Beyond Ranking: A Multi-Projection Methodology for Search Based on Transparent Representation and Confirmed Convergence

Abstract

This thesis develops an argument that begins with a narrow technical complaint about trigram-based file signatures and ends in a general epistemological and architectural position on search systems. The argument proceeds in three movements. First, a classical objection to local frequency-based trigram filtering — that it produces false negatives — is shown to rest on a mischaracterization of the method: once the procedure is correctly specified as an overlap-threshold filter rather than a containment test, the "false negatives" turn out to be a deliberate and defensible scope boundary, not a defect. Second, this reframing is generalized: no representation of data — not trigrams, not embeddings, not even "raw bytes" — occupies a neutral zero point, because every representation is already a projection that discards information. If no representation is privileged, the only defensible strategy is to implement a diverse set of representations and to declare their projection parameters transparently, rather than to search for the "correct" one. Third, this architectural commitment is extended into the interaction model: because relevance is a private function of the user, and because a system cannot rank by a criterion the user has not made explicit, ranking is replaced by a two-phase, cache-backed exploration process whose success criterion is monotone, user-confirmed convergence rather than ranking efficiency under the Probabilistic Ranking Principle.

The thesis is thus: a search system should be built as a federation of independent, parameter-transparent projections whose results are only ever unioned and contrasted — never ranked or filtered by one another — because convergence toward an unspecified information need can only be achieved through user-confirmed narrowing, not through system-side anticipation of relevance.

1. Introduction

1.1 Motivation: The Trigram Method and Its Discontents

The starting point is a well-known technique for approximate file matching. For every file, a signature is built by counting all contained trigrams (3-character sequences). The most frequent 10% of trigrams are removed; the remaining 90% form a bitmap signature.

The classical objection to this procedure is straightforward: removing the most frequent trigrams locally (i.e., per file) produces false negatives, because a trigram removed in file A as "too frequent" might be preserved in file B, where it is not locally frequent. A query built from that trigram cannot distinguish between "trigram absent" and "trigram removed" — the 0-bit is semantically overloaded.

Three classical remedies suggest themselves:

• Global document frequency (df) instead of local counting — rejected, because a global corpus-wide frequency table presupposes a linguistic/structural homogeneity across the corpus that frequently does not hold.
• Explicit removal set R carried as side information — correct, but costly in storage.
• Winnowing / minimizers — a deterministic, content-independent selection scheme with formal guarantees.

Each of these remedies treats the local removal as a flaw to be patched. This thesis argues that this framing is itself mistaken.

1.2 Thesis Statement

The trigram method is not a faulty containment test; it is a correctly scoped overlap-threshold filter. Generalizing this insight — that every retrieval representation has a legitimate, declarable scope rather than a universal correctness claim — leads to an architecture in which multiple independent projections are unioned without ranking, and in which progress toward the user's (possibly inexplicit) information need is measured by monotone, user-confirmed convergence rather than by the predictive accuracy of a system-estimated relevance ranking.

1.3 Structure of the Argument

Chapter 2 reformulates the semantics of the trigram procedure and shows that its "false negatives" are a scope boundary. Chapter 3 generalizes this into a multi-channel retrieval architecture with no filtering between channels and no ranking of results. Chapter 4 grounds the refusal to rank in a broader epistemological claim: there is no neutral representation, hence no neutral ranking. Chapter 5 formalizes the resulting space of projections as a measurable object with axes, distances, and gaps. Chapter 6 operationalizes this into a two-phase interaction protocol between client and system. Chapter 7 extends the architecture with a cache that is itself a searchable object. Chapter 8 proposes a convergence criterion as the explicit alternative to the Probabilistic Ranking Principle, together with a scaling model and testable hypotheses. Chapter 9 positions the proposal against established work. Chapter 10 concludes.

2. From Containment to Overlap-Threshold Filtering

2.1 The Decisive Reframing

The pivotal move of this thesis is to reject the implicit specification under which the trigram method was being judged. The procedure is not a containment test ("is X contained in file D?"). It is an overlap-threshold filter:

d is a candidate if |T(q) ∩ S(d)| ≥ θ — with no ranking and no sorting implied.

Under this specification, "false negatives" are not an error but a deliberate scoping of the method's domain of applicability. Queries that consist predominantly of trigrams that are "background" for the target file — i.e., locally frequent, hence removed — are, in principle, not meaningfully separable by this method. This is not a bug to be fixed by a smarter removal rule; it is a statement about what question the method can answer at all.

The starry-sky analogy. One cannot search for a star in a patch of sky that consists of nothing but stars; the background must be subtracted before a threshold can separate anything at all. The trigram method's local removal step is that subtraction — performed, necessarily, relative to the file's own background, because no universal background exists (see Chapter 4).

