# The Hemispherical Quantum Selector Concept: Michael Aaron Loftus Expanded mathematical specification HQS 1.0 · 2026-10-10 Michael Aaron Loftus’s continuous voting concept, formalized with signed rotation, volumetric weight, orthogonal priority, sensitivity equations and a deterministic interactive laboratory. Research proposal, not peer-reviewed validation or a quantum-computing implementation. > Multidimensional Quantum Customizable Selector with Size of Hemisphere relative to its respective initial size Equal to Weight of Vote Casted; Temperature Indicative of Rotation up to 360° of Approval (Clockwise) or Denial (Counterclockwise) and Rotation Across X and Z axis Given Rotation About Y=0 a Priority Function Original manuscript: https://digitalmarketingco.org/documents/quantum-selector/loftus-hemispherical-selector-original-manuscript.pdf ## 01 / A richer language of choice Inventor’s proposal · explicit formalization A binary button records a decision but usually hides its intensity, its assigned importance, and the extent to which the person is prepared to allocate a scarce preference budget. Michael Aaron Loftus proposes a different interface: a hemisphere whose scale encodes weight, whose signed rotation encodes approval or denial, and whose orthogonal tilts express priority. The central idea is not that geometry makes a preference true. It is that geometry can expose several otherwise conflated components of a choice.[^1] This white paper develops the supplied fourteen-page manuscript, “Hemispherical Projection of the Four-Dimensional Sphere for Continuous Gesture-Based Weighted Selection.” Attribution to Loftus identifies the author of the supplied concept; it is not an independent patent search, novelty opinion, priority determination, peer-review finding, or proof of quantum advantage. The five-dimensional model below is an explicit engineering extension, not a claim that the manuscript already specifies every implementation rule. The source uses C = X = Z, cubic scaling, and a signed temperature/rotation analogy, but it does not completely fix domains, units, a rotation order, a neutral state, aggregation, or the effect of negative priority. This edition supplies those missing conventions openly. In particular, it follows the latest brief by assigning signed Y rotation to approval and X/Z tilts to a distinct priority function. The manuscript’s alternative interpretation of rotational priority is acknowledged rather than silently treated as identical. “Quantum” is retained as the invention’s name and its multidimensional design metaphor. The executable model uses ordinary real-number arithmetic. No qubits, measurement postulates, physical superposition, quantum entanglement, or quantum hardware are implemented. A continuous state space is not, by itself, a quantum system. ### Equation 1: Five independent interaction variables $$ \mathbf d=(s,\theta,\alpha,\beta,\kappa)\in[0,2]\times[-2\pi,2\pi]\times[-\pi/2,\pi/2]^2\times[0,1] $$ Scale, signed approval rotation, two priority tilts, and priority gain define the proposed five-dimensional interaction state. Angles in equations are radians; controls and exported state angles are degrees. ### Equation 2: Independent state is not independent spacetime $$ \dim(\mathbf d)=5,\qquad \dim S^3=3,\qquad S^3\subset\mathbb R^4 $$ Five adjustable values do not imply five physical spatial dimensions. A constrained sphere embedded in four coordinates has only three local degrees of freedom. A three-dimensional picture cannot uniquely expose all five controls. ## 02 / From four coordinates to a hemisphere Established geometry · proposed restriction Begin with the unit 3-sphere in four-dimensional Euclidean space. Stereographic projection removes one pole and maps the remainder into three-dimensional Euclidean space. The inverse maps every finite three-vector back to a point of unit norm in four coordinates. This is classical geometry. The Harvard exposition motivates the three-sphere and Hopf construction; the projection derivation extends the familiar sphere formulas algebraically to three target coordinates.[^2][^3] Stereographic projection does not automatically produce a hemisphere. To make the interface, we deliberately select an upper hemispherical shell in the three-dimensional target, or its solid interior when discussing volume. That selection is an additional constraint, and it discards information from the unrestricted target. Likewise, displaying a shell and computing the volume enclosed by it are different operations. The live four-coordinate readout lifts the normalized tip vector v = p/r₀. It is a geometric embedding of that tip only, not an encoding of the complete ballot. In particular, the tip does not depend on the yaw angle θ or the priority gain κ. Preserve the full five-variable state in any exported record; reconstruction from the picture or the four-coordinate lift alone is impossible. At scale zero the hemisphere degenerates to a point. Its contribution is zero, its visual normal remains a remembered orientation, and the inverse projection remains defined at (0,0,0,−1). We never divide by the current radius. The excluded stereographic pole is approached only as the target norm tends to infinity; the bounded demonstrator cannot reach it. ### Equation 3: The unit three-sphere $$ S^3=\{\mathbf q\in\mathbb R^4:q_1^2+q_2^2+q_3^2+q_4^2=1\} $$ The constraint removes one degree of freedom from the four coordinates. ### Equation 4: Forward and inverse projection $$ \mathbf v=\frac{(q_1,q_2,q_3)}{1-q_4},\qquad \mathbf q=\frac{(2v_x,2v_y,2v_z,\|\mathbf v\|^2-1)}{1+\|\mathbf v\|^2} $$ Forward projection excludes q₄ = 1. The inverse is defined for every finite v. Substitution verifies unit norm; this coordinate convention fixes which pole is removed. ### Equation 5: Hemispherical shell and solid $$ H_r=\{\mathbf h:\|\mathbf h\|=r,\ h_y\ge0\},\quad B_r^+=\{\mathbf h:\|\mathbf h\|\le r,\ h_y\ge0\} $$ The visible shell is two-dimensional; the enclosed solid is three-dimensional. Their area and volume have different units. ### Equation 6: A surface parameterization $$ \mathbf h(r,\phi,\psi)=r(\sin\phi\cos\psi,\cos\phi,\sin\phi\sin\psi),\quad 0\le\phi\le\pi/2,\quad0\le\psi<2\pi $$ The polar angle is measured from +Y, not from +Z. This intentional axis convention makes the crown point r eᵧ. Longitude at the crown is redundant. ## 03 / Size is a policy, not a synonym Established volume calculus · explicit weighting rule The requested equality between relative size and vote weight requires us to say what size means. We choose relative enclosed volume as the default because the manuscript explicitly gives V = V₀s³. Doubling the radius therefore multiplies the default weight by eight, not by two. A user seeking twice the voting weight should increase radius by the cube root of two. The interface displays radius, area ratio, volume ratio, and configured weight separately so that this distinction is visible.[^4] The general exponent m describes an alternative policy: m = 1 gives relative radius, m = 2 gives relative area, and m = 3 gives relative volume. These are selectable comparison modes in the laboratory, not interchangeable descriptions of one quantity. The default paper examples use m = 3. The curved hemisphere area is 2πr²; including the base disk gives 3πr², and either area ratio still scales as s². Increasing a nonnegative scale raises the magnitude of a nonzero vote; decreasing it reduces that magnitude. It does not reverse approval into denial. There is no negative radius control. A zero-weight submission is not automatically equivalent to a neutral, present voter or to a missing record: an application must explicitly choose how each enters turnout and abstention counts. Reference radius r₀ sets the unit of the drawing, not the person’s political power. Keeping s fixed while changing r₀ changes lengths and absolute volume but leaves the normalized weight unchanged. If a system instead holds physical r fixed, changing r₀ changes s and thus weight; the calibration policy must therefore travel with the record. User-selectable weighting is appropriate for a sandbox, not an unexplained entitlement to dominate others. ### Equation 7: Volume from the spherical Jacobian $$ V=\int_0^{2\pi}\!\int_0^{\pi/2}\!