This record presents TA19 (Observable Deviation from Leakage Spectrum), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework. TA19 closes the depth architecture arc TA14 through TA18 by connecting the leakage spectrum to observable probability deviations, and feeds back into the T71 through T79 experimental prediction arc. The depth program is no longer self-contained: it now connects the internal transport hierarchy to measurable interference structure. The full mechanism chain established across TA14 through TA19 is: depth mode n produces roughness scaling as n to the fourth power, which produces leakage amplitude scaling as n squared in the low-mode regime, which drives leakage return through the complement sector, which produces observable probability deviation. Each stage has its own object, its own scaling law, and its own epistemic status. The central result is that in the perturbative low-mode regime where c times n squared is much less than one, the depth-mode observable deviation scales as approximately eta times c times n squared. The full implied structure gives a deviation proportional to eta times cn^2 divided by (1 + cn^2), which saturates to a global bound of eta as mode number increases. Observable anomaly; therefore, initially grows with depth-mode frequency before saturating; it does not grow indefinitely. The observable deviation is connected to the T71 quadrature deviation structure. The T71 interference deviation has the form involving coefficients A and B multiplying the cosine and sine of the phase angle. Under TA19, the depth-mode-dependent deviation coefficients scale as eta times n squared at low mode number, giving a depth-mode-refined interference prediction where the deviation amplitude carries an explicit n squared factor. Exact coefficients and phase prefactors are not computed here. Three lemmas establish the result: leakage return generates observable correction at leading order in the leakage coupling parameter eta (Lemma 1); the depth-mode deviation scaling follows as eta times n squared in the low-mode regime and eta times the square root of the roughness functional equivalently (Lemma 2); and the T71 quadrature deviation coefficients inherit the n squared scaling (Lemma 3). A critical epistemic note is maintained throughout: the theorem separates the robust qualitative claim from the model-dependent quantitative claim. The robust claim is that observable deviation increases monotonically with depth-mode frequency before saturating. The model-dependent claim is the specific n squared scaling, which requires the continuum depth approximation from TA15, the low-pass gate model from TA18, and the leading-order perturbative coupling assumed here. These must not be conflated in experimental interpretation. The leakage-deviation coupling is explicitly identified as a first-order perturbative ansatz rather than a structural inevitability, since probabilities are nonlinear objects and the full leakage return operator can contribute phase-sensitive cross terms at higher order in eta. Together, TA14 through TA19 form a complete emergence ladder: discrete adjacency transport, continuum evolution, smoothness selection, quartic roughness spectrum, leakage from admissibility misalignment, and perturbative observable consequences. This is a coherent multistage transport framework connecting discrete Q5 geometry to measurable interference deviations. The theorem chain progressively derives the structure of the effective transport generator \[ G_eff = Pi_Y G Pi_Y + K†BK \], from which observable phase, leakage, decoherence, and residual correction emerge as structural consequences of projected transport closure on Q5.
This record presents TA19 (Observable Deviation from Leakage Spectrum), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework. TA19 closes the depth architecture arc TA14 through TA18 by connecting the leakage spectrum to observable probability deviations, and feeds back into the T71 through T79 experimental prediction arc. The depth program is no longer self-contained: it now connects the internal transport hierarchy to measurable interference structure. The full mechanism chain established across TA14 through TA19 is: depth mode n produces roughness scaling as n to the fourth power, which produces leakage amplitude scaling as n squared in the low-mode regime, which drives leakage return through the complement sector, which produces observable probability deviation. Each stage has its own object, its own scaling law, and its own epistemic status. The central result is that in the perturbative low-mode regime where c times n squared is much less than one, the depth-mode observable deviation scales as approximately eta times c times n squared. The full implied structure gives a deviation proportional to eta times cn^2 divided by (1 + cn^2), which saturates to a global bound of eta as mode number increases. Observable anomaly; therefore, initially grows with depth-mode frequency before saturating; it does not grow indefinitely. The observable deviation is connected to the T71 quadrature deviation structure. The T71 interference deviation has the form involving coefficients A and B multiplying the cosine and sine of the phase angle. Under TA19, the depth-mode-dependent deviation coefficients scale as eta times n squared at low