Log-Periodic Signatures from Discrete Scale Invariance in the Stochastic Gravitational-Wave Background Phenomenological Template and the Walking-Technicolor Inverse Problem Jill F. Rankin Independent Researcher jill.rankin@g.austincc.edu May 2026(preprint) Abstract We develop a phenomenological framework for log-periodic signatures of discrete scale invariance (DSI) in the stochastic gravitational-wave background (SGWB) and use it to evaluate walking technicolor (WTC) as a candidate microphysical source. The phenomenological backbone is a factorization theorem: if the source unequal-time correlator carries a multiplicative DSI modulation ink, this modulation propagates to the observable energy-density spectrum at leading order inτ corr H ∗ , yielding Ω GW (f ) = Ω 0 GW (f )  1 + ε cos  2π ln(f/f ∗ ) lnb

• φ 0  , with modulation amplitudeε≪1 and discrete scaling ratiob >1. Matched-filter de- tectability of the oscillatory component scales asSNR osc ≃(ε/ √ 2)SNR baseline p N periods withN periods =ln(f max /f min )/ lnblog-periods in the detector band. This template, and the detectability landscape derived from it, applies to any DSI source. We then ask whether the DSI required by this template arises naturally in walking technicolor. Approximate continuous scale invariance is broken to DSI by a periodic modulation of the technidilaton potential, and the holographic dual of WTC gives an order-of-magnitude windowε f ∈[0.04,0.18],b ∈[1.7,2.8] for the modulation parameters at the level of the gauge propagator. The propagation from the propagator to the observable SGWB, however, is multiplied by a geometric suppression factorc geom set by the spectral support of the baseline propagator. Explicit 3D numerical evaluation of the relevant convolution with a free massive propagator, with the standard sound-shell power-law spectrum, and with the TT angular projection applied to either, gives|c geom |≲0.02 in all cases — two orders of magnitude below what would be required for LISA detectability. A closed-form saddle-point estimate shows that recoveringc geom ∼O(0.1–1) requires a propagator with relative spectral widthσ/q ∗ ≲0.2 (FWHM/q ∗ ≲0.5), about 3–5 times narrower than the natural FOPT spectrum delivers. The WTC log-periodic signal is therefore

not LISA-detectable in any of the microphysical realizations evaluated here; whether some additional narrowing mechanism (resonant pole, coherent technidilaton mode, wall-thickness localization) rescues detectability is identified as a sharp, quantitative microphysical question for future work. 2

Contents 1 Introduction4 2 Gravitational-Wave Tensor Power Spectrum5 3 Discrete Scale Invariance in the Source UETC6 3.1 DSI ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 3.2 Factorization theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 4 Observable Signatures11 4.1 DSI-modulated energy-density spectrum . . . . . . . . . . . . . . . . . . . 11 4.2 Matched-filter detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5 Ultraviolet Completion: Walking Technicolor16 5.1 Phase-transition parameter space . . . . . . . . . . . . . . . . . . . . . . . 16 5.2 Engineering discrete scale invariance . . . . . . . . . . . . . . . . . . . . . 16 5.3 Convolution for the UETC . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.4 WTC predictions and the observability gap . . . . . . . . . . . . . . . . . . 19 5.5 What would rescue WTC detectability . . . . . . . . . . . . . . . . . . . . 21 6 Discussion22 7 Conclusions23 3

1 Introduction The stochastic gravitational-wave background (SGWB) from first-order cosmological phase transitions is among the most promising observational targets for current and next-generation gravitational-wave detectors. The Laser Interferometer Space Antenna (LISA) [1] will be sensitive to phase transitions occurring at temperaturesT ∗ ∼10–10 4 GeV , covering a broad class of beyond-Standard-Model (BSM) scenarios. Pulsar-timing arrays (PTAs) have now reported evidence for a gravitational-wave background at nano-hertz frequencies [3–5], with spectra consistent with — though not yet uniquely identified as — a cosmological phase-transition origin. In this environment, spectral features that go beyond the smooth envelope predicted by conventional calculations take on special importance: they carry direct information about the microphysics of the transition and the nature of any BSM sector responsible for it. Standard calculations of the SGWB from a first-order phase transition predict a broad-band spectrum shaped by three source contributions — bubble collisions [8], sound waves [6,7], and magneto-hydrodynamic turbulence [9] — each with a characteristic broken power-law profile. A variety of beyond-standard effects can modify this picture: strong supercooling can sharpen the bubble-collision peak [8]; non-runaway walls alter the sound-wave contribution [6]; and non-equilibrium dynamics can generate additional log contributions [7]. However, none of these mechanisms generically produces a coherent log-periodic oscillation superimposed on the spectrum. Discrete scale invariance (DSI) is the symmetry that does. A system is said to possess DSI with ratiob >1 if it is invariant only under the discrete rescalingx→ b n xfor integer n, rather than under all continuous dilations [10]. DSI arises in hierarchical lattice models, fractal structures, iterated-function-system attractors, and — crucially for our purposes — near-conformal gauge theories with explicit periodic modulations. Its universal observable consequence is a log-periodic correction to any power-law observable, F (x) = x D 0  1 + A cos  2π lnx lnb

• φ  ,(1) arising from complex scaling dimensionsD n =D 0 ±2πin/ lnbin the spectrum of the dilatation operator [10]. DSI and its signatures have been studied in condensed-matter physics [10] and in financial time-series analysis [11], but its imprint on the SGWB has received comparatively little attention. Log-periodic features in the SGWB have been discussed in the context of non-standard inflationary scenarios and beyond-Einstein-gravity models [12]. In this paper we pursue a more direct route: we show that DSI in the anisotropic stress tensor of a first-order phase transition itself imprints a multiplicative log-periodic modulation on the observable SGWB. The mechanism operates at the level of the source unequal-time correlator (UETC) and is not specific to any particular BSM sector. The key technical result is a factorization theorem: under the physically well-motivated short- correlation-time approximation, valid for all realistic first-order phase transitions with β/H ∗ ≳10, the DSI modulation passes through the double time-integral of the tensor power spectrum unchanged, at the percent level. As a candidate ultraviolet (UV) completion we explore the realization of the required DSI within walking technicolor (WTC) [13], a strongly coupled hidden-sector gauge theory in the near-conformal regime. Two features motivate WTC as a host for DSI: (i) walking dynamics naturally provide approximate continuous scale invariance over a 4

