Route-TC v1, assembling the bordered certificate: the four terms in one polynomial, and the term that ran out is the one that did not exist before

Leg 53. Runner experiments/p2_route_tc_v1_assemble.py → writeup/data/p2_route_tc_v1_assemble.json → fig48 (writeup/4_p2_lottery/p2_route_tc_v1_evidence.py, rebuilds from committed data with no recomputation). Deterministic. Stage TC of plan_of_record.py, 8 pre-committed clauses.

0. The gate, and its literal answer

GATE (pre-committed). With the far-field amplitude carried as a real unknown through all four terms, does the radii polynomial close, on the a = 0 CLM object, in a class with s < 0.394?

NO.

CORRECTED AFTER VERIFIER'S REVIEW (PR #5, writeup/4_p2_lottery/VERIFY_LEG52_HEADLINE.md Part B). The gate answer and every measured number below stand and were independently confirmed. Three things in the first version of this document did not: the scope of the failure (§0 and §2), the stated mechanism (§2), and the status of TC-5b's headline control (§5). All three are corrected here, and §5b adds a normalisation ablation the first version should have run. Part C of the same review withdrew this leg's resurfacing clearance; see §6.

TERM THAT RAN OUT: Z₁, and specifically its two BLOCK-COUPLING sub-blocks, the pieces of I − A L that connect the finite block to the bordered tail. They did not exist as quantities before this leg, because before this leg the two blocks had never been in the same object. Neither the tail constant (leg 52) nor Y₀ nor Z₂ is what failed.

SCOPE, STATED EXACTLY. What this establishes is that the block-diagonal approximate inverse the standard method requires cannot close this certificate, at s = 0 and s = 0.3, under both gauges, at every split K = 4 … 64, and under all six normalisations of the augmented block ablated in §5b. What it does not establish is that no finite block can. The genuinely finite-block-independent sub-block is Z₁[tail←Γ], and its minimum over the whole sweep is 0.9961: below 1. That is precisely why the plan's next stage (MM: an approximate inverse that is not block diagonal) is a real question and not a formality.

1. What was assembled (TC-1)

The object is leg 51's: the a = 0 CLM fixed point Ω₀ = −sin θ, c_ω = −1, c_l = 1, in the compactified odd-sine basis where H, X d/dX and d/dX are exact. No new basis: leg 51 chose it and leg 52 measured the tail in it.

The linearisation's column k is, exactly,

row k+1 :  1 − k/2        row k−1 :  k/2        row 1 : −(−1)^k

so its diagonal is exactly zero for k ≥ 2 and its unbounded part is off-diagonal.

The far-field direction. ĥ is leg 52's tail_right_null: h_{K+1} = 1, propagated by the homogeneous two-term recursion, supported on modes K+1, K+3, …, decaying like m^{−2.00}. Extending it by zero onto modes 1 … K and applying the exact linearisation gives the far-field amplitude's column in the finite block, computed, not fitted:

(M = K + 1024 throughout this table.)

Two entries, both explicit: the amplitude reaches mode K through the operator's own sub-diagonal with weight (K+1)/2, and mode 1 through the rank-one H Ω term with weight ≈ K/2. The far-field amplitude is not a weakly coupled bookkeeping variable: it feeds back into the finite block with a strength that grows linearly in the split.

The matching row. a is defined as the amplitude of ĥ in the tail; the matching row states that, and its content as an equation is a coupling to the tail remainder. That is itself a finding and it is recorded as one: the far-field amplitude is fixed by the tail, not by the finite block, so the matching condition contributes to Z₁'s coupling blocks rather than sitting inside the finite matrix.

The gauge. Two are run, because if the answer turned on the gauge that would be the finding.

• dilation, leg 51's Σ_k k b_k;
• null, pin the exact dilation zero mode instead. That mode is exactly e₂: X d/dX applied to −sin θ is −(1/2) sin 2θ, and ‖L e₂‖_∞ = 0.00e+00 in float, checked, not asserted.

