Chapter 14: Open Problems
“No man ever steps in the same river twice, for it is not the same river and he is not the same man.” — Heraclitus, as reported by Plutarch (c. 500 BCE)
RUNNING EXAMPLE — THE COOLING ESPRESSO
A barista pulls a single shot at 7:02 a.m. and, on a whim, decides to taste it every sixty seconds until it is cold. At the first sip — 68°C, almost too hot to hold on the tongue — the cup is all bright acidity and a sharp, almost medicinal bitterness; the aromatics scald past the retronasal passage too fast to read. At minute three, around 55°C, the bitterness has receded and a caramel sweetness she did not notice before steps forward; the body feels rounder. By minute six, near 45°C, the espresso is at its most articulate — chocolate, dried cherry, a faint floral top note all legible at once, the famous “drinkable window” that competition tasters guard jealously. By minute ten the volatile aromatics have largely fled into the room; what remains at 30°C is a flat, sour, slightly astringent liquid, recognizably the same coffee and yet, by any honest account, a different flavor.
She has tasted one espresso. She has also tasted, by the clock, ten espressos — or rather one continuous flavor that swept through a region of the space we have spent thirteen chapters mapping. No single vector $\mathbf{f}$ describes what was in the cup, because there was never a single $\mathbf{f}$. There was a curve. And the moment we admit that the honest object is a curve and not a point, almost every clean result in this book acquires a frontier — a place where the static geometry runs out and a harder, time-dependent geometry begins.
This chapter is a tour of those frontiers. The cooling espresso is our guide, because every open problem we will name is already present, in miniature, in that single cooling cup.
Where the Static Picture Runs Out
The preceding thirteen chapters built a theory on a deliberate idealization: that a flavor is a point $\mathbf{f} \in \mathcal{F}$, a fixed location in a perceptual space of a few tens of dimensions. On that idealization rests everything — the taste simplex of Chapter 1, the receptor basis of Chapter 2, the Riemannian metric of Chapter 3, the pairing operations of Part II, the cooking transformations of Part III. The idealization earned its keep. It dissolved the combinatorial wall, it predicted real synergies, it explained why distant cuisines converge. A good idealization is not a lie; it is a coordinate chart that is accurate in a neighborhood and honest about its edges.
This chapter is about the edges. Each of the six problems below is a place where the static, point-valued picture is not merely incomplete but structurally wrong — where the missing physics is not a few more dimensions of $\mathcal{F}$ but a different kind of mathematical object: a curve instead of a point, a path-dependent metric instead of a fixed one, a deforming manifold instead of a rigid one, a joint space instead of a single observer’s. These are not failures of the theory. They are the theory pointing past itself, the way Henning’s static taste tetrahedron (1916) pointed past itself toward the high-dimensional olfactory manifold it could not hold (§1.5). The early maps failed by using too few dimensions and the wrong coordinates; the honest worry for our map is subtler — that for the deepest phenomena, the right object is not a richer space but a richer category of object.
A word on register. The previous chapters made claims and defended them; this one mostly poses problems and sketches the shape of their solutions. Where we can state a conjecture sharply enough to be wrong, we tag it [Conjecture.]; where data already constrains the answer, [Empirical.]. The goal is to phrase these questions in geometric language well enough that resolving them becomes a research program rather than a mood.
14.1 The Temporal Geometry of Flavor
We flagged this in §1.4 and again in §3.5, and deferred it each time: the honest object is not a point but a curve $\mathbf{f}(\tau)$ traced through $\mathcal{F}$ as a food is experienced over the interval $\tau \in [0, T]$. Perfumery has long encoded this as top, middle, and base notes; the same structure governs a slow-cooked stew, a glass of wine breathing, and the cooling espresso. The open problem is not whether flavor is temporal — it plainly is — but what kind of curve it is, and what intrinsic invariants of that curve the brain actually reads.
[Modeling Axiom.] Promote the flavor vector to a parametrized curve $\gamma: [0,T] \to \mathcal{F}$, $\gamma(\tau) = \mathbf{f}(\tau)$, equipped with the Riemannian metric $g$ of Chapter 3. Immediately we inherit the differential-geometric vocabulary of curves: a velocity $\dot{\gamma}(\tau)$ (how fast the percept is moving and in which direction), a speed $\|\dot\gamma\|_g$ (the rate of perceived change), an arc length $L[\gamma] = \int_0^T \|\dot\gamma(\tau)\|_g \, d\tau$ (the total perceptual distance traveled), and a geodesic curvature $\kappa_g$ (how sharply the flavor turns). The conjecture that organizes the whole problem is that hedonic and aesthetic judgments of a flavor experience are functionals of these intrinsic quantities, not of any single point on the curve.
