Paper IX: Arithmetic Packets, Dyadic Towers, and Local Polynomial Deformation

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  p=2 · 1 · 2 · 2^r

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  p=2

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• Notation table · visual_body.tex, line 34

  H · positive row step · Euler degree step 4H

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  H

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• Notation table · visual_body.tex, line 36

  j · address within a section · 0,1,2,3

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  j

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• Notation table · visual_body.tex, line 35

  k · number of differences · order k, independent of n

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  k

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• Notation table · visual_body.tex, line 68

  llll Prime branch · Allowed m · q · Row step for the original shift

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  m

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• Notation table · visual_body.tex, line 72

  p odd · mmid p-1 · p-1 · dfracp-1mp^r-1

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  m\mid p-1

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• Notation table · visual_body.tex, line 33

  n · packed row index · Euler degree 4n

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  n

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• Notation table · visual_body.tex, line 70

  p=2 · 1 · 2 · 2^r

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  p=2

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• Notation table · visual_body.tex, line 72

  p odd · mmid p-1 · p-1 · dfracp-1mp^r-1

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  \dfrac{p-1}{m}p^{r-1}

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• Notation table · visual_body.tex, line 68

  llll Prime branch · Allowed m · q · Row step for the original shift

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  q

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• Notation table · visual_body.tex, line 70

  p=2 · 1 · 2 · 2^r

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  2^r

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 15

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

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  \label{eq:arith-euler-egf}
   \sum_{d\ge0}E_d^{(N,j)}\frac{t^d}{d!}
   =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
  

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• Definition passage: Dyadic truncation · Read passage · 20_arithmetic.tex, line 880

  [Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).

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   A(n)=\frac{3^{2n+2}-1}{4(2n+1)(2n+2)},\qquad
   S_m(n)=\beta_m2^{2m}\frac{(2n)!}{(2n-2m)!}A(n-m)
         \quad(0\le m\le n),
  

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 757

  [Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

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   S_d(p)=\sum_{a=0}^{p-1}(-1)^a(2a+1)^d.
  

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 761

  [Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   \operatorname{sech}^2t=\sum_{m\ge0}\gamma_m\frac{t^{2m}}{(2m)!},
   \qquad
   \gamma_m=\sum_{b=0}^{2m}\binom{2m}{b}E_bE_{2m-b},
  

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 21

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

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   U_{N,j}(n)=E_{Nn}^{(N,j)},\qquad C_{N,j}(n)=\frac{U_{N,j}(n)}{j!}.
  

Search aliases: C_N,j

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 21

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

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   U_{N,j}(n)=E_{Nn}^{(N,j)},\qquad C_{N,j}(n)=\frac{U_{N,j}(n)}{j!}.
  

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 48

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

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   D=\frac{d}{dt},\qquad L_\lambda=\frac{D^N-\lambda}{p}.
  

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 15

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  \label{eq:arith-euler-egf}
   \sum_{d\ge0}E_d^{(N,j)}\frac{t^d}{d!}
   =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
  

Search aliases: D^N

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 48

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

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   D=\frac{d}{dt},\qquad L_\lambda=\frac{D^N-\lambda}{p}.
  

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 15

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  \label{eq:arith-euler-egf}
   \sum_{d\ge0}E_d^{(N,j)}\frac{t^d}{d!}
   =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
  

Search aliases: E_2m-b

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 761

  [Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   \operatorname{sech}^2t=\sum_{m\ge0}\gamma_m\frac{t^{2m}}{(2m)!},
   \qquad
   \gamma_m=\sum_{b=0}^{2m}\binom{2m}{b}E_bE_{2m-b},
  

Search aliases: E_b

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 761

  [Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   \operatorname{sech}^2t=\sum_{m\ge0}\gamma_m\frac{t^{2m}}{(2m)!},
   \qquad
   \gamma_m=\sum_{b=0}^{2m}\binom{2m}{b}E_bE_{2m-b},
  

Search aliases: E_d^(N,j)

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 15

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  \label{eq:arith-euler-egf}
   \sum_{d\ge0}E_d^{(N,j)}\frac{t^d}{d!}
   =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
  

Search aliases: E_Nn^(N,j)

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 21

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  
   U_{N,j}(n)=E_{Nn}^{(N,j)},\qquad C_{N,j}(n)=\frac{U_{N,j}(n)}{j!}.
  

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 100

  [Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…

  Exact mathematical passage

  f

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 42

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

  Exact mathematical passage

  
   \mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}:
                      f_d\in\mathbb Z_p\right\}
         \subset\mathbb Q_p[[t]].
  

Search aliases: H mathcal

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 42

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

  Exact mathematical passage

  
   \mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}:
                      f_d\in\mathbb Z_p\right\}
         \subset\mathbb Q_p[[t]].
  

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 100

  [Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…

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  h,k,n\in\N

Search aliases: H_p mathcal

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 42

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

  Exact mathematical passage

  
   \mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}:
                      f_d\in\mathbb Z_p\right\}
         \subset\mathbb Q_p[[t]].
  

Search aliases: L

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 107

  [Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…

  Exact mathematical passage

  v_p(x)\ge L

Search aliases: L_λ lambda

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 48

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

  Exact mathematical passage

  
   D=\frac{d}{dt},\qquad L_\lambda=\frac{D^N-\lambda}{p}.
  

Search aliases: L_λ^k lambda

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 51

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

  Exact mathematical passage

  L_\lambda^k A\in\mathcal H_p

Search aliases: M

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 767

  rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   T^{\mathrm{odd}}_{p,M}(n)=\sum_{m=0}^{\min(M,n)}
         \binom{2n}{2m}\gamma_m p^{2m}S_{2(n-m)}(p).
  

Search aliases: min

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 767

  rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   T^{\mathrm{odd}}_{p,M}(n)=\sum_{m=0}^{\min(M,n)}
         \binom{2n}{2m}\gamma_m p^{2m}S_{2(n-m)}(p).
  

Search aliases: N

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 14

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  N\ge1

Search aliases: N mathbb

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 100

  [Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…

  Exact mathematical passage

  h,k,n\in\N

Search aliases: odd mathrm

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 767

  rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   T^{\mathrm{odd}}_{p,M}(n)=\sum_{m=0}^{\min(M,n)}
         \binom{2n}{2m}\gamma_m p^{2m}S_{2(n-m)}(p).
  

Search aliases: p^2m

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 767

  rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   T^{\mathrm{odd}}_{p,M}(n)=\sum_{m=0}^{\min(M,n)}
         \binom{2n}{2m}\gamma_m p^{2m}S_{2(n-m)}(p).
  

Search aliases: p^a

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 47

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

  Exact mathematical passage

  N=p^a

Search aliases: p^L

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 110

  put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0. This finite convention makes the displayed exponents meaningful at st…

  Exact mathematical passage

  x\in p^L\Z_p

Search aliases: Q mathbb

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 42

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

  Exact mathematical passage

  
   \mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}:
                      f_d\in\mathbb Z_p\right\}
         \subset\mathbb Q_p[[t]].
  

Search aliases: Q_p mathbb

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 42

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

  Exact mathematical passage

  
   \mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}:
                      f_d\in\mathbb Z_p\right\}
         \subset\mathbb Q_p[[t]].
  

Search aliases: S

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 757

  [Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   S_d(p)=\sum_{a=0}^{p-1}(-1)^a(2a+1)^d.
  

Search aliases: S_2(n-m)

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 767

  rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   T^{\mathrm{odd}}_{p,M}(n)=\sum_{m=0}^{\min(M,n)}
         \binom{2n}{2m}\gamma_m p^{2m}S_{2(n-m)}(p).
  

Search aliases: S_d

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 757

  [Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   S_d(p)=\sum_{a=0}^{p-1}(-1)^a(2a+1)^d.
  

Search aliases: S_m

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• Definition passage: Dyadic truncation · Read passage · 20_arithmetic.tex, line 880

  [Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).

  Exact mathematical passage

  
   A(n)=\frac{3^{2n+2}-1}{4(2n+1)(2n+2)},\qquad
   S_m(n)=\beta_m2^{2m}\frac{(2n)!}{(2n-2m)!}A(n-m)
         \quad(0\le m\le n),
  

Search aliases: sech operatorname

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 760

  [Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  \operatorname{sech}^2

Search aliases: sech^2 operatorname

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 760

  [Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  \operatorname{sech}^2

Search aliases: sinh

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• Definition passage: Dyadic truncation · Read passage · 20_arithmetic.tex, line 874

  [Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).

  Exact mathematical passage

  
   \sum_{m\ge0}\beta_m z^{2m}=\left(\frac{z}{\sinh z}\right)^2,
   \qquad \beta_0=1,
  

Search aliases: T

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 767

  rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   T^{\mathrm{odd}}_{p,M}(n)=\sum_{m=0}^{\min(M,n)}
         \binom{2n}{2m}\gamma_m p^{2m}S_{2(n-m)}(p).
  

Search aliases: t

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 15

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  \label{eq:arith-euler-egf}
   \sum_{d\ge0}E_d^{(N,j)}\frac{t^d}{d!}
   =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
  

Search aliases: t^2m

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 761

  [Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   \operatorname{sech}^2t=\sum_{m\ge0}\gamma_m\frac{t^{2m}}{(2m)!},
   \qquad
   \gamma_m=\sum_{b=0}^{2m}\binom{2m}{b}E_bE_{2m-b},
  

Search aliases: t^d

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 15

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  \label{eq:arith-euler-egf}
   \sum_{d\ge0}E_d^{(N,j)}\frac{t^d}{d!}
   =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
  

Search aliases: t^Nn

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 15

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  \label{eq:arith-euler-egf}
   \sum_{d\ge0}E_d^{(N,j)}\frac{t^d}{d!}
   =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
  

Search aliases: T^odd_p,M mathrm

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 767

  rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   T^{\mathrm{odd}}_{p,M}(n)=\sum_{m=0}^{\min(M,n)}
         \binom{2n}{2m}\gamma_m p^{2m}S_{2(n-m)}(p).
  

Search aliases: T_M

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• Definition passage: Dyadic truncation · Read passage · 20_arithmetic.tex, line 886

  n] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).

  Exact mathematical passage

  \label{eq:arith-tm}
   T_M(n)=\sum_{m=0}^{\min(M,n)}S_m(n)\qquad(M,n\in\N).
  

Search aliases: U

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 21

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  
   U_{N,j}(n)=E_{Nn}^{(N,j)},\qquad C_{N,j}(n)=\frac{U_{N,j}(n)}{j!}.
  

Search aliases: u

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 111

  d Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0. This finite convention makes the displayed exponents meaningful at step zero, where all positive-order differe…

  Exact mathematical passage

  \nu_p(u)=v_p(u)

Search aliases: U_N,j

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 21

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  
   U_{N,j}(n)=E_{Nn}^{(N,j)},\qquad C_{N,j}(n)=\frac{U_{N,j}(n)}{j!}.
  

Search aliases: v

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 106

  [Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…

  Exact mathematical passage

  v_p(0)=+\infty

Search aliases: v_p

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 106

  [Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…

  Exact mathematical passage

  v_p(0)=+\infty

Search aliases: x

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 107

  [Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…

  Exact mathematical passage

  v_p(x)\ge L

Search aliases: Z mathbb

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 110

  put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0. This finite convention makes the displayed exponents meaningful at st…

  Exact mathematical passage

  x\in p^L\Z_p

Search aliases: Z mathbb

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 42

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

  Exact mathematical passage

  
   \mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}:
                      f_d\in\mathbb Z_p\right\}
         \subset\mathbb Q_p[[t]].
  

Search aliases: z

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• Definition passage: Dyadic truncation · Read passage · 20_arithmetic.tex, line 874

  [Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).

  Exact mathematical passage

  
   \sum_{m\ge0}\beta_m z^{2m}=\left(\frac{z}{\sinh z}\right)^2,
   \qquad \beta_0=1,
  

Search aliases: z^2m

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• Definition passage: Dyadic truncation · Read passage · 20_arithmetic.tex, line 874

  [Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).

  Exact mathematical passage

  
   \sum_{m\ge0}\beta_m z^{2m}=\left(\frac{z}{\sinh z}\right)^2,
   \qquad \beta_0=1,
  

Search aliases: Z_p Z mathbb

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 110

  put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0. This finite convention makes the displayed exponents meaningful at st…

  Exact mathematical passage

  x\in p^L\Z_p

Search aliases: Z_p mathbb

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 42

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

  Exact mathematical passage

  
   \mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}:
                      f_d\in\mathbb Z_p\right\}
         \subset\mathbb Q_p[[t]].
  

Search aliases: β beta

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• Definition passage: Dyadic truncation · Read passage · 20_arithmetic.tex, line 873

  [Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).

  Exact mathematical passage

  \beta_m

Search aliases: β_0 beta

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• Definition passage: Dyadic truncation · Read passage · 20_arithmetic.tex, line 874

  [Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).

  Exact mathematical passage

  
   \sum_{m\ge0}\beta_m z^{2m}=\left(\frac{z}{\sinh z}\right)^2,
   \qquad \beta_0=1,
  

Search aliases: β_m beta

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• Definition passage: Dyadic truncation · Read passage · 20_arithmetic.tex, line 873

  [Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).

  Exact mathematical passage

  \beta_m

Search aliases: γ gamma

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 760

  [Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  \gamma_m

Search aliases: γ_0 gamma

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 766

  -length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  \gamma_0=1

Search aliases: γ_m gamma

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 760

  [Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  \gamma_m

Search aliases: Δ Delta

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 101

  [Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…

  Exact mathematical passage

  
   \Delta_h f(n)=f(n+h)-f(n),\qquad
   \Delta_h^0 f(n)=f(n),\qquad
   \Delta_h^{k+1}f(n)=\Delta_h^k f(n+h)-\Delta_h^k f(n).
  

Search aliases: Δ_h Delta

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 101

  [Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…

  Exact mathematical passage

  
   \Delta_h f(n)=f(n+h)-f(n),\qquad
   \Delta_h^0 f(n)=f(n),\qquad
   \Delta_h^{k+1}f(n)=\Delta_h^k f(n+h)-\Delta_h^k f(n).
  

Search aliases: Δ_h^0 Delta

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 101

  [Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…

  Exact mathematical passage

  
   \Delta_h f(n)=f(n+h)-f(n),\qquad
   \Delta_h^0 f(n)=f(n),\qquad
   \Delta_h^{k+1}f(n)=\Delta_h^k f(n+h)-\Delta_h^k f(n).
  

Search aliases: Δ_h^k Delta

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 101

  [Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…

  Exact mathematical passage

  
   \Delta_h f(n)=f(n+h)-f(n),\qquad
   \Delta_h^0 f(n)=f(n),\qquad
   \Delta_h^{k+1}f(n)=\Delta_h^k f(n+h)-\Delta_h^k f(n).
  

Search aliases: Δ_h^k+1 Delta

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 101

  [Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…

  Exact mathematical passage

  
   \Delta_h f(n)=f(n+h)-f(n),\qquad
   \Delta_h^0 f(n)=f(n),\qquad
   \Delta_h^{k+1}f(n)=\Delta_h^k f(n+h)-\Delta_h^k f(n).
  