wide range of energies, which can be broken to DSI by a small periodic modulation of the technidilaton effective potential — motivated (but not derived from first principles) by holographic models with periodic warp factors and by RG-group limit-cycle structure near the quasi-fixed point; and (ii) the WTC phase transition is already known to generate LISA-detectable gravitational waves [13], placing the DSI-modulated prediction squarely in the observable band without requiring any new tuning. We perform an explicit convolution calculation that traces the DSI modulation from the technidilaton potential through the UETC to the observable Ω GW (f ), with every approximation quantified. The resulting prediction has two layers. At the level of the gauge propagator, the WTC parameter space maps onto a specific windowε f ∈[0.04,0.18],b 0 ∈[1.7,2.8]. The observable modulation depth in the SGWB is reduced fromε f by a geometric factor c geom set by the source-tensor convolution; explicit numerical evaluation for the natural propagator ans ̈atze (free massive, sound-shell, with and without TT angular projection) gives|c geom |≲0.02, two orders of magnitude below the value needed for LISA detectability. The WTC log-periodic signal as described here is therefore not LISA-detectable; we identify the required propagator narrowness (σ/q ∗ ≲0.2) as a sharp, quantitative target for future microphysical calculation. The phenomenological template Eq. (16) remains a clean matched-filter discriminant for any DSI source in the LISA band. A companion paper [18] demonstrates the log-periodic spectral imprinting mechanism in a controlled one-dimensional electromagnetic cavity using finite-difference time-domain (FDTD) simulations, providing a numerical proof of concept independent of gravitational- wave physics. The theoretical framework of dynamic mode-accessibility engineering that unifies both papers is developed in Ref. [19]. The paper is structured as follows. Section 2 reviews the tensor power spectrum and sets up notation. Section 3 states the DSI ansatz and derives the factorization theorem. Section 4 works out the observable signature, quadratic relic corrections, and matched-filter detectability. Section 5 develops the WTC UV completion. Section 6 discusses robustness, distinguishability, and extensions. Section 7 summarizes the main results. Throughout we use natural units c =ℏ = k B = 1 and metric signature (−, +, +, +). 2 Gravitational-Wave Tensor Power Spectrum Tensor metric perturbationsh ij in a flat Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) background satisfy h ′′ ij (k,η) + 2Hh ′ ij (k,η) + k 2 h ij (k,η) = 16πGa 2 (η) Π TT ij (k,η),(2) where primes denote derivatives with respect to conformal timeη,H=a ′ /a,a(η) is the scale factor, and Π TT ij is the transverse-traceless projected anisotropic stress sourced by the phase transition. The two-point function of the source defines the unequal-time correlator,

Π TT ij (k,η) Π TT∗ ij (k ′ ,η ′ )  = (2π) 3 δ (3) (k−k ′ ) Π(k,η,η ′ ),(3) where statistical isotropy has been used to write Π as a function ofk=|k|. Solving Eq. (2) with the retarded Green’s function G k (η,η ′ ) gives the tensor power spectrum, P h (k,η) = (16πG) 2 Z dη 1 dη 2 G k (η,η 1 )G k (η,η 2 )a 2 (η 1 )a 2 (η 2 ) Π(k,η 1 ,η 2 ).(4) 5

The fractional GW energy density per logarithmic frequency interval, referred to the critical density today, is [7] Ω GW (k,η)≃ k 3 12a 2 H 2 P h (k,η),(5) valid for sub-horizon modesk ≫ H. In what follows we work in terms of the observed frequency f = k/(2πa 0 ). 3 Discrete Scale Invariance in the Source UETC 3.1 DSI ansatz We assume that the source UETC carries a discrete scale invariance with ratiob >1 and amplitude ε≪ 1: Π(k,η,η ′ ) = Π 0 (k,η,η ′ )  1 + ε cos  2π ln(k/k ∗ ) lnb

• φ 0  ,(6) where Π 0 is the smooth DSI-free UETC,k ∗ is a reference scale, andφ 0 is an overall phase. Equation (6) is the leading-order expression consistent with invariance underk → b n kfor integern; the log-periodic modulation is the real part of the complex power-law correction associated with complex scaling dimensions [10]. 3.2 Factorization theorem For a first-order phase transition the UETC naturally separates into macroscopic (slow) and microscopic (fast) parts, Π(k,η,η ′ ) = S(η,η ′ )F (k,η− η ′ ),(7) whereS(η,η ′ ) describes the macroscopic source evolution (slowly varying on the Hubble timeH −1 ∗ ) andF(k,∆η) encodes temporal correlations (decaying onτ corr ≪ H −1 ∗ ). This separation holds when the source is stationary on timescalesτ corr ≪∆η ≪ H −1 ∗ : a good approximation for the envelope and sound-shell contributions [6,8], for whichSis approximately constant while F decays rapidly. The phase-transition source decorrelates on the bubble radius/wall-speed timescale τ corr ∼ R ∗ ∼ v w /β, giving τ corr H ∗ ∼ v w β/H ∗ ≪ 1for β/H ∗ ≳ 10.(8) In this limitF(k,∆η) is sharply peaked at ∆η= 0. To bound the error, expandFabout ∆η= 0:F(k,∆η) =F(k,0)δ τ corr (∆η) +O(τ corr H ∗ ), whereδ τ corr is a nascent delta function of widthτ corr . Substituting into the double (η 1 ,η 2 ) integral of Eq. (4), theη 2 integral is dominated by the region|η 2 − η 1 |≲ τ corr . The Green’s functionG k (η,η 2 )a 2 (η 2 ) varies on timescalek −1 . For sub-horizon modes well above the peak (k ≫ β/v w ),k −1 ≪ τ corr is automatic; at the characteristic peak (k ∼ β/v w ),kτ corr ∼1 and the slow-variation approximation is marginal, withO(1) residual corrections that we do not compute explicitly. In either case we evaluate G k (η,η 2 ) at η 2 = η 1 , giving F (k,η− η ′ )≃ F (k)δ(η− η ′ ) +O(τ corr H ∗ ),(9) 6

whereF(k)≡ R F(k,∆η)d(∆η). The relativeO(τ corr H ∗ ) error is bounded in Table 1; we emphasize, however, that the factorization theorem proved below rests ultimately on separability of the UETC in (k,∆η), not on theδ-function limit per se — for any separable Π =S(η,η ′ )F(k)g(∆η) the modulationC(k) factors out of the time integral algebraically, regardless of how rapidly the Green’s function varies (Fig. 1 validates this with the full g(∆η), not itsδ-function limit). Theδ-function limit is the simplest path to the result; the deeper property is separability. We decompose the spectral kernel asF(k) =C(k)F 0 (k), whereF 0 (k) is the smooth baseline kernel andC(k) = 1 +ε cos[2π ln(k/k ∗ )/ lnb+φ 0 ] carries the DSI modulation. Substituting into Eq. (4) and performing the η 2 integral using the delta function: P h (k,η)≃ (16πG) 2 C(k) Z dη 1 G 2 k (η,η 1 )a 4 (η 1 )F 0 (k)S(η 1 ,η 1 ).(10) C (k) depends only onk, not onη 1 , and therefore factors out of theη 1 integral exactly. The remaining integral, together with the prefactors, defines the smooth tensor power spectrum: P 0 h (k,η)≡ (16πG) 2 F 0 (k) Z dη 1 G 2 k (η,η 1 )a 4 (η 1 )S(η 1 ,η 1 ).(11) We now show explicitly thatP 0 h acquires no log-periodic structure from the Green’s function. ExpandingG 2 k (η,η 1 ) =sin 2 [k(η − η 1 )]/k 2 = [1− cos(2k(η − η 1 ))]/(2k 2 ), the integral splits as Z dη 1 G 2 k a 4 S = I 0 (η) 2k 2 − e I(k,η) 2k 2 ,(12) whereI 0 (η) = R dη 1 a 4 (η 1 )S(η 1 ,η 1 ) is strictlyk-independent, and e I(k,η) = R dη 1 cos[2k(η− η 1 )]a 4 Sis an oscillatory Fourier transform of the slowly varying envelope. Integrating e I by parts once (boundary terms vanish since S = 0 outside the source epoch):   e I(k,η)   ≤ 1 2k Z   ∂ η 1 [a 4 S]   dη 1 ≲ H ∗ k I 0 (η),(13) sincea 4 Svaries on the Hubble timescale:|∂ η 1 [a 4 S]|≲ H ∗ a 4 S. This bound assumes the macroscopic envelope is smooth (differentiable onH −1 ∗ ); ifShas sharper features at the transition boundaries varying onβ −1 , the bound is weakened to| e I|/I 0 ≲ β/k ∼ v w , which isO(1) rather than small. The deeper argument that no log-periodic contamination is generated rests on the spectral-separation property ( e Ioscillates in lineark, not inlnk) discussed below, which holds regardless of the envelope smoothness. For sub-horizon GW modes k ≫ H ∗ , the ratio | e I|/I 0 ≲ H ∗ /k for smooth envelopes. To boundH ∗ /kin terms of the factorization error, note that the characteristic GW wavenumber isk ∼ βa ∗ /(v w a 0 ), givingH ∗ /k ∼ H ∗ v w a 0 /(βa ∗ ) =v w /(β/H ∗ )≡ τ corr H ∗ . Thus | e I|/I 0 =O(τ corr H ∗ ) and Z dη 1 G 2 k a 4 S = I 0 (η) 2k 2  1 +O(τ corr H ∗ )  .(14) Thek-oscillations of e Ioccur on the linear scale ∆k ∼1/η ∗ (producing the sound-wave broken power-law spectral features [6]); in log-kspace this corresponds to ∆lnk ∼ 1/(kη ∗ )∼ τ corr H ∗ ≪ lnb. These sound-wave features are spectrally separated from the DSI modulation by the large factorlnb/(τ corr H ∗ )∼(β/H ∗ /v w )lnb≫1: there is no overlap in 7

