# RESULT — 069 conjunction-subadditivity: bits of "A and B" vs bits of A + bits of B

**Tier: T3-exploratory. DONE + SELF-HARVESTED in-session.** tmux c100_069 launched
22:55:06Z, DONE 22:58:53Z (3.8 min wall; smoke 80 s first, artifacts deleted; decode+bits
sharded phys 0/2/3). FLORES eng devtest, 5 relatedness tiers, disjoint pseudo-doc groups:
UNREL n=65 / REL n=80 / REDUN n=96 (SONAR deu/jpn back-translation paraphrase, 2 rejected)
/ SELF n=67 / LONG n=85 (2-row clauses). All ≥ MIN 64. All 5 gates PASS, all 5 frozen
predictions TRUE, **mean Brier 0.1295** (best-calibrated row so far).

## Verdict
★ **Storage of two conjoined clauses in one SONAR vector is SUBADDITIVE at every
relatedness level — R = I_spec("A, and B") / (I_spec(A)+I_spec(B)) < 1 with tight CIs
even for UNRELATED clauses — and R falls exactly as shared information rises:**

| tier | n | R_AB | 95% CI | R_BA | I_spec A | B | AB |
|---|---|---|---|---|---|---|---|
| UNREL | 65 | **0.924** | [0.911, 0.937] | 0.915 | 172.0 | 165.8 | 312.0 |
| REL   | 80 | **0.907** | [0.893, 0.920] | 0.906 | 165.4 | 168.8 | 303.2 |
| REDUN | 96 | **0.725** | [0.712, 0.738] | 0.727 | 173.8 | 161.2 | 242.9 |
| SELF  | 67 | **0.612** | [0.602, 0.624] | 0.612 | 165.8 | 165.8 | 202.9 |
| LONG  | 85 | **0.762** | [0.747, 0.777] | 0.759 | 289.5 | 295.7 | **446.1** |

The frozen ordering SELF < REDUN < REL ≤ UNREL held exactly (P1). Even unrelated clauses
compress ~8% when packed together (P2: CI excludes 1.0) — but topical relatedness buys
almost nothing beyond that (REL 0.907 vs UNREL 0.924); only genuine redundancy compresses
hard (paraphrase 0.725, verbatim copy 0.612). No order effect: R_BA ≈ R_AB everywhere
(≤0.009). LONG reproduces the ~460-bit specific-bits ceiling a THIRD time (446 vs 066's
463 / 068's 476) at demand ≈ 585 bits (P5).

## Cross-carry: how many of B's bits does z_A hold? (P4 TRUE — the cleanest gradient)
carry_spec = I(B-tokens | z_A) − I(B-tokens | z_shuf), B scored standalone:
UNREL **−2.2** bits (−1.4% of I_spec(B)) · REL **+12.9** (7.6%) · REDUN **+111.4**
(**69.1%**) · SELF **+165.8** (100.0%, = I_spec(B) exactly — encoder deterministic) ·
LONG **+3.6** (1.2%). A paraphrase's vector already carries ~2/3 of the decoder-usable
bits of its restatement; an unrelated vector carries none. This is a direct, quantitative
shared-information meter inside z-space.

## Where the compression/shortfall lands (span-split of conj I_spec; exploratory)
| tier | span A | conn | span B | deficit A | deficit B |
|---|---|---|---|---|---|
| UNREL | 153.8 | 7.1 | 150.9 | 18.2 | 14.9 |
| REL   | 153.5 | 6.2 | 143.4 | 11.9 | 25.5 |
| REDUN | 162.8 | 6.6 | 73.4 | 11.0 | **87.8** |
| SELF  | 158.6 | 5.6 | 38.6 | 7.2 | **127.2** |
| LONG  | 223.3 | 4.4 | 217.9 | **66.2** | **77.7** |
UNREL's mild subadditivity is spread symmetrically (~16 bits/clause), not paid at the
connective (~6 bits). REDUN/SELF savings sit almost entirely on the SECOND span — the
restated clause is stored nearly for free once A is in vector+prefix. LONG's ~139-bit
ceiling shortfall lands on BOTH clauses roughly equally (66/78) — capacity pressure
degrades the pair gracefully rather than dropping one clause. "and"-overhead on SELF:
I_spec(AB) − I_spec(A) = 37.1 bits; token overhead median −1 tok (tokenizer merges).

