathematical passage
S=\sum_\lambda S_\lambda.
\sum_{\lambda}Use in source · 1 passage
Search aliases: ∑_λ sum lambda
Use in source · august_development.tex, line 519
This creates a practical research program. One can refine the finite obstruction space by symmetry, monodromy, support, or arithmetic phase, obtaining Φ=∑_λΦ_λ. Since the decomposition is complete, integrability is equivalent to the simultaneous vanishing of all children. Different geometric arguments may eliminate different children. A symmetry packet may vanish by representation theory, a support packet by localization, and an arithmetic packet by compatibility across reductions. The analytic …
Exact mathematical passage
\Phi=\sum_{\lambda}\Phi_{\lambda}.
\sum_{\ell=0}^{L-1}Use in source · 2 passages
Search aliases: ∑_ℓ=0^L-1 sum ell
Use in source · appendices.tex, line 11
Polyphase assembly · S_M,r=∑_ℓ=0^L-1S_LM,r+ℓ M. These are frame sums, not sums of span projectors.
Exact mathematical passage
S_{M,r}=\sum_{\ell=0}^{L-1}S_{LM,r+\ell M}
\setminusUse in source · 3 passages
Search aliases: ∖ setminus
Use in source · august_development.tex, line 243
Support verification and multiplicity recovery. Let Z⊆ X be a closed algebraic subset and j:X∖ Z↪ X. The support exact sequence contains H_Z^2p(X,Q)⟶ H^2p(X,Q)xrightarrowj^*H^2p(X∖ Z,Q).
Exact mathematical passage
j:X\setminus Z\hookrightarrow X
\astUse in source · 1 passage
Search aliases: ∗ ast
Use in source · august_development.tex, line 1532
The desired correspondence is schematically Γ∈CH^∗(F_d× C_P× S_T)_Q, Γ_*:e_orbH^1(F_d)⟶ e_diag(M(C_P)⊗ M(S_T))(1). This formula is a target specification, not a constructed map. Its coefficient field, diagonal projector, rational multiplicities, and dimensions must be fixed before an isomorphism can be asserted. The smaller CM conductor can have a smaller rational degree than the original packet. Matching sign functions alone does not resolve that difference.
Exact mathematical passage
\Gamma\in\CH^{\ast}(F_d\times C_P\times S_T)_{\Q},\qquad \Gamma_*:e_{\mathrm{orb}}H^1(F_d)\longrightarrow e_{\mathrm{diag}}\bigl(M(C_P)\otimes M(S_T)\bigr)(1).
\inftyUse in source · 1 passage
Search aliases: ∞ infty
Use in source · august_development.tex, line 219
[Approximation becomes exact] Let P_D be the orthogonal projector onto A_D^p(X)⊗R. For a rational class α, lim_D→∞ (I-P_D)α =0 if and only if α∈ A_D^p(X) for some finite D. After stabilization, the residual is constant. A constant tail converging to zero is zero.
Exact mathematical passage
\lim_{D\to\infty}\|(I-P_D)\alpha\|=0
\capUse in source · 10 passages
Search aliases: ∩ cap
Use in source · appendices.tex, line 6
Rational Hodge space · Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X).
Exact mathematical passage
\Hdg^p(X)=H^{2p}(X,\Q)\cap H^{p,p}(X)
\cupUse in source · 1 passage
Search aliases: ∪ cup
Use in source · carrier70.tex, line 199
{y_0=y_1=y_2=y_3=0} ∪ {y_0=y_1=y_3=y_5=0} ∪ {y_0=y_1=y_4=y_5=0}.
Exact mathematical passage
\{y_0=y_1=y_2=y_3=0\} \ \cup\ \{y_0=y_1=y_3=y_5=0\} \ \cup\ \{y_0=y_1=y_4=y_5=0\}.
\intUse in source · 7 passages
Search aliases: ∫ int
Use in source · august_development.tex, line 262
Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.
Exact mathematical passage
M_{ki}=\int_X[Z_i]\smile\gamma_k, \qquad b_k=\int_X\alpha\smile\gamma_k.
\int_XUse in source · 1 passage
Search aliases: ∫_X int
Use in source · august_development.tex, line 262
Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.
Exact mathematical passage
M_{ki}=\int_X[Z_i]\smile\gamma_k, \qquad b_k=\int_X\alpha\smile\gamma_k.