log-frequency space, andP 0 h acquires no log-periodic structure at periodlnb. We note that this scale-separation argument is heuristic and does not, in itself, exclude windowing or beat phenomena inlnkthat could couple linear-koscillations to log-periodic structure; the numerical validation below provides independent support that no such coupling appears at least for the separable case tested. Hence, combining Eq. (14) with the exact factorization of C(k): P h (k,η) = C(k)P 0 h (k,η)  1 +O(τ corr H ∗ )  ,(15) whereP 0 h is the tensor power spectrum evaluated with the smooth UETC Π 0 . This is the factorization theorem: the DSI modulation transfers multiplicatively from the source to the tensor power spectrum, with the Green’s-function contribution bounded explicitly by Eq. (13). Higher-order corrections are quantified in Table 1. Scope of validity: the factorizationP h =C(k)P 0 h is exact for separable UETCs of the form Π = S(η,η ′ )F(k)g(∆η), in which case it follows algebraically fromC(k) being independent of (η 1 ,η 2 ) (cf. Fig. 1). TheO(τ corr H ∗ ) bound in Eq. (15) refers to the controlled approximations of the present section — theδ-function limit Eq. (9), the IBP bound Eq. (13) for smooth envelopes, and the bulk convolution slow variation of Sec. 5.3. Corrections from a generic non-separable UETC are not bounded by this universal estimate and would require model-specific analysis. Table 1: Relative error bound on the factorizationP h =C(k)P 0 h , forε= 0.1,v w = 1. The factorization-step (phenomenological) corrections are: (i) Delta-function approximation, Eq. (9): relative error≤ τ corr H ∗ ≡ v w /(β/H ∗ ) for smooth envelopes. (ii) Bulk convolution slow variation, Sec. 5.3: relative error≤(2π/ lnb)ε f τ corr H ∗ forb≥1.5, using the uniformly bounded absolute derivative|dδ/d lnq|≤ ε f ×2π/ lnb. (iii) Green’s function oscillations, Eq. (13): relative error≤ H ∗ /k ≡ τ corr H ∗ (Riemann–Lebesgue, smooth envelopes). Errors (i)–(iii) are quantitatively controlled and shown below. In addition, the WTC convolution carries a separateO(m V /q ∗ ) mass-gap correction (Sec. 5.3) that is model-dependent and is the dominant error for the WTC parameter band (∼10% form V /q ∗ ∼0.1); it is not included in the phenomenological budget below. β/H ∗ τ corr H ∗ Combined bound Dominant term 100.10≲ 15%δ-fn 1000.01≲ 1.5%convolution 10000.001≲ 0.15%Green’s fn Numerical validation. As an independent check on the factorization argument we evaluate Eq. (4) numerically for a separable UETC Π(k,η 1 ,η 2 ) =F 0 (k)C(k)S(η 1 ,η 2 )g(η 1 − η 2 ) with a tophat macroscopic sourceSand a Gaussian temporal correlationg(∆η) of widthτ corr . This is a toy validation of the factorization structure rather than a full WTC simulation: it tests whether the Green’s-function convolution and time integration in Eq. (4) preserve ak-dependent modulation imposed at the UETC level, and is not intended to validate the microphysics of any specific BSM source. Crucially the numerical computation uses the full Gaussian, not theδ-function limit invoked in Eq. (9). Figure 1 shows the resulting residualR(k) = (P h − P 0 h )/P 0 h alongside the analytic prediction ε cos(2π ln(k/k ∗ )/ lnb+φ 0 ). Two features confirm the theorem: (i) atτ corr H ∗ = 0.05 the numerical residual matches the analytic template to within∼10 −16 , the floor of double- precision arithmetic; and (ii) repeating the calculation atτ corr H ∗ ∈{0.01,0.05,0.20}yields 8

residuals that are pointwise identical to within numerical precision. Both observations follow from the fact that, whenC(k) depends only onkand is independent ofη 1 ,η 2 , it factors out of the double time integral algebraically — not merely up toO(τ corr H ∗ ). The error bounds in Table 1 arise instead from non-separable corrections (the convolution and Green’s-function terms): these are not probed by the present test and would require a more elaborate numerical setup. 9

10 0 10 1 wavenumber k (units with k * = 5) 0.15 0.10 0.05 0.00 0.05 0.10 0.15 R ( k ) ( P h P 0 h )/ P 0 h (a) = 0.10, b = 2.0, corr H * = 0.05, max dev. = 1.80e16 Factorization theorem: numerical residual vs.\ analytic template Analytic: cos(2ln(k/k * )/ln b + 0 ) Numerical: (P h P 0 h )/P 0 h 10 0 10 1 wavenumber k 0.15 0.10 0.05 0.00 0.05 0.10 0.15 R ( k ) (b) Independence of corr H * for separable UETCs: factorization is algebraic, not perturbative Analytic template Numerical ( corr H * = 0.01) Numerical ( corr H * = 0.05) Numerical ( corr H * = 0.2) Figure 1: Numerical validation of the factorization theorem (Sec. 3.2). The DSI-modulated tensor power spectrumP h (k) is computed by direct numerical integration of the double time integral (4) for a separable UETC Π =F 0 (k)C(k)S(η 1 ,η 2 )g(η 1 −η 2 ) with a Gaussian temporal correlationg(∆η) of widthτ corr (the full Gaussian; not theδ-function limit of Eq. (9)). (a) ResidualR(k) = (P h − P 0 h )/P 0 h (orange circles) plotted against the analytic predictionε cos(2π ln(k/k ∗ )/ lnb+φ 0 ) (gray line) atτ corr H ∗ = 0.05; maximum deviation ∼10 −16 (numerical floor). (b) The same residual evaluated atτ corr H ∗ ∈{0.01,0.05,0.20} collapses onto a single curve, confirming that the factorization is algebraic (independent ofτ corr H ∗ ) for separable UETCs. TheO(τ corr H ∗ ) corrections in Table 1 arise from non- separable structure (convolution and Green’s-function terms) that is beyond the scope of this clean test. 10

