The Reachability of the Truth: Edge Cases from Geometry to Consciousness

An expository essay — a perspective, not a research result. It offers no new theorem and claims no discovery. Its contribution is framing: a single axis that organizes a body of work built across several domains, and an honest account of why the same structure yields decisive fixes in one place and permanent silence in another. The technical results it points to are all prior art, credited in place.

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One structure, one axis

A proxy is a stand-in for something you actually care about but cannot read off directly: a metric for a value, a benchmark for a capability, a float for an exact quantity, a behavior for an inner state. Under any pressure — optimization, adversarial framing, rounding — a proxy can decouple from the truth it represents. That single shape recurs across domains that otherwise share nothing, and a family of governance tools is built around detecting it.

But detecting a decoupling requires an independent measure of the truth to check the proxy against — and that is the whole game. The interesting claim of this essay is that a wide range of seemingly unrelated problems line up on a single axis: how reachable is the ground truth? Order the domains by that one question and the right response falls out of it, from solve at one end to withhold at the other. Nothing else needs to change.

The near pole: geometry, where the truth is exactly computable

Computational geometry is where "edge case" stops being a metaphor and becomes literal. Almost every classic algorithm — convex hull, Delaunay triangulation, polygon Boolean operations — assumes general position: no three points collinear, no four concyclic, no query point exactly on an edge. The edge cases are precisely the inputs that violate this, and they are geometrically real: in the space of inputs, the degenerate configurations form measure-zero sets — lower-dimensional surfaces where the generic case gives way. They sit, literally, on the boundary between combinatorial outcomes: the point exactly on the wall between "inside" and "outside."

Underneath is a proxy-versus-truth failure in the strict sense. The core operations are sign tests — orientation ("is C left of, right of, or on line AB?"), incircle ("is D inside, on, or outside the circle through A, B, C?"). Exactly on the edge the true sign is zero; near it, a floating-point evaluation (the proxy) can return the wrong sign for the exact answer (the truth), and because these algorithms branch discretely on those signs, one wrong sign cascades into impossible topologies, infinite loops, and crashes (catalogued by Kettner, Mehlhorn, Pion, Schirra & Yap, Classroom Examples of Robustness Problems in Geometric Computations).

What makes this the near pole is that the truth is exactly computable. So the fix is not to detect the decoupling but to eliminate it: evaluate the predicate exactly. Shewchuk's adaptive-precision predicates, Yap's Exact Geometric Computation paradigm, and Edelsbrunner & Mücke's Simulation of Simplicity (which breaks ties consistently, as if the input were nudged into general position) between them close the gap completely. Here, an independent ground truth is not only available; it is cheap enough and exact enough to correct the proxy outright. The edge cases are fully solvable — because the truth is fully reachable.

The middle: metrics, evaluations, models, where the truth is partially observable

Move one step along the axis and the truth becomes real but no longer exactly on tap. A KPI proxies a value you can sometimes measure independently but not perfectly; a benchmark proxies a capability you can probe from another angle but not exhaustively; a digital twin proxies an asset whose true state you can sample but not fully observe; a safety-case evaluation proxies a real property that an independent method can partially corroborate. Here the truth is partially and independently observable, and the honest response shifts accordingly: you cannot eliminate the decoupling, but you can detect it, and you can be disciplined about the detection.

This is the register a whole governance toolkit operates in. A monitor watches a reported proxy against an independent truth signal and alerts when they diverge; an auditor asks the prior question — is that "truth" signal actually independent of the proxy, or a shadow of it? — and, crucially, returns UNVERIFIED when no independent measure exists, rather than manufacturing a false all-clear. The middle is defined by that honesty: the ground truth is reachable enough to keep the proxy accountable, and where it is not, the tool says so instead of pretending. Detection, not elimination, is the most one can claim — and claiming more would be the failure the tools are built to catch.

The far pole: consciousness, where no third-person truth exists

At the far end sits the case where the independent measure is not merely expensive or partial but absent in principle. Phenomenal experience is first-person; no third-person procedure confirms a quale (the explanatory gap; the problem of other minds). Behavioral and functional indicators are proxies that can be present in full while the truth remains out of reach — the proxy-versus-truth relation at its absolute limit, where the gap cannot be closed by any amount of third-person work.

Here the honest response is neither solve nor detect but withhold: record a first-person report as authentic testimony, treat indicators as proxies and never as proof, refuse to certify a phenomenal fact, and return a permanent UNVERIFIABLE. That is not a failure of tooling awaiting a better sensor; it is the correct output at this end of the axis. As the companion work puts it, consciousness is an edge case of the tool, not the world — the place where third-person method reaches its boundary and, done honestly, goes quiet.

The synthesis, and why the near pole matters most

Domain	Reachability of the truth	Honest response	Where it is worked out
Computational geometry	exactly computable	solve (exact predicates eliminate the decoupling)	Shewchuk; Yap; Edelsbrunner–Mücke
Metrics, evals, twins, mappings, safety cases	partially, independently observable	detect (monitor, audit independence, withhold where absent)	the governance toolkit; Keeping the Evidence Honest; Who Governs the Governor?
Consciousness / phenomenal experience	not third-person accessible, even in principle	withhold (record, respect, never verify; permanent UNVERIFIABLE)	qualia_report_governor; Recorded, Not Verified

Read top to bottom, only one thing changes — the reachability of the truth — and the honest response tracks it exactly: solve → detect → withhold. The tools are not three different ideas but one idea met with three different degrees of access to reality.

The geometry pole earns its place at the top for a reason beyond tidiness. It is the existence proof that the framework is not vacuous. It is easy to build a theory in which "the truth is unreachable" is an all-purpose excuse; what makes the far pole's UNVERIFIABLE an honest verdict rather than a shrug is that there is a near pole where an independent ground truth genuinely does close the gap — cleanly, provably, in shipping code. Because independence works where it can work, its absence elsewhere means something. The geometer's exact predicate and the qualia governor's permanent UNVERIFIABLE are the two ends of the same commitment: check the proxy against an independent truth wherever one exists, and be honest — decisively or silently — about exactly how far that "wherever" extends.

Honest positioning

This is synthesis, and it should be read as such. Every technical result it leans on is someone else's and decades established in the geometry case; the governance middle and the consciousness end are worked out in three companion papers, not here. The essay adds no theorem. What it offers is a lens — one axis, three responses — that makes a scattered body of work legible as a single stance, and a reason to trust the quiet end by pointing at the loud one. Its own claim is modest by design, which is the only way a piece about the limits of proxies has any business being written.

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Companion material: the assurance-case integrity paper (honesty about evidence), the reflexive-governance paper (honesty about oversight), and Recorded, Not Verified (honesty about minds); the governance toolkit and its ground_truth_auditor, decoupling_monitor, and qualia_report_governor.

Gergely Vámossy (2026). The Reachability of the Truth: Edge Cases from Geometry to Consciousness. https://vamossy.com/research/the-reachability-of-the-truth