M15c The Calculus of Charts - Transport, Iteration and the Comma

Abstract

The Theory of Operations studies arithmetic operations as members of structured rank hierarchies. An Aczél chart converts an associative operation into ordinary addition; a room is one level of the Symmetric Core obtained from an iterated-logarithm chart; the Transport Theorem carries classical algebra and analysis through such a chart; and the Basal-Regular Comma measures the difference between two natural Abel coordinates of the same rank-four iteration. M15c uses these ideas to determine which proposed operational calculi are genuinely new and which are classical calculus written in transformed coordinates. M15c is the phoenix replacement for M15, The Parameter Frontier, and M15b, Native Calculus for the Symmetric Core. It preserves their viable programmes, corrects their formulas and taxonomy, withdraws failed claims, and rebuilds both around the chart of an operation. M15c should therefore be used as the current source wherever it differs from M15 or M15b. The foundational result is the Transport Theorem. Every continuous, cancellative and associative operation on a real interval can, under the stated hypotheses, be represented as x op_phi y = phi^(-1)(phi(x) + phi(y)). Once the chart phi is fixed, ordinary addition, multiplication, differentiation, integration, limits, characters and special-function identities can be transported into the new arithmetic. M15c distinguishes two forms of this construction: full transport, which transforms both arguments and values, and half transport, which transforms only the arguments. It also separates the operators previously grouped under “native differentiation” into three different types: derivations, homomorphic derivatives and chart slopes. This result both validates and limits the programme of M15b. A native calculus exists not only for iterated logarithms but for every admissible chart. However, a calculus transported through one chart is isomorphic to classical calculus and cannot by itself produce new analytical theorems. The genuinely new mathematics begins only between different charts, in the choice among inequivalent Abel charts, or when an operation has no chart at all. For the Symmetric Core, room k is represented by the chart ell_k = ln^[k] and carries the operation x odot_k y = exp^ell_k(x + ell_k(y)). Each adjacent pair of rooms forms a transported field. Their invariant differential operator is the Haar derivation X_k = x ln(x) ln(ln(x)) ... ell_(k-1)(x) d/dx. This corrects the indexing of M15b and identifies its “depth-n calculus” as the calculus of the field formed by rooms n-1 and n. Gamma, trigonometric and Fourier identities survive only when every ordinary product, sum and difference is replaced by the corresponding transported operation. The logaderivative Delta^(-)f = -ln|f'| becomes the hinge of the ladder. A chart’s velocity is obtained from its logaderivative, the sequence of room fields follows from the logaderivative’s chain rule, and the linearisation of the logaderivative at a chart is the negative of that chart’s Haar derivation. This explains why several apparently independent derivative ladders in the earlier corpus repeatedly produced the Euler operator. Between charts, M15c introduces the Schwarzian of an operation as a chart invariant. For the iterated-logarithm rooms it has the explicit form S_k = (1/2) sum_(j<=k) w_j^(-2). On the span of the iterated logarithms, this invariant determines the operation. Two adjacent room fields generate the finite-dimensional affine Lie algebra, while every non-adjacent pair, and every triple, generates an infinite-dimensional Lie algebra. Continuous rooms are constructed through Kneser iteration, but they remain dependent on the selected continuation. At rank four, the same chart analysis is applied to the Half-Caterpillar and the TetraEtage. The Half-Caterpillar has two natural Abel coordinates: the Koenigs coordinate at the attracting endpoint and the Böttcher coordinate at the superattracting endpoint. In these coordinates the TetraEtage field behaves as room one at the upper end and as room two at the lower end. The earlier claim that its derivative simply forms a third generator rung is therefore replaced by an interpolation between two established rooms. The two Abel coordinates differ by the Basal-Regular Comma, a small periodic function that ordinary transport cannot detect. At the canonical base sqrt(2), the difference in function values is approximately 5 x 10^(-8), but differentiation amplifies it through the Schröder period. By the fifth height derivative, the step map alone no longer determines the derivative. Along the étage line the comma follows an exponential strip law, vanishing at the Symmetric-Core end and growing toward the TetraCore end. M15c also distinguishes three geometric classes of operations: Aczél operations, whose calculus is entirely chart transport; flat operations with a warp, including Cpow; curved webs, including Gauss’s arithmetic-geometric mean and the rank-four HyperCore mean. The obstruction is measured by web curvature. For means, flatness is equivalent to bisymmetry under the stated regularity assumptions. Operations without a suitable chart therefore require genuinely different mathematics rather than another transported calculus. The parameter programme of M15 is likewise rebuilt. A parameter root is meaningful only relative to a declared family of operations. A real rank is therefore not an intrinsic number unless a rank interpolation has first been selected. M15’s reported operational rank of (2,3,9) is recovered as log_2(9) for its particular interpolation, rather than as a universal rank coordinate. M15c also develops base derivatives of Abel functions, the Koenigs logarithm, argument rate, branch steering and finite steering horizons. The final part connects chart geometry with inference from observed dynamics. A law can be affine in at most two rooms. Only the difference between the law’s room and the observer’s reporting frame is identifiable without fixing the frame independently. For room-linear hidden systems, delay coordinates recover an affine law in the correct room, with minimal delay order equal to the observability rank. The monograph introduces mean-square, worst-case and room-valued observation defects, with the conditional quasi-arithmetic mean as the optimal room-valued report. An appendix resolves the “Symmetric-Core zeta function” of M15b. The function Z_pp(s) = product_p (1 - exp(-p s))^(-1) is not a new depth-dependent zeta function. It is the prime-partition series, equivalently the generating function for partitions into primes and a series graded by the sum of prime factors. It is zero-free on Re(s) > 0, has period 2 pi i, and has the imaginary axis as a natural boundary. Its relation to the Riemann Hypothesis is a reformulation of the classical von Koch criterion as a boundary-growth statement, not a new route to proving the hypothesis. M15c’s central conclusion is restrictive but productive: one chart reproduces classical analysis, while new mathematics occurs between charts, in the unresolved choice of chart, or where no chart exists. The monograph turns that principle into a unified calculus of transported operations, chart invariants, iteration coordinates, parameter directions and observational frames. Supersession notice M15 is superseded as the principal source for operational rank, parameter roots and angular calculus. M15b is superseded as the principal source for native Symmetric-Core calculus, transported special functions and the proposed SC Riemann programme. M15c is the authoritative replacement, retaining surviving results while correcting their indexing, operations, status and classical attribution.