\int_ZUse in source · 3 passages
Search aliases: ∫_Z int
Use in source · weil_audit.tex, line 91
2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety graph and is algebraic at J_0. On its three source factors the polarization restricts to weights (2,2,4), giving ∫_Zθ^3=3!·2·2·4=96. Pulling the explicit determinant form back to the graph gives the other two periods. In full, (∫_Zθ^3,∫_Zu,∫…
Exact mathematical passage
\int_Z\theta^3=3!\cdot2\cdot2\cdot4=96
\int_{Z_\kappa}Use in source · 2 passages
Search aliases: ∫_Z_κ int kappa
Use in source · capacity_cycles.tex, line 545
∫_Z_κres(fracQ_bΩF_m^3) =frac(2π i)^250q κ^q-b≠0.
Exact mathematical passage
\label{eq:capacity_cycles-6.5} \boxed{ \int_{Z_\kappa}\operatorname{res}\left(\frac{Q_b\Omega}{F_m^3}\right) =\frac{(2\pi i)^2}{50q}\,\kappa^{q-b}\ne0. }
\simUse in source · 4 passages
Search aliases: ∼ sim
Use in source · august_development.tex, line 424
Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees is equivalent to algebraization of topological K-classes whose Chern-character components are rational Hodge classes [source reference].
Exact mathematical passage
\operatorname{ch}_{\mathrm{top}}:K^0_{\mathrm{top}}(X)\otimes\Q\xrightarrow{\sim}H^{\mathrm{even}}(X,\Q).
\sim_{\mathbf Q}Use in source · 1 passage
Search aliases: ∼_ Q sim mathbf
Use in source · carrier70.tex, line 256
K_Y∼_ Q-3H.
Exact mathematical passage
K_Y\sim_{\mathbf Q}-3H.
\congUse in source · 1 passage
Search aliases: ≅ cong
Use in source · august_development.tex, line 1024
Weighted dyadic reconstruction. Prime 2 cannot be handled by inserting p=2 into the odd-prime orbit-mass proof. The unit group has the separate form Z_2^×≅{±1}×(1+4 Z_2). The first address is a sign packet, only afterward does the principal-unit refinement begin.
Exact mathematical passage
\mathbf Z_2^\times\cong\{\pm1\}\times(1+4\mathbf Z_2).
\neUse in source · 22 passages
Search aliases: ≠ ne
Use in source · august_development.tex, line 301
Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.
Exact mathematical passage
c_0\ne0
\subsetUse in source · 12 passages
Search aliases: ⊂ subset
Use in source · august_development.tex, line 391
Cubic fourfolds as a validation of the transfer theorem. Let X⊂ P^5 be a smooth cubic fourfold and let F(X) be its Fano variety of lines. The universal incidence variety P={(ℓ,x):x∈ℓ}⊂ F(X)× X induces the Abel-Jacobi map A=p_*q^*:H^4_prim(X,Q)→ H^2_prim(F(X),Q). Beauville and Donagi proved that this is a polarized Hodge isomorphism up to the standard twist [source reference]. The polarized adjoint is represented by an algebraic incidence return after inserting the appropriate power of the Plueck…
Exact mathematical passage
X\subset\mathbb P^5
\subseteqUse in source · 8 passages
Search aliases: ⊆ subseteq
Use in source · august_development.tex, line 15
The rational Hodge target. Statement and status. Let X be a smooth projective complex variety. For each codimension p, define Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X) and Alg^p(X)=im(CH^p(X)_QxrightarrowclH^2p(X,Q)). Every algebraic cycle class has type (p,p), so Alg^p(X)⊆Hdg^p(X). The rational Hodge conjecture asserts equality for every smooth projective X and every p [source reference].
Exact mathematical passage
\Alg^p(X)\subseteq\Hdg^p(X)
\bigoplusUse in source · 2 passages
Search aliases: ⊕ bigoplus
Use in source · august_development.tex, line 52
Primitive and coniveau reductions. Lefschetz packet decomposition. Let h∈ H^2(X,Q) be the class of an ample divisor and let L(α)=hsmileα. Hard Lefschetz and primitive decomposition yield H^2p(X,Q)=⊕_r≥0L^rH_prim^2p-2r(X,Q). The decomposition respects Hodge type.
Exact mathematical passage
H^{2p}(X,\Q)=\bigoplus_{r\ge0}L^rH_{\mathrm{prim}}^{2p-2r}(X,\Q).
