Large language models (LLMs) process text through subword tokenizers rather than directly reading characters or words. Because these tokenizers are trained predominantly on English-centric corpora, they introduce a systematic and often overlooked disadvantage for many non-English languages. In this work, we quantify this tokenizer tax for Indian languages using the FLORES-200 parallel corpus, measuring tokenization fertility across six widely used tokenizers and fourteen languages. Under cl100k_base (used by GPT-3.5 and GPT-4), Indian languages experience an average 8.0x tokenization tax relative to English, reaching 13.0x for Malayalam, reducing the effective context window to as little as 12% of that available to English users for equivalent semantic content. We identify the primary mechanism behind this disparity: failed byte-pair merges that leave text fragmented into single-byte tokens, with merge failure strongly correlating with tokenizer tax (Pearson r = 0.89). We further show that this phenomenon is not an inherent property of Indic scripts but a consequence of tokenizer design. Multilingual tokenizers such as XLM-R and OpenAI's o200k_base reduce the average Indic tokenizer tax by 73%, demonstrating that the disparity is largely remediable. Beyond token statistics, we quantify a practical consequence by showing that, under fixed context budgets, Indian-language documents preserve substantially less original content than equivalent English documents. Finally, we examine the relationship between tokenizer fertility and reading comprehension performance on the Belebele benchmark, finding that the apparent correlation is largely explained by language resource availability rather than tokenizer behavior alone.
Some limits on what language models know are not gaps in data coverage but structural properties of learning from text. We introduce the information shadow: the region of phenomena that a text-trained learner cannot acquire regardless of scale, comprising (I) structures language cannot express, (II) functions that are statistically non-identifiable from the training distribution, and (III) functions that are representable but unreachable by gradient-based training. We give each type a probe that is decisive because the premise of the shadow is, in that setting, provable. For Type I, Language Compression Residuals compare a text learner, which sees only a lossy text-like encoding of the signal, against a full-signal learner, which sees the underlying signal directly. The text learner sits at a computable expressibility ceiling while the full-signal learner pulls away by a gap that stays flat across 300x more data, so the deficit is a property of the channel, not of training. For Type II, the Counterfactual Distinction Test trains models on data exactly consistent with two incompatible rules. Across a provable string task and a language-like agreement task, behavior on counterfactuals is set by the model's inductive bias, while 5% disambiguating data steers the learned rule bidirectionally to either target (r = +/-1.0, p<1e-10). For Type III, Basin Escape Mapping exhibits a function that is representable at 100% (by hand construction) yet reached 0% of the time by standard training and instantly from a nearby initialization, with width scaling providing no help (p = 1.6 x 10^-14). Each effect is isolated by a control that rules out a capacity or modality artifact. We release the probe suite and discuss implications for benchmark design, capability auditing, and shadow-aware uncertainty.