B62 (prototype): Does the rule spend the traffic's rank, and is the agreement count just rank in another basis?

[NGL321]

Part of #532. Opened by B57 (#629), which built the graded agreement instrument, read it on three arms, and found the rank it needs present at construction and gone by the horizon.

Question

Is the traffic's rank collapse caused by the training rule, and does the graded agreement count measure anything the traffic's rank does not?

What B57 established, and the two things it could not

B57 read the instrument B52 (#622) specified. It passed its acceptance check decisively, it moves under training, and it reads N(0.25) = 1.0000 on every arm from 1,000 taught ticks on. Two facts it turned up are this ticket's:

1. The rank is spent, not absent. The traffic is not born at rank one. Effective rank at construction → 20k, on the same window:

arm	uncentered	centred
baseline	2.766 → 1.0045	24.912 → 1.611
winner	2.786 → 1.0320	27.270 → 9.318
flat	2.544 → 1.0034	2.938 → 2.865

B57 measured this while the rule ran and deliberately did not attribute it: no arm was run with the rule off past construction, so training spends the rank and the rank decays on its own are not separated. That is one cheap arm.

2. The count may be a re-description of the rank. q_i is a normalised Rayleigh quotient, and B57's own excess is +1.0000 against a matched-generic null reading 0.0000 at every level — which the surprise ledger (#520, row 8) files with two readings it cannot separate. Either the surviving direction is one the built maps genuinely share and generic ones do not, or q_i sits far from 1 wherever the traffic is rank one and near 1 otherwise. On the second reading the instrument is measuring traffic rank in another basis and B52's whole construction reduces to B60 (#633)'s object.

What to read

1. A frozen-map arm past construction. Same dome, same seed, same window, the transport and prediction rules off, the world running. If the traffic's rank collapses anyway, the rule is exonerated and the collapse is the body's or the sandbox's; if it holds near 2.77 uncentered, the rule spends it. Report both forms.
   Sweep the window at the horizon, not only at construction. B57's amendment found centred traffic ER growing monotonically with T (13.014 / 18.465 / 24.912 at T = 250 / 500 / 1000) while the uncentered form is stable to the fourth decimal, so a centred rank quoted at one window means nothing on its own. The 24.9 figure above is therefore not a target to hold against — establish the window behaviour first and state the comparison in terms it survives.
2. A rule-by-rule split, only if item 1 says the rule spends it. PredictionRule and TransportRule separately — they are separable in the rig and B57 ran them together.
3. Whether q_i separates from rank. Read N(θ) against a synthetic traffic of stated rank on the built maps — rank 1, 2, 4, 8, drawn inside the maps' own row space and outside it — and say whether the count tracks the rank or the alignment. This is the falsifier for the ledger's second reading, and it needs no run at all.
4. The joint report on every row, per B48's standing constraint. Not optional here either.

Branch table

Proceed without asking on any of these.

• The rank collapses with the rules off → the collapse is not the objective's and no objective term can be aimed at it. Say so plainly; the lever moves to whatever writes the stalks.
• The rank holds with the rules off → training spends it, and item 2's split says which rule. This makes the collapse an objective-design fact and B60: Does the transport rule need the traffic's full metric, or does the mean suffice? #633 inherits it.
• N(θ) tracks stated rank and not alignment → the instrument is traffic rank in another basis. Do not repair it here — file it, and it is B60: Does the transport rule need the traffic's full metric, or does the mean suffice? #633's to rule on whether the map keeps a second name for one quantity.
• N(θ) separates from rank → B52's construction is vindicated on its own terms and the ledger's row 8 resolves to its first reading.
• Anything outside every branch → the surprise ledger (#520), one row, and carry on.

Named risks

• Do not carry a number across B49's two-objects line. Everything here is on node stalks. Audience differentiation is on the transport operator and is tabled beside, never argued from.
• Name the surface (Write #437's ruling: amend ADR-0032's ledger in place, and give domain.md the surface rule #455). B57 read c925866 after B8: Write #540's ruling: per-edge m_e, lateral lanes at 1, and the invariant doubled #548 and B15: Write #556's reserve mask: k_v = n - p, and the bound it supersedes #562; inherit no figure.
• The window is T = 1,000, T0's own, and B57 defended it against T = 250 / 500. Keep it unless there is a reason, and state the reason.
• The centred and uncentered forms disagree qualitatively — uncentered the count is 1, centred it is 0. Report both on every row; B53's advisory on B57 (prototype): First reading of the graded agreement instrument — acceptance check, matched-generic null, and does it move? #629 is why, and B60: Does the transport rule need the traffic's full metric, or does the mean suffice? #633 is downstream of it.