[Conjecture.] The temporal-invariance conjecture. Two flavor experiences that trace curves related by an orientation-preserving reparametrization $\tau \mapsto \phi(\tau)$ — same path through $\mathcal{F}$, different timing — are judged similar in quality but different in character, and the difference is carried entirely by the speed profile $\|\dot\gamma\|$. A wine that “opens slowly” and a wine that “opens fast” trace nearly the same path; what distinguishes them is $\phi$. If true, this separates flavor experience into a shape invariant (the unparametrized path, an equivalence class under reparametrization) and a tempo invariant (the speed profile) — exactly the separation that Volume 7, Geometric Aesthetics, found between structural form and its unfolding in music and narrative.
The difficulty is measurement. Time-intensity (TI) and the more recent temporal-dominance-of-sensations (TDS) methods give us, at best, a few coordinates of $\gamma(\tau)$ sampled coarsely — a panelist tracks “the dominant sensation right now,” a projection of $\gamma$ onto its largest component, not the curve itself. We can measure that the dominant sensation in the espresso shifts bitter → sweet → aromatic-complex → sour-flat, but we cannot yet reconstruct the full trajectory, and so cannot yet test whether the brain’s judgment is a functional of arc length, curvature, peak intensity, or endpoint. [Empirical.] What the TDS literature does establish is that the temporal order of sensations changes liking even when the time-averaged composition is held fixed — sweet-first and sweet-last versions of the same total stimulus are not judged equal. That single robust finding is enough to kill the time-average model $\bar{\mathbf{f}} = \frac{1}{T}\int_0^T \mathbf{f}(\tau)\,d\tau$: the brain is not integrating the curve to a point. It is reading the curve.
14.2 Context Dependence: The Path-Dependent Metric
Section 14.1 lets the stimulus vary in time. The deeper problem is that the space itself changes with what came before. The same compound, presented twice, does not land at the same perceived point; where it lands depends on the path the palate has already traveled. We met this once already as cross-adaptation (§3.4): taste something intensely sweet and the next thing tastes less sweet; the artichoke compound cynarin makes plain water taste sweet. Those were treated as local metric distortions. The open problem is that the distortion is not a fixed feature of the space — it is a function of the trajectory, which means the metric is path-dependent, which means $\mathcal{F}$ is not a Riemannian manifold at all in the usual sense.
[Modeling Axiom.] Replace the fixed metric $g$ with a history-dependent metric $g_{[\gamma]}(\mathbf{f})$, a functional of the entire prior trajectory $\gamma|_{[0,\tau]}$. The perceived distance between the current stimulus and the previous one is then $\langle \dot{\mathbf{f}}, \dot{\mathbf{f}}\rangle_{g_{[\gamma]}}$, and because $g$ depends on the path, the perceived “distance” from a neutral palate to a sip of espresso is genuinely different from the distance to the same sip after a bite of dark chocolate. Geometrically, this is the move from a Riemannian manifold to something closer to a system with memory — the natural objects are those of non-equilibrium physics, where the state of the medium records its history.
There are two clean candidate formalisms, and choosing between them is itself an open problem. The first is hysteresis: the metric relaxes back toward a baseline $g_0$ with some time constant after a perturbation, so that recent stimuli weigh more than distant ones — a fading memory. The second is a genuine connection with non-zero curvature, in which the change induced by a closed loop of stimuli (taste A, then B, then A again, ending where you started in stimulus-space) does not return the percept to its starting value. [Conjecture.] The holonomy conjecture. Transport the palate around a closed loop in stimulus space — a “palate cleanser” that returns the chemistry exactly to its initial state — and the perceived flavor will in general be rotated relative to its start, by an amount equal to the holonomy of the adaptation connection around that loop. The wine-pairing tradition is, on this reading, an empirical atlas of holonomies: the maxim that a sorbet “clears” the palate is a claim that certain loops have near-zero holonomy and can be used to return the metric to baseline. We have, as yet, no laboratory measurement of a flavor holonomy. Designing the closed-loop experiment that would measure one — and distinguish true holonomy from mere hysteresis — is, in my view, the single most tractable of the six problems in this chapter.