mode number, giving a depth-mode-refined interference prediction where the deviation amplitude carries an explicit n squared factor. Exact coefficients and phase prefactors are not computed here. Three lemmas establish the result: leakage return generates observable correction at leading order in the leakage coupling parameter eta (Lemma 1); the depth-mode deviation scaling follows as eta times n squared in the low-mode regime and eta times the square root of the roughness functional equivalently (Lemma 2); and the T71 quadrature deviation coefficients inherit the n squared scaling (Lemma 3). A critical epistemic note is maintained throughout: the theorem separates the robust qualitative claim from the model-dependent quantitative claim. The robust claim is that observable deviation increases monotonically with depth-mode frequency before saturating. The model-dependent claim is the specific n squared scaling, which requires the continuum depth approximation from TA15, the low-pass gate model from TA18, and the leading-order perturbative coupling assumed here. These must not be conflated in experimental interpretation. The leakage-deviation coupling is explicitly identified as a first-order perturbative ansatz rather than a structural inevitability, since probabilities are nonlinear objects and the full leakage return operator can contribute phase-sensitive cross terms at higher order in eta. Together, TA14 through TA19 form a complete emergence ladder: discrete adjacency transport, continuum evolution, smoothness selection, quartic roughness spectrum, leakage from admissibility misalignment, and perturbative observable consequences. This is a coherent multistage transport framework connecting discrete Q5 geometry to measurable interference deviations. The theorem chain progressively derives the structure of the effective transport generator [ G_eff = Pi_Y G Pi_Y + K†BK ], from which observable phase, leakage, decoherence, and residual correction emerge as structural consequences of projected transport closure on Q5.
This record presents TA20 (Depth-Mode Leakage Signature), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework. TA20 is a control theorem rather than a prediction theorem. TA19 established that depth-mode observable deviations scale as eta times n squared in the perturbative low-mode regime. TA20 identifies the diagnostic signature that distinguishes genuine Q5 depth-mode leakage from ordinary experimental noise, systematic offset, or generic drift. It does not add new content to the leakage mechanism but provides the control structure needed for experimental interpretation of TA19. The signature consists of three necessary conditions that must hold simultaneously. The first is monotone scaling: deviation amplitude shows a trend-level monotonic increase with depth-mode frequency in the low-mode regime, up to experimental uncertainty. The second is smooth-mode null: for the gate-aligned mode with n equal to zero, the deviation vanishes or is negligible, because the constant mode has maximal overlap with the gate and minimal leakage. The third is phase-locked quadrature: the deviation appears in the cosine-phi and sine-phi quadrature basis of the T71 interference correction, with amplitude consistent with the n-squared scaling. These are necessary conditions, not sufficient ones. A deviation pattern satisfying all three is compatible with the Q5 depth-mode leakage mechanism but does not constitute proof of it. This distinction is stated explicitly and is one of the central epistemic contributions of the theorem. The three conditions test different layers of the mechanism. Monotone scaling tests spectral dependence. The smooth-mode null tests admissibility alignment and is particularly powerful because it is tied directly to the gate geometry rather than being a free parameter. Phase-locked quadrature tests phase structure and is the strongest discriminator experimentally, since a generic systematic effect would need to accidentally reproduce the specific T71 quadrature decomposition with coherent cosine and sine components and correlated scaling across mode numbers. Ordinary experimental noise generically fails at least one of the three conditions. Generic noise does not respect n-squared scaling in mode frequency. Systematic offsets do not vanish when n is equal to zero. Noise-driven deviations are not generically phase-locked to the T71 quadrature basis. The conjunction of all three conditions, therefore, provides a false-positive rejection criterion. The phase-locking condition includes a caveat: the intrinsic mechanism generates quadrature structure, but observational averaging, decoherence, or detector integration could wash out phase structure in practice. A phase-independent observation is therefore not consistent with the intrinsic mechanism at the signal level, but this must be interpreted with care in the presence of measurement imperfections. TA20 provides the control conditions that TA21 (Experimental Quadrature Extraction Protocol) will operationalize into a concrete experimental procedure. The theorem chain progressively derives the structure of the effective transport generator Geff=PiYGPiY+K†BK, from which observable phase, leakage, decoherence, and residual correction emerge as structural consequences of projected transport closure on Q5.