2.2 The Effect of Removal, Reframed as Benefit

Reframed in this way, the removal step is not information loss to be minimized but a mechanism with a positive, quantifiable function:

• It lowers the random overlap between files that share a common corpus "baseline tone," making true hits stand out more clearly against noise.
• It is especially effective for files drawn from a similar corpus, because in that setting local and class-wide frequency are strongly correlated — the similarity of the corpus implicitly supplies the consistency that a global stop-list would otherwise need to supply explicitly.
• The removal rate (10%) has an interior optimum: too little removal yields no gain in discriminability; too much removal begins to strike selective, Zipf-tail trigrams and damages true hits more than it damages noise.

This reframing dissolves the original objection without requiring any of the three classical remedies: global df is unnecessary because the method never claimed corpus-wide uniformity; an explicit removal set R is unnecessary because the scope boundary is accepted rather than patched; winnowing remains available as an alternative point in the design space, not as a correction.

3. Architecture: Multi-Channel Search Without a Filter Chain

3.1 Five Independent Retrieval Channels

The reframing of Chapter 2 generalizes into an architectural principle: rather than seeking a single corrected procedure, the system runs several independent, parallel projections (retrieval channels), each scoped to what it can legitimately answer:

1. Exact text search — containment, no tolerance.
2. Trigram/10% signature — set overlap with threshold θ, frequency suppression.
3. Multi-vector late interaction (ColBERT family) — learned token similarity (MaxSim), knowledge external to the corpus (encoder training).
4. AST structural search — syntactic tree structure, invariant under identifier renaming and layout.
5. Corpus-trained vectors — structurally like (3), but the knowledge source is the corpus itself rather than an externally trained model.

3.2 Union, Not Filtering

No channel filters another; all run independently over the full corpus. The output is the union of all candidate sets, and every element carries a provenance declaration: which channel produced it, under which parameters, and which raw features support it (overlap value, MaxSim score, structural match).

3.3 Why No Ranking

The system does not rank the unioned results. Three arguments support this:

• Relevance is a private function of the person issuing the query. Any system-side ranking is only an estimator of that function, and cannot outperform the user's own application of it when the user is capable of applying it directly.
• Ranking is impossible when the relevance function is not explicit even to the user (the Anomalous State of Knowledge / berrypicking condition). Any anticipation of an unknown dimension is logically impossible, because the very measure of approximation to that dimension would already presuppose the dimension itself.
• The sole exception is a query that supplies its own evaluation function (an explicit scoring mechanism). In that case, "ranking" is merely delegated evaluation of a function the client already possesses — not anticipation of one it does not.

4. No Neutral Ground: An Epistemology of Representation

4.1 There Is No Zero Point

The central epistemological claim of this thesis is that the refusal to rank is a special case of a more general refusal: the refusal to privilege any single representation. Not only ranking, but every representation — the trigram, the bitmap, even "raw data" conceived as bytes or file boundaries — is already a projection, an anticipation of what matters. There is no neutral intermediate stage and no zero point; raw data is itself an encoding.

4.2 Consistency Instead of Absolute Soundness

A direct consequence is that the correctness of an index cannot be defined absolutely, only relative to a comparison between projection levels. One can speak of consistency between representations, but not of soundness in an absolute sense, because soundness would require a ground truth outside all representation — which does not exist.

4.3 Transparency as the Achievable Property

If neutrality is unattainable, the only property that remains achievable is transparency of the projection: an explicit declaration of operator, parameters, and the order of steps applied. Transparency replaces neutrality as the epistemic standard the architecture holds itself to.

4.4 Diversity as Strategy

Since every single representation necessarily discards something, the only consistent strategy is to implement as diverse a set of representations as possible, covering the theoretically available space of projections, rather than searching for "the right one." This is the epistemological justification for the five-channel architecture of Chapter 3, and it motivates the formal treatment of that space in Chapter 5.

5. The Space of Projections

5.1 A First Axis: Tolerance

The implemented procedures can be placed on axes. A first, one-dimensional axis is "tolerance / degree of abstraction from the literal character sequence":

| Method | Tolerance | Invariant | |---|---|---| | Exact | none | order and identity of characters | | Trigram/10% | local, set-like | character adjacency, not position/order | | Multi-vector | learned, continuous | token meaning, not spelling |

5.2 Four Axes

Adding AST search and corpus-trained vectors yields at least a four-dimensional overview:

| Axis | Exact | Trigram/10% | Multi-vector | AST | Corpus-vectors | |---|---|---|---|---|---| | Tolerance | none | set-like | learned | structural | learned | | Structural binding | character | character | token | tree | segment | | Knowledge source | none | none | foreign | grammar | corpus | | Background suppression | no | explicit | implicit | no | implicit |

5.3 Measuring Diversity

"Even coverage" presupposes a metric. Since no privileged metric over projections exists, one is constructed from the behavior of the methods themselves: the distance between two methods is defined as the Jaccard distance of their result sets over a set of samples and the corpus. A new method is valuable to the extent that it is maximally distant from all existing ones (maximin / farthest-point selection). The rank of the resulting distance matrix estimates the effective dimensionality of coverage; redundancy shows up as low rank.