The answer does not turn on the gauge (both fail), but the dilation gauge is separately inadmissible, see §4.

2. Z₁ decomposed by sub-block (TC-4)

The radii-polynomial method needs an approximate inverse of the shape A = Γ⁻¹ ⊕ A_tail (block diagonal: a computed finite inverse, an explicit tail estimate). Γ is the augmented finite block above; A_tail is leg 52's bordered tail inverse. The four sub-blocks of I − A L are then separate, computable objects, and a sub-block's norm is a lower bound on Z₁ for that A.

All values in the weighted ℓ¹ operator norm ‖M‖_w = max_j (1/w_j) Σ_i w_i |M_ij|, with w_k = 1 (flat) or (1+k)^s. M = K + 1024. Border = analytic (leg 52's far-field pair).

Flat, s = 0, null gauge:

Algebraic s = 0.3, null gauge:

The dilation-gauge rows are worse throughout (Z₁[Γ←tail] larger by a factor 1.52 to 8.73 (worst at K = 4, flat)) and are in the JSON.

Read the two diagonal blocks first. Z₁[ΓΓ] is the float inverse's own defect, 10⁻¹⁶ … 10⁻¹². Z₁[tail tail] is 10⁻¹³ … 10⁻¹¹, but that number cannot report what it looks like it reports: with a border on, A_tail is inverted from the bordered matrix itself, so A_tail · B_tail = I identically and the quantity measures float round-off, not the tail's truncation defect at M. It only makes the NO stronger, so nothing turns on it; it should not be read as "the tail–tail block is fine". (Also: Z1_total sums all four sub-blocks, whereas the norm the block system actually induces is max(z_GG + z_tG, z_Gt + z_tt). The reported Z₁ is therefore a slight over-estimate: 44.539 against 43.151 at the best row. Conservative; changes nothing.) Leg 52's bordered tail constant appears here as ‖A_tail‖ = 2.19 … 10.3 and behaves exactly as leg 52 reported.

The two off-diagonal blocks are the whole result.

Z₁[tail←Γ] = ‖A_tail L_{tail,Γ}‖. L_{tail,Γ} has a single structural entry: mode K feeds residual mode K+1 with coefficient 1 − K/2. This block does not involve Γ⁻¹ at all, so it is a lower bound on Z₁ for every choice of finite block, however clever. Its value is ≈ (K/2 − 1) · ‖A_tail e_{K+1}‖_w / w_K, the second factor is O(1) because leg 52's bordered tail inverse is a constant rather than a decaying multiplier, and so it grows exactly ×2 per doubling of K (0.996, 2.977, 6.892, 14.545, 29.176). But its minimum over the whole sweep is 0.9961, which is below 1, so this term alone does not forbid closure, and the structural claim has to be stated as what it is: the rate is structural, the failure at the best split is not from this term.

Z₁[Γ←tail] = ‖Γ⁻¹ L_{Γ,tail}‖ (this is the term that actually exceeds 1 everywhere, and it does involve Γ⁻¹. Measured across every row of the sweep,

Z1[Gamma<-tail] / ||Gamma^-1||  =  1.9667, 1.9921, 1.9980, 1.9995, 1.9999   (flat)
                                   1.9028, 1.9503, 1.9712, 1.9826, 1.9896   (s = 0.3)

so under the shipped normalisation Z₁[Γ←tail] = 2‖Γ⁻¹‖ and the whole K-dependence lives in ‖Γ⁻¹‖, not in the coupling entry. The coupling contributes a factor 2, and its dominant column is the rank-one row-1 term) −(−1)^m into residual mode 1, weight 1 from every tail mode, not the (K+1)/2 sub-diagonal.