\int_0^r\rho^2\sin\phi\,d\rho\,d\phi\,d\psi=\frac{2\pi r^3}{3} $$ The volume element contains ρ² sin φ; omitting the Jacobian would produce the wrong scaling law. ### Equation 8: Radius, area and volume ratios $$ s=\frac r{r_0},\quad\frac{A}{A_0}=s^2,\quad\frac V{V_0}=s^3,\quad W_m=s^m $$ W is dimensionless. Physical volume is not silently used as a dimensionless vote. ### Equation 9: Target weight and elasticity $$ s=W^{1/m},\quad\frac{dW}{ds}=ms^{m-1},\quad\frac{d\log W}{d\log s}=m\quad(s>0) $$ For m = 3, a small relative radius change produces approximately three times that relative weight change. The logarithmic formula is undefined at s = 0. ### Equation 10: Finite rather than infinitesimal scaling $$ \Delta W=(s+\Delta s)^m-s^m;\qquad m=3:\ \Delta W=3s^2\Delta s+3s(\Delta s)^2+(\Delta s)^3 $$ The cubic terms matter for large pinches and spreads. The new scale must remain in the allowed interval. ### Equation 11: Calibration sensitivity at fixed physical radius $$ \left.\frac{\partial W}{\partial r_0}\right|_r=-\frac{mW}{r_0},\qquad \left.\frac{\partial W}{\partial r_0}\right|_s=0 $$ These apparently conflicting derivatives describe different controlled experiments. The reference radius in the laboratory uses the second convention. ## 04 / Approval has a sign and a memory Proposed bounded signed channel Define clockwise as seen from the positive local Y axis looking toward the origin, with the X–Z reference dial shown before any tilt. Positive θ means approval and negative θ means denial. In standard right-handed rotation matrices, the corresponding geometrical Y rotation is −θ. The on-screen camera can change the apparent direction of a tilted arc; the signed control and reference convention, not a viewer’s impression, determine the stored value. The normalized approval a runs continuously from −1 to +1 over a signed one-turn range. The temperature index T is 100a in illustrative index units: it is not thermodynamic temperature, degrees Celsius, energy, agitation inferred from a sensor, or emotional certainty. Its warm/cool color is redundant decoration; text and signed numbers carry the meaning for color-blind and nonvisual users. A 360° turn has the same orientation matrix as 0°, but it must not have the same vote. Store the signed unwrapped θ, clamped to a one-turn interval, independently of the mesh orientation. Never recover the vote from atan2 alone. At ±360°, additional motion in the same direction is clamped rather than silently wrapping through neutrality. Dragging back changes the stored value continuously. The manuscript’s proportional expression sign(T)·θ becomes nonnegative if both T and θ are already negative. Applying a sign twice would turn denial into approval. This implementation uses a = θ/(2π) exactly once and derives T from a. A positive priority multiplier then changes the magnitude without reversing that sign. Editing temperature would simply be an alternative input for the same angle, not a sixth independent dimension. ### Equation 12: Signed approval and illustrative temperature $$ a=\frac{\theta}{2\pi}\in[-1,1],\qquad T=100a,\qquad\theta=\frac{2\pi T}{100} $$ The degree equivalent is a = θ_deg/360. Positive and negative endpoints remain distinct state values. ### Equation 13: Bounded state update $$ \theta_{t+1}=\operatorname{clip}(\theta_t+g_\theta\Delta x,-2\pi,2\pi) $$ Horizontal displacement uses a documented gain. Clipping is a many-to-one operation: motion outside the bounds is intentionally discarded. ### Equation 14: Orientation does not retain turn history $$ R_y(0)=R_y(2\pi)=R_y(-2\pi),\quad a(0)=0,\quad a(2\pi)=1,\quad a(-2\pi)=-1 $$ This is the key reason an orientation-only data model is insufficient. ### Equation 15: Temperature sensitivity $$ \frac{\partial a}{\partial\theta}=\frac1{2\pi},\qquad\frac{\partial T}{\partial\theta}=\frac{100}{2\pi} $$ Derivatives are per radian; divide the full index span by 360 for derivatives per degree. ## 05 / A reproducible three-axis frame Established rotations · selected composition order Rotations do not generally commute. We declare column vectors and the composition R = Rz(β) Rx(α) Ry(−θ), with the rightmost operation applied first. The ordered inputs are not an unspecified “Euler rotation”; another order yields different coordinates even with identical sliders. Exporting the matrix alongside the named angles and model version makes the convention inspectable.[^5] The hemisphere’s material crown starts on +Y. Rotation around its own Y axis changes the surface meridian but not the crown. The tip p consequently isolates X/Z tilt, while a separate meridian point b reveals yaw. Reading only p would miss the entire signed approval channel. The normal n retains unit length even when the scale becomes zero; the material position p becomes the origin. The closed tilt bounds keep the crown’s Y coordinate nonnegative. That statement does not mean every vertex of the tilted hemisphere stays above the global Y = 0 plane: a rigidly tilted hemisphere includes vertices below it. We rotate the whole object without clipping, because clipping would change its volume and violate the chosen weight law. “Hemisphere” here is a body-relative shape. A unit quaternion is exported as another representation of orientation, scalar component first. It is not the same object as the four-coordinate stereographic lift. Both can have unit norm, but their meanings differ. The sign-paired quaternions q and −q represent the same rotation, and neither preserves the signed full-turn voting history. The original angles remain authoritative. ### Equation 16: Declared rotation matrices $$ R_x(\alpha)=\begin{pmatrix}1&0&0\\0&c_\alpha&-s_\alpha\\0&s_\alpha&c_\alpha\end{pmatrix},\quad R_y(\gamma)=\begin{pmatrix}c_\gamma&0&s_\gamma\\0&1&0\\-s_\gamma&0&c_\gamma\end{pmatrix},\quad R_z(\beta)=\begin{pmatrix}c_\beta&-s_\beta&0\\s_\beta&c_\beta&0\\0&0&1\end{pmatrix} $$ c denotes cosine and s with an angle subscript denotes sine, not the scale factor. Here γ = −θ. ### Equation 17: Rotated surface and invariants $$ \mathbf p_h=R_z(\beta)R_x(\alpha)R_y(-\theta)\mathbf h,\quad R^TR=I,\quad\det R=1,\quad\|R\mathbf h\|=\|\mathbf h\| $$ Rigid rotation preserves length, area and volume; it does not alter W. ### Equation 18: Crown coordinates $$ \mathbf n=(-\sin\beta\cos\alpha,\cos\beta\cos\alpha,\sin\alpha),\qquad\mathbf p=r_0s\mathbf n $$ At zero tilts p = (0,r₀s,0). Positive α moves the tip toward +Z. At α = 0, positive β moves it toward −X. ### Equation 19: Yaw-sensitive meridian marker $$ \mathbf b=r_0s(\cos\beta\cos\theta+\sin\beta\sin\alpha\sin\theta,\ \sin\beta\cos\theta-\cos\beta\sin\alpha\sin\theta,\ \cos\alpha\sin\theta) $$ This point starts on +X and is affected by yaw. It separates rotation around the crown axis from a change of the crown position. ### Equation 20: Ordered rotation quaternion $$ q_R=q_z(\beta)\otimes q_x(\alpha)\otimes q_y(-\theta),\quad q_i(t)=(\cos(t/2),\mathbf e_i\sin(t/2)) $$ Quaternion multiplication follows the same rightmost-first order as the matrix product. It is an orientation coordinate, not a probability amplitude. ## 06 / Priority without a hidden reversal Proposed policy · neutral-Y evaluation To implement “rotation across X and Z given rotation about Y = 0,” compute the priority frame R₀ = Rz(β)Rx(α), independently of θ. This zero-Y condition means zero Y rotation, not a zero Y coordinate. The tip at neutral tilt has its maximum positive Y coordinate. Conflating those two meanings would invert the geometry. Assign α and β explicit application meanings, such as urgency and personal relevance, before gathering input. Their normalized average u is signed priority: positive tilts raise it and negative tilts lower it. If one tilt increases while the other decreases by the same amount, u is unchanged even though the hemisphere moves. This is an intentional many-to-one priority map, not evidence that the two directions are physically equivalent. The fifth continuous control κ determines how strongly priority modulates a vote. P = 1 + κu lies between zero and two. At κ = 0, tilts only change the displayed orientation; at κ = 1, the most negative pair suppresses contribution completely. Keeping P nonnegative prevents an important denial from becoming an approval merely because of an additional negative factor. Increasing priority makes an approving vote more positive and a denying vote more negative. Increasing κ amplifies this effect when u is positive and attenuates magnitude when u is negative. At u = 