Search aliases: λ lambda

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 47

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

  Exact mathematical passage

  \lambda\in\mathbb Z_p

Search aliases: ν nu

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 111

  d Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0. This finite convention makes the displayed exponents meaningful at step zero, where all positive-order differe…

  Exact mathematical passage

  \nu_p(u)=v_p(u)

Search aliases: ν_p nu

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 111

  d Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0. This finite convention makes the displayed exponents meaningful at step zero, where all positive-order differe…

  Exact mathematical passage

  \nu_p(u)=v_p(u)

Search aliases: ∈ in

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 100

  [Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…

  Exact mathematical passage

  h,k,n\in\N

Search aliases: ∑ sum

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 15

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  \label{eq:arith-euler-egf}
   \sum_{d\ge0}E_d^{(N,j)}\frac{t^d}{d!}
   =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
  

Search aliases: ∑_a=0^p-1 sum

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 757

  [Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   S_d(p)=\sum_{a=0}^{p-1}(-1)^a(2a+1)^d.
  

Search aliases: ∑_b=0^2m sum

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 761

  [Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   \operatorname{sech}^2t=\sum_{m\ge0}\gamma_m\frac{t^{2m}}{(2m)!},
   \qquad
   \gamma_m=\sum_{b=0}^{2m}\binom{2m}{b}E_bE_{2m-b},
  

Search aliases: ∑_d≥0 sum ge

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 15

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  \label{eq:arith-euler-egf}
   \sum_{d\ge0}E_d^{(N,j)}\frac{t^d}{d!}
   =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
  

Search aliases: ∑_d≥0 sum geq

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 42

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

  Exact mathematical passage

  
   \mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}:
                      f_d\in\mathbb Z_p\right\}
         \subset\mathbb Q_p[[t]].
  

Search aliases: ∑_m=0^min(M,n) sum min

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 767

  rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   T^{\mathrm{odd}}_{p,M}(n)=\sum_{m=0}^{\min(M,n)}
         \binom{2n}{2m}\gamma_m p^{2m}S_{2(n-m)}(p).
  

Search aliases: ∑_m≥0 sum ge

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• Definition passage: Odd-length source and inverse coefficients · Read passage · 20_arithmetic.tex, line 761

  [Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed only to interpret the result as a prime-adic congruence ladder.

  Exact mathematical passage

  
   \operatorname{sech}^2t=\sum_{m\ge0}\gamma_m\frac{t^{2m}}{(2m)!},
   \qquad
   \gamma_m=\sum_{b=0}^{2m}\binom{2m}{b}E_bE_{2m-b},
  

Search aliases: ∑_n≥0 sum ge

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 15

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  \label{eq:arith-euler-egf}
   \sum_{d\ge0}E_d^{(N,j)}\frac{t^d}{d!}
   =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
  

Search aliases: ∞ infty

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• Definition passage: Differences and valuation conventions · Read passage · 20_arithmetic.tex, line 106

  [Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…

  Exact mathematical passage

  v_p(0)=+\infty

Search aliases: ≠ ne

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 20

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  1/j!\ne0

Search aliases: ≤ le

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 14

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  0\le j<N

Search aliases: ≥ ge

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• Definition passage: Congruential Euler coefficients · Read passage · 20_arithmetic.tex, line 14

  [Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.

  Exact mathematical passage

  N\ge1

Search aliases: ≥ geq

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 42

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

  Exact mathematical passage

  
   \mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}:
                      f_d\in\mathbb Z_p\right\}
         \subset\mathbb Q_p[[t]].
  

Search aliases: ⊂ subset

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• Definition passage: Hurwitz integral series · Read passage · strengthening.tex, line 42

  [Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.

  Exact mathematical passage

  
   \mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}:
                      f_d\in\mathbb Z_p\right\}
         \subset\mathbb Q_p[[t]].
  

Search aliases: *

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• Introduction passage · Read passage · 72_recovered.tex, line 2041

  Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.

  Exact mathematical passage

  c=a*b

Search aliases: 1 mathbf

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• Introduction passage · Read passage · 10_foundations.tex, line 129

  In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.

  Exact mathematical passage

  
  (f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad
  (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
  

Search aliases: 1_n mathbf

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• Introduction passage · Read passage · 10_foundations.tex, line 129

  In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.

  Exact mathematical passage

  
  (f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad
  (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
  

Search aliases: A_r

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• Introduction passage · Read passage · 72_recovered.tex, line 2041

  Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.

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  A_r

Search aliases: A_x

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• Introduction passage · Read passage · continuation_01.tex, line 22

  Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d; the cokernel below is presented directly using its integer-domain image.

  Exact mathematical passage

  
  r_x=F(x)/q^L\in\mathbb Z^e,
  \qquad A_x=DF(x):\mathbb Z^d\longrightarrow\mathbb Z^e,
  \qquad J_x=\rho_q\circ A_x.
  

Search aliases: B

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• Introduction passage · Read passage · 72_recovered.tex, line 2041

  Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.

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  B_r

Search aliases: B_1

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• Introduction passage · Read passage · strengthening.tex, line 411

  Let B_m denote the Bernoulli numbers with B_1=-1/2. The identity obtained by substituting u=e^t-1 into log(1+u)/u is B_m=∑_r=0^m(-1)^rfracr!r+1 {m

  Exact mathematical passage

  B_1=-1/2

Search aliases: B_4^s

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• Introduction passage · Read passage · supplement_11.tex, line 78

  Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.

  Exact mathematical passage

  B_4^sF\in\mathcal H_{\mathrm{int}}

Search aliases: B_j

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• Introduction passage · Read passage · strengthening.tex, line 1089

  [Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j, there is one residue and the least row period is one.

  Exact mathematical passage

  
   \widetilde C_j(\mathbb Z_2)=1+2^{b_j}\mathbb Z_2,
   \qquad j!\widetilde C_j(\mathbb Z_2)=j!+2^{B_j}\mathbb Z_2.
   \label{eq:four-interpolation-images-final}
  

Search aliases: b_j

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• Introduction passage · Read passage · strengthening.tex, line 1085

  [Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j, there is one residue and the least row period is one.

  Exact mathematical passage

  
   T_j(x)=\frac{\widetilde C_j(x)-1}{2^{b_j}}
  

Search aliases: B_m

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• Introduction passage · Read passage · strengthening.tex, line 411

  Let B_m denote the Bernoulli numbers with B_1=-1/2. The identity obtained by substituting u=e^t-1 into log(1+u)/u is B_m=∑_r=0^m(-1)^rfracr!r+1 {m

  Exact mathematical passage

  B_m

Search aliases: B_r

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• Introduction passage · Read passage · 72_recovered.tex, line 2041

  Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.

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  B_r

Search aliases: c

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• Introduction passage · Read passage · 72_recovered.tex, line 2041

  Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.

  Exact mathematical passage

  c=a*b

Search aliases: C^N

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• Introduction passage · Read passage · supplement_06.tex, line 69

  Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands pol…

  Exact mathematical passage

  D=C^N

Search aliases: c_a

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• Introduction passage · Read passage · supplement_06.tex, line 70

  Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands pol…

  Exact mathematical passage

  
   \chi_D(X)=\det(XI-D)=X^d+\sum_{a=0}^{d-1}c_aX^a
  

Search aliases: C_r

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• Introduction passage · Read passage · 72_recovered.tex, line 2041

  Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.

  Exact mathematical passage

  C_r

Search aliases: cosh

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• Introduction passage · Read passage · 72_recovered.tex, line 66

  The central change in this edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weigh…

  Exact mathematical passage

  
  H_p(t)=\cosh^2(pt)\operatorname{sech}t,
  \qquad
  K_p(t)=\operatorname{sech}^2(pt).
  

Search aliases: cosh^2 cosh

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• Introduction passage · Read passage · 72_recovered.tex, line 66

  The central change in this edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weigh…

  Exact mathematical passage

  
  H_p(t)=\cosh^2(pt)\operatorname{sech}t,
  \qquad
  K_p(t)=\operatorname{sech}^2(pt).
  

Search aliases: cosh_1 cosh

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• Introduction passage · Read passage · supplement_02.tex, line 36

  r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e^at, cosh_a(t)=frace^at+e^-at2, sech(t)=cosh_1(t)^-1. The constant term of cosh_a is proved to be one, so the reciprocal used here is a genuine unit i…

  Exact mathematical passage

  
   \exp_a(t)=e^{at},\qquad
   \cosh_a(t)=\frac{e^{at}+e^{-at}}2,\qquad
   \sech(t)=\cosh_1(t)^{-1}.
  

Search aliases: cosh_a cosh

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• Introduction passage · Read passage · supplement_02.tex, line 36

  r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e^at, cosh_a(t)=frace^at+e^-at2, sech(t)=cosh_1(t)^-1. The constant term of cosh_a is proved to be one, so the reciprocal used here is a genuine unit i…

  Exact mathematical passage

  
   \exp_a(t)=e^{at},\qquad
   \cosh_a(t)=\frac{e^{at}+e^{-at}}2,\qquad
   \sech(t)=\cosh_1(t)^{-1}.
  

Search aliases: D^a

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• Introduction passage · Read passage · supplement_06.tex, line 74

  packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands polynomial evaluation into a finite sum and proves ∑_a=0^dχ_D[a]D^a=0, χ_D[d]=1. Here χ_D[a] denotes the coefficient at index a. The identities hold over any commutative coefficient ring; the …

  Exact mathematical passage

  
   \sum_{a=0}^{d}\chi_D[a]D^a=0,
   \qquad \chi_D[d]=1.
  

Search aliases: D^d

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• Introduction passage · Read passage · 10_foundations.tex, line 129

  In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.

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  (f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad
  (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
  

Search aliases: D_T

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• Introduction passage · Read passage · 40_regularity.tex, line 17

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  
   J=D_YF(a)\in M_e(\mathbb K),\qquad
   K=D_TF(a)\in M_{e\times k}(\mathbb K).
   \label{eq:regularity-blocks}
  

Search aliases: D_Y

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• Introduction passage · Read passage · 40_regularity.tex, line 17

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  
   J=D_YF(a)\in M_e(\mathbb K),\qquad
   K=D_TF(a)\in M_{e\times k}(\mathbb K).
   \label{eq:regularity-blocks}
  

Search aliases: det

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• Introduction passage · Read passage · supplement_06.tex, line 70

  Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands pol…

  Exact mathematical passage

  
   \chi_D(X)=\det(XI-D)=X^d+\sum_{a=0}^{d-1}c_aX^a
  

Search aliases: e

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• Introduction passage · Read passage · 40_regularity.tex, line 10

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  e,k\in\N

Search aliases: E^(N,j)_Nn

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• Introduction passage · Read passage · supplement_07.tex, line 184

  Normalization is also an explicit implication. Using the existing definition E^(N,j)_Nn=j!C_N,j(n), the source proves that multiplication by j! commutes with every iterated difference. If the unnormalized difference has precision A, the normalized difference has precision A-v_p(j!). Precision is defined as x=0 hboxor v_p(x)≥ A, so zero and negative factorial-loss thresholds are handled correctly. For j<p, the loss is zero.

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  E^{(N,j)}_{Nn}=j!C_{N,j}(n)

Search aliases: e^-at

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• Introduction passage · Read passage · supplement_02.tex, line 36

  r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e^at, cosh_a(t)=frace^at+e^-at2, sech(t)=cosh_1(t)^-1. The constant term of cosh_a is proved to be one, so the reciprocal used here is a genuine unit i…

  Exact mathematical passage

  
   \exp_a(t)=e^{at},\qquad
   \cosh_a(t)=\frac{e^{at}+e^{-at}}2,\qquad
   \sech(t)=\cosh_1(t)^{-1}.
  

Search aliases: e^at

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• Introduction passage · Read passage · supplement_02.tex, line 36

  r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e^at, cosh_a(t)=frace^at+e^-at2, sech(t)=cosh_1(t)^-1. The constant term of cosh_a is proved to be one, so the reciprocal used here is a genuine unit i…

  Exact mathematical passage

  
   \exp_a(t)=e^{at},\qquad
   \cosh_a(t)=\frac{e^{at}+e^{-at}}2,\qquad
   \sech(t)=\cosh_1(t)^{-1}.
  

Search aliases: e^t

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• Introduction passage · Read passage · strengthening.tex, line 412

  Let B_m denote the Bernoulli numbers with B_1=-1/2. The identity obtained by substituting u=e^t-1 into log(1+u)/u is B_m=∑_r=0^m(-1)^rfracr!r+1 {m

  Exact mathematical passage

  u=e^t-1

Search aliases: E_h

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• Introduction passage · Read passage · 10_foundations.tex, line 271

  Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients vanish when their lower index exceeds n, which proves the interpolation claim. Equality of every point of the stencil proves equality of differences. …

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  E_h f(n)=f(n+h)

Search aliases: E_j

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• Introduction passage · Read passage · supplement_10.tex, line 5

  ntCheckpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical; the final core records their resolution. Scope and final statement. Let p be a prime, let m>0 divide p-1, put N=mp and h=(p-1)/m, and let 0≤ j<N. The original rational packet is defined by ∑_q=0^nbinomNn+jNqC_N,j(q)=δ_n0, E_j(n)=j! C_N,j(n). For every n,k,u∈ N, the new formal result establishes E_j(n)∈ Z_p, Δ_uh^kE_j(n)∈ p^min(k,mn) Z_p. He…

  Exact mathematical passage

  
   \sum_{q=0}^{n}\binom{Nn+j}{Nq}C_{N,j}(q)=\delta_{n0},
   \qquad E_j(n)=j!\,C_{N,j}(n).
  

Search aliases: e_n+h^(m,p)

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• Introduction passage · Read passage · 72_recovered.tex, line 55

  Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are h periodic after their constant term. The two arguments expose different parts of the same finite character structure.

  Exact mathematical passage

  
  e_{n+h}^{(m,p)}=e_n^{(m,p)}
  \qquad(n\geq1).
  

Search aliases: e_n+uh^(m,p)

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• Introduction passage · Read passage · supplement_07.tex, line 109

  The positive-tail period. Write h=(p-1)/m. For n≥1, evaluate U^pf_m=Uf_m at the nonnegative degree mn-1: f_m(mn+p-1)=f_m(mn). Since mn+p-1=m(n+h), the identified sequence satisfies e_n+h^(m,p)=e_n^(m,p) (n≥1). Induction gives e_n+uh^(m,p)=e_n^(m,p) for every u≥0. This proves a period, not necessarily the least positive period. The proof also includes p=2, where m=1 and h=1.

  Exact mathematical passage

  e_{n+uh}^{(m,p)}=e_n^{(m,p)}

Search aliases: e_n^(m,p)

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• Introduction passage · Read passage · 72_recovered.tex, line 55

  Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are h periodic after their constant term. The two arguments expose different parts of the same finite character structure.

  Exact mathematical passage

  
  e_{n+h}^{(m,p)}=e_n^{(m,p)}
  \qquad(n\geq1).
  

