Prove the sharp joint dirty-word interval envelope (no new κ)

research/chronological-dirty-intervals/PROOF.md

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+# The sharp joint interval envelope for the balanced dirty word
+
+Prepared for @CubeJerry with substantial OpenAI Codex assistance.
+Apache-2.0. This is a scoped written proof, conditional on the precisely
+identified predecessor lemmas below; it is not a supplier-admission proof.
+
+## Contract and result
+
+Use Chafik Boukhalfa's unchanged #628 balanced word on n >= 1 formal
+noncommuting factors. There is a protected source x, an unread target y,
+and h = ceil(log2 n) distinct dirty scratch registers z1,...,zh, one per
+depth. Siblings reuse the same deeper scratch. Every cut is at a complete
+ideal leaf, `target += sign A_j source`.
+
+For a matrix over the free noncommutative integer algebra, its coefficient
+row mass is the sum of the absolute integer coefficients of all monomials
+in that row, after collecting equal words. Define
+
+```text
+F(0) = 2,
+F(h) = floor(h^2/2) + h + 2,  h >= 1.
+```
+
+For every ordered pair of cuts s,t in either word orientation, every row
+of `P_t P_s^-1` has mass at most F(h). This is the sharp uniform envelope
+at each height: Arham Tahir's #1374 family at n=2^h attains F(h), including
+with commuting real scalar factors in {+1,-1}. At h=0 one leaf attains two.
+Sharpness is over all n at a fixed height, not a claim that every n attains it.
+
+Consequently the maximum joint-register operator norm is at most F(h) for
+arbitrary contractive factors, without commutation. It is at most F(h) G
+when factors commute and every subset product, including the empty product,
+has norm at most G >= 1. The latter implication uses #1278's squarefree
+interval theorem. The contraction implication does not require it.
+
+The predecessor upper envelope was h^2+2h+3 for h>=1 (Nick Petalas,
+#5872); the new gain is ceil(h^2/2)+h+1. #1374 already supplied both the
+matching lower family and the explicit sharp-constant question. The new
+content here is the upper proof, specifically the inverse-prefix sparsity
+invariant and its use in chronological phase products.
+
+## Imported premises and notation
+
+Hanson Sun's #1085 proves that every prefix and inverse prefix, for both
+orientations, has entries zero or one signed monomial; its source row and
+target column are identity. For a positive call of complete product T,
+the target/source coefficient of a prefix is 0 or T, and that of its
+inverse is 0 or -T. A suffix has the same entry-size bound, because it is
+the inverse of a prefix of the inverse word. Each such child row has mass
+at most its number of registers. We use these assertions as written premises.
+
+Write B for the ordered left product, A for the ordered right product,
+and z for the parent's first scratch. The completed child maps are
+
+```text
+U = I + e_z B e_x^T,   V = I + e_y A e_z^T,
+F = V^-1 U^-1 V U = I + e_y AB e_x^T.
+```
+
+Let P_a be a positive prefix of the left child and Q_b a positive prefix
+of the right child. They act on the same deeper scratch W. #1278's four
+positive-parent phase formulas are
+
+```text
+0: P_a;   1: Q_b U;   2: P_a U^-1 V U;   3: Q_b F.
+```
+
+At a child boundary either adjacent representation gives the same map.
+The last representation includes the final boundary. Negative prefixes
+have the same maps after complementary cuts and full-map multiplication.
+All algebra below uses matrix action on column vectors, with chronological
+execution multiplying on the left. No factor commutation is used.
+
+## Lemma 1: scratch support has at most one direction per pair
+
+For every positive prefix and its inverse, each scratch diagonal entry is
+one, and for every pair of distinct scratch registers at most one of the
+two opposite off-diagonal entries is nonzero.
+
+Induct on the call. Completed outer maps only change z by a multiple of x
+or y by a combination of x,z. Their left or right multiplication does not
+change the scratch-to-scratch block of the active child, in any of the four
+phase formulas or their inverses. For a left child z is its unread target:
+its column in this block is the identity column. For a right child z is
+its protected source: its row in this block is the identity row. The block
+on W satisfies the induction hypothesis. Thus z cannot participate in
+both directions of any pair, nor can any pair inside W. Its diagonal and
+the deeper diagonals are one. This also proves the claim for an unused
+reserved depth, whose row and column are identity. The leaf case is empty.