4 Observable Signatures 4.1 DSI-modulated energy-density spectrum Combining Eq. (6) with the factorization (15) and using Eq. (5), the observable GW energy-density spectrum is Ω GW (f ) = Ω 0 GW (f )  1 + ε cos  2π ln(f/f ∗ ) lnb

• φ 0  .(16) The fractional residual R(f )≡ Ω GW (f )− Ω 0 GW (f ) Ω 0 GW (f ) = ε cos  2π ln(f/f ∗ ) lnb
• φ 0  (17) is a sinusoid inlnfwith period ∆lnf=lnb, amplitudeε, and phaseφ 0 at leading order inτ corr H ∗ . By construction, its Pearson correlation coefficient with a fixed-period cosine template equalsr= 1.00 for any bandwidth spanning complete log-periods, modulo the O(τ corr H ∗ ) factorization corrections quantified in Table 1. This analyticr= 1 should be distinguished from the empiricalr= 0.81±0.04 reported in the companion FDTD paper [18]: in that setting, finite time-series length, Hann-window spectral leakage, and imperfect power-law envelope subtraction all reduce the observed correlation below unity. The FDTD value quantifies detection efficiency in a realistic finite-bandwidth experiment; r= 1 is the leading-order property of the underlying physics, recovered in the limit of infinite bandwidth and exact envelope knowledge. For the smooth baseline Ω 0 GW we adopt the standard sound-wave contribution [6, 7], Ω sw (f )h 2 = 2.65× 10 −6  H ∗ β  2  κ sw α 1 + α  2  100 g ∗  1/3 v w S sw (f ),(18) S sw (f ) =  f f sw  3  7 4 + 3(f/f sw ) 2  7/2 ,(19) with peak frequency f sw = 1.9× 10 −5 Hz 1 v w β H ∗ T ∗ 100 GeV  g ∗ 100  1/6 .(20) Hereαis the transition strength,κ sw is the fraction of the released latent heat converted to fluid bulk motion,g ∗ is the number of relativistic degrees of freedom atT ∗ , andv w is the wall velocity. We set the DSI reference scalef ∗ ∼ f sw . Figure 2 shows the spectrum, residual, and log-period spacing for representative parameter values. 4.2 Matched-filter detectability The oscillatory component of the signal is δΩ GW (f ) = ε Ω 0 GW (f ) cos  2π ln(f/f ∗ ) lnb
• φ 0  .(21) The squared matched-filter signal-to-noise ratio for a search with fixed template parameters (b,φ 0 ) is SNR 2 osc = Z [δΩ GW (f )] 2 σ 2 (f ) d lnf = ε 2 Z [Ω 0 GW (f )] 2 σ 2 (f ) cos 2  2π ln(f/f ∗ ) lnb
• φ 0  d lnf.(22) 11

10 3 10 2 10 1 10 0 10 1 Frequency f [Hz] 10 13 10 12 10 11 10 10 10 9 10 8 10 7 h 2 GW ( f ) (a) = 0.10,b = 2.0, 0 = 0 Smooth baseline 0 GW DSI-modulated GW 10 3 10 2 10 1 10 0 10 1 Frequency f [Hz] 0 + R ( f ) / 0 (b) f * Analytic: r = 1.00 (see text; cf. FDTD: r = 0.81 ± 0.04) ( GW 0 )/ 0 Fixed-period fit: cos(2ln(f/f * )/ln b) 10 3 10 2 10 1 10 0 10 1 Frequency f [Hz] ln f = ln b f * (c) log-period spacing Figure 2: Log-periodic modulation of the SGWB. (a) Power spectrumh 2 Ω GW (f) (orange, solid) and smooth baselineh 2 Ω 0 GW (f) (blue, dashed) versus frequency, forε= 0.1,b= 2, φ 0 = 0. (b) Fractional residualR(f)≡[Ω GW (f)−Ω 0 GW (f)]/Ω 0 GW (f), showing the clean sinusoidal oscillation inlnfpredicted by Eq. (16). The orange curve is the fixed-period cosine fit. (c) Log-period spacing: vertical ticks mark frequencies where the modulation peaks (cos = +1), equally spaced by ∆ lnf = lnb. The reference scale f ∗ is indicated. 12

Over N periods complete log-periods ⟨cos 2 ⟩ = 1/2, giving SNR 2 osc

ε 2 2 N periods SNR 2 bin ,(23) where SNR bin is the baseline SNR per log-period of width lnb: SNR 2 bin ≡ Z lnb [Ω 0 GW ] 2 σ 2 d lnf.(24) Hence SNR osc

ε √ 2 p N periods SNR bin ,(25) with N periods

ln(f max /f min ) lnb .(26) Throughout this paperSNR baseline ≡ SNR bin denotes the per-log-period baseline SNR; the total-band baseline SNR isSNR total

p N periods SNR bin . For LISA with effective band [f min ,f max ] = [10 −4 ,1]Hz(ln(f max /f min )≈9.21) the per-log-period baseline SNR at the WTC signal level isSNR bin ≈20–25; the factor p N periods / √ 2ranges from 2.6 atb= 2 to 1.7 atb= 5, providing meaningful amplification. We writeSNR osc ≈(ε/ √ 2)SNR bin p N periods in what follows; figures use SNR bin = 20 to set contours. The detectability plane (bvs.ε) is shown in Figs. 3 and 4, with SNR contours at {1, 5, 10, 20} and the WTC prediction band overlaid. 13

23456 Discrete scaling factor b 0.01 0.05 0.10 0.50 Modulation amplitude SNR base = 20; LISA band [10 4 , 1] Hz Forecast SNR contours for DSI oscillations in the SGWB WTC [0.04, 0.18] b[1.7, 2.8] SNR=1 SNR=5 SNR=10 SNR=20 Figure 3: Forecast matched-filter SNR contours for the DSI oscillatory component in the (b,ε) plane, assumingSNR baseline = 20 and a LISA frequency band [10 −4 , 1]Hz. Contours are shown atSNR osc = 1,5,10,20. The orange shaded region is the WTC propagator-level forecastε f ∈[0.04,0.18],b 0 ∈[1.7,2.8] from the holographic mapping of Sec. 5.2. The observable modulation depth isε∼ c geom ε f , wherec geom is the geometric suppression factor of Sec. 5.3; for the propagator structures evaluated in this paper|c geom |≲0.02 (Fig. 5), which would shift the band by roughly two decades to the lower left and place it well below the SNR osc = 5 contour. 14

