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 \[ G_eff = Pi_Y G Pi_Y + K†BK \], 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 [ 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 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.
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 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.
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.
T79 completes the Q5 interferometric falsification framework by introducing a structured null-control protocol for quadrature-cycling residuals. Earlier theorems established the residual form, the Z4 quadrature-cycling structure, statistical thresholds, magnitude estimates, and systematics discrimination criteria. T79 addresses the final degeneracy problem: the cycling defectIQ5=∑k=03∣vk+1−Jvk∣2can vanish trivially when all quadrature vectors are zero. The theorem, therefore, establishes the joint operational requirementIQ5≈0and∣vk∣≈0,ensuring that the detected structure corresponds to nontrivial quadrature transport rather than the simple absence of signal. The theorem then defines four explicit null-control experiments designed to isolate ordinary instrumental artifacts from candidate Q5-consistent cycling structure. These include blocked-path operation, detector-only measurements, intentional laser-power modulation, and a classical phase-reference channel. Each control specifies a concrete pass condition under which stable, nonzero quarter-cycle transport behaviour must disappear or fail to persist. In particular, blocked-path operation forces P1P2=0, eliminating the interference residual derived in T72, while detector-only and reference-channel measurements test whether the observed cycling behaviour originates from instrumentation rather than coherent transport structure. T79 reframes the experimental objective as layered falsification rather than direct proof. A candidate signal survives only if coherent two-path interference exhibits nonzero stable quadrature vectors satisfying the quarter-cycle rotation law, while all null controls fail to reproduce the same structured behaviour. The theorem, therefore, converts the earlier invariant framework into a hardened experimental protocol capable of rejecting broad classes of ordinary systematic effects without claiming definitive proof of Q5 transport itself. Passing the protocol establishes only that the observed cycling behaviour survives the identified null controls and remains structurally consistent with the Q5 quadrature-transport prediction. Status: solid for the joint nontriviality condition IQ5≈0 together with ∣vk∣≈0; solid for the blocked-path null implied by the T72 interference structure; solid that the four null controls define experimentally testable falsification operations; conditional on whether the listed controls exhaust all relevant systematic classes for a given experimental platform; not a claim that passing the full null-control protocol would uniquely establish Q5 transport physics.
T75 develops a model-dependent prediction for the observable Q5 fringe-bias scale arising from leakage into hidden quarter-phase transport classes. Using the weight parameterization\[w_0=1-\epsilon,\qquadw_1=\epsilon_1,\qquadw_2=\epsilon_2,\qquadw_3=\epsilon_3,\]withϵ1+ϵ2+ϵ3=ϵ≪1,The theorem derives the exact bias relationη2=(ϵ+ϵ2)2+(ϵ3−ϵ1)2.The resulting fringe bias is therefore controlled not only by total leakage magnitude but also by the structure of the leakage distribution across the quarter-phase classes. The cosine-sector deviation arises from the overall leakage displacement term (ϵ+ϵ2), while the sine-sector deviation is governed by the antisymmetric imbalance mode (ϵ3−ϵ1). T75 is structurally important because it converts the earlier qualitative leakage architecture into a quantitative interferometric prediction. Under the Q5 suppression estimates inherited from T57, the theorem predicts a characteristic fringe-bias scaleηQ5∼10−4for single-bridge leakage, andηQ5∼10−7under an additional return-mediated suppression layer. The theorem, therefore, supplies the missing magnitude estimate needed to complete the experimental prediction arc developed through T72-T74: signal form, structural discriminator, detection threshold, and expected signal scale. T75 does not claim a universal numerical prediction or guaranteed observability across interferometer designs. The predicted magnitude depends on the leakage-path architecture, the symmetry properties of the leakage distribution, and the mapping between the structural suppression parameter ϵ and experimentally accessible optical leakage modes. The resulting prediction is therefore conditional rather than uniquely derived. Status: solid for the exact leakage-weight decompositionη2=(ϵ+ϵ2)2+(ϵ3−ϵ1)2and for the associated order-of-magnitude scaling analysis under the stated assumptions; conditional on the leakage-path architecture, symmetry structure, and suppression mapping inherited from T57; speculative for the existence or observability of any physical Q5 fringe residual prior to experimental detection.