\bigoplus_{c\mid m}Use in source · 1 passage
Search aliases: ⊕_cmid m bigoplus mid
Use in source · august_development.tex, line 809
endcases Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is itself algebraic.
Exact mathematical passage
\label{eq:legacy-8.3} \boxed{ H^4_{\mathrm{prim}}(X_m^4,\Q) =\bigoplus_{c\mid m}\operatorname{im}Q_{m,c}. }
\bigoplus_{r\ge0}Use in source · 1 passage
Search aliases: ⊕_r≥0 bigoplus ge
Use in source · august_development.tex, line 52
Primitive and coniveau reductions. Lefschetz packet decomposition. Let h∈ H^2(X,Q) be the class of an ample divisor and let L(α)=hsmileα. Hard Lefschetz and primitive decomposition yield H^2p(X,Q)=⊕_r≥0L^rH_prim^2p-2r(X,Q). The decomposition respects Hodge type.
Exact mathematical passage
H^{2p}(X,\Q)=\bigoplus_{r\ge0}L^rH_{\mathrm{prim}}^{2p-2r}(X,\Q).
\otimesUse in source · 8 passages
Search aliases: ⊗ otimes
Use in source · appendices.tex, line 8
Real obstruction · O=P_Hdg⊗R-P_Alg⊗R. No algebraic realization of these orthogonal projectors is presumed.
Exact mathematical passage
\mathcal O=P_{\Hdg\otimes\R}-P_{\Alg\otimes\R}
\otimes_{\Q}Use in source · 3 passages
Search aliases: ⊗_Q otimes Q
Use in source · august_development.tex, line 17
Choose a positive inner product on H^2p(X,R). The orthogonal projectors P_Hdg^p and P_Alg^p act on the real scalar extensions Hdg^p(X)⊗_QR and Alg^p(X)⊗_QR. These real orthogonal projectors provide a reformulation. They are not asserted to be rational matrices or algebraic correspondences. Since the algebraic space is nested in the Hodge space, P_Hdg^pP_Alg^p=P_Alg^p.
Exact mathematical passage
\Hdg^p(X)\otimes_{\Q}\R
\perpUse in source · 3 passages
Search aliases: ⊥ perp
Use in source · august_development.tex, line 27
[Master obstruction projector] The operator O_X^p=P_Hdg^p-P_Alg^p is the orthogonal projector onto (Hdg^p(X)⊗_QR)∩(Alg^p(X)⊗_QR)^⊥. Therefore O_X^p=0 Longleftrightarrow Hdg^p(X)=Alg^p(X). For nested orthogonal projectors Q≤ P, the difference P-Q is self-adjoint and (P-Q)^2=P-PQ-QP+Q=P-Q. Its range is im P∩(im Q)^⊥.
Exact mathematical passage
(\Hdg^p(X)\otimes_{\Q}\R)\cap(\Alg^p(X)\otimes_{\Q}\R)^\perp
\bigcapUse in source · 3 passages
Search aliases: ⋂ bigcap
Use in source · august_development.tex, line 160
[Positive child-kernel intersection] For any subspace V, ker(S|_V)=⋂_λker(S_λ|_V). If Sv=0, positivity gives 0=⟨ v,Sv⟩=∑_λ⟨ v,S_λ v⟩. Each nonnegative summand vanishes, which implies S_λ v=0. The converse is immediate.
Exact mathematical passage
\ker(S|_V)=\bigcap_\lambda\ker(S_\lambda|_V).
\bigcap_rUse in source · 2 passages
Search aliases: ⋂_r bigcap
Use in source · august_development.tex, line 322
Positive multi-channel form. Assume the polarized adjoint A_r^† is realized by an algebraic return correspondence B_r. This is an additional geometric hypothesis, not an automatic consequence in arbitrary settings. Then T=∑_rA_r^† A_r is positive and ⟨ v,Tv⟩=∑_r A_rv ^2. Therefore ker T=⋂_rker A_r. Joint injectivity of the lowering channels is enough.
Exact mathematical passage
\ker T=\bigcap_r\ker A_r.
\bigcap_\lambdaUse in source · 1 passage
Search aliases: ⋂_λ bigcap lambda
Use in source · august_development.tex, line 160
[Positive child-kernel intersection] For any subspace V, ker(S|_V)=⋂_λker(S_λ|_V). If Sv=0, positivity gives 0=⟨ v,Sv⟩=∑_λ⟨ v,S_λ v⟩. Each nonnegative summand vanishes, which implies S_λ v=0. The converse is immediate.