0, changing κ has no effect on the numerical vote. The gain is a declared comparison control in this laboratory; a real consultation should normally fix it for all participants rather than let people choose their own scoring policy. ### Equation 21: Priority evaluated at neutral yaw $$ R_0=R_z(\beta)R_x(\alpha),\quad u=\frac{\alpha+\beta}{\pi}\in[-1,1],\quad P=1+\kappa u\in[0,2] $$ In exported degrees, u = (tiltX + tiltZ)/180. u is signed; P is a nonnegative multiplier. ### Equation 22: Individual directional priority effects $$ \frac{\partial P}{\partial\alpha}=\frac{\partial P}{\partial\beta}=\frac\kappa\pi,\quad\frac{\partial P}{\partial\kappa}=u,\quad\frac{\partial P}{\partial\theta}=\frac{\partial P}{\partial s}=0 $$ Priority is independent of approval and size by construction; that separation is a product decision. ### Equation 23: Cancellation and policy alternatives $$ u(\alpha,-\alpha)=0,\quad u_\lambda=\frac{2}{\pi}[\lambda\alpha+(1-\lambda)\beta],\quad0\le\lambda\le1 $$ The laboratory fixes λ = 1/2. A different λ is a new, versioned policy—not another hidden tuning parameter. Opposite tilts only cancel when λ = 1/2. ## 07 / From a geometric gesture to a number Proposed output semantics · dimensional consistency The baseline weighted expression is B = Wa. The priority-adjusted contribution is C = WaP. Keep both in the report: a disagreement about P should not erase the original approval or the assigned weight. C is a score under the stated rule, not a probability, a truth estimate, a unit of wellbeing, or a calibrated intensity comparable across people without empirical validation. The source’s C = X = Z is implemented as three named output channels that copy the same score: C, X_vote and Z_vote. Physical tip coordinates are exported separately as tip[0], tip[1], tip[2]. In general pₓ ≠ p_z ≠ C, and the units are different. This resolves the source’s identification without falsely asserting a geometric theorem. If physical equality X = Z is genuinely required, impose it as an additional surface constraint. On a sphere the equality selects a curve, not the whole hemisphere. Normalizing two arbitrary values until they agree is a lossy projection. A diagram cannot convert that loss into a one-to-one representation. The demonstrator therefore preserves the unconstrained spatial tip and duplicates only the explicitly labeled score channels. A neutral angle gives C = 0 at every scale and tilt. A zero weight gives the same number for a different reason. A zero multiplier produces yet another zero. The explanation panel identifies these cases; a single score alone cannot distinguish neutrality, abstention, suppression and opposing aggregate votes. ### Equation 24: Baseline and final contribution $$ B=s^m\frac\theta{2\pi},\qquad C=s^m\frac\theta{2\pi}\left(1+\kappa\frac{\alpha+\beta}{\pi}\right) $$ Default m = 3 honors the manuscript’s volumetric weight. C is dimensionless. ### Equation 25: Range and sign preservation $$ |C|\le2\,2^m,\qquad \operatorname{sgn}(C)=\operatorname{sgn}(\theta)\quad(s>0,\ P>0) $$ The configured scale limit is 2. At s = 0 or P = 0 the contribution vanishes, so the strict sign relation does not apply. ### Equation 26: Score aliases, not spatial coordinates $$ X_{\mathrm{vote}}:=C,\quad Z_{\mathrm{vote}}:=C,\qquad (p_x,p_y,p_z)\text{ remains a separate vector} $$ The colonequals symbol declares an assignment. It does not claim that a physical coordinate equals a vote. ### Equation 27: Physical equality is a lower-dimensional restriction $$ p_x=p_z=t,\quad p_y=\sqrt{r^2-2t^2},\quad |t|\le r/\sqrt2 $$ On the world-upper spherical shell this equality traces a semicircle in a plane. It is not satisfied by arbitrary tilts. ## 08 / Every variable, in both directions Analytical consequences · interior derivatives Differentiate the declared scoring function before interpreting its controls. All derivatives below hold with other inputs fixed in the interior of their domains. At a clamp, outward motion has zero effect; inward motion uses the interior slope. A hard clamp is not differentiable at its threshold. The laboratory reports sensitivities per degree for angles to match its controls, while these displayed equations use radians. Positive scale changes amplify either sign when aP is nonzero; negative scale changes attenuate either sign. Positive θ changes always move C upward or leave it fixed; negative changes move it downward. Positive α or β changes increase an approval and deepen a denial; negative changes soften each. Increasing κ follows the sign of au, not simply the sign of a. These interactions are why a slider label alone is not a sufficient explanation. For the tip geometry, α raises z monotonically over the bounded interval, while the sign of its effect on x depends on β and the sign of its effect on y depends on α. Increasing β moves x toward the negative side while cos α is nonnegative, but lowers y for positive β and raises y for negative β. Scale expands every nonzero coordinate away from the origin; a negative coordinate becomes more negative. “All coordinates increase during spread” is therefore false as an algebraic statement, although their magnitudes scale together. The mixed partials quantify coupling. An angle becomes more influential at larger scale; priority gain matters more at stronger signed approval. The displayed derivatives are properties of this formula, not measured human responses. A local uncertainty approximation assumes small independent zero-mean input errors; systematic bias and correlated gestures require the covariance form and empirical measurements. Quantization places a practical floor on precision. Floating-point computation and a finely stepped slider do not provide infinite psychological resolution. Report enough digits to reproduce calculations, but do not present six decimal places as six decimal places of validated human certainty. The public export retains raw finite values; the screen rounds for readability. ### Equation 28: Complete score gradient $$ \nabla C=\left(ms^{m-1}aP,\ \frac{s^mP}{2\pi},\ \frac{s^ma\kappa}{\pi},\ \frac{s^ma\kappa}{\pi},\ s^ma u\right) $$ The component order is (s, θ, α, β, κ). A negative approval makes both tilt sensitivities nonpositive. ### Equation 29: Scale curvature $$ \frac{\partial^2C}{\partial s^2}=m(m-1)s^{m-2}aP\quad(s>0) $$ The scale response is convex for approval, concave for denial, and linear in radius-weight mode m = 1. At zero, evaluate the underlying polynomial rather than a singular-looking rearrangement. ### Equation 30: Mixed effects $$ \frac{\partial^2C}{\partial s\partial\theta}=\frac{ms^{m-1}P}{2\pi},\quad\frac{\partial^2C}{\partial\theta\partial\alpha}=\frac{s^m\kappa}{2\pi^2},\quad\frac{\partial^2C}{\partial\alpha\partial\kappa}=\frac{s^ma}{\pi} $$ The β analogues equal the α expressions. Nonzero mixed partials describe genuine interaction in the selected scoring rule. ### Equation 31: Tip Jacobian with respect to tilt $$ \frac{\partial\mathbf p}{\partial\alpha}=r(\sin\beta\sin\alpha,-\cos\beta\sin\alpha,\cos\alpha),\quad\frac{\partial\mathbf p}{\partial\beta}=r(-\cos\beta\cos\alpha,-\sin\beta\cos\alpha,0) $$ Equivalently the α column is r(sin β sin α, −cos β sin α, cos α). The x response is not universally positive or negative. ### Equation 32: Other tip derivatives $$ \frac{\partial\mathbf p}{\partial s}=r_0\mathbf n,\quad\frac{\partial\mathbf p}{\partial r_0}=s\mathbf n,\quad\frac{\partial\mathbf p}{\partial\theta}=\frac{\partial\mathbf p}{\partial\kappa}=\mathbf0 $$ Yaw and gain leave the crown unchanged even while they can change C; spatial coordinates alone are insufficient. ### Equation 33: Local error propagation $$ \operatorname{Var}(C)\approx\nabla C^T\Sigma\nabla C;\quad \Sigma\text{ diagonal}\Rightarrow\sum_j(\partial_j C)^2\sigma_j^2 $$ Σ is an empirically estimated input-error covariance, not provided by the demonstrator. Do not invent confidence intervals without it. ### Equation 34: Quantization bound $$ |\delta C|\lesssim\frac12\sum_j|\partial_jC|\Delta_j $$ Δⱼ is the input grid spacing. This is a first-order local bound, not a global guarantee for large errors. ### Equation 35: Forward sensitivity of the lift $$ \frac{\partial q_i}{\partial v_j}=\frac{2\delta_{ij}}{1+v^Tv}-\frac{4v_iv_j}{(1+v^Tv)^2},\quad\frac{\partial q_4}{\partial v_j}=\frac{4v_j}{(1+v^Tv)^2} $$ For i,j ∈ {1,2,3}, the projection’s Jacobian explains how