And ‖Γ⁻¹‖ grows like K², not K (and the K² is created by this leg's own augmentation. For the augmented block it is exactly 2(K² − 1) (30, 126, 510, 2046, 8190 at K = 4 … 64, ×4.00 per doubling); re-running the same code path with far_field=False) the same block without the amplitude column and matching row: gives exactly 4(K − 1) (12, 28, 60, 124, 252, ×2.02). The inflation is the weight pairing: the amplitude column carries W_a = ‖ĥ‖_w ≈ K/2 while the matching row carries ρ = w_{K+1}, so in weighted coordinates the matching equation has coefficient ≈ 2/K, a deliberately weak equation, and inverting it costs a factor K. That is a normalisation choice, and §5b ablates it.

(The first version of this document cited leg 51's A_norm 165 → 359 → 769 → 1633 as "‖Γ⁻¹‖ grows like K". That is a different matrix, leg 51's unaugmented finite block, and at K = 64 the two differ by 23×. The citation was wrong and so was the exponent.)

The structural fact, which survives all of the above. The standard radii-polynomial tail estimate works because the unbounded part of the operator is a multiplier: the split cuts through an entry of size Λ_M, and the tail inverse is 1/Λ_M, so the product is O(1) and can be made small. Here the unbounded part is off-diagonal, so any split cuts through an entry of size K/2, and bordering, which fixed the tail block's invertibility, does nothing to the size of the tail inverse: it returns a constant, not 1/K. Constant times K/2 diverges, and that is exactly the ×2-per-doubling growth measured in Z₁[tail←Γ], reaching 38.2 by K = 64. What that argument does not deliver on its own is failure at the smallest split, where Z₁[tail←Γ] = 0.9961; the failure there comes from Z₁[Γ←tail] = 2‖Γ⁻¹‖, which §5b shows is not a normalisation artifact either.

This is banked lesson 75 in action: two defects in the same problem are not the same defect. Leg 51 found the tail block non-invertible; leg 52 fixed that; the coupling is a different defect and it survives the fix.

3. The four terms in one polynomial (TC-2)

Y₀ = 0 exactly, in every class and at every split, including the new matching row. The matching row's residual at the anchor is exactly 0.0, and the reason is the same degeneracy that gives leg 51's Y₀ = 0: the a = 0 CLM anchor is one basis mode, its tail is identically zero, so the far-field amplitude it implies is exactly zero and the matching condition is satisfied with nothing left over.

With Y₀ = 0 the polynomial Z₂ r² − (1 − Z₁) r + Y₀ ≤ 0 always has the root r = 0. That root certifies nothing: it is the statement that an exact anchor is exact. The reportable quantity is whether the inequality holds on a positive interval, which needs Z₁ < 1. It does not, anywhere.

Assembled, with the bounded gauge:

The full ten rows are in TC2_polynomial, each also carrying the rigorous finite-block Z₁, the two coupling blocks, ‖A‖, the Banach-algebra quadratic bound Q with Z₂ = 2‖A‖·Q, the discriminant and both roots.

And the counterfactual is in the same row, because it is the number that would have been reported if the terms had never been assembled: poly_leg51_only repeats the polynomial with leg 51's finite-block Z₁ and ‖A‖ alone. That one has a positive interval in every row: r_max = 2.142e−02 at s = 0.3, K = 4, down to 4.642e−04 at flat K = 64. The difference between those two columns is exactly the content of this leg: three terms of four close comfortably, and the fourth is the one that only exists once they are assembled.

4. The border's own defect (TC-3)

Three magnitudes, and what each dominates.

(a) The matching row's residual at the anchor: exactly 0.0. It dominates nothing. It is zero for a degenerate reason (§3) and it is reported so that a later leg on a real profile knows the quantity exists and where it enters.

