Generalized DCCQ: From Binary Quotients to Multinomial Simplex Geometry and Critical-Strip Coordinates

Notation at a glance

• Bold symbols denote vectors, ≔ introduces definitions, and logarithms are natural; m counts independent probability degrees of freedom among m+1 categories.The notation also names the simplex, open critical strip, full map Φ_m, and one-versus-rest map R_m.Source: Notation at a glance · p. 1

1 Introduction

• The question is how to extend binary DCCQ to the m-dimensional simplex while retaining every multinomial degree of freedom and determining the resulting complex-strip geometry.The paper takes established count-ray/simplex identification and log-ratio coordinates as its starting point.Source: §1 · p. 2
• The paper also introduces a barycentrically anchored one-versus-rest map to a vertical line and an integer-lattice realization on rational simplex points.These are listed as distinct contributions alongside the full coordinate map.Source: §1 · p. 2
• The dimension ladder gives a critical line for m=1, the full open critical strip for m=2, and the strip with m−2 additional real coordinates for m>2.The ternary simplex is real-analytically diffeomorphic to the open strip.Source: §1 · p. 2

2.1 Ordered multinomial kernel

• The ordered-word kernel omits the multinomial multiplicity coefficient and separates the distinguished mass from the unrestricted conditional composition of the remaining categories.The uniform conditional split is imposed only in the later one-versus-rest submodel.Source: §2.1 · p. 3

2.2 Full multinomial DCCQ coordinate

• The full coordinate map combines the distinguished mass and one oriented log-ratio contrast as a complex coordinate, retaining the other m−2 contrasts as real coordinates.This complex-coordinate choice is not forced by the kernel, but the inverse-map proof establishes that it loses no probability information.Source: §2.2 · p. 4

3.1 Count-ray quotient and simplex reconstruction

• Positive count-ray classes correspond to rational simplex points, dense in the open simplex, while normalization discards common integer scale.The Euclidean closure of the rational simplex points is the full closed simplex.Source: §3.1 · p. 5

3.2 The DCCQ coordinate as a simplex chart

• For m ≥ 2, Φ_m is a real-analytic diffeomorphism, combining a bounded mass and one oriented log-ratio into a complex coordinate while retaining m−2 real contrasts.The underlying real tuple and simplex-to-Euclidean equivalence are standard; the DCCQ-specific step is the complex construction and its dimensional and symmetry interpretation.Source: §3.2 · pp. 5–6
• The chart depends on the distinguished coordinate, reference component, and first contrast; category permutations give equivalent choices, but no canonical choice is asserted.A preferred permutation-equivariant construction is left as a separate question.Source: §3.2 · p. 6

4 One-versus-rest DCCQ map and characteristic lines

• The one-versus-rest family fixes the conditional split among the remaining categories, making it a proper submodel for m > 1 and the full binary model for m = 1.On the equal-pair ternary family, grouping the last two categories is reversible because their conditional split is fixed; this differs from literal restriction of the full map.Source: §4 · pp. 7–8
• For every m ≥ 1, R_m maps (0, 1) real-analytically and bijectively onto the line ℜs = 1/(m+1), with the simplex barycenter at zero height.The map shares the binary logit dependence; m shifts the horizontal anchor and the height origin.Source: §4 · p. 7

4.1 Reflection, conjugation, and binary case

• For σ ≠ 1/2 and t ≠ 0, conjugation and reflection generate four points, exchanging the lines L_σ and L_{1−σ}.The two operations coincide only when ℜs = 1/2.Source: §4.1 · p. 8
• Only the binary member has its characteristic line coincide with its reflected partner; for m > 1, the lines ℜs = 1/(m+1) and ℜs = m/(m+1) are distinct.These are coordinate-geometric statements, not claims about zeros.Source: §4.1 · p. 8