This record presents TA18 (Curvature-Induced Leakage Law), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework. TA17 established that depth modes carry roughness cost scaling as the fourth power of the mode number. TA18 connects this roughness spectrum to the leakage amplitude, where the leakage operator K=I−PiY measures the component of a state that fails to align with the admissible gate mode. This is the first theorem in the depth architecture series to establish an explicit mechanistic chain: depth roughness drives gate misalignment, which drives leakage into the complement sector. The central structural assumption is that the admissible gate projection Pi_Y acts effectively as a low-pass selector in the depth basis. This is motivated by two independent considerations: the Riemann-Lebesgue decay of Fourier coefficients for smooth functions (overlap with high-frequency modes decays as mode frequency increases), and the T122 structure of the gate mode, which selects a balanced slowly-varying phase configuration. Under this model, gate overlap scales approximately as 1/(1+c∗n2) for depth mode number n, where c is a structural constant determined by the Q5 geometry of the gate mode. The leakage amplitude then satisfies the exact expression ell_n = cn^2 / (1 + cn^2). In the low-mode regime where c∗n2 is small, this gives ell_n approximately proportional to n^2. Combined with the TA17 result that roughness scales as n^4, the two quantities are related by ell_n proportional to the square root of the roughness functional in the low-mode regime. Leakage therefore grows monotonically with depth roughness but more slowly: quadratically rather than quartically. An important correction is stated explicitly: an earlier qualitative claim that leakage is proportional to curvature is too strong. The correct relationship is that leakage grows monotonically with roughness but sub-linearly in the roughness functional. Monotonicity and proportionality are distinct, and the distinction matters for the scaling hierarchy. The leakage saturates to ell_n approaching 1 for high modes. This saturation is structurally significant: beyond a critical roughness scale, additional oscillation does not meaningfully increase misalignment because the state is already almost entirely outside the admissible sector. This behaviour is architecturally adjacent to the rupture regime identified in TA27 and the bear-clause destabilization in T88 and T89, though the full saturation-to-rupture transition is not derived here. The quadratic leakage scaling is model-dependent and should be verified against the explicit Q5 construction of Pi_Y when available. The qualitative conclusion that leakage increases monotonically with mode frequency is robust across model variations. TA18 provides the leakage spectrum that feeds into TA19 (Observable Deviation from Leakage Spectrum), which connects the mode-dependent leakage amplitude to deviations in interference observables and probability outputs, closing the chain back to the T71 through T79 experimental prediction arc. The theorem chain progressively derives the structure of the effective transport generator Geff=PiYGPiY+K†BK, from which observable phase, leakage, decoherence, and residual correction emerge as structural consequences of projected transport closure on Q5.
This record presents TA21 (Experimental Quadrature Extraction Protocol), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework. TA20 identified the three-part diagnostic signature for Q5 depth-mode leakage: monotone mode scaling, smooth-mode null, and phase-locked quadrature deviation. TA21 converts this control logic into a concrete operational protocol specifying preparation steps, measurement procedure, fitting procedure, and control experiments. The depth coordinate arising in the Q5 framework from the 320-slot transport graph structure does not have a direct laboratory counterpart. The protocol, therefore, works with a structural analog: depth modes of frequency n are simulated by inserting a phase modulator in one arm of a standard two-path interferometer and driving it with a structured phase profile equal to n times a fixed reference profile. The modulation frequency n plays the role of the depth-mode index, with n equal to zero giving a flat constant phase corresponding to the smooth gate-aligned mode, and increasing n giving increasing phase oscillation corresponding to higher depth modes. The protocol tests mode-dependent transport sensitivity rather than literal realization of the depth coordinate. The spectral scaling of leakage with modulation frequency is the object under test, not the geometric origin of that scaling. The measurement procedure scans the global inter-arm phase and records the interference intensity