(b) The asymptotic expansion's own truncation defect. ‖(L ĥ)|_{tail rows}‖_w / ‖ĥ‖_w, i.e. how far the far-field mode is from being annihilated once its recursion is cut at M. At K = 64, M = K + 1024: 6.15e−02 (flat), 1.10e−01 (s = 0.3). It falls with the truncation: fitted exponent −1.0000 (flat) and −0.7794 (s = 0.3) in M, ladder and fits in the JSON. It is a genuine defect of the border, it is 10⁻¹–10⁻², and it dominates Y₀ (which is zero) and nothing else: it is two to three orders below the coupling blocks.

(c) The gauge row's entry on the far-field column: and this one does not exist. Σ_{m>K} m h_m with h_m ~ C m^{−2} is a harmonic sum. Measured at K = 64:

M − K =   256     512    1024    2048
        3347.4  4556.6  5862.2  7222.1        →  +1865 per e-fold in M

Logarithmically divergent. The underlying statement is not numerical: the dilation gauge Σ_k k b_k has dual norm max_k k / w_k on ℓ¹_w, which is infinite for every s < 1, i.e. for every class in which the target profile has finite norm. The far-field column has an entry that does not exist, and it was invisible until the amplitude had a column at all.

What it dominates: it makes ‖Γ‖ itself unbounded under the natural gauge, hence ‖A‖ and hence Z₂. The repair is available and is not a tuning of s: pin the exact null direction e₂ instead (a bounded functional of norm 1/w₂). Both gauges are run above, and with the bounded gauge ‖Γ⁻¹‖ improves by 1.78–1.84× and Z₁[Γ←tail] by 1.52–8.73×: and the gate still answers NO, which is why the gauge is reported as a defect of the assembly rather than as the cause of the failure.

5. Controls

Positive control (TC-5). Same code path with Λ¹ dissipation (−μk on the diagonal), turning the unbounded part from a shift into a multiplier. The control uses the unbordered tail inverse and drops the far-field column, because a dissipative tail has no kernel and needs no amplitude unknown: bordering it with one would charge the control a defect the dissipative problem does not have. Values are in TC5_positive_control.

The instrument reports the other answer (K = 16, M = K + 1024):

The coupling blocks fall like 1/μ, exactly as the mechanism predicts, and the assembled Z₁ crosses below 1 at μ = 2. A method that reported "does not close" for everything would not be measuring; this one reports closure as soon as the operator has a diagonal.

Negative controls (TC-5b): rewired so they can fail. The first version of this leg computed the coupling from Γ⁻¹ and L_{Γ,tail}, neither of which references the border, and then reported "Z₁[Γ←tail] is 546.57 for all four borders" as the sharpest form of the result. VERIFIER was right that this is a tautology of the code, not a measurement. The border direction is now wired through the amplitude column as well (the extra column is L applied to whichever direction the border names) so a wrong amplitude direction changes Γ itself and the control genuinely can come out differently. At K = 16, s = 0.3:

analytic / SVD = 1.0004, the border a proof can write down matches the most favourable one-dimensional choice that exists, reproducing leg 52's T-2 result one level up. The two deliberately wrong directions are worse by 12.6× (random) and 3.84e+13× (second); the second singular pair is not the kernel, so using it as the amplitude direction makes the augmented block essentially singular. The controls can fail, and they do.

What remains true, and is now stated as a structural observation rather than as a control: L_{Γ,tail} and L_{tail,Γ} are pieces of the operator, so the coupling exists whatever direction the amplitude column carries, the border chooses how badly it is conditioned, not whether it is there.

(TC-8) NORMALISATION ABLATION: because ‖Γ⁻¹‖'s K² is a choice, not a fact. §2 shows the augmentation inflates ‖Γ⁻¹‖ from 4(K−1) to 2(K²−1) through the weight pairing between the amplitude column (W_a) and the matching row (ρ). Five conventions for that pair, plus the un-augmented block as the lower envelope, null gauge:

The smallest Z₁ lower bound over every normalisation tried is 20.47) one to two orders above 1, and still growing in K. Note also that row_like_col drops ‖Γ⁻¹‖ from 22.68 to 8.90 at s = 0.3, K = 4 while leaving Z₁[Γ←tail] at 43.15 unchanged: Z₁[Γ←tail] = 2‖Γ⁻¹‖ is a fact about the shipped convention, not an identity. The gate answer survives the renormalisation that the corrected mechanism invites.

