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ZetaCosmology: correct image-structure of other universes in CMB
Add Zeta Cosmology page
MetaplecticChirp: fix matrix row separators and rotation rendering
MetaplecticChirp: fix math delimiters for proper rendering
Add Metaplectic Chirp page
add: restricted SU(2) Yang-Mills via spectral whitening — referee report on issue #964 transfer template; flags 7.1 torus reduction, 7.2 gap location, 7.3 zeros of S, 7.4 Akhiezer transfer as fixable, not established
fix: break markdown link parsing of 𝒯[…](ξ) by inserting U+2009 thin space
rewrite: Fresnel regularizer e^{-i(ηλ)Θ²}, Backlund-exact mean density, J_{-½}/J₀ split via S(T) — product ηλ with η, λ independent in iterated limit
rewrite: two-parameter (η, λ) structure with iterated limit η↓0 then λ↓0 The single regularizer σ collapsed regularization and displacement together; in the σ↓0 limit both vanished simultaneously, leaving the Hadamard pole at λ=δ unregularized. The two parameters now act independently: - η > 0: Gaussian regularizer width, controls e^{−η t²} - λ ≥ 0: probe-line displacement at ½ + λ + iℝ Lorentzian half-width w_λ = |λ − δ| is independent of η. Iterated limit (η ↓ 0 first, then λ ↓ 0) preserved throughout.
add §5.4: RH as strict Cramer-Karhunen convergence; Theorem 5.6 RH⇔strict convergence
add Lorentzian profile definition and theorem (literal Breit-Wigner identity)
rewrite: pure Unicode math, no LaTeX
fix: escape inline math _ as \_ for markdown italic protection
fix: brace-form _{X} in inline math to defeat markdown italic pairing
fix: strip \_ inside math, put $$ blocks on own lines (MathJax)
fix: replace [/]/<>/\_ with \lbrack/\rbrack/\lt/\gt/_{...} for MathJax
add: WarpedCramerTransformAndOffLineZeroDetection wiki page
fix: escape underscores in inline math; rephrase 'linear-in-X' patterns; remove italic in blockquote
fix: split dense inline-math line into display math to ensure rendering
fix: drop \mathrm{B} and 'in-$t$' patterns that broke MathJax tokenization
fix: add space after \lbrace/\rbrace control words for MathJax tokenization
fix: replace \{...\} with \lbrace...\rbrace in inline math for MathJax
PadeMuntzFractionalRiccati: full markdown port of the LaTeX paper
PadeMuntzFractionalRiccati: replace with verbatim LaTeX source
Add MuntzCompletenessAndCoefficientDivergence: duality between Muntz-Szasz completeness and Riccati coefficient divergence, role of Pade-Muntz acceleration
Sharpen WSS hypothesis (binary, on Y not Z) and reframe higher moments via Cramer isomorphism Section 2.1 rewritten: - WSS is binary; strike all 'approximately' qualifiers. - The c.2023 numerical observation on Z is finite-window translation-stability of the normalized kernel up to a discrepancy that vanishes as window length grows -- not exact translation-invariance, which is impossible at finite window length. - The exact stationarity is the central HYPOTHESIS, stated as a binary proposition about the pullback Y(u) = Z(Theta^{-1}(u)) along the shifted Riemann-Siegel theta function. - Connection: WSS-of-Y on the u-axis predicts exactly the finite-window translation-stability fingerprint observed on Z, since the inverse warp has slowly-varying Jacobian. Section 6 rewritten: - Frame Z as a sample path of a non-stationary Gaussian process; Y as a sample path of a stationary Gaussian process (under the hypothesis). - Cramer representation theorem stated explicitly as a HILBERT SPACE ISOMORPHISM between H_Y (time-domain L^2 closure) and L^2(R, dF_Y) (spectral L^2). Sample path Y and random orthogonal measure Phi_Y are two coordinatizations of the same Hilbert space. - Inversion (6.6) is a deterministic Lebesgue-Stieltjes integral against the explicit sample path Z(t) Theta'(t) dt. Section 7 rewritten: - Gaussian + orthogonal => full law of Phi_Y determined by dF_Y alone (standard Gaussian theory, theorem). - Isserlis-Wick is then a CONSEQUENCE of Gaussianity, not an extra ansatz. Higher moments are entirely encoded in dF_Y. - Keating-Snaith is the corollary of the Gaussian-WSS hypothesis. Status table 7.4 updated: - Cramer isomorphism: theorem (conditional on WSS hypothesis). - Wick: theorem (conditional on Gaussian-WSS hypothesis). - Y Gaussian and WSS with autocovariance J_0(2s): hypothesis. - Keating-Snaith for all k: corollary of the hypothesis. - Spectral support [-1,1]: still conjectural (band-edge calculation is heuristic).