14.3 The Satiety Transformation: Alliesthesia as a Deforming Manifold
The first two problems let the stimulus and then the metric vary. This one lets the hedonic landscape itself deform, and it does so on a slow timescale set by the body’s internal state. The phenomenon is one of the most robust in all of food science. [Empirical.] Michel Cabanac’s classic demonstrations of alliesthesia (1971) showed that the pleasantness of a fixed stimulus changes with the organism’s internal state: a sugar solution that is delightful when you are hungry becomes neutral and then actively unpleasant as you are sated, even though the stimulus on the tongue is chemically identical. Sensory-specific satiety (Rolls and colleagues) sharpens this: satiety is not a uniform dimming of all pleasure but is specific to the sensory properties just consumed — eat your fill of the savory course and the dessert is still, somehow, fully desirable.
This last fact is the crux, and it is exactly what a scalar model of “appetite” cannot represent. If satiety were a single dial turning down a global hunger scalar, dessert would suffer along with everything else. It does not. The deformation is directional: it suppresses the hedonic value of the region of $\mathcal{F}$ you have been occupying while leaving distant regions nearly untouched.
[Modeling Axiom.] Recall the hedonic field $H: \mathcal{F} \to \mathbb{R}$ of §3.6 and §5.6, which assigns to each flavor point a pleasantness. Promote it to a time-and-history-dependent field $H_t(\mathbf{f})$ that deforms as a function of cumulative intake. Sensory-specific satiety is then a local depression of the field carved at the point you have been eating: roughly, $$ H_t(\mathbf{f}) = H_0(\mathbf{f}) - \int_0^t \rho(s)\, K\!\big(\mathbf{f}, \mathbf{f}(s)\big)\, ds, $$ where $\mathbf{f}(s)$ is the flavor consumed at time $s$, $\rho(s)$ is an intake rate, and $K$ is a kernel — almost certainly the same kernel that defines perceptual nearness in the Chapter 3 metric — that determines how far the depression spreads from the consumed point. A narrow kernel gives strongly sensory-specific satiety (only the exact flavor cloys); a broad kernel gives global satiety (everything dims together). The buffet, and the perennial room for dessert, are then a single geometric fact: the meal digs a well in $H_t$ around the savory region, and dessert sits on a part of the manifold the well has not reached.
[Conjecture.] The satiety kernel $K$ is, up to a slowly varying scale, the heat kernel of the Chapter 3 metric — meaning that satiety diffuses across flavor space along the same geodesics that govern perceptual similarity, and that the rate of this diffusion is an organism-level parameter that varies with metabolic state. If so, the deforming hedonic landscape and the static perceptual metric are not two theories but one: satiety is heat flow on the flavor manifold, and the “dynamics of appetite” become a well-posed PDE on the space we have already built. This connects forward to Volume 6, Geometric Cognition, where the same heat-kernel machinery describes the decay of attention and the spread of activation in semantic memory.
THE COOLING ESPRESSO, REVISITED — THREE CLOCKS AT ONCE
The barista’s experiment runs three clocks simultaneously, and the open problems of this chapter are exactly the problems of keeping them separate. The fast clock (seconds) is §14.1: the curve $\gamma(\tau)$ the flavor traces as volatiles flee and the tongue’s thermoreceptors recalibrate — bitter giving way to sweet to aromatic complexity to flat sourness. The medium clock (the span of the tasting, minutes) is §14.2: each sip adapts the palate for the next, so the metric by which she judges minute six was set by minutes one through five; her ninth sip is read on an instrument the previous eight have bent. And the slow clock (had she been drinking, not spitting) is §14.3: caffeine and volume slowly deform $H_t$, the sixth cup of the morning never tasting like the first. One cooling cup, three time-dependent geometries, none of them expressible as a point. The static theory described the cup. These three clocks describe the drinking.
14.4 The Social Geometry of Eating
Every problem so far has assumed a single perceiver. But the central human fact about food is that we almost never eat alone by preference — the meal is the oldest shared ritual, and “breaking bread together” names the act, not the bread. A theory that stops at one observer’s $\mathbf{f}$ has not yet reached the table. The open problem is to build the geometry of joint tasting: what is the right object when two or more people explore flavor space together, talking as they go?
[Modeling Axiom.] Each diner $i$ carries their own perceptual chart — recall from §2.5 that receptor polymorphisms (the TAS2R38 bitter alleles, for instance) effectively rotate each person’s flavor space relative to a reference. A shared meal is then not one trajectory but a family of trajectories $\{\gamma_i\}$, one per diner, through charts related by individual rotations $R_i$. The social object of interest lives in the fiber bundle whose base is the shared physical dish and whose fibers are the individual perceptual spaces. Conversation at the table — “do you get the cherry?” “no, but there’s something smoky” — is the act of attempting to construct transition maps between fibers: of agreeing on a shared vocabulary that aligns one person’s coordinates with another’s.