Split-point sweep (TC-6). K = 4 … 64. The fair question was never whether the largest split works; it was whether any split works. None does, and the trend is monotone in the wrong direction.

6. Novelty (TC-0), run before the construction

Verdict PROCEED_NARROW, six queries, four ledger entries, all committed in the runner and in LITERATURE_CHECK.md.

• arXiv:1503.06315 (Breden–Desvillettes–Lessard, DCDS-A 35(10) 4765–4789). Leg 52 left open whether their construction reaches a tridiagonal operator with an exactly zero diagonal. This pass fetched the publisher's abstract page: the stated hypothesis is a tridiagonal dominant linear part. Our tail does not satisfy it. That was evidence, not proof (a reading of an abstract, not the full text) and it has since been superseded: LIT's ninth pass extracted the PDF and settled the question outright (assumption (4) needs C1 ≤ μ_k/ω_k^{s_L} ≤ C2 with C1 > 0, assumption (5)'s ratios are undefined at μ_k = 0, the LU construction divides by μ_k, and a vanishing diagonal is not on their own future-work list). Read LIT's item 2, not this entry. The general observation remains a re-derivation, exactly as leg 52 recorded it.
• arXiv:2604.01868 (Bojin Chen, De Huang, Xiangyuan Li, April 2026). CLEARANCE WITHDRAWN: THE FLAG STANDS. This pass originally reported leg 52's search-index flag as cleared, because query 4 returned a summary naming the paper's title and authors correctly. VERIFIER adjudicated that against LIT's ninth pass and LIT wins. Query 4 prepends the literal arXiv ID, so it tests retrieval by ID, which was never in dispute, and not the topical recall without an identifier the flag was actually raised against. LIT re-ran leg 52's query verbatim and reproduced the null result, and enumerated the links returned by an ID-bearing query: all other papers, with the correct title appearing only in the prose summary, i.e. the model answering from its own knowledge rather than from a surfaced link. This pass logged counts, not links, so its clearance could not be audited against its own record. The accurate statement: fetching by ID has always worked; topical recall still fails; the flag stands.
• arXiv:2406.16597 / CPA (2026), self-similar blowup for cubic NLS. Bordering a certificate to kill a symmetry-induced kernel is standard, and no novelty is claimed for the move. What is measured here is what happens when the kernel is the far field of an unbounded off-diagonal operator, so the border acquires a coupling term.
• SIADS doi:10.1137/23M1607507, semilinear PDEs on unbounded domains via spectral methods. Semilinear: the unbounded part is a multiplier, so the standard tail estimate applies. Confirms leg 51's reading of the field's shape; supplies no construction for an unbounded off-diagonal part.

7. Ceiling (TC-7), pre-committed before the numbers existed

Measured on the a = 0 CLM fixed point: one mode, analytic, in every weight class considered.

Y₀ is exactly zero there, including the new matching row, because the anchor is one basis mode and therefore has no far field at all. The polynomial's root r = 0 is available for a degenerate reason and certifies nothing; the reportable quantity is the positive interval, and there is none.

Nothing is claimed about HL_S2_nonsymmetric. On the gate's own terms that run happens only if the polynomial closes here, and it did not. A wall measured on the easiest available object bounds the difficulty for the real target from below, not from above.

No link of the L1→L4 chain moved. Clay unchanged at ~0.05%.

8. Reproduce

.venv/bin/python -u experiments/p2_route_tc_v1_assemble.py     # regenerate the JSON
.venv/bin/python writeup/4_p2_lottery/p2_route_tc_v1_evidence.py   # rebuild fig48 only

Every number quoted above is present in writeup/data/p2_route_tc_v1_assemble.json.