Plan, don't do. A prototype may build and read. It may not score a candidate architecture and it may not set a threshold a later ticket would inherit as a bar.

[NGL321]

Resolution

The prediction rule spends the traffic's rank, the transport rule buys the agreement, and the two pull opposite ways — and N(θ) is not rank in another basis. Two of the branch table's branches fired and neither is the ambiguous one.

Surface c925866, main — B57's own, so the comparison against its ladder is row for row. Seed 42, one seed. δ_P held against Diagnostics.whole_graph on every arm before anything was read: rank 3000, dim H⁰ 1800 of 4800, agreeing to the integer on all four runs. Readout: prototypes/cold-start/T6/READOUT-635.md on worktree-b62-rank-635.

1. The rank holds with the rules off

The control is exact and costs no argument: T0/run.py's teaching_read is patchworks.agent.run plus bias.step() and transport.step(), so dropping the two rule steps leaves the same world, dome, seed stream and stimulus with nothing learning. Uncentered traffic ER at T = 1,000:

taught ticks	rules off	rules on (B57)
construction	2.7798	2.7656
500	2.7750	2.9825
1,000	2.7786	1.0967
2,000	2.7819	1.0132
5,000	2.7812	1.0130

Flat to the third decimal across every rung — a spread of 0.0069 — while the rules-on arm rises to 2.9825 by 500 and collapses by 1,000. N(θ) stays 0.0000 at every θ at every rung: the count never switches on without training either. The collapse is not the body's and not the sandbox's.

2. Which rule — and they are antagonistic

rung	frozen	bias only	transport only	both (B57)
ER @5,000	2.7812	1.0089	3.0024	1.0130
q of leading direction @5,000	0.8835	0.5495	0.0543	0.1830
A(0.25) @5,000	0.0000	0.0039	0.5769	0.9935
exposure (of 32)	9.05 → 9.05	9.05 → 9.05	9.05 → 7.02	9.05 → 8.62
audience differentiation	0.5307 → 0.5307	0.5307 → 0.5307	0.5307 → 0.5313	0.5307 → 0.4826

• PredictionRule spends the rank and buys no agreement. It reproduces the collapse alone, on the same interval (2.7798 → 1.0087 by 2,000 against both-on's 1.0132), and its leading direction stops at q = 0.5495 carrying A(0.25) = 0.0039. Nothing is agreed at the end of it — and it costs exactly zero in both joint columns, bit-identical to the frozen arm at every rung, because it writes K and never touches a restriction map.
• TransportRule buys the agreement and raises the rank. ER climbs to 3.4018 at 1,000 and settles at 3.0024; its leading direction reaches q = 0.0543 — better alignment than both rules together manage — holding differentiation flat and paying 2.03 dimensions of exposure per cell.

So the map has spoken of the rule as one lever; on this quantity it is two, pulling opposite ways, and the trained surface is their compromise — whose differentiation cost (→ 0.4826) is neither rule's own price.

This is a reading of the shipped rules, not a candidate: no arm is scored, and transport-only reaches better q is a measurement, not a proposal.

3. N(θ) separates from rank — row 8 resolves to its first reading

Traffic of stated rank in bands of stated level, through B57's agreement() unmodified. The levels are not chosen: q is the generalized Rayleigh quotient of (δ_Pᵀδ_P, GᵀG), so on the visible subspace the attainable levels are a pencil's eigenvalues, read off the maps with no traffic in them. q is a Rayleigh quotient, so on the span of the r lowest directions every direction has q ≤ q_r — a stated band stays stated however the instrument's own SVD mixes it.

band	r = 1	r = 2	r = 4	r = 8
low_q, q ∈ [0.0022, 0.0092]	1.0000	1.9422	3.4905	6.6410
mid_q, q = 1.0000	0.0000	0.0000	0.0000	0.0000
high_q, q ∈ [1.9905, 1.9975]	0.0000	0.0000	0.0000	0.0000

Both halves of the ledger's second reading fail. A rank-one traffic reads 0.0000 when generic — against q_i sits far from 1 wherever the traffic is rank one — and a rank-eight traffic reads 6.6410 when low-q. The band decides whether anything is counted; the rank decides how much.

The spectrum says why a generic direction cannot reach the band by luck: median visible q is exactly 1.0000 (B52's no-relationship point, measured rather than asserted) and 441 of 3,000 directions sit at q ≤ 0.25, but a random direction is a mixture of all 3,000 and concentrates at the mean — which the isotropic family confirms at every rank (q̄ 1.0130–1.0244, N 0.0000 throughout). That is why B57's matched-generic null reads 0.0000 everywhere.

