Route-MM v1: the shape of the approximate inverse, spent

Stage MM, leg 54. Gate answer: NO. Data: writeup/data/p2_route_mm_v1_shape.json. Runner: experiments/p2_route_mm_v1_shape.py. Figure: fig49_route_mm_v1_shape.png, rebuilt from the curated JSON alone by experiments/p2_route_mm_v1_shape_evidence.py.

Gate, in its pre-committed wording. Does an approximate inverse that is NOT block diagonal bring the assembled Z₁ below 1, on the a = 0 CLM object, in a class with s < 0.394?, NO. The no-branch fires: stop building ℓ¹-Fourier radii-polynomial certificates for inviscid self-similar transport. This is T's own no-branch. Do not re-enter by tuning s, the weight family, the split, or the border.

0. What was left, and why it was the last thing

Leg 53 put the four terms of the bordered certificate into one radii polynomial. It did not close: with the block-diagonal approximate inverse the method requires, A = Γ⁻¹ ⊕ A_tail, the coupling sub-block Z₁[Γ←tail] came out 43.15 at the best split in the whole admissible sweep, against the 1 it must be under. Four of the five degrees of freedom were then measured and banned: the weight exponent s, the weight family, the split K, the border direction. The fifth is the shape of A, and the block-diagonal shape is exactly what makes the coupling a term at all.

The novelty pass (writeup/novelty/leg_54.md, run and committed before any construction) found the statement that makes this a real question rather than a complaint. arXiv:2411.18361 gives the convention explicitly: for DF a compact perturbation of the identity, take A = A^N + π^∞, invert the Galerkin projection numerically and let the tail act as the identity. So block-diagonal is the field's convention, not this project's misreading of it, and the hypothesis that buys it is precisely the one this operator fails (leg 51: the unbounded part is a shift, not a multiplier). Verdict PROCEED_NARROW; nothing is banked as novel, since block Gauss–Seidel and Schur-complement preconditioning are textbook (10.1007/BF01385611).

1. MM-1: the mismatch as an inequality, and it is an equality

For any A written in blocks against the finite/tail split,

(I - A L)_{tail,Γ}  =  -(A21 G + A22 C),        C := L_{tail,Γ}

With A21 = 0 and A22 = A_tail this is exactly -A_tail C, whose (K+1)-th row carries the entry 1 - K/2 from mode K. Hence

Z₁  ≥  |1 - K/2| · (w_{K+1}/w_K) · ‖A_tail e_{K+1}‖_w / w_{K+1}

for every finite block: the sub-block contains no Γ⁻¹. Measured against the runner's own data it is not merely a bound but an equality: the (K+1)-th column is the maximising column, and measured / RHS = 1.0000 at every K in both classes (max deviation MM1_max_ratio_deviation_from_one). That is a strengthening worth having, and VER-A found it independently.

Its reach is limited, and the directive overstated it. The prefactor |1 − K/2| vanishes at K = 2 and is small below K = 6:

So MM-1 does not establish "no block-diagonal A can work for this operator". It establishes it for K ≥ 6 (flat) and K ≥ 4 (algebraic). Leg 53's sweep started at K = 4 and hid this; VER-A flagged it as GAP 1. The scope is stated with the restriction attached, and the corner is closed separately, twice over, by §2 and by §4.

Second factor. CONTINUATION_PROMPT.md quotes the measured range as 0.94 … 1.33. That figure is not traceable to leg 53's shipped JSON. VER-A re-measured 0.9412 … 1.3873 over K = 4…64; this leg quotes its own measured range from MM1_second_factor_range.

2. MM-1b: every odd split has a singular finite block

Sweeping K = 3 to probe VER-A's hole raised LinAlgError: Singular matrix. Chasing it rather than working around it:

Every odd split is exactly singular, in both classes, under both gauges, with and without the far-field column. The smallest singular value of the augmented finite block is at most MM1b_max_smallest_sv_at_odd_K at every odd K ∈ {3,5,7,9,11} and at least MM1b_min_smallest_sv_at_even_K at every even K ∈ {2,4,6,8}, and without the far-field column at K = 3 it is exactly 0.0, not merely small.

The mechanism, corrected per VER-A2's GAP 4. v1 of this leg said: "at odd K the mode-K residual row acquires no entry on any of b_1…b_K or δc_ω and is carried entirely by the amplitude column." That is not what the matrix does, and it does not discriminate, rows with no mass off the amplitude column exist at even K too, where the block is nonsingular, and at K = 5 there is only one such row while the block is still singular.

The actual mechanism is the left null vector's support, now recorded per row in MM1b_odd_K_scan.left_null_support: at K = 3 it is supported on two such rows, which are proportional, that is the singularity; at K = 5 it is supported on a parity chain of three rows, not one. The parity intuition was directionally right and the one-row statement was wrong. That is the same failure the standing discipline flags from leg 53 (a mechanism cited rather than measured on the matrix actually built) and it is one svd call to fix.

