"""Exact spectral, resonance and reciprocal parameter mechanisms. CC0-1.0."""
from pathlib import Path
from html import escape
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<title id="title">Spectral, resonant and reciprocal parameter equivalence</title>
<desc id="desc">The spectral list records exponent and block length. A common spectral shift is allowed in B1 because vertical dilations belong to it, whereas A1 permits only exponent scaling. The polynomial extension coefficient transforms by t plus r+1 minus kappa times d. At resonance zero and nonzero coefficients are distinct. Exchanging coordinate foliations identifies kappa and its reciprocal.</desc>
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text(42,49,'The exact parameter equivalence mechanisms','title')
text(42,87,'Theorem 7.15: all exponents move together; polynomial resonance retains a class.','')
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text(66,151,'Spectral list: (αᵢ, nᵢ) means e^(αᵢx) P_(nᵢ−1)','sub math')
for y,vals in [(226,[0,1,3]),(360,[2,3,5])]:
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text(1120,233,'α ∈ R','small');text(1120,367,'real slice','small')
text(640,300,'ỹ = e^(2x)y:  Σ → Σ+2;  p → p+2Y;  Y → Y.','math','middle')
text(66,435,'B₁: common affine change α → aα+b, a ≠ 0; block lengths are retained.','')
text(66,473,'A₁: common scaling α → aα only; a zero exponent stays zero.','')
text(66,514,'Thus these two lists are B₁-equivalent and A₁-inequivalent. Exercise 7.','small')
box(42,578,1196,386,'#ecfdf5')
text(66,619,'Resonance of the affine quotient lift','sub')
text(66,660,'H = xp + κY + t x^R q,   R = r+1,   kernel = Pᵣq.','math')
text(66,699,'ỹ = y + d x^R  ⇒  t → t+(R−κ)d.','math')
text(66,735,'p changes by Rd x^(R−1)q, which lies in the kernel.','small')
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text(337,807,'κ ≠ R','sub math','middle');text(337,851,'Choose d = −t/(R−κ).','math','middle');text(337,889,'Every t is equivalent to zero.','small','middle')
text(923,807,'κ = R','sub math','middle');text(923,851,'Translation leaves t unchanged.','', 'middle');text(923,889,'Constant y scale: two classes, t=0 and t≠0.','small','middle')
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text(66,1027,'Exchanging the two invariant coordinate foliations','sub')
text(66,1070,'Tκ = ⟨p, q, xp+κY⟩,   κ ≠ 0,1.','math')
text(66,1112,'X = y,  ỹ = x:   diag(1,κ) → diag(κ,1) → diag(1,1/κ).','math')
text(66,1154,'The last step rescales the isotropy generator by 1/κ.','small')
text(66,1196,'The complete point-equivalence relation is κ ~ 1/κ.','')
text(42,1291,'Local holomorphic coordinates; no classification of singular orbits or restricted projective conjugacy is asserted.','small')
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