changing target coordinates changes the four-dimensional representation. δᵢⱼ is the Kronecker delta. ## 09 / Aggregation is a separate invention decision Proposed aggregation · no social-choice guarantee A single gesture is not yet a voting system. To compare several submissions, distinguish the total contribution S from a normalized mean M. The mean divides by total effective nonnegative weight WP and remains in [−1,1] when its denominator is positive. The total grows with the number and strength of submissions; the mean does not measure turnout. When the denominator is zero, return no result, not zero approval. Two equally weighted opposed groups can yield M = 0 despite intense polarization. Report the denominator, number of eligible submissions, abstention policy, distribution of approval, and dispersion alongside the mean. Never interpret cancellation as indifference. Store the raw channels under an appropriate privacy policy so an announced aggregation can be audited without silently replacing participants’ inputs. Unrestricted self-selected weight invites everyone to choose the maximum. A bounded per-person allocation budget can make weight a tradeoff across proposals, but it does not establish strategy-proofness or interpersonal cardinal comparability. A quadratic-cost rule is another policy: its norm makes concentration expensive, but coupling it to cubic radius requires a transparent conversion. Neither policy is implemented as an election backend in this paper. Normalization also changes the effect of one person’s weight. Raising a contributor’s weight moves M toward that contributor’s approval, not necessarily upward. Under fixed weights, raising their approval moves M upward. These are statements about the formula, not normative judgments about whose preferences deserve greater weight. ### Equation 36: Total and normalized mean $$ S=\sum_iW_iP_i a_i,\quad D=\sum_iW_iP_i,\quad M=\begin{cases}S/D&D>0\\\text{undefined}&D=0\end{cases} $$ Nonnegative effective weights ensure the normalized mean stays between the least and greatest submitted approval. ### Equation 37: Influence on the aggregate $$ \frac{\partial M}{\partial a_i}=\frac{W_iP_i}{D},\quad\frac{\partial M}{\partial W_i}=\frac{P_i(a_i-M)}{D} $$ These partials hold P fixed and require D > 0. Increasing the weight of a below-mean opinion lowers the mean. ### Equation 38: Polarization alongside the mean $$ \sigma_a^2=\frac{\sum_iW_iP_i(a_i-M)^2}{D} $$ This dispersion distinguishes a split electorate from shared neutrality. It is not uncertainty of the mean or a population sampling confidence interval. ### Equation 39: Two possible budget constraints $$ \sum_jW_{ij}\le B_i\quad\text{or}\quad\sum_j C_{ij}^{\,2}\le B_i $$ These are alternative proposed budgets, not equivalent mechanisms. The second bounds the squared score rather than the allocated weight. Any implementation needs eligibility, accounting, and declared policy. ## 10 / An instrument everyone can operate Implementable interaction · evaluation still required The laboratory combines a projected three-dimensional hemispherical mesh with a separate control panel. The mesh is rendered from deterministic SVG polygons; it needs neither a graphics driver nor a model API. A fixed orthographic camera is a presentation choice, not another vote dimension. Approval mode uses horizontal drag; tilt mode uses vertical motion for X and horizontal motion for Z. Two-pointer pinch changes scale. A visible mode label makes the current mapping explicit.[^6] Every continuous control also has a labeled numeric field, a native slider, and decrement/increment buttons. Presets and reset can be reached by keyboard. These alternatives matter because dragging and multipoint gestures cannot be the only route to a function. Native scrolling remains available outside the clearly marked interaction surface. No device-motion permission is requested. A production interface should evaluate motor error, accidental mode changes, comprehension of cubic scaling, full-turn memory, and task completion against familiar rating controls. Counterbalance tasks and order; include users with motor and visual disabilities; report learning effects, error distributions, and dropouts. Do not infer usability superiority from the mathematical model or the beauty of the artwork. This is a local demonstration, not a submitted vote. Control changes do not create accounts, transmit selector state, call AI, or write a server record. JSON and source exports are generated in the browser. The surrounding website may still have ordinary page analytics; therefore we do not claim that visiting the entire site creates no network traffic. Runtime arithmetic is local and does not use a metered inference service. ### Equation 40: Pinch update with a nonzero gesture anchor $$ s_{\mathrm{new}}=\operatorname{clip}\left(s_{\mathrm{anchor}}\frac{d_{\mathrm{new}}}{d_{\mathrm{anchor}}},0,2\right),\qquad d_{\mathrm{anchor}}>0 $$ At s = 0 a pinch ratio alone cannot restore size. The numeric field or plus button restores a positive scale; this exception is explained rather than hidden. ### Equation 41: Screen projection is not the vote $$ \mathbf z_{\mathrm{screen}}=A_{2\times3}\,R\mathbf h+\mathbf o $$ A has two rows defining a fixed camera basis. Screen pixels are a lossy two-dimensional display; exported model coordinates are computed before this projection. ## 11 / Real-world applications, with boundaries Hypothetical application designs · not deployments The examples below are proposals for web applications, not claims of existing deployments or measured benefits. They keep approval, weight, and priority separate because each application has a different reason to collect them. A preference can inform a recommendation or consultation without becoming a binding vote.[^7] In dating, signed interest is not consent. A large positive score never licenses contact that the other person has not agreed to receive, and a negative or withdrawn response must override any numerical match. Do not expose identifiable rankings, infer protected characteristics, or punish people for refusing to participate. Mutual opt-in is a separate gate, not a derivative of C. Pre-legislative consultation is a plausible use because intensity, urgency and relevance can be displayed before a bill is drafted. It must be clearly labeled as consultation, not an official election, a representative sample, or a constitutional allocation of voting power. Publish the eligibility method, weighting policy and disaggregated distributions. The U.S. Election Assistance Commission’s guidelines illustrate why actual voting systems require a much broader security, accessibility and testing framework; this laboratory is not EAC certified. ### Equation 42: Mutual permission is a gate, not a high score $$ \mathrm{connect}(i,j)=\mathbf1[\mathrm{optIn}_{ij}\land\mathrm{optIn}_{ji}\land\neg\mathrm{blocked}_{ij}] $$ A matching application must not substitute geometric affinity for the participants’ explicit permission. The gate is illustrative pseudomathematics, not a consent or identity system. ### Equation 43: Consultation distribution, not only a headline $$ F(t)=\frac{\sum_iW_iP_i\mathbf1[a_i\le t]}{D},\qquad D>0 $$ A weighted empirical distribution makes a polarized consultation visible even when its mean is near zero. It does not correct selection bias in who responds. ### 1. Dating discovery Interest in a profile; allocate a limited attention budget; tilts mean readiness and relevance. Boundary: Require mutual opt-in, easy withdrawal, no public person rankings, and no inference that intensity is consent. ### 2. Pre-legislative consultation Support or oppose a draft clause; allocate a declared issue budget; tilts mean urgency and impact. Boundary: Nonbinding, voluntary, and not a population-representative election; disclose sampling and weighting. ### 3. Participatory budgeting Support a local project; assign scarce planning points; tilts mean urgency and neighborhood relevance. Boundary: Keep actual monetary budgets separate and publish accessible alternatives and eligibility rules. ### 4. Product roadmaps Approve a feature direction; allocate a team priority budget; tilts mean urgency and user impact. Boundary: Preserve minority needs and accessibility fixes even when popular features dominate. ### 5. Design review Support a visual direction; assign confidence in