T73 develops a phase-cycling discriminator for Q5-consistent deviations that cannot be identified from a single interference scan. Building on T72, which shows that any first-harmonic residual can be absorbed by refitting the ordinary fringe parameters, T73 demonstrates that the lost information can survive in the constrained transformation law across multiple controlled phase settings. The theorem considers four scans separated by externally controlled \(\pi/2\) phase offsets and replaces independent per-scan refitting with a single jointly constrained fit across all four settings. For these offsets, the ordinary fringe model occupies one two-dimensional phase-cycling chirality subspace, while the opposite chirality subspace is exactly orthogonal to it. Consequently, the joint fit removes the ordinary \(J^{-1}\)-chirality component while preserving the orthogonal \(J\)-chirality component unattenuated. This yields the phase-cycling invariant \[ \mathcal I_{Z_4} = \sum_{k=0}^{3} \left|v_{k+1}-Jv_k\right|^2, \] where \(v_k\) is the residual quadrature vector extracted after the joint fit. A nontrivial signal detectable by this construction must exhibit the surviving quarter-turn transformation law \(v_{k+1}=Jv_k\), giving \(\mathcal I_{Z_4}\approx0\) with nonzero residual amplitude. The result is a single-chirality test rather than a choice between two equivalent rotation signs, because the opposite chirality is structurally absorbed by the joint null model. T73 does not establish that Q5's physical residual necessarily carries the surviving chirality. That identification requires a separate residual-to-observable bridge connecting the later Q5 residual structure to the experimental phase-cycling coordinates. Nor does the theorem establish immunity to correlated experimental systematics; those require separate noise and control analysis. T73 establishes the mathematical distinguishability mechanism and operational phase-cycling protocol under the stated common-reference and joint-fit assumptions. Status: solid for the exact chirality separation, joint-fit discriminator, and \(\mathbb Z_4\) phase-cycling invariant under the stated four-offset protocol; conditional on a common external phase reference and jointly constrained null model; unresolved as to whether the physical Q5 residual occupies the surviving chirality; no claim of experimental detection or uniqueness against all correlated systematics.
T73 develops a phase-cycling discriminator for Q5-consistent deviations that cannot be identified from a single interference scan. Building on T72, which shows that any first-harmonic residual can be absorbed by refitting the ordinary fringe parameters, T73 demonstrates that the lost information can survive in the constrained transformation law across multiple controlled phase settings. The theorem considers four scans separated by externally controlled (\pi/2) phase offsets and replaces independent per-scan refitting with a single jointly constrained fit across all four settings. For these offsets, the ordinary fringe model occupies one two-dimensional phase-cycling chirality subspace, while the opposite chirality subspace is exactly orthogonal to it. Consequently, the joint fit removes the ordinary (J^{-1})-chirality component while preserving the orthogonal (J)-chirality component unattenuated. This yields the phase-cycling invariant [ \mathcal I_{Z_4} = \sum_{k=0}^{3} \left|v_{k+1}-Jv_k\right|^2, ] where (v_k) is the residual quadrature vector extracted after the joint fit. A nontrivial signal detectable by this construction must exhibit the surviving quarter-turn transformation law (v_{k+1}=Jv_k), giving (\mathcal I_{Z_4}\approx0) with nonzero residual amplitude. The result is a single-chirality test rather than a choice between two equivalent rotation signs, because the opposite chirality is structurally absorbed by the joint null model. T73 does not establish that Q5's physical residual necessarily carries the surviving chirality. That identification requires a separate residual-to-observable bridge connecting the later Q5 residual structure to the experimental phase-cycling coordinates. Nor does the theorem establish immunity to correlated experimental systematics; those require separate noise and control analysis. T73 establishes the mathematical distinguishability mechanism and operational phase-cycling protocol under the stated common-reference and joint-fit assumptions. Status: solid for the exact chirality separation, joint-fit discriminator, and (\mathbb Z_4) phase-cycling invariant under the stated four-offset protocol; conditional on a common external phase reference and jointly constrained null model; unresolved as to whether the physical Q5 residual occupies the surviving chirality; no claim of experimental detection or uniqueness against all correlated systematics.
T74 establishes the statistical detection threshold for the Q5 phase-cycling signal developed in T72 and T73. Starting from the single-scan fringe residualϵ(ϕ)=2P1P2[(A−1)cosϕ+Bsinϕ],The theorem defines the intrinsic bias magnitudeη=(A−1)2+B2,and proves that the maximum balanced-path residual amplitude satisfiesϵmax=η.A key refinement is introduced: measurement of a first-harmonic bias alone is not operationally sufficient for Q5 detection, because such a bias can be absorbed into standard interferometric fitting parameters. The correct discriminator is therefore the phase-cycling invariantIQ5=∑k=03∣vk+1−Jvk∣2,which tests whether the measured quadrature evolution obeys the expected Z4 cycling structure established in T73. T74 proves that Q5 detectability requires two simultaneous conditions. First, the intrinsic signal scale must exceed the statistical shot-noise floor,η≳NNσ,where N is the photon count and Nσ is the required detection significance. Second, the observed phase evolution must remain consistent with the Q5 phase-cycling law through sufficiently small IQ5, distinguishing genuine Q5-compatible structure from ordinary calibration artifacts. Under shot-noise scaling, the minimum detectable bias obeysηmin∼NNσ. T74 is structurally important because it completes the minimal experimental prediction arc initiated in T72 and T73 by separating signal shape, structural discriminator, and statistical detectability. The theorem converts the earlier transport-cycling framework into a falsifiable interferometric prediction program with explicitly identified observables, structural consistency conditions, and detection scaling laws. Status: solid for the fringe-residual amplitude relations and shot-noise detection-threshold scaling under the stated assumptions; solid for the necessity of combining signal magnitude with structural phase-cycling discrimination; conditional on the T73 Z4 cycling framework and practical extraction of quadrature vectors; speculative for the existence or magnitude of any physical Q5 signal prior to empirical observation.
