Paper IXStatements and proofs
Statements and proofs
Find a statement anywhere in this work by title, printed number, type or source label. Entries follow the chapter order. Each proof link opens the written proof passage.
Statements by chapter
Arithmetic packets and coefficient transport
- Theorem 2.1: Residue reconstruction Read proof
- Theorem 2.2: Constructed unit-triangular coordinates Read proof
- Theorem 2.3: Hurwitz product rules Read proof
- Theorem 2.4: Forward substitution and denominator clearing Read proof
- Theorem 2.5: Filtered finite reconstruction Read proof
- Theorem 2.6: Finite Newton–Mahler identities Read proof
- Theorem 2.7: Weighted difference bound Read proof
- Theorem 2.8: Differences of unit modes Read proof
- Theorem 2.9: Finite unit coordinates Read proof
- Theorem 2.10: Companion jumps and packet recurrences Read proof
- Theorem 2.11: Dyadic–odd packet factorization Read proof
Congruential Euler packets and their arithmetic
- Definition 3.1: Congruential Euler coefficients
- Proposition 3.2: A denominator bound for every address Read proof
- Example 3.3: The Bernoulli boundary address
- Definition 3.4: Differences and valuation conventions
- Proposition 3.5: Lucas reduction and address independence Read proof
- Theorem 3.6: Nishibuchi congruence
- Theorem 3.7: All-address differences on the divisor branch Read proof
- Example 3.8: Why the factorial loss is necessary
- Corollary 3.9: A common period in the standard window Read proof
- Lemma 3.10: Prime-power period from finite differences Read proof
- Theorem 3.11: All-row dyadic higher differences Read proof
- Corollary 3.12: Step-sensitive divisor-branch bound Read proof
- Corollary 3.13: Dyadic periods from row zero Read proof
- Lemma 3.14: Finite Newton dominance Read proof
- Theorem 3.15: Exact four-packet displacement law Read proof
- Corollary 3.16: All positive periods and their least element Read proof
- Definition 3.17: Odd-length source and inverse coefficients
- Theorem 3.18: Classical odd-length reconstruction Read proof
- Corollary 3.19: Odd-prime truncation ladder Read proof
- Definition 3.20: Dyadic truncation
- Lemma 3.21: Exact reconstruction and source integrality Read proof
- Lemma 3.22: Exact weighted inverse coefficient Read proof
- Theorem 3.23: Optimal uniform dyadic tower Read proof
- Proposition 3.24: Row-dependent summand weights Read proof
- Theorem 3.25: Isometric packet bijections Read proof
- Corollary 3.26: Exact image of the original coefficients Read proof
Threshold-free reciprocal transfer at prime-power section orders
Factorial precision at section order four
Sharp higher-difference families
Exact first differences, least periods, and dyadic images
- Lemma 7.1: A first-difference criterion Read proof
- Theorem 7.2: Exact four-address first differences Read proof
- Corollary 7.3: Exact image and residue enumeration Read proof
- Theorem 7.4: Lower-half exact laws Read proof
- Proposition 7.5: Small-rank complete isometry table Read proof
- Remark 7.6: Failure of a universal upper-half exact law
Analytic interpolation and exact Taylor radii
The final congruence in the original Euler convention
Finite obstructions, compatible lifts, and deformation
- Lemma 10.1: Polynomial Taylor remainder Read proof
- Theorem 10.2: Finite polynomial obstruction Read proof
- Proposition 10.3: Affine depth zero Read proof
- Proposition 10.4: Independence of the integer representative Read proof
- Theorem 10.5: Common correction and fiber product Read proof
- Corollary 10.6: Simultaneous polynomial lifting Read proof
- Theorem 10.7: Torsor and lift cardinalities Read proof
- Theorem 10.8: Compatible polynomial tower Read proof
- Theorem 10.9: Completion and smooth branching Read proof
- Remark 10.10
- Proposition 10.11: Smooth coefficient stability Read proof
- Theorem 10.12: Continuous charts with a selected Jacobian minor Read proof
- Theorem 10.13: Prescribed-parameter obstruction and count Read proof
- Theorem 10.14: Every compatible integer parameter path Read proof
- Theorem 10.15: Arbitrary p-adic parameter endpoints Read proof
- Theorem 10.16: Weighted nonlinear lifting Read proof
- Example 10.17: Strict threshold and displacement boundary
- Theorem 10.18: Weighted coefficient deformation Read proof
- Theorem 10.19: Joint coefficient and parameter deformation Read proof
- Example 10.20: First-order sign and error precision
Analytic regularity of the local solution map
- Lemma 11.1: Derivative of polynomial evaluation Read proof
- Lemma 11.2: The augmented derivative Read proof
- Theorem 11.3: Analytic local root graph Read proof
- Theorem 11.4: Regularity of a supplied root selection Read proof
- Corollary 11.5: Regularity throughout an open parameter set Read proof
- Corollary 11.6: Integral coefficients and centers Read proof
- Example 11.7: The derivative sign and a nonlinear control
A. Exact classical Euler tails and the remaining simplicity problem
Hurwitz series and finite field subgroup inversion
- Historical definition D.1: Hurwitz series ring
- Historical definition D.2: Exponential mode
- Historical proposition D.3: Addition law for modes
- Historical theorem D.4: Finite field mode algebra
- Historical proposition D.5: Augmentation criterion
- Historical corollary D.6: Finite inverse formula
- Historical definition D.7: Multiple selector
- Historical lemma D.8: Subgroup average
- Historical theorem D.9: Support and positive degree period
- Historical definition D.10: Compressed inverse
- Historical proposition D.11: Compressed inverse recurrence
- Historical corollary D.12: Uniform compressed period