as a function of phase for each modulation frequency. The intensity pattern is fit to a quadrature form, extracting coefficients A_n and B_n. The deviation magnitude is then computed as the square root of the sum of squares of A_n minus one and B_n, providing a scalar observable that can be plotted against n. The TA20 signature is tested by checking three conditions. First, smooth-mode null: the deviation magnitude at n equal to zero should vanish or be negligible. Second, monotone scaling: the deviation magnitude should show a trend-level monotonic increase with n up to experimental uncertainty, with the stronger model-dependent form predicting scaling as eta times n squared. Third, phase-locked quadrature: the deviation must remain in the cosine-phi and sine-phi basis rather than appearing as random fringe drift, amplitude noise, or uncontrolled phase wandering. Three control experiments provide false-positive rejection. Power variation changes beam intensity without changing phase structure and should leave the deviation magnitude unchanged, ruling out intensity artifacts. Noise injection adds random phase noise and should reduce visibility without producing clean n-dependent scaling, ruling out noise-driven drift. Symmetric modulation applies identical phase modulation to both arms and should suppress the deviation to near zero, because the symmetric modulation cancels in interference. The symmetric modulation control is the strongest of the three because it directly tests whether asymmetry is essential to the signal, which matches the leakage architecture. The measurement machinery is standard: two-path interferometry and quadrature extraction are established experimental techniques. The interpretation is not standard and is classified separately. The mapping from phase modulation to depth-mode analog is a structural analogy, not a formal derivation. The TA18 through TA19 depth-leakage model is falsified at the protocol level if the deviation magnitude shows no n-dependence, if the deviation is random or phase-unlocked, or if the symmetric control fails to suppress the deviation. A positive result compatible with all three conditions does not constitute proof of the Q5 mechanism but constitutes a result consistent with it. The theorem chain progressively derives the structure of the effective transport generator Geff=PiYGPiY+K†BK, from which observable phase, leakage, decoherence, and residual correction emerge as structural consequences of projected transport closure on Q5.
T104 computes the explicit observable interference correction generated by Q5 leakage-return transport. T101 established the admissible structure of a hidden-half correction term but left the correction function unspecified. T102 derived the leakage-return operator C=L†BL, and T103 identified the leakage generator B=iAL from the explicit Q5 leakage adjacency. T104 completes the chain by directly computing the resulting interference correction. The central structural result is that first-order mixing vanishes identically: L†BL=0. This occurs because the leakage generator transports amplitude through the helical chain e3↔e4↔e5, while the observable return map extracts only the boundary modes e3 and e5. A single application of B moves amplitude into the middle gate-null mode e4, preventing any first-order return contribution. Observable mixing therefore begins only at second order. The first nontrivial return operator is C2=L†B2L=−801(I−iσy), which is Hermitian and generates the leading observable correction. Applying the T100 renormalization procedure yields normalized probability shifts Δ+=−160sinϕsin(4θ),Δ−=+160sinϕsin(4θ), with Δ++Δ−=0. The resulting observable probabilities are P+obs=cos2θ−2η2160sinϕsin(4θ)+O(η3),P−obs=sin2θ+2η2160sinϕsin(4θ)+O(η3). The correction is therefore second order in the leakage coupling and possesses a characteristic angular envelope sin(4θ). This replaces the earlier minimal placeholder structure of T101 and introduces a richer node pattern, with nulls at θ=0,4π,2π, together with maxima at θ=8π,83π. The phase dependence is proportional to sinϕ, arising from the e±iπ/4 phases contained in the leakage generator AL. This shifts the correction into quadrature relative to the leading Born interference fringe. T104 establishes that the Q5 leakage-return mechanism produces a fully explicit interference correction that is antisymmetric between output channels, conserves total probability, vanishes at the required endpoints, and emerges naturally from the two-step transport structure of the leakage chain. The theorem replaces the phenomenological correction function of T101 with a direct Q5-derived prediction and identifies sinϕsin(4θ) as the characteristic observable signature of leakage-return transport.