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.
This work presents a minimal, experimentally testable deviation from standard two-path interference. The deviation arises from an underlying discrete phase structure and appears as a small first-harmonic bias in the interference fringe. The predicted effect does not introduce higher harmonics or alter the qualitative form of the fringe. Instead, it manifests as a coupled visibility rescaling and phase offset that can be absorbed into standard sinusoidal fits. For this reason, the deviation is not directly visible at the level of fringe shape alone. To distinguish the effect from conventional systematics, a phase-cycling discriminator is introduced. Under controlled quarter-phase shifts, the residual is predicted to exhibit a structured quadrature rotation that is not generically reproduced by noise, detector imbalance, or calibration drift. A set of null-control conditions is provided to isolate this behavior experimentally. A characteristic magnitude for the effect is estimated under a class of structural assumptions, yielding a candidate scale on the order of 10−410^{-4}10−4, with a more strongly suppressed regime near 10−710^{-7}10−7. These values lie near the sensitivity threshold of high-precision interferometric measurements. The result is a concrete, falsifiable prediction: either a stable first-harmonic residual with the specified phase-cycling structure is observed, or the corresponding parameter regime is constrained. This document isolates the observable prediction and experimental test, independent of the broader theoretical framework from which it is derived.
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.
T86 establishes the reduced barrier cycle operator for the Q5 transport framework. Let\[A=E_d=i\sigma_y,\qquadA^2=-I,\]and letBdenote the full Q5 barrier crossing operator containing the orientation-reversal component Θ. Using the canonical phase projectionΠphase,introduced in T26-T29, the theorem identifies the reduced barrier operatorBred=Πphase(B),under the conditional assumption that the barrier crossing acts as a quarter-turn on the reduced defect plane. Under this assumption,Bred=e(π/4)A. Using the exponential identityeθA=Icosθ+Asinθ,The theorem proves\[B_{\mathrm{red}}^4=-I,\qquadB_{\mathrm{red}}^8=I.\] The reduced barrier operator therefore, produces the spinor state cycleψn+4=−ψn,ψn+8=ψn,giving a Z8 state period and a Z4 observable period through projective sign identification. T86 is structurally important because it isolates the reduced coarse barrier operator explicitly within the Q5 framework. Earlier theorems established:- the rotational generator A,- the barrier crossing structure,- and the projection architecture. T86 now combines these ingredients into a single reduced operator form. The theorem also sharply isolates the remaining open step. The algebraic structuree(π/4)Acorrectly reproduces the required Z8 cycling, but the geometric origin of the quarter-turn angleπ/4has not yet been derived from first-principles Q5 barrier geometry. Accordingly, T86 proves:Πphase(B)=e(π/4)Aconditional on the quarter-turn assumption, while T87 is designated as the theorem responsible for deriving the angle geometrically. The theorem further clarifies that the identification applies only after projection onto the reduced defect plane. The full barrier operatorBcontains orientation-reversal structure and scalar/symmetric dressing removed byΠphase. Thus, the clean exponential forme(π/4)AIt is a reduced-sector statement rather than a claim about the full raw barrier operator globally. T86 therefore serves as the algebraic bridge connecting:- the barrier crossing structure from T47-T49,- the Z8 spinor cycling from T68,- and the dual-scale generator structure established in T85. Status: solid fore(π/4)Aproducing\[B_{\mathrm{red}}^4=-I,\qquadB_{\mathrm{red}}^8=I,\]and for the resulting Z8/Z4 cycling structure; solid for the projection properties ofΠphase;conditional for the identificationΠphase(B)=e(π/4)A,which depends on deriving the quarter-turn angleπ/4from Q5 barrier geometry in T87.
T66 develops a discrete holonomy-layer interpretation within the reduced Q5 transport architecture while explicitly separating intrinsic transport structure from observable effective phase structure. The theorem identifies an internal holonomy sector with quarter-turn rotational residue,ΔΩhol∈{0,2π,π,23π},producing an intrinsic Z4-type phase structure associated with the mediated rotational generator algebra. However, the theorem further shows that the observable effective phase operator is not identical to the intrinsic holonomy contribution alone. Instead, observable phase structure is modelled through a first-order decomposition of the formΔΩobs≈ΔΩhol+ΔΩsym,where the symmetric contribution smooths the observable phase geometry into an effectively continuous structure. T66 is structurally important because it resolves a potentially fatal overprediction problem in the earlier framework. Without the distinction between intrinsic and observable phase sectors, the architecture would incorrectly predict directly observable discrete quantum phase quantization. The theorem instead establishes that discrete holonomy survives as an internal transport-layer structure while observable interference behaviour remains effectively continuous after mediated reduction and symmetric-sector averaging. The theorem, therefore, introduces the concept of a hidden discrete phase layer embedded within a continuous effective observable geometry. Status: solid for the intrinsic holonomy-sector decomposition and the distinction between internal and observable phase structure under the stated operator assumptions; conditional on the first-order decomposition framework and mediated observable reduction model; speculative for any physical claim regarding hidden discrete phase layers in actual quantum systems.