one’s review; tilts mean impact and implementation readiness. Boundary: Self-reported confidence is not objective expertise; document who decides. ### 6. Research agenda setting Prefer a research question; allocate review effort; tilts mean feasibility and potential value. Boundary: Do not confuse enthusiasm with evidence quality or predicted discoveries. ### 7. Classroom formative feedback Express comprehension or confusion; weight a self-assessment; tilts mean urgency and topic relevance. Boundary: Not a grade, diagnosis, or ranking of students; offer private accessible responses. ### 8. Course planning Support a proposed module; allocate elective-interest points; tilts mean relevance and timing. Boundary: Avoid inferring that silence means opposition or excluding small specialist cohorts. ### 9. Team retrospectives Support a process change; weight attention available; tilts mean urgency and practical impact. Boundary: Protect against retaliation and small-group reidentification. ### 10. Meeting agendas Prefer an agenda item; allocate available discussion time; tilts mean urgency and decision readiness. Boundary: Minutes are a separate scarce resource; keep an override for safety-critical issues. ### 11. Community moderation policy Support a proposed rule; allocate consultation points; tilts mean urgency and perceived impact. Boundary: Never let popularity alone determine individual punishments or remove appeal rights. ### 12. Creative commissioning Prefer a concept; weight a client’s declared brief priorities; tilts mean timing and brand fit. Boundary: Record qualitative reasons and preserve the designer’s ability to explain tradeoffs. ### 13. Music discovery Approve a track; allocate listening attention; tilts mean mood fit and novelty interest. Boundary: A preference sample is not a psychological diagnosis; keep personalization optional. ### 14. Film and book clubs Prefer a selection; assign available participation; tilts mean relevance and readiness. Boundary: Make content boundaries and accessibility requirements separate veto constraints. ### 15. Travel planning Support an itinerary; allocate planning interest; tilts mean timing and personal relevance. Boundary: Budget, disability access, visas and safety are hard constraints, not scores to average away. ### 16. Restaurant group choice Approve an option; assign attention to the choice; tilts mean urgency and convenience. Boundary: Allergies and dietary restrictions must be enforced before preference aggregation. ### 17. Customer advisory boards Support a service change; allocate advisory points; tilts mean business impact and urgency. Boundary: Do not mistake a self-selected board for the entire customer base. ### 18. Open-source maintenance Prefer an improvement; allocate actual review capacity; tilts mean severity and readiness. Boundary: Security reports and maintainer judgment cannot be reduced to popularity. ### 19. Urban design charrettes Support a street design; allocate an equal consultation budget; tilts mean impact and urgency. Boundary: Provide offline participation and disaggregated results to avoid digital exclusion. ### 20. Environmental project review Prefer an intervention; allocate consultation effort; tilts mean urgency and local relevance. Boundary: Technical evidence and legal protections remain distinct from expressed preference. ### 21. Museum exhibition planning Prefer an exhibition theme; allocate curatorial attention; tilts mean relevance and readiness. Boundary: Avoid treating popular appeal as the only cultural or educational value. ### 22. Nonprofit strategy Support a program; allocate a declared planning budget; tilts mean urgency and mission fit. Boundary: Donor wealth should not silently become beneficiary voice. ### 23. Personal decision journaling Compare options privately; assign one’s own attention; tilts mean urgency and importance. Boundary: Useful as reflection, not a mental-health assessment or automated life advice. ### 24. Accessibility feature prioritization Support an improvement; assign planning effort; tilts mean barrier severity and readiness. Boundary: Accessibility obligations cannot be voted away by a majority that does not experience the barrier. ## 12 / Evidence, privacy and the research frontier Limitations · falsifiable next steps A future production service would need authentication appropriate to the setting, rate limits, anti-duplication rules, announced eligibility and budgets, a versioned scoring policy, consent and withdrawal mechanisms, and a threat model covering coercion and compromised clients. A signed or hashed record can detect changes under specific assumptions; it does not prove that a preference was freely expressed or a human was unique.[^8] Preference vectors can reveal intimate or political information. Use purpose limitation, data minimization, documented retention, restricted access and a clear deletion policy. Aggregate views can still expose people in small groups. The NIST Privacy Framework is a risk-management reference, not an automatic compliance certificate. This page deliberately does not collect real ballots or persist your laboratory inputs. The method’s open questions are empirical: do participants understand the separation of a, W and P; does cubic weighting encourage overconfidence; can the same person reproduce a preference reliably; do tilt semantics transfer across devices; and does a familiar two-dimensional control achieve the same result with fewer errors? A rigorous evaluation would preregister tasks, hypotheses, exclusion criteria and success thresholds before data collection. The current contribution is an explicit, inspectable mathematical and software realization of an attributed concept. Its equations can be reproduced and falsified; its product assumptions can be changed under a new version. It does not establish a quantum mechanism, a patent claim, better collective decisions, universal human preference measurement, or a safe remote-election architecture. Beautiful visualization should invite scrutiny, not replace it. ### Equation 44: A reproducible record $$ \mathcal R=(\mathrm{modelVersion},\mathbf d,m,r_0,\mathrm{units},\mathrm{rotationOrder},\mathrm{policy}) $$ Recompute derived outputs from this record and compare within a stated floating-point tolerance. A future ballot service would add provenance and safeguards, not overwrite the raw inputs. ### Equation 45: A testable prediction error $$ \mathrm{MAE}=\frac1N\sum_{i=1}^N|C_i^{\mathrm{retest}}-C_i^{\mathrm{initial}}| $$ Test–retest error is one possible evaluation metric. No experimental data are supplied, so this paper reports no numerical reliability claim. ## Variable atlas | Identifier | Term | Units | Domain | Meaning | |---|---|---|---|---| | s | Radius scale | dimensionless | 0–2 | Relative radius; zero is a degenerate, zero-weight state. | | θ | Signed approval angle | radians; UI degrees | −2π to +2π | Positive clockwise in the reference frame; retain full-turn sign. | | α | X-axis tilt | radians; UI degrees | −π/2 to +π/2 | First priority tilt; applied before Z tilt. | | β | Z-axis tilt | radians; UI degrees | −π/2 to +π/2 | Second priority tilt; applied last. | | κ | Priority gain | dimensionless | 0–1 | Strength of priority modulation; fifth interaction variable. | | r₀, r | Reference and current radii | model length units | r₀ > 0; r = r₀s | The demo permits r₀ up to 1000; physical calibration is not measured. | | m | Weight-policy exponent | dimensionless | 1, 2 or 3 | Radius, area or volume policy; default 3. | | V₀, V | Enclosed hemisphere volumes | model length cubed | nonnegative | V₀ = 2πr₀³/3 and V = V₀s³. | | A₀, A | Reference and current area | model length squared | nonnegative | Curved area or base-included area; their ratio is s². | | W | Configured vote weight | dimensionless | 0 to 2ᵐ | Equals volume ratio only when m = 3. | | a, T | Approval and temperature index | dimensionless; index units | a: −1 to +1; T: −100 to +100 | T is redundant encoding of a, not a physical temperature. | | u, P | Signed priority and multiplier | dimensionless | u: −1 to +1; P: 0 to 2 | A signed diagnostic and a nonnegative scoring factor. | | B, C | Baseline and adjusted scores | dimensionless | policy-bounded | B = Wa; C = WaP. | | X_vote, Z_vote | Duplicated output channels | dimensionless | equal to C | Aliases for the score, not physical