Exact mathematical passage
\ker(S|_V)=\bigcap_\lambda\ker(S_\lambda|_V).
\cdotsUse in source · 14 passages
Search aliases: ⋯ cdots
Use in source · august_development.tex, line 206
Finite saturation and exact rational closure. For each degree cutoff D, let A_D^p(X) be the rational span of codimension-p cycle classes of degree at most D. Then A_1^p⊆ A_2^p⊆⋯⊆ H^2p(X,Q).
Exact mathematical passage
A_1^p\subseteq A_2^p\subseteq\cdots\subseteq H^{2p}(X,\Q).
\rangle_{70}Use in source · 1 passage
Search aliases: ⟩_70 rangle
Use in source · carrier70.tex, line 58
∑_i=0^5⟨ ta_i⟩_70=210.
Exact mathematical passage
\sum_{i=0}^{5}\langle ta_i\rangle_{70}=210.
\rangle_cUse in source · 1 passage
Search aliases: ⟩_c rangle
Use in source · august_development.tex, line 780
orem[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balance equation descends. The maximal common divisor is unique. The converse follows by scaling.
Exact mathematical passage
\langle ugb_i\rangle_{gc}=g\langle \bar u b_i\rangle_c,
\rangle_{gc}Use in source · 1 passage
Search aliases: ⟩_gc rangle
Use in source · august_development.tex, line 780
orem[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balance equation descends. The maximal common divisor is unique. The converse follows by scaling.
Exact mathematical passage
\langle ugb_i\rangle_{gc}=g\langle \bar u b_i\rangle_c,
\rangle_qUse in source · 6 passages
Search aliases: ⟩_q rangle
Use in source · august_development.tex, line 1398
[Four-phase CRT reduction] The six-packet A_b,c,d is balanced modulo 5q if and only if the four-packet Γ_b,c,d is balanced modulo q. For a unit u, let s=⟨ ub⟩_q and z=⟨ u5^-1b⟩_q. The complete five-element fiber over s has sum 5s+10q, its phase-zero lift is 5z. The four nonzero phases therefore sum to 5s+10q-5z. Adding the last two entries and imposing the middle value 15q gives s+⟨ uc⟩_q+⟨ ud⟩_q-z=q. Since ⟨ ue⟩_q=q-z, this is exactly the balance equation for (b,c,d,e).
Exact mathematical passage
s=\langle ub\rangle_q
\rangle_{Y_r}Use in source · 1 passage
Search aliases: ⟩_Y_r rangle
Use in source · august_development.tex, line 333
[Finite Gram certificate] Choose a rational basis e_1,…,e_d of V. If G_ij=∑_r⟨ A_re_i,A_re_j⟩_Y_r has nonzero determinant, every rational Hodge class in V is algebraic.
Exact mathematical passage
G_{ij}=\sum_r\langle A_re_i,A_re_j\rangle_{Y_r}
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- research/hodge-reader/README.md · 0 mathematical passages
- research/hodge-reader/evidence/formal/FORMAL_STATUS.md · 0 mathematical passages
- research/hodge-reader/paper/appendices.tex · 36 mathematical passages
- research/hodge-reader/paper/august_development.tex · 673 mathematical passages
- research/hodge-reader/paper/bibliography.tex · 3 mathematical passages
- research/hodge-reader/paper/capacity_cycles.tex · 185 mathematical passages
- research/hodge-reader/paper/carrier70.tex · 158 mathematical passages
- research/hodge-reader/paper/edition_audit.tex · 29 mathematical passages
- research/hodge-reader/paper/main.tex · 0 mathematical passages
- research/hodge-reader/paper/opening.tex · 1 mathematical passage
- research/hodge-reader/paper/tables/cm_audit.tex · 0 mathematical passages
- research/hodge-reader/paper/tables/dyadic_classification.tex · 1 mathematical passage
- research/hodge-reader/paper/tables/fermat_named_checks.tex · 0 mathematical passages
- research/hodge-reader/paper/tables/odd_prime_spots.tex · 1 mathematical passage
- research/hodge-reader/paper/tables/residual_degrees_100.tex · 0 mathematical passages
- research/hodge-reader/paper/transfer210.tex · 103 mathematical passages
- research/hodge-reader/paper/weil_audit.tex · 90 mathematical passages
- research/hodge-reader/shared/preamble.tex · 0 mathematical passages
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