5 The ternary simplex and the full critical strip

• The ternary map Φ2(p) = p0 + i log(p+/p−) is a real-analytic diffeomorphism from the open simplex to the open critical strip.Its inverse is positive and sums to one, and both directions are real analytic.Source: §5 · p. 9
• A fixed-p0 slice maps to the vertical line ℜs = p0, while changing p+/p− changes the height; the ternary barycenter maps to 1/3 + i0.The mass p0 is the real-coordinate role, while p+ and p− form the oriented contrast.Source: §5 · p. 9
• The map is related to log-ratio coordinates but is not an isometric log-ratio transformation: its real component is mass and its imaginary component is one log-ratio.Source: §5 · p. 9
• The ternary case has no residual contrast coordinates; for m > 2, the full geometry retains m−2 additional real contrasts.The full coordinate space is represented as 𝒮 × ℝ^(m−2).Source: §5 · p. 9

6 Integer-lattice realization of the ternary strip coordinate

• Positive integer counts map to a lattice point whose doubled subareas are n(k0, k+, k−), so count and area vectors represent the same positive ray class.The standard triangle is T_n = conv{(0, 0), (n, 0), (0, n)} and x_k = (k+, k−).Source: §6 · p. 10, Figure 3 · p. 11
• For k = (2, 3, 1), both count and area coordinates give Φ2 = 1/3 + i log 3.The doubled subareas are (12, 18, 6).Source: §6 · p. 10
• Scaling counts by g scales lattice areas by g² while Φ2 stays fixed, so this ray-based realization does not preserve absolute lattice area.Direct count substitution computes the coordinate without an area calculation; the lattice theorem supplies its exact geometric realization.Source: §6 · p. 10

Hyperbolic representation

• The hyperbolic construction re-expresses the ternary log-ratio rather than adding a degree of freedom.For integer representatives, positive pair-ray classes correspond to rational ratios; extending the ratio to all positive reals gives the full right branch by density and continuity.Source: §7 · p. 11
• For the same ternary ray, the inverse ratio r = (X + Y)/(X − Y) returns 3; exchanging k+ and k− leaves X fixed, negates Y, and sends r to 1/3, making orientation explicit.Source: §7 · p. 11
• A fixed-mass slice corresponds to one full hyperbola branch; the whole simplex requires both factors, and signed recovery needs (X, Y) or an orientation label.The coordinate identifications are real-analytic, not claims of lattice or metric isometry.Source: §7 · p. 12
• The signed sector bounded by OE, OP, and the hyperbola arc has area A_H(h) = h/2, whereas the straight triangle OEP has signed area sinh(h)/2.The common factor of two follows from the planar area formula; the curved and straight regions are not the same, and the correspondence is not area-preserving.Source: §7 · p. 12, Figure 4 · p. 12
• Exchanging k+ and k− negates the log-ratio and signed sector area, while the even coordinate X = cosh(t/2) is proposed for further study outside this paper’s scope.The figure marks the straight triangle OEP with a dashed chord for comparison with the curved sector.Source: §7 · p. 12, Figure 4 · p. 12

Transverse symmetry defect

• D(s) = 0 exactly when Re s = 1/2, where the conjugation and reflection images coincide.The defect is the unsigned separation of the two symmetry images, rather than a failure of either symmetry.Source: §8 · p. 13
• The defect’s minimum is 0 on the critical line, and its unattained supremum is 1 as σ approaches either strip boundary.Height cancels because (1 − s) − conjugate(s) = 1 − 2σ.Source: §8 · p. 13
• For the barycentric family, D1 = 0, D2 = 1/3, D3 = 1/2, and Dm approaches 1 as m grows.In the ternary strip, D = |1 − 2p0|; the measure is independent of t and does not record which side of the critical line contains the point.Source: §8 · p. 13
• The one-versus-rest characteristic lines approach a strip boundary in this normalized defect scale, with Dm = 2am.The defect and barycentric offset encode the same displacement; for the ternary case, the reflection axis is p0 = 1/2.Source: §8 · p. 13