5.4 Interpolation

Within a single parameter family (removal rate, θ, n-gram length), interpolation is well-defined, since the parameter forms a continuum. Between method families, no canonical intermediate step exists; "interpolation" there means only that the union or contrast of result sets approximates an intermediate projection — valid only if the result space is sufficiently smooth, which is itself measurable: does set-distance grow with parameter distance?

5.5 Gaps in the Space

Gaps in the space become visible as pairs of files that are indistinguishable under every implemented method but differ in raw content. Identified open axes include:

• Edit distance / fuzzy matching (between Exact and Trigram),
• Token normalization / BM25-style overlap (between Trigram and Multi-vector),
• Layout / compressibility (Normalized Compression Distance),
• Graph relations (imports, references),
• Time / provenance (metadata, history).

6. The Two-Phase Interaction Model

The architecture is operationalized through a two-phase protocol between client and system.

6.1 Phase 1 — Exploration (unspecific)

• The client supplies a sample or a vague query, together with a target volume N and a distribution w over methods.
• The system returns, per method i, at most ni = wi · N candidates, at sampling rate ni / |Ci|.
• If |Ci| ≤ ni, the output is complete (rate = 1); this is explicitly reported, because completeness is qualitatively different from a sample — statements about absence become possible.
• Feedback per method includes |Ci|, ni, rate, and seed, so that even incomplete sets remain interpretable.
• Residual budget rule (when a method does not exhaust its allotment): the default is lapse (the distribution is preserved); the alternative is redistribution (the volume is preserved, the distribution is skewed). The choice belongs to the client.
• Sampling mode is selectable: uniform (reveals dominant clusters) vs. diversity-oriented / farthest-point over signature distance (reveals the margins).
• Deterministic sampling (hash with seed) is used instead of randomness — reproducible and traceable ("more" yields a continuation of the same order).

6.2 Phase 2 — Specification (after sufficient explicability has been reached)

The client chooses among four paths:

1. Refine the query (new samples as contrast, positive or negative).
2. Formulate scoring (a client-authored evaluation function over delivered raw features; the system either evaluates it or returns raw data for client-side evaluation).
3. Restrict to specific methods, axes, or parameter ranges.
4. Continue exploring (other samples, resolution levels, signature maps).

6.3 Presentation Rule

Results are presented in a fixed, meaningless order (e.g., alphabetical by method identifier), with equal rank given to every entry. Any form of highlighting — including sorting or emphasis on multiply-confirmed hits — would already constitute a weighting introduced through the back door, in violation of the principle established in Chapter 3.3 and Chapter 4.

7. The Cache as an Inspection Space

7.1 Separating Search from Sampling

The cache stores complete search runs — candidate sets, raw features, provenance — not merely the selected sample. This separates:

• the search run (expensive, performed once), from
• sampling and scoring (cheap, repeatable arbitrarily often against the cache).

7.2 Consequences

• Search runs themselves become a searchable object — the same projection principles (contrast, overlap, mapping) can be applied to earlier runs.
• The cache key is (corpus state, method version, parameters); otherwise projections would silently intermix. Staleness is flagged, never silently replaced.
• The retention policy (what is discarded under storage pressure) is itself an anticipation and must therefore be either declared explicitly or delegated to the client (pinning).
• User judgments (confirmation / rejection) are recorded with timestamp and round, and remain revisable without requiring recomputation.

8. Convergence as a Criterion for Abandoning Ranking

8.1 Against the Probabilistic Ranking Principle

The Probabilistic Ranking Principle justifies ranking as optimal under uncertainty and bounded user attention. This thesis proposes a competing criterion that permits the exhaustive user evaluation that the architecture of Chapters 3–7 implies:

Exhaustive evaluation by the user is acceptable as long as every iteration refines the search space by a step, rather than anticipating something the user has not confirmed.

8.2 Requirements for Legitimate Convergence

• Monotone refinement: what has been confirmed stays confirmed; what has been rejected stays rejected (a guarantee enforced by the cache of Chapter 7).
• Confirmation is the only legitimate source of narrowing. Contrasts derived from user judgments ("these features separate your confirmed from your rejected samples") are suggestions, never automatic applications.
• Stagnation detection: the system can detect and report when confirmed and rejected samples remain indistinguishable across all views — a finding about missing coverage, not a value judgment.