Search aliases: e_q^(m,p)

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• Introduction passage · Read passage · 72_recovered.tex, line 47

  Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are h periodic after their constant term. The two arguments expose different parts of the same finite character structure.

  Exact mathematical passage

  
  \sum_{q=0}^{n}
  \binom{mn}{mq}e_q^{(m,p)}
  =
  \delta_{n0}
  \quad\text{in }\FF_p.
  

Search aliases: exp

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• Introduction passage · Read passage · supplement_02.tex, line 36

  r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e^at, cosh_a(t)=frace^at+e^-at2, sech(t)=cosh_1(t)^-1. The constant term of cosh_a is proved to be one, so the reciprocal used here is a genuine unit i…

  Exact mathematical passage

  
   \exp_a(t)=e^{at},\qquad
   \cosh_a(t)=\frac{e^{at}+e^{-at}}2,\qquad
   \sech(t)=\cosh_1(t)^{-1}.
  

Search aliases: exp_a exp

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• Introduction passage · Read passage · supplement_02.tex, line 36

  r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e^at, cosh_a(t)=frace^at+e^-at2, sech(t)=cosh_1(t)^-1. The constant term of cosh_a is proved to be one, so the reciprocal used here is a genuine unit i…

  Exact mathematical passage

  
   \exp_a(t)=e^{at},\qquad
   \cosh_a(t)=\frac{e^{at}+e^{-at}}2,\qquad
   \sech(t)=\cosh_1(t)^{-1}.
  

Search aliases: F

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• Introduction passage · Read passage · 40_regularity.tex, line 12

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  
   F\in\mathbb K[Y_1,\ldots,Y_e,T_1,\ldots,T_k]^e,
   \qquad a=(y_0,t_0),
  

Search aliases: F_0

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• Introduction passage · Read passage · 72_recovered.tex, line 2892

  [Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.

  Exact mathematical passage

  F_0=0

Search aliases: F_1

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• Introduction passage · Read passage · 72_recovered.tex, line 2638

  Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.

  Exact mathematical passage

  
  F=(F_1,\ldots,F_r):\Zp^d\longrightarrow\Zp^r
  

Search aliases: F_e

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• Introduction passage · Read passage · continuation_01.tex, line 16

  Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d; the cokernel below is presented directly using its integer-domain image.

  Exact mathematical passage

  
  F=(F_1,\ldots,F_e)\in\mathbb Z[X_1,\ldots,X_d]^e,
  \quad q\ge1,\quad L\ge1,
  \quad x\in\mathbb Z^d,
  \quad F(x)\equiv0\pmod{q^L}.
  

Search aliases: F_i

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• Introduction passage · Read passage · 72_recovered.tex, line 2641

  Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.

  Exact mathematical passage

  F_i\in\Z_p[X_1,\ldots,X_d]

Search aliases: f_m

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• Introduction passage · Read passage · supplement_07.tex, line 101

  The positive-tail period. Write h=(p-1)/m. For n≥1, evaluate U^pf_m=Uf_m at the nonnegative degree mn-1: f_m(mn+p-1)=f_m(mn). Since mn+p-1=m(n+h), the identified sequence satisfies e_n+h^(m,p)=e_n^(m,p) (n≥1). Induction gives e_n+uh^(m,p)=e_n^(m,p) for every u≥0. This proves a period, not necessarily the least positive period. The proof also includes p=2, where m=1 and h=1.

  Exact mathematical passage

  U^pf_m=Uf_m

Search aliases: F_n

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• Introduction passage · Read passage · 72_recovered.tex, line 2892

  [Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.

  Exact mathematical passage

  F_{n+2}=F_{n+1}+F_n

Search aliases: F_N(k+1)+j

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• Introduction passage · Read passage · 72_recovered.tex, line 2893

  [Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.

  Exact mathematical passage

  
  \boxed{
  F_{N(k+2)+j}
  =
  L_NF_{N(k+1)+j}
  -(-1)^NF_{Nk+j}.
  }
  

Search aliases: F_N(k+2)+j

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• Introduction passage · Read passage · 72_recovered.tex, line 2893

  [Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.

  Exact mathematical passage

  
  \boxed{
  F_{N(k+2)+j}
  =
  L_NF_{N(k+1)+j}
  -(-1)^NF_{Nk+j}.
  }
  

Search aliases: F_n+1

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• Introduction passage · Read passage · 72_recovered.tex, line 2892

  [Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.

  Exact mathematical passage

  F_{n+2}=F_{n+1}+F_n

Search aliases: F_n+2

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• Introduction passage · Read passage · 72_recovered.tex, line 2892

  [Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.

  Exact mathematical passage

  F_{n+2}=F_{n+1}+F_n

Search aliases: f_n+d

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• Introduction passage · Read passage · 10_foundations.tex, line 129

  In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.

  Exact mathematical passage

  
  (f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad
  (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
  

Search aliases: F_Nk+j

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• Introduction passage · Read passage · 72_recovered.tex, line 2893

  [Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.

  Exact mathematical passage

  
  \boxed{
  F_{N(k+2)+j}
  =
  L_NF_{N(k+1)+j}
  -(-1)^NF_{Nk+j}.
  }
  

Search aliases: F_p

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• Introduction passage · Read passage · supplement_02.tex, line 31

  rence theorem remain pending. No continuum convergence or analytic summability theorem is inferred from the formal power-series equalities in this block. The historical numerical study is not being relabelled as a fresh experiment. pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; s…

  Exact mathematical passage

  
   S_n(p)=\sum_{a=0}^{p-1}(-1)^a(2a+1)^n\in\Z,
   \qquad F_p(t)=\sum_{n\ge0}S_n(p)\frac{t^n}{n!}.
  

Search aliases: F_r

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• Introduction passage · Read passage · 72_recovered.tex, line 2638

  Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.

  Exact mathematical passage

  
  F=(F_1,\ldots,F_r):\Zp^d\longrightarrow\Zp^r
  

Search aliases: f_r

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• Introduction passage · Read passage · 10_foundations.tex, line 129

  In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.

  Exact mathematical passage

  
  (f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad
  (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
  

Search aliases: FF F mathbb

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• Introduction passage · Read passage · 72_recovered.tex, line 47

  Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are h periodic after their constant term. The two arguments expose different parts of the same finite character structure.

  Exact mathematical passage

  
  \sum_{q=0}^{n}
  \binom{mn}{mq}e_q^{(m,p)}
  =
  \delta_{n0}
  \quad\text{in }\FF_p.
  

Search aliases: FF_p F_p FF mathbb

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• Introduction passage · Read passage · 72_recovered.tex, line 47

  Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are h periodic after their constant term. The two arguments expose different parts of the same finite character structure.

  Exact mathematical passage

  
  \sum_{q=0}^{n}
  \binom{mn}{mq}e_q^{(m,p)}
  =
  \delta_{n0}
  \quad\text{in }\FF_p.
  

Search aliases: FF_p^+ F_p^+ FF mathbb

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• Introduction passage · Read passage · 72_recovered.tex, line 175

  Let p be prime. Write F_p^+ for the additive group of the finite field.

  Exact mathematical passage

  \FF_p^+

Search aliases: FF_p^r F_p^r FF mathbb

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• Introduction passage · Read passage · 72_recovered.tex, line 2522

  Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.

  Exact mathematical passage

  
  \rho_L(x_L)
  =
  \frac{Mx_L-b}{p^L}
  \pmod p
  \in\FF_p^r.
  

Search aliases: g

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• Introduction passage · Read passage · 10_foundations.tex, line 129

  In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.

  Exact mathematical passage

  
  (f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad
  (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
  

Search aliases: g_n-r

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• Introduction passage · Read passage · 10_foundations.tex, line 129

  In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.

  Exact mathematical passage

  
  (f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad
  (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
  

Search aliases: H_2

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• Introduction passage · Read passage · 72_recovered.tex, line 72

  edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weights. This is the correct sen…

  Exact mathematical passage

  
  H_2(t)=\frac{\sinh(2t)\sinh t}{2t^2},
  \qquad
  K_2(t)=\frac{4t^2}{\sinh^2(2t)}.
  

Search aliases: H_int mathcal mathrm

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• Introduction passage · Read passage · supplement_11.tex, line 78

  Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.

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  \mathcal H_{\mathrm{int}}

Search aliases: H_p

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• Introduction passage · Read passage · 72_recovered.tex, line 61

  The central change in this edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weigh…

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  H_p

Search aliases: hbox

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• Introduction passage · Read passage · supplement_07.tex, line 185

  Normalization is also an explicit implication. Using the existing definition E^(N,j)_Nn=j!C_N,j(n), the source proves that multiplication by j! commutes with every iterated difference. If the unnormalized difference has precision A, the normalized difference has precision A-v_p(j!). Precision is defined as x=0 hboxor v_p(x)≥ A, so zero and negative factorial-loss thresholds are handled correctly. For j<p, the loss is zero.

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   x=0\quad\hbox{or}\quad v_p(x)\ge A,
  

Search aliases: I

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• Introduction passage · Read passage · 10_foundations.tex, line 272

  Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients vanish when their lower index exceeds n, which proves the interpolation claim. Equality of every point of the stencil proves equality of differences. …

  Exact mathematical passage

  I+\Delta_1

Search aliases: i

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• Introduction passage · Read passage · 72_recovered.tex, line 2641

  Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.

  Exact mathematical passage

  F_i\in\Z_p[X_1,\ldots,X_d]

Search aliases: im operatorname

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• Introduction passage · Read passage · 30_lifting.tex, line 170

  s over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K).

  Exact mathematical passage

  \lambda_J:\im J\to C(J,K)

Search aliases: int mathrm

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• Introduction passage · Read passage · supplement_11.tex, line 78

  Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.

  Exact mathematical passage

  \mathcal H_{\mathrm{int}}

Search aliases: J

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• Introduction passage · Read passage · 30_lifting.tex, line 162

  The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K).

  Exact mathematical passage

  J:U\to V

Search aliases: J^-1

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• Introduction passage · Read passage · continuation_11.tex, line 36

  bseteq K^k, V⊆ K^e such that t_0∈ U, y_0∈ V, ψ(t_0)=y_0, ψ(U)⊆ V. The map is analytic at every point of U, and its graph describes all roots in the product neighborhood: ∀ t∈ U ∀ y∈ V, F(y,t)=0 Longleftrightarrow y=ψ(t). At the base parameter it has the strict Fréchet derivative Dψ(t_0)=-J^-1K. Consequently it is differentiable on U. The map is defined on the ambient parameter space, but its root, analyticity and uniqueness conclusions are restricted to the specified neighborhood. Uniqueness is …

  Exact mathematical passage

  
  \boxed{D\psi(t_0)=-J^{-1}K.}
  

Search aliases: J_F

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• Introduction passage · Read passage · 72_recovered.tex, line 2652

  rrow Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.

  Exact mathematical passage

  J_F(x_L)

Search aliases: J_x

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• Introduction passage · Read passage · continuation_01.tex, line 22

  Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d; the cokernel below is presented directly using its integer-domain image.

  Exact mathematical passage

  
  r_x=F(x)/q^L\in\mathbb Z^e,
  \qquad A_x=DF(x):\mathbb Z^d\longrightarrow\mathbb Z^e,
  \qquad J_x=\rho_q\circ A_x.
  

Search aliases: K

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• Introduction passage · Read passage · 30_lifting.tex, line 162

  The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K).

  Exact mathematical passage

  K:U\to W

Search aliases: K mathbb

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• Introduction passage · Read passage · 40_regularity.tex, line 10

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  \mathbb K

Search aliases: K^e mathbb

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• Introduction passage · Read passage · 40_regularity.tex, line 11

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  (Y,T)\in\mathbb K^e\times\mathbb K^k

Search aliases: K^k mathbb

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• Introduction passage · Read passage · 40_regularity.tex, line 11

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  (Y,T)\in\mathbb K^e\times\mathbb K^k

Search aliases: K_2

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• Introduction passage · Read passage · 72_recovered.tex, line 72

  edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weights. This is the correct sen…

  Exact mathematical passage

  
  H_2(t)=\frac{\sinh(2t)\sinh t}{2t^2},
  \qquad
  K_2(t)=\frac{4t^2}{\sinh^2(2t)}.
  

Search aliases: K_p

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• Introduction passage · Read passage · 72_recovered.tex, line 61

  The central change in this edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weigh…

  Exact mathematical passage

  K_p

Search aliases: ker

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• Introduction passage · Read passage · 30_lifting.tex, line 164

  The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K).

  Exact mathematical passage

  \label{eq:common-quotient}
   C(J,K)=U/(\ker J+\ker K).
  

Search aliases: L mathcal

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• Introduction passage · Read passage · continuation_08.tex, line 49

  Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero dimensions. Neither J nor the full reduced Jacobian is assumed surjective.

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  \mathcal L_v

Search aliases: L_N

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• Introduction passage · Read passage · 72_recovered.tex, line 2892

  [Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.

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  L_N

Search aliases: L_v mathcal

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• Introduction passage · Read passage · continuation_08.tex, line 49

  Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero dimensions. Neither J nor the full reduced Jacobian is assumed surjective.

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  \mathcal L_v

Search aliases: log

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• Introduction passage · Read passage · root_scope.tex, line 91

  Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coefficientwise that mathscr A∈ Z_2⟨ x⟩^×, mathscr A≡1pmod2, and mathscr A(N)=A(N).

  Exact mathematical passage

  
   \mathscr A(x)=\frac{9^{x+1}-1}{8(x+1)(2x+1)},
   \qquad \mathscr A(-1)=-\frac{\log9}{8}.
  

Search aliases: Longleftrightarrow

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• Introduction passage · Read passage · continuation_11.tex, line 33

  Local analytic graph theorem. There exist a map ψ: K^k⟶ K^e and open sets U⊆ K^k, V⊆ K^e such that t_0∈ U, y_0∈ V, ψ(t_0)=y_0, ψ(U)⊆ V. The map is analytic at every point of U, and its graph describes all roots in the product neighborhood: ∀ t∈ U ∀ y∈ V, F(y,t)=0 Longleftrightarrow y=ψ(t). At the base parameter it has the strict Fréchet derivative Dψ(t_0)=-J^-1K. Consequently it is differentiable on U. The map is defined on the ambient parameter space, but its root, analyticity and uniqueness co…

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  \forall t\in U\;\forall y\in V,
  \qquad F(y,t)=0\quad\Longleftrightarrow\quad y=\psi(t).
  

Search aliases: Longrightarrow

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• Introduction passage · Read passage · continuation_08.tex, line 53

  Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero dimensions. Neither J nor the full reduced Jacobian is assumed surjective.

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  o_x(v)=0\quad\Longrightarrow\quad\#\mathcal L_v=\#\ker J.
  