23456 Discrete scaling factor b 0.01 0.05 0.10 0.50 Modulation amplitude SNR base = 20; LISA band [10 4 , 1] Hz Forecast SNR contours with LISA 5 sensitivity and WTC prediction WTC [0.04, 0.18] b[1.7, 2.8] LISA 5 threshold (SNR osc = 5) LISA accessible (SNR osc 5) SNR=1 SNR=5 SNR=10 SNR=20 Figure 4: Same as Fig. 3, with the approximate LISA 5σdetection threshold (blue line, SNR osc = 5 forSNR baseline = 20) and LISA-accessible region (purple shading) overlaid. As in Fig. 3, the orange band is the WTC propagator-level forecast; the observable band ε ∼ c geom ε f inherits a further suppression from the convolution geometry and, for the propagator ans ̈atze evaluated here, falls below the LISA-accessible region. Identifying a propagator structure that supplies the missing narrowness (Sec. 5.5) is the principal microphysical question this paper leaves open. 15

5 Ultraviolet Completion: Walking Technicolor Before developing the WTC embedding it is worth stating explicitly which claims in this paper rest on what kind of argument. The phenomenological backbone (Sec. 3–4) follows from the UETC ansatz and the controlled-approximation bounds of Sec. 3.2; the UV-completion layer developed in this section is a candidate realization motivated by holography and near-conformal dynamics but not derived from a complete microscopic model. Table 2 makes this hierarchy explicit. Table 2: Scope and epistemic status of the principal claims of this paper. Rows 1–2 form the phenomenological backbone and are derived under explicit, quantified approximations. Rows 3–4 form the UV-completion layer and should be read as a candidate realization rather than a first-principles derivation. ClaimStatusSection DSI in UETC ⇒ log-periodic SGWB template derived3.2, 4.1 Factorization in short-correlation regimecontrolled approximation3.2, Table 1 WTC as DSI hostcandidate UV completion 5.2–5.4 Holographic origin of periodic warp factorconjectural motivation5.2 We emphasize that the cited literature [10,13,16,17] motivates the individual ingredi- ents — DSI in near-conformal systems, walking dynamics, holographic warp factors, radion potentials — but the complete chain from a microscopic WTC Lagrangian to the periodic technidilaton potential (29) is not, to our knowledge, established in the literature. The construction below should therefore be read as a plausibility argument for a WTC-style UV completion, not as a derivation from first principles. 5.1 Phase-transition parameter space We adopt the benchmark large-N f QCD realization of walking technicolor [13]. The hidden sector is anSU(N c ) gauge theory withN f fundamental techniquarks in the near-conformal windowN f /N c ≳4–8. Near this window the gauge coupling walks — evolves slowly over many decades of energy scale — providing approximate scale invariance; the theory is attracted toward a quasi-fixed point (the Banks–Zaks fixed point) before condensing at Λ TC . Benchmark values areN c = 8,N f = 8, technidilaton decay constantF φ ≈1TeV, with an ultra-supercooled first-order phase transition (FOPT) characterized by [13] α≈ 0.73–0.83, β/H ∗ ≈ 100–1000, v w ≈ 1.(27) These give a sound-wave-dominated SGWB with h 2 Ω 0 GW (f peak )∼ 10 −9 –10 −8 at f peak ∼ 0.1–10 Hz,(28) comfortably within the LISA sensitivity band [1, 2], and satisfy τ corr H ∗ ≲ 0.01≪ 1. 5.2 Engineering discrete scale invariance Walking dynamics provide approximate continuous scale invariance: the technidilatonφis the pseudo-Nambu–Goldstone boson of the approximate scale symmetry, and its effective 16

potential is of Coleman–Weinberg form [14]. DSI arises when this symmetry is broken from continuous to discrete. We realize this by adding a small explicit periodic modulation, V (φ) = V CW (φ)  1 + ε f cos  2π ln(φ/φ 0 ) lnb 0  , ε f ≪ 1, b 0

1,(29) whereV CW is the Coleman–Weinberg potential [14]. Such modulations are motivated by two independent sources. First, in AdS/CFT dual descriptions of near-conformal dynamics, periodic warp factors in the extra dimension generate exactly this type of potential modulation in the 4D effective theory. Second, near the quasi-fixed point the RGβ-function has no zero; instead the integrated RG flow over one cycle inφ-space is zero, corresponding to a limit cycle rather than a fixed point — the RG-flow realization of DSI [10]. Holographic origin ofε f andb 0 . We do not derive Eq. (30) from a complete 5D action, nor address dynamical stabilization of the periodic perturbation against deformation toward generic non-periodic warp factors. The construction in this paragraph is a constructive existence proof : it demonstrates how a small periodic warp-factor perturbation would, if present, produce the 4D modulation Eq. (29) with the inherited parameter ranges we use in Sec. 5.4. Identifying the dynamical origin — and the radiative stability beyond the discrete-shift-symmetry argument below — is a separate question deferred to future work. Equation (29) arises in the holographic dual of WTC without requiring fine-tuning of five-dimensional parameters. Modelling the technidilaton as the radion field in a Randall– Sundrum-type AdS 5 geometry [16], a small periodic modulation of the standard warp factor A(y) = ky of the form A(y) → ky + δA 0 sin(n p ky), δA 0 ≪ 1,(30) generates, at linear order inδA 0 , the multiplicative log-periodic correction to the 4D technidilaton potential [17] V 4 (φ)≈ V CW (φ)  1 + 4δA 0 cos  2π ln(φ/φ 0 ) lnb 0  +O(δA 2 0 )  ,(31) withε f = 4δA 0 andlnb 0 =kL/n p , whereLis the proper length of the extra dimension andn p is the number of warp-factor oscillation periods. For the ETC hierarchykL ≈ ln (Λ ETC /Λ TC )≈2–3 andn p = 3–4 (bothO(1) integers in AdS units), one obtains b 0 =e kL/n p ∈[1.7,2.8]. The required amplitudeε f ∈[0.04,0.18] corresponds toδA 0 ∈ [0.01,0.045], a 1–5% warp-factor perturbation that is technically natural (protected by the approximate discreteφ→ φ+L/n p shift symmetry of the periodic modulation) and requires no independent fine-tuning. Both DSI parameters therefore emerge fromO(1) choices of the 5D geometry. The modulation in Eq. (29) induces a log-periodic correction to the gauge-field propa- gator at momentumq. We derive this at leading order inε f . In the near-conformal WTC regime, the technidilaton VEV⟨φ(q)⟩at renormalisation scaleqis related to its UV value by ⟨φ(q)⟩=⟨φ UV ⟩(q/q 0 ) −∆ φ , where ∆ φ is the technidilaton scaling dimension (∆ φ ≈ 1 near the quasi-fixed point). The gauge-boson mass is generated viam 2 V (φ) =y 2 ⟨φ⟩ 2 ; a modulation δV ∝ ε f cos(2π lnφ/ lnb 0 ) shifts the mass asδm 2 V /m 2 V =ε f cos(2π ln(q/q ∗ )/ lnb 0 ) +O(ε 2 f ) by the chain rule. Propagating to the full propagator at leading order in ε f : D(q; ∆η) = D 0 (q; ∆η) [1 + δ(q)], δ(q) = ε f cos  2π ln(q/q ∗ ) lnb 0