And B57's account of the real reading is rebuilt from stated ingredients. A low-q mean under a mid-q variety gives uncentered ER 1.1423 with N = 1.0000 and centred 6.8614 with N = 0.0000 — the real baseline's 1.0045 / 1.611 and 1.0000 / 0.0000. It also breaks the last tie: uncentered ER runs 1.02 → 1.06 → 1.14 → 2.85 while N(0.25) stays pinned at 1.0000, because only the mean is low-q.

4. The window, swept at the horizon

Frozen arm at every rung, uncentered / centred: T = 250 reads 2.7383 / 13.4027, T = 2,000 reads 2.7849 / 31.0925. Uncentered holds to ~2% across an eightfold window; centred doubles and keeps climbing with no plateau. A sample covariance over T ticks has rank at most T against 4,800 columns, so the centred column is bounded by the window and not by the surface — it reports how long you looked. Every comparison above is stated uncentered. 24.9 is correctly no target, and neither is 31.1 — the same quantity at another window on the same untrained surface.

5. Three findings outside the branch table → surprise ledger (#520), one row each

1. The two rules are antagonistic on the rank (§2 above). Not a shape the map anticipated.
2. ker(G) = ker(δ_P) exactly. ‖G‖ = 1.4607 and δ_P restricted to ker(G) has top singular value 1.75e-15; both kernels are 1800-dimensional. Exact agreement is unavailable to any visible direction by construction — so B48's earned = 0 has an operator-side proof, not only a counting-side measurement, and B52's case for a graded statistic arrives independently.
3. N(θ) has a vacuous regime. It is a participation ratio inside the counted band, so it reads 1.0000 on a direction carrying 0.4% of the traffic — the bias arm at 5,000, where the direction carrying 99.5% sits at q = 0.5493. B57's own conclusions are untouched (its A(0.25) runs 0.9547 → 0.9977). Offered as a standing constraint: an agreement count is quoted with the weight it counts — A(θ) beside N(θ) — or it is not a reading. The profile already carries both; complying costs no run.

What this hands #633

#633 asks whether TransportRule needs the traffic's full metric C⁻¹, and its whole case rests on how much structure C has. §2 says the near-rank-one C it would whiten is manufactured by the prediction rule, not by the transport rule whose update #633 proposes to change — under TransportRule alone the traffic keeps effective rank 3.0024. Whether that relocates the lever is #633's to rule on; this ticket does not rule it. It also does not inherit a second name for one quantity: §3 refuses that branch.

What this does not establish

• One seed, one arm (baseline, seed 42). The rules-off control exists only there.
• The split arms stop at 5,000, not 20,000. The rules-on collapse is complete by 1,000 and settled by 2,000, so 5,000 brackets it — but the long horizon is unread on the split, and B38's stamp is per-run.
• The split arms carry no matched-generic null. against_generic re-assembles three full δ_P/G pairs per reading and is the rig's dominant memory cost; this box's low-memory guard killed three runs before it was dropped. The excess is inherited from B57 on those two arms rather than re-measured.
• §2's antagonism is measured, not explained. That one rule writes K and the other writes the maps is a location, not a mechanism.
• No threshold set, no candidate scored, the mask unread.

Decision provenance

• Ceded to the agent: the choice of q bands (read off the pencil rather than chosen), the amplitude ratios in the composite, stopping the split arms at 5,000, and dropping the matched-generic null on those two arms. Each is stated above with its reason.
• Taken from the ticket: the branch table, the T = 1,000 window, the joint report on every row, both forms on every row, plan don't do.
• Taken from the map: B49's two-objects line (everything here is on node stalks), B48's joint rule, the Write #437's ruling: amend ADR-0032's ledger in place, and give domain.md the surface rule #455 surface rule.
• Not taken: no repair of the instrument (§5.3 is filed as a constraint, not applied), no ruling on B60: Does the transport rule need the traffic's full metric, or does the mean suffice? #633's lever, no reading of the mask.

[NGL321]

Amendment: item 1 is now read to the full horizon, and the answer is unchanged

The resolution above read the frozen arm to 5,000 ticks — the 10,000 and 20,000 rungs having been killed three times by this box's low-memory guard. Re-run alone and without the matched-generic null, it reaches B57's own horizon:

taught ticks	rules off	rules on (B57)
construction	2.7798	2.7656
5,000	2.7797	1.0130
10,000	2.7801	1.0064
20,000	2.7817	1.0045

A spread of 0.0072 over the whole 20,000 ticks — smaller than the window-to-window spread of the same quantity — with N(θ) at 0.0000 for every θ at every rung, and exposure and audience differentiation bit-stable at 9.05/32 and 0.5307 throughout.