Search aliases: M_e

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• Introduction passage · Read passage · 40_regularity.tex, line 17

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  
   J=D_YF(a)\in M_e(\mathbb K),\qquad
   K=D_TF(a)\in M_{e\times k}(\mathbb K).
   \label{eq:regularity-blocks}
  

Search aliases: M_e× k times

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• Introduction passage · Read passage · 40_regularity.tex, line 17

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  
   J=D_YF(a)\in M_e(\mathbb K),\qquad
   K=D_TF(a)\in M_{e\times k}(\mathbb K).
   \label{eq:regularity-blocks}
  

Search aliases: M_r,d

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• Introduction passage · Read passage · 72_recovered.tex, line 2512

  Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.

  Exact mathematical passage

  
  M\in M_{r,d}(\Zp),
  \qquad
  b\in\Zp^r.
  

Search aliases: mathscr

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• Introduction passage · Read passage · root_scope.tex, line 91

  Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coefficientwise that mathscr A∈ Z_2⟨ x⟩^×, mathscr A≡1pmod2, and mathscr A(N)=A(N).

  Exact mathematical passage

  
   \mathscr A(x)=\frac{9^{x+1}-1}{8(x+1)(2x+1)},
   \qquad \mathscr A(-1)=-\frac{\log9}{8}.
  

Search aliases: o

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• Introduction passage · Read passage · continuation_08.tex, line 53

  Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero dimensions. Neither J nor the full reduced Jacobian is assumed surjective.

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  o_x(v)=0\quad\Longrightarrow\quad\#\mathcal L_v=\#\ker J.
  

Search aliases: o_x

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• Introduction passage · Read passage · continuation_08.tex, line 53

  Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero dimensions. Neither J nor the full reduced Jacobian is assumed surjective.

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  o_x(v)=0\quad\Longrightarrow\quad\#\mathcal L_v=\#\ker J.
  

Search aliases: overline J overline

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• Introduction passage · Read passage · 72_recovered.tex, line 2652

  tbfEditorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.

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  \overline J_F(x_L)

Search aliases: overline J_F overline

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• Introduction passage · Read passage · 72_recovered.tex, line 2652

  tbfEditorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.

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  \overline J_F(x_L)

Search aliases: overline M overline

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• Introduction passage · Read passage · 72_recovered.tex, line 2529

  Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.

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  \overline M

Search aliases: p^(p-1)x+j

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• Introduction passage · Read passage · 72_recovered.tex, line 1881

  The analytic function is defined by its unit source series. The expression p^(p-1)x+j in the interpolation identity is evaluated only at nonnegative integers. No analytic function x↦ p^x on all of Z_p is used.

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  p^{(p-1)x+j}

Search aliases: p^min(k,mn) min

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• Introduction passage · Read passage · 20_arithmetic.tex, line 193

  [All-address differences on the divisor branch] Let p be prime, m>0, mmid p-1, N=mp, and 0≤ j<N. Write h=(p-1)/m. For every u,k,n∈ N, U_N,j(n) · ∈ Z_p, Δ_uh^kU_N,j(n) · ∈ p^min(k,mn) Z_p, v_p(Δ_uh^kC_N,j(n)) · ≥min(k,mn)-v_p(j!). The factorial correction is part of the normalized statement, including addresses j≥ p.

  Exact mathematical passage

  
   U_{N,j}(n)&\in\Z_p,\label{eq:arith-divisor-integral}\\
   \Delta_{uh}^{k}U_{N,j}(n)&\in p^{\min(k,mn)}\Z_p,\label{eq:arith-divisor-bound}\\
   v_p\bigl(\Delta_{uh}^{k}C_{N,j}(n)\bigr)
     &\ge\min(k,mn)-v_p(j!).\label{eq:arith-divisor-normalized}
  

Search aliases: p^x

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• Introduction passage · Read passage · 72_recovered.tex, line 1881

  The analytic function is defined by its unit source series. The expression p^(p-1)x+j in the interpolation identity is evaluated only at nonnegative integers. No analytic function x↦ p^x on all of Z_p is used.

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  x\mapsto p^x

Search aliases: pmod

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• Introduction passage · Read passage · 72_recovered.tex, line 2518

  Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.

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  Mx_L\equiv b\pmod {p^L}
  

Search aliases: Q mathbb

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• Introduction passage · Read passage · 72_recovered.tex, line 2641

  Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.

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  \Q_p

Search aliases: q^L

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• Introduction passage · Read passage · continuation_01.tex, line 16

  Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d; the cokernel below is presented directly using its integer-domain image.

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  F=(F_1,\ldots,F_e)\in\mathbb Z[X_1,\ldots,X_d]^e,
  \quad q\ge1,\quad L\ge1,
  \quad x\in\mathbb Z^d,
  \quad F(x)\equiv0\pmod{q^L}.
  

Search aliases: Q_2 Q mathbb

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• Introduction passage · Read passage · supplement_11.tex, line 78

  Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.

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  \Q_2[[t]]

Search aliases: Q_p Q mathbb

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• Introduction passage · Read passage · 72_recovered.tex, line 2641

  Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.

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  \Q_p

Search aliases: Qp Q_p mathbb

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• Introduction passage · Read passage · 00_scope.tex, line 19

  Throughout, N={0,1,2,…}. When p is prime, Z_p denotes the p-adic integers and Q_p their fraction field. We normalize v_p(p)=1. For a rational or p-adic quantity, a displayed lower valuation bound is understood with the mathematical convention v_p(0)=+∞. The corresponding formal predicates are zero-safe: the quantity is zero or its valuation meets the stated bound. An exact finite valuation assertion requires a nonzero quantity, and this is stated where used. Congruence in Z_p means ideal divisib…

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  \Qp

Search aliases: R

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• Introduction passage · Read passage · 30_lifting.tex, line 162

  The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K).

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  R

Search aliases: R_q

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• Introduction passage · Read passage · continuation_01.tex, line 21

  Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d; the cokernel below is presented directly using its integer-domain image.

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  R_q=\mathbb Z/q\mathbb Z

Search aliases: R_q^d

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• Introduction passage · Read passage · continuation_01.tex, line 29

  et ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d; the cokernel below is presented directly using its integer-domain image.

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  R_q^d

Search aliases: R_q^e

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• Introduction passage · Read passage · continuation_01.tex, line 28

  F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d; the cokernel below is presented directly using its integer-domain image.

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  R_q^e

Search aliases: r_x

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• Introduction passage · Read passage · continuation_01.tex, line 22

  Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d; the cokernel below is presented directly using its integer-domain image.

  Exact mathematical passage

  
  r_x=F(x)/q^L\in\mathbb Z^e,
  \qquad A_x=DF(x):\mathbb Z^d\longrightarrow\mathbb Z^e,
  \qquad J_x=\rho_q\circ A_x.
  

Search aliases: s

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• Introduction passage · Read passage · 10_foundations.tex, line 274

  Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients vanish when their lower index exceeds n, which proves the interpolation claim. Equality of every point of the stencil proves equality of differences. …

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  \Delta_1^s f

Search aliases: S_n

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• Introduction passage · Read passage · supplement_02.tex, line 31

  rence theorem remain pending. No continuum convergence or analytic summability theorem is inferred from the formal power-series equalities in this block. The historical numerical study is not being relabelled as a fresh experiment. pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; s…

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   S_n(p)=\sum_{a=0}^{p-1}(-1)^a(2a+1)^n\in\Z,
   \qquad F_p(t)=\sum_{n\ge0}S_n(p)\frac{t^n}{n!}.
  

Search aliases: sech operatorname

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• Introduction passage · Read passage · supplement_02.tex, line 36

  r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e^at, cosh_a(t)=frace^at+e^-at2, sech(t)=cosh_1(t)^-1. The constant term of cosh_a is proved to be one, so the reciprocal used here is a genuine unit i…

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   \exp_a(t)=e^{at},\qquad
   \cosh_a(t)=\frac{e^{at}+e^{-at}}2,\qquad
   \sech(t)=\cosh_1(t)^{-1}.
  

Search aliases: sinh^2 sinh

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• Introduction passage · Read passage · 72_recovered.tex, line 72

  edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weights. This is the correct sen…

  Exact mathematical passage

  
  H_2(t)=\frac{\sinh(2t)\sinh t}{2t^2},
  \qquad
  K_2(t)=\frac{4t^2}{\sinh^2(2t)}.
  

Search aliases: t^2

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• Introduction passage · Read passage · 72_recovered.tex, line 72

  edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weights. This is the correct sen…

  Exact mathematical passage

  
  H_2(t)=\frac{\sinh(2t)\sinh t}{2t^2},
  \qquad
  K_2(t)=\frac{4t^2}{\sinh^2(2t)}.
  

Search aliases: t^n

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• Introduction passage · Read passage · supplement_02.tex, line 31

  rence theorem remain pending. No continuum convergence or analytic summability theorem is inferred from the formal power-series equalities in this block. The historical numerical study is not being relabelled as a fresh experiment. pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; s…

  Exact mathematical passage

  
   S_n(p)=\sum_{a=0}^{p-1}(-1)^a(2a+1)^n\in\Z,
   \qquad F_p(t)=\sum_{n\ge0}S_n(p)\frac{t^n}{n!}.
  

Search aliases: t_0

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• Introduction passage · Read passage · 40_regularity.tex, line 12

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  
   F\in\mathbb K[Y_1,\ldots,Y_e,T_1,\ldots,T_k]^e,
   \qquad a=(y_0,t_0),
  

Search aliases: T_1

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• Introduction passage · Read passage · 40_regularity.tex, line 12

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  
   F\in\mathbb K[Y_1,\ldots,Y_e,T_1,\ldots,T_k]^e,
   \qquad a=(y_0,t_0),
  

Search aliases: T_j

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• Introduction passage · Read passage · strengthening.tex, line 1085

  [Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j, there is one residue and the least row period is one.

  Exact mathematical passage

  
   T_j(x)=\frac{\widetilde C_j(x)-1}{2^{b_j}}
  

Search aliases: T_k

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• Introduction passage · Read passage · 40_regularity.tex, line 12

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  
   F\in\mathbb K[Y_1,\ldots,Y_e,T_1,\ldots,T_k]^e,
   \qquad a=(y_0,t_0),
  

Search aliases: U^p

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• Introduction passage · Read passage · supplement_07.tex, line 101

  The positive-tail period. Write h=(p-1)/m. For n≥1, evaluate U^pf_m=Uf_m at the nonnegative degree mn-1: f_m(mn+p-1)=f_m(mn). Since mn+p-1=m(n+h), the identified sequence satisfies e_n+h^(m,p)=e_n^(m,p) (n≥1). Induction gives e_n+uh^(m,p)=e_n^(m,p) for every u≥0. This proves a period, not necessarily the least positive period. The proof also includes p=2, where m=1 and h=1.

  Exact mathematical passage

  U^pf_m=Uf_m

Search aliases: V

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• Introduction passage · Read passage · 30_lifting.tex, line 162

  The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K).

  Exact mathematical passage

  U,V,W

Search aliases: v_2

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• Introduction passage · Read passage · root_scope.tex, line 96

  Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coefficientwise that mathscr A∈ Z_2⟨ x⟩^×, mathscr A≡1pmod2, and mathscr A(N)=A(N).

  Exact mathematical passage

  v_2(\log9)=3

Search aliases: W

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• Introduction passage · Read passage · 30_lifting.tex, line 162

  The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K).

  Exact mathematical passage

  U,V,W

Search aliases: widetilde C widetilde

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• Introduction passage · Read passage · strengthening.tex, line 1083

  [Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j, there is one residue and the least row period is one.

  Exact mathematical passage

  \widetilde C_j

Search aliases: widetilde C_j widetilde

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• Introduction passage · Read passage · strengthening.tex, line 1083

  [Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j, there is one residue and the least row period is one.

  Exact mathematical passage

  \widetilde C_j

Search aliases: X

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• Introduction passage · Read passage · 72_recovered.tex, line 2641

  Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.

  Exact mathematical passage

  F_i\in\Z_p[X_1,\ldots,X_d]

Search aliases: X^a

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• Introduction passage · Read passage · supplement_06.tex, line 70

  Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands pol…

  Exact mathematical passage

  
   \chi_D(X)=\det(XI-D)=X^d+\sum_{a=0}^{d-1}c_aX^a
  

Search aliases: X^d

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• Introduction passage · Read passage · supplement_06.tex, line 70

  Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands pol…

  Exact mathematical passage

  
   \chi_D(X)=\det(XI-D)=X^d+\sum_{a=0}^{d-1}c_aX^a
  

Search aliases: X_1

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• Introduction passage · Read passage · 72_recovered.tex, line 2641

  Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.

  Exact mathematical passage

  F_i\in\Z_p[X_1,\ldots,X_d]

Search aliases: X_d

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• Introduction passage · Read passage · 72_recovered.tex, line 2641

  Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.

  Exact mathematical passage

  F_i\in\Z_p[X_1,\ldots,X_d]

Search aliases: x_L

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• Introduction passage · Read passage · 72_recovered.tex, line 2517

  Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.

  Exact mathematical passage

  x_L\in\Zp^d

Search aliases: Y

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• Introduction passage · Read passage · 40_regularity.tex, line 11

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

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  (Y,T)\in\mathbb K^e\times\mathbb K^k

Search aliases: y

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• Introduction passage · Read passage · 40_regularity.tex, line 12

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  
   F\in\mathbb K[Y_1,\ldots,Y_e,T_1,\ldots,T_k]^e,
   \qquad a=(y_0,t_0),
  

Search aliases: y_0

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• Introduction passage · Read passage · 40_regularity.tex, line 12

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  
   F\in\mathbb K[Y_1,\ldots,Y_e,T_1,\ldots,T_k]^e,
   \qquad a=(y_0,t_0),
  

Search aliases: Y_1

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• Introduction passage · Read passage · 40_regularity.tex, line 12

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  
   F\in\mathbb K[Y_1,\ldots,Y_e,T_1,\ldots,T_k]^e,
   \qquad a=(y_0,t_0),
  

Search aliases: Y_e

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• Introduction passage · Read passage · 40_regularity.tex, line 12

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

  Exact mathematical passage

  
   F\in\mathbb K[Y_1,\ldots,Y_e,T_1,\ldots,T_k]^e,
   \qquad a=(y_0,t_0),
  

Search aliases: Z^d mathbb

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• Introduction passage · Read passage · continuation_01.tex, line 16

  Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d; the cokernel below is presented directly using its integer-domain image.

  Exact mathematical passage

  
  F=(F_1,\ldots,F_e)\in\mathbb Z[X_1,\ldots,X_d]^e,
  \quad q\ge1,\quad L\ge1,
  \quad x\in\mathbb Z^d,
  \quad F(x)\equiv0\pmod{q^L}.
  

Search aliases: Z^e mathbb

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• Introduction passage · Read passage · continuation_01.tex, line 22

  Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d; the cokernel below is presented directly using its integer-domain image.

  Exact mathematical passage

  
  r_x=F(x)/q^L\in\mathbb Z^e,
  \qquad A_x=DF(x):\mathbb Z^d\longrightarrow\mathbb Z^e,
  \qquad J_x=\rho_q\circ A_x.
  

Search aliases: Z_2 Z mathbb

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• Introduction passage · Read passage · supplement_11.tex, line 78

  Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.