+
+In particular, with k scratch registers the scratch-to-scratch block has
+coefficient mass at most k(k+1)/2. #1085's unit entry-size premise is used
+here: counting occupied entries is then counting their coefficient mass.
+
+## Lemma 2: inverse-positive-prefix source occupancy is nonadjacent
+
+For an inverse of a positive prefix, record which entries in its source
+column are nonzero, in row order `(target,z1,...,zk)`. This binary string
+has no consecutive ones. The leaf strings have length one, so the claim
+holds initially.
+
+For an active left child, let f in {0,1} indicate its inverse-prefix
+target/source entry, which is `-f B`; let v be its deeper source-occupancy
+string. Its induction string is `(f,v)`. Inverting the four phase formulas
+gives the following parent strings:
+
+| Phase | Inverse-parent source occupancy `(y,z,W)` |
+| --- | --- |
+| 0 | `(0,f,v)` |
+| 1 | `(0,1,0,...,0)` |
+| 2 | `(1-f,f,v)` |
+| 3 | `(1,0,...,0)` |
+
+For phase 1, `U^-1 Q_b^-1 e_x = e_x-e_z B`, because x is external
+to the right child. For phase 3, `F^-1 Q_b^-1 e_x = e_x-e_y AB`.
+For phase 2, applying `U^-1 V^-1 U` to `P_a^-1 e_x` leaves the z
+coefficient `-f B`, changes the y coefficient to `-(1-f)AB`, and leaves
+the W coefficients unchanged. These are exact free-algebra identities.
+Since `(1-f)f=0`, all four strings remain nonadjacent. Padding unused
+deepest registers adds zero bits, so unequal child sizes cause no exception.
+
+There are therefore at most ceil(k/2) occupied source entries among the k
+scratch rows. Their target entries vanish. Lemma 1 now gives the new bound
+
+```text
+H(k) := k(k+1)/2 + ceil(k/2) = floor((k+1)^2/2)
+```
+
+on the *sum of the masses of all scratch rows* of an inverse-positive
+prefix. For k=0 this is zero. Left multiplication by the full positive
+map changes only the child's target row. Hence a positive suffix
+`U P_a^-1` or `V Q_a^-1` has the same scratch rows and the same H(k) bound.
+
+## Lemma 3: cross-child products with the correct right factor
+
+Embed a left-child map L on `(x,z,W)` and a right-child map R on `(z,y,W)`;
+there are k shared scratch registers W. Every entry has mass at most one.
+Assume that the rightmost factor in a product is either an inverse-positive
+prefix or a positive suffix, so its W-row mass sum is at most H(k).
+All its rows have mass at most k+2.
+
+For R L, the y row is the sum of its own identity row, at most one copy of
+L's z row, and at most one monomial multiple of each W row of L. Its mass
+is at most `1+(k+2)+H(k)`. A W row is bounded by `(k+2)+H(k)`; the z row
+is L's z row, and x is fixed. Thus every row of R L is bounded by
+
+```text
+H(k)+k+3 = F(k+1).
+```
+
+For L R, the z row is at most one x term, the unchanged z row, and the
+weighted sum of W rows of R: mass at most `H(k)+2`. A W row has at most
+one x term plus that W-row sum; the y row is unchanged from R. Thus its
+maximum row mass is at most `max(H(k)+2,k+2) <= F(k+1)`.
+The products use shared W throughout; no fresh scratch is introduced.
+
+## All six phase pairs, both cut orders
+
+Induct on n, keeping both cut orders. Both children have height at most
+h-1; pad them to k=h-1 shared scratch registers as needed. Lemma 2 holds
+with this padding. Same-phase intervals reduce to child intervals, including
+the reversed order, and are bounded by F(h-1) <= F(h).
+
+For distinct phases, the three adjacent pairs and their inverses are:
+
+| Pair | Forward interval | Reverse interval |
+| --- | --- | --- |
+| 0,1 | `Q_b (U P_a^-1)` | `(P_a U^-1) Q_b^-1` |
+| 1,2 | `(P_b U^-1) (V Q_a^-1)` | `(Q_a V^-1) (U P_b^-1)` |
+| 2,3 | `(Q_b V^-1) P_a^-1` | `P_a (V Q_b^-1)` |
+
+Every factor is a child prefix, inverse prefix or suffix. In each product
+the rightmost factor is specifically an inverse-positive prefix or positive
+suffix. Lemma 3 applies and gives F(h). This chronological restriction is
+the point that the generic product of two arbitrary prefix bounds misses.