This reframes a great deal of culinary culture as the social engineering of those transition maps. A sommelier’s tasting note is a published chart meant to align many strangers’ fibers onto common coordinates; a wine flight is a guided geodesic that a roomful of people walk together so that their trajectories $\gamma_i$ stay close enough to talk about. [Conjecture.] The shared-exploration conjecture. The pleasure of a shared meal exceeds the sum of the diners’ private pleasures by a term that scales with the successful alignment of fibers — the mutual information between diners’ trajectories once transition maps are applied. We enjoy eating together not merely because company is pleasant but because converging on a shared description of a flavor is itself rewarding, a small joint act of map-making. Spence’s work on the social and multisensory modulation of flavor (2015 and after) supplies the empirical edge: the same dish is rated more pleasant in company, and pleasantness tracks conversational engagement. The geometric content we conjecture — that the bonus is specifically a fiber-alignment term and not a generic mood effect — is sharply falsifiable: it predicts that meals which fail to align (a supertaster and a non-taster, given no shared vocabulary) should show no social bonus or even a penalty.
14.5 The Evolution of Flavor Preferences
The hedonic field $H$ has been treated, until now, as fixed for a given person. Over a lifetime it is anything but. The most familiar instance is also the most revealing. [Empirical.] Human infants show innate hedonic responses present essentially at birth: sweet and umami are accepted, bitter and sour are rejected, with the bitter rejection especially strong and stereotyped (Steiner’s classic gustofacial studies; Mennella and colleagues on the developmental trajectory). The adaptive logic is transparent — sweetness signals calories, bitterness signals the alkaloids of toxic plants — and it is no accident that the foods adults learn to love (coffee, dark chocolate, hops, bitter greens, fermented everything) cluster precisely in the region of $\mathcal{F}$ that the infant is built to refuse. The open problem is to model how the hedonic field reshapes itself, over years, to invert a sign that began as a survival reflex.
[Modeling Axiom.] The lifetime hedonic field is a slowly evolving object $H^{(\text{life})}_t(\mathbf{f})$ deforming on the timescale of years rather than the minutes of §14.3. The initial condition $H^{(\text{life})}_0$ is the innate, evolutionarily tuned field: a sharp negative peak over the bitter region, positive over sweet and umami. Acquired taste is the local lifting of the bitter well, driven by repeated exposure paired with positive context — the well does not vanish (a truly bitter compound still warns) but is locally deformed and, crucially, re-signed in the specific sub-region associated with safe, culturally endorsed foods. This is manifold deformation under a learning rule, and it is the same mathematics as the metric learning of expertise we invoked for the line cook in Chapter 1: the surface that perception lives on is plastic, and experience reshapes it.
Two features make this more than a metaphor. First, the deformation is path-dependent and irreversible in a way §14.2’s fast adaptation is not: you cannot un-acquire a taste with a palate cleanser; the well stays lifted. The right formalism is therefore not the relaxing hysteresis of §14.2 but a slow plastic deformation with a near-permanent component — the flavor-space analogue of how Volume 6 models the consolidation of a learned skill. Second, the deformation is culturally steered: which regions of the bitter well get lifted is set by what a culture eats, so adults from different food cultures carry systematically different deformations of the same innate initial condition. [Conjecture.] Adult cross-cultural variation in liking is well-approximated as a small number of distinct deformation flows acting on a shared infant field — a finite atlas of “acquired-taste vector fields,” one per major food tradition — whose principal components should recover the very cuisine eigenvectors that Chapter 9 extracted from recipe databases, now derived from the demand side (preference) rather than the supply side (what is cooked). If the two routes to the cuisine PCA agree, it would be strong evidence that preference and tradition co-evolve along a shared geometry.
14.6 Artificial Flavor Perception
The final problem is the one that would make this book’s central object measurable rather than merely modeled. Throughout, $\mathbf{f}$ has been a perceptual vector recovered indirectly — from sensory panels, from receptor recordings, from the structure of pairing data. The standing question of computational gastronomy (Chapter 10) is whether we can build an instrument that reads it: a sensor array that ingests a food’s chemistry and outputs a flavor vector matching what a human would perceive. Electronic noses and electronic tongues exist; they are arrays of broadly tuned chemical sensors, and on the surface their architecture is an engineered echo of the biological combinatorial code of Chapter 2 — many low-specificity sensors whose joint response pattern encodes the stimulus. So why is the problem still open?