Section 2 rewrite: translation stability of the kernel across [10^2, 10^4] is the empirical anchor The c.2023 numerical computation observed two distinct, mutually-reinforcing facts: (a) Translation stability: computing the normalized autocovariance on translated windows across [100, 10,000] gave the same curve with the same constants. This is the empirical signature of wide-sense stationarity in the Cesaro sense: the kernel is a property of Z, not of the window. (b) Six-to-seven-decimal zero-set agreement with J_0 across multiple zeros: the empirical signature of an identification, not a shape match. Section 2 now splits these as 2.1 (translation stability), 2.2 (J_0 identification), 2.3 (why the combination is decisive). Working hypothesis upgraded to the boxed identity rho(s) = J_0(2s), with only reproduction-on-current-arb4j and rigorous-derivation as the remaining open items. Status table in 7.1 split correspondingly: translation stability and J_0 identification get separate rows.
Recalibrate J_0 kernel confidence: empirical match is 6-7 decimals across multiple zeros, not just shape - Section 2 rewritten: the c.2023 numerical computation matched the rescaled J_0 zero pattern to 6-7 decimal places simultaneously across multiple zeros, which is identification, not coincidence. Reproduction on current arb4j framework is in progress. - Section 3.7 rewritten: derive K(s) = (pi c_1 / 2) J_0(2s) explicitly via the chirp-slope-1/2 -> Jacobian-factor-2 mechanism; first zero at j_{0,1}/2 matches empirics. Add a status sub-table separating empirical match (strong) from rigorous determination of constants (open). - Section 7.1 status table updated to reflect the high-precision empirical evidence and the heuristic-vs-rigorous split for the J_0 derivation.
Strip cosmetic LaTeX spacing commands GitHub Wiki MathJax renders \; and \, as literal semicolons and commas instead of as whitespace, which corrupted every equation in the document. Removed 132 instances of \;, 64 of \,, and 12 of \! throughout. Math semantics unchanged.
Rewrite ThetaWarpedHardyZStationaryPullback with full rigor - Replace forbidden \operatorname with \text{Im}, \text{Re} - All math in 878 ... 878 blocks (no fence-block ambiguity) - Define time-averaged autocovariance precisely with T, t_0, limits - Detailed chirp calculation: Riemann-Siegel, non-stationary phase bound on second cosine, Mellin substitution v = vartheta'(t), band-edge xi = vartheta'/(vartheta'+C) - Prove the warp produces stationarity (3.16): phase becomes s independent of t after warping - Identify J_0 form via integral representation and band-edge square-root weight; connect to chirp slope 1/2 giving first node at j_{0,1}/2 - Distinguish three regularization candidates; show why only log-power damping balances Hardy-Littlewood log T variance - Closed form W_{a1,a2}(t) elementary (no special functions) - Lemma 5.1 statement and explicit proof showing pointwise L^2 fails, then correct Lemma 5.2 with renormalized autocovariance and explicit (log T)^{2beta} (log log T)^2 / T factor - Cramer inversion (6.3)-(6.4) as deterministic Lebesgue-Stieltjes - Status table separating classical theorems from numerical observations from open conjectures - Wick/Keating-Snaith connection at k^2 exponent - Computational cross-check program against arb4j sampling