axes. | | ρ, φ, ψ | Spherical coordinates | length; radians; radians | ρ ≥ 0; φ: 0 to π/2; ψ: 0 to 2π | Radial integration coordinate, polar angle and longitude. | | h, p_h | Body and rotated surface points | model length | on the shell | p_h = Rh. | | n, p, b | Unit crown normal, crown tip, meridian marker | unitless; length; length | geometry-constrained | n and p ignore yaw; b reveals it. | | R, R₀, q_R | Orientation matrix, neutral-yaw matrix, quaternion | dimensionless | R ∈ SO(3); unit quaternion | q_R uses scalar-first order and is distinct from the stereographic lift. | | v, q | Normalized tip and four-coordinate lift | dimensionless | v ∈ ℝ³; q ∈ S³ | The lift does not encode all five interaction variables. | | γ, c_t, s_t | Geometric yaw and trig abbreviations | radians; dimensionless | γ = −θ | c_t = cos t; s_t = sin t, not radius scale. | | λ | Alternative tilt balance | dimensionless | 0–1 | Fixed at 1/2 in the demo; changing it requires a new policy. | | Σ, σⱼ, Δⱼ | Input covariance, error deviation, grid spacing | appropriate input units | estimated, not supplied | Needed for uncertainty analysis; no invented confidence intervals. | | δᵢⱼ, eᵢ, I | Kronecker delta, basis vector, identity matrix | dimensionless | algebraic symbols | δᵢⱼ is 1 when indices agree and 0 otherwise. | | S, D, M, σₐ² | Aggregate numerator, denominator, mean, dispersion | dimensionless | D ≥ 0; M undefined at D = 0 | Keep total strength, mean and polarization separate. | | Bᵢ, N, i, j | Allocation budget, sample count and indices | policy units; count | context-dependent | Bᵢ is a budget, distinct from baseline score B. | | gθ, Δx, d | Gesture gain, horizontal motion, pointer distance | radians/pixel; pixels; pixels | documented calibration | Presentation-to-state mapping; no inference about emotion. | | A₂×₃, o, z_screen | Camera projection, screen offset, pixel position | display units | fixed camera | Different A from surface area; the subscript identifies the camera matrix. | | ℛ, MAE | Versioned record and retest error | record; score units | defined in context | A reproducibility contract and proposed evaluation metric, not observed evidence. | ## Computed examples ### Neutral baseline ```json { "model": "hqs-1.0", "state": { "scale": 1, "angle": 0, "tiltX": 0, "tiltZ": 0, "gain": 0.5 }, "config": { "radius": 1, "exponent": 3 }, "radius": 1, "volume": 2.0943951023931953, "referenceVolume": 2.0943951023931953, "volumeRatio": 1, "areaRatio": 1, "weight": 1, "approval": 0, "temperatureIndex": 0, "directionalPriority": 0, "priority": 1, "weighted": 0, "contribution": 0, "outputChannels": { "C": 0, "X_vote": 0, "Z_vote": 0 }, "normal": [ 0, 1, 0 ], "tip": [ 0, 1, 0 ], "meridian": [ 1, 0, 0 ], "matrix": [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ], "rotationQuaternion": [ 1, 0, 0, 0 ], "sphere4": [ 0, 1, 0, 0 ], "derivatives": { "scale": 0, "angle": 0.002777777777777778, "tiltX": 0, "tiltZ": 0, "gain": 0 }, "interpretation": "Neutral approval: priority can change, but the signed contribution remains zero.", "limits": "Classical deterministic demonstration; not a quantum computer, validated preference measure or certified ballot system." } ``` ### Full clockwise approval ```json { "model": "hqs-1.0", "state": { "scale": 1, "angle": 360, "tiltX": 0, "tiltZ": 0, "gain": 0.5 }, "config": { "radius": 1, "exponent": 3 }, "radius": 1, "volume": 2.0943951023931953, "referenceVolume": 2.0943951023931953, "volumeRatio": 1, "areaRatio": 1, "weight": 1, "approval": 1, "temperatureIndex": 100, "directionalPriority": 0, "priority": 1, "weighted": 1, "contribution": 1, "outputChannels": { "C": 1, "X_vote": 1, "Z_vote": 1 }, "normal": [ 0, 1, 0 ], "tip": [ 0, 1, 0 ], "meridian": [ 1, 0, -2.4492935982947064e-16 ], "matrix": [ [ 1, 0, 2.4492935982947064e-16 ], [ 0, 1, 0 ], [ -2.4492935982947064e-16, 0, 1 ] ], "rotationQuaternion": [ -1, 0, -1.2246467991473532e-16, 0 ], "sphere4": [ 0, 1, 0, 0 ], "derivatives": { "scale": 3, "angle": 0.002777777777777778, "tiltX": 0.002777777777777778, "tiltZ": 0.002777777777777778, "gain": 0 }, "interpretation": "Approval contributes 1.000000 weighted units; priority preserves its magnitude.", "limits": "Classical deterministic demonstration; not a quantum computer, validated preference measure or certified ballot system." } ``` ### Full counterclockwise denial ```json { "model": "hqs-1.0", "state": { "scale": 1, "angle": -360, "tiltX": 0, "tiltZ": 0, "gain": 0.5 }, "config": { "radius": 1, "exponent": 3 }, "radius": 1, "volume": 2.0943951023931953, "referenceVolume": 2.0943951023931953, "volumeRatio": 1, "areaRatio": 1, "weight": 1, "approval": -1, "temperatureIndex": -100, "directionalPriority": 0, "priority": 1, "weighted": -1, "contribution": -1, "outputChannels": { "C": -1, "X_vote": -1, "Z_vote": -1 }, "normal": [ 0, 1, 0 ], "tip": [ 0, 1, 0 ], "meridian": [ 1, 0, 2.4492935982947064e-16 ], "matrix": [ [ 1, 0, -2.4492935982947064e-16 ], [ 0, 1, 0 ], [ 2.4492935982947064e-16, 0, 1 ] ], "rotationQuaternion": [ -1, 0, 1.2246467991473532e-16, 0 ], "sphere4": [ 0, 1, 0, 0 ], "derivatives": { "scale": -3, "angle": 0.002777777777777778, "tiltX": -0.002777777777777778, "tiltZ": -0.002777777777777778, "gain": 0 }, "interpretation": "Denial contributes 1.000000 weighted units; priority preserves its magnitude.", "limits": "Classical deterministic demonstration; not a quantum computer, validated preference measure or certified ballot system." } ``` ### Half-size, half approval ```json { "model": "hqs-1.0", "state": { "scale": 0.5, "angle": 180, "tiltX": 0, "tiltZ": 0, "gain": 0.5 }, "config": { "radius": 1, "exponent": 3 }, "radius": 0.5, "volume": 0.2617993877991494, "referenceVolume": 2.0943951023931953, "volumeRatio": 0.125, "areaRatio": 0.25, "weight": 0.125, "approval": 0.5, "temperatureIndex": 50, "directionalPriority": 0, "priority": 1, "weighted": 0.0625, "contribution": 0.0625, "outputChannels": { "C": 0.0625, "X_vote": 0.0625, "Z_vote": 0.0625 }, "normal": [ 0, 1, 0 ], "tip": [ 0, 0.5, 0 ], "meridian": [ -0.5, 0, 6.123233995736766e-17 ], "matrix": [ [ -1, 0, -1.2246467991473532e-16 ], [ 0, 1, 0 ], [ 1.2246467991473532e-16, 0, -1 ] ], "rotationQuaternion": [ 6.123233995736766e-17, 0, -1, 0 ], "sphere4": [ 0, 0.8, 0, -0.6 ], "derivatives": { "scale": 0.375, "angle": 0.00034722222222222224, "tiltX": 0.00017361111111111112, "tiltZ": 0.00017361111111111112, "gain": 0 }, "interpretation": "Approval contributes 0.062500 weighted units; priority preserves its magnitude.", "limits": "Classical deterministic demonstration; not a quantum computer, validated preference measure or certified ballot system." } ``` ### Priority amplification ```json { "model": "hqs-1.0", "state": { "scale": 1, "angle": 180, "tiltX": 45, "tiltZ": 45, "gain": 0.5 }, "config": { "radius": 1, "exponent": 3 }, "radius": 1, "volume": 2.0943951023931953, "referenceVolume": 2.0943951023931953, "volumeRatio": 1, "areaRatio": 1, "weight": 1, "approval": 0.5, "temperatureIndex": 50, "directionalPriority": 0.5, "priority": 1.25, "weighted": 0.5, "contribution": 0.625, "outputChannels": { "C": 0.625, "X_vote": 0.625, "Z_vote": 0.625 }, "normal": [ -0.5, 0.5000000000000001, 0.7071067811865475 ], "tip": [ -0.5, 0.5000000000000001, 0.7071067811865475 ], "meridian": [ -0.7071067811865475, -0.7071067811865476, 8.659560562354934e-17 ], "matrix": [ [ -0.7071067811865475, -0.5, -0.5 ], [ -0.7071067811865476, 0.5000000000000001, 0.4999999999999999 ], [ 8.659560562354934e-17, 0.7071067811865475, -0.7071067811865476 ] ], "rotationQuaternion": [ 0.1464466094067263, 0.3535533905932738, -0.8535533905932737, -0.3535533905932738 ], "sphere4": [ -0.5, 0.5000000000000001, 0.7071067811865475, 0 ], "derivatives": { "scale": 1.875, "angle": 0.003472222222222222, "tiltX": 0.001388888888888889, "tiltZ": 0.001388888888888889, "gain": 0.25 }, "interpretation": "Approval contributes 0.625000 weighted units; priority amplifies its magnitude.", "limits": "Classical deterministic demonstration; not a quantum computer, validated preference measure or certified ballot system." } ``` ### Opposition, lower priority ```json { "model": "hqs-1.0", "state": { "scale": 1.5, "angle": -120, "tiltX": -30, "tiltZ": 0, "gain": 0.6 }, "config": { "radius": 1, "exponent": 3 }, "radius": 1.5, "volume": 7.0685834705770345, "referenceVolume": 2.0943951023931953, "volumeRatio": 3.375, "areaRatio": 2.25, "weight": 