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 \( K = I - Pi_Y \) 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 \( 1/(1 + c*n^2) \) 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 \( c*n^2 \) 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 \[ G_eff = Pi_Y G Pi_Y + K†BK \], from which observable phase, leakage, decoherence, and residual correction emerge as structural consequences of projected transport closure on Q5.
This record presents 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 - Pi_Y ) 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 + cn^2) ) 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 ( cn^2 ) 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 [ 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 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 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.
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 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.
T94 extends the leakage-state framework of T93 by deriving the leading-order return dynamics of phase amplitude that leaves the Y-admissible crossing sector. Rather than treating leakage as a terminal loss mechanism, the theorem models leakage as a redistribution process in which rejected amplitude enters a secondary sector, circulates internally, and subsequently returns to the observable crossing sector. The extended state space is Hext=C2⊕HL, with dynamics governed by the coupled generator G=(AK−K†B), where K transfers amplitude into the leakage sector, K† returns amplitude from the leakage sector, and B governs internal circulation within HL. The theorem establishes that leakage is redistribution rather than destruction. For an initially coherent crossing state ψPQ(0), amplitude enters the leakage sector according to λ(t)=tKψPQ(0)+O(t2), and subsequently evolves through the internal circulation operator etB. The resulting leading-order return contribution to the crossing sector is R(t)=K†etBKψPQ(0). This return term is identified as the residual surface return of the extended Q5 architecture. A central result of the theorem is that residual return is not automatically restorative. The effect of the returned phase is determined entirely by its alignment with the gate-admissible crossing component ψY(t)=ΠYΨ(t). The coherence response is governed by the real overlap ℜ⟨ψY(t),R(t)⟩. Three regimes emerge: ℜ⟨ψY,R⟩>0 produces reinforcement of coherent crossing, ℜ⟨ψY,R⟩<0 produces dephasing, and ℜ⟨ψY,R⟩=0 produces a neutral contribution. The theorem, therefore, refines the decoherence picture introduced in T93. Decoherence is not interpreted as the disappearance of phase information but as the return of redistributed phase in a state that is no longer aligned with the admissible crossing mode. Misaligned return decreases gate-admission weight and reduces interference visibility even when the total norm remains conserved. Several corollaries follow. Generic residual return does not remain phase-locked to the crossing mode and therefore fails to restore coherence automatically. Exact recovery requires reconstruction of the full leakage-sector evolution etB, making reversibility information-demanding even when the underlying dynamics remain unitary. Effective irreversibility is thus attributed to the information cost of reconstruction rather than to fundamental non-unitarity. T94 serves as the first formal return-transport theorem in the Q5 sequence. Together with T93, it provides a complete first-order picture of leakage dynamics: crossing sectorKleakage sectoretBsurface circulationK†return transport. The theorem establishes that the coherence consequences of leakage are determined not by the existence of returned phase, but by the alignment of that returned phase with the unique Y-admissible crossing mode.
T93 extends the reduced Q5 interference framework of T91–T92 to an enlarged leakage-state space Hext=C2⊕HL, where C2 is the coherent crossing sector and HL collects non-crossing leakage modes. The theorem distinguishes between visible coherent weight C(t)=∥ΠPQΨ(t)∥2, leakage weight D(t)=∥(I−ΠPQ)Ψ(t)∥2, and gate-admission weight G(t)=∥ΠYΨ(t)∥2, where ΠY projects onto the Y-admissible crossing mode. A key result is that visible coherence and gate alignment are distinct quantities. A state may retain nonzero amplitude in the observable P/Q sector while failing admission into the Y-crossing mode. This leads to a geometric definition of decoherence: DY(t)=1−∥ΠYΨ(t)∥2. Decoherence is therefore interpreted as loss of overlap with the admissible crossing channel rather than destruction of phase information. The theorem further establishes that interference visibility is governed by the gate-admitted component of the state: V(t)=∥ΠYΨ(t)∥, yielding the extended interference law Pr(P)=21(1+V(t)sin(2t)cosϕ). Perfect visibility corresponds to complete gate alignment, while reduced visibility reflects either leakage into non-crossing modes or misalignment with the crossing sector. An extended generator G=(AK−K†B) is introduced to couple the coherent crossing sector to the leakage sector. Under the skew-adjoint conditions \[A^{\dagger}=-A,\qquadB^{\dagger}=-B,\] the evolution operator etG preserves total norm, implying C(t)+D(t)=1. Leakage therefore represents redistribution of phase weight rather than irreversible loss. The theorem provides a geometric interpretation of decoherence within the Q5 framework: phase information may remain globally conserved while becoming inaccessible to coherent crossing dynamics. In this picture, coherence is determined not merely by the presence of phase amplitude, but by its alignment with the unique Y-admissible transport mode.