• φ 0  ,(32) 17

withq ∗ ∼ q 0 . The modulation inherits the same log-periodb 0 as the potential, up to the conformal-dimension factor ∆ φ which isO(1) near the fixed point. Higher-order corrections enter at O(ε 2 f ). Note that |δ(q)|≤ ε f ≪ 1 so D is positive definite for all q. 5.3 Convolution for the UETC The transverse-traceless anisotropic stress is bilinear in the gauge fields, so the UETC is the convolution Π(k,η,η ′ )∝ Z d 3 p (2π) 3 P TT D(p; ∆η)D(|k−p|; ∆η),(33) whereP TT projects onto the transverse-traceless sector. Expanding to linear order inε f and retaining only the cross-term (the self-term is O(ε 2 f )), Π(k)⊃ Z d 3 p (2π) 3 P TT D 0 (p)D 0 (|k−p|)  δ(p) + δ(|k−p|)  .(34) We bound the two terms separately. Termδ(p). The baseline propagatorD 0 (p) is sharply peaked atp∼ q ∗ ∼ β/v w with relative half-width ∆p/p∼ τ corr H ∗ ≪1. The variation ofδacross this peak is bounded using the absolute derivative:     dδ d lnq     = ε f 2π lnb 0   sin

2π lnq/ lnb 0

• φ 0
    ≤ ε f 2π lnb 0 ,(35) uniformly bounded for allq, including near the zeros ofδ(where the logarithmic derivative d lnδ/d lnqwould diverge, but the divergence is integrable since the weightD 0 is smooth and |δ|→ 0). Integrating over the support | ln(p/k)|≲ τ corr H ∗ : |δ(p)− δ(k)|≲ 2πε f lnb 0 τ corr H ∗ ≲ 13ε f τ corr H ∗ (b 0 ≥ 1.5),(36) giving δ(p) = δ(k)[1 +O(ε f τ corr H ∗ )] over the support of D 0 (p). Termδ(|k−p|). Nearp ≈k,|k−p| →0 andδ(|k−p|) oscillates rapidly. For a massive gauge-boson propagatorD 0 (q) = 1/(q 2 +m 2 V ), this region hasD 0 (|k−p|)∼1/m 2 V , which is in fact larger than the bulk valueD 0 (q ∗ )∼1/q 2 ∗ by the factorq 2 ∗ /m 2 V ≫1 when q ∗ ≫ m V . The integrated contribution of the small-|k−p|region is nevertheless suppressed by the three-dimensional phase-space measurep 2 dp: contributions from|k−p|≲ m V scale asm 3 V ×D 0 (0)×D 0 (q ∗ )∼ m V /q 2 ∗ , compared to the bulk contribution at|k−p|∼ q ∗ scaling asq 3 ∗ × D 0 (q ∗ ) 2 ∼1/q ∗ . The ratio is (boundary)/(bulk)∼ m V /q ∗ ≡ ρ 1/2 , with ρ≡ m 2 V /q 2 ∗ ≪ 1. The combined bound is δ(p)≈ δ(|k−p|)≈ δ(k)  1 +O(ε f τ corr H ∗ ) +O(m V /q ∗ )  ,(37) withm V /q ∗ ≪1 in the WTC regime. For typical WTC benchmark parametersm V /q ∗ ∼ 0.1, the boundary contribution sets a residual relative error of order 10% on the convolution factorization, exceeding theO(τ corr H ∗ ) bulk correction. We retainρ 1/2 ≡ m V /q ∗ in error budgets below. The cross-term contributionsδ(p) andδ(|k−p|) each factor asδ(k)×Π 0 (k,η,η ′ ) over the dominant support, giving a total cross-term contribution of 2δ(k)Π 0 from the two 18

linear terms. This means the UETC modulation depthεinherited from the propagator modulationδ=ε f cos(···) satisfiesε= 2ε f × c geom at leading order, wherec geom is a geometric coefficient arising from the angular average ofP TT over the convolution support and from the spectral support of the baseline propagatorD 0 (q). The explicit numerical evaluation in the ”Quantitative geometric requirement” paragraph below shows thatc geom is not genericallyO(1) for the natural FOPT propagator structures; we keep the symbolic factor here and compute it quantitatively in Fig. 5. Hence Π(k,η,η ′ ) = Π 0 (k,η,η ′ )  1 + ε cos  2π ln(k/k ∗ ) lnb

• φ 0   1 +O(ετ corr H ∗ ) +O(m V /q ∗ )  , (38) withε∼ ε f andb=b 0 at leading order (up toO(1) model-dependent factors). For WTC benchmark parametersβ/H ∗ ≳100 andm V /q ∗ ∼0.1, the combined relative correction is dominated by the mass-gap term at ∼ 10%, not the ∼ 1% bulk error. Quantitative geometric requirement. The factorization of the DSI modulation under the convolution Eq. (34) is conditional on the spectral support of the baseline propagator D 0 (q). To quantify the requirement we evaluate the cross-term convolution numerically along two axes. First, we use a Gaussian propagatorD 0 (q) =exp[−(q− q ∗ ) 2 /(2σ 2 )] and scan the widthσ/q ∗ . A saddle-point expansion of the angular integral gives the closed-form c geom (σ/q ∗ ) ≈ exp " − 2π 2 ln 2 b 0  σ q ∗  2

,(39) which the numerics reproduce to a few percent acrossσ/q ∗ ∈[0.01,0.25] (Fig. 5a). Eq. (39) makes the requirement explicit:c geom ≥0.5 requiresσ/q ∗ ≲0.13 (FWHM/q ∗ ≲0.31); c geom ≥ 0.1 requires σ/q ∗ ≲ 0.24 (FWHM/q ∗ ≲ 0.56). Second, we evaluate the convolution with two physically motivated propagator struc- tures that one might hope concentrate the support nearq ∗ . (a) For the standard sound- shell-model spectral shapeD 0 (q)∝(q/q ∗ ) a [7/(4 + 3(q/q ∗ ) 2 )] 7/2 [6? ] witha= 3 (matching S sw (f) of Eq. (19)), the relative FWHM is≈1.4, equivalent toσ/q ∗ ≈0.6. Direct evalua- tion givesc geom ≃5×10 −3 — consistent with Eq. (39), and smaller than the free-massive toy because of the additional UV phase-space weighting. (b) Adding the TT angular projector appropriate to a longitudinal (v i v j ) stress source —F TT ∝ p 2 (1−cos 2 θ) 2 /|k−p| 2 or its variants — shuffles the sign ofc geom but not its magnitude:|c geom |remains in the range [2×10 −4 ,1.3×10 −2 ] across all combinations of IR slope and angular form tested. The TT factor emphasizes configurations perpendicular tok, at which|k−p| ∼ √ 2q ∗ falls at a half-integer multiple oflnb 0 inlnq— a maximum of the modulation’s antiphase — so the angular weight acts as an additional cancellation pathway rather than a focusing one. The cumulative diagnostic is summarized in Fig. 5. None of the propagator structures naturally produced by a strong first-order phase transition deliver the spectral narrowness required to keepc geom ≳0.1. In the optimistic limit of the holographic WTC band (ε f ∼0.18), this means an observable modulationε ∼ c geom ε f ≲2×10 −3 , roughly two orders of magnitude below LISA’s 5σ matched-filter threshold (cf. Fig. 4). 5.4 WTC predictions and the observability gap The WTC parameter space [13], spanned byF φ ≈1TeV, Λ ETC ∼ 5–10TeV, and soft massesm p ∼1–100GeV, maps via the holographic construction of Sec. 5.2 onto a window 19

