The Nature of Knowledge: six axes, and what each licenses alone

Two of the six describe the claim itself. On U you commit to a shape: a bare point, an interval, a distribution, a set you can’t narrow. On I you ask whether the quantity you’re talking about can be recovered from evidence at all, however carefully you stated it.

The other four are about the edge between the claim and whatever produced it. Between them they cover the check that produced it, the instrument that read it, whether the checker was independent, and how far the evidence’s own region reaches. Scoring an edge doesn’t require settling what the claim is worth, so we fold those four together with I into a single bound at the end.

At U5 (Partially identified (set-valued, non-shrinking)) the width doesn’t shrink as n grows. That non-shrinking is the diagnostic: the width belongs to the identification problem rather than to the sample. If your band narrows as data accumulates, it isn’t showing you the identification gap. At U6 (Residual-bounded (named unmodeled terms)) the bound isn’t a probability at all:

Nothing further - but the bound itself is NOT a probability. It is a corroboration claim: a track record of how often this model class has blown through its stated bounds. You cannot bound the residual from inside the model, because a model's residual estimate covers exactly the errors it can represent.

Neither limit gets bought off with a larger sample. That leaves an out-of-model track record as a routinely available promoter at the top of this ladder. It’s why we ship a cap at the last level instead of a probability.

A cap bounds the consequence rather than the distribution, and the two quantify over different things. A distributional bound quantifies over outcomes: every draw from the law lands inside the stated region, a claim you can only make from inside a model class. A cap quantifies over exposure. It names the worst realized loss you’ve agreed to carry and the action that stops the loss there, and says nothing about the shape of the residual. The level-6 pricing line already carries one:

"Competitor repricing is unmodeled and could contribute 4 volume points, +/- $36k on this SKU; our exposure is capped by a 30-day revert clause."

That 30-day revert clause doesn’t predict the elasticity and doesn’t bound the competitor the model never represented. It bounds what a wrong elasticity can cost: thirty days of mispriced margin on one SKU, then the price goes back. A stated cap is valid only if some owner and some system can actually execute the stopping action. If the revert depends on a decision nobody owns, or on a system that can’t reprice inside a month, you never had a cap.

Estimating the error on the error doesn’t escape the regress. The ladder’s U3 entry (Calibrated predictive (error on error)) says why:

A hierarchical model. Dist(Dist(A)) collapses to Dist(A) under the Giry monad multiplication - integrate out the mixing measure - so stacking hyperpriors produces fatter tails and never an escape from the regress. (1) This is monad multiplication, not the tower property; the tower property is the conditional-expectation identity that makes the collapse decision-irrelevant, which is a different fact. (2) The collapse is decision-irrelevant only because Bayes risk is AFFINE in the predictive measure, so a decision maker sees the marginal and nothing else. That is exactly the assumption rung 4 abandons: a credal set has no mixing measure to integrate, so there is no marginal to collapse to, and the regress genuinely stops being a regress.

U and I don’t substitute for each other, and the reason is definitional rather than a theorem. U types the shape of what a report asserts; I types whether the quantity it’s about can be recovered. Sharpening the shape can’t change which question the report answers, so refining a claim along U moves it up the grid below and never right.

Pearl’s Causal Hierarchy Theorem gives that empirical bite: a rung-k quantity isn’t determined by rung-(k-1) data alone, absent causal assumptions, except on a measure-zero set of structural models. Drop that qualifier and the theorem forbids this ladder’s own I2 through I5 - Ignorability asserted, Partially identified causal, Point-identified, local and Point-identified, target population. Recovering any of those from observational data means importing an assumption the data can’t check. Randomize on the population the decision is about and you need no such import. That imported assumption is itself a claim, scorable on these same six axes.

Climbing U can’t rescue a confounded estimand, and the failure is worse than no improvement. The trap "Calibrated elasticity band" above is the worked case. Calibration passes there, because it’s scored against the estimand you happen to have. More data makes it worse: the variance goes to zero and the bias stays. Under confounding, any trust metric that falls as posterior variance falls is anti-correlated with the truth.

obs-slope / negcontrol in the catalog. Preregistered, blinded, powered, independently run - and zero identification, because nothing was intervened on. S 5 ("live + powered") and S 6 ("durable") presuppose a deployed randomized intervention, so among empirical-domain kinds S >= 5 forces I >= 5 for every kind except negcontrol, which reaches S 5 at I 2. Outside that domain the presupposition does not apply and maximal S with zero I is directly constructible: refutation and hash both sit at S 6 with I 0.

Severity on its own still doesn’t justify a claim, because a severe test of an unidentified quantity is a severe test of the wrong thing.

Breaks if realized severity is not discounted by search: a test with one-shot severity 0.95 has realized severity near zero if it is the surviving member of an unlogged specification search.