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  \Z_2

Search aliases: Z_2 mathbb

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• Introduction passage · Read passage · root_scope.tex, line 89

  Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coefficientwise that mathscr A∈ Z_2⟨ x⟩^×, mathscr A≡1pmod2, and mathscr A(N)=A(N).

  Exact mathematical passage

  \mathbb Z_2\langle x\rangle

Search aliases: Zp Z_p mathbb

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• Introduction passage · Read passage · 00_scope.tex, line 18

  Throughout, N={0,1,2,…}. When p is prime, Z_p denotes the p-adic integers and Q_p their fraction field. We normalize v_p(p)=1. For a rational or p-adic quantity, a displayed lower valuation bound is understood with the mathematical convention v_p(0)=+∞. The corresponding formal predicates are zero-safe: the quantity is zero or its valuation meets the stated bound. An exact finite valuation assertion requires a nonzero quantity, and this is stated where used. Congruence in Z_p means ideal divisib…

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  \Zp

Search aliases: Zp^d Z_p^d Zp mathbb

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• Introduction passage · Read passage · 72_recovered.tex, line 2517

  Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.

  Exact mathematical passage

  x_L\in\Zp^d

Search aliases: Zp^r Z_p^r Zp mathbb

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• Introduction passage · Read passage · 72_recovered.tex, line 2512

  Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.

  Exact mathematical passage

  
  M\in M_{r,d}(\Zp),
  \qquad
  b\in\Zp^r.
  

Search aliases: × times

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• Introduction passage · Read passage · 40_regularity.tex, line 11

  Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.

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  (Y,T)\in\mathbb K^e\times\mathbb K^k

Search aliases: δ delta

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• Introduction passage · Read passage · 72_recovered.tex, line 47

  Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are h periodic after their constant term. The two arguments expose different parts of the same finite character structure.

  Exact mathematical passage

  
  \sum_{q=0}^{n}
  \binom{mn}{mq}e_q^{(m,p)}
  =
  \delta_{n0}
  \quad\text{in }\FF_p.
  

Search aliases: Δ_1 Delta

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• Introduction passage · Read passage · 10_foundations.tex, line 272

  Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients vanish when their lower index exceeds n, which proves the interpolation claim. Equality of every point of the stencil proves equality of differences. …

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  I+\Delta_1

Search aliases: Δ_1^s Delta

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• Introduction passage · Read passage · 10_foundations.tex, line 274

  Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients vanish when their lower index exceeds n, which proves the interpolation claim. Equality of every point of the stencil proves equality of differences. …

  Exact mathematical passage

  \Delta_1^s f

Search aliases: δ_n0 delta

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• Introduction passage · Read passage · 72_recovered.tex, line 47

  Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are h periodic after their constant term. The two arguments expose different parts of the same finite character structure.

  Exact mathematical passage

  
  \sum_{q=0}^{n}
  \binom{mn}{mq}e_q^{(m,p)}
  =
  \delta_{n0}
  \quad\text{in }\FF_p.
  

Search aliases: Δ_u Delta

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• Introduction passage · Read passage · 00_scope.tex, line 32

  Write Δ_u f(n)=f(n+u)-f(n) and use Δ_u^k for its k-fold iterate. Euler coefficients indexed by ordinary degree and by packet row are distinguished explicitly. A packet displacement by u rows is a degree displacement by Nu in a packet of width N.

  Exact mathematical passage

  \Delta_u f(n)=f(n+u)-f(n)

Search aliases: Δ_u^k Delta

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• Introduction passage · Read passage · 00_scope.tex, line 32

  Write Δ_u f(n)=f(n+u)-f(n) and use Δ_u^k for its k-fold iterate. Euler coefficients indexed by ordinary degree and by packet row are distinguished explicitly. A packet displacement by u rows is a degree displacement by Nu in a packet of width N.

  Exact mathematical passage

  \Delta_u^k

Search aliases: Δ_uh^k Delta

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• Introduction passage · Read passage · 20_arithmetic.tex, line 193

  [All-address differences on the divisor branch] Let p be prime, m>0, mmid p-1, N=mp, and 0≤ j<N. Write h=(p-1)/m. For every u,k,n∈ N, U_N,j(n) · ∈ Z_p, Δ_uh^kU_N,j(n) · ∈ p^min(k,mn) Z_p, v_p(Δ_uh^kC_N,j(n)) · ≥min(k,mn)-v_p(j!). The factorial correction is part of the normalized statement, including addresses j≥ p.

  Exact mathematical passage

  
   U_{N,j}(n)&\in\Z_p,\label{eq:arith-divisor-integral}\\
   \Delta_{uh}^{k}U_{N,j}(n)&\in p^{\min(k,mn)}\Z_p,\label{eq:arith-divisor-bound}\\
   v_p\bigl(\Delta_{uh}^{k}C_{N,j}(n)\bigr)
     &\ge\min(k,mn)-v_p(j!).\label{eq:arith-divisor-normalized}
  

Search aliases: λ_J lambda

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• Introduction passage · Read passage · 30_lifting.tex, line 170

  s over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K).

  Exact mathematical passage

  \lambda_J:\im J\to C(J,K)

Search aliases: λ_K lambda

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• Introduction passage · Read passage · 30_lifting.tex, line 170

  ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K).

  Exact mathematical passage

  \lambda_K:\im K\to C(J,K)

Search aliases: ρ rho

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• Introduction passage · Read passage · 72_recovered.tex, line 2522

  Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.

  Exact mathematical passage

  
  \rho_L(x_L)
  =
  \frac{Mx_L-b}{p^L}
  \pmod p
  \in\FF_p^r.
  

Search aliases: ρ_L rho

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• Introduction passage · Read passage · 72_recovered.tex, line 2522

  Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.

  Exact mathematical passage

  
  \rho_L(x_L)
  =
  \frac{Mx_L-b}{p^L}
  \pmod p
  \in\FF_p^r.
  

Search aliases: ρ_L^F rho

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• Introduction passage · Read passage · 72_recovered.tex, line 2646

  Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.

  Exact mathematical passage

  
  \rho_L^F(x_L)
  =
  \frac{F(x_L)}{p^L}
  \pmod p.
  

Search aliases: ρ_q rho

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• Introduction passage · Read passage · continuation_01.tex, line 21

  Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d; the cokernel below is presented directly using its integer-domain image.

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  \rho_q

Search aliases: χ chi

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• Introduction passage · Read passage · supplement_06.tex, line 70

  Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands pol…

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   \chi_D(X)=\det(XI-D)=X^d+\sum_{a=0}^{d-1}c_aX^a
  

Search aliases: χ_D chi

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• Introduction passage · Read passage · supplement_06.tex, line 70

  Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands pol…

  Exact mathematical passage

  
   \chi_D(X)=\det(XI-D)=X^d+\sum_{a=0}^{d-1}c_aX^a
  

Search aliases: ψ psi

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• Introduction passage · Read passage · continuation_11.tex, line 26

  Local analytic graph theorem. There exist a map ψ: K^k⟶ K^e and open sets U⊆ K^k, V⊆ K^e such that t_0∈ U, y_0∈ V, ψ(t_0)=y_0, ψ(U)⊆ V. The map is analytic at every point of U, and its graph describes all roots in the product neighborhood: ∀ t∈ U ∀ y∈ V, F(y,t)=0 Longleftrightarrow y=ψ(t). At the base parameter it has the strict Fréchet derivative Dψ(t_0)=-J^-1K. Consequently it is differentiable on U. The map is defined on the ambient parameter space, but its root, analyticity and uniqueness co…

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  \psi:\mathbb K^k\longrightarrow\mathbb K^e
  

Search aliases: … ldots

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• Introduction passage · Read passage · 00_scope.tex, line 18

  Throughout, N={0,1,2,…}. When p is prime, Z_p denotes the p-adic integers and Q_p their fraction field. We normalize v_p(p)=1. For a rational or p-adic quantity, a displayed lower valuation bound is understood with the mathematical convention v_p(0)=+∞. The corresponding formal predicates are zero-safe: the quantity is zero or its valuation meets the stated bound. An exact finite valuation assertion requires a nonzero quantity, and this is stated where used. Congruence in Z_p means ideal divisib…

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  \N=\{0,1,2,\ldots\}

Search aliases: → to

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• Introduction passage · Read passage · 30_lifting.tex, line 162

  The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K).

  Exact mathematical passage

  J:U\to V

Search aliases: ↦ mapsto

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• Introduction passage · Read passage · 72_recovered.tex, line 1881

  The analytic function is defined by its unit source series. The expression p^(p-1)x+j in the interpolation identity is evaluated only at nonnegative integers. No analytic function x↦ p^x on all of Z_p is used.

  Exact mathematical passage

  x\mapsto p^x

Search aliases: ∀ forall

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• Introduction passage · Read passage · continuation_11.tex, line 33

  Local analytic graph theorem. There exist a map ψ: K^k⟶ K^e and open sets U⊆ K^k, V⊆ K^e such that t_0∈ U, y_0∈ V, ψ(t_0)=y_0, ψ(U)⊆ V. The map is analytic at every point of U, and its graph describes all roots in the product neighborhood: ∀ t∈ U ∀ y∈ V, F(y,t)=0 Longleftrightarrow y=ψ(t). At the base parameter it has the strict Fréchet derivative Dψ(t_0)=-J^-1K. Consequently it is differentiable on U. The map is defined on the ambient parameter space, but its root, analyticity and uniqueness co…

  Exact mathematical passage

  
  \forall t\in U\;\forall y\in V,
  \qquad F(y,t)=0\quad\Longleftrightarrow\quad y=\psi(t).
  

Search aliases: ∑_a=0^d sum

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• Introduction passage · Read passage · supplement_06.tex, line 74

  packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands polynomial evaluation into a finite sum and proves ∑_a=0^dχ_D[a]D^a=0, χ_D[d]=1. Here χ_D[a] denotes the coefficient at index a. The identities hold over any commutative coefficient ring; the …

  Exact mathematical passage

  
   \sum_{a=0}^{d}\chi_D[a]D^a=0,
   \qquad \chi_D[d]=1.
  

Search aliases: ∑_a=0^d-1 sum

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• Introduction passage · Read passage · supplement_06.tex, line 70

  Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands pol…

  Exact mathematical passage

  
   \chi_D(X)=\det(XI-D)=X^d+\sum_{a=0}^{d-1}c_aX^a
  

Search aliases: ∑_h≥0 sum ge

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• Introduction passage · Read passage · root_scope.tex, line 97

  Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coefficientwise that mathscr A∈ Z_2⟨ x⟩^×, mathscr A≡1pmod2, and mathscr A(N)=A(N).

  Exact mathematical passage

  
   \mathscr A(x)=\frac{\log9/8}{1+2x}
     \sum_{h\ge0}\frac{(\log9)^h}{(h+1)!}(x+1)^h.
   \label{eq:root:source}
  

Search aliases: ∑_q=0^n sum

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• Introduction passage · Read passage · 72_recovered.tex, line 47

  Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are h periodic after their constant term. The two arguments expose different parts of the same finite character structure.

  Exact mathematical passage

  
  \sum_{q=0}^{n}
  \binom{mn}{mq}e_q^{(m,p)}
  =
  \delta_{n0}
  \quad\text{in }\FF_p.
  

Search aliases: ∑_r=0^m sum

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• Introduction passage · Read passage · strengthening.tex, line 414

  Let B_m denote the Bernoulli numbers with B_1=-1/2. The identity obtained by substituting u=e^t-1 into log(1+u)/u is B_m=∑_r=0^m(-1)^rfracr!r+1 {m r}, where the braces are Stirling numbers of the second kind. Each 2r!/(r+1) is dyadically integral. Indeed, if s=v_2(r+1)>0, then r≥2^s-1 and v_2(r!)≥ v_2((2^s-1)!)=2^s-1-s≥ s-1. The last inequality is 2^s≥2s; the case s=0 is immediate. Consequently 2B_m∈ Z_2 (m≥0). Thus the transfer factors below require no appeal to the von Staudt--Clausen theorem.

  Exact mathematical passage

  
   B_m=\sum_{r=0}^m(-1)^r\frac{r!}{r+1}
                                    \left\{\begin{matrix}m\\r\end{matrix}\right\},
   \label{eq:bernoulli-stirling-denominator}
  

Search aliases: ∑_r=0^n sum

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• Introduction passage · Read passage · 10_foundations.tex, line 129

  In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.

  Exact mathematical passage

  
  (f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad
  (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
  

Search aliases: ∘ circ

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• Introduction passage · Read passage · continuation_01.tex, line 22

  Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d; the cokernel below is presented directly using its integer-domain image.

  Exact mathematical passage

  
  r_x=F(x)/q^L\in\mathbb Z^e,
  \qquad A_x=DF(x):\mathbb Z^d\longrightarrow\mathbb Z^e,
  \qquad J_x=\rho_q\circ A_x.
  

Search aliases: ≅ cong

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• Introduction passage · Read passage · continuation_08.tex, line 52

  Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero dimensions. Neither J nor the full reduced Jacobian is assumed surjective.

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  \mathcal L_v\cong\ker J

Search aliases: ≡ equiv

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• Introduction passage · Read passage · 72_recovered.tex, line 2518

  Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.

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  Mx_L\equiv b\pmod {p^L}
  

Search aliases: ⊆ subseteq

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• Introduction passage · Read passage · continuation_11.tex, line 28

  Local analytic graph theorem. There exist a map ψ: K^k⟶ K^e and open sets U⊆ K^k, V⊆ K^e such that t_0∈ U, y_0∈ V, ψ(t_0)=y_0, ψ(U)⊆ V. The map is analytic at every point of U, and its graph describes all roots in the product neighborhood: ∀ t∈ U ∀ y∈ V, F(y,t)=0 Longleftrightarrow y=ψ(t). At the base parameter it has the strict Fréchet derivative Dψ(t_0)=-J^-1K. Consequently it is differentiable on U. The map is defined on the ambient parameter space, but its root, analyticity and uniqueness co…

  Exact mathematical passage

  U\subseteq\mathbb K^k

Search aliases: ⋆ star

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• Introduction passage · Read passage · 10_foundations.tex, line 129

  In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.

  Exact mathematical passage

  
  (f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad
  (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
  

Search aliases: ⟨ langle

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• Introduction passage · Read passage · root_scope.tex, line 89

  Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coefficientwise that mathscr A∈ Z_2⟨ x⟩^×, mathscr A≡1pmod2, and mathscr A(N)=A(N).

  Exact mathematical passage

  \mathbb Z_2\langle x\rangle

Search aliases: ⟩ rangle

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• Introduction passage · Read passage · root_scope.tex, line 89

  Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coefficientwise that mathscr A∈ Z_2⟨ x⟩^×, mathscr A≡1pmod2, and mathscr A(N)=A(N).

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  \mathbb Z_2\langle x\rangle

Search aliases: ⟶ longrightarrow

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• Introduction passage · Read passage · 72_recovered.tex, line 2638

  Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.

