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: \[ partial_t(psi) = c_d * partial_d(psi) + D_d * partial_d^2(psi) + omega * A * psi + O(epsilon^2) \] where \[ c_d * partial_d(psi) \] 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 \[ G_eff = Pi_Y G Pi_Y + K†BK \], from which observable phase, leakage, decoherence, and residual correction emerge as structural consequences of projected transport closure on Q5.
This record presents 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 = nepsilon ). The result is: [ partial_t(psi) = c_d * partial_d(psi) + D_d * partial_d^2(psi) + omega * A * psi + O(epsilon^2) ] where [ c_d * partial_d(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 = isigma_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 [ 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.
T84 establishes the discrete sampling law for continuous Q5 phase transport. Starting from the reduced phase transport operator\[T_{\mathrm{phase}}=I+\epsilon A+O(\epsilon^2),\qquadA^2=-I,\qquad\epsilon=\frac1{160},\]The theorem defines the discrete observable phase sequenceμk=L(Tphasek),where L is the normalized phase-readout functional satisfying\[L(I)=0,\qquadL(A)=1.\] Using the T80 continuous-limit result\[T_{\mathrm{phase}}^k\toe^{k\epsilon A},\]together withekϵA=Icos(kϵ)+Asin(kϵ),The theorem provesμk=sin(kϵ)+O(ϵ). Substitutingϵ=1601,gives the discrete sampled phase lawμk=sin(160k)+O(1601). T84 is structurally important because it converts the continuous phase-flow lawL(etA)=sintfrom T83 into an indexed observable transport sequencek↦μk. The theorem, therefore, establishes the sampled observable structure inherited from the continuous Q5 transport flow. Several structural consequences follow immediately. Pass-through occurs exactly whensin(kϵ)=0. Maximal pass-through occurs nearkϵ=2π+nπ,Although exact maxima are generally inaccessible because the discrete sampling lattice does not usually intersect those irrational phase locations exactly. For small accumulated transport,k≪160,The small-angle approximation givesμk≈160k,so early-stage phase readout grows linearly with transport depth. Together with T80-T83, the transport/readout chain now becomes\[T_{\mathrm{raw}}\toT_{\mathrm{phase}}\toT_{\mathrm{phase}}^k\toe^{k\epsilon A}\to\mu_k=\sin(k\epsilon).\] Status: solid for the sampled phase lawμk=sin(kϵ)+O(ϵ)given the T80 continuous limit and bounded accumulated phase t=kϵ; solid for the small-step linear regimeμk≈k/160;conditional for higher-order corrections inherited from the unresolved higher-order structure ofTphase.
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 paper develops the quantum dynamical layer of the Quantum Lattice Model (QLM) by showing how the standard wave equations of quantum mechanics arise as continuum descriptions of coherent phase-action transport on a discrete Planck-scale lattice. The framework is constructed from the per-radian Planck primitives {ℏ, ℓP, tP}, where each lattice tick advances physical phase by one radian and transports one quantum of reduced action. The corresponding phase-action and phase-flow relations are dS = ℏ dθ and E = ℏ dθ/dτ, while the invariant lattice transport identity c = ℓP/tP fixes the maximal causal phase-transport speed of the lattice. Starting from symmetric nearest-neighbor phase transport between Planck cells, the continuum limit yields the relativistic wave operator (1/c²)∂t² − ∇². Introducing intrinsic rest-phase rotation with frequency ω0 = mc²/ℏ produces the Klein-Gordon equation as the continuum description of synchronized spatial phase transport and local rest-phase rotation. A first-order relativistic factorization yields the Dirac equation, while extraction of the rapidly rotating rest-phase carrier produces the nonrelativistic Schrödinger equation governing slow envelope dynamics. Within this framework, the quantum wavefunction is interpreted as a coherent phase-envelope amplitude describing phase-action transport across the lattice. The conserved density ρ = |φ|² arises from the continuity equation of Schrödinger evolution, and the associated current takes the phase-flow form J = (ℏ/m)ρ∇θ. The paper also shows how interaction sectors enter the same transport framework as modular coupling layers. Scalar potentials contribute additional local phase rotation to the slow envelope, while routing-weighted transport modifies the effective capacity of lattice links. Coulomb coupling and weak-field gravitational coupling are presented as examples: the Coulomb potential appears as a scalar phase-rotation term, and gravitational routing modulation of the rest-phase carrier yields the weak-field potential V = mΦ. These results organize the Schrödinger, Klein-Gordon, and Dirac equations as successive continuum limits of deterministic phase-action transport on the Planck lattice, providing the quantum dynamical layer of the Quantum Lattice Model.
TA11 closes the first Transport Architecture block (TA3–TA11) by establishing the primitive one-way transport scale ε = 1/320 and fixing the phase axis A = iσ_y, both from the same oriented two-leg transport mechanism. The oriented distance-2 transport layer E₂(Q₅) contains exactly 160 directed channels (80 unordered pairs × 2 orientations). Normalizing a unit closure defect against this layer gives δfull=1/160. The branch-exchange involution Θ, established in T47/T50, splits this defect equally across two oriented legs, fixing ε = 1/320. By TA5, the observable round-trip phase coefficient is μ=2ε=1/160. The phase-doubling lemma establishes that the same two-leg mechanism that produces magnitude doubling also forces the phase axis. Each mediated leg contributes π/4, accumulating π/2 total. Purely real symmetric coupling produces no phase residue, Q=QT∈R implies Im(Q)=0 implies µ=½Tr(QR)=0, so nonzero phase requires oriented complex transport contributions. The two-leg accumulation forces the generator Aiσy. The T8–TA11 Bridge Lemma identifies the spectral ratio β/α with the primitive transport scale under three constraints: same defect source (δfull=1/160), same two-branch splitting (ε=1/320), and same perturbative normalization. This gives β/α=εone−way=1/320, structurally derived rather than numerically inserted. The continuous phase closure theorem shows that iterating the one-way transport operator gives Tphasek=e(k/320)A+O(k/3202), with observable readout µk=sin(k/320)+O(k/3202), controlled for k≪3202. The primitive scale 1/320 thereby links the spectral ratio, one-way transport, phase generator, continuous-limit evolution, and observable sine-law readout in a single unified structure. Geometry supplies the defect scale. Transport supplies the phase structure.
T65 develops an emergent effective-time framework for the reduced Q5 transport architecture. The theorem interprets repeated admissible barrier transport steps as a discrete iterative evolution process and introduces a coarse-grained effective time parametert=nτ,where n counts admissible transport iterations and τ is the effective iteration scale. The resulting reduced evolution operator takes the formUeff=e−iτHeff,with the reduced generator structure emerging through the mediated reduction chain\[A\to\mathcal A_L\toC=iK^\dagger \mathcal A_L K\toQ_\perp\to\mu.\]The theorem, therefore, establishes that the observable reduced rotational dynamics arise through mediated admissibility reduction rather than direct projection from the original transport generator. T65 is structurally important because it reframes the reduced rotational invariant μ as part of an effective iterative evolution system on the reduced defect sector. The theorem further demonstrates, through explicit second-order mediated calculations, that antisymmetric rotational structure emerges dynamically from leakage-sector circulation. The resulting effective-time parameter is a coarse-grained description of repeated admissible transport evolution rather than a derivation of fundamental physical time. Reversibility remains preserved at the operator level, and no intrinsic thermodynamic arrow of time is claimed. Status: solid for the effective iterative evolution structure and mediated reduced-generator construction under the stated reduction assumptions; conditional on the effective-unitary embedding and global closure properties of the reduced evolution operator; speculative for any identification with fundamental physical time or irreversible temporal dynamics.