10 1 10 0 /q * (Gaussian propagator width) 10 3 10 2 10 1 10 0 c geom c geom = 0.5 c geom = 0.1 (a) Geometric suppression of the DSI modulation in the convolution Sound-shell equivalent ( /q * 0.59) Saddle-point: exp[ 2 2 ( /q * ) 2 /ln 2 b 0 ] Numerical 10 0 k/q * 0.2 0.1 0.0 0.1 0.2 R ( k ) = ( 0 )/ 0 (b) Modulation is recovered only when the propagator is narrow enough Analytic: 2 f cos(2ln(k/q * )/ln b 0 ) Numerical, /q * = 0.05 Numerical, /q * = 0.20 Numerical, /q * = 0.60 Figure 5: Geometric suppression of the DSI modulation in the WTC convolution. (a) Em- piricalc geom ≡ R peak /(2ε f ) versus Gaussian propagator widthσ/q ∗ , with the saddle-point analytic formula Eq. (39) (gray line; reproduces the data to a few percent overσ/q ∗ ≲0.25). The vertical dashed line marks the standard sound-shell spectral width (σ/q ∗ ≈0.60, FWHM/q ∗ ≈1.4), at whichc geom is several orders of magnitude below the values required for LISA detectability (c geom ∼0.3–1). (b) ResidualR(k) = (Π−Π 0 )/Π 0 vs.k/q ∗ for three propagator widths overlaid on the analytic prediction 2ε f cos(2π ln(k/q ∗ )/ lnb 0 ). For a narrow Gaussian (σ/q ∗ = 0.05) the modulation is recovered cleanly; at the sound-shell width (σ/q ∗ = 0.60) it is essentially washed out. 20

for the propagator-level modulation parameters ε f ∈ [0.04, 0.18], b 0 ∈ [1.7, 2.8].(40) This range follows fromε f = 4δA 0 withδA 0 ∈[0.01,0.045] andb 0 =exp(kL/n p ) with kL≈ ln(Λ ETC /Λ TC )≈1.6–2.3 andn p = 3–4. The observable amplitude in the SGWB,ε, is related toε f by the geometric factorc geom of Sec. 5.3:ε∼ c geom ε f . As shown in Fig. 5, for the free-massive, sound-shell, and TT-projected sound-shell propagator ans ̈atze tested in this paper, |c geom |≲ 0.02 throughout. The implied observable band is therefore ε≲ c geom ε max f ∼ (2× 10 −2 )(0.18)∼ 4× 10 −3 ,(41) roughly two orders of magnitude below LISA’s 5σmatched-filter threshold atSNR bin = 20. In other words, while the propagator-level DSI window Eq. (40) is squarely in the high-SNR region of Figs. 3–4 if the modulation propagates withc geom ∼1, the explicit convolution gives c geom too small for this to occur with any of the standard propagator structures. Approximation hierarchy and model-dependent factors. The chain from a WTC- type Lagrangian to the observable Ω GW (f) runs through several steps that we have sketched but not derived from first principles. In particular: (i) The Goldberger–Wise-type relation ε f = 4δA 0 between the holographic warp-factor perturbation and the 4D technidilaton potential modulation [Eq. (31)] depends on the specific radion–dilaton identification in the 5D dual. (ii) The chain-rule transfer of the periodic modulation fromV(φ) to the gauge propagatorD(q) [Eq. (32)] involves the technidilaton scaling dimension ∆ φ , which isO(1) near the quasi-fixed point but not exactly unity, shifting the inherited log-periodb relative tob 0 by anO(1) factor. (iii) The mass-gap and angular factors set the magnitude ofc geom ; Sec. 5.3 computes this for the natural propagator ans ̈atze and finds|c geom |≲0.02. Taken together, the propagator-level band Eq. (40) carriesO(1) uncertainties from (i)–(ii) and is therefore an order-of-magnitude forecast; the observable band ε∼ c geom ε f inherits an additional two-decade suppression from (iii), unless physics beyond the scope of this paper restores a propagator narrowness that the natural ans ̈atze do not deliver. 5.5 What would rescue WTC detectability The closed-form Eq. (39) converts the question “is the WTC signal LISA-detectable?” into a sharp microphysical inverse problem: what propagator structure at FOPT scales givesσ/q ∗ ≲0.2 (FWHM/q ∗ ≲0.5)? Each of the following candidates would, if realized in WTC, narrow the support enough to bring c geom into the O(0.1–1) regime: (a) Resonant thermal pole. If the WTC finite-temperature gauge propagator develops a high-Qresonance at the FOPT scale, the effectiveD 0 (q) would be a Breit–Wigner of width Γ/q ∗ ∼1/Q.Q∼5–10 gives Γ/q ∗ ∼0.1–0.2, sufficient to satisfy the narrow-support criterion. Whether such a resonance exists in the WTC finite-Tpropagator is a calculation we do not perform. (b) Coherent technidilaton oscillation. Post-FOPT coherent oscillations of the tech- nidilaton field atm φ would source a narrow-band signal at the dilaton mass. If this signal dominates over the bubble-collision/sound-wave contribution in the LISA band, the effectiveD 0 (q) acquires a sharp peak nearq ∼ m φ , providing the required spectral narrowness. Quantifying the relative amplitude of coherent vs. thermal contributions is again outside the present scope. 21

(c) Bubble-wall thickness localization. The bubble wall thicknessδ w ∼1/m φ sets a UV scale for the source momentum distribution. For sufficiently largem φ /T ∗ , the relevant convolution kernel may concentrate near a single dynamical scale rather than smearing across the full sound-shell spectrum. A definitive WTC prediction therefore awaits a microphysical calculation that quantifies the FOPT propagator structure including (at minimum) finite-temperature self-energy corrections and the specific spin/orbital structure of the source. In the meantime, the phenomenological backbone of Sec. 3–4 provides the template against which any future microphysical prediction can be tested. 6 Discussion Robustness of the factorization. The key approximation is the short-correlation-time limitτ corr H ∗ ≪1. Its validity requiresβ/H ∗ ≫1, i.e. a transition that completes rapidly compared to the Hubble time. This is satisfied for the WTC benchmark (β/H ∗ ∼100– 1000) and is a generic property of strong first-order transitions. Slow transitions with β/H ∗ ≲10 would require higher-order corrections, which can be computed systematically as an expansion inτ corr H ∗ . The separate factorization condition|dδ/d lnq|·(∆q/q)≪1 — using the absolute derivative|dδ/d lnq| ≤ ε f ×2π/ lnb 0 , which is uniformly bounded for allq(the logarithmic derivatived lnδ/d lnqwould diverge at the zeros ofδ, but the absolute derivative does not; see Sec. 5.3) — is equally well controlled and introduces no additional tuning. FDTD analogy and cosmological causal structure. The companion FDTD pa- per [18] demonstrates log-periodic spectral imprinting in a controlled electromagnetic cavity; the correspondence to the cosmological FOPT warrants explicit comment. In the cavity, rigid static boundaries enforce global mode selection via discrete standing-wave conditions (Dirichlet or absorbing boundary conditions at the walls): the mode spectrum is shaped by the entire geometry simultaneously. In a FOPT no global boundary condition exists: bubbles nucleate independently within their past light cones, and causal horizons preclude global mode coherence. The structural role of the geometric boundary is instead played by the characteristic bubble spacingR ∗ ∼ v w /β, which acts as a local, dynamic filter. Modes withk ≫ R −1 ∗ are exponentially suppressed by the decay of the temporal correlation functionF(k,∆η) at large separations; modes withk ≪ R −1 ∗ see a nearly homo- geneous source and are coherently accessible. In the language of the companion framework paper [19],R ∗ plays the role of the boundary-conditioned mode density cutoff, with the plasma mean free path providing the dynamic spectral participation filter. Crucially, the factorization theorem of Sec. 3.2 relies only on the local conditionτ corr H ∗ ≪1 — set by β/H ∗ ≫1, independent of any global causal horizon structure. The FDTD result therefore validates the mathematical mechanism of DSI imprinting (that a log-periodically structured boundary parameter transfers its signature multiplicatively to the power spectrum), while the factorization theorem independently establishes the validity of that transfer in the cosmological context via purely local causal arguments. Distinguishability from other spectral features. The log-periodic modulation (16) produces a coherent, phase-stable sinusoid inlnf, persisting overN periods ∼6–13 full oscillations across the LISA band forb∈[1.7,2.8]. This is qualitatively distinct from other 22