T95 derives the first quantitative prediction of the extended Q5 leakage-state framework by determining the short-time decay of interference visibility for a gate-aligned crossing state coupled to leakage modes. Building on the leakage architecture of T93 and the return-transport mechanism of T94, the theorem establishes that the initial loss of coherence is governed entirely by the leakage coupling strength and is independent of the internal leakage dynamics at leading order. The extended state space is Hext=C2⊕HL, with generator \[\mathcal{G}=\begin{pmatrix}A & -K^{\dagger} \\K & B\end{pmatrix},\qquadA=i\sigma_y.\] An initially gate-aligned state Ψ(0)=(ψ00),ΠYψ0=ψ0,∥ψ0∥=1, begins entirely within the admissible crossing mode, with no initial leakage-sector occupation. The theorem first establishes that leakage amplitude enters the auxiliary sector at first order in time: λ(t)=tKψ0+O(t2). Consequently, the associated leakage weight is second order: ∥λ(t)∥2=t2∥Kψ0∥2+O(t3). Because the full evolution is norm preserving, the visible crossing-sector weight must decrease by the same amount. This leads directly to the short-time gate-admission weight ∥ΠYΨ(t)∥2=1−t2∥Kψ0∥2+O(t3). Taking the square root yields the principal result: V(t)=1−21∥Kψ0∥2t2+O(t3), where V(t)=∥ΠYΨ(t)∥ is the interference visibility introduced in T93. Using the gate decoherence measure DY(t)=1−V(t)2, The theorem obtains DY(t)=∥Kψ0∥2t2+O(t3). Thus, decoherence begins quadratically in time rather than linearly. Leakage amplitude appears immediately, but observable coherence loss emerges only after that displaced amplitude acquires measurable weight. A central consequence is that the leading decoherence coefficient depends solely on ∥Kψ0∥2, the leakage strength of the initial state. The internal leakage-sector circulation operator B does not contribute to the leading-order visibility loss and enters only at higher orders through the return dynamics derived in T94. The theorem, therefore, separates the immediate coherence-loss mechanism from the subsequent circulation and return processes. Several corollaries follow. States satisfying Kψ0=0 experience no short-time decoherence from leakage. Stronger leakage coupling produces faster visibility decay, and the coefficient ∥Kψ0∥2 may in principle be inferred from the curvature of an interference fringe at short times. Combining the result with the reduced interference law of T91 shows that leakage suppresses the interference term through a quadratic visibility factor. The theorem provides the first explicit dynamical prediction of the extended leakage architecture. It demonstrates that phase information is not destroyed when coherence is lost. Rather, phase leaves the Y-admissible crossing mode at a rate determined by the leakage coupling, producing a universal quadratic onset of decoherence in the short-time regime.
This record presents TA15 (Continuum Limit of Depth Evolution), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework. TA14 established that the effective transport generator G acts as a local adjacency operator on the 320-slot depth graph, inducing weighted averaging over neighbouring depth values rather than deterministic stepping, making depth a continuum-inducing parameter. TA15 performs the continuum limit explicitly. Taking the elementary depth spacing epsilon=1/320 as a small parameter, and noting that after choosing any ordering of the 320 slots compatible with local transport adjacency, the induced transport law reduces locally to nearest-neighbour form up to higher-order corrections, a standard Taylor expansion yields a differential evolution equation in the continuous depth coordinate d=n∗epsilon. The result is: partialt(psi)=cd∗partiald(psi)+Dd∗partiald2(psi)+omega∗A∗psi+O(epsilon2) where cd∗partiald(psi) is the oriented transport term arising from the Mobius/Gray directional bias c_d = vepsilon, D_d * partial_d^2(psi) is the symmetric spreading term from symmetric adjacency averaging (D_d = kappaepsilon^2), omega * A * psi is the local rotational phase term from the reduced generator A = i*sigma_y, and leakage-return corrections from the K†BK block enter at higher order. Three lemmas support the derivation: the symmetric continuum limit (equal forward and backward rates produce a diffusion equation in depth), the oriented continuum limit (Mobius/Gray directional bias produces a first-order transport equation), and the phase rotation term (the A-block contributes independently of the depth derivative structure). Two regime corollaries identify the transport-dominated case (strong directional bias gives directed propagation along depth) and the diffusion-dominated case (symmetric transport gives depth diffusion). A third corollary identifies the continuum equation as the starting point for TA16 (Smooth-Path / Minimum-Curvature Principle). The theorem does not claim a physical wave equation, Schrodinger equation, gravitational field equation, or any observable identification of the depth coordinate. The continuum limit is a structural result about the 320-slot transport lattice. Explicit coefficient values for c_d, D_d, and omega remain conditional on the full Q5 construction of the leakage generator (open from T126). Together, TA14 and TA15 mark a conceptual transition point in the architecture: the depth coordinate moves from a combinatorial indexing label to an emergent coarse coordinate with local generator flow, continuum approximation, and transport/diffusion decomposition. This transition occurs without abandoning the discrete substrate. The theorem chain progressively derives the structure of the effective transport generator Geff=PiYGPiY+K†BK, from which observable phase, leakage, decoherence, and residual correction emerge as structural consequences of projected transport closure on Q5.