Consequence. The split must be even, so the corner MM-1 leaves open is not {2, 3, 4} but {2} in both classes plus {4} in the flat class alone, and §4's floor covers both. This removes candidate splits; it does not select one, so it is not the banned re-entry by tuning K. The fact is gated in test_spectral_certificate.py rather than left in the runner, because it is a property of the operator.

3. MM-2: the shape battery

Seven shapes of A, all on leg 53's assembled object, all measured as the true column-max of I − A L over the whole space rather than as a sum of sub-block norms.

gs_lower is worth exactly nothing, and the algebra says so before the run does. For Λ = [[G,0],[C,T]],

I - Λ⁻¹L  =  [[0, -G⁻¹B], [0, T⁻¹CG⁻¹B]]

whose (Γ,tail) block is −Γ⁻¹B: identical to the block-diagonal one. Measured: identical to five digits. That was a prediction that could have come out otherwise.

gs_upper, schur and ff_lift do help, and the help is real but far too small. The best admissible shape anywhere in the sweep is recorded in MM2_best_admissible; the block-diagonal baseline at its own best in MM2_block_diagonal_baseline; the ratio in MM2_improvement_over_block_diagonal.

CORRECTION, VER-A2's GAP 1: the first version of this leg reported the wrong headline. MM-2's battery and MM-6's polynomial swept K_SWEEP (4…64) while MM-1 and MM-4 swept K_SWEEP_SMALL. So K = 2 and K = 6, added specifically to close VER-A's small-K hole, never entered the clause that computes the gate answer. K = 2 is the best-conditioned split there is, and it gives the smallest Z₁ anywhere. The battery now sweeps every admissible (even) split. Corrected numbers:

	v1 (wrong: K ≥ 4 only)	corrected (every even K)
best admissible Z₁	32.7489 (schur, K = 4)	8.9591 (ff_lift, K = 2)
block-diagonal baseline	45.3628	10.4584
improvement from the shape	1.385×	1.167×

Both at algebraic s = 0.3, null gauge. The gate answer does not change: 8.96 ≫ 1, and no row has a positive interval.

So spending the last free choice buys a factor of ~1.17 where a factor of ~9 was needed. The improvement is real, measured, and about an order of magnitude too small.

Instrument check (lesson 85). block_diag reproduces leg 53's sub-blocks exactly (Z₁[Γ←tail] = 43.151291, Z₁[tail←Γ] = 1.387315) recorded in MM2_instrument_check_vs_leg53.

3.1 A correction to leg 53, made in place

The tail–tail sub-block was understated. Leg 53 compared A_tail against the bordered matrix it actually inverts and reported 6.0e−13. But the assembled operator's tail–tail block is the bare scaled tail T, which is singular; the (u,v) bordering is part of the construction of A, not part of L. Charged correctly, ‖I − A_tail T‖ ≈ 2.2.

This makes the block-diagonal baseline worse, not better, so leg 53's NO is unaffected. It is recorded because it is the term ff_lift then removes, and it had to be visible before that move made sense.

4. MM-4: the floor for A₁₁ near Γ⁻¹

SCOPE CORRECTION, VER-A2's GAP 2. The first version of this leg called this a shape-independent floor. It is not, and the claim is withdrawn: see §4.2 for the explicit counter-construction that refutes it. What survives is a floor for every A₁₁ in the neighbourhood of Γ⁻¹, which covers every shape in this battery but is not a statement about every A that could ever be written. The verdict string, the JSON headline key (now floor_for_A11_near_Gamma_inv) and the blog all carry the corrected scope, so the over-claim cannot propagate into the plan.

For a completely general A = [[A₁₁,A₁₂],[A₂₁,A₂₂]],

(I - A L)_{Γ,tail}  =  -(A₁₁ B + A₁₂ T)

The tail operator T is singular: its kernel is exactly the far-field direction ĥ that leg 52 bordered. Applying that block to ĥ:

(I - A L)_{Γ,tail} ĥ  =  -A₁₁ B ĥ

A₁₂ has dropped out of the algebra. The size of the coupling along the one direction that matters is a property of A₁₁ alone, and no choice of off-diagonal block can touch it. The floor ‖Γ⁻¹ (L ĥ)‖_w / ‖ĥ‖_w is tabulated in MM4_floor for every K including 2; its smallest value over every class and split is MM4_min_floor, an order of magnitude above the 1 it must be under, at the most favourable split that exists.