  Exact mathematical passage

  
  F=(F_1,\ldots,F_r):\Zp^d\longrightarrow\Zp^r
  

Search aliases: 1_d≡ r (N) mathbf equiv

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• Use in source · Read passage · supplement_09.tex, line 83

  Modules. mcodeCyclotomicFiltration, mcodePrimeBlockPolynomial, and mcodeCyclicAmiceCoefficients. 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F_r'=F_r-1, and hence the coefficient recurrence (d+1)coe…

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   H_r(t)=\sum_{d\ge0}\mathbf1_{d\equiv r\ (N)}\frac{t^d}{d!},
   \qquad F_r(T)=H_r(\log(1+T)).
  

Search aliases: 1_r=0 mathbf

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• Use in source · Read passage · supplement_09.tex, line 96

  For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F_r'=F_r-1, and hence the coefficient recurrence (d+1)coefTd+1F_r =coefTdF_r-1-dcoefTdF_r. The initial values are F_r(0)=1_r=0.

  Exact mathematical passage

  F_r(0)=\mathbf1_{r=0}

Search aliases: A mathcal

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• Use in source · Read passage · 20_arithmetic.tex, line 927

  Its exponential generating function is A(t)=∑_n≥0A(n)fract^2n(2n)! =fraccosh(3t)-cosh t4t^2. The quotients by powers of t are well defined because the required initial coefficients vanish. The hyperbolic identity A(t)cosh t=(fracsinh(2t)2t)^2 then gives the exact formal-series product (∑_m≥0β_m(2t)^2m) A(t) =secht. Taking the factorial-scaled coefficient of t^2n produces the stated finite sum. Full depth leaves no omitted terms. The relevant declarations are leanKomorebiPaperIX.Dyadic.beta_integ…

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   \mathcal A(t)=\sum_{n\ge0}A(n)\frac{t^{2n}}{(2n)!}
               =\frac{\cosh(3t)-\cosh t}{4t^2}.
  

Search aliases: a'

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• Use in source · Read passage · 71_native.tex, line 238

  R(a,b), R(a',b), R(a',b') Longrightarrow R(a,b')

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  R(a,b),\ R(a',b),\ R(a',b')\quad\Longrightarrow\quad R(a,b')

Search aliases: a^(m)

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• Use in source · Read passage · 72_recovered.tex, line 259

  [Multiple selector] Define a^(m)∈ H( F_p) by a_n^(m) = 1, · mmid n,

  Exact mathematical passage

  a^{(m)}\in\Hcal(\FF_p)

Search aliases: A^-1

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• Use in source · Read passage · strengthening.tex, line 63

  Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1^dbinom di a_i b_d-i. If A is supported at degrees divisible by N, this recurrence also proves that B has the same support.

  Exact mathematical passage

  B=A^{-1}

Search aliases: a^-2

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• Use in source · Read passage · strengthening.tex, line 730

  ties give sinh t-sin t =2asinh(at)cosh(bt)+2bcosh(at)sinh(bt), cosh t-cos t=2sinh(at)sinh(bt). Consequently D_3(t):=G_4,3(t)F_2(t) =frac32(fraccoth(at)at +fraccoth(bt)bt). This expression is a power series: its two polar terms cancel because a^-2+b^-2=0. Its constant is one, only degrees divisible by four survive, and the Bernoulli expansion gives D_3(t)=∑_i≥0d_ifract^4i(4i)!, d_i=3(-frac14)^i frac2^4i+2B_4i+2(4i+2)(4i+1). Again v_2(d_i)≥2i, and the inverse coefficients h_i have the same bound. …

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  a^{-2}+b^{-2}=0

Search aliases: A^2

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• Use in source · Read passage · root_scope.tex, line 467

  One sufficient new arithmetic input would be an odd-denominator projective quadratic A Z^2+2BZ+C≡ A W_r(Z)pmod2^P, A odd, 0<|B^2-AC|<2^P, at a proved precision for every admitted ordinary r. Then (B^2-AC)/A^2≡δ_rpmod2^P cannot vanish modulo 2^P. No depth-independent theorem supplying this inequality has been established. The tested recurrence and exterior-contact identities retain balancing terms and do not force a contradiction. Finite exponential models approximate the actual target at every d…

  Exact mathematical passage

  (B^2-AC)/A^2\equiv\delta_r\pmod{2^P}

Search aliases: a^2

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• Use in source · Read passage · continuation_01.tex, line 51

  For every P∈ Z[X_1,…,X_d], integer a, and x,t∈ Z^d, Lean proves a^2mid P(x+at)-P(x)-a∑_i=1^d(∂_iP)(x)t_i. The proof uses polynomial induction: constants, addition, and multiplication by a variable. The product rule for the actual partial derivatives controls the multiplication step. It does not assume a Taylor identity or a liftability condition.

  Exact mathematical passage

  
  \boxed{
   a^2\mid P(x+at)-P(x)-a\sum_{i=1}^{d}(\partial_iP)(x)t_i.
  }
  

Search aliases: a^k

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• Use in source · Read passage · supplement_11.tex, line 54

  Define a Kummer sequence over a commutative ring by P_a(f) Longleftrightarrow a^kmidΔ_1^k f(n) for all k,n. The exact discrete product rule Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^i f(n) Δ_1^s-ig(n+i) is proved by induction. It proves closure under products. Applied to fg=1, the same formula isolates Δ_1^s g(n)f(n+s); induction and multiplication by g(n+s) prove closure under actual pointwise inverses.

  Exact mathematical passage

  
  \PP_a(f)\quad\Longleftrightarrow\quad
   a^k\mid\Delta_1^k f(n)\ \text{for all }k,n.
  

Search aliases: a^p^L

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• Use in source · Read passage · 10_foundations.tex, line 364

  [Finite unit coordinates] Let p be prime, L≥0, and a∈( Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ϵ(a)∈{1,-1}, ⟨ a⟩_4≡1pmod4.

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  \omega_L(a)=a^{p^L},\qquad \langle a\rangle_L=a\omega_L(a)^{-1}.
  

Search aliases: a^p^ℓ-1 ell

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• Use in source · Read passage · supplement_05.tex, line 81

  ffPadic_coe, .standard_window_integral targetWindowIndependence.residue_independence, .reduced_unique targetWindowIndependence.two_window_addresses_congruent pageheading2. Explicit finite prime-power decompositionProposition 6.2; odd-prime branch Let p be prime, ℓ≥1, and a∈( Z/p^ℓ Z)^×. Define actual units in the same ring by ω_ℓ(a)=a^p^ℓ-1, ⟨ a⟩_ℓ=a ω_ℓ(a)^-1. The Lean source writes ℓ=L+1, so zero precision is excluded without truncated natural-number subtraction.

  Exact mathematical passage

  
   \boxed{\omega_\ell(a)=a^{p^{\ell-1}},\qquad
   \langle a\rangle_\ell=a\,\omega_\ell(a)^{-1}.}
  

Search aliases: a^p^ℓ-1(p-1) ell

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• Use in source · Read passage · supplement_05.tex, line 89

  The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtained directly. The proof does not assume a Hensel lift or an already supplied root representative.

  Exact mathematical passage

  
   \omega_\ell(a)^{p-1}
   =a^{p^{\ell-1}(p-1)}=a^{\varphi(p^\ell)}=1.
  

Search aliases: a^φ(p^ℓ) varphi ell

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• Use in source · Read passage · supplement_05.tex, line 89

  The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtained directly. The proof does not assume a Hensel lift or an already supplied root representative.

  Exact mathematical passage

  
   \omega_\ell(a)^{p-1}
   =a^{p^{\ell-1}(p-1)}=a^{\varphi(p^\ell)}=1.
  

Search aliases: A_-c

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• Use in source · Read passage · 20_arithmetic.tex, line 525

  d product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting from the independently established integrality of the reciprocal G, proves that inversion negates the charge. Finally A_c+2=A_c-1, so increasing …

  Exact mathematical passage

  
   A_{-c}G=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G').
  

Search aliases: A_0

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• Use in source · Read passage · edition_summary.tex, line 16

  The strongest common-step conclusion in width four is particularly compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and positive step. It is exact in address two at every power-of-two order. The proof chapters also give mixed-step forms, the cancellation criterion in addre…

  Exact mathematical passage

  
   A_0(k)=A_1(k)=k,\qquad
   A_2(k)=\max\{k+4,\,2k+\lfloor k/2\rfloor+1\},\qquad
   A_3(k)=k+2.
  

Search aliases: A_0 mathcal

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• Use in source · Read passage · strengthening.tex, line 339

  Let Γ_R(P,Q) be the R-subalgebra generated by all γ_m(P) and γ_r(Q). Adjoin E, F with E^2=1+2P, F^2=1+2Q, and call the resulting algebra A_0. Its elements are finite expressions; no completion is being presumed. It is generated as a Γ_R(P,Q)-module by 1, E, F, E F. The functional maps this module to R. For an even basis term this is [source reference] and independence. For a basis term with an odd Euler factor the moment is zero, since P and Q are even polynomials in their respective variables.

  Exact mathematical passage

  \mathcal A_0

Search aliases: a_0

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• Use in source · Read passage · 10_foundations.tex, line 12

  The original arithmetic seed notation S=(A,a_0, O, R,{P_N,j},w,τ_p) records a coefficient ring or module, a seed, allowed operators, relations, residue selectors, a depth weight, and a topology. This notation describes the ingredients of the construction; a category of seed presentations and a notion of equivalence still require separate definitions. We therefore state each theorem for a concrete object. Section~[source reference] discusses the additional structure needed for the original native…

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  \mathcal S=(A,a_0,\mathcal O,\mathcal R,\{P_{N,j}\},w,\tau_p)
  

Search aliases: a_0,1

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• Use in source · Read passage · 20_arithmetic.tex, line 672

  The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvement. The recurrence proves the following exact valuations: cll Address j · Further orders k · v_2(a_j,k) in that order

  Exact mathematical passage

  
   a_{0,1}=-2,\quad a_{1,1}=-\frac65,\quad
   a_{2,1}=-\frac{16}{15},\quad a_{3,1}=-\frac{36}{35}.
  

Search aliases: a_0^-1

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• Use in source · Read passage · strengthening.tex, line 64

  Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1^dbinom di a_i b_d-i. If A is supported at degrees divisible by N, this recurrence also proves that B has the same support.

  Exact mathematical passage

  
   b_0=a_0^{-1},\qquad
   b_d=-a_0^{-1}\sum_{i=1}^d\binom di a_i b_{d-i}.
   \label{eq:hurwitz-reciprocal-recurrence}
  

Search aliases: a_0^[M]

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• Use in source · Read passage · 72_recovered.tex, line 1977

  For a sequence a=(a_n)_n≥0 and M≥2, define a_r^[M](q)=a_Mq+r (0≤ r<M). The analysis map is P_Ma = (a_0^[M],…,a_M-1^[M]). Synthesis interleaves the branches by the rule a_Mq+r=a_r^[M](q).

  Exact mathematical passage

  
  \mathcal P_Ma
  =
  \left(a_0^{[M]},\ldots,a_{M-1}^{[M]}\right).
  

Search aliases: A_1

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• Use in source · Read passage · edition_summary.tex, line 16

  The strongest common-step conclusion in width four is particularly compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and positive step. It is exact in address two at every power-of-two order. The proof chapters also give mixed-step forms, the cancellation criterion in addre…

  Exact mathematical passage

  
   A_0(k)=A_1(k)=k,\qquad
   A_2(k)=\max\{k+4,\,2k+\lfloor k/2\rfloor+1\},\qquad
   A_3(k)=k+2.
  

Search aliases: a_1

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• Use in source · Read passage · 20_arithmetic.tex, line 615

  [Finite Newton dominance] Let f: N→ Q and write a_k=Δ_1^kf(0). Suppose that c∈ N, a_1≠0, v_2(a_1)=c, and v_2(a_k)≥ c+1+lfloorlog_2krfloor (k≥2). Then for every n≥0 and T>0, f(n+T)-f(n)≠0, v_2(f(n+T)-f(n))=v_2(T)+c.

  Exact mathematical passage

  a_1\ne0

Search aliases: a_1,1

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• Use in source · Read passage · 20_arithmetic.tex, line 672

  The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvement. The recurrence proves the following exact valuations: cll Address j · Further orders k · v_2(a_j,k) in that order

  Exact mathematical passage

  
   a_{0,1}=-2,\quad a_{1,1}=-\frac65,\quad
   a_{2,1}=-\frac{16}{15},\quad a_{3,1}=-\frac{36}{35}.
  

Search aliases: A_1^s

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• Use in source · Read passage · 20_arithmetic.tex, line 541

  Packing g_j into degrees divisible by four gives K_4,j. Its source property is precisely charge-one flatness. Inversion gives charge -1, and the charge shift restores charge one for the reciprocal P_4,j. If dec(F)(d)=d![t^d]F, the exact coefficient identity is dec(A_1^sF)(d) =2^-sΔ_4^sdec(F)(d). At d=4n, packing turns the right side into 2^-sΔ_1^sC_j(n). This extraction requires no negative degree and therefore holds at every n, including zero. It proves the unit-step bound leanKomorebiPaperIX.D…

  Exact mathematical passage

  
   \operatorname{dec}(A_1^sF)(d)
         =2^{-s}\Delta_4^s\operatorname{dec}(F)(d).
  

Search aliases: A_2

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• Use in source · Read passage · edition_summary.tex, line 16

  The strongest common-step conclusion in width four is particularly compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and positive step. It is exact in address two at every power-of-two order. The proof chapters also give mixed-step forms, the cancellation criterion in addre…

  Exact mathematical passage

  
   A_0(k)=A_1(k)=k,\qquad
   A_2(k)=\max\{k+4,\,2k+\lfloor k/2\rfloor+1\},\qquad
   A_3(k)=k+2.
  

Search aliases: A_2 mathcal

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• Use in source · Read passage · strengthening.tex, line 640

  Let A_2 be the divided-power algebra over R generated by P,Q, adjoining B, D with B^2=1+4P, D^2=1+4Q. The functional is integral on its module basis 1, B, D, B D by the Bernoulli lemma, independence, and vanishing of odd moments. Every component in [source reference] has integral divided powers, by the same finite identities as before; P^2 is even easier than P^2/2. Consequently L((U^4-1)^m)∈4^m m! Z_2, v_2(Δ_1^kC_2(n))≥2k+v_2(k!). The starting-row conclusion follows from the finite base expansi…

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  \mathcal A_2

Search aliases: a_2

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• Use in source · Read passage · strengthening.tex, line 659

  The low Newton coefficients of the actual reciprocal are C_2(0)=1, C_2(1)=-frac115, C_2(2)=frac1345, a_1=-frac1615, a_2=frac6445, where a_m=Δ_1^m C_2(0). Thus v_2(a_1)=4 and v_2(a_2)=6. For m≥3, [source reference] bounds a_m by 2m+v_2(m!). The finite shifted Newton formula Δ_1^kC_2(n)=∑_m=k^n+ka_mbinom nm-k therefore gives v_2(Δ_1^kC_2(n))≥ k+v_2(k!)+3 for every k≥1. For k≥3 this is 2k≥ k+3; for k=1,2 it uses the exact initial coefficients and the higher tail bound. The finite step expansion pro…

  Exact mathematical passage

  
   C_2(0)=1,\quad C_2(1)=-\frac1{15},\quad C_2(2)=\frac{13}{45},
   \qquad a_1=-\frac{16}{15},\quad a_2=\frac{64}{45},
   \label{eq:address-two-initial-newton}
  

Search aliases: a_2,1

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• Use in source · Read passage · 20_arithmetic.tex, line 672

  The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvement. The recurrence proves the following exact valuations: cll Address j · Further orders k · v_2(a_j,k) in that order

  Exact mathematical passage

  
   a_{0,1}=-2,\quad a_{1,1}=-\frac65,\quad
   a_{2,1}=-\frac{16}{15},\quad a_{3,1}=-\frac{36}{35}.
  