Because, as §10.6 warned, the map is not the territory: chemical composition is not perceived flavor. The honest statement of the problem makes the gap precise. [Modeling Axiom.] We seek a learned map $\Phi: \mathcal{C} \to \mathcal{F}$ from chemical-sensor space $\mathcal{C}$ to perceptual flavor space $\mathcal{F}$ that commutes with human perception — that is, $\Phi$ should agree with the biological encoder $P: \mathcal{C} \to \mathcal{F}$ (the receptors-plus-brain pipeline of Chapters 2 and 12) not pointwise on chemistry but on perceptual structure: $\Phi$ should preserve perceptual distances, so that $d_\mathcal{F}(\Phi(c_1), \Phi(c_2))$ tracks how different two foods taste, not how different their chromatograms are. The instrument’s job is not to measure chemistry accurately; gas chromatography already does that superbly (Appendix C). Its job is to measure chemistry the way a human mis-measures it — to inherit the same projections, the same suppressions, the same concentration-dependent nonlinearities that turn ten thousand compounds into a few tens of perceived dimensions.
[Conjecture.] The matched-encoder conjecture. A sensor array can match human flavor perception if and only if it is trained to reproduce the human metric (perceptual distances and the off-diagonal interactions of §1.1, §3.2) rather than human labels (named descriptors). The reason is the recurring lesson of the whole geometric series: labels are scalars, a lossy projection of the structure that matters, while the metric is the structure. An array trained to predict the words “fruity, smoky, bitter” will fail to generalize because it is fitting the projection; an array trained so that its output distances match human discrimination data will succeed, because it is fitting the geometry. This is testable now, and it is the most directly engineering of the six problems — which is why I suspect it will be the first to fall. The obstacles are not mysteries of principle but the known hard parts of §10.6: matrix effects (the same compound reads differently in fat versus water), concentration nonlinearities, and the temporal and contextual dependencies that are themselves §§14.1–14.2. A static artificial nose will, at best, read a single point of the cooling espresso’s curve. Reading the curve — an instrument that tastes the cup the way the barista did, across all three of its clocks — is the problem that contains all the others.
The Frontier as a Map
It is worth seeing how tightly these six problems interlock, because their unity is itself the chapter’s main claim. Each is the same move — relaxing one fixity of the static theory — applied to a different piece of the apparatus. Section 14.1 lets the stimulus vary in time (point → curve). Section 14.2 lets the metric vary along the path (fixed $g$ → history-dependent $g_{[\gamma]}$). Section 14.3 lets the hedonic field deform with internal state on a medium timescale (fixed $H$ → diffusing $H_t$). Section 14.5 lets that same field deform on the slow timescale of a life (fixed $H$ → plastic $H^{(\text{life})}_t$). Section 14.4 lets the number of observers exceed one (single chart → fiber bundle of charts). And §14.6 asks for an instrument that respects all of the above. The static theory was the limit in which every one of these is frozen; the frontier is the systematic thawing.
Seen this way, the open problems are not a ragged list of things the theory cannot do. They are a single coordinated direction of generalization, and they share their mathematics with the rest of the series: the history-dependent metric of §14.2 is the curved connection of Geometric Reasoning; the diffusing hedonic field of §14.3 and the plastic field of §14.5 are the heat-kernel and consolidation dynamics of Geometric Cognition (Volume 6); the shape-versus-tempo split of §14.1 and the alignment bonus of §14.4 are the compressibility and structural-channel results of Geometric Aesthetics (Volume 7). This is the suspicious coincidence that runs through the whole Geometric Series, stated once more from the side of the tongue: the same small inventory of objects — manifolds, metrics, connections, fields, bundles — keeps being the right description, whether the domain is ethics, economics, cognition, or a cooling cup of coffee. We argued throughout this volume that the combinatorial explosion of food was an illusion of the wrong coordinates. The open problems say something quieter and one register deeper: that even the time, the history, the appetite, the company, and the life of a flavor — everything that seemed to lie outside a clean geometry of taste — is most likely geometry too. We have not yet measured it. But the espresso has already cooled, tracing its curve through the space, indifferent to whether we have the mathematics ready to follow. The geometry was always real. The scalars — and now, we can add, the static points — were always insufficient.