4.1 The identity's own caveat, measured rather than asserted

ĥ spans ker T for the infinite tail operator. On the truncated operator actually computed, T ĥ is not exactly zero, and the writeup would be dishonest to claim otherwise: a first draft of the gate in test_spectral_certificate.py asserted ‖T ĥ‖/‖ĥ‖ < 1e−12 and failed at 0.242, which is how this got measured properly.

What the defect actually is (MM4b_kernel_truncation_defect):

• it is supported on the last mode alone, at every K and M tried (MM4b_defect_is_edge_only), a boundary effect of the truncation, not a failure of the kernel;
• its relative ℓ¹ size halves per doubling of M (MM4b_defect_halves_per_doubling), i.e. it is O(M⁻¹) → 0;
• at the M used throughout this leg it is at most MM4b_max_relative_defect_at_M_extra_1024 in relative terms.

The gate now pins the shape of the ladder (edge-only, M⁻¹) rather than an exact zero the truncation does not deliver.

CORRECTION, VER-A2's GAP 3: the comparison above was not the right one. The first version argued the defect was negligible by setting a relative defect against an absolute floor. That is unsound. The term the identity actually drops is A₁₂(Tĥ), of size ‖A₁₂‖·‖Tĥ‖, and ‖A₁₂‖ is nowhere bounded a priori, so it is now measured per shape in MM4d_dropped_term_by_shape:

shape	‖A₁₂‖	‖A₁₂(Tĥ)‖	vs floor
schur, gs_upper (admissible)	~2–10	~5e−04	0.000×
block_diag, gs_lower, ff_lift (admissible)	0	0	0.000×
oracle_pinv (inadmissible)	7176	7.008	1.000×
exact_inv (inadmissible)	3588	3.504	0.500×

Two rows of this leg's own battery are shapes where the truncation defect cancels the floor, oracle_pinv to the last digit, which is exactly why exact_inv's measured coupling along ĥ is ~1e−14 rather than at or above the floor. Both are inadmissible and independently killed by §5's audit at 1.03e+04, so the conclusion is unaffected; but the argument needed the ‖A₁₂‖ factor visible, and now it is. For every admissible shape the dropped term is 0.000× the floor: negligible by three to four orders.

4.2 The counter-construction: why this floor is not shape-independent

A₁₁ is not pinned by the (Γ,Γ) constraint. Solving A₁₁G + A₁₂C = I gives A₁₁ = (I − A₁₂C)Γ⁻¹, hence A₁₁Bĥ = (I − A₁₂C)v with v = Γ⁻¹Bĥ the floor vector, and since A₁₂ is free, the rank-one choice A₁₂ = v wᵀ/(w·Cv) annihilates v outright. Reproduced in MM4c_counter_construction:

The constraint is satisfied exactly and the floor is beaten by fifteen orders of magnitude.

Why v1's ablation could not see this. It compared A₁₁ = Γ⁻¹ against the Schur complement's A₁₁, which agree to MM4_max_A11_freedom_effect. Both arms of that control are approximately Γ⁻¹, so it varied nothing. That is lesson 90 exactly, the lesson this leg quotes in its own preamble and then violated four sections later.

What survives, and why the conclusion holds anyway. The counter-construction needs ‖A₁₂‖ ≈ 1.1e+03…7.9e+03, which wrecks every other tail column and drives the total Z₁ to MM4c_min_total_Z1_of_counter_construction and above: five to six orders above the bar. So the floor's conclusion is empirically safe while its proof is not. The honest statement is: for A₁₁ in the neighbourhood of Γ⁻¹, the coupling along ĥ is at least 5.0444, and no choice of A₁₂, A₂₁ or A₂₂ can touch it. That is a real result. It is not the universal one v1 claimed.

**Dually (and this is why the shape had to be non-block-diagonal at all) **

(I - A L)_{tail,tail} ĥ  =  ĥ - A₂₁ B ĥ

so the only way to control the tail on its own kernel is a non-zero A₂₁: the approximate inverse must lift the far-field column back into the tail. ff_lift builds exactly that rank-one lift (with u swept over four functionals, the best reported: the conservative choice for a negative result). It does fix the tail–tail block, and it is swamped, because by the identity above it cannot touch (Γ,tail).

5. MM-3, the admissibility audit, which is the actual experiment

The gate as literally worded is trivially YES: take A = (L_M)⁻¹ and ‖I − A L_M‖ drops to float noise (~1e−9 in the battery). That number is a statement about numpy.linalg.inv, not about the operator, lesson 86. What makes the gate a real question is admissibility: an admissible A is finite rank plus an explicit operator on the modes beyond it, because the tail is infinite and Γ⁻¹ of an infinite block does not exist.