Search aliases: a_2,d

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• Use in source · visual_body.tex, line 456

  At every center x_0∈ Z_2, the interpolations admit ordinary Taylor series C_j(x_0+z)=∑_d≥0a_j,d(x_0)z^d. For positive degrees, v_2(a_j,d(x_0)) · =d · · (j=0,1), v_2(a_2,d(x_0)) · ≥max{2d+lfloor d/2rfloor, d+3}, v_2(a_3,d(x_0)) · ≥ d. At address two equality with 5d/2 holds along every power-of-two degree d≥4.

  Exact mathematical passage

  
  \begin{aligned}
   v_2(a_{j,d}(x_0))&=d &&(j=0,1),\\
   v_2(a_{2,d}(x_0))&\ge\max\{2d+\lfloor d/2\rfloor,\ d+3\},\\
   v_2(a_{3,d}(x_0))&\ge d.
  \end{aligned}
  

Search aliases: A_3

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• Use in source · Read passage · edition_summary.tex, line 16

  The strongest common-step conclusion in width four is particularly compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and positive step. It is exact in address two at every power-of-two order. The proof chapters also give mixed-step forms, the cancellation criterion in addre…

  Exact mathematical passage

  
   A_0(k)=A_1(k)=k,\qquad
   A_2(k)=\max\{k+4,\,2k+\lfloor k/2\rfloor+1\},\qquad
   A_3(k)=k+2.
  

Search aliases: A_3,1

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• Use in source · Read passage · 72_recovered.tex, line 1277

  [Exact change between local source charts] Editorial range correction. This comparison holds for the even source subsequences, equivalently the evenized generating functions H_p. It does not identify their full odd-degree source values. For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero; the even kernel L_3K_2 cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity H_p=L_pK_ℓ H_ℓ…

  Exact mathematical passage

  A_{3,1}=S_1(3)=3

Search aliases: a_3,1

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• Use in source · Read passage · 20_arithmetic.tex, line 672

  The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvement. The recurrence proves the following exact valuations: cll Address j · Further orders k · v_2(a_j,k) in that order

  Exact mathematical passage

  
   a_{0,1}=-2,\quad a_{1,1}=-\frac65,\quad
   a_{2,1}=-\frac{16}{15},\quad a_{3,1}=-\frac{36}{35}.
  

Search aliases: a_3,d

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• Use in source · visual_body.tex, line 456

  At every center x_0∈ Z_2, the interpolations admit ordinary Taylor series C_j(x_0+z)=∑_d≥0a_j,d(x_0)z^d. For positive degrees, v_2(a_j,d(x_0)) · =d · · (j=0,1), v_2(a_2,d(x_0)) · ≥max{2d+lfloor d/2rfloor, d+3}, v_2(a_3,d(x_0)) · ≥ d. At address two equality with 5d/2 holds along every power-of-two degree d≥4.

  Exact mathematical passage

  
  \begin{aligned}
   v_2(a_{j,d}(x_0))&=d &&(j=0,1),\\
   v_2(a_{2,d}(x_0))&\ge\max\{2d+\lfloor d/2\rfloor,\ d+3\},\\
   v_2(a_{3,d}(x_0))&\ge d.
  \end{aligned}
  

Search aliases: a_B^s f

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• Use in source · Read passage · supplement_08.tex, line 91

  The earlier coefficient criterion is recovered exactly. Define the literal coefficient defect (δ a)(d)=a(d+p^2)-a(d+p). The new source proves, for every s,d, a_B^s f(d)=p^-s(δ^s a_f)(d). For an integral coefficient x, it proves the bridge p^-sx∈ Z_p Longleftrightarrow p^smid x in Z_p. Thus the differential definition is equivalent to the existing coefficient-level Dwork criterion. Division occurs in Q_p, while the assertion after division is membership in Z_p. It is not divisibility in the field…

  Exact mathematical passage

  
   a_{B^s f}(d)=p^{-s}(\delta^s a_f)(d).
  

Search aliases: A_c

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• Use in source · Read passage · 20_arithmetic.tex, line 216

  Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios directly to the family under study.

  Exact mathematical passage

  
   H_c(t)=\sum_{n\ge0}\frac{t^{Nn+c}}{(Nn+c)!},\qquad
   A_c(t)=\sum_{n\ge0}\frac{t^{Nn}}{(Nn+c)!},\qquad
   R_c(t)=\frac{A_{c-1}(t)}{A_c(t)}\quad(1\le c<N).
  

Search aliases: A_c+2

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• Use in source · Read passage · 20_arithmetic.tex, line 530

  atives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting from the independently established integrality of the reciprocal G, proves that inversion negates the charge. Finally A_c+2=A_c-1, so increasing a charge by two preserves flatness. These are leanKomorebiPaperIX.ChargedDworkClosure.B_mul, leanKomorebiPaperIX.Cha…

  Exact mathematical passage

  A_{c+2}=A_c-1

Search aliases: A_c+d

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• Use in source · Read passage · 20_arithmetic.tex, line 517

  For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting…

  Exact mathematical passage

  \label{eq:arith-charged-product}
   A_{c+d}(FG)=(A_cF)G+F(A_dG)
         +2F'G'''+3F''G''+2F'''G'.
  

Search aliases: A_c-1

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• Use in source · Read passage · 20_arithmetic.tex, line 216

  Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios directly to the family under study.

  Exact mathematical passage

  
   H_c(t)=\sum_{n\ge0}\frac{t^{Nn+c}}{(Nn+c)!},\qquad
   A_c(t)=\sum_{n\ge0}\frac{t^{Nn}}{(Nn+c)!},\qquad
   R_c(t)=\frac{A_{c-1}(t)}{A_c(t)}\quad(1\le c<N).
  

Search aliases: A_c^s

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• Use in source · Read passage · 20_arithmetic.tex, line 515

  For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting…

  Exact mathematical passage

  A_c^s

Search aliases: A_d

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• Use in source · Read passage · 20_arithmetic.tex, line 517

  For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting…

  Exact mathematical passage

  \label{eq:arith-charged-product}
   A_{c+d}(FG)=(A_cF)G+F(A_dG)
         +2F'G'''+3F''G''+2F'''G'.
  

Search aliases: a_d

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• Use in source · Read passage · 20_arithmetic.tex, line 231

  At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive blocks of length p then gives an integral residue-coordinate factor p^v_p(lfloor d/prfloor!)…

  Exact mathematical passage

  
   \mathcal W_p=\left\{\sum_{d\ge0}a_dT^d:
            p^{\lfloor d/p\rfloor}a_d\in\Z_p\right\}.
  

Search aliases: a_d+1

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• Use in source · Read passage · 20_arithmetic.tex, line 272

  To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d; whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-b^p. Weighted monomials factor as a_dT^d=(p^lfloor d/prfloora_d) (T^p/p)^lfloor d/prfloorT^dbmod p. The operator preserves these monomials and hence f…

  Exact mathematical passage

  (d+1)a_{d+1}+da_d

Search aliases: a_d+pk

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• Use in source · Read passage · 20_arithmetic.tex, line 333

  Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n); every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^u-I), whose kth power for u>0 is an integral combination of powers of Δ_h of order at least k. The case u=0…

  Exact mathematical passage

  
   d![t^d]B_p^kF=p^{-k}\Delta_{p(p-1)}^ka_{d+pk}.
  

Search aliases: a_f

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• Use in source · Read passage · supplement_08.tex, line 66

  pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_r=0^dbinom dr a_f(r)a_g(d-r), a_Uf(d)=a_f(d+1). These identities prove multiplication and derivative stability of the lattice. Membership does not req…

  Exact mathematical passage

  
   a_f(d)=d![t^d]f,\qquad
   I_p=\{f\in\Q_p[[t]]:a_f(d)\in\Z_p\text{ for every }d\}.
  

Search aliases: a_f^-1

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• Use in source · Read passage · supplement_08.tex, line 79

  The reciprocal is genuinely integral. For f∈ I_p with constant coefficient one, the usual power-series reciprocal is used. Its inverse equation yields the exact coefficient recurrence a_f^-1(d)=δ_d0-∑_r=0^d-1 binom dr a_f^-1(r)a_f(d-r). Strong induction proves that all of these coefficients are integral. Combining this construction with the preceding all-order theorem proves flatness of the actual reciprocal. No flatness or integrality of that reciprocal is assumed in the normalized-series speci…

  Exact mathematical passage

  
   a_{f^{-1}}(d)=\delta_{d0}-\sum_{r=0}^{d-1}
       \binom dr a_{f^{-1}}(r)a_f(d-r).
  

Search aliases: a_fg

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• Use in source · Read passage · supplement_08.tex, line 71

  pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_r=0^dbinom dr a_f(r)a_g(d-r), a_Uf(d)=a_f(d+1). These identities prove multiplication and derivative stability of the lattice. Membership does not req…

  Exact mathematical passage

  
   a_{fg}(d)=\sum_{r=0}^{d}\binom dr a_f(r)a_g(d-r),
   \qquad a_{Uf}(d)=a_f(d+1).
  

Search aliases: a_g

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• Use in source · Read passage · supplement_08.tex, line 71

  pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_r=0^dbinom dr a_f(r)a_g(d-r), a_Uf(d)=a_f(d+1). These identities prove multiplication and derivative stability of the lattice. Membership does not req…

  Exact mathematical passage

  
   a_{fg}(d)=\sum_{r=0}^{d}\binom dr a_f(r)a_g(d-r),
   \qquad a_{Uf}(d)=a_f(d+1).
  

Search aliases: a_H_ax+j

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• Use in source · Read passage · 72_recovered.tex, line 2393

  , v_0,w∈ Z_p^d, H>0, and a,j,n,r∈ N. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H_a=I+p^L+aB_a for an integral matrix B_a. Every refined packet b_a,j(x)=a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Equivalently, Δ_H_a^ ra_n ≡0 pmod p^(L+a)r.

  Exact mathematical passage

  
  b_{a,j}(x)=a_{H_ax+j}
  

Search aliases: a_Hx+j

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• Use in source · Read passage · 72_recovered.tex, line 1681

  means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not an as-written certificate. The full original proof follows under the corrected hypotheses. Assume C^H=I+p^LB with B∈ M_d( Z_p). For every address j and every x∈ Z_p, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^TC^jB^rv_0. The series converges on Z_p, and Δ_H^ra_n ≡0 pmod p^Lr.

  Exact mathematical passage

  
  \boxed{
  a_{Hx+j}
  =
  \sum_{r\geq0}
  p^{Lr}\binom xr
  w^TC^jB^rv_0.
  }
  

Search aliases: a_i

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• Use in source · Read passage · 10_foundations.tex, line 337

  [Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).

  Exact mathematical passage

  a_i\in\Zp

Search aliases: A_j

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• Use in source · Read passage · edition_summary.tex, line 11

  The strongest common-step conclusion in width four is particularly compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and positive step. It is exact in address two at every power-of-two order. The proof chapters also give mixed-step forms, the cancellation criterion in addre…

  Exact mathematical passage

  
   \sum_{i=0}^k(-1)^{k-i}\binom ki E_{4(n+iH)}^{(4,j)}
   \equiv0\pmod{2^{\,v_2(k!)+kv_2(H)+A_j(k)}},
  

Search aliases: a_j

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• Use in source · Read passage · 40_regularity.tex, line 40

  ts strict derivative is the continuous linear map represented by the matrix of evaluated formal partial derivatives. For the derivative statement, use polynomial induction on constants, addition, and multiplication by a variable. Constants have zero derivative. The sum rule agrees with the additivity of formal partial differentiation. For PX_j, the product rule gives d(PX_j)_a(v)=a_j dP_a(v)+P(a)v_j, which agrees with the evaluated formal product rule for every coordinate. The finite product rul…

  Exact mathematical passage

  
   d(PX_j)_a(v)=a_j\,dP_a(v)+P(a)v_j,
  

Search aliases: a_j,d

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• Use in source · visual_body.tex, line 455

  At every center x_0∈ Z_2, the interpolations admit ordinary Taylor series C_j(x_0+z)=∑_d≥0a_j,d(x_0)z^d. For positive degrees, v_2(a_j,d(x_0)) · =d · · (j=0,1),

  Exact mathematical passage

  C_j(x_0+z)=\sum_{d\ge0}a_{j,d}(x_0)z^d

Search aliases: a_j,k

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• Use in source · Read passage · 20_arithmetic.tex, line 677

  The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvement. The recurrence proves the following exact valuations: cll Address j · Further orders k · v_2(a_j,k) in that order

  Exact mathematical passage

  v_2(a_{j,k})\ge k

Search aliases: a_k

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• Use in source · Read passage · 20_arithmetic.tex, line 614

  [Finite Newton dominance] Let f: N→ Q and write a_k=Δ_1^kf(0). Suppose that c∈ N, a_1≠0, v_2(a_1)=c, and v_2(a_k)≥ c+1+lfloorlog_2krfloor (k≥2). Then for every n≥0 and T>0, f(n+T)-f(n)≠0, v_2(f(n+T)-f(n))=v_2(T)+c.

  Exact mathematical passage

  a_k=\Delta_1^kf(0)

Search aliases: a_k+i

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• Use in source · Read passage · 20_arithmetic.tex, line 628

  Finite Newton reconstruction at an arbitrary row gives Δ_1^kf(n)=∑_i=0^nbinom ni a_k+i. The monotonicity of lfloorlog_2krfloor therefore preserves the higher-coefficient bound after translation. In addition, Δ_1f(n)-a_1∈2^c+1 Z_2, so the first difference is nonzero of exact valuation c.

  Exact mathematical passage

  
   \Delta_1^kf(n)=\sum_{i=0}^{n}\binom ni a_{k+i}.
  

