the flip result

Equation

Book equation 4.5.

\[\text{flip}:\quad \mathrm{task}(O)>\mathrm{task}(R)\ \ \text{and}\ \ \operatorname{tr}M^{O}_\delta>\operatorname{tr}M^{R}_\delta,\qquad \mathrm{task}(\text{anti})<\mathrm{task}(R),\qquad \text{bits}(O)=\text{bits}(R).\]

Book equation 3.1.

\[\begin{gathered} d_O(u)=u^{\top}\Sigma\,u, \qquad u=(\cos15^\circ,\ \sin15^\circ), \\ \Sigma_1=\operatorname{diag}(0.3,1.7),\ \Sigma_2=\operatorname{diag}(1.7,0.3), \qquad d_O=0.394\ \text{vs}\ 1.606. \end{gathered}\]

Book equation 0.10.

\[d_O=\operatorname{tr}(P_C\,M_\delta)=\mathbb E\!\left[\delta^{\top}P_C\,\delta\right],\qquad M_\delta=\mathbb E\!\left[\delta\delta^{\top}\right],\qquad P_C=I\ \Rightarrow\ d_O=\operatorname{tr}M_\delta.\]

Conditions

• Matched bits, and reconstruction matched or reported, for every comparison of two codes.
• A read operator misaligned with where the signal has its energy. The alignment of book equation 4.6 is the dial, and near one, the coupling null, no flip is possible.
• The read operator is the average of local linearizations, and the ordering it predicts is the ordering on average over the workload.

Conditions are curated in entries.toml rather than read from a record.

Ledger

• measures. GO-2 (neg. half: not reconstruction) [demonstrated]. At matched bits, downstream preservation is not controlled by reconstruction error. geometric-observation/claims/LEDGER.md:63.
• measures. GO-2 (pos. half: consumer-projected covariance controls) [replicated]. Downstream preservation is controlled by the error covariance projected on the consumer’s read subspace, tr(P_C·Σ_δ). geometric-observation/claims/LEDGER.md:64.
• measures. GO-4 [replicated]. Fixed-budget verdicts invert under budget-matched observation, per the wavelength mechanism. geometric-observation/claims/LEDGER.md:66.
• refutes or corrects. NEG-16 (KV serving, end-task) [refuted]. At matched bits and matched reconstruction error, steering KV quantization error into the attention read subspace degrades LongBench task score by the registered floors on a deployed-class model. Refuted at its registered effect sizes on … geometric-observation/claims/LEDGER.md:92.

First stated

Volume 14, chapter 8, geometric-observation/chapters/ch08_value.md:1-30, DOI 10.5281/zenodo.21776291. The two-dimensional case is the cover and equation 3.1 of Data Mining as Observation.

Measurements

Failures and corrections

• NEG-16 (KV serving, end-task), [refuted]. At matched bits and matched reconstruction error, steering KV quantization error into the attention read subspace degrades LongBench task score by the registered floors on a deployed-class model. Refuted at its registered effect sizes on … geometric-observation/claims/LEDGER.md:92.
• geometric-observation/chapters/ch16_honest_negatives.md:28-30 at b2626a6. NEG-4 — lightweight online (Lloyd) key calibration beats the default. [refuted]. It improves reconstruction and is worse on the consumer metric — reconstruction-is-not-the- target, in its purest, most anti-correlated form.
• geometric-observation/chapters/ch16_honest_negatives.md:32-34 at b2626a6. NEG-5 … NEG-9 — the KV-key ladder that found the right statistic. Five refuted proxies, and together the most instructive sequence in the ledger, because each refutation located the next hypothesis:
• geometric-observation/chapters/ch16_honest_negatives.md:36-50 at b2626a6. - NEG-5: the originally-registered test missed (2/4 bars); the apparent reversal was a matched-bits confound (an uncounted per-block codebook, ~+25% bits). Refutes the test, not GO-2 — and forced a bit-matched redesign. - NEG-6: a demeaned-error-norm proxy fails to order the arms. The controlling quantity is the across-token variance of the query-projected error, not an error-magnitude scalar. Magnitude ≠ downstream-relevant structure (Chapter 6’s lemma, learned here). - NEG-7: at fixed reconstruction, the downstream ranking flips with the consumer. Invariant-preservation is a property of the (compressor, consumer) pair, not the compressor — the sharpest form of GO-2’s negative half. - NEG-8: the variance-ratio proxy reaches only Spearman 0.80; it is a noisy linear estimator of \(\operatorname{tr}(P_C\Sigma_\delta)\). Gate on the direct projection instead. - NEG-9: even the direct trace \(\operatorname{tr}(P_C\Sigma_\delta)\), while nailing the discriminating flip 12/12, is not a complete rank statistic — it misorders a middle pair whose per-token error has structure the trace misses. The mechanism is real; the scalar is not exhaustive.
• geometric-observation/chapters/ch16_honest_negatives.md:56-61 at b2626a6. NEG-10 — the flip appears on any independent representation (prospective). [refuted]. On embedding retrieval, arm b reconstructed strictly better and Pareto-dominated downstream — no flip — so reconstruction was not falsified there. Precondition discovered: the flip is observable only for reconstruction-matched arms. The anti-probe half still transferred. This negative is why every later flip row is built on recon-matched arms; the miss defined the protocol.
• geometric-observation/chapters/ch16_honest_negatives.md:106-112 at b2626a6. - Identifiability (fixable): NEG-5, NEG-10, legal-035, NEG-12 — the read operator was mis-estimated or the arms weren’t recon-matched. Fixed by the blind probe and by protocol. - Coupling (a true boundary): D4 — read and signal energy intrinsically aligned; not rehabilitatable by any read-operator recovery. - Precondition (needs a working consumer): moral-on-frozen-embeddings — no read direction to protect until the consumer is competent. - Mechanism-absent (a genuine refutation): NEG-11 — the claimed effect does not exist.

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/Flip.lean, theorems trace_eq, reading_difference, reader_swap_reverses, flip, cos_sq_pi_div_twelve, sin_sq_pi_div_twelve, reading_15_first, reading_15_second, sqrt_three_bounds, readings_15_to_three_decimals, reader_75_is_swap, four_to_one_flip, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation primer S, chapters 2, 3, 4, 6, 7, 11, 12, 13, 14.

Related

read distortion; alignment; anti arm; rank certificate; coupling null.

See also

none

Status

Generated 2026-09-10 by encyclopedia/generate.py; book at observation-data-mining f3914f0; the commit of every record is listed in the encyclopedia’s provenance.