known spectral features: (i) The kink at the crossover from sound-wave to turbulence domination is a single discontinuity in the spectral slope, not a periodic oscillation. (ii) A sharp bubble-collision peak is a feature of limited frequency extent, not a multi-period sinusoid. (iii) Stochastic backgrounds from astrophysical sources produce spectra that are smooth inlnfto high accuracy. A likelihood-ratio test between the smooth template Ω 0 GW and the DSI-modulated template (16) provides the optimal discriminant. The three-parameter family (b,ε,φ 0 ) can be mapped from the data by standard matched-filter techniques [2]. Parameter degeneracies. The phaseφ 0 merely shifts the oscillation inlnfand does not affect detectability;εandbcan be independently constrained from the oscillation depth and period respectively. The frequency resolution needed to resolve individual oscillations is ∆f/f ∼ lnb/(2π); forb= 2 this is ∆f/f ≈0.11, well within LISA’s capabilities over its four-year nominal mission. Alternative UV completions. The factorization result and the observable template (16) are model-independent consequences of DSI in the UETC, requiring onlyτ corr H ∗ ≪1. Walking technicolor is one concrete realization; other BSM models with approximate conformal symmetry and explicit periodic modulations — extended Higgs sectors with Coleman–Weinberg potentials modified by threshold corrections, Randall–Sundrum–type models with periodic radion potentials, or clockwork models [15] — are equally valid candidates and will produce the same spectral template with different (b,ε) values. A detection of log-periodic oscillations in the SGWB would uniquely fixbandε, allowing discrimination among UV completions. Multi-messenger signatures. Beyond gravitational waves, the DSI in the WTC potential generates log-periodic modulations in the technidilaton production rate and hence in the energy density of any dark-radiation component coupled to the hidden sector, providing in principle an independent observational handle on the same (ε,b) parameters. 7 Conclusions We have developed a phenomenological framework for log-periodic signatures of discrete scale invariance in the stochastic gravitational-wave background, and used it to evaluate walking technicolor as a candidate microphysical source. The main results are: 1.Factorization theorem. In the physically motivated short-correlation-time limit (τ corr H ∗ ≪1, satisfied forβ/H ∗ ≳10), a DSI modulation of the source UETC propagates to the observable Ω GW (f) at the per-cent level:P h =C(k)P 0 h [1 + O(τ corr H ∗ )]. The theorem is exact for separable UETCs (Fig. 1); theO(τ corr H ∗ ) bound applies to the controlled approximations of Sec. 3.2. 2. Universal spectral template. The observable signature is Ω GW = Ω 0 GW [1 + ε cos(2π ln(f/f ∗ )/ lnb+φ 0 )] — a sinusoid inlnfsuperimposed on the smooth baseline, characterized at leading order inτ corr H ∗ by three parameters (ε,b,φ 0 ). This template is independent of the microphysical origin of the DSI. 23

3.Matched-filter detectability.SNR osc ≃(ε/ √ 2)SNR baseline p N periods , withN periods

6–13 oscillations in the LISA band. This gives a useful enhancement over the naive εsuppression and defines the detectability landscape (Figs. 3–4) for any DSI source. 4.WTC as a candidate source: an empirical obstruction. Walking technicolor breaks approximate continuous scale invariance to DSI through a holographically motivated periodic modulation of the technidilaton potential, giving a propagator- level forecastε f ∈[0.04,0.18],b 0 ∈[1.7,2.8]. The observable amplitude in the SGWB is reduced fromε f by a geometric factorc geom from the source-tensor convolution. Explicit 3D numerical evaluation of the convolution (Fig. 5) for a free massive propagator, for the standard sound-shell spectral shape, and with the TT angular projector applied to either, gives|c geom |≲0.02 throughout. The implied observable bandε≲4×10 −3 falls about two orders of magnitude below LISA’s 5σmatched-filter threshold. The WTC log-periodic signal is therefore not LISA-detectable in any of the propagator structures evaluated here. 5.A sharp inverse problem. The saddle-point relationc geom (σ/q ∗ )≈ exp[−2π 2 (σ/q ∗ ) 2 / ln 2 b 0 ] makes the rescue condition explicit: a propagator with relative spectral width σ/q ∗ ≲0.2 (FWHM/q ∗ ≲0.5) is required to keepc geom ≳0.1. Candidate microphys- ical mechanisms that could deliver such a narrowing (Sec. 5.5) — a high-Qthermal resonance, a coherent technidilaton oscillation, or a bubble-wall-thickness localization — are identified as the natural targets for future microphysical calculation. A non-detection of log-periodic structure by LISA would place sharp upper limits on εas a function ofb, constraining the allowed parameter space for any DSI source. A detection would simultaneously reveal the discrete scaling ratio, the DSI amplitude, and the phase of the modulation, providing a unique window into the self-similar structure of the hidden-sector dynamics. In the meantime, the matched-filter template Eq. (16) is simple, well-defined, and implementable in any LISA data-analysis pipeline — ready to test any future microphysical prediction against the phenomenological backbone established here. Acknowledgments The author thanks the gravitational-wave and beyond-Standard-Model communities for stimulating discussions. No external funding was received for this work. References [1] P. Amaro-Seoane et al. (LISA Collaboration), “Laser Interferometer Space Antenna,” (2017) [arXiv:1702.00786]. [2] C. Caprini et al., “Science with the space-based interferometer eLISA. II: Grav- itational waves from cosmological phase transitions,” JCAP 1604, 001 (2016) [arXiv:1512.06239]. [3]G. Agazie et al. (NANOGrav Collaboration), “The NANOGrav 15 yr Data Set: Evidence for a Gravitational-Wave Background,” Astrophys. J. Lett. 951, L8 (2023) [arXiv:2306.16213]. 24

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