3.375, "approval": -0.3333333333333333, "temperatureIndex": -33.33333333333333, "directionalPriority": -0.16666666666666666, "priority": 0.9, "weighted": -1.125, "contribution": -1.0125, "outputChannels": { "C": -1.0125, "X_vote": -1.0125, "Z_vote": -1.0125 }, "normal": [ 0, 0.8660254037844387, -0.49999999999999994 ], "tip": [ 0, 1.299038105676658, -0.7499999999999999 ], "meridian": [ -0.7499999999999997, -0.649519052838329, -1.1250000000000002 ], "matrix": [ [ -0.4999999999999998, 0, 0.8660254037844387 ], [ -0.4330127018922193, 0.8660254037844387, -0.24999999999999986 ], [ -0.7500000000000001, -0.49999999999999994, -0.4330127018922192 ] ], "rotationQuaternion": [ 0.48296291314453427, -0.1294095225512604, 0.8365163037378078, -0.22414386804201336 ], "sphere4": [ 0, 0.7994080650317895, -0.46153846153846145, 0.38461538461538464 ], "derivatives": { "scale": -2.025, "angle": 0.0084375, "tiltX": -0.0037499999999999994, "tiltZ": -0.0037499999999999994, "gain": 0.1875 }, "interpretation": "Denial contributes 1.012500 weighted units; priority attenuates its magnitude.", "limits": "Classical deterministic demonstration; not a quantum computer, validated preference measure or certified ballot system." } ``` ### Equal and opposite tilts ```json { "model": "hqs-1.0", "state": { "scale": 1, "angle": 90, "tiltX": 45, "tiltZ": -45, "gain": 1 }, "config": { "radius": 1, "exponent": 3 }, "radius": 1, "volume": 2.0943951023931953, "referenceVolume": 2.0943951023931953, "volumeRatio": 1, "areaRatio": 1, "weight": 1, "approval": 0.25, "temperatureIndex": 25, "directionalPriority": 0, "priority": 1, "weighted": 0.25, "contribution": 0.25, "outputChannels": { "C": 0.25, "X_vote": 0.25, "Z_vote": 0.25 }, "normal": [ 0.5, 0.5000000000000001, 0.7071067811865475 ], "tip": [ 0.5, 0.5000000000000001, 0.7071067811865475 ], "meridian": [ -0.49999999999999983, -0.5, 0.7071067811865476 ], "matrix": [ [ -0.49999999999999983, 0.5, -0.7071067811865476 ], [ -0.5, 0.5000000000000001, 0.7071067811865475 ], [ 0.7071067811865476, 0.7071067811865475, 4.329780281177467e-17 ] ], "rotationQuaternion": [ 0.5, 5.551115123125783e-17, -0.7071067811865475, -0.5 ], "sphere4": [ 0.5, 0.5000000000000001, 0.7071067811865475, 0 ], "derivatives": { "scale": 0.75, "angle": 0.002777777777777778, "tiltX": 0.001388888888888889, "tiltZ": 0.001388888888888889, "gain": 0 }, "interpretation": "Approval contributes 0.250000 weighted units; priority preserves its magnitude.", "limits": "Classical deterministic demonstration; not a quantum computer, validated preference measure or certified ballot system." } ``` ### Zero weight ```json { "model": "hqs-1.0", "state": { "scale": 0, "angle": 360, "tiltX": 0, "tiltZ": 0, "gain": 0.5 }, "config": { "radius": 1, "exponent": 3 }, "radius": 0, "volume": 0, "referenceVolume": 2.0943951023931953, "volumeRatio": 0, "areaRatio": 0, "weight": 0, "approval": 1, "temperatureIndex": 100, "directionalPriority": 0, "priority": 1, "weighted": 0, "contribution": 0, "outputChannels": { "C": 0, "X_vote": 0, "Z_vote": 0 }, "normal": [ 0, 1, 0 ], "tip": [ 0, 0, 0 ], "meridian": [ 0, 0, 0 ], "matrix": [ [ 1, 0, 2.4492935982947064e-16 ], [ 0, 1, 0 ], [ -2.4492935982947064e-16, 0, 1 ] ], "rotationQuaternion": [ -1, 0, -1.2246467991473532e-16, 0 ], "sphere4": [ 0, 0, 0, -1 ], "derivatives": { "scale": 0, "angle": 0, "tiltX": 0, "tiltZ": 0, "gain": 0 }, "interpretation": "Zero weight: this state contributes nothing.", "limits": "Classical deterministic demonstration; not a quantum computer, validated preference measure or certified ballot system." } ``` ### Priority cancellation ```json { "model": "hqs-1.0", "state": { "scale": 1, "angle": -360, "tiltX": -90, "tiltZ": -90, "gain": 1 }, "config": { "radius": 1, "exponent": 3 }, "radius": 1, "volume": 2.0943951023931953, "referenceVolume": 2.0943951023931953, "volumeRatio": 1, "areaRatio": 1, "weight": 1, "approval": -1, "temperatureIndex": -100, "directionalPriority": -1, "priority": 0, "weighted": -1, "contribution": 0, "outputChannels": { "C": 0, "X_vote": 0, "Z_vote": 0 }, "normal": [ 6.123233995736766e-17, 3.749399456654644e-33, -1 ], "tip": [ 6.123233995736766e-17, 3.749399456654644e-33, -1 ], "meridian": [ 3.061616997868383e-16, -1, 1.4997597826618576e-32 ], "matrix": [ [ 3.061616997868383e-16, 6.123233995736766e-17, 1 ], [ -1, 3.749399456654644e-33, 3.061616997868383e-16 ], [ 1.4997597826618576e-32, -1, 6.123233995736766e-17 ] ], "rotationQuaternion": [ -0.5000000000000002, 0.5000000000000001, -0.49999999999999983, 0.49999999999999994 ], "sphere4": [ 6.123233995736766e-17, 3.749399456654644e-33, -1, 0 ], "derivatives": { "scale": 0, "angle": 0, "tiltX": -0.005555555555555556, "tiltZ": -0.005555555555555556, "gain": 1 }, "interpretation": "Priority multiplier is zero: this configured state suppresses the contribution.", "limits": "Classical deterministic demonstration; not a quantum computer, validated preference measure or certified ballot system." } ``` ## Implementation commission ENGINEERING COMMISSION — HEMISPHERICAL QUANTUM SELECTOR, HQS 1.0 ROLE AND STANDARD Act as a mathematical interface engineer, numerical analyst, accessibility specialist and adversarial software tester. Implement Michael Aaron Loftus’s attributed hemispherical-selector concept as the explicitly defined classical model below. Intellectual depth means falsifiable definitions, correct units, reproducible computation and candid limits—not claims of genius, scientific rank, patent novelty or quantum computation. Read the source manuscript and this specification. Keep inventor-provided ideas distinct from the implementation choices. Do not silently substitute an attractive sphere for the required hemisphere. DELIVERABLE Build an interactive hemispherical mesh, independent controls, a complete numerical report, an interpretation panel, a reproducible JSON export and runnable source. Deliver the mathematics and code, not merely a rendering. Use ordinary local JavaScript/TypeScript and SVG, Canvas or an existing graphics library. The selector must work without an AI model, paid API, hosted inference, external calculation service or backend vote submission. Pre-generated editorial artwork is optional and must not be represented as evidence of physical hardware. CANONICAL STATE Keep exactly five named continuous interaction channels: scale s in [0,2], signed approval angle theta in [-360,360] degrees, X tilt alpha and Z tilt beta each in [-90,90] degrees, and priority gain kappa in [0,1]. Initial state is s=1, theta=alpha=beta=0, kappa=0.5. Separate configuration includes a positive reference radius r0 and weight exponent m in {1,2,3}; default r0=1 and m=3. These configuration values are not additional hidden votes. Reject nonfinite/out-of-range imported inputs. Clamp only documented direct gesture updates. Store degrees in the public JSON and convert once to radians for trigonometry. SEMANTICS AND SIGN Positive theta is clockwise when viewed from +localY toward the origin in the untilted X–Z reference frame. Use a right-handed geometric rotation Ry(-theta), NOT Ry(theta). Positive/negative approval endpoints are distinct even though both orientations coincide with the starting orientation. Retain signed unwrapped theta; never reconstruct it solely from atan2, a matrix or a quaternion. Clamp beyond one signed revolution. Approval a=theta/360; illustrative temperature T=100a in index units, not Celsius. Apply the sign only once. Color must never be the only carrier of sign. WEIGHT AND PRIORITY Compute r=r0*s; V0=2*pi*r0^3/3; V=V0*s^3; area ratio=s^2; volume ratio=s^3; configured W=s^m. Explain m=1 radius, m=2 area, m=3 volume. Default W=V/V0. Never equate doubled radius with doubled volume. Evaluate priority at zero Y rotation: R0=Rz(beta)Rx(alpha). Set u=(alpha+beta)/180 and P=1+kappa*u. Thus P lies in [0,2] and cannot reverse the sign. Baseline B=W*a; final C=W*a*P. X_vote and Z_vote are explicit aliases of C, not physical x and z. Export all three and keep physical coordinates separate. Increasing priority deepens an opposition score; do not describe it as increasing approval. GEOMETRY AND ORIENTATION Use a Y-up local hemispherical shell h=r(sin(phi)cos(psi),cos(phi),sin(phi)sin(psi)), phi in [0,pi/2], psi in [0,2pi). Column-vector rotation order is R=Rz(beta)Rx(alpha)Ry(-theta), rightmost first. Apply R to every mesh point. Do not clip the tilted hemisphere against the global Y=0 plane; clipping would