Relation to standard critical-strip symmetry

• A non-real zero of xi away from the axis has reflected and conjugate partners, while the Riemann Hypothesis asserts that all nontrivial zeros lie on Re s = 1/2.These are stated as standard xi symmetries and the RH claim, not as consequences of the coordinate construction.Source: §9 · p. 13
• The DCCQ construction is a coordinate model: R2(p) = 1/3 + it does not imply that 1/3 + it is a zero of xi.Neither the lattice realization nor the hyperbolic representation identifies a DCCQ object with xi.Source: §9 · p. 13
• Reflection symmetry pairs Lσ with L1−σ; selecting the critical axis as a zero-bearing set requires analytic information beyond coordinate symmetry.The special property of L1/2 is its coincidence with its reflected partner and, pointwise, C(s) = J(s).Source: §9 · p. 13

10 Positioning, scope, and limitations

• The proposed contribution organizes established ingredients into a complex-strip coordinate construction and a reflection/conjugation interpretation, without making an exhaustive priority claim.The paper treats simplex, closure, log-ratio, lattice, area, hyperbolic, and zeta/xi-symmetry ingredients as established.Source: §10 · p. 14
• The paper does not claim zeros on the one-versus-rest lines, an identification with xi, or a proof of RH; integer-ray sets are not the full continuous model.The coordinate correspondence also does not preserve lattice area or an unspecified metric.Source: §10 · p. 14

11 Discussion and further directions

• Alphabet-indexed barycentric offsets approach 1/2, while the ternary model realizes every unsigned offset |p0 − 1/2| in [0, 1/2).The discussion also leaves preferred strip coordinates or a permutation-equivariant atlas as possible later applications.Source: §11 · p. 14
• Integer normalized-volume realizations cover rational interior compositions, a dense subset of the continuous chart image; real compositions give the full target.The paper frames later extensions as requiring choices of scale information, observable, and a mechanism for selecting a spectral set; Φm fixes none of these.Source: §11 · p. 14

12 Conclusion

• Integer compositions modulo common scaling determine rational simplex points; a primitive representative records the ray, while its gcd multiplier restores the original integer scale.The Euclidean closure of the rational simplex is the closed simplex.Source: §12 · p. 15
• The dimensional hierarchy is m = 1 critical-line coordinates, m = 2 the full strip, and m > 2 the strip with m − 2 additional real contrasts.The one-versus-rest family is a Bernoulli submodel with a fixed conditional split; its map, the full-map restriction, and the grouped binary map are distinct operations.Source: §12 · p. 15
• In the ternary case, lattice subareas recover the rational point and log-ratio, while the hyperbolic representation retains mass and orientation using area ratios rather than absolute areas.For ordinary signed planar sector area, t = 4A_H; analytic zero selection remains separate from these coordinate interfaces.Source: §12 · p. 15

Declaration of generative AI use

• The author reports using ChatGPT as a research assistant and states that they reviewed and verified the mathematics, citations, and final text.The author takes full responsibility for the publication.Source: Declaration of generative AI use · p. 15

• The parent triangle T_n has doubled area n^2.Source: §A.2 · p. 15
• The opposite subtriangles have doubled areas nk+, nk−, and n(n−k+−k−) = nk0, respectively.Source: §A.2 · p. 15

• The dimension m indexes m+1 categories, while n is the total count and lattice side-length scale; count vectors are identified by normalized composition.Source: B Notation reference · pp. 15–16
• Probability vectors lie in the closed simplex, whose relative interior has positive components; conditional probabilities q_j yield log-ratios u_j = log(p_j/p_m).Source: B Notation reference · p. 16
• For m ≥ 2, Φ_m maps to the open critical strip times ℝ^(m−2), with strip coordinate s = p0 + i u1.Source: B Notation reference · p. 16
• The one-versus-rest embedding assigns p to category 0 and (1−p)/m to each other category; R_m uses its shifted logit as the imaginary coordinate.Source: B Notation reference · p. 16
• The reference also defines conjugation and reflection involutions, symmetry defects, the right hyperbola branch and sector area, and standard zeta-zero and permutation notation.Source: B Notation reference · pp. 16–17
• The lattice triangle T_n uses scale n, and its doubled parent and opposite subtriangle areas sum to the parent area.Source: B Notation reference · p. 16