So the rank does not decay on its own is read to the horizon rather than bracketed, and the ruling is unchanged. f78377d… branch worktree-b62-rank-635, commit 84a97d9.

What this does not extend. The split arms still stop at 5,000. Whether PredictionRule and TransportRule stay antagonistic at 20,000 is unread, and it is B68 (#645)'s first item, pre-registered there as that ticket's own falsifier: if TransportRule's ER falls to the both-on arm's ~1.01 by 20,000, the antagonism is a transient of the first 5,000 ticks and the premise is gone.

[NGL321]

Correction to the amendment above — the ladder now comes from one arm

The amendment's table mixed two runs: its construction row came from the through-5,000 frozen arm and its 10,000 and 20,000 rows from the 20,000 arm. That is a column no single arm produced, and it should not have been posted as one.

The cause is worth recording because it nearly cost more than a table: this box's low-memory guard killed the runner script while its python child survived and kept writing, so two frozen 20,000-tick arms ran concurrently and interleaved their lines into the shared console log, which is what the rungs were read off. The JSON records were never at risk — each is rewritten whole at every checkpoint, so a file is always one run — and the surviving record has monotonic elapsed times. The log is kept, renamed 635-frozen-20k-INTERLEAVED.log, and no number in the readout comes from it.

The 20,000-tick frozen arm, from its own record, throughout:

taught ticks	rules off	rules on (B57)
construction	2.7802	2.7656
100	2.7807	2.8073
200	2.7808	2.8378
500	2.7757	2.9825
1,000	2.7787	1.0967
2,000	2.7816	1.0132
5,000	2.7785	1.0130
10,000	2.7821	1.0064
20,000	2.7825	1.0045

Spread 0.0068 over the full horizon.

And the accident left a replication rather than a gap. The earlier through-5,000 arm is a genuinely separate run, and on its own seven rungs it reads 2.7798 / 2.7801 / 2.7804 / 2.7750 / 2.7786 / 2.7819 / 2.7812 — spread 0.0069. Two independent frozen arms, the same flatness, neither spread reaching a thousandth of the collapse being compared against. It is reported beside the table rather than merged into it.

§2's frozen column is the through-5,000 arm throughout, which the readout now states. The ruling is unchanged in every particular — the resolution's substantive claims never rested on the third decimal of a flat line — and the correction is to how the number was quoted, not to what it says. Branch worktree-b62-rank-635, commit e595bfe.

[NGL321]

Second correction — the transport figures, re-taken from the surviving record

Same cause as the correction above and one more instance of it, which I should have checked for the moment I found the first. The transport arm was first run with the matched-generic null switched off to survive this box's low-memory guard; the runner chain's own transport step then ran the same arm to the same output path and overwrote it before the commit — its parent script had been killed, but its python child had not. The figures I published came from the first run; the committed record is the second.

	published	record
ER @5,000	3.0024	3.1347
q of leading direction @5,000	0.0543	0.0636
A(0.25) @5,000	0.5769	0.5552
exposure	7.02	7.09
audience differentiation	0.5313	0.5505

The full transport ladder, from the record: ER 2.7798 → 2.8109 → 2.9705 → 3.1619 → 3.1837 → 3.3307 → 3.1347, q of the leading direction 0.8832 → 0.1779 → 0.1133 → 0.0636.

The reading is unchanged. Transport-only never collapses, ends far above construction, drives its leading direction well below the both-on arm's 0.1830, and pays in exposure. The two runs differ in the second decimal.

Two things change in kind rather than degree, and both run in the finding's favour.

1. Transport does not hold differentiation flat — it raises it, 0.5307 → 0.5505, above construction. So against B48's joint rule (agreement rising while differentiation stays nonzero) the transport rule does not merely pass the differentiation clause; it moves that column the right way, and its entire price is the 1.96 dimensions of exposure per cell.
2. The "split arms carry no matched-generic null" caveat was wrong and is struck. All three surviving arms carry it, three redraws each, reading N_gen = A_gen = 0.0000. The excess is measured on every arm here rather than inherited from B57.

Readout rebuilt from the records at commit 2248ce0; the map entry and ledger row 9 are corrected to match. Nothing in the ruling, the branch table, or items 1 and 3 is affected — those rest on the collapse-versus-flat contrast and on the synthetic bands, neither of which involves these figures.

The process lesson, since it cost three corrections. A killed runner script does not kill its python child on this box, so a "sequential" chain can leave a survivor writing to a path a later run also targets. Two arms then interleave a console log and race a JSON. b62_run.sh's header carries the warning; the durable fix is a per-run output path, which this rig does not have and #645 should not inherit.