- Historical corollary D.13: Rational ordinary generating function
- Historical theorem D.14: Exponential form and minimal period
- Historical example D.15: A compressed inverse modulo seven
Unit triangular coefficient transport
Congruential reciprocal packets
- Historical definition D.20: Congruential reciprocal packet
- Historical theorem D.21: Triangular inverse system
- Historical proposition D.22: A denominator bound
- Historical example D.23: Euler and Bernoulli boundaries
- Historical standard fact D.24: Lucas congruence
- Historical theorem D.25: Standard window reduction
- Historical corollary D.26: Standard window integrality
- Historical corollary D.27: Address independence modulo the prime
- Historical theorem D.28: Uniform standard window period
- Historical corollary D.29: Rational residual generating function
- Historical theorem D.30: Fourier invariance of the standard window operator
- Historical corollary D.31: A second proof of the standard window period
- Historical remark D.32
One local Euler source and kernel calculus
- Historical lemma D.33: Odd source generating function
- Historical lemma D.34: Zigzag inverse kernel
- Historical theorem D.35: Odd modulus Euler coordinate pair
- Historical corollary D.36: Odd modulus truncation
- Historical proposition D.37: Odd row weight
- Historical proposition D.38: Dyadic inverse coefficients
- Historical theorem D.39: Dyadic Euler coordinate pair
- Historical theorem D.40: Unified local Euler transfer
- Historical theorem D.41: Unified inverse local transfer
- Historical corollary D.42: Exact change between local source charts
- Historical example D.43: The first odd reconstruction
- Historical example D.44: The first nontrivial dyadic row
- Historical remark D.45
- Historical theorem D.46: Exact dyadic kernel weight
- Historical theorem D.47: Optimal dyadic truncation
Mahler transfer and local analytic branches
- Historical theorem D.48: Finite Mahler reconstruction
- Historical corollary D.49: Mahler valuation ledger
- Historical standard fact D.50: Local unit decomposition
- Historical theorem D.51: One unit mode
- Historical example D.52: Paired principal unit expansions
- Historical corollary D.53: Basic odd and dyadic difference gains
- Historical theorem D.54: Polynomial exponential Mahler transfer
- Historical theorem D.55: Matrix Carmichael and Mahler transfer
- Historical theorem D.56: Finite adelic transfer
- Historical example D.57: A simultaneous Fibonacci jump
- Historical definition D.58: Amice pullback
- Historical theorem D.59: Derivative conjugacy
- Historical proposition D.60: Weighted convergence
- Historical theorem D.61: Depleted odd prime reconstruction
- Historical theorem D.62: Odd prime analytic Euler branch
- Historical remark D.63
- Historical lemma D.64: Analytic dyadic source
- Historical theorem D.65: Direct dyadic Euler interpolation
- Historical theorem D.66: Unified local interpolation statement
Polyphase refinement of local Mahler coordinates
- Historical theorem D.67: Exact discrete polyphase reconstruction
- Historical theorem D.68: Nested discrete refinement
- Historical proposition D.69: One carry convolution law
- Historical theorem D.70: Polyphase shift matrix
- Historical theorem D.71: Exact local branch reconstruction
- Historical proposition D.72: Edge coefficients of the affine kernel
- Historical theorem D.73: Mahler polyphase transfer
- Historical theorem D.74: Kernel composition
- Historical theorem D.75: Polyphase valuation ledger
- Historical theorem D.76: Prime power sparsification
- Historical theorem D.77: Scalar polyphase precision amplification
- Historical lemma D.78: Matrix power amplification
- Historical theorem D.79: Matrix polyphase precision amplification
- Historical example D.80: Refined dyadic Fibonacci ladder
- Historical proposition D.81: Local phase lattice
- Historical corollary D.82: Polyphase refinement of the Euler branches
- Historical remark D.83
Arithmetic lift obstructions
- Historical definition D.84: Linear lift obstruction
- Historical theorem D.85: One level linear lifting
- Historical proposition D.86: Representative independence
- Historical corollary D.87: Exact number of linear children
- Historical example D.88: A visible linear obstruction
- Historical definition D.89: Polynomial lift obstruction
- Historical theorem D.90: One level polynomial lifting
- Historical proposition D.91: Polynomial representative independence
- Historical corollary D.92: Simple root lifting
- Historical example D.93: A singular polynomial obstruction
- Historical theorem D.94: Packet decomposition of a lift obstruction
- Historical remark D.95
Recurrence packets and a common integral spine
- Historical theorem D.96: Companion jump recurrence
- Historical theorem D.97: Characteristic polynomial of the jump
- Historical theorem D.98: Residue subsequences of a second order recurrence
- Historical corollary D.99: Fibonacci residue packets
- Historical theorem D.100: Periodicity of an invertible recurrence modulo a prime power
- Historical corollary D.101: Residue packet periods
- Historical theorem D.102: Base change of an integral recurrence
- Historical corollary D.103: Prime independent recurrence spine
- Historical remark D.104
Chinese remainder packet refinement
- Historical definition D.105: Residue projector
- Historical proposition D.106: Complete residue packet system
- Historical theorem D.107: Coprime packet factorization
- Historical corollary D.108: Several coprime moduli
- Historical standard fact D.109: Finite local reconstruction
- Historical proposition D.110: Dyadic and odd residue refinement
- Historical remark D.111