An agent searching specifications runs the search far faster than anyone does by hand, and it can log every branch it took, which turns a judgement call into a log. We haven’t built it (G4).

telemetry in the catalog: T 5, I 0. The record matches the population and period the decision lives in, and identifies nothing. It cannot also serve as the I witness, since a record sitting at I 0 witnesses identification only by its absence; obs-slope is the I witness, preregistered and blinded and powered and independently run and still I 0, because nothing was intervened on.

Breaks in reflexive domains, where the sign inverts: more live exposure CONSUMES the regularity rather than confirming it. Pricing is reflexive by construction.

You act on the estimate, and acting on it changes the thing it estimated. Competitors reprice against the move; customers adapt to the new level. Every period the estimate sets a price, the market that generated that number stops existing in the form it was measured in. Your next reading comes off a population already conditioned by your last decision.

We run a gate before any of the five gets weighted. It asks whether the evidence’s construct domain - empirical, formal, synthetic, or any - is one your decision admits. A mismatch doesn’t get a discount. The gate drops the support, and a warrant left with nothing admitted is refused outright. A machine-checked proof about code doesn’t by itself provide evidence about a demand curve, whatever its reliability.

Call an aggregator sound when it never exceeds any of its five inputs. Soundness alone doesn’t single out the minimum: the function that always returns zero is sound too, and it separates nothing. So we ask for both properties at once, sound and greatest among the sound ones. Only the componentwise minimum has both, since a sound aggregator is bounded above by its smallest input and the minimum attains that bound. An average has neither. It can exceed its smallest coordinate, and it scores a claim strong on reliability and weak on severity exactly like those two numbers reversed, though the axis that binds is different.

We exclude U from the fold by construction. It measures how much a claim commits to, not how well the claim survived checking. A claim reported below what its evidence supports is under-sold rather than untrustworthy.

The fold compares level indices, so it’s safe only under a reparameterization that relabels all five ladders the same way. Re-cut one ladder’s cut points and you can move which axis sits at the minimum with no change in the underlying evidence. Every level the composition section names in prose or a caption is printed beside the ladder’s own words for it. The figures print the index alone, because a graph node gets one short line.

8 of the catalog’s 33 kinds have more than one axis tied at the minimum. One of them is llm1, a single unmeasured LLM-as-judge verdict: its meet is level 0, with I (Association) and rho (unmeasured) tied there together. Break that tie by whichever axis sits first in an array and you advise “fix identification” when the instrument reading the claim is equally binding. So the fold returns the full set of axes at the minimum.

In a conjunctive chain every step has to hold for the conclusion to hold, so the weakest link caps the whole chain however strong the others are. Two of the gate’s checks are specific to chaining. Its four edge coordinates have to equal the fold of the steps it declares. On every edge axis, that fold also has to sit at or below the fold of what its admitted supports license. Attach one more in-domain support that is weaker on some edge axis and the right-hand side drops, so a chain that passed can start failing. Adding evidence can lower what a set licenses, because the bound is a minimum. A chain that clears both checks can still be refused by the identification cap, the U ceiling, the decision’s axis gates or its meet floor, each on its own.

Parallel support pools by the same minimum fold, as a deliberately conservative placeholder: it credits independence nowhere. Three supports that vary in three different ways pool to the componentwise minimum, which sits at or below all three of them, exactly as a chain of three steps would. Take the componentwise maximum instead and a support strong on one axis paired with a support strong on another buys back coverage neither one earned. Shared source data do discount the effective count, without changing the score vector. Supports tracing back to the same underlying panel are close to being one support. A review that counts citations instead of sources overstates its own base, always in the optimistic direction.

Severity itself doesn’t compose under this rule (G1), and the fix that gap names is an e-process. Optional stopping is the ordinary practice of watching evidence accumulate and deciding, from what you see, whether to keep collecting. It invalidates a fixed-sample p-value outright. That guarantee assumes the sample size was fixed in advance, and a stopping decision made in light of the running result breaks it however innocuous the peek felt. An e-value is a nonnegative statistic whose expected value under the null is at most one. A running product of sequentially valid e-values is a nonnegative supermartingale. Ville’s inequality bounds it at every stopping time at once, so you stay valid under exactly the monitoring that breaks p-values.

Each claim you fold this way is a node in a content-addressed proof graph, hashed to the code and data that produced it.

We score and order 33 kinds of evidence next door, by dominance on every axis (componentwise dominance). It names no decision, and no positive weighting of the 5 scored axes can reverse a pair it decides. 262 of the 528 pairs order that way and 266 do not, so 49.6% is decidable before you pick a loss function. The rest need weights, and the 4 scored decisions there each carry a different set. Read the evidence catalog.