8.3 A Scaling Model

Two quantities govern convergence:

1. Corpus size N: a lower bound on the number of rounds is ≈ log₂(N/k) bits, if each round divides the candidate set by a roughly constant factor (clean separation by the projections). If this fails, the number of rounds grows linearly with N.
2. Representation distance d: the distance between the dimension the user actually means and the nearest implemented projections, measured via result-set difference.
3. small d → convergence within few rounds,
4. medium d → the target set is only approximable as an intersection/combination of several projections (the interpolation case of §5.4),
5. large d → stagnation regardless of N (no combination of implemented projections separates the dimension).

Interaction: large N amplifies small deficits in d — at large corpus size, the hit fraction within any sample shrinks, so the precision of individual methods becomes more important. At small corpus size, a user can compensate for even weak projections through exhaustive inspection.

8.4 Core Hypothesis to Be Tested

Round count scales as ~log N when d is small or medium, with a clear stagnation signal when d is large.

• If the first part turns out to be linear in N, the method is viable only for small-to-medium corpora.
• If the stagnation signal is absent, the user cannot distinguish between "keep exploring" and "demand a new method" — undermining the entire convergence criterion of §8.1–8.2.

9. Related Work and Positioning

| Aspect of the Concept | Established Approach | Key Difference | |---|---|---| | Multiple representations, contrast instead of weighting | Principle of Polyrepresentation (Ingwersen) | there: overlap as a relevance indicator; here: overlap only as contrast | | Unsorted sets with provenance | Metasearch / Federated Search | there: usually fused into a ranking (CombSUM, RRF) | | Raw features instead of scores | Vespa, Solr/Elasticsearch feature logging, PyTerrier | there: scoring is part of the system, exchangeable rather than mandatorily client-side | | Exploration instead of query–response | Exploratory Search (Marchionini), Berrypicking (Bates), ASK (Belkin), Information Foraging (Pirolli/Card), Scatter/Gather | typically delivers clusters/facets, not method diversity | | Sample as query | Query by Example, Relevance Feedback (Rocchio), AIDE | feedback there is usually converted back into a ranking | | Volume/distribution, sampling | Approximate Query Processing (BlinkDB), Online Aggregation (Hellerstein), Technology-Assisted Review | covers budget/feedback, not multi-projection coverage | | Diversity selection over methods | Result Diversification (MMR, DPP), Quality-Diversity / MAP-Elites, Novelty Search | typically operates at the result level, here at the method level | | Cached runs as corpus | TREC Session Track, Scroll/PIT (Elasticsearch), notebook provenance | as a searchable object, rarely implemented |

Individual methods this thesis builds on:

• Trigram signatures: Signature Files (Faloutsos), BitFunnel (Bing, frequency-aware bit allocation), Google Code Search / Zoekt, Winnowing/MOSS, MinHash/SimHash.
• AST search: Deckard, SourcererCC, NiCad, Semgrep/ast-grep, Comby.
• Multi-vector: ColBERT, ColBERTv2/PLAID.
• Structure-free-yet-structural: Normalized Compression Distance (Cilibrasi/Vitányi).

Where this thesis goes beyond established work:

• Treating the set of methods itself as a measured, coverable space (result-set distance as a metric).
• Grounding the refusal of any system-side ordering in an epistemological argument (the impossibility of anticipating unknown dimensions), not merely a pragmatic one.
• Taking the contrast between projections, not the quality of any single projection, as the unit of epistemic gain.
• Proposing a convergence criterion (monotone, confirmed refinement) as an explicit replacement for the Probabilistic Ranking Principle's efficiency criterion, rather than merely declining to implement ranking for lack of resources.

10. Conclusion and Open Questions

This thesis has traced a single argumentative thread from a narrow technical dispute to a general architectural and epistemological position. The trigram method's "false negatives" dissolve once the method is correctly specified as an overlap-threshold filter with a legitimate scope boundary rather than a universal containment test. Generalizing the lesson — that every representation has a legitimate scope and no representation occupies neutral ground — motivates an architecture of independent, transparently parameterized projection channels whose outputs are unioned, never filtered or ranked against each other. Because relevance is a private and often inexplicit function of the user, ranking is replaced by a two-phase exploration protocol backed by a cache that preserves full search runs, and the system's success is measured not by ranking efficiency but by monotone, user-confirmed convergence.

The thesis leaves open, and explicitly proposes as testable, the central scaling hypothesis of Chapter 8.4: that round count grows logarithmically with corpus size when representation distance is small or medium, accompanied by a detectable stagnation signal when representation distance is large. Confirming or refuting this hypothesis — rather than any further argument about ranking in the abstract — is the decisive empirical test of the methodology proposed here.