Search aliases: a_m

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• Use in source · Read passage · strengthening.tex, line 664

  The low Newton coefficients of the actual reciprocal are C_2(0)=1, C_2(1)=-frac115, C_2(2)=frac1345, a_1=-frac1615, a_2=frac6445, where a_m=Δ_1^m C_2(0). Thus v_2(a_1)=4 and v_2(a_2)=6. For m≥3, [source reference] bounds a_m by 2m+v_2(m!). The finite shifted Newton formula Δ_1^kC_2(n)=∑_m=k^n+ka_mbinom nm-k therefore gives v_2(Δ_1^kC_2(n))≥ k+v_2(k!)+3 for every k≥1. For k≥3 this is 2k≥ k+3; for k=1,2 it uses the exact initial coefficients and the higher tail bound. The finite step expansion pro…

  Exact mathematical passage

  a_m=\Delta_1^m C_2(0)

Search aliases: a_M(Nq+s)+r

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• Use in source · Read passage · 72_recovered.tex, line 2031

  Direct substitution gives (a_r^[M])_s^[N](q) = a_M(Nq+s)+r. Iterate this identity along the mixed radix path.

  Exact mathematical passage

  
  \left(a_r^{[M]}\right)_s^{[N]}(q)
  =
  a_{M(Nq+s)+r}.
  

Search aliases: a_M(q+1)

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• Use in source · Read passage · 72_recovered.tex, line 2094

  For r<M-1, (Ea)_Mq+r=a_Mq+r+1=a_r+1^[M](q). For r=M-1, (Ea)_Mq+M-1=a_M(q+1)=E_qa_0^[M](q). After M shifts every phase has advanced by one coarse index.

  Exact mathematical passage

  
  (Ea)_{Mq+M-1}=a_{M(q+1)}=E_qa_0^{[M]}(q).
  

Search aliases: a_M-1^[M]

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• Use in source · Read passage · 72_recovered.tex, line 1977

  For a sequence a=(a_n)_n≥0 and M≥2, define a_r^[M](q)=a_Mq+r (0≤ r<M). The analysis map is P_Ma = (a_0^[M],…,a_M-1^[M]). Synthesis interleaves the branches by the rule a_Mq+r=a_r^[M](q).

  Exact mathematical passage

  
  \mathcal P_Ma
  =
  \left(a_0^{[M]},\ldots,a_{M-1}^{[M]}\right).
  

Search aliases: a_Mq+r

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• Use in source · Read passage · 10_foundations.tex, line 485

  The recovered Annals text develops a broader polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplicati…

  Exact mathematical passage

  a_r^{[M]}(q)=a_{Mq+r}

Search aliases: a_Mq+r+1

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• Use in source · Read passage · 72_recovered.tex, line 2090

  For r<M-1, (Ea)_Mq+r=a_Mq+r+1=a_r+1^[M](q). For r=M-1, (Ea)_Mq+M-1=a_M(q+1)=E_qa_0^[M](q). After M shifts every phase has advanced by one coarse index.

  Exact mathematical passage

  
  (Ea)_{Mq+r}=a_{Mq+r+1}=a_{r+1}^{[M]}(q).
  

Search aliases: A_n

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• Use in source · Read passage · 72_recovered.tex, line 484

  [Exact coefficient transport] Let A(t)=∑_n≥0A_nfract^nn!, S(t)=H(t)A(t). Then S_n = ∑_r=0^n binom nr h_rA_n-r. The inverse coordinate formula is A_n = ∑_r=0^n binom nr k_rS_n-r. Both transforms are lower triangular with diagonal one.

  Exact mathematical passage

  
  A(t)=\sum_{n\geq0}A_n\frac{t^n}{n!},
  \qquad
  S(t)=H(t)A(t).
  

Search aliases: a_n

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• Use in source · Read passage · 10_foundations.tex, line 78

  Over a characteristic-zero field K, put E(a)=∑_n≥0fraca_nn!t^n, D(A)_n=n![t^n]A. These inverse maps between sequences and K[[t]] provide formal exponential coordinates. No analytic convergence is needed for the coefficient identities below.

  Exact mathematical passage

  
  \mathcal E(a)=\sum_{n\geq0}\frac{a_n}{n!}t^n,
  \qquad \mathcal D(A)_n=n![t^n]A.
  

Search aliases: a_n+1

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• Use in source · Read passage · 71_native.tex, line 161

  Consider the recurrence a_n+1=a_n with integer initial value. Its global parameter space is X= Z. Its local parameter spaces are X_p= Z_p, and every local realization has the same prime-independent recurrence. Give the local factors their usual p-adic topologies and the product its product topology. For these integral local factors, the restricted product relative to all Z_p is the ordinary product. Write

  Exact mathematical passage

  a_{n+1}=a_n

Search aliases: A_N,j

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• Use in source · Read passage · strengthening.tex, line 174

  [All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν),

  Exact mathematical passage

  A_{N,j}(n)=(-1)^n C_{N,j}(n)

Search aliases: A_n-r

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• Use in source · Read passage · 72_recovered.tex, line 490

  [Exact coefficient transport] Let A(t)=∑_n≥0A_nfract^nn!, S(t)=H(t)A(t). Then S_n = ∑_r=0^n binom nr h_rA_n-r. The inverse coordinate formula is A_n = ∑_r=0^n binom nr k_rS_n-r. Both transforms are lower triangular with diagonal one.

  Exact mathematical passage

  
  \boxed{
  S_n
  =
  \sum_{r=0}^{n}
  \binom nr h_rA_{n-r}.
  }
  

Search aliases: a_n-r

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• Use in source · Read passage · 10_foundations.tex, line 95

  [Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1. The transported selectors widehat P_N,j=U_HP_N,jU_H^-1 are idempotent and pairwise orthogonal, and their sum is the identity.

  Exact mathematical passage

  
  (U_Ha)_n=\sum_{r=0}^{n}\binom nr h_r a_{n-r},\qquad
  (U_H^{-1}a)_n=\sum_{r=0}^{n}\binom nr k_r a_{n-r}.
  

Search aliases: a_n^(m)

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• Use in source · Read passage · 72_recovered.tex, line 260

  [Multiple selector] Define a^(m)∈ H( F_p) by a_n^(m) = 1, · mmid n, 0, · mnmid n.

  Exact mathematical passage

  
  a_n^{(m)}
  =
  \begin{cases}
  1,&m\mid n,\\
  0,&m\nmid n.
  \end{cases}
  

Search aliases: A_p,2m

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• Use in source · Read passage · 72_recovered.tex, line 1188

  [Unified local Euler transfer] For every prime p and every even n≥0, E_n = ∑_r=0^n/2 binom n2r κ_p,r A_p,n-2r. If H_p(t) = ∑_m≥0A_p,2mfract^2m(2m)! and K_p(t) = ∑_r≥0κ_p,rfract^2r(2r)!, then K_p(t)H_p(t)=secht. For odd p, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). For p=2, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t).

  Exact mathematical passage

  
  H_p(t)
  =
  \sum_{m\geq0}A_{p,2m}\frac{t^{2m}}{(2m)!}
  

Search aliases: A_p,n

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• Use in source · Read passage · 72_recovered.tex, line 1156

  For a prime p, define A_p,n = S_n(p), · p odd, R_n, · p=2 and n even, 0, · p=2 and n odd, and κ_p,r = (-1)^rZ_2r+1p^2r, · p odd,

  Exact mathematical passage

  
  A_{p,n}
  =
  \begin{cases}
  S_n(p),&p\text{ odd},\\
  R_n,&p=2\text{ and }n\text{ even},\\
  0,&p=2\text{ and }n\text{ odd},
  \end{cases}
  

Search aliases: A_p,n-2r

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• Use in source · Read passage · 72_recovered.tex, line 1177

  [Unified local Euler transfer] For every prime p and every even n≥0, E_n = ∑_r=0^n/2 binom n2r κ_p,r A_p,n-2r. If H_p(t) = ∑_m≥0A_p,2mfract^2m(2m)! and K_p(t) = ∑_r≥0κ_p,rfract^2r(2r)!, then K_p(t)H_p(t)=secht. For odd p, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). For p=2, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t).

  Exact mathematical passage

  
  \boxed{
  E_n
  =
  \sum_{r=0}^{n/2}
  \binom n{2r}
  \kappa_{p,r}
  A_{p,n-2r}.
  }
  

Search aliases: a_q

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• Use in source · Read passage · 10_foundations.tex, line 113

  The factorials are nonzero in K, so encoding and decoding cancel coefficientwise. The constant coefficient of H is 1; hence its formal power-series inverse exists. Multiplication by H and H^-1 gives the two inverse laws. The ordinary product coefficient formula, multiplied by n!, gives the displayed binomial sums. In row n, the coefficient of a_n is h_0=1 and coefficients of a_q with q>n vanish. Conjugating the identities of Theorem~[source reference] by the constructed equivalence proves the tr…

  Exact mathematical passage

  a_q

Search aliases: a_r

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• Use in source · Read passage · 72_recovered.tex, line 34

  This article begins with the Hurwitz product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_rb_n-r. Over a field of characteristic zero it is multiplication of exponential generating functions. Over a finite field it realizes the additive group algebra through the exponential modes ϵ_u=(u^n)_n≥0. This finite algebra gives an exact proof of the standard window period. Hurwitz series rings are treated in [source reference]. The finite difference and umbral background may be found in [source reference].

  Exact mathematical passage

  
  (a\starh b)_n
  =
  \sum_{r=0}^{n}
  \binom nr a_rb_{n-r}.
  

Search aliases: a_r+1^[M]

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• Use in source · Read passage · 72_recovered.tex, line 2090

  For r<M-1, (Ea)_Mq+r=a_Mq+r+1=a_r+1^[M](q). For r=M-1, (Ea)_Mq+M-1=a_M(q+1)=E_qa_0^[M](q). After M shifts every phase has advanced by one coarse index.

  Exact mathematical passage

  
  (Ea)_{Mq+r}=a_{Mq+r+1}=a_{r+1}^{[M]}(q).
  

Search aliases: a_r+Ms^[MN]

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• Use in source · Read passage · 10_foundations.tex, line 488

  polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a …

  Exact mathematical passage

  
  A(z)=\sum_{r=0}^{M-1}z^rA_r(z^M),\qquad
  \bigl(a_r^{[M]}\bigr)_s^{[N]}=a_{r+Ms}^{[MN]}.
  

Search aliases: A_r^[M]

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• Use in source · Read passage · 72_recovered.tex, line 1993

  [Exact discrete polyphase reconstruction] The analysis map P_M is a bijection between sequences and ordered families of M sequences. If A(z)=∑_n≥0a_nz^n and A_r^[M](w)=∑_q≥0a_Mq+rw^q, then A(z)=∑_r=0^M-1z^rA_r^[M](z^M).

  Exact mathematical passage

  
  A_r^{[M]}(w)=\sum_{q\geq0}a_{Mq+r}w^q,
  

Search aliases: a_r^[M]

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• Use in source · Read passage · 10_foundations.tex, line 485

  The recovered Annals text develops a broader polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplicati…

  Exact mathematical passage

  a_r^{[M]}(q)=a_{Mq+r}

Search aliases: A_u

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• Use in source · Read passage · 10_foundations.tex, line 494

  a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a carry and a term with one carry: C_r(w)=∑_u+v=rA_u(w)B_v(w) +w∑_u+v=r+MA_u(w)B_v(w), 0≤ u,v<M. Adding the two phase addresses gives the formula. It describes ordinary Cauchy multiplication; Hurwitz convo…

  Exact mathematical passage

  
  C_r(w)=\sum_{u+v=r}A_u(w)B_v(w)
  +w\sum_{u+v=r+M}A_u(w)B_v(w),\qquad 0\leq u,v<M.
  

Search aliases: a_Uf

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• Use in source · Read passage · supplement_08.tex, line 71

  pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_r=0^dbinom dr a_f(r)a_g(d-r), a_Uf(d)=a_f(d+1). These identities prove multiplication and derivative stability of the lattice. Membership does not req…

  Exact mathematical passage

  
   a_{fg}(d)=\sum_{r=0}^{d}\binom dr a_f(r)a_g(d-r),
   \qquad a_{Uf}(d)=a_f(d+1).
  

Search aliases: A_z_n

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• Use in source · Read passage · continuation_03.tex, line 49

  The one-step Taylor theorem already supplies the exact correction equation b_n+A_z_nt=0, b_n=F(z_n)/p^L+nbmod p. Surjectivity supplies a correcting digit for every residual b_n. The chosen correction gives an actual new integer point z_n+1, using the proved nonlinear Taylor remainder.

  Exact mathematical passage

  
  b_n+A_{z_n}t=0,\qquad
  b_n=F(z_n)/p^{L+n}\bmod p.
  

Search aliases: A_z_n+1

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• Use in source · Read passage · continuation_03.tex, line 58

  Every coordinate change is divisible by p^L+n, hence by p. Each formal partial derivative is itself an integer polynomial, so its value is unchanged modulo p. Therefore A_z_n+1=A_z_n=A_x. The proof assumes no separate surjectivity or solvability statement at later levels. A dependent recursion chooses each next point from the proved one-step existence theorem. It produces one compatible sequence, rather than unrelated witnesses at different precisions.

  Exact mathematical passage

  
  A_{z_{n+1}}=A_{z_n}=A_x.
  

Search aliases: A_ℓ,n ell

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• Use in source · Read passage · 72_recovered.tex, line 1278

  . For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero; the even kernel L_3K_2 cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity H_p=L_pK_ℓ H_ℓ and its proof remain valid. For any two primes p and ℓ, the source sequences A_p,n and A_ℓ,n are related by an exact unit lower triangular transformation obtained by composing the inverse kernel L_p with the reconstruction kernel K_ℓ.

  Exact mathematical passage

  A_{\ell,n}

Search aliases: act mathrm

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• Use in source · Read passage · root_scope.tex, line 374

  ion K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a dyadic isometry. It therefore has a unique critical point c_r∈ Z_2. Set Φ_r^act=K_r(c_r). The earlier independent-undepletion notation for this same scalar is Φ_r(2^2r-25). Excluding a multiple root on the surviving chart is exactly the assertion Φ_r^act≠0.

  Exact mathematical passage

  \Phi_r^{\mathrm{act}}=K_r(c_r)

Search aliases: aug operatorname

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• Use in source · Read passage · 72_recovered.tex, line 210

  [Augmentation criterion] Let x=∑_u∈ F_pc_u[u] ∈ F_p[ F_p^+]. Then x is a unit if and only if aug(x) = ∑_uc_u is nonzero. If J is the augmentation ideal, then J^p=0.

  Exact mathematical passage

  
  \operatorname{aug}(x)
  =
  \sum_uc_u
  

Search aliases: B mathcal

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• Use in source · Read passage · supplement_10.tex, line 58

  The normalized defect is not assumed to be a derivation. Its proved product rule contains the two middle Leibniz sums: B_Θ(FG)=(B_Θ F)G+F(B_Θ G)+ B_p(F,G). The coefficients of B_p are the actual integral quotients binomp^2a/p and binom pa/p. Derivative stability and this correction formula show that the elements F∈ W_p satisfying B_Θ F∈ W_p are closed under the finite ring operations.

  Exact mathematical passage

  
   B_\Theta(FG)=(B_\Theta F)G+F(B_\Theta G)+\mathcal B_p(F,G).