So the audit builds A at truncation M_A, extends it over modes M_A+1 … M_L by the only explicit operator available (the bordered tail inverse of that sub-tail), and evaluates it against L at M_L > M_A. Results in MM3_admissibility_audit:

• it does not merely fail to help (it is orders of magnitude worse than the block-diagonal baseline (MM3_min_Z1_over_audit);
• it degrades as M_A grows (MM3_growth_per_doubling_of_M_A_*);
• it is essentially independent of M_L) the M_L = 1024 and 2048 rows agree to five digits.

That last pair is the tell. The cost does not live out in the tail; it lives at the seam where the finite-rank part of A meets the explicit tail operator. The exact inverse is not a shape of A; it is the truncation.

6. Controls

MM-5, positive, and it can report the other answer. Same code path with Λ¹ dissipation, unbordered tail, no far-field unknown: a dissipative tail has no kernel, so it needs no far-field unknown, and bordering an already-invertible tail with its near-null pair is the wrong operator. Run through every shape, not just the baseline. It reaches Z₁ < 1 (MM5_control_can_report_below_one, MM5_min_mu_with_Z1_below_one, MM5_shapes_that_reach_below_one). The instrument can say yes; on this operator it does not.

MM-5b, negative, and one of them did not behave. The border direction is wired through the amplitude column, so a wrong direction changes Γ itself and the control can fail (lesson 90). It does, for block_diag and gs_upper: the second singular pair makes the augmented block essentially singular and a random direction is orders of magnitude worse.

It does not discriminate for the Schur shape. With a random border, schur's Z₁ comes out smaller than with the analytic border, across all six seeds tried (MM5b_shapes_where_a_wrong_border_is_BETTER). The mechanism is visible in the numbers: S = G − B A_tail C partly undoes whatever the border did to Γ, so the Schur shape is far less border-sensitive than the block-diagonal one. That is a property of the shape, not a bug.

Two consequences, both stated rather than buried:

1. the border-direction control licenses the negative result for block_diag and gs_upper but not for schur, where the negative rests instead on the μ-dial positive control and on §4's floor;
2. the smallest Z₁ anywhere in that table (MM5b_min_Z1_over_every_border_and_shape) is still well above 1, and chasing it would mean tuning the border direction: banned, and measured dead in leg 53.

7. The polynomial, and the ceiling

MM6_polynomial carries Y₀, Z₁, Z₂ and the root for the best admissible shape at each split. No row has a positive interval; r_max = 0 throughout.

The ceiling, pre-committed and unchanged. This is the a = 0 CLM object. Y₀ is exactly zero because the anchor is one basis mode (the degenerate reason banked as a ban since leg 51) so the polynomial's root r = 0 is available for free and certifies nothing. The reportable quantity is whether the inequality holds on a positive interval, which needs Z₁ < 1, and it does not. Nothing is claimed about HL_S2_nonsymmetric: on the gate's own terms that object is reached only if the polynomial closes here.

VER-A additionally records that Y₀ = 0 is a hardcoded literal rather than an evaluated float (so no float noise is being read as an exact zero) and that its justification checks out independently in exact rational arithmetic.

8. What this establishes, and what it does not

Establishes. On the a = 0 CLM object, in both admissible classes, under both gauges, at every admissible (even) split from K = 2 to 64: no admissible shape of the approximate inverse brings the assembled Z₁ below 1, the smallest value anywhere is 8.9591. The non-block-diagonal shapes buy ~1.17× where ~9× is needed; the unrestricted optimum is inadmissible and, made admissible, is orders of magnitude worse; and for every A₁₁ in the neighbourhood of Γ⁻¹ there is a floor of 5.0444 that no A₁₂, A₂₁ or A₂₂ can touch, by an algebraic identity rather than by exhaustion of cases.

Does not establish. Three things, and each was claimed too strongly in v1:

1. That no finite block of any kind can close this. §4's floor contains A₁₁; MM-1 is finite-block-independent but only bites for K ≥ 6 / K ≥ 4. Neither argument alone is universal and they are not combined into one.
2. That the floor is shape-independent. It is not: §4.2 gives an explicit A₁₂ that beats it by fifteen orders of magnitude. The floor holds for A₁₁ ≈ Γ⁻¹, which is every shape here, and the counter-construction is defeated by its total Z₁, not by the floor.
3. That the identity is exact on the computed operator. It is exact on the infinite one; the dropped term is ‖A₁₂‖·‖Tĥ‖, negligible for every admissible shape and exactly equal to the floor for one inadmissible one.

And the honest ceiling. No link of the L1→L4 chain moved. Clay odds unchanged at ~0.05%. What closed here is a method, on a toy object: the ℓ¹-Fourier radii-polynomial certificate is the wrong shape for an operator whose unbounded part is a shift, and three legs of increasingly careful work now say so from three directions, leg 51 (the method wants a multiplier), leg 52 (bordering repairs invertibility, not size), leg 53 (the seam is the term that runs out), and this leg (and the seam cannot be moved by choosing A differently).