Euler Signal Bridge

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1 KOMOREBI: EXACT EULER / SIGNAL BRIDGE
2 
3 This is a bounded, reproducible demonstration of a shared operator law.
4 It does not claim that the underlying FIR identities are new.
5 
6 Let S a(n) = a(n+1), Delta = S-I, and R_(M,s) a(q)=a(Mq+s).
7 Then
8 
9   Delta_q^k R_(M,s) = R_(M,s) (S^M-I)^k.
10 
11 Proof: advancing one position in the sampled lane advances M positions in
12 the original sequence. Thus S_q R_(M,s)=R_(M,s) S^M; subtract the identity
13 and take powers. This is an exact multirate filter identity.
14 
15 If c_r=Delta^r a(0), then the lane coefficients are
16 
17   d_(s,k) = sum_r [t^r](1+t)^s ((1+t)^M-1)^k c_r.
18 
19 Proof: evaluate S^s(S^M-I)^k a at index zero, and substitute S=I+Delta.
20 For M>=1 and s>=0 the sum is finite, supported on k<=r<=Mk+s.
21 This identity is valid for arbitrary integer sequences without an
22 analytic interpolation assumption.
23 
24 The same filter has frequency response H_(M,k)(omega)=(exp(i M omega)-1)^k.
25 For k>=1 its comb zeros lie at omega=2 pi j/M. This follows by applying
26 the operator to exp(i omega n). The growing Euler sequence itself has
27 no ordinary convergent DTFT; the frequency response belongs to the filter.
28 The causal form (I-S^(-M))^k differs by a shift and has the same gain.
29 
30 SIGNED EULER CONTROL
31 
32 Use cosh(t)*sech(t)=1 and a(n)=E_(2n). The initial values are
33 1,-1,5,-61,1385. Initial difference coordinates are 1,-2,8,-80,1664.
34 
35 M=2: S^2-I=2 Delta+Delta^2, giving d_1=2(-2)+8=4.
36 The next coefficient is d_2=4(8)+4(-80)+1664=1376.
37 M=4: S^4-I=4 Delta+6 Delta^2+4 Delta^3+Delta^4.
38 Its first output is -8+48-320+1664=1384=8*173.
39 Successive even refinements give 2 d_1+d_2=1384, agreeing exactly.
40 
41 SIGNAL / ARITHMETIC INTERPRETATIONS
42 
43 The signal reading evaluates the shift polynomial on the unit circle.
44 The arithmetic reading expands that same polynomial in Delta and measures
45 the exact valuations of its integer outputs. A frequency notch and p-adic
46 divisibility are distinct properties connected through this common operator.
47 
48 For prime p the residual is explicit:
49 
50   S^p-I = Delta^p + p B_p(Delta),
51   B_p(t) = sum_(j=1)^(p-1) (binom(p,j)/p) t^j in Z[t].
52 
53 Modulo p the correction disappears. At higher precision it must be kept.
54 For p=2, B_2(Delta)=Delta. Thus the residual 2 c_1=-4 cancels part
55 of c_2=8 to give the output 4. Discarding the residual would give 8.
56 This is a precise place to retain a residual packet in the calculus.
57 
58 The general identities are proved algebraically above. The supplied exact
59 computation checks independent implementations in declared finite ranges.
60 Whole-branch congruences need bounds on all remaining coefficients.
61 
62 REPRODUCTION
63 
64 Run python verify_bridge.py in this folder. Python 3, NumPy and Matplotlib
65 are required. Inspect exact_results.json and computation_manifest.json.
66 The source hash is recorded in SOURCE_SHA256.txt.
67 
68 PROJECT SOURCE
69 
70 The Euler example extends the finite computations in
71 Euler_Deep_Polyphase_Refinement_Example.pdf, especially its first and second
72 refinements and exact kernel-composition law. This demonstration supplies
73 the explicit FIR interpretation and prime-residual identity.

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