## Per-clause fidelity: positional interference is CAPACITY-GATED (exploratory)
In-regime (short tiers, conj ~55-68 tok, below the ~73-tok knee): interference tiny
(clause recall in conj vs alone: −1.4 to −5.7 pts) and position effect ≈ 0 (UNREL −1.2,
REL −0.0, SELF 0.0, REDUN +2.5). Past the knee (LONG, conj ~119 tok): interference −23.5
pts and a large FIRST-clause advantage — first 60.2 vs second 47.4 recall, **position
effect +12.6 pts, sign consistent under AB↔BA swap**. Front-loading (066d, 068) is not an
intrinsic property of conjunction — it appears only when the bits budget is exceeded.

## Gates (all PASS)
- G1 positive control: UNREL singles I_spec 168.9 bits ∈ [130,240] (066 eng c1 = 182 —
  in-regime reproduction, lesson-5); round-trip chrF 90.5 ≥ 75.
- G2 null: |I_shuf|/I_mu = 0.133 < 0.15 (certified; margin modest, note below).
- G3 norm profile: max/min mean ‖z‖ = 1.341 < 5 (conj norms ~0.27 vs singles ~0.23).
- G4 spm span alignment 95.67% ≥ 95% (marginal); ntok overhead median −1 ∈ [−2,6].
- G5 REDUN validity: 96 pairs, median chrF(A,B) = 69.1 ∈ [30,92] (target 70 — near-perfect).

## Predictions → Brier (frozen in PREREG_LITE)
| pred | P | outcome | Brier |
|---|---|---|---|
| P1 R ordering SELF<REDUN<REL≲UNREL | 0.55 | **TRUE** (.612<.725<.907<.924) | 0.2025 |
| P2 R_UNREL<0.95, CI upper<1.0 | 0.70 | **TRUE** (0.924, CI hi 0.937) | 0.0900 |
| P3 R_SELF ∈ [0.45,0.68] | 0.55 | **TRUE** (0.612) | 0.2025 |
| P4 carry REDUN≥50 & |UNREL|≤20 | 0.75 | **TRUE** (111.4 / −2.2) | 0.0625 |
| P5 LONG I_spec(AB) ≤ 520 | 0.70 | **TRUE** (446.1) | 0.0900 |

**Mean Brier = 0.1295.**

## Limitations / what did NOT run
- I_spec is decoder-extractable bits (lower-bound proxy), not true MI; "subadditive" means
  subadditive in bits USABLE BY THIS DECODER. The ~8% UNREL compression could be encoder
  exploitation of shared English structure OR mild lossy packing — span deficits + small
  fidelity interference (−5.7) lean toward lossy packing; not disentangled here.
- REDUN paraphrases are SONAR round-trips — generated by the very model under test, so
  shared info is model-conditional; an external paraphrase source would be cleaner.
- Span attribution credits a position's bits to that token's clause; gains at B-span
  positions can reflect information about the A-prefix context. zA-carry-in-conj is
  prefix-confounded (flagged in prereg); the clean carry uses standalone B (headline).
- LONG "clauses" are 2-sentence discourse pastes joined by ", and" (grammatically
  awkward); single join template; eng only; single seed; n varies 65–96 per tier;
  G2 margin (0.133) and G4 margin (95.67%) are thin; greedy decode only.
- Shuffle null is within-tier (shares register/length); I_shuf slightly positive at LONG
  (+50.6 bits) as in 066 — I_spec conservative there.

## Follow-up worth funding? **Y (narrow).**
1. Relatedness as a CONTINUUM: regress per-pair R on chrF(A,B)/embedding-sim across many
   intermediate levels — is there a sharp redundancy threshold or smooth exchange rate
   between shared bits and stored bits?
2. Capacity-gated position effect dose-response: sweep conj length through the knee and
   watch the first-clause advantage switch on (ties 066d/068; clean causal handle on
   front-loading).
3. Operator cost: swap ", and" for "but/or/;/because" (ties 033 composition operators) —
   is the ~6-bit connective span constant across operators, and does contrastive "but"
   change per-clause budgets?

## Provenance / hygiene
Repo: PREREG_LITE.md (frozen pre-launch), src/{run_069.py,run_069.sh}, out/{results,
bits×3, decodes×3, items, run.log}. Box dir ~7 MB (z npz box-only). GPUs verified free at
claim AND exit (35/15/15 MiB); box tmux server empty both ends; session self-ended; no
foreign process touched. Smoke passed first try (all gates at n=6; artifacts deleted).
Single experiment in flight throughout. Local commit, no push. T3.