CP-H1-INT-001: Integral-Cohomology and Flux-Quantization Audit for a Carrier-Derived Exterior-Moment H1 Line on a Locked 6D-to-3D Cut-and-Project Incidence Complex v0.1.0

Abstract

CP-H1-INT-001 v0.1.0 presents a theory-neutral analytical and deterministic audit of a necessary condition for an integral-cohomology or flux-quantization closure of the carrier-derived exterior-moment H1 line on the locked DG-001 6D-to-3D cut-and-project incidence complex. The primary locked finite source is: HLV-DG-001 Locked Confirmatory Results v0.1.1 DOI: 10.5281/zenodo.22107618 The immediate upstream analytical records are: CP-H1-ORI-SCALE-001 DOI: 10.5281/zenodo.22207132 CP-H1-SRC-UNI-001 DOI: 10.5281/zenodo.22206661 CP-H1-SRC-001 DOI: 10.5281/zenodo.22180728 CP-H1-FLAG-001 DOI: 10.5281/zenodo.22180467 CP-H1-SEL-001 DOI: 10.5281/zenodo.22180275 The locked incidence complex has (N0,N1,N2,N3) = (1110,5345,6960,2826) and beta1 = 99. For a connected finite cell complex, the universal coefficient theorem gives H^1(X;Z) = Hom(H_1(X;Z),Z), so H^1(X;Z) is torsion-free. In the present case it is a free abelian group of rank 99. The audit asks whether the real harmonic line selected by the previously defined carrier-derived exterior-moment source contains a nonzero integral cohomology class. An exact period criterion is established. A nonzero real line L = span{h} in H^1(X;R) contains a nonzero integral class if and only if there exists a nonzero scalar a such that a is an integer for every integral 1-cycle c. Equivalently, the additive subgroup generated by all periods of h must be cyclic. After primitive normalization all cycle periods must be integer multiples of one common period unit. A deterministic spanning tree of the connected locked one-skeleton produces N1 - N0 + 1 = 4236 fundamental graph cycles. These cycles generate the integral cycle group of the one-skeleton. Because h is closed, its periods on the fundamental cycles determine its cohomological period homomorphism. The source-selected harmonic line was independently reconstructed using two deterministic shift-invert routes. The two selected lines agree with absolute cosine equal to 1.0000000000 to displayed precision. Their L2 difference is approximately 1.88 x 10^-11, and the relative difference between their fundamental-cycle period vectors is approximately 1.91 x 10^-11. A conservative normalized-period comparison tolerance of 1 x 10^-8 is declared. Continued-fraction rational reconstruction was then used to test bounded arithmetic heights. At the declared tolerance, primitive integral period vectors with maximum absolute fundamental-cycle integer at most 10^4 are excluded. For example, one normalized period is approximately 0.2520321060. The nearest rational with denominator at most 10^4 is 2512/9967, whose distance is approximately 4.01 x 10^-7, well above the declared numerical tolerance. At denominator bound 10^5, the numerical separation is no longer sufficient for a certified exclusion under the same tolerance. The finite-height calculation therefore does not prove irrationality of the selected line at arbitrary arithmetic height. This limitation is fundamental: projective integrality is an exact arithmetic property, and rational directions of increasing height are dense in real projective space. Floating-point agreement alone cannot provide an all-height irrationality proof. The audit therefore derives an exact lower-dimensional decision route. Let A denote the 15-channel parent-bivector edge source and P_H the counting-metric harmonic projector. Since the locked incidence matrices are integer, the centered parent-coordinate source is rational, and the harmonic subspace admits a rational basis, the exact operator G = A^T P_H A is a 15 x 15 matrix over Q. The nonzero eigenvalues of the full sign-free harmonic source operator K_wedge = P_H A A^T P_H are exactly the eigenvalues of G. The projected source has full channel rank 15 and its leading eigenvalue is simple. Under these conditions, the source-selected harmonic line contains a nonzero rational vector if and only if the simple leading eigenvalue of G is rational. Therefore a rigorous proof that the leading eigenvalue of exact G is irrational would exclude any nonzero rational or integral cohomology representative on the selected line. Conversely, if the leading eigenvalue is rational, an exact rational eigenvector can be constructed and the existence of a primitive integral representative can then be tested directly. The current double-precision leading value, lambda1 approximately 3336.59358698, is not itself evidence of irrationality. The principal status is: CPH1INT001_INTEGRAL_FLUX_CLOSURE_NOT_CERTIFIED_LOW_HEIGHT_RATIONALITY_EXCLUDED_NUMERICALLY The result therefore does not establish an integral source-selected cohomology class and does not establish native flux quantization. It also does not prove all-height non-integrality. A physical flux quantum, gauge coupling, dimensional scale, physical gauge field, Standard-Model interaction, physical six-dimensional spacetime, continuum limit, or empirical validation is not derived. No parameter search, threshold optimization, target-versus-null specificity simulation, post-result rescue, or empirical fitting is performed. The next exact mathematical step is to construct the 15 x 15 Gram operator G over Q using exact or modular sparse arithmetic, compute and factor its exact characteristic polynomial, and determine whether its simple leading eigenvalue is rational. No flux quantum or physical coupling may be introduced merely to force an integral result.