+
+For repeated left phases, #1278 gives
+
+```text
+I_02 = P_b P_a^-1 + e_y A [e_z^T U P_a^-1].
+```
+
+The added row is external to the child. Its mass including its identity
+entry is at most `1+(k+2)=h+2`; the other rows are child-interval rows.
+For the inverse interchange a,b and negate the added term. Thus both orders
+are bounded by `max(F(h-1),h+2) <= F(h)`.
+
+For repeated right phases,
+
+```text
+I_13 = Q_b Q_a^-1 - [(Q_b V^-1)e_z] B e_x^T.
+```
+
+The new column x is external to the child. Its contribution to each row has
+mass at most one, by the prefix entry-size premise for `Q_b V^-1`.
+The reverse formula interchanges a,b and changes minus to plus. Hence both
+orders are bounded by `F(h-1)+1 <= F(h)`, also at h=1.
+
+The final pair is
+
+```text
+I_03 = Q_b P_a^-1 + e_y AB e_x^T,
+I_03^-1 = P_a Q_b^-1 - e_y AB e_x^T.
+```
+
+For the forward direction, only the y row requires a refinement to Lemma 3.
+Write `Q_b[y,z]=e A` and `P_a^-1[z,x]=-f B`, e,f in {0,1}.
+If e=0, the target row has mass at most `H(k)+2` (its identity, the W-row
+sum, and the added AB). If e=1, combine the added AB with the z-row
+contribution before applying the triangle inequality. Its x entry is
+`(1-f)AB`; its other entries are unchanged from A times the z row. The
+combined row has mass at most k+2: when f=1 the x terms cancel, and when
+f=0 the original z-row x entry was absent, leaving at most k+1 other
+entries before the one addition. The y row is therefore bounded by
+`1+H(k)+(k+2)=F(h)`. Other rows already obey Lemma 3.
+
+For the reverse direction, Lemma 3 bounds the z and W rows of
+`P_a Q_b^-1`; the extra term changes only the external y row. That row
+is the y row of Q_b^-1 plus one x entry, of mass at most `k+3=h+2`.
+This too is at most F(h).
+
+The n=1 base consists only of identity and the single leaf shear in either
+sign and has envelope two. This completes the induction for positive words
+and both cut orders. An interval of the negative word is a positive-word
+interval at the complementary cuts in the opposite order, so both word
+orientations follow. The proof never expands a rounded leaf internally.
+
+## Sharpness and operator substitution
+
+#1374's pinned lower theorem uses n=2^h, scalar factors all +1 except
+`A_(n-2^j-1)=-1` for 0<=j<=h-2, and chronological cuts
+
+```text
+a_h = floor(3*4^(h-1)/5),   b_h = (4^h+2)/3.
+```
+
+Its target row is exactly
+
+```text
+y + (-1)^(h-1)(1+floor(h/2)) x
+  + sum_(j=1..h) j*(-1)^(h+j) z_j.
+```
+
+The absolute row sum is F(h). This predecessor theorem supplies all-height
+attainment; our verifier independently corroborates that target row by
+literal replay at small sizes and compressed exact scalar prefix matrices
+at larger heights. No priority is claimed for the lower family or the old
+n<=16 maximum census.
+
+For arbitrary contractive factors every formal word evaluates to an operator
+of norm at most one, so coefficient row mass bounds the maximum joint norm.
+For commuting factors with all subset products bounded by G, first invoke
+#1278's squarefree conclusion in the free algebra, then commute each
+surviving word into a subset product and apply the triangle inequality.
+Bounding individual factors by G alone is not sufficient. Sharpness of the
+coefficient envelope and contraction-domain envelope does not claim equality
+F(h)G for every fixed nonminimal choice of G.
+
+## Trust boundary
+
+The all-size conclusion is the written induction using #1085 and #1278
+as stated premises, and #1374 for sharpness. Exact finite checks corroborate
+the new invariants, six phase identities, interval bounds and scalar witnesses;
+they are not the basis for extrapolation. Formal proof-kernel verification
+and independent human review are unclaimed. This theorem concerns arbitrary
+joint disturbances; Philippe Van Velzen's separate destination-only #2177
+result is unchanged. Internal rounding, inverse-leaf preimages, precision,
+physical movement, compiler, stock, analytic recovery and a complete caller
+invoice remain outside scope. No selected framework assumption or kappa changes.