change its volume. R must be orthonormal with determinant +1. Export the 3x3 matrix and scalar-first quaternion qR=qz(beta)*qx(alpha)*qy(-theta). Preserve the original theta because orientation alone loses full-turn sign. COORDINATE OUTPUT Export crown normal n=(-sin(beta)cos(alpha),cos(beta)cos(alpha),sin(alpha)), crown position p=r*n, and yaw-sensitive meridian b=R(r,0,0). Normalize v=p/r0, not p/r: the zero-scale case must remain safe. Export the stereographic lift q=(2vx,2vy,2vz,|v|^2-1)/(1+|v|^2). Verify |q|=1 and inverse projection q_xyz/(1-q4)=v. Explain that q is a point on S3, NOT the rotation quaternion and NOT a complete encoding of all five channels. The tip ignores theta and kappa. Five controllable dimensions are not five spacetime dimensions or qubits. SENSITIVITY REPORT Return the complete gradient of C with respect to scale, theta-degrees, alpha-degrees, beta-degrees and kappa: (m*s^(m-1)*a*P, W*P/360, W*a*kappa/180, W*a*kappa/180, W*a*u). Confirm these against centered finite differences at interior inputs, using one-sided reasoning at bounds. Show both signs of perturbation for each input. Check mixed partials and analytical coordinate derivatives where claimed. Distinguish derivatives per radian from derivatives per degree. Do not claim infinite precision; state display rounding and floating-point tolerance. INTERACTION AND ACCESSIBILITY Offer visibly labeled approval-drag and tilt-drag modes. Two-pointer pinch can control s; anchor distance must be nonzero. At s=0 pinch alone cannot recover a positive scale, so provide a plus button and numeric input. Provide a labeled slider, editable number field and increment/decrement buttons for every channel, plus keyboard-accessible reset and examples. Touch behavior must not block scrolling outside the instrument. No automatic rotation should alter the stored vote. Focus indicators, reduced-motion handling, signed text, readable contrast and narrow-screen layouts are required. A visualization is supplementary: the report must be usable without seeing or dragging it. RESULT INTERPRETATION Show radius, actual model volume, all three relative size measures, approval, temperature index, signed priority, multiplier, baseline score, final score, crown and meridian coordinates, orientation matrix, orientation quaternion, S3 lift and five sensitivities. Explain zero approval, zero weight and zero priority as different cases. Use the state/model version/configuration as the source of truth for exported results. Do not send selector inputs to site analytics, query strings, logs, cookies or server endpoints. Plain page analytics, if any, must be disclosed separately. WORKED FIXTURES Verify neutral baseline; theta=+360 and -360 with matching visual orientations and opposite score; s=0.5,theta=180,neutral tilt under m=3; s=1,theta=180,alpha=beta=45,kappa=0.5; s=1.5,theta=-120,alpha=-30,beta=0,kappa=0.6; alpha=-beta cancellation; s=0; kappa=1 with both tilts=-90 giving P=0. Calculate expected results from the equations, never invent them. Test exponent alternatives, radius calibration, quaternion/matrix equivalence, deterministic repeatability, finite outputs, geometry norms and invalid input rejection. AGGREGATION AND ETHICS If showing a collection, compute S=sum(WPa), D=sum(WP), M=S/D only for D>0; otherwise return no result. Report disagreement as well as mean. Treat dating affinity as separate from mutual consent; refusal and withdrawal override any score. Label pre-legislative examples as nonbinding consultation. Never claim electoral certification, representative sampling, strategy-proofness, fairness, human-level precision or quantum advantage from the formula. Production identity, coercion resistance, audit trails, privacy and budgets are additional systems, not implied features of this demo. ACCEPTANCE EVIDENCE Supply executable unit tests; browser evidence that all controls update the report; real JSON export and source download; endpoint sign preservation; narrow-screen layout checks; and explicit statements of what was not tested. Verify that control changes trigger no inference/API requests. Retain the source material and cite external geometry, accessibility and voting-system standards with numbered Chicago notes in first-appearance order plus an alphabetized bibliography. Do not claim tests passed before running them. ## Notes [^1]: Michael Aaron Loftus, “Hemispherical Projection of the Four-Dimensional Sphere for Continuous Gesture-Based Weighted Selection” (author-supplied manuscript, October 10, 2026), 1–14. https://digitalmarketingco.org/documents/quantum-selector/loftus-hemispherical-selector-original-manuscript.pdf [^2]: Oliver Knill, “Hopf Fibration,” Math 22a, Harvard College, Fall 2018, accessed October 10, 2026. https://people.math.harvard.edu/~knill/teaching/math22a2018/exhibits/threesphere/index.html [^3]: David W. Lyons, “1.3: Stereographic Projection,” in Introduction to Groups and Geometries, Mathematics LibreTexts, accessed October 10, 2026. https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/01:_Preliminaries/1.03:_Stereographic_projection [^4]: MIT OpenCourseWare, “Session 77: Triple Integrals in Spherical Coordinates,” 18.02SC Multivariable Calculus, Fall 2010, accessed October 10, 2026. https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/pages/4.-triple-integrals-and-surface-integrals-in-3-space/part-a-triple-integrals/session-77-triple-integrals-in-spherical-coordinates/ [^5]: Kevin M. Lynch and Frank C. Park, “3.2.1. Rotation Matrices (Part 1 of 2),” Modern Robotics video supplements, Northwestern University, accessed October 10, 2026. https://modernrobotics.northwestern.edu/nu-gm-book-resource/3-2-1-rotation-matrices-part-1-of-2/ [^6]: World Wide Web Consortium, “Understanding SC 2.5.7: Dragging Movements (Level AA),” Understanding WCAG 2.2, accessed October 10, 2026. https://www.w3.org/WAI/WCAG22/Understanding/dragging-movements.html [^7]: U.S. Election Assistance Commission, “Voluntary Voting System Guidelines,” accessed October 10, 2026. https://www.eac.gov/voting-equipment/voluntary-voting-system-guidelines [^8]: National Institute of Standards and Technology, NIST Privacy Framework: A Tool for Improving Privacy through Enterprise Risk Management, Version 1.0 (January 16, 2020). https://www.nist.gov/publications/nist-privacy-framework-tool-improving-privacy-through-enterprise-risk-management ## Bibliography Knill, Oliver. “Hopf Fibration.” Math 22a, Harvard College. Fall 2018. Accessed October 10, 2026. https://people.math.harvard.edu/~knill/teaching/math22a2018/exhibits/threesphere/index.html Loftus, Michael Aaron. “Hemispherical Projection of the Four-Dimensional Sphere for Continuous Gesture-Based Weighted Selection.” Author-supplied manuscript, October 10, 2026. https://digitalmarketingco.org/documents/quantum-selector/loftus-hemispherical-selector-original-manuscript.pdf Lynch, Kevin M., and Frank C. Park. “3.2.1. Rotation Matrices (Part 1 of 2).” Modern Robotics video supplements. Northwestern University. Accessed October 10, 2026. https://modernrobotics.northwestern.edu/nu-gm-book-resource/3-2-1-rotation-matrices-part-1-of-2/ Lyons, David W. “1.3: Stereographic Projection.” In Introduction to Groups and Geometries. Mathematics LibreTexts. Accessed October 10, 2026. https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/01:_Preliminaries/1.03:_Stereographic_projection MIT OpenCourseWare. “Session 77: Triple Integrals in Spherical Coordinates.” 18.02SC Multivariable Calculus. Fall 2010. Accessed October 10, 2026. https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/pages/4.-triple-integrals-and-surface-integrals-in-3-space/part-a-triple-integrals/session-77-triple-integrals-in-spherical-coordinates/ National Institute of Standards and Technology. NIST Privacy Framework: A Tool for Improving Privacy through Enterprise Risk Management, Version 1.0. January 16, 2020. https://www.nist.gov/publications/nist-privacy-framework-tool-improving-privacy-through-enterprise-risk-management U.S. Election Assistance Commission. “Voluntary Voting System Guidelines.” Accessed October 10, 2026. https://www.eac.gov/voting-equipment/voluntary-voting-system-guidelines World Wide Web Consortium. “Understanding SC 2.5.7: Dragging Movements (Level AA).” Understanding WCAG 2.2. Accessed October 10, 2026. https://www.w3.org/WAI/WCAG22/Understanding/dragging-movements.html